Research Report

KSTS/RR-22/001
December 20, 2022

History of Western Philosophy fromthe quantum theoretical point of view; [Ver. 5]

byShiro Ishikawa

Shiro Ishikawa
Department of Mathematics
Keio University

Department of Mathematics
Faculty of Science and Technology
Keio University
https://www.math.keio.ac.jp/academic/research/
c
⃝2022
KSTS
3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522 Japan

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History of Western Philosophy from the quantum
theoretical point of view; [Ver. 5]
Shiro ISHIKAWA*1 (Mail: ishikawa@math.keio.ac.jp)

Abstract: In this text I will concentrate only on the scientiﬁc aspects of the ”History ofWestern Philosophy”. Thus, the title of this text could have been ”Philosophy of Scienceand its History”. Recently I proposed ”quantum language” which was characterized as
the metaphysical and linguistic turn of quantum mechanics. This turn from physics
to language does not only realize the remarkable extension of quantum mechanics but
also yield the quantum mechanical world view. And thus, the turn makes us expectthat Western philosophies (i.e., Parmenides, Zeno, Plato, Anselmus, Thomas Aquinas,
Descartes, John Locke, Leibnitz, Berkeley, Hume, Kant, Wittgenstein, Hempel, etc.) can
be fully understood in the quantum mechanical worldview. Upon reading this text, thereader will be convinced that the history of Western philosophy can be characterized asa history of the search for quantum language (≈ the language of science), if limited only
to the scientiﬁc point of view. That is, the philosophy from Greece to the present day
was the history of the growth of quantum language. Therefore, quantum language has
power to solve many unsolved philosophical problems. In this text, I will prove most
all the famous unsolved problems of traditional philosophies, e.g., (a): Zeno’s paradox(Greek philosophy), (b): the problem of universals (Scholastic philosophy), (c): the mindbody problem and the subjectivity problem (Descartes-Kant epistemology), (d): How isanalytic philosophy related to Descartes-Kant epistemology? (Analytic philosophy), (e):Why do logic and statistics both work in our world? (Analytic philosophy), etc.

*1 Department of mathematics, Faculty of science and Technology, Keio University, 3-14-1, Hiyoshi,

Kouhokuku, in Yokohama, 223-8522, Japan,
html

HP:https://ishikawa.math.keio.ac.jp/indexe.

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Preface:0.1: Philosophy has progressed toward quantum language
First, let me say that, this text will concentrate only on the scientiﬁc part of thephilosophy.
Quantum language (i.e., QL (=MT = measurement theory), the linguistic Copenhagen
interpretation of quantum mechanics, the quantum mechanical worldview) proposed bymyself is a scientiﬁc language that is inspired by quantum mechanics, and moreover, it
has a great power to describe classical systems as well as quantum systems.
My lecture for graduate students in the faculty of science and technology in Keio university has been continued, with a gradual improvement, for about 20 years.
Figure 0.1: The location of QL in the history of western philosophy(realistic) monism (i.e., without measurement axiom)
relativity
4 −→ theory−−−−−−−−−−→○(monism)(unsolved)3○
−−→ Newton ○6 TOE:theory of(realism)−−→ everything
quantum
5 −→ mechanics −−−−−−−−→○(quantum phys.)

Socrates
Aristotle
0○:Greek

7○

Schola-8○−−−−−−−−−−−−−−−−−→
8
○



quantization

7
○

9
−
○
−
−
−
−
−
−
−
−
→

statistics−−−−−−−−−→

dualism
applied math.
system theoryclassical mechanical worldview


(=MT)



(mind-body dualism)(Copernican revolution)) ○QL:quantum
15
linguistic−−→
language
Descartes
Kant scientiﬁc
10
○

11 
−−→ Spinoza, Locke,... −−→−−−−−−→○
quantummechanicalturn
epistemological worldview
worldview

12
○



linguistic turn



?

(logic in philo.))(logic, set theory)Wittgensteinmechanical
Saussure −−−−−−−→○

Boole, Frege,
14
13
○
turn
Zadeh
Russell, Cantor −−→philosophy ○1−2−−−−→○
sticism
Parmenides, Zeno
Platolinguistic turnmath.

logical worldview(idealistic) dualism (i.e., with measurement axiom)
Figure 0.1 (in Preface): The history of the world-descriptions
Philosophy (≈ dualistic idealism) has progressed toward QL
0
1
2
10
12
14
15 )
(i.e., ○→
○→
○→
○→
○→
○→
○





The main part of quantum language 3 - 8 - 15 and statistics 7 - 9 - 15 in Figure
0.1 (in Preface) were already studied in the following:(A1 ) ref. [74]: S. Ishikawa, ”Linguistic Copenhagen interpretation of quantum meii

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chanics: Quantum Language [Ver. 5]”, Dept. Math. Keio University, 2019,
KSTS/RR-19/003, 473 pages(http://www.math.keio.ac.jp/academic/research_pdf/report/2019/19003.
pdf)

Therefore, in this text I concentrate on part 0 - 1 - 2 - 10 - 12 - 14 - 15 , which isalmostequal to the history of western philosophy. This part 0 - 1 - 2 - 10 - 12 - 14 
was not detailed in my lecture, but it will be a good preparation to read (A1 ) (since
15(A1 ) is mathematically graduate level). That is, this text is written as the supplementary
reader in my lecture, and thus it can be read without the mathematical preparation. Evenso, this text is not an adjunct to ref. [74]. The content of this text is at the heart of thephilosophy of science, and I am convinced that the level of the content is worthy of a
graduate course.
Note that Figure 0.1 (in Preface) asserts that
(A2 ) statistics, quantum mechanics, Scholasticism, Descartes=Kant philosophy and analytic philosophy are each one aspect of quantum language(or, immature states of quantum language) .
Also, I think that the following is one of the most fundamental problems of western
philosophy:
The progress problem (B) : Has philosophy made progress?

- 1 - 2 - 10 - 12 - 14 - 15
Preface) mean the progress of philosophy?

(B) Do the time series0in Figure 0.1 (inThe same thing, in other words(B1 ) What is the goal of the pseudoscientiﬁc time series 0 - 1 - 2 - 10 - 12 - 14?
(B2 ) Or is it literature that has no goal, but merely reﬂects the prevailing conditionsof the time?

Thoughout this text, I conclude that(B3 ) Has this pseudo-scientiﬁc timeline already come to an end? Will it be perpetuated in the future?
(B4 ) In Eastern philosophy, we seldom hear the question, ”Has philosophy progressed?
What is the diﬀerence between the two?
In this text I conclude thatiii

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(C): My answer; the progress of philosophy (≈ dualistic idealism)(C1 ) If ”to make progress” is deﬁned by ”to come near quantum language” (i.e.,”becoming more and more like quantum
language”)† we can say that thetime series 0 - 1 - 2 - 10 - 12 - 14 can be regarded as progress, that is,

This will be proved throughout this text.
†

This deﬁnition is somewhat ambiguous, but I consider it suﬃcient for the purposes of this text.

Here, in this text, it suﬃces to consider that ”idealism”≈”linguistic theory”≈”metaphysics”(cf. Note 1.4). Thus, we conclude that
(C2 ) a scientiﬁc perfection of dualistic idealism is realized by quantum language(QL).
which is surely one of the most important assertions in western philosophy.
Also recall that quantum language is intended as a language of science. Therefore,(C3 ) the scientiﬁc purpose of philosophy of dualistic idealism was to makethe language of science.

The conventional theory is to think of statistics as the ”language of science,” but in thistext, we propose quantum language. This is because statistics is in the halfway positionof being both idealistic and monistic, acting as a kind of applied mathematics.

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0.2: Almost all unsolved problems in traditional philosophy can be solved in QL: Wittgenstein’s dream cametrue
It should be noted that most unsolved problems in the history of western philosophy
are caused by insuﬃcient understanding of idealism and dualism. Thus,• if the (C2 ) is true (i.e., if quantum language is overwhelmingly powerful compared toother immature philosophies), I can expect that most unsolved problems in dualistic
idealism can be solved in the framework of quantum language.
In this text, I will show that the following unsolved philosophical problems are easily
solved as corollaries of the (C2 ):(D1 ): The list of unsolved problems of dualistic idealism to be solved in this text
Wittgenstein’s dream came true
It is said that
• Wittgenstein carried out one of the most ambitious attempts in the history ofphilosophy to solve all the problems of philosophy at once in [125]: ”Tractatus
Logico-Philosophicus (abbreviated as TLP)”.
However, 100 years later, we can’t say that he succeeded in his challenge. However, as
the following list shows, QL realized his dream.

Almost famous unsolved problems concerning dualistic idealism can be solved in QL• What is probability (or, measurement, causality)? cf. Sec. 1.2.1)• What is ”Copenhagen interpretation”? (cf. Sec. 1.3)
(In fact, answering this question is the main purpose of this text. This should be
realized when the reader ﬁnishes reading this text.)• Zeno paradox (Flying arrow, Achilles and a tortoise), (cf. ref. [47], or Sec. 2.4)( ”to solve Zeno paradox” = ”to use DST-formula” (cf. Sec. 2.4.3 )• the measurement theoretical understanding of Plato’s allegory of the sun , (cf.
Sec. 3.3.2)v

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• Plato’s Idea theory≈Zadeh’s fuzzy theory≈Sausuure’s linguistic theory (cf. Sec.
4.1.3)• Syllogism does not always holds in quantum systems (cf. Sec. 4.3.3)
Syllogism always holds in classical and quantum systems (cf. Remark 12.9 in Sec.
12.1)
(The two above seem contradictory, but they are not contradictory because they
are formulated diﬀerently.)• Only the present exists (cf. Sec. 6.1.2: Augustines)• Anselmus’ proof of God’s existence is a wordplay (cf. Sec. 6.4.3)• We proposed the deﬁnition of ”universal” and clariﬁed the meaning of ”the problem of universals” (cf. ref. [83], Sec. 6.5).

Thus, the problem of universals is regarded as a kind of ”monism vs. dualism”.
• What is Geocentrism vs. Heliocentrism? Ultimately, this is a dispute between
diﬀerent worldviews. (cf. Sec. 7.4)

• Two (scientiﬁc or non-scientiﬁc) interpretations of ”I think, therefore I am” .(cf.
Sec. 8.2.2)• Leibniz-Clark correspondence (i.e., what is space-time?), (cf. Sec. 9.2)
• The existence theorem of the kinetic function (cf. Sec. 9.2)

• Brain in a vat argument (cf. Sec. 9.4.2)• The solution of Hume’s problem of induction (cf. Sec. 9.6)

• The grue paradox does not exist in quantum language (cf. Sec. 9.7)
• What is causality? (cf. Sec. 10.2)• Kantian synthesis of continental rationalism and British empiricism is pseudoscientiﬁc and is rewritten as QL synthesis. (cf. ref, [84], Sec. 10.3)

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• The mind-body problem (i.e., How are mind and body connected?), (cf. ref. [70],Sec. 12.8)• The subjectivity problem (≈ the problem of qualia (cf. Sec. 9.4.1)• What is Peirce’s abduction? (cf. Sec. 11.4.1)• Five-minute hypothesis (cf. Sec. 11.5.2)• McTaggart’s paradox (cf. Sec. 11.5.3)• quantitative representation of ”Signiﬁer” and ”signiﬁed” (cf. Sec. 12.1)• The quantum linguistic understanding of ”Tractatus Logico-Philosophicus(=TLP)” and ”Zadeh’s fuzzy sets”, in other words, QL can answer the question,”What is a proposition?” (cf. ref. [81], or Sec. 12.1)

That is, Wittgenstein took over the baton of ”dualistic idealism (i.e., the mainstream of philosophy)” from Kant. This is essentially important, since Wittgenstein’s theory (i.e., TLP) must belong to dualistic idealism in order to assert (C)vii

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(or, Figure 0.1 (in Preface)).

• The proposal of QL logic (= fuzzy logic); Why does logic work in our world? (cf.
Sec. 12.1)
This question is proved within quantum language as a set of ”why does statistics
work in this world?”. Statistics is regarded as a kind of ”multi-valued logic”,”
which allows for more elaborate arguments than ordinary logic, which is a ”binary
logic,” but both are kinds of measurement.

• Solving analytic-synthetic distinction problem (Carnap-Quine controversy) (cf.
Sec. 12.3)• Lewis Carroll’s paradox (cf. Sec. 12.5)• Flagpole problem, (cf. Sec. 12.6)• Hempel’s raven paradox (cf. Sec. 12.7)

These are quantum linguistic solutions to unsolved problems in philosophy, but of course,
these are just a few of the most representative ones, and there are many more that could
be solved.
For the quantum linguistic solutions of unsolved problems in quantum mechanics,
statistical mechanics, statistics and probability theory, see List (D2 ) in Chap. 13: ”Postscript.”

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0.3: Why is philosophy strange?
As will be mentioned in Chap 1, QL is composed with two axioms (Axiom 1 concerning
measurement and Axiom 2 concerning causal relation) and the manual to use Axioms 1
and 2 (called ”Copenhagen interpretation”, or precisely, ”linguistic Copenhagen interpretation”). That is, (cf. refs [48, 74] ),(=quantum language)(E)

measurement theory[the manual to use Axioms 1 and 2]

Axiom 1: measurement
Axiom 2; causality

linguistic Copenhagen interpretation(Empirical theory)(Axiomatic theory)
Throughout this text, I assert the following.
=

+

(0.1)

(F) The scientiﬁc part of the history of Western philosophy was the history of theconstruction of QL (= Axioms 1 and 2 + the linguistic Copenhagen interpretation).

More precisely, we say thatAs will be shown throughout this text, the following can be seen from the ﬁgure
above.
(F1 ) Western philosophy is the history of the debate on Plato’s ”Idea”(F2 ) The history of Western philosophy has always been a mechanism to keep things
fresh (to avoid getting stuck in a rut) by alternating between the ”axiomatic”
and ”Copenhagen Interpretation” aspects.

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The discovery of QL changes the view of modern philosophy as follows. In other words,
rejecting ”Kant’s synthesis” and advocating the following ”QL synthesis” .
(G)







Develop Descartes theory Leibniz
Spinoza(=mind-body dualism)Monadology
Pantheism



•
, •



Solve






the mind-body problem
• −−−−−−−−−−−−−−−−−−−−−→

Axiomatic theoryQL WittgensteinContinental rationalism 


Axiom
1

Descartes theory





TLP


•
,
Axiom
2



mind-body dualism 
Derived from cogito















How
to
use
Locke,
Berkley,
QLDescartes theory
epistemology
linguistic




Solve
the
subjectivity(Copernican
Copenhagen
problem, etc.





•
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
→
revolution)≈interpretationEmpirical
theoryHume, Kant

British empiricism




QL

Figure 9.1 (in Chap. 9): QL synthesis

That is,(H) Western philosophies (i.e., the dualistic idealism from Plato to Wittgenstein) is
incomplete and immature, namely, pseudoscientiﬁc. And thus, they can only beunderstood from the perspective of quantum language. That is,
Without seeing the ﬁnished product, it is impossible to know exactly what it is.

(I1 ) Axiomatic theory (i.e., Axioms 1 and 2 ) in QL is too diﬃcult mathematically. Thus,
philosophies that were studied by Anselmus, Thomas Aquinas, Spinoza, Leibniz,
Wittgenstein, etc.) were very incomplete. However, their high aspirations wereadmired by philosophy enthusiasts.
(I2 ) The Copenhagen interpretation of quantum mechanics is tricky and exciting, andone does not have to be an expert to enjoy it. Thus, philosophies that were studied
by Augustinus, Descartes, Locke, Berkeley, Hume, Kant could introduce the generalpublic to the pleasures of philosophy. Therefore, I agree to the established opinion
such that the mainstream of western philosophy (after Descartes) is as follows.

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(I3 ) The readers will understand the question ”Why is always philosophy strange?” Thatis, the strangeness of philosophy (i.e., dualistic idealism) is due to the strangenessof the Copenhagen interpretation. In other words, it is due to the strangeness ofquantum mechanics (e.g. Schrödinger’s cat, etc.).

Thus,(J) Western philosophy has evolved, with refreshing changes, into a quantum language.
This refresh was also made possible by the alternating emergence of axiomatictheories and Copenhagen interpretation.
I think this is why Western philosophy has ﬂourished (and not fallen into a rut).

0.4: Supplementary information
Finally, I’d like to add a few things
This text is intended as a supplement to the main text of the graduate course, ref. [74],i.e.
(A1 ) ref. [74]: S. Ishikawa, ”Linguistic Copenhagen interpretation of quantum mechanics: Quantum Language [Ver. 5]”, Dept. Math. Keio University, 2019,
KSTS/RR-19/003, 473 pages(http://www.math.keio.ac.jp/academic/research_pdf/report/2019/19003.
pdf)
and thus, this text can be read without the knowledge of quantum theory. Also, themathematical arguments were left to ref. (A1 ). The low level of mathematics does notxiKSTS/RR-22/001
December 20, 2022

imply a low level of philosophy, and I am proud to say that the level of philosophy ofscience in this text is about the same as that of a graduate school.
This text is the revised version of ref. [82]:(K) ref. [82]: S. Ishikawa, History of Western Philosophy from the quantum theoreticalpoint of view [Ver. 4], Research Report (Department of mathematics, Keio
university, Yokohama), (KSTS-RR-21/001, 2021, 306 pages)(http://www.math.keio.ac.jp/academic/research_pdf/report/2021/21001.
pdf)
In [ver. 4] of this (K), I wrote in a way that could be read as ”ref. [74] in (A1 ) is primaryand ref. [82] in (K) is secondary,” but in this text, I emphasized that ”ref. [74] and thistext are equivalent in scholarly value”.
Philosophy, like literature, may be written for the general public as intellectual entertainment, but this does not necessarily diminish its academic value.
In this text, for this reason, the use of mathematical formulas has been kept to aminimum. This is because I believe that quantum mechanics without formulas is incomprehensible, but quantum language without formulas is comprehensible. This is because
quantum language without formulas is mostly Copenhagen interpretation.
In order to make the book as enjoyable to read as a picture book, we have used manyﬁgures and pictures. Moreover, as English is not my ﬁrst language, I believe that theﬁgures and pictures help me to convey my true meaning.
I hope many readers will enjoy this text.*2 .
December 2022
Shiro Ishikawa*2 For the further information concerning quantum language (e.g.

an improved version of this text
may be published), see my home page: https://ishikawa.math.keio.ac.jp/indexe.html

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xv

Contents
xx
Chapter 1
1.1
1.2

1.3

1.4

Chapter 2
2.1

2.2

2.3

2.4

The outline of quantum language (=measurement theory)
About Worldviewism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The overview of quantum language(=measurement theory) . . . . . . . . .
1.2.1 Axiom 1; What is probability (or, measurement)?) and Axiom 2(What is causality?) . . . . . . . . . . . . . . . . . . . . . . . . . .
Linguistic Copenhagen interpretation (or in short, Copenhagen interpretation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 What is the Copenhagen interpretation? . . . . . . . . . . . . . . .
1.3.2 Descartes ﬁgure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.3 The linguistic Copenhagen interpretation [ (E 0 )-(E7 ) ] . . . . . . . .
1.3.4 Descriptive power of quantum language . . . . . . . . . . . . . . . .
Appendix: The mathematical foundations of quantum language . . . . . .
1.4.1 Mathematical preparations (the C ∗ -algebraic formulation of QL) .
1.4.2 Axiom 1 [Measurement] and Axiom 2 [Causality] . . . . . . . . . .
1.4.3 Product observable, Product measurement . . . . . . . . . . . . . .
1.4.4 Inference; Fisher inference and Bayesian inference in classical QL .
1.4.5 Precise measurement, Tensor precise measurement . . . . . . . . . .
1.4.6 Inﬁnite Tensor Product Observable, Inﬁnite Tensor Product Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ancient Greek philosophy (before Socrates)
Thalēs (BC.640 - BC.546) . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Thalēs: the ﬁrst philosopher: ” the arche (= the ﬁrst thing of allthings) is water” . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Thalēs’ ability at math . . . . . . . . . . . . . . . . . . . . . . . . .
Pythagoras (BC.582 - BC.496) . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 The mathematical ability of Pythagorean religious organization . .
2.2.2 The arche (= the ﬁrst principle of all things) is number . . . . . . .
Hērakleitos and Parmenides . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Hērakleitos(BC.540 - BC.480) . . . . . . . . . . . . . . . . . . . . .
2.3.2 Parmenides; the ancestor of the Copenhagen interpretation . . . . .
2.3.3 Parmenides; The ancestor of the Copenhagen interpretation . . . .
2.3.4 Motion function method as a worldview . . . . . . . . . . . . . . .
Zeno (BC490 - BC430): The Motion Paradox . . . . . . . . . . . . . . . .
2.4.1 What is Zeno’s paradoxes?
Without a worldview, we cannot say
anything . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2* The solution⋆ about Zeno’s paradoxes (e.g., Flying arrow) in the
motion function method . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 DST formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
2
6
6
11
11
13
16
22
23
23
25
32
34
37
38
39
41
41
44
46
46
51
54
54
56
56
60
63
63
65
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2.4.4

Chapter 3
3.1

3.2
3.3

3.4

3.5

Chapter 4
4.1

4.2

4.3

Chapter 5
5.1
5.2

5.3

5.4

Appendix: The discussion about Zeno’s paradoxes (e.g., Achilles anda tortoise]) in the motion function method . . . . . . . . . . . . . .

67

The Big Three in Greek Philosophy (Socrates, Plato)Protagoras and Socrates . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Ethics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Magic proposition: I know I know nothing . . . . . . . . . . . . . .
Plato (BC.427 - BC.347) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 The theory of Ideas – Asserted ﬁction – . . . . . . . . . . . . . . . .
Three allegories about the Idea (the cave, anamnesis, the sun ) . . . . . . .
3.3.1 Allegory of the cave . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 The theory of anamnesis . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 The allegory of the sun: Measurement theoretical aspect of Idea theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.4* The measurement theoretical understanding of the allegory of the sun
Plato: The worldview called Idea theory; dualistic idealism . . . . . . . . .
3.4.1 Worldviewism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 The necessity of the worldview . . . . . . . . . . . . . . . . . . . . .
Appendix: How could Plato’s philosophy be the source of Western philosophy? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 History of development from ideas (Plato) to observables (quantum
language) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2 The history of Western philosophy is the history of the pursuit of”What is Idea?” . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The Big Three in Greek Philosophy (Aristotle)
Aristotle (BC.384 - BC.322) . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Realistic worldview vs. idealistic worldview . . . . . . . . . . . . .
4.1.2 Edios and Hyle . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Appendix: The history of Aristotle (Monistic realism) vs.Plato (Dualistic idealism) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Why does the motion happen? . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 From purpose to causality: Modern science started from the discoveryof ”causality” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Practical logic; Why does logic hold in this world? . . . . . . . . . . . . .
4.3.1 Practical logic; Aristotle’s syllogism in ordinary language . . . . . .
4.3.2 Mathematical logic; due to Boole, Frege . . . . . . . . . . . . . . .
4.3.3 Aristotle’s syllogism is not a syllogism . . . . . . . . . . . . . . . .
4.3.4 Set-theoretical worldview (= logical worldview) . . . . . . . . . . .
4.3.5 Appendix: Syllogism does not always hold in quantum systems? . .

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Around Alexandria; Hellenistic period
Around Alexandria; Hellenistic period . . . . . . . . . . . . . . . . . . . .
Euclid(BC.330 - BC.275) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Euclid geometry - Parallel postulate . . . . . . . . . . . . . . . . .
5.2.2 non-Euclidean revolution . . . . . . . . . . . . . . . . . . . . . . . .
Aristarchus (BC.310 - BC.230) . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 the diameter of the moon: the diameter of the sun . . . . . . . . .
5.3.2 Ancient Heliocentrism . . . . . . . . . . . . . . . . . . . . . . . . .
Archimedes (BC.287 - BC.212) . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Buoyancy (Archimedes’ principle) . . . . . . . . . . . . . . . . . . .
5.4.2 The tomb of Archimedes . . . . . . . . . . . . . . . . . . . . . . . .
5.4.3 Principle of leverage . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.5
5.6

Chapter 6
6.1

6.2

6.3

6.4

6.5

6.6

6.7

Chapter 7
7.1
7.2

7.3

7.4

7.5

Eratosthenes (BC.275 - BC.194) . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 The biggest ancient observer . . . . . . . . . . . . . . . . . . . . . .
Claudius Ptolemaios (AD.83 - AD.168) . . . . . . . . . . . . . . . . . . . .
5.6.1 A culmination of Ancient Astronomy . . . . . . . . . . . . . . . . .

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The Middle Ages
Augustinus(AD. 354 - AD.430) . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1 The benefactor who saved ”Idea theory” . . . . . . . . . . . . . . .
6.1.2* ”Confessions” by Augustinus: Only the present exists . . . . . . . .
6.1.3 ”Subjective time” is a magic word which excites our delusion . . .
Scholasticism –Bamboo (=Aristotle) cannot be grafted to a tree (=Plato) –6.2.1 Aristotle’s philosophy spread to the Islamic world . . . . . . . . . .
6.2.2 Crusade expedition and Inﬂow of Islamic culture . . . . . . . . . .
The discovery of zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Positional notation (= the discovery of zero): Arabic numerals . .
6.3.2 Arabic numerals and Roman numerals . . . . . . . . . . . . . . . .
6.3.3 The explosion of mathematics . . . . . . . . . . . . . . . . . . . . .
Anselmus; the problem of universals and the proof of the existence of God
6.4.1 Aristotelian Philosophy’s Inﬂow from Islam . . . . . . . . . . . . .
6.4.2 Review: the worldviewism . . . . . . . . . . . . . . . . . . . . . . .
6.4.3 The proof of the existence of God . . . . . . . . . . . . . . . . . . .
6.4.4 Overview of the problem of universals; see next section for details.
Scholasticism; Problem of universals . . . . . . . . . . . . . . . . . . . . .
6.5.1 What is the problem of universals? . . . . . . . . . . . . . . . . . .
6.5.2 Review of Sec 1.4 (Appendix: The mathematical foundations of quantum language) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.3 Scientiﬁc deﬁnition of ”universal” . . . . . . . . . . . . . . . . . . .
6.5.4 Examples: universals in Classical QL and Abstract Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.5 Detailed discussion: Problem of Universals in science . . . . . . . .
6.5.6 Primary universals in incomplete worldview . . . . . . . . . . . . .
The problem of universals in history of western philosophy . . . . . . . . .
6.6.1 Problem of universals in Scholasticism; Anselmus, Thomas Aquinas,
etc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ockham’s razor and Plato’s beard . . . . . . . . . . . . . . . . . . . . . . .
6.7.1 Statistics or quantum language: which will Occam’s razor cut
oﬀ?How sharp is Ockham’s razor? . . . . . . . . . . . . . . . . . . .

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Early modern – Scientiﬁc revolution: From Geocentrism to Heliocentrism
Paradigm shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Francis Bacon (1561 - 1626): The father of empiricism, Inductive reasoning
7.2.1 How to create science: The exclusion of idols (=prejudice, preconception) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
From Geocentrism to Heliocentrism . . . . . . . . . . . . . . . . . . . . . .
7.3.1 What is ”Geocentrism vs. Heliocentrism”? . . . . . . . . . . . . . .
7.3.2 Somehow ”from Geocentrism to Heliocentrism” . . . . . . . . . . .
7.3.3 ”Geocentrism vs. Heliocentrism” is the problem of the worldview .
7.3.4 The Galileo legend; Leaning Tower of Pisa, Trial of Galileo . . . . .
Principia; Newtonian worldview . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1 Principia (1687) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.2* What is ”Geocentrism vs. Heliocentrism”? After all, the worldviewism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix; About ”Dialogues concerning two new sciences” by Galileo Galilei

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Chapter 8
8.1

8.2

8.3

8.4

Chapter 9
9.1

9.2

9.3

9.4

9.5

9.6

9.7

Chapter 10
10.1

Modern philosophy: Continental rationalism: Descartes, Spinoza, Leibniz
Popularization of philosophy; Descartes=Kant philosophy . . . . . . . . .
8.1.1 Background of Descartes=Kant philosophy . . . . . . . . . . . . . .
8.1.2 Philosophy for the general public (≈ literary philosophy≈ science
without formulas≈ Copenhagen interpretation) . . . . . . . . . . .
The cogito proposition ”I think, therefore I am” is a catchphrase of modern
philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.1 Discourse on the Method (1637) . . . . . . . . . . . . . . . . . . . .
8.2.2* Regrettably, the cogito proposition is not a (scientiﬁc) proposition.
Descartes’ genius strategy . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.1 From the cogito proposition to mind-body dualism . . . . . . . . .
8.3.2 Descartes problem (= mind-body problem + subjectivity problem .
8.3.3 Two Geniuses: Thomas (Theory of Heaven) to Descartes (Theory of
Earth) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Continental Rationalism; Descartes, Spinoza, Leibniz . . . . . . . . . . . .
8.4.1 Spinoza: Pantheism . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.2 Leibniz: Monadology . . . . . . . . . . . . . . . . . . . . . . . . . .

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211
211
212
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215
218
221
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223

Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, Hume
Locke (1632 - 1704): The father of British Empiricism . . . . . . . . . . .
9.1.1 ”An Essay Concerning Human Understanding” by Locke (1689) . .
9.1.2 Tabula·Rasa (British empiricism) vs Nativism (Continental rationalism) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.3 Secondary quality ( and primary quality ) . . . . . . . . . . . . . .
* Leibniz-Clarke Correspondence: What is space-time?
. . . . . . . . . . .
9.2.1 “What is space?” and “What is time?” . . . . . . . . . . . . . . . .
9.2.2 Leibniz-Clarke Correspondence . . . . . . . . . . . . . . . . . . . .
9.2.3 Appendix: motion function method ( in classical system ) . . . . .
Subjective idealism: Berkeley, ”To be is to be perceived” . . . . . . . . . .
9.3.1 Priest: Berkeley . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.2 (A3 ): To be is to be perceived . . . . . . . . . . . . . . . . . . . . .
9.3.3 ”Einstein-Tagore Meeting” and ”Bohr-Einstein debates” . . . . . .
9.3.4 Bohr-Einstein debates: Do the laws of physics require measurement?
Qualia problem and Brain in a vat argument . . . . . . . . . . . . . . . . .
9.4.1 * The problem of qualia (= the subjectivity problem) . . . . . . . .
9.4.2* Brain in a vat argument . . . . . . . . . . . . . . . . . . . . . . . .
Hume; skeptic who didn’t measure, ”A Treatise of Human Nature” . . . .
9.5.1 Hume’s straying Brain pseudo-science (≈Less than brain science) ;Hume’s wordplay . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.5.2 Hume; The causality problem . . . . . . . . . . . . . . . . . . . . .
Hume’s problem of induction in the quantum mechanical worldview . . . .
9.6.1 If discussions are not based on a certain worldview, the discussions
will turn into chit-chat, and will not be scientiﬁc . . . . . . . . . .
9.6.2 The solution of Hume’s problem of induction; Without a worldview,nothing can be said . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.6.3 Examples (Coin toss, Emerald) . . . . . . . . . . . . . . . . . . . .
* Grue paradox cannot be represented in quantum language . . . . . . . . .
9.7.1 Grue paradox (explanation without quantum language) . . . . . . .
9.7.2 Grue paradox (explanation with quantum language) . . . . . . . .

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Kant: Copernican revolution
Critique of Pure Reason . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1.1 Three Critiques . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

10.2
10.3
Chapter 11
11.1

11.2

11.3
11.4
11.5

11.6

11.7

Chapter 12
12.1

12.2

12.3
12.4

10.1.2 The purpose of ”Critique of Pure Reason” (1781) . . . . . . . . . .
10.1.3 Thing-in-itself, Copernican revolution; from copy theory to constitution theory, Kantian synthesis . . . . . . . . . . . . . . . . . . . . .
10.1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
* Kant’s Copernican revolution: What is causality? . . . . . . . . . . . . .
10.2.1 Four answers to ”what is causality?” . . . . . . . . . . . . . . . . .
From Kantian synthesis to quantum linguistic synthesis . . . . . . . . . . .
Linguistic philosophy (Before TLP)
Elementary knowledge of symbolic logic in mathematics . . . . . . . . . .
11.1.1 Propositional logic and predicate logic . . . . . . . . . . . . . . . .
11.1.2 Which do you trust more, logic or statistics? . . . . . . . . . . . . .
Probabilistic symbolic logic in mathematics . . . . . . . . . . . . . . . . .
11.2.1 Easy example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2.2 Quasi-product probability . . . . . . . . . . . . . . . . . . . . . . .
11.2.3 Logic symbols and logical operations . . . . . . . . . . . . . . . . .
George Boole, Gottlob Frege and Bertrand Russell: Mathematics or philosophy?
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Peirce (1839-1914): Abduction . . . . . . . . . . . . . . . . . . . . . . . . .
11.4.1 * What is Peirce’s abduction? . . . . . . . . . . . . . . . . . . . . .
Bertrand Russell: ﬁve-minute hypothesis, McTaggart’s paradox, Moore’s
paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.5.1 Russell’s paradox in set theory . . . . . . . . . . . . . . . . . . . . .
11.5.2 * Five-minute hypothesis . . . . . . . . . . . . . . . . . . . . . . . .
11.5.3 * McTaggart’s paradox . . . . . . . . . . . . . . . . . . . . . . . . .
11.5.4 Moore’s paradox: ”It is raining, but I do not believe it is raining” .
Saussure: Copernican revolution in language . . . . . . . . . . . . . . . . .
11.6.1 Saussure’s linguistics: What comes ﬁrst, things or words? . . . . .
11.6.2 * The quantitative expressions of ”signiﬁed” . . . . . . . . . . . . .
Several Copernican revolutions [II] . . . . . . . . . . . . . . . . . . . . . .
11.7.1 Copernican revolution as a method of making idealism from realism
11.7.2 From Descartes to Kant: It is a progress? . . . . . . . . . . . . . .
Linguistic philosophy ( After TLP)
QL-logic (i.e., Fuzzy logic in QL); My scientiﬁc understanding of TLP . .
12.1.1 Wittgenstein: Tractatus Logico Philosophicus (=TLP) . . . . . . .
12.1.2 Zadeh and Kalman; The problem of universals in 20th century . . .
12.1.3 QL proposition (= fuzzy proposition in QL = TF-measurement) inclassical system . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1.4 Fuzzy logic in QL . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1.5 Calculations in QL-proposition (= TF-measurement ) . . . . . . . .
12.1.6 Fundamental theorem in QL logic . . . . . . . . . . . . . . . . . . .
12.1.7 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wittgenstein; the biggest star of analytic philosophy. . . . . . . . . . . . .
12.2.1 TLP attempted to answer the question, ”What is a proposition?” .
12.2.2 The power of Wittgenstein’s word: Linguistic turn . . . . . . . . . .
12.2.3 Philosophical Investigations (1953) and Wittgenstein’s paradox . .
Quine’s analytic-synthetic distinction in QL . . . . . . . . . . . . . . . . .
12.3.1 Analytic and Synthetic Propositions in QL . . . . . . . . . . . . . .
Popper’s falsiﬁability in the Copenhagen interpretation . . . . . . . . . . .
12.4.1 Popper’s falsiﬁability . . . . . . . . . . . . . . . . . . . . . . . . . .
12.4.2 Falsiﬁability for worldviews . . . . . . . . . . . . . . . . . . . . . .
12.4.3 Falsiﬁability in the Copenhagen interpretation . . . . . . . . . . . .

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12.5
12.6
12.7

12.8

Chapter 13
13.1
13.2

13.3
13.4

Lewis Carroll’s logical paradox . . . . . . . . . . . . . . . . . . . . . . . .
Flagpole problem: What is a scientiﬁc explanation? . . . . . . . . . . . . .
12.6.1 * The quantum linguistic solution of Flagpole problem . . . . . . . .
* Hempel’s raven paradox: A milestone in the philosophy of science . . .
12.7.1 Is ”the set of all tyrannosaurus” meaningful? : the set theoreticalworldview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.7.2 Hempel’s raven problem in the set theoretical worldview; What is the
problem? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.7.3 Hempel’s raven problem in the quantum mechanical worldview . .
Three approaches to the mind-body problem . . . . . . . . . . . . . . . . .
12.8.1 The mind-body problem . . . . . . . . . . . . . . . . . . . . . . .
12.8.2 The ﬁrst approach; Cognitive scientiﬁc approach . . . . . . . . . . .
12.8.3 The second approach; Illusory problem? . . . . . . . . . . . . . . .
12.8.4 * The third approach; Quantum linguistic solution to the mind-bodyproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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404
406
408
414
414
416
417
417

Postscript: Philosophy (of worldviews) has progressed towards quantum language : Philosophy is one
421
What is the core of philosophy of science? . . . . . . . . . . . . . . . . . . 421
Reviews from Plato to Wittgenstein . . . . . . . . . . . . . . . . . . . . . . 423
13.2.1 (C1 ): Plato . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
13.2.2 (C2 ): Augustines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
13.2.3 (C3 ): Anselmus and Thomas . . . . . . . . . . . . . . . . . . . . . . 424
13.2.4 (C4 ): Descartes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
13.2.5 (C5 ): Kant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
13.2.6 (C6 ): Wittgenstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
Almost all open questions of dualistic idealism have been solved; Wittgenstein’s dream came true . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
Philosophy of science is the scientiﬁc part of Western philosophy . . . . . 430
13.4.1 Dr. Hawking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
13.4.2 The purpose of philosophy of science is to create a language of science 430
13.4.3 QL wants rivals! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

References

435

Index

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1

Chapter 1

The outline of quantum language(=measurement theory)
Abstract: Readers who are completely unfamiliar with quantum languages (or, quantum mechanics) would be rather encouraged to skip this chapter and start reading in thenext chapter.
I have written this text in such a way that no prior knowledge of ”quantum language” isrequired to read Chapters 2 and beyond. If there are serious readers who are concerned
about this, it is enough to read through Section 1.1sim1.3. If you are a graduate student,
however, please read Section 1.4 ”Appendix. Mathematical Foundations of Axioms 1 and
2” and ”Quantum Language (ref. [74])” are recommended for graduate students.
Many parts of philosophy are enlightening, literary, and literary, and the domain of thescientiﬁc part is not that large. However, I believe:(♯) Western philosophy has always been dominated by scientiﬁc thought. To put it morebluntly,
Western philosophy’s scientiﬁc thinking = dualistic idealism = quantum language=the central thema of philosophy of science
The purpose of this text is to explain this (♯).
Suppose there were aliens whose civilization is more advanced than ours. They would
be more advanced than we are when it comes to math, physics, statistics, and AI. But
they might not have any concept of ”ethics” at all. Then again, they may not have any
so-called philosophy. But I think they do have a ”theory of dualistic idealism”. Because I
think that

[The theory of dualistic idealism] = [quantum language]≈[quantum mechanics] ≈ [statistics]
The contents of this chapter are as follows
Sec. 1.1: About worldviews
Sec. 1.2: quantum language ( Axiom 1 ( measurement) and Axiom 2 (causality)
Sec. 1.3: Copenhagen interpretation
Sec. 1.4: Appendix . Mathematical Foundations of Axioms 1 and 2

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1.1 About Worldviewism

1.1 About Worldviewism
Philosophy is usually based on worldviewism. Here, worldviewism is a way of stating
philosophy in the following form:
Worldviewism:
the world is so

Let’s think sotherefore

Worldview −−−−−→ Consideration, discussion(A)

For example, let’s take a very good worldview (Newtonian mechanics and statistics) toillustrate the above.
Kinetic equation + universal gravitation(i)

Newtonian mechanics
realistic worldview

therefore−−−−−→


heliocentrismcf. Chap.7

Heavy and light balls fall the same way cf. Sec.7.5



···

or,
The world can be represented in a sample space with parameters(ii)

Fisher statistics (cf. Note 1.3)
idealism

therefore−−−−−→

The person in the distance
wearing a skirt must be a woman.

cf. Sec.1.4.4
I found ﬁsh fossils in this mountain,
so I estimated that this placeused to be a sea cf. Sec.11.4



···

This text will focus on the following three types of worldviews (in particular, the idealistic
worldview).
(B) Worldview Typeworldview(i): realistic worldview (physics, Aristotle philosophy, etc.)(=Aristotelian worldview)







 (ii): idealistic worldview (Statistics, Plato, Descartes, Kant , etc. )(Plato worldview)







(logical worldview)(iii):
mathematics(=mathematicalthinking)(Set theory, etc. )*1(set thoretical worldview)2

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Chap. 1 The outline of quantum language (=measurement theory)
Let me explain [(i): realistic worldview (=realism)], [(ii): idealistic worldview (=idealism)] and [(iii): mathematics].
(i) A realist worldview (=realism)] is ”a statement that describes the laws of the (concrete) world as accurately as possible” and we can assume physics. As a matter ofcourse, ”the world comes ﬁrst, the worldview comes after”. this realistic worldview
can be pointed out by experiments to be wrong.
the truth that ” the world is so”

Let’s consider sotherefore−−−−−→ consideration, discussion

realistic worldview(ii) [Idealistic worldview (=idealism)] is ”a way of trying to understand the world
through sutra-like statements. We can assume statistics (which can be applied to
medicine, economics, engineering, or anything else). Thus, ”the worldview comesﬁrst and the world comes after”. The Idealistic worldview (since it is worded like asutra) makes no sense to determine right or wrong by experimentation, but ”usefulness (usefulness?)” is evaluated by ”usefulness”. In fact, statistics is very useful,as it is said to be the most powerful science.
the ﬁction (spell) that ”the world is so”

idealistic worldview

therefore−−−−−−−−−−−−−→
habitual thinking

Let’s consider so

consideration, discussion(iii) We do not consider [math] to be a worldview, but for convenience we will assume
its position here. I wrote an excuse in Note1.1.
The worldview that this text mainly deals with is ”quantum language (= quantum mechanical worldview),” which is a dualistic version of ”statistics. Therefore, quantum language is a dualistic idealism, the same as the mainstream of Western philosophy: Plato→ Augustine → Anselmus → Thomas Aquinas → Descartes → Kant → Wittgenstein.
Thus, throughout this text, I will explore the relationship between the genealogy ofidealism and quantum language in the ﬁgure below.

*1 It is impossible to consider mathematics as a kind of worldview, but I will classify it here in thesense of ”mathematical thinking.” (cf. Sec.4.3.4: Set-theoretic Worldview, Logical Worldview)3

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1.1 About Worldviewism

Now, the following are nearly identical.
• To understand the physics (worldview) of Newtonian mechanics• To master the language of Newtonian mechanics• To understand the logic (propositions) of Newtonian mechanics
Thus, ”worldview” and ”language” are almost synonymous.
Wittgenstein said in [ref. [125]: ”Tractatus Logico-Philosophicus (abbreviated as
TLP)”(1921)],(C) ”The limits of my language mean the limits of my world.”
This implies that(D) What cannot be spoken in my language (i.e., my worldview)

does not exist.
Namely,(E1 )Without a worldview, there is no logic or proposition.

In the same sense,(E2 )Without a worldview, nothing can be said(or, only chit-chat can be said).

♠Note 1.1. (i): The (scientiﬁc) worldview is made up of statements that relate the world tomathematics, but as Hempel’s paradox ”All crows are black” shows, the relation between
set theory and the world must be somewhat unnatural. The question, ”Can mathematics
itself be a worldview? is a somewhat tricky question. In this text, mathematics alone is
not considered a worldview. However, since we usually think of ”natural numbers” and”space” as existing in the real world when we actually do mathematics, it is not necessarily
wrong to think that mathematics and the world are connected even without a worldview.
(ii): The question, ”Is there a complete worldview? is a serious question (cf. Sec. 12.2.3).
Wittgenstein’s paradox). However, since we cannot move on if we dwell on these fundamental issues, we will proceed in the spirit of ”keeping a lid on what smells bad.
♠Note 1.2. [the deﬁnition of ”existence”]
The word ”existence” is polysemous and diﬃcult to use. For example.
• Does God exist?
• Do Atoms Exist?
• Does Beauty exist?
• Does Mathematics Exist?
• Does Self exist?
• Does the mind exist?
• Does the past exist?
• Does ”what is in front of us” exist?
These show the diﬃculty to use the word ”existence”.

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Chap. 1 The outline of quantum language (=measurement theory)
In this text, I assume that
• When we are discussing a certain worldview, we assume that the keywords
for that worldview exist.
For example, with Descartes’ mind-body dualism, ”mind,” ”body,” and ”thing” exist.

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1.2 The overview of quantum language(=measurement theory)

1.2 The overview of quantum language(=measurement theory)
1.2.1 Axiom 1; What is probability (or, measurement)?) and Axiom 2 (Whatis causality?)
The idea of quantum language (also known as ”measurement theory”) is inspired andstructured by quantum mechanics, in which microphenomena are analyzed. Quantum
language is a language, by which we cannot only describe quantum mechanics but also
most sciences (e.g., economics, psychology, engineering, etc.). However, it should be notedthat quantum language is not almighty, for example, the theory of relativity is beyond
the description of quantum language. Nevertheless, I believe that quantum language will
be widely accepted as the language of science.
The framework of measurement theory (=quantum language) is simple, that is, it iscomposed with two axioms (Axiom 1 concerning measurement and Axiom 2 concerningcausal relation) and the manual to use Axioms 1 and 2 (called the linguistic Copenhagen
interpretation). That is, (cf. refs [48, 74] ),(=quantum language)[Axiom 1][Axiom 2]

measurement theory = measurement + causal relation
linguistic Copenhagen interpretation+[the manual to use Axioms 1 and 2](1.1)
That is,Although you do not need to fully understand Axiom 1 and Axiom 2 to read this text, I,
for completeness, mention them as follows. (For the mathematical foundations of Axioms
1 and 2, see Section 1.4 (Appendix: the mathematical foundations of quantum language) later. The following axiom can be regarded as the mathematical generalization of theprobabilistic interpretation of quantum mechanics (due to M. Born (cf. ref. [6])).

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Chap. 1 The outline of quantum language (=measurement theory)
Axiom 1 [measurement] (in Section 1.2.1)(This will be explained more precisely in Section 1.4 )With any system S, a C ∗ -algebraic basic structure [A ⊆ B(H)] can be associatedin which measurement theory of that system 
can be formulated. When the observerXtakes a measurement M(O=(X, 2 , F ), S[ρ] ) i.e., a measurement of an observable(or, by a measuring
instrument) O=(X, 2X , F ) for a system S[ρ] i.e. a system Swith a state ρ) , the probability that a measured value x (∈ X) is obtained by themeasurement is given by ρ(F ({x}))(≡ A∗ (ρ, (F ({x}))A ).

♠Note 1.3. [The relation between Quantum Language and Statistics](i): In the above, let ρ(F ({x})) be denoted by Pρ ({x}). And let (X, 2X , Pρ (·)) be regarded
as a sample space with parameter. If we start from this point (i.e., (X, 2X , Pρ (·) ), we will
be in statistics. This is because”statistics = theory of probability space with parameters”
That is,quantum language
X

M(O=(X, 2 , F ), S[ρ] ) )philosophy

statistics−−−−−−−−−−−−−−−−−−−−−−−→
Elimination of observable O(X, 2X , Pρ (·))
applied mathThe elimination of an observable O implies the elimination of a philosophy. That is,• Philosophy = ”The Theory of Plato ideas (= observable)”(ii): Let me be a little more speciﬁc. Statistics is mostly a theory of normal distributions,so let’s use the following example of a normal distribution.
Z(x − µ)2
1√
Nµ,σ [Ξ] =]dx (Ξ ∈ B(R) =the Borel ﬁeld of the real line R)
exp[−
2σ 2
2πσ 2
Ξ
Then, we have the sample space with the paramers µ, σ(∈ R

× R+ ) such that

(R, B(R), Nµ,σ [ · ])This is the starting point of statistics.

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1.2 The overview of quantum language(=measurement theory)
The following is a mathematical generalization of Heisenberg’s kinetic equation (≈Schrödinger equation).
Axiom 2 Causality [Axiom 2; causality](This will be explained more precisely in Section 1.4 )Let T be a tree (i.e., semi-ordered tree structure). For each t(∈ T ), a C ∗ -algebraic
basic structure [At ⊆ B(Ht )] is associated. Then, the causal chain is represented
by a sequential causal operator {Φt1 ,t2 : At2 → At1 }(t1 ,t2 )∈T 2 , where it holds that≦

Φt1 ,t2 Φt2 ,t3 = Φt1 ,t3 (t1 ≤ t2 ≤ t3 )..

Note that this axiom is the mathematical generalization of Schrödinger equation (i.e.,
Heisenberg’s kinetic equation).

Here, note that(A) the above two axioms are kinds of spells (i.e., incantation, magic words, metaphysical statements), and thus, it is impossible to verify them experimentally. What
we should do is not to understand the two, but to learn the spells (i.e.,
Axioms 1 and 2) by rote.
In this sense, quantum language is a kind of metaphysics (≈ idealism). Therefore,(B) The formation of quantum languages depends on human marvelous language ability♠Note 1.4. (i): In this text, we assume that [metaphysics] = [idealism] = [theory nottestable by experiment]. For example, most philosophy, mathematics, statistics, etc. aremetaphysics.
On the other hand, [realism] = [theory that can be tested]. Examples include Newtonianmechanics, relativity, and quantum mechanics. Thus, quantum mechanics is a realism,
but quantum language, which is a verbalization of quantum mechanics (or, the Copernican
revolution), is a metaphysics*2 .

*2 For the Copernican revolution, see Sec.10.2 and Sec.11.7.

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Chap. 1 The outline of quantum language (=measurement theory)(ii): If metaphysics did something wrong in the history of science, it is because metaphysics
attempted to answer the following questions seriously in ordinary language:(♯1 ) What is ⃝⃝ (e.g., ⃝⃝ = measurement, probability, causality, space-time, etc.)?
Although the question (♯1 ) looks attractive, it is not productive. What is important is
to create a language to deal with the keywords. So the (♯1 ) has to be replaced by thefollowing (♯2 ).
(♯2 ) How should ⃝⃝ (e.g., ⃝⃝ =measurement, probability, causality) be used?
The problem (♯1 ) will now be solved in the sense of (♯2 ). If there are some failure in
the history of philosophy, philosophers failed to propose a suitable language. It should
be noted that Newton’s success is due to the proposal of ”the language called Newtonianmechanics”. Recall that Newton mechanics does not answer the question ”What are mass,force, acceleration?”, but ”How should they (i.e., mass, force, acceleration) be used?” orequivalently, ”How do they relate to each other?”. That is,”[mass] · [acceleration] = [force].”Thus, I think that Newton’s most important accomplishment is the following three (which
are equivalent):(♭1 ) to create physics called Newtonian mechanics.
(♭2 ) to create the worldview called Newtonian mechanics.
(♭3 ) to create the language called Newtonian mechanics.
where these are equivalent: i.e. (♭1 )≈(♭2 )≈(♭3 ).
Similarly I think that the greatest task of philosophy is the following three: from theidealistic point of view,(♮1 ) to create a philosophy called □ □(♮2 ) to create a worldview called □ □(♮3 ) to create a language called □ □where these are equivalent: i.e. (♮1 )≈(♮2 )≈(♮3 ). Our purpose is to show that □ □ =quantum language.

Thus we can answer the following problem from the quantum linguistic point of view:
Problem 1.1. Scientiﬁcally please answer to the problem:(♯) What is measurement, probability, causality?
[Answer]:
As mentioned in Note 1.4, the question ”What are measurement, probability, and causality?” should be interpreted as the question ”How do we use the terms measurement,
probability, and causality?” If so, it suﬃces to propose a worldview [=language) withthe words measurement, probability, and causality as keywords. But if so, since we havealready proposed ”quantum language (= Axiom 1(measurement) + Axiom 2 (causality)]” and know that ”measurement, probability, and causality are keywords of quantum
language”, we already know that the answer to the question (♯) is a ”quantum language”.
Of course, the most important thing is ”conﬁrmation of the great descriptive power ofquantum language”, which I think you can understand by reading this text and ref. [74].
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1.2 The overview of quantum language(=measurement theory)
If there is a reader who does not accept this, he/she should create a ”language more
powerful than quantum language”.
♠Note 1.5. (i): The above problem is the most diﬃcult unsolved problem in the history ofphilosophy. The readers may be surprised at how easily this can be solved. I think many
readers already know how powerful it is to create a proper language as the great successof Newtonian mechanics. Again, recall that Newtonian mechanics never answer to ”What
are forces, accelerations, and masses?”. What Newtonian mechanics answers to is ”Howare used the terms: forces, accelerations, and masses?”(ii): Recall the quotes in ”Tractatus Logico-Philosophicus” ( by Wittgenstein ) such that(♯) The limits of my language mean the limits of my world.
which says that ”world=language”. However, he could not create ”his language”.

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Chap. 1 The outline of quantum language (=measurement theory)

1.3

Linguistic Copenhagen interpretation (or in short,
Copenhagen interpretation)

1.3.1 What is the Copenhagen interpretation?
In the previous section, an overview of quantum language [Axiom 1 (measurement) andAxiom 2 (causality)] was outlined.
(A)(=measurement theory)[Axiom 1][Axiom 2]

quantum language(=QL) = measurement + causal relation+(linguistic) Copenhagen interpretation[the manual to use Axioms 1 and 2](1.1)
In this section, the ”Copenhagen interpretation ((linguistic) Copenhagen interpretation)”
will be explained. Before doing so, let us reiterate the following(B1 ) Axioms are a kind of incantation (spell, magic word, metaphysical statement) andcannot be experimentally veriﬁed
Further,(B2 ) Quantum language is a language, and you may not be able to use it well at ﬁrst.
You can only acquire the ability to use it through practice and trial and error.
♠Note 1.6. In Mermin’s book [101], he said
• If I were forced to sum up in one sentence what the Copenhagen interpretation says
to me, it would be “Shut up and calculate”• Stop being bothered!
Also, D. Howard said in [34]:• Even within the Copenhagen School, there was a wide range of opinion on the Copenhagen interpretation. For example, there was disagreement about ”wave function collapse” which is supposed to be the central theme of the Copenhagen interpretation.

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1.3 Linguistic Copenhagen interpretation (or in short, Copenhagen interpretation)(See ref. [65] for my opinion on ”wave function collapse.”)This means that
”What is the Copenhagen interpretation?” is unsolved problem.
We believe that this is one of the most important unsolved problems in science. Thus, I
can say that one of the purposes of this text is to answer the unsolved problem: ”What is
the Copenhagen interpretation?”. I assert that the linguistic Copenhagen interpretation is
the true Copenhagen interpretation (cf. ref. [74] ). That is, we assert that the Copenhagen
interpretation is justiﬁed in philosophy (i.e., language) and not in physics.
Thus. in this text, ”Copenhagen interpretation” is identiﬁed with ”linguistic Copenhagen
interpretation”.

If so,(C1 ) It is essential to acquire a habitual thinking to master the axioms (Axioms 1 and
2). In order to master the quantum language as quickly as possible, you will needa good manual for mastering the axioms (Axioms 1 and 2)
Thus, we get the following deﬁnition,
Deﬁnition 1.2. [Copenhagen Interpretation (=Linguistic Copenhagen Interpretation)](C2 )Copenhagen Interpretation:= Manual for using spells (Axiom 1 and Axiom 2)
def.

The above can be rephrased as drawing a line between ”good use (i.e., scientiﬁc use)”and ”bad use (i.e., pseudo-scientiﬁc use)” as follows.
(C3 ) Rules for drawing the line between scientiﬁc and pseudoscientiﬁc propositions
Checking the validity of this deﬁnition is one of the key objectives of this text.
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Chap. 1 The outline of quantum language (=measurement theory)

1.3.2 Descartes ﬁgure
Now, let’s go on to explain the Copenhagen Interpretation.
Since Axiom 1 includes the term ”measurement”, the concept of ”measurement” shouldbe, at ﬁrst, understood in dualism (i.e., ”observer” and ”measuring object”) as illustrated
in Figure 1.3.
a ○)”b in dualism
Figure 1.3. [Descartes Figure]: Image of ”measurement(=○+In the ﬁgure, ”measurement” is characterized as interaction between ”observer” and”system”(matter or object to be measured, measuring object), composed ofa projection of light onto the object (i.e., someone, not necessarily an observer,(D1 ) ○
shines the light.)
b perception of the reaction of the object (i.e., the observer receives the reaction.)○

However, I want to emphasize that the interaction cannot be represented by kinetic
equations. Therefore,(D2 ) in measurement theory (= quantum language), we use the term ”measurement”
instead of ”interaction”. Therefore, we won’t say the above (D1 ) outright.
After all, we think that:(D3 ) there is no measured value without observer.
Thus, measurement theory is composed of three keywords:
measured value ,(observer, brain, mind)

observable (≈ measuring instrument)(telescope, thermometer, eye, ear, body, polar star)

,

state
(matter, measuring object)

In view of the above ﬁgure, it might be called ”ternary relation (or, trialism)” instead of”dualism”. But, following the convention, we use ”dualism” throughout this text.
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1.3 Linguistic Copenhagen interpretation (or in short, Copenhagen interpretation)♠Note 1.7.
(i): Descartes’ dualistic idealism has the following form:[A](mind) ←− [B(body, sensory organ)] −→[C](matter)(medium)

This corresponds to measurement theory (= quantum language) as follows.
\[A](= mind)[B](Mediating of A and C)(body)[C](= matter)quantum language

measured value

observablestate[system](ii): If the interaction of (D1 ) could be described by the equation of motion, the relationshipbetween ”observer” and ”object” would become obvious, and we would end up with amonism of ”observer + object”. Thus, we assume that (D1 ) can not be described bykinetic equation.

(iii): The concept of ”observable” (which can be identiﬁed with ”measuring instrument”)
is not easy. For example, telescopes, glasses and eyes are a type of measuring instrument.
A directional magnet is, of course, a measuring instrument. If so, then the polar star isalso a type of measuring instrument.

(iv): The following argument, called the Bohr-Einstein debates, is extremely profound (cf.
Sec.9.3.3 Bohr-Einstein debates).
• For example, assume that an apple is put on food scale. We are not sure if he is trying
to weigh an apple or make sure the food scale isn’ t broken (see Figure below). Thus,
the diﬀerence between ”body” and ”matter” is arbitrary. This is determined by theobserver. Both ”body” and ”matter” obey the laws of physics. However, the laws
governing ”mind” are assumed to be unknown. Maybe there is no such thing. This is
the position of monism. On the other hand, the dualist position is that measurement
is essential.

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Chap. 1 The outline of quantum language (=measurement theory)

Einstein is the former position (i.e., monism) and Bohr is the latter (i.e., dualism). It’s thebiggest scientiﬁc debate of the 20th century, but it’s not yet settled. My standing point is
as follows.
• These two (monism and dualism) are not to be uniﬁed. As mentioned in Figure 0.1
in Preface, they should be compatible.
(v): By the way, Merleau-Ponty (1908 - 1961) might think in the following manner.
• I shake hands with my right hand and the left hand. In this case, if I regard theright hand as the measuring instrument, I feel the existence of my left hand. On thecontrary, if I regard the left hand as the measuring instrument, I feel the existence of
my right hand.

Thus, ”observable” is a treasure trove of wordplay. In fact, as will be shown throughout
this text, I think that
• The history of Western philosophy is the story of the maturation of ”Plato’s Idea”
into ”observables” through twists and turns.
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1.3 Linguistic Copenhagen interpretation (or in short, Copenhagen interpretation)

1.3.3 The linguistic Copenhagen interpretation [ (E0 )-(E7 ) ]The (linguistic) Copenhagen Interpretation is ”a manual for using Axiom 1 (measurement) and Axiom 2 (causality). If that were the case (if it were a manual), wouldn’t wehave to list all sorts of miscellaneous things and ”there would be no end to the explanations”? Even car driving manuals are endless in detail. There is no such thing as acomplete rulebook for baseball or soccer, either. The author believes that the Copenhagen
Interpretation may have such a fear (cf. Sec. 12.2.3 : Wittgenstein’s paradox).
Now, below [(E0 )∼(E7 )], I will brieﬂy explain the Copenhagen interpretation. The mostimportant of these, and especially important, is,(E4 ) Only one measurement is permitted.
(E):The linguistic Copenhagen interpretation

With Descartes ﬁgure 1.3 (in Sec. 1.3, i.e.

and the following (E0 )-(E7 ) in mind,
describe every phenomenon in terms of Axioms 1 and 2!
(E0 ) (i); If you don’t measure it, you don’t know anything. Don’t talk about things
that cannot be measured. This seems to be Berkeley’s saying ”To be is to beperceived.” On the other hand, Einstein, a monistic realist of the anti-Copenhagen
interpretation, said ”The moon is there even when we are not looking”.

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Chap. 1 The outline of quantum language (=measurement theory)(ii); [Popper’s Falsiﬁability; ref. [105]]
In order to guarantee the objectivity of a scientiﬁc theory, there must be a possibility that the hypothesis will be disproved by experiment or observation. Thatis, truth must always be subjected to experiments that deny its truth. And if thedenying experiment is conﬁrmed, then the truth must be rejected.
♠Memo 1.1. It is impossible to consider Popper’s falsiﬁability as a ”foundational principleof science”. This is because statistics (and quantum language) do not have disfalsiﬁabilit.
Therefore, it is more acceptable for scientists to consider disprovability as one rule in
the Copenhagen interpretation (cf. Sec. 12.7.3).

(E1 ) Consider the dualism composed of ”observer” and ”matter (= object to be measured)”, where ”observer” and ”matter (= measuring object)” must be absolutely
separated. Figuratively speaking, ”Audience should not go on stage”, or”the observer cannot measure the observer himself”
or”The measurement is not dependent on the observer”
That is, the following qualia problem is non-sense.

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1.3 Linguistic Copenhagen interpretation (or in short, Copenhagen interpretation)
To be more speciﬁc, the words ”I”, ”Here”, ”Now” are forbidden . Hence, ”I
think, therefore I am” is non-sense.
♠Note 1.8. Consider the followings:(♯1 ) I measure my body temperature with a thermometer.
(♯2 ) I feel my body feverish.
and(♭1 ) The doctor measures my body temperature with a thermometer.
(♭2 ) The doctor feels my body feverish.
In terms of measurement, (♯1 ) and (♭1 ) are the same. On the other hand, (♯2 )and (♭2 )are diﬀerent. Thus, in the strictest sense, we consider that (♯2 ) cannot be regarded asa measurement. However, the (♭2 ) seems to be a measurement. This example will help
you understand that cogito proposition ”I think, therefore I am” in Chapter 8.

(E2 ) While ”matter” is in the space-time, the observer is not (Sec.9.2: Leibniz’s relationalism concerning space-time). That is, observer’s space-time does not exist.
Thus, the question: ”When, where and by whom was the measured value obtained?” is out of the scope of measurement theory. Thus, words such as ”now,””here,” and ”I” should not be used in a scientiﬁc proposition. If you are going touse it, you need to be very careful. That is,there is no tense either in measurement theory or in science.
The ”tense” is a treasure trove of word play (cf. Augustinus ”Only the present
exists”(Sec. 6.1), McTaggart’s paradox, Russell’s ﬁve-minute hypothesis in Chap.
11).

(E3 ) In measurement theory, ”observable(=measuring instrument ≈ body)” is the mostimportant than ”measured value(≈mind)” and ”state(≈matter)” in (D3 ). The
prototype of observables is Plato’s Idea. Also, statistics is not philosophical because it does not have ”observables” (cf. Note 1.3).
(E4 ) Only one measurement is permitted.
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Chap. 1 The outline of quantum language (=measurement theory)

Thus, the state after measurement (or, wave function collapse, the inﬂuence ofmeasurement) is meaningless.

For the virtual wave function collapse, see:
• ref.[65] S. Ishikawa, Linguistic interpretation of quantum mechan5, No.4, 150-155, 2015, DOI:
ics; Projection Postulate, JQIS, Vol.
10.4236/jqis.2015.54017(http://www.scirp.org/Journal/PaperInformation.aspx?PaperID=
62464)♠Note 1.9. Ref.[65] is powerful as follows. The Schrödinger cat is the most famousparadox in quantum mechanics. However, we are not bothered by this paradox since
the state after measurement is not described in quantum language.

(E5 ) There is no probability without measurement. Also, the measurement cannot bemeasured.
There is no logic without measurement. See Figure 0.1 in Preface such as
⃝14Analytic philosophy ( logic) −−−−−−−−−−−→ (See Chapter 12)  ⃝15−−→ QL (= Measurement theory)


⃝
9


statistics ( probability) −−−−−−−−−−−→(E6 ) There is one state and it never moves. Therefore, there is no time (time is just anordered parameter cf. Sec. 9.2). Therefore, we always use the Heisenberg picture(basically we do not use the Schrödinger picture), etc. It is still surprising that
Parmenides mentioned almost all of the Copenhagen interpretations 2500 years
ago (cf. Sec. 2.3).

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1.3 Linguistic Copenhagen interpretation (or in short, Copenhagen interpretation)and so on.
If we believe that quantum language is the ﬁnal destination of dualistic idealism (cf.
11 and 115○○ in Figure 0.1 (in Preface)), it seems natural to think as follows(E7 ) most maxims of the philosophers (particularly, the dualistic idealism) can be regarded as expressions in linguistic Copenhagen interpretation.
♠Note 1.10. To become skillful in the Copenhagen interpretation, it is not enough tomemorize (E0 )∼(E7 ), but rather to do a lot of exercises. This may be the quickest way toread ref. [74].

Remark 1.4. [Why ”philosopher’s golden rule = Copenhagen Inter-

pretation”?]
Why ”philosopher’s golden rule = Copenhagen Interpretation”? I insist on the diagram
below. That is, we reject the pseudoscientiﬁc ”Kantian synthesis” and insist on thefollowing ”QL synthesis”(cf. Chap. 8).






Develop Descartes theory Leibniz
Spinoza(=mind-body
dualism)
Monadology• Pantheism , •



Solve






the mind-body problem
• −−−−−−−−−−−−−−−−−−−−−→

Axiomatic theory

QL





WittgensteinContinental
rationalism
Axiom 1Descartes theory





TLP

, Axiom 2•
mind-body dualism 





Derived from cogito















How to use
Locke,
Berkley,
QLDescartes theory
epistemology
linguistic




Solve
the
subjectivity(Copernican
Copenhagen problem, etc.







revolution)• −−−−−−−−−−−−−−−−−→≈ interpretationEmpirical
theoryHume, Kant

British empiricism




QL

For example, Descartes’ ”I think therefore I am”, Locke’s ”Secondary quality”,
Berkeley’s ”To be is to be perceived”, Kant’s ”Copernican revolution”, Wittgenstein’s
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Chap. 1 The outline of quantum language (=measurement theory)”The limits of my language mean the limits of my world”, ”What we cannot speak about
we must pass over in silence” are the products of the pursuit of dualistic idealism. If
so, it is a matter of course that the ideas of British empiricism and the Copenhagen
interpretation become similar, so [QL integration] is natural (cf. Sec. 10.3).
The above says that
(F1 ) Both Cartesian-Kantian philosophy and the linguistic Copenhagen interpretation
are products of the pursuit of dualistic idealism.
Then, it is natural to consider
maxims of philosophers ≈ the linguistic Copenhagen interpretation.
(F2 ) : Now we can answer the following interesting question:

Why is philosophy (= dualistic idealism) preposterous?
BecauseThe linguistic Copenhagen interpretation is preposterous.

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1.3 Linguistic Copenhagen interpretation (or in short, Copenhagen interpretation)

1.3.4 Descriptive power of quantum language
We assert the following.

Assertion 1.5. [descriptive power of quantum language](cf. ref. [74])The descriptive power of quantum language (=QL) is quite large. The following can besaid about it.
Descriptive power of quantum language≈descriptive power of statistics+ descriptive power of quantum mechanics
Therefore, we assert that(G1 ) quantum language is a language, by which most sciences (e.g., economics, psychology, engineering, etc.) are described

Quantum language is a language, by which we cannot only describe quantum mechanics but also most sciences (e.g., economics, psychology, engineering, etc.). Thus, as a
modiﬁed version of ”statistics is the language of science”, we consider that(G2 ) Quantum language is the language of science,or,
to do sciences is to describe phenomena by quantum language
However, it should be noted that quantum language is not almighty, for example, thetheory of relativity is beyond the description of quantum language.

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Chap. 1 The outline of quantum language (=measurement theory)

1.4

Appendix: The mathematical foundations of quantum
language

Literary enjoyment is also one of the charms of philosophy (e.g., references [23, 116]). If
you want to read this book as literary, it might be recommended to skip this section.
Therefore, I have tried to write this book in such a way that you can enjoy it even if you
skip this section (the mathematics part).
However, since we are interested in the ”scientiﬁc side of philosophy”, it is necessary tomention some scientiﬁc theory.
Although we will explain ”quantum language (= measurement theory”) in this section,
I may have omitted too much. This section alone may not be suﬃcient to read themathematical part of this text. For the precise explanation, see the references [46, 74].
Also, ref. [48] is easy to read.
We have two formulations of QL such that
 (A1 ): the C ∗ -algebraic formulation (in Section 1.4)(A) the formulation of QL(A2 ): the W ∗ -algebraic formulation (cf. ref. [74])(A1 ) is simple, on the other hand, (A2 ) is rather mathematical. In this text, we usually
use the C ∗ -algebraic formulation (A1 ).

1.4.1 Mathematical preparations (the C ∗ -algebraic formulation of QL)
Following refs. [46, 48, 49, 74] (all our results until present are summarized in ref. [74]),
we will review quantum language, which has the following form:
Quantum language = measurement + causality + Copenhagen interpretation(Axiom 1)(= measurement theory)(Axiom 2)(how to use Axioms 1 and 2)(1.2)

which asserts that ”measurement” and ”causality” are the most important concepts inscience.
Consider an operator algebra B(H) (i.e., an operator algebra composed of all boundedlinear operators on a Hilbert space H with the norm kGkB(H) = sup∥u∥H =1 kGukH . Let
A(⊆ B(H)) is a C ∗ -algebra, (i.e., norm-closed subalgebra of B(H)) (cf. refs. [121], [118],
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1.4 Appendix: The mathematical foundations of quantum language[128]). Our purpose of this section is not to explain QL in general situation but to explain
QL in an understandable setting. Thus, from here, we devote ourselves to the following
simple cases:(B1 ): quantum QL (when A = B(Cn ), where H = Cn )
i.e. the C ∗ -algebra composed of all (n × n) complex matrixes(B) QL =(B2 ): classical QL (when A = C(Ω)),

i.e. the space of all continuous functions on a compact space Ω
The pair [A, B(H)] is called a C ∗ -algebraic basic structure. Let A∗ be the dual Banach
space of A. That is, A∗ = {ρ | ρ is a continuous linear functional on A }, and the normkρkA∗ is deﬁned by sup{|ρ(G)(= A∗ hρ, GiA )| | G ∈ A such that kGkA (= kGkB(H) ) ≤ 1}.
Deﬁne the mixed state ρ (∈ A∗ ) such that kρkA∗ = 1 and ρ(L) ≥ 0 for all L ∈ A suchthat L ≥ 0. And deﬁne the mixed state space Sm (A∗ ) such that
Sm (A∗ )={ρ ∈ A∗ | ρ is a mixed state}.
A mixed state ρ(∈ Sm (A∗ )) is called a pure state (or simply, state) if it satisﬁes that“ρ = θρ1 + (1 − θ)ρ2 for some ρ1 , ρ2 ∈ Sm (A∗ ) and 0 < θ < 1” implies “ρ = ρ1 = ρ2 ”.
Put
Sp (A∗ )={ρ ∈ Sm (A∗ ) | ρ is a pure state},
which is called a state space. It is well known (cf. ref. [118]) that∗
 [Case (C1 )]; Sp (Bc (H) ) = {ρ = |uihu| (i.e., the Dirac notation) | kukH = 1}(C)∗[Case (C2 )]; Sp (C(Ω) ) = {ρ = δω0 | δω0 is a point measure at ω0 ∈ Ω} ≈ Ω.
In [Case (C2 )], under the following identiﬁcation:∗

Sp (C(Ω) ) ∈ δω ↔ ω ∈ Ω,
ω and Ω is respectively called a state and a state space.
Deﬁnition 1.6. [Observable, Image observable] According to the noted idea (cf.
refs. [14], [33]), an observable O =(X, P(X), G) in A is deﬁned as follows:(i) X is a ﬁnite set, P(X)(= 2X , the power set of X).
(ii) [Additivity] G is a mapping from P(X) to A satisfying: (a): for every Ξ ∈ P(X),
G(Ξ) is a non-negative element in A such that 0 ≤ G(Ξ) ≤ I, (b): G(∅) = 0
and G(X) = I, where 0 and I is the 0-element and the identity in A respectively.
(c):[additivity]G(Ξ1 ) + G(Ξ2 ) = G(Ξ1 ∪ Ξ2 )T
for all Ξ1 , Ξ2 ∈ P(X) such that Ξ1 Ξ2 = ∅.
If G(Ξ) = G(Ξ)2 (∀Ξ ∈ P(X)), then O =(X, P(X), G) in A is a projective observable (or,crisp observable). Also, O =(X, P(X), G) in A is also called an X-valued observable. Let
Y be a ﬁnite set, and let Θ : X → Y be a map. Then, Θ(O) =(Y, P(Y ) , G(Θ−1 (·)) in A
is also an observable in A ( which is called an image observable).
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Chap. 1 The outline of quantum language (=measurement theory)♠Note 1.11. We have simpliﬁed the setup considerably.
• C ∗ -algebraA is assumed to have the identity I (In particular, for classicalsystems, the state space Ω is assumed to be a compact space).
In thecase that Ω is a locally compact space (e.g., Ω = R), it is usual to use
C0 (Ω) = {f : Ω → C, | f is continuous, limω→∞ f (ω) = 0}. In this text, weS
use the compactiﬁcation Ω(e.g., Ω = Ω {∞}).
• We assume that X is a ﬁnite set. This may be too elementary.
• Perhaps we should have adopted the W ∗ -algebraic formulation. In general, the exact
measurement is impossible in C ∗ -algebraic formulation (cf. Sec. 1.4.5).
Despite the above diﬃculties, the C ∗ -algebraic formulation was adopted since it was handy.

Example 1.7. C ∗ -algebraic formulation, [Classical QL]:

H = L2 (Ω, m), where m is the Lebesgue measure, A = C(Ω) (⊆ B(H)),[C(Ω), B(L2 (Ω, m))]: C ∗ -algebraic basic structure.
X = {C, W, H}, Observable O = (X, 2X , G), where [G({x})] : Ω → R is continuous(∀x ∈ X), such that 0 ≤ G({x})](ω) ≤ 1 (∀x ∈ X, ω ∈ Ω,
[G({C})](ω) + [G({W })](ω) + [G({H})](ω) = 1(∀ω ∈ Ω)

Also, a state is deﬁned by δω0 ≈ ω0 (∈ Ω).

1.4.2

Axiom 1 [Measurement] and Axiom 2 [Causality]

With any system S, a a C ∗ -algebra A(⊆ B(H)) can be associated in which the measurement theory (2) of that system can be formulated. A state of the system S is represented by an element ρ(∈ Sp (A∗ )) and an observable is represented by an observable
O =(X, P(X), G) in A. Also, the measurement of the observable O for the system S with

the state ρ is denoted by MA (O, S[ρ] ) or more precisely, MA (O =(X, P(X), G), S[ρ] ) .
An observer can obtain a measured value x (∈ X) by the measurement M(O, S[ρ] ).
The Axiom 1 presented below is a kind of mathematical generalization of Born’s probabilistic interpretation of quantum mechanics. And thus, it is no longer a law of physics,
but an axiom of metaphysics
Now we can present Axiom 1 as follows.

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1.4 Appendix: The mathematical foundations of quantum language
Axiom 1 [State, Measurement] in the C ∗ -algebraic formulation•[General (i.e., quantum and classical ) case]:(i) [State]: With any system S, a C ∗ -algebra A(⊆ B(H)) can be associated. A
state of the system S is represented by an element ρ(∈ Sp (A∗ )).
(ii) [ Measurement ]: An observable is represented by an observable
O =(X, P(X), G) in A. And, the measurement of the observable O for thesystem S with the state ρ is denoted by MA (O, S[ρ] ) or more precisely,
MA (O =(X, P(X), G), S[ρ] ) . An observer can obtain a measured value x(∈ X) by the measurement M(O, S[ρ] ). The probability that a measured
value x (∈ X) obtained by the measurement MA (O =(X, P(X), G), S[ρ] ) is
given by ρ(G({x}))(≡ A∗ hρ, G({x})iA ).

The above includes both classical and quantum cases. In the classical case, theabove axiom can be written as follows.
•[Classical case, i.e. the case that A = C(Ω)]: The above (i) and (ii) can besummarized for the classical system as follows(i) [Classical state]: A classical state of the system S is represented by anelement ω(∈ Ω).
(ii) [Classical measurement ]: The probability that a measured value x(∈ X)
is obtained by the measurement MC(Ω) (O( =(X, P(X), G)), S[ω] ) is given by[G({x})](ω).

Example 1.8. C ∗ -algebraic formulation, [Classical QL]:
For simplicity, assume that Ω = [0, 60] in Example 1.7. Thus, the observable is represented
by O = (X = {C, W }, 2X , G)

Similar to Example 1.7, we have a measurement MC(Ω) (O = (X = {C, W }, 2X , G), S[ω0 ] ).
Thus Axiom 1 says that26

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Chap. 1 The outline of quantum language (=measurement theory)(♯1 ) When an observer measures the observable O = (X = {C, W }, 2X , G) for a system
S with the state ω0 (∈ Ω), the probability that a measured value x(∈ X) is obtained
is given by [G({x})](ω0 ).
Using everyday language, we say:(♯2 ) Suppose 100 subjects are asked to drink this glass of ω0 ◦ C water. One person is
randomly selected from these 100 and asked, ”Was this water cold, warm?

”it was cold”[G({C})](ω0 )The probability that this subject answers
is”it was warm”[G({W })](ω0 )////Example 1.9. [Quantum measurement( Schtern–Gerlach experiment (1922))]
Assume that we examine the beam (of silver particles(or simply, electrons) after passing
through the magnetic ﬁeld. Then, as seen in the following ﬁgure, we see that all particles
are deﬂected either equally upwards or equally downwards in a 50:50 ratio.

S

[↑]

N

[↓]

U
○

electron estate ω =

α1
α2



D
○

ScreenStern–Gerlach experiment (1922)
Consider the two dimensional Hilbert space H = C2 , And therefore, we get the noncommutative basic algebra B(H), that is, the algebra composed of all 2 × 2 matrices.
Thus, we have the quantum basic structure:[C(H) ⊆ B(H) ⊆ B(H)] = [B(C2 ) ⊆ B(C2 ) ⊆ B(C2 )]since the dimension of H is ﬁnite.
The spin state of an electron P is represented by ρ(= |ωihω|), where ω ∈ C2 such that
α1
( where, ||ω||2 = |α1 |2 + |α2 |2 = 1 ).
kωk = 1. Put ω =α2Deﬁne Oz ≡ (Z, 2Z , Fz ), the spin observable concerning the z-axis, such that, Z = {↑, ↓}
and




1 0
0 0
Fz ({↑}) =, Fz ({↓}) =
,
(1.3)
0 0
0 1


0
Fz (∅) =
0


0
,
0



1
Fz ({↑, ↓}) =
0


0
.
1
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1.4 Appendix: The mathematical foundations of quantum language
Here, Born’s quantum measurement theory (the probabilistic interpretation of quantum
mechanics) says that(♯) When a quantum measurement MB(C2 ) (O, S[ρ] ) is taken, the probability thata measured valueThat is, putting ω (=

↑
↓




is obtained is given by hω, F z ({↑})ωi = |α1 |2hω, F ({↓})ωi = |α2 |
z




2


α1
), we says that
α2When the electron with a spin state ρ progresses in a magnetic ﬁeld,U○the probability that the Geiger countersounds
D
○



 

 1 0
α1
2
= |α1 | 
 α1 α2 0 0α2is give by 



 
 0 0

 
α1
α1 α2
= |α2 |20 1
α2Also, we can deﬁne Ox ≡ (X, 2X , F x ), the spin observable concerning the x-axis, suchthat, X = {↑x , ↓x } and
1/2 1/2
1/2 −1/2xx
F ({↑x }) =, F ({↓x }) =.
1/2 1/2−1/2 1/2
And furthermore, we can deﬁne Oy ≡ (Y, 2Y , F y ), the spin observable concerning they-axis, such that, Y = {↑y , ↓y } and

1/2F ({↑y }) =−i/2√
where i = −1.
Here, putting
y


i/2
,
1/2

Ŝx = Fx ({↑}) − Fx ({↓}),
1/2 −i/2
F ({↓y }) =,i/2 1/2yŜy = Fy ({↑}) − Fy ({↓}),

Ŝz = Fz ({↑}) − Fz ({↓})

we have the following commutation relation:Ŝy Ŝz − Ŝz Ŝy = 2iŜx ,Ŝz Ŝx − Ŝx Ŝz = 2iŜy ,Ŝx Ŝy − Ŝy Ŝx = 2iŜz

Next, we explain Axiom 2 ( which is rarely used in this text).
Let (T, ≤) be a tree-like semi-ordered set. The following two are typical.
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Chap. 1 The outline of quantum language (=measurement theory)
s2

s4

s1
s0
s6
s7

s3

s5

Tree T

real line (i.e., linear ordered set T)

Assume that, for each t ∈ T, a C ∗ -algebra At is associated.
A continuous linear operator Φ1,2 : A2 → A1 is called a homomorphism (or, causal
operator), if it satisﬁes that(♯1 ) Φ1,2 (L) ≥ 0 (∀L ∈ A2 such that L ≥ 0)(♯2 ) Φ1,2 (I2 ) = I1 , where I1 and I2 is identity maps in A1 and A2 respectively.
(♯3 ) Φ1,2 (L2 · M2 ) = Φ1,2 (L2 )Φ1,2 (M2 ) and Φ1,2 (L∗2 ) = (Φ1,2 (L2 ))∗ (∀L2 , M2 ∈ A2 ))The dual operator Φ∗1,2 : A1 ∗ → A2 ∗ of a causal operator causal operator Φ1,2 : A2 → A1
satisﬁesΦ∗1,2 (Sp (A∗1 )) ⊆ Sp (A∗2 )(1.4)

In classical case (i.e., At = C(Ωt ) = C(Ω), t ∈ T), a homomorphism Φt1 ,t2 : C(Ωt2 )→ C(Ωt1 ) is characterized as follows:(D) there exists a continuous map ϕt1 ,t2 : Ωt1 → Ωt2 such that[Φt1 ,t2 (gt2 )](ωt1 ) = gt2 (ϕt1 ,t2 (ωt1 ))(∀ωt1 ∈ Ωt1 , ∀gt2 ∈ C(Ωt2 )),Φ1,2 g2

Ω1

(1.5)

g2

ω1 ................................................................................................................................................................................................................................ ϕ1,2 (ω1 )Ω2Causal map ϕ1,2 and causal operator Φ1,2
Thus, the family of continuous maps { ϕt1 ,t2 : Ωt1 → Ωt2 , t1 , t2 ∈ T (t1 ≤ t2 ) } is aclassical causal relation.
Now we can propose Axiom 2 (i.e., causality). (For details, see ref. [74].)

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1.4 Appendix: The mathematical foundations of quantum language
Axiom 2 [ Causality ] in the C ∗ -algebraic formulation•[General case (i.e., quantum case (Ati = Bc (Hti )) and classical case (Ati =C(Ωti ))) ]:Assume [t1 ≤ t2 ].
(♯1 ): Causality is represented by a causal operator Φt1 ,t2 : At2 → At1
In the same sense,(♯2 ): Causality is represented by a dual causal operator Φ∗t1 ,t2 : A∗t1 → A∗t2•[Classical case (Ati = C(Ωti )]Assume [t1 ≤ t2 ]:(♭1 ): Classical causality is represented by a causal operator Φt1 ,t2 : C(Ωt2 ) →C(Ωt1 )In the same sense,(♭2 ): Classical causality is represented by a dual causal operator Φ∗t1 ,t2 : C(Ωt1 )∗→ C(Ωt2 )∗Also,(♭3 ): Classical causality is represented by a continuous map { ϕt1 ,t2 : Ωt1 → Ωt2 ,(t1 ≤ t2 ) }Remark 1.10. [Heisenberg picture (= moving observation) and Schrödinger picture (=moving state)](i): The above (♯1 ) and (♭1 ) are called the Heisenberg pictures. And (♯2 ), (♭2 ) and (♭3 )are called the Schrödinger pictures.
(ii):In most cases, the Heisenberg picure (=observable-moving) and Schrödinger picture(=state-moving) are equivalent. However, in complex situations (especially in quantum
systems), the Schrödinger picture (=state-moving) is sometimes powerless. Therefore,
it is recommended to always use the Heisenberg picture according to the Copenhagen
interpretation (E6 ) ”the state is ﬁxed and does not move”.

Remark 1.11. So far, the following three theories (= scientiﬁc language) have beenformulated.
(i): Quantum language (including classical and quantum systems): This is the theory (=30

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Chap. 1 The outline of quantum language (=measurement theory)
scientiﬁc language) with the most powerful descriptive power.
(how to use Axioms 1 and2)Copenhagen
interpretationstate
QL = measurement + causality +(Axiom 1)(1.6)(Axiom 2)(ii): Quantum language of classical systems: only the classical part of quantum language,
but as descriptive as statistics(how to use Axioms 1 and2)classical QL =statemeasurement+

causality+

Copenhagen
interpretation(Axiom 1[Classical case (i), (ii)]) (Axiom 2[Classical case])(1.7)(iii): Abstract Classical Mechanics: Embodies the spirit of ”Think of Newtonian mechanics as a model for all motion! and is roughly equivalent to dynamical systems theory(which does not include the concept of probability).
abstract classical mechanics =

state

+

(Axiom 1 [Classical case] (i))

31causality



(1.8)

(Axiom 2 [Classical case])

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1.4 Appendix: The mathematical foundations of quantum language

1.4.3 Product observable, Product measurement
Deﬁnition 1.12. [(i): Product observable (= simultaneous observable), product measurement (= simultaneous measurement)]: Let Oi =(Xi , P(Xi ) , Gi ) (i = 1, 2, ..., N ) becommutative observables in A. Deﬁne a product observable (= simultaneous observable)
×i=1,2,...,n Oi = (×ni=1 Xi , P(×ni=1 Xi ), ×i=1,2,...,n Gi ) such that[×

i=1,2,...,n

Gi ](

×

j=1,2,...,nΞj ) =

×

j=1,2,...,nGj (Ξj ) (∀Ξj ∈ P(Xj ), j = 1, 2, ..., n)Also, MA ( ×i=1,2,...,n Oi = (×i=1 Xi , P(×i=1 Xi ), ×i=1,2,...,n Gi ), S[ρ] ) is called a product measurement (= simultaneous measurement) of MA ( Oi = (Xi , P(Xi ), Gi ) , S[ρ] )(i = 1, 2, ..., n).
[(ii): quasi-product observable, quasi-product measurement]: Let Oi =(Xi , P(Xi ) ,Gi ) (i = 1, 2, ..., N ) be commutative observables in A. Deﬁne a quasi-product observable
n
n
qp
×qp
i=1,2,...,n Oi = (×i=1 Xi , P(×i=1 Xi ), ×i=1,2,...,n Gi ) such that
n

n

qp

[

×

i=1,2,...,nGi ](X1 ×X2 ×...×Xj−1 ×Ξj ×Xj+1 ×...×Xn ) = Gj (Ξj )(∀Ξj ∈ P(Xj ), j = 1, 2, ..., n)Also, MA ( ×i=1,2,...,n Oi = (×i=1 Xi , P(×i=1 Xi ), ×i=1,2,...,n Gi ), S[ρ] ) is called a quasiproduct measurement of MA ( Oi = (Xi , P(Xi ), Gi ) , S[ρ] ) (i = 1, 2, ..., n).
[(iii): Tensor C ∗ -algebra, tensor product observable, tensor product measurement]: Let
Oi = (Xi , P(Xi ) , Gi ) be observables in Ai , (i = 1, 2, ..., n). Deﬁne a tensor product obNNnnservable i=1,2,...,n Oi = (×i=1 Xi , P(×i=1 Xi ), i=1,2,...,n Gi ) in the tensor C ∗ -algebra
Nn
i=1 Ai such that
qp

[

O

n

Gi ](

i=1,2,...,n×

j=1,2,...,nΞj ) =nO

Gj (Ξj )qp(∀Ξj ∈ P(Xj ), j = 1, 2, ..., n)

j=1,2,...,n

N
N
n
n
Nn
Also, MNni=1 Ai (i=1,2,...,n Oi = (×i=1 Xi , P(×i=1 Xi ),i=1,2,...,n Gi ), S[ i=1 ρi ] ) is
called a tensor product measurement of MAi ( Oi = (Xi , P(Xi ), Gi ) , S[ρi ] ) (i = 1, 2, ..., n).
.

Example 1.13.

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Chap. 1 The outline of quantum language (=measurement theory)

Assume that(♯) water at ω ◦ C at time t = 0 becomes ϕ0,1 (ω)◦ C at time t = 1, where ϕ0,1 : [0, 60] →[0.60] is deﬁned such that ϕ0,1 (ω) = ω2 (0 ≤ ω ≤ 60).
Let’s measure whether the water is cold or warm at time 0 and time 1. Using the notationin Examples 1.8, we consider two measurements MC(Ω) (O = (X = {C, W }, 2X , G), S[ω0 ] )≈ MC(Ω) (Φ1,0 O = (X =and MC(Ω) (O = (X = {C, W }, 2X , G), S[ϕ0,1 (ω0 )] )X{C, W }, 2 , Φ1,0 G), S[ω0 ] ) .
However, the Copenhagen interpretation says that ”Only one measurement is permitted”. Thus, combine the two measurements above into one product measurement (=simultaneous measurement) as follows
2

MC(Ω) (O × Φ1,0 O = (X 2 = {C, W }2 , 2X , G × Φ1,0 G), S[ω0 ] )Then, Axiom 1 says that(♭1 ) The probability that the measured value (x0 , x1 ) ∈ {C, W }2 is obtained by theabove measurement is given by[G({x1 })](ω0 ) · [G({x2 })](ω0 /2)
Or, using ordinary language, we see(♭2 ) the probability that one of the subjects (chosen at random) answers it was”cold” at time 0 and ”cold” at time 1






”cold” at time 0 and ”warm” at time 1
 ”warm” at time 0 and ”warm” at time 1 ”warm” at time 0 and ”cold” at time 1
[G({C})](ω)·[G({C})](ω
/2)


0
0




[G({C})](ω0 ) · [G({W })](ω0 /2)
is equal to[G({W })](ω0 ) · [G({W })](ω0 /2) 





[G({W })](ω0 ) · [G({C})](ω0 /2)

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1.4 Appendix: The mathematical foundations of quantum language

1.4.4 Inference; Fisher inference and Bayesian inference in classical QL

We begin with the following notation:
Notation 1.14. [MC(Ω) (O, S[∗] ) and MC(Ω) (O, S[∗] (ρ)) ] : Consider a classical measurement MC(Ω) (O:=(X, F, F ), S[ω0 ] ). Here,(♯1 ) It is usual that a state ω0 (∈ Ω) is unknown when a measurement MC(Ω)(O:=(X, F, F ), S[ω0 ] ) is taken.
That is because(♯2 ) we usually take a measurement MC(Ω) (O, S[ω0 ] ) for the purpose to know a stateω0 (∈ Ω)
Thus,
(E1 ) when we want to emphasize that ”we don’t know the state ω0 at all”, we usually
use the notation MC(Ω) (O:=(X, F, F ), S[∗] ) instead of MC(Ω) (O:=(X, F, F ), S[ω0 ] )(E2 ) when we want to emphasize that ”we don’t know the state ω0 , but know that the
distribution is ρ(∈ Sm (C(Ω)∗ ))(see Sec. 1.4.1), we usually use the notation MC(Ω)(O:=(X, F, F ), S[∗] (ρ)) instead of MC(Ω) (O:=(X, F, F ), S[ω0 ] )////Theorem 1.15. [Inference (I); Fisher’s maximal likelihood method ]Assume that a measured value x(∈ X) is obtained by a measurement MC(Ω)(O:=(X, 2X , F ), S[∗] ). Then, the unknown state [∗] can be inferred to be ω0 (∈ Ω)such that[F ({x})](ω0 ) = max[F ({x})](ω)ω∈Ω[Inference (II); Bayesian inference ]Assume that a measured value x(∈ X) is obtained by a measurement MC(Ω)(O:=(X, 2X , F ), S[∗] (ρ0 )).
Then, the distribution ρ1 (∈ Sm (C(Ω)∗ ) of unknown
state [∗] can be inferred as follows.
Zρ1 (D) =[F ({x})](ω)ρ0 (dω)(∀D ⊆ Ω, where D is any open set in Ω))D

For the proof, see ref. [44] or §5.2 in ref. [74]).
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Remark 1.16. [The relation between statistics (inference) and dynamical system theory(control)]MC(Ω) (O:=(X, F, F ), S[ω0 ] ) has two aspects as follows.
•• [inference]:
when we know an observable O and a measured value x(∈ X), infer the stateω0 (∈ Ω)!
• [control]:when we know an observable O and we want to get a measured value x(∈ X), howshould the state ω0 (∈ Ω) be set up?
Inference and control are in diﬀerent moods, but the substance (= Fisher’s maximum
likelihood method) is the same.

1.4.4.1 Exercise: Fisher’s maximum likelihood method and Bayesian inference (ref. [44])Problem 1.17. (i): I can’t tell if it is a man or a woman, but I see someone in the
distance. He/she is wearing a skirt. I need you to estimate whether this person is maleor female.
(ii):Actually, this area is very male dominated.
Number of men : Number of women = 49 : 1
If so, is he/she wearing a skirt male or female?

Answer (i): Fisher inference
Let Ω = {ωM , ωW } be a state space, where ωM [ resp. ωW ] is implied by ”men” [ resp.
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1.4 Appendix: The mathematical foundations of quantum language”women”]. The observable O = ({y, n}, 2{y,n} , F ) in C(Ω) is deﬁned byNumber of men wearing skirts< 0.01, [F ({n})](ωM ) = 1 − [F ({y})](ωM )Number of men
Number of women wearing skirts[F ({y})](ωW ) => 0.5, [F ({n})](ωW ) = 1 − [F ({y})](ωW )Number of women[F ({y})](ωM ) =Assume that a measured value y is obtained by the measurement MC(Ω) (O:=({y, n}, 2{y,n} , F ),S[∗] ), that is, He/she is wearing a skirt. Then,
Note that[F ({y})](ωW ) = max{[F ({y})](ωM ), [F ({y})](ωW )}Thus, Fisher’s maximal likelihood metheod says that• the person is a woman.

Answer (ii): Bayesian inference
Consider the measurement MC(Ω) (O:=({y, n}, 2{y,n} , F ), S[∗] ρ0 ). Note that
ρ({ωM }) = 49/50,

ρ({ωW }) = 1/50

Thus, Bayesian inference says that
Probability of being male
0.01 49/50
49=
×=Probability of being female
0.5
1/50
50
That is,Probability of being male =

49
,
99

Probability of being female =50
99

Thus, the probability of being a woman is a little higher.

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Chap. 1 The outline of quantum language (=measurement theory)

1.4.5 Precise measurement, Tensor precise measurement
The C ∗ -algebraic formulation is easier than the W ∗ -algebraic formulation. The in thistext, C ∗ -algebraic formulation is adopted. However, C ∗ -algebraic classical measurement
has the weakness that ”precision measurement cannot be formulated naturally.” Having
said that, it is useful to have precision measurements, so we deﬁne them here for convenience.
Deﬁnition 1.18. [precision measurement]: Let Ω be a compact space, and let(Ω, B(Ω), ν) be a measure space such that(i) B(Ω) is the Borel ﬁeld (i.e., it is a smallest σ-ﬁeld that contains all open sets), andµ(Ω) = 1(ii) ν({ω}) = 0 (∀ω ∈ Ω)(iii) ν(D) > 0 (∀D ⊆ Ω, D is open)
The exact observable OEΩ = (Ω, B(Ω), EΩ ) is deﬁned by
1 (ω ∈ D)[EΩ (D)](ω) =(∀D ∈ B(Ω))
0 (ω ∈/ D)
Putting X = Ω, OEΩ is sometimes denoted by (X, B(X), EΩ ).
////We see(F1 ) the measured value x(∈ X = Ω) obtained by the precise measurement MC(Ω) (OEΩ =(X, B(X), EΩ ), S[ω] ) surely belongs to any open set including ω, that is, x(∈ X =Ω) = ω.
Deﬁnition 1.19. [Tensor precise measurement]: As above, let (Ω, B(Ω), ν) be acompact measure space. and N
let OEΩ = (Ω, B(Ω), EΩ ) be the precise observable in C(Ω)*3
Let Λ be a inﬁnite set. Let λ∈Λ C(Ω) be the inﬁnite tensor space of C(Ω). Then, wesee:OC(ΩΛ ) =C(Ω)λ∈Λ

Also, the tensor precise observable in C(ΩΛ ) is deﬁned byO
OEΩ = (ΩΛ , B(ΩΛ ), EΩΛ )λ∈Λ

Putting X = Ω, we sometimes denoteNλ∈Λ OEΩ = (XΛ

, B(X Λ ), EΩΛ ).
////We have the following theorem.
Proposition 1.20. [inﬁnite tensor precise measurement]:*3 It is not guaranteed that EΞ ∈ C(Ω)

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1.4 Appendix: The mathematical foundations of quantum languageN(F2 ) The inﬁnite tensor precise measurementMC(ΩΛ ) ( λ∈Λ OEΩ = (X Λ , B(X Λ ), EΩΛ ),S[(ωλ )λ∈Λ ] ) is taken. Then, the probability that the measured value (xλ )λ∈Λ (∈ X Λ =ΩΛ ) belongs to any open set which includes (ωλ )λ∈Λ (∈ ΩΛ ) is equal to 1. In thesame sense, the measured value (xλ )λ∈Λ (∈ X Λ = ΩΛ ) is surely equal to (ωλ )λ∈Λ .
////1.4.6 Inﬁnite Tensor Product Observable, Inﬁnite Tensor Product Measurements
In the C ∗ -formulation of the measurement, the inﬁnite tensor product measurement requires some ingenuity.
This is explained below. The same argument can be made for quantum systems, but
we will limit our discussion to classical systems.
Let Λ be an arbitrary set. For each λ(∈ Λ), consider the measures
N MC(Ωλ ) (Oλ (=(Xλ , P(Xλ ), Fλ )), S[δωλ ] ). Then, these inﬁnite tensor product measures λ∈Λ MC(Ωλ ) (Oλ S[δΩλ ] ).
That is,O
MC(Ωλ ) (Oλ , S[δωλ ] )
λ∈Λ

=MC(×
=MC(×

λ∈Λ

(
Ω )

λ∈Λ

(
Ω )

O

λ

Oλ , S[⊗λ∈Λ δωλ ] )

λ∈Λ

O

λ

λ∈Λ

Oλ = ( × Xλ , P0 ( × Xλ ),
λ∈Λ

λ∈Λ

O

Fλ ), S[⊗λ∈Λ δωλ ] )λ∈ΛLet me explain what this means. In the above,(♯1 ) C(ΩΛ ) and ⊗λ∈Λ δωλ are not a problem. because Ω is compact and ΩΛ is also
compact (by Tychonoﬀ’s theorem (cf ref. [128])).
(♯2 ) ] P0 (×λ∈Λ Xλ ) is determined by devising the following.
P0 ( × Xλ )
n
=

λ∈Λ

× Ξλ Ξλ ⊆ Xλ (λ ∈ Λ),λ∈Λ

{λ ∈ Λ | Ξλ 6= Xλ } is ﬁniteoHere, we have the following, the generalization of Axiom 1.
Axiom’ 1 [Inﬁnite tensor product measurement]: C ∗ -algebraic formulationN
MC(Ωλ ) (Oλ , S[δωλ ] ) (=Suppose that the inﬁnite tensor measurement
λ∈Λ N
N
MC(×
( λ∈Λ Oλ = (×λ∈Λ Xλ , P0 (×λ∈Λ Xλ ), λ∈Λ Fλ ), S[⊗λ∈Λ δωλ ] ))
Ω )
λ∈Λ

λ

yields the measured value (xλ )λ∈Λ ( where xλ ∈ Xλ (∀λ ∈ Λ ) ). The probability that this measured value belongs to ×λ∈Λ Ξλ (∈ P0 (×λ∈Λ Xλ )) is× [Fλ (Ξλ )](ωλ ).

λ∈Λ

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39

Chapter 2

Ancient Greek philosophy (before
Socrates)
Readers can start reading from this chapter (i.e., you can skip Chapter 1).
In Ancient Greek philosophy (before Socrates), the phrase ”the arche (= the ﬁrst
thing of all things) is” is standard. Here ”” is, for example, as follows.
Thalēs· · · · · · water

Democritus· · · · · · atom

Pythagoras· · · · · · Number is within all things

Hērakleitos· · · · · · motion, ﬁreParmenides· · · · · · logic, motion

Zeno· · · · · · logic, motion

Worldviewism:
Let’s think so

the arche is sotherefore

Worldview −−−−−→ Consideration, discussion
This argument has shaped Western philosophy for 2,500 years.
Firstly we examine Pythagoras’ saying ” Number is within all things”. But unfortunately, this golden rule has never been followed in all the history of philosophy.
• The reason that quantum language has a strong descriptive power is due to thefact that quantum language adhered to Pythagoras’ saying.

Also, we note that Parmenides’ claim is very similar to the Copenhagen interpretation of quantum mechanics. And thus,• we treated Parmenides like the ancestor of the Copenhagen interpretation
Next, we examine Zeno’s paradoxes (the ﬂying arrow, Achilles and the tortoise)from a quantum linguistic perspective and present a new view of Zeno’s paradoxes.

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And we conclude that
• The confusion about Zeno’s paradox lies in the lack of a world description method.
This paradox can be easily solved under the worldview of the motion function
method, which is a subclass of quantum language. It is not a mathematicalproblem about geometric series, as is often claimed.

This success has led us to worldviewism (i.e., the spirit that ”Start from worldview!”). Throughout this text, we always argue thatwithout worldview, there is no logic, no proposition.
It should be noted that this spirit is hardly accepted even today. That is because
this is the reason that Zeno’s paradoxes can not be solved in analytic philosophy.

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Chap. 2 Ancient Greek philosophy (before Socrates)

2.1 Thalēs (BC.640 - BC.546)
2.1.1 Thalēs: the ﬁrst philosopher: ” the arche (= the ﬁrst thing of all things)
is water”
Every race had its own ”mythology”. Myths are the literature that explains the world by
reason of God. Myths have been handed down from generation to generation from ourancestors by oral and written word. There is a great deal of maritime trade, and many
ethnic groups come together to interact, but it is rare to ﬁnd a region that is not united
by force of arms because it is not a vast plain. This situation was realized on the eastern
Mediterranean coast (now called Greece, Turkey, Syria, and Israel (Egypt)) around 1000
BC. In this region, various civilizations and cultures have merged to create a new culture.
Particularly noteworthy was the realization of uniﬁcation not by force of arms, but by
culture. In other words, ”philosophy as a synthesis of several myths” and ”the alphabet
as a synthesis of several letters” were born. That is,=⇒ philosophy
 integration of several myths(A)
integration of the several characters =⇒ Phoenician alphabet =⇒ alphabet
The alphabet is a phonetic alphabet because it was created with the intention of being acommon letter between diﬀerent ethnic groups. Egypt had an advanced civilization (suchas the pyramids), but it was so uniﬁed in thought that philosophy did not develop.
In Aristotle’s book ”Metaphysics,” Thales Is called the ”ﬁrst philosopher”. For example,
to the question ”why does an earthquake occur?” Most myths will say ”because God isangry”. Thalēs appeared in Miletus in ancient Greece. He said, in accordance with hisworldview,(B) The arche (= the ﬁrst principle of all things) is water. Therefore, the earthquake
is caused by the vibration of the water
the arche is so

Let’s think sotherefore

Worldview −−−−−→ Consideration, discussion
This may be childish, but is an explanation that does not brought out the ”God” . Thisis the reason why Aristotle said Thalēs was ”the ﬁrst philosopher.”41

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2.1 Thalēs (BC.640 - BC.546)♠Note 2.1. S. Weinberg (1933 -2021), An American physicist, won a Nobel Prize of physics
in 1979, said in his book [123] ”To explain the world; The discovery of modern science”
as follows:(♯) It seems to me that to understand these early Greeks, it is better to think of them notas physicists or scientists or even philosophers, but as poets.
I agree with him. As mentioned earlier, I believe that the main purpose of the ancient
Greek philosophers was not to pursue truth but to provide a common topic (i.e., a common
myth, a universal myth) in order to avoid conﬂict between diﬀerent peoples. This paper,
however, may be a bit more radical. As far as to describing the world (not ethics), Plato,

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Chap. 2 Ancient Greek philosophy (before Socrates)
Descartes, Kant, etc. all should be regarded as poets or ﬁction writers. As we will discusslater, I believe that the purpose of Kant’s Critique of Pure Reason is not the pursuit oftruth, but a proposal for a symbiosis with science. That is because the rule of philosophy
is as follows:(♯1 ) Only discussion, no experimentation.
(♯2 ) Winners are determined by the popularity of the general public (or philosophical enthusiasts).

In short, I think most of the philosophy is ”enlightenment and ethics”. However, this text
focuses on the scientiﬁc part of philosophy.
♠Note 2.2. Who is the ﬁrst philosopher? Of course there may be a lot of opinions for thisquestion. As mentioned above, Aristotle said Thalēs was ”the ﬁrst philosopher.” Also,
A.N.Whitehead (1861 - 1947) said that(♯) Western philosophy is characterized as a series of footnotes to Plato
Although I do not know Whitehead’s intension, I want to think that this means ”Plato is
the ﬁrst philosopher”, which is the same as the spirit of this text.

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2.1 Thalēs (BC.640 - BC.546)

2.1.2 Thalēs’ ability at math
By the statement:”the ﬁrst principle of all things is water”, we cannot judge Thalēs’knowledge, However, the following is known as Thalēs’ theorem, which shows his abilityof math:Theorem 2.1. [central angle = β, inscribed angle= α] =⇒ β = 2αβαProof. It suﬃces to draw the additional line through the center.
Although it is a problem within the scope of junior high school mathematics, since
training in elementary geometry is neglected in middle and high schools, only about 60
percent of graduate students in mathematics at an average level can answer this proof
immediately.
However, if the next theorem is the discovery of Thales, we can trust Thales’ mathematicalability.
Theorem 2.2.
• The vertical angles are equal. That is, in the ﬁgure below, it holds
that a = c, b = d

The model answer may be as follows.
a + b = 180◦ , b + c = 180◦
thus,

(2.1)
a = 180◦ − b = c////
However, I think this is too obvious and diﬃcult to prove. The fact that Thales said that”the vertical angles are equal” means that he was aware that ”it is worth speaking.” This
fact makes us believe in Thales’ (or at that time) mathematical ability. This tradition ofancient Greek philosophy may have given rise to Euclid’s Elements.
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Chap. 2 Ancient Greek philosophy (before Socrates)♠Note 2.3. (i): When Thales visited Egypt, there is a story that the king of Egypt was
impressed by Thales’ measuring the height of the pyramid in the way of measuring called
triangulation. But, I think it is unreliable. Three great pyramids in the Egyptian Giza
desert (deceased persons are Khufu, Khafre, Menkaure) erecting time of is the 2500 B.C.
Thus I guess that the triangulation was known in those days (2000 years before Thalēs) in
Egypt. If so, Thalēs’ theorem should be doubt whether it is due to Thalēs. However, evenas a true prover was unknown, the ability of mathematics at the time (i.e., the discoveryof the concept of ”proof”) should be surprising. This led to Euclid’s ”Elements” (due toEuclid (275 BC - 330 BC)).
(ii): Anyone who is interested in the history of mathematics will be interested in the”origin of area”. However, I don’t know anything about this.

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2.2 Pythagoras (BC.582 - BC.496)

2.2 Pythagoras (BC.582 - BC.496)
2.2.1 The mathematical ability of Pythagorean religious organization

Pythagoras appears to have been the son of a gem-engraver on the island of Samos.
Although contemporary scholars disagree about the origins of Pythagoras, they do agree
that, around 530 BC, he travelled to Croton in southern Italy, where he founded a schoolfor mathematics. Pythagoras was a leader in the mathematics study group, which may
be regarded as a kind of religious organization called Pythagorean religion.

As the mathematical achievements, the followings are known:• the discovery of irrational numbers, the Pythagorean theorem, the construction ofa regular pentagon
and so on.
Theorem 2.3. Finding of irrational members, e.g.,
46√

2 is an irrational number.

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Chap. 2 Ancient Greek philosophy (before Socrates)
Theorem 2.4. (Pythagorean theorem): In 4ABC, the followings are equivalent:(♯1 ) ∠A = 90◦(♯2 ) AB2 + CA2 = BC2
Proof. The proof of Pythagoras himself is unknown, but the next proof will be learned injunior high school.

Thus,
4×

ab
+ c2 = (a + b)2
2

And thus, a2 + b2 = c2 .
////♠Note 2.4. However, this proof presupposes the additivity of the area. Therefore, there
may be room for consideration.

Construction 2.5. the construction of a regular pentagon

Explanation: In a regular pentagon as shown in the ﬁgure below (left), put AB =BC = CD = DE = EA = 1. Then, we see√
1+ 5
AC = AD =2√√√
Hence, it suﬃces to construct 1+2 5 . By the Pythagorean theorem, 5(= 12 + 22 ) can√
be constructed as follows (the ﬁgure below (right)). Thus, we easily get 1+2 5 .
47
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2.2 Pythagoras (BC.582 - BC.496)
A

√
B

E

5

1
C

D

C

D

2
♠Note 2.5. (i):If you are a university student in mathematics, you must know at least three
proofs (of Pythagorean theorem). Supplement the two proofs as follows:.

(a)(b)
where (a) is due to Euclid ( (b) is due to Einstein?).
(ii): It could have been something extra, but I wrote the explanation of the construction of a regular pentagon for beginners. Even university students in the department ofmathematics sometimes don’t know this.
♠Note 2.6. The above two (the discovery of irrational numbers and the Pythagorean theorem) are one of the most important discoveries in mathematics. If the following episodes
are true, we can trust his mathematical ability.
(♯1 ) Pythagoras was killing the disciple, who found the irrational number, in order to hide
the existence of irrational numbers.
(♯2 ) When Pythagoras discovered the Pythagorean theorem, he celebrated it, oﬀered thesacriﬁce of the bull.
The two theorems, about 2000 years from the originally discovered to the scientiﬁc revolution (Descartes’ Analytical Geometry, etc.; 17th century), not been used even once withan essential meaning. Nevertheless, their importance had been recognized in Pythagorean
organization. This suggests a very high mathematical level of Pythagorean organization.
Even if I had been a member of the Pythagorean school, I don’t think I could have recognized the importance of these two theorems to the extent that they deserve in (♯1 ) and(♯2 ) above.
♠Note 2.7. (i): Pythagoras is said to have known that the earth was round. And he consider
the following astronomical system:48

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Chap. 2 Ancient Greek philosophy (before Socrates)

I love this astronomical system and could watch it for an hour and never get tired of it. par
At the time, astronomy might be believed to be a part of mathematics. Greek astronomy
develops in the following way.
geocentrism 1

the earth is round(♭1 )Pythagoras−→

BC582-496

Eudoxusgeocentrism 2−→

BC400-347

geocentrism 3

Aristotle−→ Ptolemaios

BC384-322

AD83-168(Although there are several geocentrism k (= 1, 2, 3), the explanation is omitted.)
where Eudoxus(BC.400 – BC.347) is a Greek astronomer and mathematician (called thegreatest of ancient Greek mathematicians), a student of Plato. He proposed a kind of
geocentrism and the method of exhaustion (as the quadrature). It is said that many of
his theorems are written in Euclid’s Elements.
Even in ancient China which had the great culture, the prevailing belief was that the Earthwas ﬂat and square, while the heavens were round, until the introduction of European
astronomy in the 17th century(ii): Also, ancient Greek mathematics should be praised. That is,
Pythagorean theorem(♭2 )Pythagoras
BC582-496

method of exhaustion−→

EudoxusBC400-347−→

Elements

quadrature

Euclid−→ Archimedes

BC330-275

BC287-212

The above two (♭1 ) and (♭2 ) are the greatest achievements in ancient Greek philosophy.

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2.2 Pythagoras (BC.582 - BC.496)

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Chap. 2 Ancient Greek philosophy (before Socrates)

2.2.2 The arche (= the ﬁrst principle of all things) is number

The main spirit of Pythagorean religious organization is ”the ﬁrst principle of all things isnumber”. Now let us explain the following principle (called Pythagoreanism in this text):(A):Pythagoreanism

the ﬁrst principle of all things is number.
That is, ”Describe the world using mathematics!”, or”Number is within of all things””To make a scientiﬁc theory is to parameterize things”
Although this spirit was never realized in Western philosophy, I believe that this is themost important claim throughout Western philosophy. This spirit was ﬁrst realizedin physics (i.e., Newtonian mechanics and quantum mechanics) as follows:

The phrase: ”The arche (= ﬁrst principle) of all things is” is a fashion in those days.
Note that ”water”, ”ﬁre” etc. are visible, but ”number” is not.

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2.2 Pythagoras (BC.582 - BC.496)♠Note 2.8. (i): After about 2000 years from Pythagoras, Galileo was talking about a similar
thing. That is, the universe is written in the language of mathematics.
In fact, Galileo wrote the universe in the language of mathematics.

(ii): Descartes discovered that number is the within of all points of space as the Cartesian
coordinates.

y 6(2, 1)

1O2x(iii): At the beginning of his book ”ref. [125]: Tractatus Logico-philosophicus”, Wittgenstein said
• (1.1): The world is the totality of facts, not of things.
Since ”fact”=”state” (or, precisely, ”state”=”quantiﬁcation of fact”), he was merely emphasizing the orthodoxy of Pythagoreanism. But unfortunately he didn’t perform parameterization. Thus, his work had become literature.
♠Note 2.9. As mentioned in Note 2.7, Pythagoras is said to have known that the earth
was round. It may be a joke, but apparently he thought the earth was round because themost beautiful shape was a sphere. It is natural that the pure Pythagoreanism such as(♯) The world is written in only the language of mathematics.
is not true. I believe that the world and mathematics are not directly related (thoughreligious people might not think so). If it can be written in only the language of mathematics, it is just mathematics. However, this pure Pythagoreanism has carried over tothe present day and is inhibiting the healthy development of our worldview (e.g., see ”The
theory of probability” in [74], or ”Analytic philosophy” in Chap. 12).

Hence we have the following problem (i.e., the problem of worldview), which is the main
problem in this text:
Problem 2.6. The problem of worldview is as follows.
• A scientiﬁc worldview has the form: ”scientiﬁc worldview= mathematics + α”. If
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Chap. 2 Ancient Greek philosophy (before Socrates)so,what is α?
As mentioned later, let us say here conclusion now. For example, α is ”motion”, ”causal
relation”, ”probability”, ”measurement”, etc. From the quantum theoretical point of view,that is, in this text, we devote ourselves to ”measurement (Axiom 1) and causality (Axiom
2) (cf. Sec.1.2.1).
♠Note 2.10. (i): In this period, the distinction between mathematics and science may nothave been clear. The clear answer to the question ”What is mathematics? we had to wait
for Cantor’s set theory and axiomatic set theory by Zermelo and Frenkel (around 1900
A.D.)
(ii): The reader may wonder, ”Why bother to raise something like Problem 2.6 here?” I
think(♯) In analytic philosophy, mathematics (number theory, set theory) is too strong (sostrong that it could be called a ”set-theoretic worldview”), and ”alpha” is weak.
As will be discussed in Chapters 11 and 11, Wittgenstein’s attempt to strengthen the”alpha” in ”TLP), ref.[125]” was not successful. As a foreshadowing of this, I wrote
Problem 2.6 here

In science, Pythagoreanism ﬂourished in the early modern period (during the age of the
Scientiﬁc Revolution with Galileo-Newtonian mechanics and Bernoulli’s law of large numbers, etc.). In philosophy, however, Pythagoreanism did not gain a mainstream position
throughout the history of Western philosophy, as follows(B) :Pythagoreanism (♭1 ) in Western philosophyZeno,Anselmus,Thomas,Descartes−−−−−−−−−−−−−−−−−−−−−−→ Axioms 1 and 2 in QL Spinoza, Leibniz, Wittgenstein,
Number is within of all things(Continental rationalism)(♭1 ):

Pythagoras(♭2 ): Parmenides
there exists only ”one”

Plato,Augustinus,Locke,−−−−−−−−−−−−−−−−→
Berkley, Hume, Kant

(British empiricism)linguistic Copenhagen
interpretation in QL












QL

Looking at (B) above, we can see that Pythagoreanism (♭1 ) never ﬂourished in the mainstream of Western philosophy. In fact, the universal problem (by Amselmus and Thomas)
was understood only by some philosophers. Spinoza and Leibniz, who tried to deepen
Descartes’ theory, failed theoretically, but were diverted to the role of reinforcing ethics(or God). Wittgenstein also failed as a theory (though he succeeded in literature andfashion). In short, Pythagoreanism (♭1 ) was too diﬃcult. However, Locke, Hume, andKant, who tried to apply Descartes’ theory, succeeded in their own way.

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2.3 Hērakleitos and Parmenides

2.3 Hērakleitos and Parmenides
2.3.1 Hērakleitos(BC.540 - BC.480)

Hērakleitos said the following.
(A):Hērakleitos(BC.540 - BC.480) in Ephesus

The arche (= the ﬁrst principle of all things) is ﬁreAnd, further,

Everything ﬂows.

Although ”Everything ﬂows” and ”You cannot step into the same river twice” are interesting, everyone may be able to say similar thing. Hence, in this text, we interpret
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Chap. 2 Ancient Greek philosophy (before Socrates)”Everything ﬂows” as follows.
(B1 ) ”motion” is the most fundamental key-word in science.
If so, we can relate the (B1 ) to Parmenides (or even Newton).
That is,(B2 ) Both Hērakleitos and Parmenides said thatthe arche of all things is motion
which will be discussed in the following section:

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2.3 Hērakleitos and Parmenides

2.3.2 Parmenides; the ancestor of the Copenhagen interpretation
If so, we can relate the (B) to Parmenides.

2.3.3 Parmenides; The ancestor of the Copenhagen interpretation
In the same period of Hērakleitos, Parmenides said the exact opposite of words ofHērakleitos. That is,(C):Parmenides(BC.515 - unknown)
Parmenides said:(C1 ) Everything does not change. There is no motion and no change. Time does
not exist. There exists only ”one”, and not ”many”.
Also,(C2 ) We should not rely on our senses to understand the world, but should thinklogically with reason. Even if it appears to be moving, it’s just that the human being has the sense to see it. It does not guarantee the existence of the
movement.
(Notice): Remark the similarity between (C1 ) and the Copenhagen interpretation(cf. Sec. 1.3), i.e.
• only one measurement is permitted• State does not moveAlso, in case of quantum mechanics, its object is too small, is not seen. Thus,
we cannot rely on the sense, but only calculation. We can completely consent toParmenides’ assertion (C2 ) in case of quantum mechanics.

Thus, it is no exaggeration to say that Parmenides is the founder of the Copenhagen interpretation.

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Chap. 2 Ancient Greek philosophy (before Socrates)
Zeno,Anselmus,Thomas,Descartes−−−−−−−−−−−−−−−−−−−−−−→ Axioms 1 and 2 in QL Spinoza, Leibniz, Wittgenstein,
Number is within of all things(Continental rationalism)(♭1 ):

Pythagoras(♭2 ): Parmenides
there exists only ”one”

Plato,Augustinus,Locke,−−−−−−−−−−−−−−−−→
Berkley, Hume, Kant

linguistic Copenhagen
interpretation in QL(British empiricism)












QL

The Parmenides genealogy has always been involved in the Copenhagen Interpretationand has formed the mainstream of Western philosophy all the way up to Kant.

♠Note 2.11. It is certain that the Copenhagen interpretation of quantum mechanics wasborn in Niels Bohr institution of the university of Copenhagen.

However, I may have an opinion such that(♯) It is not too much to say that Parmenides and Kolmogorov are the founders of thelinguistic Copenhagen interpretation of quantum mechanics.
( where Kolmogorov is the founder of the modern theory of probability (cf. ref. [91]). It
is known that Kolmogorov’s extension theorem is the most fundamental in the theory ofprobability.) That is because(♭1 ) Parmenides says that there exists only ”one”, and not ”many”, there is no motion .
(♭2 ) Linguistic Copenhagen interpretation says that only one measurement is permitted .
(♭3 ) Kolmogorov’s extension theorem says that only one probability space is permitted .
Roughly speaking, I think that (♭1 )=(♭2 )=(♭3 ).

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2.3 Hērakleitos and Parmenides
If Parmenides didn’t think motion was important, there’s no way he could say ”motion
doesn’t exist”. Thus, I consider that Parmenides believed in the importance of motion.
Parmenides’ assertion is similar to Hērakleitos’, that is,(D) ”motion” is the most fundamental key-word in science.
The following (E1 ) and (E2 ) are my ﬁction about the diﬀerence between Hērakleitos andParmenides:(E1 ) Since Hērakleitos said ”The ﬁrst principle of all things is ﬁre”, he seems to premise
the realistic world. Thus, his motion is similar to the motion in physics.
(E2 ) Parmenides might study the abstract motion in the linguistic worldview. For example, his motion is ”vegetable growth”, ”increase of the population”, ”economic
growth”, ”Achilles’ motion” and so on.
♠Note 2.12. Many would think the following.
(♯1 ) Hērakleitos is ordinary, and thus understandable, hence, scientiﬁc, therefore, realistic.
(♯2 ) Parmenides is ridiculous, and thus incomprehensible, hence, philosophical, therefore,
idealistic (= linguistic).

Table 2.1:(= Table 4.1)
Philosophical controversy that has been ongoing for 2,500 years[(monistic) realistic worldview] vs. [(dualistic) linguistic/idealistic world view]
dispute ⧹ [R] vs. [L]

Realistic worldview

Idealistic worldview(monism, realism, no measurement)(dualism, idealism, measurement)a motion○:Hērakleitosb○:Ancient Greece
Aristotle
c Problem of universals
Ockham○:d space-time
Newton ( Clarke)○:e quantum theory○:
Einsteinf○:philosophyof science
Carnapg fuzzy sets○:
KalmanFor the precise argument, see Table 4.1.

Parmenides
Plato
AnselmusLeibniz
Bohr
Quine
Zadeh

a ○
g In the table above, we can think of it as○∼• The dualistic idealism (Platonism) says things that are absurd and incomprehensible, and the monistic realism (Aristotelianism) says things that are obvious
and uninteresting.

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Chap. 2 Ancient Greek philosophy (before Socrates)

The reason why I think this way is that, as I have said many times before, a curious
Copenhagen interpretation sits at the core of the idealistic worldview.

Now, since Parmenides is the ”founder of the Copenhagen Interpretation,” it is reasonable to claim the following as the mainstream of Western philosophy.

But Parmenides was too much of a genius to be generally understood. There is a partial
impression that he was a philosopher who dreamed up a bunch of nonsensical ideas. It
is diﬃcult to say that Parmenides is the source of Western philosophy. Therefore, in thistext, the following will be considered ”the main stream of Western philosophy.

Of course, in making this claim, we must show that• the central theme of Western philosophy has always been Plato’s Idea for 2500
years.
This text does this. Even so, Parmenides’ achievement should always be honored.

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2.3 Hērakleitos and Parmenides

2.3.4 Motion function method as a worldview
As mentioned in the previous section, Pythagoras said ”The arche (= the ﬁrst principleof all things) is number”, Hence,(F) Mathematics (or, parameterization) is indispensable to describe the world. However, we need words (or, concepts) to connect mathematics and the world.
And further, we want to consider the following ﬁction:(G) As an inﬂuential candidate of the key-words, Parmenides (and Hērakleitos) thoughtof ”motion”
As a kind of mechanical worldviews, we introduce the following ”motion function method”,
which is assumed to be due to Parmenides in this text though the true discoverer cannot
be speciﬁed, (cf. Note 2.15 ).
(H): (Scientiﬁc linguistic) motion function method (due to Parmenides?)Principle 2.7. [The motion function method]
Let T be time axis, and let X be space axis. A function f : T → X is called amotion function.
Then, the motion function method (in the mechanical worldview) is proposed asfollows:(H1 ) ”motion” should be described by the motion function f : T → X.
X 6

f (t)

- T

♠Note 2.13. In the above, we tshould note that(♯1 ) ”Moving feeling” is erased since the above graph is ﬁxed.
If it is so, as Parmenides says, we think that(♯2 ) if we devote ourselves to logic or mathematics without relying on the sense,
then we cannot look at ”motion”.
Also, the motion function method belongs to the realistic worldview as well as the
linguistic worldview. In fact, it is easy to see that the motion function method canbe derived from Newton mechanics. However, motion functions are not limited tothose derived from Newtonian mechanics. Thus, in this text we usually consider
that it belongs to the linguistic worldview.
(♯3 ) For the derivation of motion function method from quantum language, seeref. [47], or Chap. 14 in ref.[74].

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Chap. 2 Ancient Greek philosophy (before Socrates)
The motion function method is easy, and it is usually studied in elementary school as the
DST formula:D(distance)

=

S × T
(speed)(♣)(time)

Exercise 2.8. An A spot and a B spot are 1400 meters away. Amy left the A spot for a
B spot by 80 m per minute. Betty left the B spot for A spot at the same time by 60 m
per minute. How many minutes later will Amy and Betty meet?

[Answer] Amy’s motion function fA : R(time axis) → R(one dimensional space axis) is
deﬁned by fA (t) = 60t, Betty’s motion function fB : R(time axis) → R(one dimensional space axis)
is deﬁned by fB (t) = 1400 − 80t, Thus, solving fA (t) = fB (t), we see
60t = 1400 − 80t

then,t = 10Hence, after 10 minutes later, the two meet.
♠Note 2.14. Some may think that to consider two motion functions fA and fB is notconsistent with Parmenides’ saying: there exists only ”one” and not ”many”(cf. Sec.1.2.1:
linguistic Copenhagen interpretation). However, if so, it suﬃces to consider the following
motion function:(♯) (fA , fB ) : R(time axis) → R2 (two dimensional space axis).
This technique is common practice in quantum mechanics and analytical mechanics.
♠Note 2.15. (a): Note that the motion function method is derived from quantum language(cf. ref. [45, 46, 74]). Namely,
Sec. 9.2.3 in this text

[quantum language] −−−−−−−−−−−−−−−→ [motion function]
derivation
That is,(♯) the motion function method is one aspects of the quantum mechanicalworldview (i.e., quantum language)
Although I do not know, from the historical point of view, the discoverer of the motionfunction method, I want to assume that Parmenides is the main character, since he wasa teacher of Zeno (cf. next section). Strictly speaking, the discovery might not be inAncient Greece since the complete understanding of the concept of ”function” is after
Leibniz. However, we think that the spirit of the motion function method was understoodby Pythagoras, Aristotle, Archimedes, etc.
(b): Of course, the above ”motion function method” is incomplete and temporary. The
motion function f : T (time) → X(space) is not suﬃcient without the answers to thequestions ”What is time?” and ”What is space?” (the Leibniz=Clarke correspondence (cf.
Sec. 9.2.2). For the quantum linguistic answers to this questions, see ref. [74]61

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2.3 Hērakleitos and Parmenides(c): The reader should want to ask the following questions.
• What is the motion function method? Isn’t it in physics? Please answer brieﬂy (without quantum language).
To put it simply, it is not in physics, but
a kind of ”habitual thinking” (≈ ”idealistic (or, linguistic) worldview” ).
Such an idea may have come from D. Hume and I Kant (cf. Sec. 9.6, Chap.10).
♠Note 2.16. As the scientiﬁc worldviews before Newtonian mechanics, the most importantis”the motion function method” and ”Archimedes’ principle of leverage and buoyant force”
Some may have a question such as• Why isn’t the importance of the motion function method emphasized?
Why can’t the discoverer of the motion function method be speciﬁed?
Although I have no clear answer, I consider as follows:• The realistic worldview (i.e., physics) was usually discovered by one genius, for example, Archimedes, Newton, Maxwell, Einstein and so on. On the other hand, theidealistic worldview is discovered by plural persons. For example, the discoverer of thetheory of probability (e.g., Pascal, J. Bernoulli, Laplace, Kolmogorov, etc.) cannot
be speciﬁed. Probability theory was imperceptibly formed by a number of people. In
this text (cf. ref. [74]), we want to regard quantum theory, discovered by Heisenberg, Schrödinger and Born, as a kind of mechanical worldview (and not realistic
worldview).
• The realistic worldview (i.e., physics) was usually discovered by one genius, for example, Archimedes, Newton, Maxwell, Einstein and so on. On the other hand, theidealistic worldview is discovered by plural persons. For example, the discoverer of thetheory of probability (e.g., Pascal, J. Bernoulli, Laplace, Kolmogorov, etc.) cannot
be speciﬁed. Probability theory was imperceptibly formed by a number of people. In
this text (cf. ref. [74]), we want to regard quantum theory, discovered by Heisenberg, Schrödinger and Born, as a kind of mechanical worldview (and not realistic
worldview).
Finally, let us discuss the importance of the motion function.
• There are only two ways to think about motion. That is, ”motion function method” or”Newtonian mechanics (diﬀerential equation method). When we think about motion
in our daily lives, 99% of the time we use the motion function method. We rarely use”Newtonian mechanics. Newtonian mechanics” is used only by some people in thescience ﬁeld.

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Chap. 2 Ancient Greek philosophy (before Socrates)

2.4 Zeno (BC490 - BC430): The Motion Paradox
In this section, from the quantum linguistic point of view, we study Zeno’s paradoxes, theoldest paradox in science.

2.4.1 What is Zeno’s paradoxes?
anything

Without a worldview, we cannot say

Zeno is a disciple of Parmenides. Thus, Zeno’s Paradoxes may a collaboration between
the two. Although Zeno’s paradox has some types (i.e., ”ﬂying arrow”, ”Achilles anda tortoise”, ”dichotomy”, ”stadium”, etc.), I think that these are essentially the sameproblem. And I think that the ﬂying arrow expresses the essence of the problem exactly
and is the best masterpiece in Zeno’s paradoxes. As we will see in the next section,”Achilles and a tortoise” is a trick question.

Now we present Zeno’s paradoxes (i.e., ﬂying arrow) as follows: Please enjoy the literarypleasure.

Paradox 2.9. [Zeno’s paradoxes: The literature-like antinomy]
The literature-like proof of [Flying arrow is at rest]• Consider a ﬂying arrow. In any one instant of time, the arrow is not moving.
Therefore, if the arrow is motionless at every instant, and time is entirely composedof instants, then motion is impossible.
The literature-like proof of [Flying arrow is not at rest]• We have to accept that an arrow passes there. However, ”to pass there” is not
equivalent ”to exist there”. What is ”to pass there”? ”To pass there” is both ”toexit there” and ”not to exist there”. Therefore, ﬂying arrow is not at rest.

Therefore,
if we believe in such ”logic” as above, we fall into a paradox.
Thus, our present problem is ”How do we deﬁne ’logic’ ?”63

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2.4 Zeno (BC490 - BC430): The Motion Paradox
Now we can answer the question ”What is Zeno’s paradoxes?”. That is, we consider
that(A) Zeno’s paradoxes say ”Don’t trust on ‘logic’ unconditionally”, that is,”Start from a worldview and not logic”.
since Paradox 2.9 shows that ”antinomy” happens in the logic in ordinary language. If
so, we have to obey the worldviewism in Sec. 1.1. that is,• First declare the worldview, and discuss the world in the worldview. Namely,
world is soconclusion
thereforeworldview −−−−−→ discussions, calculation, practical logic, properties
premisesubject

Therefore, to solve Zeno’s paradoxes is to solve the following problem
Problem 2.10. Propose a certain worldview, in which Zeno’s paradoxes (e.g., Flying
arrow) can be discussed.
[Answer]; This is answered in Answer 2.11 below.
Slogan-wise, we say(B) ”Without a worldview, there is no logic (or precisely, practical logic)”.
Again see ”the worldviewism (A)” in Sec. 1.1.
♠Note 2.17. The great tradition of philosophy from Plato to Kant was always based on
the worldviewism(cf. Chaps. 3-10). I believe that the above (B) is the fundamental spiritof philosophy. As mentioned in Chaps. 11 and 12 later, the spirit of analytic philosophy
is ”Start from logic” or ”Think logically !”. Thus, I am somewhat skeptical of analytic
philosophy

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Chap. 2 Ancient Greek philosophy (before Socrates)

2.4.2* The solution⋆ about Zeno’s paradoxes (e.g., Flying arrow) in the motionfunction method
If we obey the motion function method, we can easily solve Zeno’s paradoxes (e.g., Flying
arrow) as follows.
Answer 2.11. [An answer to Problem 2.10(scientiﬁc answer)] Under the motionfunction method (cf. Section 2.3.4), we discuss ”Flying arrow” as follows.
• Consider the motion function x(t), that is, for each time t, the position x(t) of
the arrow is corresponded. It is obvious that(♯) ”for each time t, the position x(t) of the arrow is corresponded” does not
imply that the motion function x(t) is a constant function.
Therefore, the arrow is not necessarily at rest.

6

x(t)

t

♠Note 2.18. Recall that we were confused in Paradox2.9[Zeno’s paradoxes]. However, we
could easily solve it in Answer 2.11. Thus we should be surprised at the power of the
motion function method. If a certain worldview is determined, Zeno’s paradoxes can besolved. Thus, the motion function method is not necessarily determined uniquely. For
example, it is a good exercise to solve Zeno’s paradoxes under Newtonian mechanics orthe theory of relativity.

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2.4 Zeno (BC490 - BC430): The Motion Paradox

2.4.3 DST formula

In this section, we deﬁned ”motion function method”, which was a metaphysical method.
However, it is not diﬃcult, and it is similar to the ”DST formula” learned in elementary
school. The DST formula was the following formula.

D(distance)

S × T

=

(speed)(♣)(time)

This ”DST formula” may be the most diﬃcult formula/concept in elementary schoolmathematics. It took me a long time to be able to use this formula when I was a child.
Rather, I sometimes get confused even now. The following memory method, which was notavailable when the author was in elementary school, seems to be quite popular nowadays.

Here, the Pythagorean Theorem is a mathematical formula. The law of conservation of
momentum is a formula of Newtonian mechanics. Now let’s ask,
(C) In which ﬁeld is the DST formula (♣) a formula?
This problem is equivalent to the following problem:(D) Solve Zeno’s paradoxes!
That is, as the answer for (C), we think that• the DST formula (♣) is a formula in quantum language.
Thus, I think the DST formula (♣) is one of the most diﬃcult formulas taught in juniorhigh school, high school, and university.

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Chap. 2 Ancient Greek philosophy (before Socrates)

2.4.4 Appendix: The discussion about Zeno’s paradoxes (e.g., Achilles and a
tortoise]) in the motion function method
The idea of associating Zeno’s paradox (”Achilles and a tortoise”) with an inﬁnite series
misses the essence of Zeno’s paradox, because Parmenides and Zeno’s interest should bea worldview. That is, ”Flying arrow” is the most important paradox in Zeno’s paradoxes.
However, since geometric series’ method is most famous, I add ”Achilles and a tortoise”
as an appendix (cf. ref. [47], or Chap. 14 in ref. [74]). Readers should also taste theliterary pleasure in the following.
Paradox 2.12. [Zeno’s paradoxes(the literature-like answer)][Achilles and a tortoise]
Zeno’s paradox (Achilles and a tortoise) is as follows.
• Consider the competition of Achilles (a quick runner) and a tortoise (a late runner). Consider the competition of Achilles and a tortoise. Achilles’ starting point
will be behind the turtle’s starting point. Suppose that both started simultaneously. If Achilles tries to pass a tortoise, Achilles has to go to the place in which a
tortoise is present now. However, then, the tortoise should have gone ahead more.
Achilles has to go to the place in which a tortoise is present now further. Evenif Achilles continues to do this inﬁnite times, he will not be able to keep up withthe turtle.

[The scientiﬁc answer to Zeno’s paradox (Achilles and a tortoise) by the motionfunction method ]For example, assume that the velocity vq (= v) [resp. vs (= γv)] of the quickest [resp.
slowest] runner is equal to v(> 0) [resp. γv (0 < γ < 1)]. And further, assume that theposition of the quickest [resp. slowest] runner at time t = 0 is equal to 0 [resp. a (> 0)].
Thus, we can assume that the position q1 (t) of the quickest runner and the position q2 (t)of the slowest runner at time t (≥ 0) is respectively represented by the following motionfunction:q1 (t) = vt
(2.2)
q2 (t) = γvt + a
Thus, it suﬃces to calculate formula (2.2).
Although it can be solved by various method, I present two methods as follows(i.e., (i) or(ii)):
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2.4 Zeno (BC490 - BC430): The Motion Paradox[(i): Algebraic calculation of (2.2)]:
Solving q1 (s0 ) = q2 (s0 ), that is,vs0 = γvs0 + aaa
we get s0 = (1−γ)v. That is, at time s0 = (1−γ)v, the fast runner catches up with the slow
runner.
[(ii): Iterative calculation of (2.2)]:
Deﬁne tk (k = 0, 1, ...) such that, t0 = 0 andtk+1 = γvtk + a(k = 0, 1, 2, ...)k)aThus, we see that tk = (1−γ(1−γ)v (k = 0, 1, ...). Then, we have that(1 − γ k )a (1 − γ k+1 )a ,
q1 (tk ), q2 (tk ) =
1−γ
1−γ
a
a 
→
,
1−γ 1−γ(2.3)

aas k → ∞. Therefore, the quickest runner catches up with the slowest at time s0 = (1−γ)v.

q1 , q26

a
1−γ

..
.

..
..
.

3

(1−γ )a1−γq2 (t) = γvt + a?

2(1−γ )a1−γ······

6aq1 (t) = vt0(= t0 )

av(= t1 )

2

3(= t2 )(= t3 )

(1−γ )a (1−γ )a(1−γ)v(1−γ)v···a(1−γ)vt(= s0 )

Graph: q1 (t) = vt, q2 (t) = γvt + a[(iii): Conclusion]: After all, by the above (i) or (ii), we can conclude thata.
(♯) the quickest runner can overtake the slowest at time s0 = (1−γ)v♠Note 2.19. (a): Note that the above (ii) [= the solution using a geometric series] is anothersolution of (i). Of course, there was no need to use geometric series. The point is that there
is a diﬀerence in the position of whether one considers the Achilles and a turtle paradox
to be a mathematical problem or a philosophical problem. Many philosophers have gotten

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Chap. 2 Ancient Greek philosophy (before Socrates)
into dead ends by confusing the two positions. Philosophers should have considered it asa problem of world description.
(b): From the philosophical point of view, ”ﬂying arrow” is deﬁnitely better than ”Achilles
and a tortoise”. However, as far as quizzes go, ”Achilles and a tortoise” is well done.
”Achilles and a tortoise” is well done in the sense that the trick is designed to make iteasy for solvers to fall into it. In fact, for 2,500 years, most solvers have fallen for thistrick.

♠Note 2.20. (i): As mentioned in Preface, the purpose of this text is to understand thehistory of western philosophy from the quantum linguistic point of view. Thus,(♯) We aren’t interested about how Zeno himself considered his paradoxes. The established theory may say that Zeno might study the inﬁnite division of time in physics(and space)
However, if so, Zeno’s paradoxes are the problem in physics and not philosophy. Thenthe problem should be entrusted to physicists. However, in this text, we assume that
Zeno’s paradoxes belong to philosophy and not physics. Also, for the answer to ”What isspace-time in quantum language?”, see Sec. 9.2.2: (Leibniz=Clarke correspondence).
(ii): Zeno’s paradox attracted the interest of many philosophers because, in the author’s
opinion, the kinetic function method is a metaphysical method. However, Zeno’s paradox
is mentioned by many philosophers, including Aristotle, Spinoza, Bertrand Russell, andBergson. But I always think, ”Even a wise philosopher looks like a fool when talking
about Zeno’s paradox.” So, readers may think I am a fool. I don’t mind if they think so,
but I would like them to continue reading to the next chapter, not to stop reading here.
♠Note 2.21. It would not be interesting to solve an unsolved problem that is 2,500 years
old if this did not aﬀect various issues in modern philosophy. I have some doubts about
analytic philosophy, as I will discuss in Chapters 11 and 12.
• The reason for this is that Zeno’s paradox cannot be solved in analytic philosophy.
More speciﬁcally, it is because the problems in the list (D1 ) of Preface and the list (D2 )of Postscript cannot be solved by analytic philosophy.
Since it was ”solving Zeno’s paradox” that started my questioning of analytic philosophy,
I devoted a considerable number of pages of this chapter to explaining ”Zeno’s paradox”.

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71

Chapter 3

The Big Three in Greek Philosophy(Socrates, Plato)
By the appearance of The Big Three in Greek Philosophy (Socrates, Plato, Aristotle), theorigin of western philosophy was formed as follows.
(♭1 ) Ethics, etc.(Socrates)*1(♭2 ) worldview(♭21 ) : dualism and idealism (Plato, Augustinus, Anselmus, Thomas Aquinas,Descartes, Locke, Berkeley, Hume, Kant,Wittgenstein*2 ,...,quantum language))(♭22 ) : monism and realism (Aristotle → Newton)
Whitehead(1861 - 1947) said:• Western philosophy is characterized as a series of footnotes to Plato
I agree to him, thus in this text I will devote myself to the above (♭21 ).

Socrates was the teacher of Plato, and Plato was the teacher of Aristotle.

*1 Here in the text, we will concentrate only on the scientiﬁc part of Plato*2 I am of the opinion that Wittgenstein should belong to dualistic idealism. (cf. Chaps. 11,12)

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3.1 Protagoras and Socrates

3.1 Protagoras and Socrates
3.1.1 Ethics
The philosophy of worldview aimed at the following problems(A) How is the world described?
How is the world understood?
By what kind of language is the world described?
But, there is another philosophy (i.e., philosophy of ethics) diﬀerent from the worldview.

Ethics, morals [How should we live?]
Many people in the sciences (no, even the liberal arts) may think that ethical philosophy is”debating skills”. In fact, Protagoras (490 BC. - 420 BC.), a central ﬁgure of the Sophists,
preached that ”man is the measure of all things”. He argued for relativism, which holds
that there is no such thing as objective truth. In other words, he argued that only the subjective
judgment of each person is essential. Since the common sense of capitalism is”the average of subjective value” = ”price”
one might say that the sophist’s claim is reasonable and modern common sense.
However, Socrates (BC.469 - BC.399) had objected to this idea. He investigated that(B) How should we live?
And, he clariﬁed the following words:(C) ”goodness”, ”happiness”, ”virtue”, ”justice”, ”courage”, ”love” · · · · · ·That is, Socrates asserted that the investigation of the above words is also the central themeof philosophy. In the following dispute:”relativism (rational sophists) ” vs. ”absolutism (a man of faith: Socrates) ”
Socrates advocated the ethical philosophy.

Note that three philosophers (Socrates (BC.469 - BC.399), Buddha (BC.565 - BC.486), Confucius (BC.551 - BC.479) and Mozi (BC.470 - BC.390)) were contemporary, and investigated thesame problem (B). In this sense, we can say that
(D1 ) The mathematics of Pythagoras was unparalleled in the world. Compared to this, Socrates
was common sense and mediocre.
It is a matter of course that
(D2 ) If these words mentioned in the above (C) didn’t spread, the human race might have been72

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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)ruined. At least, we wouldn’t be able to form ”human society”. Maybe the mankind
perished.
Therefore, I cannot overemphasize the importance of ethical philosophy. Also, the philosophy of
ethics is worldwide. When it isn’t so, we’re in trouble. That is, when it isn’t so, ”world peace”
isn’t achieved. As emphasized throughout this text, I believe that•

The main theme of philosophy is ethics (and not worldview). Even if we
didn’t have a philosophy of world description, we would have done it reasonably well.

Hence I agree that Socrates is called the father of philosophy .
♠Note 3.1. In general, Ethical philosophy does not have any truth or universality. Societyof ants has ”ethics and morals of ants”, and apes must have ”ethics and morals of apes”.
The ethical philosophy of the Neanderthals must have been quite diﬀerent from ours.
Therefore, our ancestors must have destroyed the Neanderthals. Communication andcontact with civilized aliens will happen in the future. However, it is too optimistic to
expect the aliens to be unconditionally friendly at this time. I remembered that the lateDr. Hawking had emphasized the same opinion.

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3.1 Protagoras and Socrates

3.1.2 Magic proposition: I know I know nothing♠Note 3.2. Socrates did not leave a book. The ”Sophists vs. Socrates” is Plato’s ﬁction.
Since Plato is a disciple of Socrates, it’s not fair. For example. The strongest logic, ”Iknow that I know nothing” goes something like this.
• Sophists: something is asserted · · · · · ·• Socrates: deny Sophists’ assertion• Sophists and Socrates: debate (called Socratic Method)· · · · · · Sophists and Socrates tell eloquently, and thus, they get tired. · · · · · ·• At that time, Socrates says ”Your ignorance is now revealed. I know that I know
nothing, but you do not know that you know nothing. Thus, I am superior to you”.
This is Socrates’ strongest logic ”I know that I know nothing”. If we, without sticking
to an established theory, read Plato’s novel which makes Socrates a main character, we
may have a variety of opinions on the following issues.
• Which is playing with sophistry, Socrates or sophists?
I may feel that Socrates uses more sophistry.

In physics, you can draw conclusions with experiments, in mathematics you can provethem, and in engineering you just need to employ something useful. However, ethicalphilosophy does not solve the problem in that way. So Socrates (= Plato) came up withthe magic word ”I know that I know nothing” (self-referential proposition, antiCopenhagen interpretive proposition) as a way to end the discussion.
(E) Since Socrates is the main character of Plato’s novel, he must always be undefeated.
Suppose Socrates had said to Protagoras (whose name he borrowed on behalf of theSophists)(F) ”I know that I am ignorant, while you do not know that you are ignorant. You, on
the other hand, do not know that you are ignorant.”
If this is a scientiﬁc proposition, then Socrates must provide a way to experimentally
verify the following(G1 ) Socrates knows that Socrates himself knows nothing.
(G2 ) Protagoras does not know that Protagoras himself knows nothing
If Socrates cannot present a method to experimentally verify this, then it is a pseudoscientiﬁc proposition. Therefore, Protagoras must have a ”point” to make, but due to thecircumstances described in (E) above, Protagoras was not given an opportunity to argue
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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)
his case.
In the ﬁeld of argumentation, pseudo-scientiﬁc propositions are often eﬀective. Since
the meaning of pseudo-scientiﬁc statements is unclear, listeners (i.e., readers) often takethem as profound propositions, and they often become ”decisive words”. Descartes also
used the pseudo-scientiﬁc proposition ”I think, therefore I am” with success. A good
philosopher, a good man of letters, and a good sophist have much in common. This is not
surprising. Philosophy is a kind of literature, not science. Therefore, it is not only good,
but also recommended for philosophy to make skillful use of pseudo-scientiﬁc propositions
in order to advance the argument.
As described above, philosophy and pseudoscience have a very high aﬃnity. This highaﬃnity led to the emergence of the ”Copenhagen Interpretation” in philosophy. This isbecause,(H) the Copenhagen Interpretation is ”a rule for drawing a line between scientiﬁc andpseudo-scientiﬁc propositions (taboo language) (cf. Sec. 1.3 (C3 )).
Recall the following:where,••


Plato · · · Idea
Anselmus, Thomas· · · universal
Axiomatic theory
Descartes · · · I think therefore I am · · · mind-body problem
Wittgenstein · · · practical logic
Copenhagen  Augustinus · · · Only the present exists.
Descartes · · · I think therefore I am · · · subjectivity problem
interpretationKant · · · thing-in-itselt

I believe that the history of Western philosophy is the history of the highest intellects ofthe time, who have been thinking about these pseudo-scientiﬁc terms and propositions formore than 2,000 years.
I will demonstrate this throughout this text.

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3.2 Plato (BC.427 - BC.347)

3.2 Plato (BC.427 - BC.347)

3.2.1 The theory of Ideas – Asserted ﬁction –In the binary opposition (in ethics):(A)”relativism (rational sophists)” vs. ”absolutism (a man of faith: Socrates)”

Plato, a student of Socrates, established ”the theory of Ideas” as the foundation of absolutism in order to support Socrates.
If the propositions such as ”Man’s life is heavier than the Earth”, ”Love is forever”,”Love always overcomes money”, etc. are the objective truth, the occupation of theinsurance company does not hold. However, Socrates wanted to believe so. To help
Socrates, Plato proposed the occult heavenly world (i.e., the world of Idea). That is,(B) the theory of Ideas is a reckless attempt to derive ethics (i.e., ”How should welive?”) from worldview (i.e., ”How is the world?”), that is,
the ﬁction (spell) that ”the world is so”

idealistic worldview

Let’s consider sotherefore−−−−−−−−−−−→ consideration, discussion
habitual thinking

This method (= the form of philosophy) has been accepted as the standard form of ”howto tell philosophy” in the history of two thousands hundreds of years.

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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)(C): The ﬁction called ”the theory of Ideas”
Theory 3.1. [Theory of Ideas]:
The theory of Ideas is as follows• In the real world, it is impossible to say that ”love is better than money.
However, there exists another world (i.e., the Idea world) in which ”love isbetter than money” can be believed as an objective truth. And the existenceof the Ideal world can be realized through the Idea.
This is the theory of Ideas.
Then, the real world is a shadow picture, hence, in the real world,(D1 ) love sometimes loses money(D2 ) We can’t live on justice alone.
(D3 ) Good man is sometimes unhappy.
That is, Plato wanted to say that”love always overcomes money” is the objective truth in the world of Idea,
therefore, ”Believe in love!”
Whether you believe in this reasoning or not, this idea, i.e. the discovery of two key
words ”Idea world” and ”reality world”, was the beginning of ”dualistic idealism” andhas always been the mainstream of Western philosophy, despite the twists and turns that
followed. The theory of ideas, which was supposed to be a logistic support of ethicalphilosophy (Socrates), became the mainstream of the philosophy of worldview.
(E) Our human DNA prefers logic (or reasoning) as if the philosophy of ethics were
derived from the philosophy of worldview. That is,”the world is so”, therefore ”we should live so”
namely,(F1 ) (=(B))
the ﬁction (spell) that ”the world is so”

idealistic worldview

Let’s consider sotherefore−−−−−−−−−−−→ consideration, discussion
habitual thinking

Also, for example,
don’t steal a glance

invisible man exists•

therefore

idealistic worldview −−−−−→ practical logic, ethics·morals(ﬁctional)premisemain subject

andultimate beauty exists•

she is an embodiment of beauty.
therefore

idealistic worldview −−−−−→ practical logic, ethics, morals, aesthetic sense
premise, ﬁction

main assertion

This is, of course, irrational since this is a reckless attempt such that the problem ”How77
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3.2 Plato (BC.427 - BC.347)
should we live?” is answered from the objective point of view. However, the human brain
does not operate on logic alone. ”Logic” cannot function without some kind of sensuouscommon soil. Logic alone is not enough, what is important is ”logic in a common
sensory soil” (= ”practical logic”).
In other word, ”without common sensory soil, logic cannot work”, therefore”the ﬁrst thing to do is to form a common sensory soil. If so, Plato’s way oftelling philosophy (F1 ) may not be reckless. That is because the (F1 ) means
we should live soworld is so(F2 )thereforeﬁctional worldview −−−−−→ practical logic, Ethics · morals
introduction·preface·ﬁctioncommon sensory soil

main subject

That is, ”logic” depends on ”worldview”. Slogan-wise, we say(G) (=Sec.2.4 (B)) ”Without a worldview, there is no logic (= practical logic)”.
Recall that this slogan plays an essential role in the solution of Zeno’s paradox
Sec.2.4 (B), Sec. 1.1).

(cf.

♠Note 3.3. Here, ”logic in the common soil of the senses” has a similar meaning to ”logic
under a certain worldview”. In a broader sense, this ”logic” is the logic usually used byordinary people, such as the logic of Newton mechanics, the logic of a court case, the logic
of politics, the logic of family life, the logic of insurance solicitation, and so on. One ofthe various kinds of logic is the logic of mathematical logic (= symbolic logic), which is souniversal that even aliens may know it. However,(♯) this does not mean that mathematical logic experts can master the logic of otherdisciplines well.
Naturally, a good understanding of economics means that one can use the logic of thecommon soil of economics. Judges and lawyers don’t have to learn mathematical logic.
It is natural to expect that logic under the theory of Ideas promote our understanding of
Socrates and Plato ethics.
As mentioned in Chap. 12, we have two types of logic (i.e., symbolic logic and practical
logic), that is,symbolic logic (=mathematical logic), i.e., logic in mathematics•
practical logic (= non-mathematical logic), i.e., logic induced by worldview
Practical logic plays an important role in worldviews. On the other hand, symbolic logic
plays an important role in mathematics. I am skeptical of analytic philosophy, in which
symbolic logic and practical logic may be confused. I feel that the above spirit (♯) is scarcein analytic philosophy. Analytic philosophy is not mathematics.
♠Note 3.4. (i): Aesop (BC.620 - BC.510): Idea theory is similar to Aesop’s fable in someways. It is natural to be hesitant to teach morality to others face-to-face. I think Aesop
was preaching ethics indirectly through the animals. Aesop’s fables were already wellknown before the late 5th century BC. Of course, Plato must have been aware of thepersuasive power of Aesop’s fables. And further, Plato might think that

A worldview is an abstraction of many allegories
In this sense, Plato might discover the power of abstraction.

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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)(ii): In the book ”Sapiens: A Brief History of Humankind” (cf. [30], 2016), Harari,
the author, describes three revolutions which brought about dramatic changes to humansocieties.

1 : the Cognitive revolution, about 70,000 years ago ○2 : the Agricultural revolution, about 11,000 years ago○

3 : the Scientiﬁc revolution, beginning a mere 500 years ago○
1 implies that
The above ○• we can rule the world because we are the only animals that can create and believe inﬁctions like God, the state, money and human rights.
Buddha (BC.565 BC.486), Confucius (BC.551 BC.479), and Mozi (BC.470 BC.390) wereactive on similar themes at the same time as Socrates and Plato. The reason is probably
because the technology to record it as a character has been established.
////

Plato’s say is as follows: No matter how much we argue, the issue of ”relativism (rational
sophistry) vs. absolutism (man of faith: Socrates)” can’t be concluded. It is not a truththat we seek. There may not be the truth.
What people want is an ”asserted ﬁction”, not the truth.
And this is the philosophy. I think that’s what Plato would have thought. In a simple
analogy, it’s next.

Say!, ”I love you,” even if it’s a lie.
♠Note 3.5. S. Weinberg (1933 -2021), a physicist at the University of Texas, Austin, won
a Nobel Prize in 1979 for work that became a cornerstone of particle physics, said in his
book [123] ”To explain the world; The discovery of modern science” as follows:(♭) [ in Chapter 1] There is an important feature of modern science that is almost completely missing in all the thinkers I have mentioned, from Thales to Plato: none ofthem attempted to verify or even (aside perhaps from Zeno) seriously to justify their
speculations.
Plato’s philosophy is a representative of idealism (cf. for the deﬁnition of ”idealism”, seeSec. 1.1), which is a completely diﬀerent style of philosophy from (experiment-oriented)
science. As noted above, Plato’s philosophy is a philosophy that begins with ”asserted
ﬁction”. Also, it is a time of F. Bacon (i.e., the father of British Empiricism) of thescientiﬁc revolution (the 17th century) that the importance of observation was, for theﬁrst time, emphasized. See Section 7.2
Recall the rule of philosophy (≈ idealism) is as follows:79

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3.2 Plato (BC.427 - BC.347)(♯1 ) Only discussion, no experimentation.
(♯2 ) Winners are determined by the popularity of the general public (or philosophical enthusiasts).

♠Note 3.6. In order to avoid eternal argument, Socrates invented ”Socratic method”, thatis, the magic phrase ”I know that I know nothing”. Plato also invented the theory of
Ideas.
For completeness, we add the following:(♯1 ) I think that Plato did not believe in the existence of the world of Idea. If he believed
in it, he was not a philosopher but a founder of religion. He also understood that thetheory of Ideas is sophistry, and there is no truth in ethics.
Even so, there may be a reason to consider that(♯2 ) Something such as the prototype of the sense of ethics of the human commonness
may be printed in the arrangement of a human DNA,
However, this idea may not be within philosophy.

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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)

3.3 Three allegories about the Idea (the cave, anamnesis, thesun )The ”Ideal World” is like heaven. With religion, you can repeat it over and over until
they believe it. Plato, however, tried to explain this with ”three allegories (the allegoryof the cave, the allegory of anamnesis, and the allegory of the sun).
This Platonic contrivance was ”Idea”, which became the ”magic word” that has always
remained at the center of Western philosophy’s 2,000-plus year history.

3.3.1 Allegory of the cave
The allegory of the cave was presented by Plato in his work ”Republic” in order to promote
the understanding of ”the theory of Ideas”.

Allegory 3.2. [Allegory of the cave]

A group of prisoners are looking at the shadows on the back wall of a cave. The
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3.3 Three allegories about the Idea (the cave, anamnesis, the sun )shadows just are reality since they have never experienced anything other than shadows.
The shadows represent physical reality. One day, one of the prisoners gets free. Namely,
he is the philosopher, the lover of wisdom. Let’s assume he is Socrates. He turns aroundand the ﬁrst thing he sees is objects of stone and wood made to resemble the shapesof real things such as a tree. Further along, generating the light that hits these objects
that then produce shadows on the back wall of the cave, is a ﬁre. Beyond the ﬁre is theentrance/exit of the cave. The philosopher exits the cave and is temporarily blinded by
the light. The ﬁrst thing he sees is a real tree. Finally, the philosopher sees the sun which
Plato called The Idea of the Good. The freed prisoner (who is assumed to be Socrates)
would think that the world outside the cave was superior to the world he experienced in
the cave. And he would want to bring his fellow cave dwellers out of the cave and into thesunlight. The returning prisoner, whose eyes have become accustomed to the sunlight,
would be blind when he re-enters the cave, just as he was when he was ﬁrst exposed tothe sun. The prisoners would infer from the returning man’s blindness that the journey
out of the cave had harmed him and that they should not undertake a similar journey.
Hence, there is a possibility that the prisoners would therefore reach out and kill anyone
who attempted to drag them out of the cave. In fact Socrates was killed.

Remark 3.3. Descartes’ dualistic idealism has the following form:[A](mind)←− [B : body] −→[C](matter)(medium)

This suggests a correspondence between the Allegory of the Cave and measurement
theory (= quantum language), as follows.
\[A](= mind)[B](Mediating of A and C)(body)

Plato(Allegory of the cave)actual world

sunlight

quantum language

measured value

observable[C](= matter)/[Idea world]state[system]

In essence, he brought up the instrument of visualization (the sun = Idea) to make us
believe in an ”ideal world”.

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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)

3.3.2 The theory of anamnesis
Plato uses various metaphors to explain the Idea Theory. Besides the ”Allegory of theCave” in the previous section, they are the ”The theory of anamnesis” (this section) andthe ”Allegory of the Sun” (next section).
As a mediator between the real world and the idea world, Plato presents the conceptof ”anamnesis”. Namely,• We had seen the Idea before we were born. But we forget that when we are born.
Therefore, to know an Idea is to recall the Idea. In other words, learning is nothing
less than recalling (= anamnesis).

In the history of western philosophy, the theory of anamnesis is located as follows:[modern philosophy]

Descartes
Plato[anamnesis]−→[Locke, Berkeley, Hume]














[a priori, nativism]






 Continental Rationalism






















[tabula rasa, experience]

British Empiricism[Leibniz][”Critique of Pure Reason”]−→ *3

Kant philosophy[summing-up, compromise]

However, I think that ”tabula rasa vs. nativism” is pseudoscientiﬁc (cf. Sec. 10.3), andthus, the above is not our interest.
Remark 3.4. Descartes’ dualistic idealism has the following form:[A](mind)←− [B : body] −→[C](matter)(medium)

Adding the recall theory to the table in Remark 3.3, we obtain the following*3 In our style, ”Kant’s Synthesis” can be rewritten as ”QL Synthesis”(cf. Chap. 10).

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3.3 Three allegories about the Idea (the cave, anamnesis, the sun )\[A](= mind)[B](Mediating of A and C)(body)

Plato(Allegory of the cave)actual world

sunlight

actual world

anamnesis

measured value

observable

Plato (anamnesis)
quantum language[C](= matter)/[Idea world]/[Idea world]state[system]

In short, in order to make the ”Idea World” seem familiar, Plato said, ”Before we wereborn, we lived in the Idea World. we just forgot about it.”84

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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)

3.3.3 The allegory of the sun: Measurement theoretical aspect of Idea theory
The analogy of the sun is found in ”Republic”, written by Plato. Upon being urged by
Glaucon (Plato’s elder brother) to deﬁne goodness, a cautious Socrates professes himself
incapable of doing so. Instead he draws an analogy and oﬀers to talk about ”the childof goodness”. For the answer to ”Why a child?”, see Note 3.7.
Allegory 3.5. [The Allegory of the Sun]: The Allegory of the Sun explains whatthe ”Idea of Good” is all about. No matter how much you open your eyes, you cannot
see anything ”visible” in the ”visible” world, such as a ﬂower, a tree, or a dog, without
the light of the sun.

In the beginning, when it’s dark as shown on the left ﬁgure, you can’t see anything, even
if your vision is normal. However, by developing the skills to use a measuring instrument
called the ”Sun [= Idea of Light],” you can see that it is a ”tree”. Thus, in this fable,
the ternary relation (”the beholder,” ”the sun (the mediating thing),” ”the thing to beseen”) are clear and very easy to understand.

In the same way, things that exist in the invisible realm, such as virtue and courage,
cannot be known unless one has the skill to use the measuring instrument called the”idea of the good”, even if one has reason.

In other words, if you learn how to use a measuring instrument called the sun, you canmake the object ”visible”. In the same way, the mastery of the measuring instrument,
which is called the ”idea of Good,” makes clear virtue and courage.
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3.3 Three allegories about the Idea (the cave, anamnesis, the sun )Remark 3.6. The polar star can be regarded as a measuring instrument such as a kind ofcompass (cf. Note 1.7 (iii)). Thus, it is reasonable to regard the sun (Idea) as a measureof ”Good”.

That is, I want to think:”Idea” = ”instrument to make Ideal world visible”.
Therefore, three key-words in Plato philosophy:[A](mind)←− [B] −→[C](matter)(medium)

correspond to as follows:
Adding the allegory of the sun to the table in Remark 3.4, we obtain the following\[A](= mind)[B](Mediating of A and C)(body)

Plato(Allegory of the cave)actual world

sunlight

Plato (anamnesis)

actual world

anamnesis

actual world

Idea (Sun)measured value

observable

Plato (Sun)quantum language[C](= matter)/[Idea world]/[Idea world]/[Idea world]state[system]

Of course, this is somewhat too forced. That is because the theory of Idea was not put
forward with the intention of being a theory of measurement (or epistemology).

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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)

3.3.4* The measurement theoretical understanding of the allegory of the sun
Recall Figure 1.3 [ Descartes ﬁgure] in Sec. 1.2.1, namely,a ○)”b in dualism
Figure 1.3 ; [Descartes Figure]: Image of ”measurement(=○+This is very similar to the ”the allogory of the sun such as :Not only are the ﬁgures similar, but it is tempting to assert the following relationship.
Figure B [Idea of good]in Allegory 3.5

science→======←philosophy

Figure A: [Sun]in Allegory 3.5

87equivalent======

Descartes Figure
in Figure 1.3

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3.3 Three allegories about the Idea (the cave, anamnesis, the sun )Here, it should be noted that this is essentially the same as the ﬁgure [Allegory of thesun].
Summary 3.7. We see:Figure B [Idea of good]in Allegory 3.5xDeﬁnegoodness

science→======←philosophy

Figure A: [Sun]equivalent======

Descartes Figure
in Figure 1.3

in Allegory 3.5xIllustration

Plato philosophy (Theory 3.1)

science

Theory of IdeasAllegoryy

Axiom 1 [measurement]

Allegory of the cave(Allegory 3.2)

+ AnamnesisyAnamnesis(Sec. 3.3.2)

philosophy

science

We consider that• the understanding of the ﬁrst line above, namely,
Figure B [Idea of good]in Allegory 3.5

science→======←philosophy

Figure A: [Sun]in Allegory 3.5

equivalent======

Descartes Figure
in Figure 1.3

This is equal to the measurement theoretical understanding of Allegory 3.5 [Allegory of
the sun]So, of the three explanations of Idea, ”the allegory of the cave,” ”the theory of Anamnesis,” and ”the allegory of the sun,” I think the one that comes closest to quantum language(= measurement theory) is the allegory of the sun.
♠Note 3.7. However, I do not think ”The Allegory of the Sun” is a suitable example of
Idealism because it is too scientiﬁc and easy to understand. Perhaps Plato gave too
scientiﬁc an explanation in order to make the Idea Theory understandable to his readers.
I believe that Plato was not satisﬁed with the parables in The Allegory of the Sun, so
he called The Allegory of the Sun ”The Son of Good.” In B. Russell’s work ”History ofWestern Philosophy,” he writes (cf. ref. [116])• Aristotle’s metaphysics, roughly speaking, may be described as Plato diluted by common sense. He is diﬃcult because Plato and common sense do not mix easily.
If we believe Russell’s words, then Idealism and the ”the allegory of the sun” cannot easily
be mixed. In other words.
• This ”Allegory of the Sun” is too scientiﬁc and may interfere with a philosophicalunderstanding of Idealism.
However, in Scholastic philosophy, more than a thousand years after Plato, Anselmus was

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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)
the man who took the Idea theory even more scientiﬁcally. This was the beginning of aseries of revelations about the scientiﬁc aspects of the Ideas. A summary of this is written
in Sec. 3.5.

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3.4 Plato: The worldview called Idea theory; dualistic idealism

3.4 Plato: The worldview called Idea theory; dualistic idealism
3.4.1 Worldviewism
Review the Platonic worldview (= Idealistic worldview)(cf. Sec. 1.1).
(B) Worldview Typeworldview(i): realistic worldview (physics, Aristotle philosophy, etc.)(=Aristotelianworldview)







 (ii): idealistic worldview (Statistics, Plato, Descartes, Kant , etc. )(Plato worldview)







(logical worldview)(iii):
mathematics(=mathematicalthinking)(Set theory, etc. )*4(logical worldview)

Above, our interest is in (ii), the idealistic worldview (= Platonic worldview). In otherwords.
(ii) [Idealistic worldview (=idealism)] is ”a way of trying to understand the world
through sutra-like statements”. We can assume statistics (which can be applied to
medicine, economics, engineering, or anything else). Thus, ”the worldview comesﬁrst and the world comes after”. The Idealistic worldview (since it is worded like asutra) makes no sense to determine right or wrong by experimentation, but ”usefulness (usefulness?)” is evaluated by ”usefulness”. In fact, statistics is very useful, as
it is said to be the most powerful science. The idealistic worldview is quite broad,and dualistic idealism is the mainstream of Western philosophy [Plato, Scholasticism, Descartes-Kant’s epistemology, analytic philosophy (in the author’s opinion)].
the ﬁction (spell) that ”the world is so”

idealistic worldview

Let’s consider sotherefore−−−−−−−−−−−→ consideration, discussion
habitual thinking

There are rather non-scientiﬁc idealistic worldviews, and in fact, Plato, Schola, DescartesKant’s epistemology, and analytic philosophy are ”rather spurious idealistic worldviews.
In Plato’s philosophy, Idealism (= ﬁctional worldview) is nothing but a ﬁction (= parable
or allegory).
• Idealism is nothing more than a quibble (a parable, a fashion) to justify
Socrates Plato’s ethics.
Plato’s way of speaking philosophy (i.e., ”a rather fanciful conceptual worldview”) is
common to all philosophies of the dualist-idealist lineage, as follows.

*4 It is impossible to consider mathematics as a kind of worldview, but I will classify it here in thesense of ”mathematical thinking.” (cf. Sec.4.3.4: Set-theoretic Worldview, Logical Worldview)90

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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)(A) Plato (the theory of Ideas)−→Augustinus−→Thomas Aquinas −→Descartes−→Kant(epistemology)−→analytic philosophy
There are, of course, reasons why a ”rather frowned upon conceptual worldview” could
become the mainstream of Western philosophy as described above. I shall write thisbelow.

3.4.2 The necessity of the worldview
Even if the theory of Ideas is a fairy tale, Plato’s idea has a point.
(B) Ethics · morals is dependent on the world (=environment around).
It is a matter of course that there is a diﬀerence between the ancient Greek ethics and themodern American ethics. In this sense, strictly speaking, the following (=Plato’s way oftelling philosophy) is true:(C) (= (F2 ) in Sec. 3.2.1)
the ﬁction (spell) that ”the world is so”

idealistic worldview

Let’s consider sotherefore−−−−−−−−−−−→ Ethics · morals, etc.
habitual thinking

common sensory soil

main subject

3.4.2.1 The necessity of idealism (= metaphysical world)
If the (D) is strictly put into practice, this is not philosophy but life consultation. For
example,• If you are really depressed, go to a psychiatrist and ask him to prescribe somemedication. Philosophers are not to be trusted.
Thus, philosophers have to assume an unrealistic world (metaphysical world). This is because if the settings can be veriﬁed by experiment, mistakes may be pointed out. Therefore,(D) Metaphysical worldview (i.e., idealism) is desirable
This is the reason to adopt the idealism (i.e., metaphysical world) in Plato’s way of telling
philosophy

3.4.2.2 The necessity of dualism
Also,(E) since the goal is ethics and morality, a world that reﬂects human beings in some way
is preferable. In other words, the dualism of ”things” and ”people” is preferable.
For example,
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3.4 Plato: The worldview called Idea theory; dualistic idealism• We don’t usually take moral lessons from monism, such as Newtonian mechanics.
This is the reason to adopt ”mind-body dualism” in philosophy.
After all, we conclude that(F) in the philosophy of worldview, dualistic idealism is desirable.

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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)

3.5 Appendix: How could Plato’s philosophy be the source ofWestern philosophy?
Let us consider ”How could Plato’s philosophy be the source of Western philosophy?”

In other words, ”How did the above ﬁgure come about?” is what we would like to consider.
Since this question is to be considered throughout this text, I do not intend to draw ahasty conclusion in this section, but there may be readers who would like to know theconclusion here and now. For such readers, I would like to conclude.
• The word ”Idea” just happens to be the best ”magic word” in the history of philosophy*5 .
This will be explained as follows.

3.5.1 History of development from ideas (Plato) to observables (quantum language)
Descartes’ dualistic idealism has the following form:[A](mind)←− [B : body] −→[C](matter)(medium)

We then consider the position of ”Plato’s Idea” as shown in the table below.
\[A](= mind)[B](Mediating of A and C)(body)

Plato (Sun)actual world

Idea

quantum language

measured value

observable[C](= matter)/[Idea world]state[system]Let’s preview here the history of the evolution of ”Ideas” to become ”Observables.” As
will be explained throughout this text, the following can be said in summary.

*5 However much of a genius Plato was, I think it was ”accidental.

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3.5 Appendix: How could Plato’s philosophy be the source of Western philosophy?
(A)
a Idea○:linguistic version−−−−−−−−−−−→

d signiﬁer, signiﬁed○:

Plato’s idea theoryChristianyversion

Saussure’s linguistic theoryb universal○:

e membership function○:

Anselmus, Thomas

Zadeh’s fuzzy theoryquantiﬁcationycognitiveyversion

c secondary quality○:yidentiﬁcation

linguistic version−−−−−−−−−−−→

John Lockef observable○:
quantum language

I will describe it in more detail later, but here I will draw a picture to illustrate theimage.
a Idea○the meter standard of goodness, the meter standard of beauty, , · · · ”=⇒
Device for visualizing goodness, Device for visualizing beautyb universal○
mankind, Americans, hero, good people, · · ·c secondary quality (=sensations of inherent nature (=primary qualities))○”sweet, spicy”, ”hot, cold”, ”beautiful, ugly” · · ·d Saussure’s Signiﬁant and Signiﬁe○
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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)e fuzzy set (≈ membership function)○f observable(≈ measuring instrument)○
position, momentum, energy· · · (in quantum mechanics)

B({y})1•[B({y})](ω0 )
•
ω0

Ω

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3.5 Appendix: How could Plato’s philosophy be the source of Western philosophy?

3.5.2 The history of Western philosophy is the history of the pursuit of ”Whatis Idea?”
As mentioned in (A), I think(B) (=(A)) The history of Western philosophy is the history of the pursuit of ”What is
Idea?”
That is,(C) (=(A))
a Idea○:linguistic version−−−−−−−−−−−→

d signiﬁer, signiﬁed○:

Plato’s idea theoryChristianyversion

Saussure’s linguistic theoryb universal○:

e membership function○:

Anselmus, Thomas

Zadeh’s fuzzy theoryquantiﬁcationycognitiveyversion

c secondary quality○:
John Lockeyidentiﬁcation

linguistic version−−−−−−−−−−−→

f observable○:
quantum language

Thus, we may consider the following.
(D) The history of Western philosophy is the history of the pursuit of ”What is Idea?”
Even so, if many genius philosophers did not continue to hold the belief that there is ascientiﬁc answer to the question ”What is an idea?” Therefore, the answer to question(B) is as follows.
(D1 ) ”Idea” was the basic concept of dualistic idealism(D2 ) As you can see from (C)(=(A)), ”idea” has various aspects and is interesting enoughto never get tired of.
3.5.2.1 Guardian of christianity
Now, even if (F)(=(F1 )+(F2 )) above is the right answer, it does not mean that the”pursuit of ideas” can continue for more than 2,000 years.
Another reason, I think, is Christianity. Eastern philosophy has continued for 2,500
years in a delicately tense relationship, with various styles of Buddhism, Confucianism,
and Taoism mixed up. In the West, on the other hand, Christianity was too strong.
Therefore, the main theme (= ethics, morality) was mostly based on ”Christianity +Socrates”, and various styles were not born. Moreover, it can be said that the positionof Plato’s philosophy became immovable as a result of Augustine’s adoption of Plato’s
philosophy as the theoretical pillar of Christianity.
Also, according to worldviewism (Plato’s philosophical narrative), the introductory part(= the ﬁctional worldview of dualistic idealism) can be changed. Therefore, the progressof Western philosophy (which could not be realized in the East) can be realized in thefollowing way.
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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)• Plato−→Augustinus−→Thomas Aquinas −→Descartes−→Locke−→...−→Kant
That is,By reason of (D2 ), western philosophy could be kept fresh
To use an analogy, we think that the eﬀect was like car model change. The Platonic
way of telling philosophy is almost the mainstream of Western philosophy. This device(i.e., model change) brought about the rise of Western philosophy.

The pursuit of ”Idea” is, after all, equivalent to the pursuit of ”dualistic idealism.
Therefore, in fact, it is not a ”mere model change. The following diagram shows this.

In the case of Eastern philosophy, the founder was always great, and no mechanism wascreated to pass the baton to his successor.
(E1 ) There were so many diﬀerent aspects to ”Idea” that it wasnever boring or interesting.
(E2 ) Western philosophy kept itself fresh by alternating between”axiomatic theories” and the Copenhagen Interpretation.
3.5.2.2 Thanks to Aristotle (monistic realism), ”Idea Theory” became stronger
Compared to Eastern philosophy, a distinctive characteristic of ancient Greek philosophy is its emphasis on ”mathematics and logic” represented by Pythagoras.
The entrance gate of the school founded by Plato (the Platonic Academy) reads.
”The person who does not know the geometry should not pass through this
gate”

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3.5 Appendix: How could Plato’s philosophy be the source of Western philosophy?

Although Platonic philosophy has nothing to do with mathematics, the signboard was
eﬀective enough, as it led Aristotle and other gifted students to pass through the doorsof the Platonic Academy.
In fact, I believe that one of the factors that allowed Plato’s philosophy (dualistic idealism) to persist and remain in the mainstream of philosophy was the existence of its rival,
Aristotle’s philosophy (monistic realism). Since monistic realism is connected to Newton(physics), its prosperity has kept it in a ﬁrm position until today. In comparison, Plato’s
philosophy (dualistic idealism) has never given the general public a sense of ”usefulness.2
And yetAristotle (monistic realism) vs. Plato (dualistic idealism)
has been the ”greatest scientiﬁc controversy (see Sec. 4.1.3)” for 2500 years.
While Aristotle’s philosophy (monistic realism) was ﬁrmly established by Newton,
Plato’s philosophy (dualistic idealism) continued to boldly challenge Aristotle’s philosophy (monistic realism) despite its lack of a solid foundation. By continuing to challenge,
dualistic idealism became stronger and stronger. In the East, dualistic idealism became
religious because a powerful rival, Newton, did not emerge.

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99

Chapter 4

The Big Three in Greek Philosophy(Aristotle)
Although Aristotle was a disciple of Plato, he did not understand Plato’s philosophy. He
then proposed an realistic worldview that was quite diﬀerent from Plato’s philosophy.
He is called the father of all sciences (≈ the father of the realistic worldview). It is no
exaggeration to say that philosophy was started by these two men. That is, we see:the realistic worldview (physics)
Aristotle + Hellenism+Islam

Plato + Augustinus















⇒

Abelard, Ockham
Scholaticism(A.D.1000∼1500)
Anselmus,Thomas

⇒



















Newton → theory of relativity

Descartes·Kant
Wittgenstein→quantum
languagethe idealistic worldview (no experiment)

The main theme of this text is the ”idealistic worldview,” however, it is recommended
that the ”idealistic worldview” be understood in contrast to the ”realistic worldview”.

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4.1 Aristotle (BC.384 - BC.322)

4.1 Aristotle (BC.384 - BC.322)
4.1.1 Realistic worldview vs. idealistic worldview

Aristotle (BC.384 - BC.322), the student of Plato, is called the father of all sciences (≈the father of the realistic worldview). He could not accept Plato’s theory of Ideas(=
asserted ﬁction ). Namely,• Philosopher Plato preferred asserted ﬁction (without experiment) totruth (with experiment)
On the other hand,• Scientist Aristotle preferred truth (with experiment) to asserted ﬁction(= without experiment)♠Note 4.1. S. Weinberg (1933 -2021), a physicist at the University of Texas, Austin, won
a Nobel Prize in 1979 for work that became a cornerstone of particle physics, said in his
book [123] ”To explain the world; The discovery of modern science” as follows:(♯) [ in Chapter 3] I confess that I ﬁnd Aristotle frequently tedious, in a way that Plato is
not, but although often wrong Aristotle is not silly, in the way that Plato sometimes
is.
Plato was not aiming for science, that is, Plato’s purpose is to support Socrates’s ethicalphilosophy. Therefore, from the scientiﬁc point of view, some may feel Plato silly. Namely,(♭1 ) a ﬁction that was asserted (by Plato) over 2000 years ago is somewhat silly, if nottedious.
On the other hand, Aristotle might be aim for science. Thus, from the modern point ofview, some may feel Aristotle tedious. Namely,(♭2 ) truths discovered (by Aristotle) over 2000 years ago are tedious and often wrong, if
not silly
The above (♭1 ) and (♭2 ) are merely statements of the commonplace. And thus, he may
not be saying anything negative. Science progresses, so you’ll ﬁnd the old science boring.
But literature does not fade away after more than two thousand years. However, one of
our purposes of this text is to show that100

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Chap. 4 The Big Three in Greek Philosophy (Aristotle)

Although Aristotle does not appear in this diagram, it is impossible to understand themeaning of this diagram without an understanding of Aristotle.

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4.1 Aristotle (BC.384 - BC.322)

4.1.2 Edios and Hyle
Aristotle proposed the concepts such as ”eidos” and ”hyle” as follows.
(A): Edios(Aristotle’s Idea) and hyle
Aristotle said that
• Edios (= Aristotle’s Idea = true form) is not in the heaven, but in hyle (=matter = particle).
In short, Aristotle simply stated that”the nature of a thing is in the thing (not in the heavens).

The above is the same as the Pythagorean claim.

I think the reason for this is clear from the ﬁgure below left
The ﬁnal forms of monism and dualism wereThus, dualism (Plato) is considerably more complex than monism (Aristotle). With
this in mind, please read the following.
Summary 4.1. [The key-words of the realistic worldview] The meaning of ”hyle”
and ”edios” in (A) above is diﬃcult to understand. However, note that• Aristotle is the founder of the realistic worldview (= physics)

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Chap. 4 The Big Three in Greek Philosophy (Aristotle)
Aristotle philosophy was completed by Netwonian mechanics. That is,hylepoint mass

Aristotle −−−−−−→ Newton
progress
state
eidos
Thus, we see:\[B](Mediating of A and C)(body)[A](= mind)

Aristotle

————

————

Newton————————[C](= matter)hyle[eidos]point mass
[state][Note]: The above table should be compared to the following table in Remark 3.3 [ i.e.
Plato’s Idea theory ]:\[A](= mind)[B](Mediating of A and C)(body)

Plato: Sun
Remark 3.6

actual world

Idea

measured value

observable

quantum language[C](= matter)
Idea world
system[state]♠Note 4.2. The book : ”History of western philosophy” due to B. Russell says that (cf.
ref. [116]),• Aristotle’s metaphysics, roughly speaking, may be described as Plato diluted by common sense. He is diﬃcult because Plato and common sense do not mix easily.
I think this representation is misleading since the two men have very diﬀerent areas ofexpertise. Aristotle proposed a monistic realist philosophy that is quite diﬀerent from the
Platonic philosophy of dualistic idealism.

the realistic worldview (physics)
Aristotle + Hellenism+Islam

Plato + Augustinus















⇒

Abelard, Ockham
Scholaticism(A.D.1000∼1500)
Anselmus,Thomas

⇒























Newton → theory of relativity

Descartes·Kant
Wittgenstein
statistics→

quantum
languagethe idealistic worldview (no experiment)
However, modern common sense is also suspect. This is because statistics, the mostsuccessful branch of idealism, is generally considered to belong to applied mathematics
and is not generally recognized as a type of idealism.
Immediately when Newtonian mechanics was proposed, realism was generally understood.
However, I believe that idealism is still not generally understood. In short, scientiﬁc

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4.1 Aristotle (BC.384 - BC.322)
understanding of idealism is as diﬃcult as understanding quantum mechanics.

For ”realistic worldview vs, idealistic worldview”, see Sec 4.1.3.

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Chap. 4 The Big Three in Greek Philosophy (Aristotle)

4.1.3 Appendix: The history of Aristotle (Monistic realism) vs.Plato (Dualistic
idealism)”Realistic Worldview or Idealistic Worldview? ” is one of the greatest debates in thehistory of philosophy. From our point of view, however, the two are not in a duel, but
coexist, as shown in the ﬁgure below (Figure 0.1 in Preface).
Figure 0.1 ( in Preface) : The location of QL in the history of western philosophy(realistic) monism (i.e., without measurement axiom)
relativity
4 −→ theory−−−−−−−−−−→○(monism)(unsolved)3○
−−→ NewtonTOE:
theory of6○(realism)−−→
everything
quantum
5 −→ mechanics −−−−−−−−→○(quantum phys.)

Socrates
Aristotle
0○:Greek

7○

Schola-8○


linguistic turn−−−−−−−−−−−−−−−−−→

8
○



quantization
7○9−−−−−−−−−→○
−−−−−−−−−→ statistics

dualism
applied math.
system theoryclassical mechanical worldview


(=MT)



(mind-body dualism)(Copernican revolution)) ○QL:quantum
15
linguistic−−→
language
Descartes
Kant
10○
scientiﬁc
11 −−→ Spinoza, Locke,... −−→−−−−−−→○
quantum mechanicalturn
epistemological worldview
worldview

12
○



linguistic turn



?

(logic in philo.))(logic, set theory)Wittgensteinmechanical
Saussure −−−−−−−→○

Boole, Frege,
14
13
○
turn
Zadeh
Russell, Cantor −−→philosophy ○1−2−−−−→○
sticism
Parmenides, Zeno
Plato

math.

logical worldview(idealistic) dualism (i.e., with measurement axiom)
Figure 0.1 (in Preface): The history of the world-descriptions
Philosophy (≈ dualistic idealism) has progressed toward QL
0
1
2
10
12
14
15 )
(i.e., ○→
○→
○→
○→
○→
○→
○

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4.1 Aristotle (BC.384 - BC.322)
Assertion 4.2. [Debates: realistic worldview vs. idealistic worldview]

Table 4.1 (=Table 2.1) : realistic worldview (Aristotle) vs. idealistic worldview (Plato)
dispute ⧹ [R] vs. [L]a motion○:b○:Ancient
Greecec Problem of universals○:
d space-time○:
e quantum theory○:f○:philosophyof scienceg fuzzy sets○:

Realistic worldview

Idealistic worldview(monism, realism, no measurement)(dualism, idealism, measurement)

Hērakleitos
Aristotle
OckhamNewton ( Clarke)
Einstein
CarnapKalmanParmenides
Plato
AnselmusLeibniz
Bohr
Quine
Zadeh

♠Note 4.3. The following is a brief explanation of the above table, with some caveats.
Although the table above is divided by ”Realism vs. Idealism,” the distinction is notalways clear.
a this ○a was too long ago to know what Parmenides really meant.
○:e Bohr was a physicist and would have considered himself a realist. Thus, the Bohr○:
Einstein controversy is a controversy within physics. If so, it is a controversy within the
TOE of Fig. 1 of the Foreword. Therefore, we will have to wait for the emergence of agenius of Einstein’s caliber to resolve it.
f As for Carnap and Quine, I only know their conclusions. I did not understand their○:position or the content of their arguments.
g Since Kalman and Zadeh’s specialty is dynamical system theory (= statistics), they○:both belong to idealism in the main textual sense. With the advent of quantum language,
I believe that this controversy has been resolved.

a Heraclitus argued plausibly that ”all things are in ﬂux. Parmenides, on the other○
hand, argued that• everything is unchanging. There is no movement, no change. Time does notexist. There is only ”one,” not ”many.
and other statements similar to the Copenhagen Interpretation.

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Chap. 4 The Big Three in Greek Philosophy (Aristotle)b This was practically the beginning of the ”monistic realism (no need for humans)○
vs. dualistic idealism (need for humans),” and this is where science began.
c The Problem of universals was a 250-year debate between the Aristotelians and○the Platonists, with the Platonists gaining the upper hand at ﬁrst, and then the
Aristotelians gradually gaining the upper hand. Although it was not a quantitative
scientiﬁc debate, the problem of universals shows the high potential of Scholastic
philosophy (both Aristotelian and Platonist).

Since ”universals” = ”observable,” from the standpoint of quantum language (Platonic/dualism), we can conclude that ”universals” exist.
d The Leibniz-Clarke correspondence (cf. ) is a dispute between two geniuses, ”New○ton (and Clarke) vs. Leibniz. Without knowing quantum language, many mightjudge Newton the winner, but quantum language gives it to Leibniz.

e The ”Bohr-Einstein controversy” in quantum mechanics is the biggest scientiﬁc○
debate of the 20th century. In a nutshell, this is the debate such that ”Is quantum
mechanics Aristotelian or Platonian? ”
In order to settle this issue, I proposed a quantum language. A quantum language,
which is a generalized and more powerful modiﬁcation of quantum mechanics, isclearly not physics. Quantum language, which is a generalized and more pow107

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4.1 Aristotle (BC.384 - BC.322)
erful modiﬁcation of quantum mechanics, is clearly not physics. (Bohr-Einsteindebates(cf. ref. [74])). However, one may argue that this settlement is probably
a superﬁcial settlement. However, if there is a ”real settlement,” only a genius ofEinstein’s level would be able to achieve it.

f The ”Carnap-Quine controversy” in the philosophy of science is a controversy over○the distinction between analytical and synthetic propositions. In quantum language,”all scientiﬁc propositions are synthetic propositions”, so we give them to Quine.
Because the spirit of Copenhagen interpretation is ”if you don’t measure, you don’tknow anything.” (cf. Sec. 12.3).

g The ”fuzzy set” proposed by Zadeh was overwhelmingly supported by some systems○
engineers. Zadeh’s paper (cf. ref. [129], 1965) is the most cited paper of the 20th
century. In response, Kalman (the most respected systems engineer of his timeand proponent of the Kalman ﬁlter) objected on the grounds that ”fuzzy sets”
did not ﬁt within the framework of dynamical systems theory (classical dynamical
worldview).(cf. [130]). Now, half a century has passed since then, and the feverof fuzzy sets has cooled down completely, leaving only the impression that Kalman
was right after all. However, if we consider that[fuzzy set]=[universal]=[observable]
Zadeh’s proposal ﬁts within the quantum language (quantum mechanical worldview). Therefore, I think that the engineers who enthusiastically supported ”fuzzysets” at the time had a good point. From the standpoint of quantum language,
dynamical systems theory (classical mechanical worldview) is a monistic idealism,
and I feel that it is half-baked.

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4.2 Why does the motion happen?

4.2 Why does the motion happen?
4.2.1 From purpose to causality: Modern science started from the discovery of”causality”
When a certain thing happens, the cause always exists. This is called causality(=causal
relation). You should just recall the next proverb:
Smoke is not located on the place which does not have ﬁre.

However the situation is not so simple as you think. Consider, for example,• This morning I feel good.
(♯1 ) Is it because that I slept sound yesterday?
or(♯2 ) Is it because I go to favorite golf from now on?
You will ﬁnd the diﬃculty in using the word ”causality”. In daily conversation, the word”causality” is used in many contexts, mixing up ”a cause (past)”, ”a reason (implication)”,and ”the purpose and a motive (future)”.
As mentioned in Sec. 2.3, the pioneers in the study of movement and change areHērakleitos and Parmenides:• Hērakleitos(BC.540 -BC.480): ”Everything ﬂows.”



(A)
• Parmenides (born around BC. 515): ”There is no movement.”(Zeno’s teacher)
I think the reader will have the following question.
• Why are their names still there, even though it was 2,500 years ago?
As I mentioned before, ”motion and change” is the most important keyword in science(= ”worldview”), that is, I consider:(B)[The beginning of World description]= [The discovery of movement and change ] = HērakleitosParmenides

This is why their names are still there.
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Chap. 4 The Big Three in Greek Philosophy (Aristotle)
Thus, from the standpoint of this text (i.e., the scientiﬁc standingpoint), we say(B2 ) Heraclitus and Parmenides are the correct answers to the main themeof ancient Greek philosophy, ”What is the origin of all things?”(cf. Sec.3.1).

However, Aristotle (BC384–BC322) pursued an even more fundamental problem:(C) What is the essence of movement and change?
and concluded as follows.
(D):Purpose (Aristotle)
Aristotle asserted that all the movements had the ”purpose”.
• For example, a stone falls because it has the purpose to go downward, andsmoke rises because it has the purpose to go upward.
A heavy stone falls fast because it has a strong purpose of ”falling fast”.

4.2.1.1 The discovery of science: From purpose to causality
Under the inﬂuence of Aristotle, ”Purpose” had remained as a mainstream idea of ”Movement” for a long period of 1500 years or more.
We were freed from the spell of ”Purpose”, only after Galileo, Bacon, Descartes, andNewton et al. discovered the essence of movement and change lies in “Causality”.
Scientiﬁc revolution from ”Purpose” to ”Causality”
is the greatest paradigm shift in the history of science. It is not an exaggeration even if
we call the shift ”birth of modern science”.
the birth of worldview(E)

Movement(Hērakleitos, Parmenides, Zeno, Aristotle)
the birth of modern science”purpose” (and no experiment)−−−−−−−−−−−−−−−−−−−−−−−−→ Causality (and experiment)
Aristotle’s spirit :(About 1500 years)

111( Bacon, Galileo, Descartes, Newton)

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I cannot emphasize too much the importance of the discovery of the term: ”causality”.
That is,(♯) Science is the discipline about phenomena that can be represented by the term”causality”. (i.e., ”No smoke without ﬁre” )Thus, I consider that the discovery of ”causality” is equal to that of science. In therealistic worldview, Newtonian kinetic equation (i.e., the equation of the chain of causality)
was ﬁnal in a sense. However, in the idealistic worldview, the problem ”What is causality?”
is not solved yet. For the complete answer to the problem, we had wait for the appearanceof quantum language (Axiom 2 (causal relation) in Sec.1.2.1, also, see ref. [74]).
Summary 4.3. [Solutions to the causality problem] For example, we see:(F1 ) The causality is represented by Newtonian kinetic equation in Newtonian mechanics(F2 ) The causality is represented by Maxwell’s equations in electromagnetism(F3 ) The causality is represented by Schrödinger equation (or equivalently, Heisenberg’s
kinetic equation) in quantum mechanics(F4 ) The causality is represented by Axiom 2 (in Section 1.2) in quantum language(Continued to Sec. 10.2: What is causality?).

♠Note 4.4. S. Weinberg (1933 -2021), a physicist at the University of Texas, Austin, won
a Nobel Prize in 1979 for work that became a cornerstone of particle physics, said in his
book [123] ”To explain the world; The discovery of modern science” as follows:• [ in Chapter 3]:We can agree with the classical scholar R. J. Hankinson that ”wemust not lose sight of the fact that Aristotle was a man of his time and for thattime he was extraordinarily perspicacious, acute, and advanced.” Nevertheless, there
were principles running all through Aristotle’s thought that had to be unlearned in
the discovery of modern science. For one thing, Aristotle’s work was suﬀused withteleology: things are what they are because of the purpose they serve.
Recall the above (E). I think that only people after the scientiﬁc revolution (17th century)
understand the meaning of ”science”.
Aristotle is also the father of biology. I think the reason Aristotle failed to discovercausality is that biology was one of his major research topics. In biology, we tend to
think in terms of purpose theory, and causality is very diﬃcult to understand. It is112

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Chap. 4 The Big Three in Greek Philosophy (Aristotle)
in astronomy and physics that the causal relationship is very easy to see. In addition,
causality is relatively easy to experimentally verify. Thus, the scientiﬁc revolution was bornout of [(i) geodynamics, (ii) causality (Newtonian mechanics), and (iii) the importance ofexperimentation (i.e., British empiricism)]. That is,(i) Geocentrism(ii) causality (Newtonian mechanics)=⇒ scientiﬁc revolution(iii) the importance of experimentation
Even so, British empiricism (Descartes-Kant epistemology) has become ”brain science
without experimentation,” and the story is not simple.

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4.3 Practical logic; Why does logic hold in this world?

4.3 Practical logic; Why does logic hold in this world?
♠Note 4.5. In 1991, I discovered the mathematical expression of ”Heisenberg’s uncertainty
principle (γ line thought experiment)” (cf. ref. [38], or, Sec. 4.3 in ref. [74]). However,I have not been able to invent a practical application for this theory (no one has done ityet).

Believing that the reason for the lack of progress in my research was my ”immature
understanding of the Copenhagen Interpretation”, I started a serious research on the
Copenhagen Interpretation and chose the following problem (A 0 ) as my next goal. In
answer to (A0 ), I obtained the result (cf.[41, 42, 43, 44]), which was not well evaluated.
I knew I would need to do some pretty extensive theory to get an evaluation. In orderto gain recognition, I thought it was necessary to create a rather extensive theory. It is”quantum language” and this is summarized in refs. [74] and [82]. The revised version of[82] is this text. In this sense, problem (A0 ) is the starting point of this study.

The following is one of the biggest problems in philosophy.
(A0 ) Why does logic hold in this world?

We will answer this question in Chapter 12. To answer this we must be prepared formuch. This section is one of these preparations.

We consider that two kinds of logic exist such as(A1 ) mathematical logic (= symbolic logic)(A2 ) non-mathematical logic (=practical logic)
Here, mathematical logic ( due to Boole, Frege) is elementary and well known. A propo114

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Chap. 4 The Big Three in Greek Philosophy (Aristotle)
sition in mathematical logic is usually considered to be a mathematical proposition. Also,
A proposition in practical logic is a non-mathematical proposition, written by ordinary
language. For example, the (B2 ) below is non-mathematical logic.

4.3.1 Practical logic; Aristotle’s syllogism in ordinary language
If the ecology of various animals is observed, it will be clear that the base of language wasdue to intimidation, solidarity, reproduction. Language was one of the strongest arms forthe survival and breeding. Such a time have continued for millions of years. Of course, thebiggest events in the ”history of language” happened one after the other. For example,(B1 ) ”rhythm and song”, ”logical structure”, ”quantity concept”, ”grammar”, ”tense”,
etc.
However, it was done gradually by many people, tens of thousands of years ago, and it
is not possible to identify the names of the contributors. But the surprise that ordinary
language had a logical structure is carried on by the term ”Aristotelian syllogism.” Namely,
Aristotle’s syllogism in ordinary language
The following is famous Aristotole’s syllogism(B2 ) Since Socrates is human being,
and human being is mortal,
it follows that Socrates is mortal.

Although this is quite famous, the reader may have several questions concerning this asfollows,(C1 ) Syllogism is essential for mathematical proofs. Therefore, it is natural to assume
that Pythagoras already knew syllogism.
(C2 ) Also, it is natural to consider that syllogism was frequently used in the debate
between Socrates and sophists.
Thus I guess that(D) The knowledge of syllogism of the time was summarized in Aristotle’s book:”Organon”, which was compiled by his followers about B.C. 40. And, syllogism
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4.3 Practical logic; Why does logic hold in this world?
was endorsed by Aristotle and remained authoritative for almost 2,000 years.
In fact, Immanuel Kant said that there was nothing else to invent after the work ofAristotle.
♠Note 4.6. (i): Here,(♯1 ) Who is the discoverer of the syllogism?
Thalēs and Pythagoras would naturally have known about the syllogism, so it is certainly
not Aristotle. It is not possible to identify the names of the discoverer. Similarly,(♯2 ) Who is the discoverer of the motion function method?
(♯3 ) Who is the discoverer of probability theory (or, statistics)?
As mentioned in Note 2.16, it is not possible to identify the names of the discoverers.
Compared to the clear discoverers of monistic realism (classical mechanics and relativity),
the work of dualistic idealism often takes the form of gradual discovery by several. If so,(♭) it is tempting to formulate (♯1 )∼(♯3 ) in quantum language

all at once.
This was the motivation for my research.

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4.3.2 Mathematical logic; due to Boole, Frege
The following ”Elementary Logic” is suﬃcient to read this text.
Postulate 4.4.
[Symbolic logic; due to G. Boole, G. Frege]
For any proposition P , the truth function ϕ(P ) is determined such that
1 (if P is true)
ϕ(P ) =
0 (if P is wrong (i.e., not true))(E1 ) Assume that P1 , P2 are propositions. Then, P1 ∧ P2 , P1 ∨ P2 , ¬P1 , P1 → P2 arepropositions.
And it holds that ϕ(P1 ∧ P2 ) = min{ϕ(P1 ), ϕ(P2 )}, ϕ(P1 ∨ P2 ) =max{ϕ(P1 ), ϕ(P2 )}, ϕ(¬P ) = 1 − ϕ(P ).
where ∧, ∨, ¬, → respectively is called ”and”, ”or”, ”not”, ”implies”. Note that P1 → P2
is deﬁned by ¬P1 ∨ P2 ”.
Also, assume that Pθ is a proposition (θ ∈ Θ ≡ {1, 2, ..., n}), then it holds(i): P1 ∧ P2 ∧ ... ∧ Pn (denoted by ∧ Pθ , or ∀θ(∈ Θ)[Pθ ]) is a propositionθ∈Θ(ii): P1 ∨ P2 ∨ ... ∨ Pn (denoted by ∨ Pθ , or ∃θ(∈ Θ)[Pθ ]) is a proposition.
θ∈ΘHere, ϕ(P1 ∧ P2 ∧ ... ∧ Pn ) = mini=1,...,n ϕ(Pi ), ϕ(P1 ∨ P2 ∨ ... ∨ Pn ) = maxi=1,...,n ϕ(Pi ).
(E2 ) The above ﬁnite set Θ(≡ {1, 2, ..., n}) can be extended to an inﬁnite set Θ.
[Predicate logic (This text does not go deep into predicate logic.)](i): ∧ Pθ is denoted by ∀θ[[θ ∈ Θ] → [Pθ ]].
θ∈Θ(ii): ∃θ[Qθ ] is deﬁned by ¬[∀θ[¬[Qθ ]]].
Therefore,∨ Pθ = ¬[ ∧ ¬Pθ ] = ¬[∀θ[[θ ∈ Θ] → [¬Pθ ]] = ∃θ[[θ ∈ Θ] ∧ [Pθ ]]

θ∈Θ

θ∈Θ

♠Note 4.7. The most important problem in Postulate 4.4 is the ”proposition P ” in line 3 .
In the case of mathematical logic, there is no problem because ”proposition = mathematical proposition”. However, if it is non-mathematical logic (practical logic), there is nodeﬁnition of ”proposition P ,” which causes a problem. For example,(♯) Is ”I think, therefore I am” a proposition?
I naturally assumed that ”What is a proposition?” was written in Wittgenstein’s ”Tractatus Logico-Philosophicus (widely abbreviated and cited as TLP), and I started reading it,
but I remember being disappointed because I did not ﬁnd the deﬁnition of a proposition
anywhere in the TLP. But as he writes in Sec. 12.2 (H1 ), he honestly asks ”What is
a proposition?” in Philosophical Inquiry (§38 in [126]), he wrote that he was unable topropose the deﬁnition of ”Proposition”. . Thus, I understood TLP as a book that poses
the question, ”What is a proposition?” In any discipline, not only philosophy, it is often
said that ”it is more important to pose a problem than to solve it.” TLP is a book that117

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raises the question, ”Logic is not only about mathematical logic!” and ”Someone deﬁne a(non-mathematical) proposition!”

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4.3.3 Aristotle’s syllogism is not a syllogism
Exercise 4.5. The truth table below is well known. Wittgenstein was one of those who
contributed to the popularization of the truth table.
(i): The proof of syllogism: [(p → q) ∧ (q → r)] → (p → r)
p
1
1
1
1
0
0
0
0

q
1
1
0
0
1
1
0
0

r
1
0
1
0
1
0
1
0

p→q
1
1
0
0
1
1
1
1

q→r
1
0
1
1
1
0
1
1

p→r
1
0
1
0
1
1
1
1

Truth Table(p → q) ∧ (q → r)
1
0
0
0
1
0
1
1

(p → q) ∧ (q → r) → (p → r)
1
1
1
1
1
1
1
1

Therefore, syllogism: [(p → q) ∧ (q → r)] → (p → r) always holds.
(ii):Modus ponens: ((p → q) ∧ p) → q
Proof of (ii)p q p → q (p → q) ∧ p (p → q) ∧ q (p ∧ (p → q)) → q ((p → q) ∧ q) → p
modus ponens
1 1
1
1
1
1
1
1 0
0
0
0
1
1
0 1
1
0
1
1
0
0 0
1
0
0
1
1
For example, put p := ”it rains”, q := ”the ground is wet”. The, modus ponens says thefollowings.
[”it rains”∧[”it rains” → ”the ground is wet”]]→ ”the ground is wet”.

(iii):[Since Socrates is a human being]∧[every human being is mortal]→[Socrates is mortal].
Using set theory, we prove it as follows.
Assume that H is the set of all humans and M is the set of all mortals.
. Then,[Socrates is a human being]⇔ [Socrates ∈ H]
[every human being is mortal]⇔ ∀x[x ∈ H → x ∈ M ]119

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4.3 Practical logic; Why does logic hold in this world?
Thus,[Socrates ∈ H] ∧ ∀x[x ∈ H → x ∈ M ]→[[Socrates ∈ H] ∧ [Socrates ∈ H → Socrates ∈ M ]](by Modus ponens)→[Socrates ∈ M ]That is,h(F)[Socrates ∈ H]

(Socrates is a human being)∧∀x[x ∈ H → x ∈ M ](every human being is mortal)i→ [Socrates ∈ M ](Socrates is mortal)

Thus,• The above (F) says: Aristotle’s syllogism:[Socrates is a human being]∧[every human being is mortal]→[Socrates is mortal]
is not a syllogism but a variant of modus ponens.

♠Note 4.8. It seems that the term ”three-stage argument” is sometimes used in multiple
senses, but here we will assume that it means (i) above. Aristotle’s syllogism (iii)” is often
cited as a typical example of syllogism, but this usage should be used with caution because
it may cause confusion. In the next section, I will use ”Aristotle’s syllogism (iii)” as an
example of ”the danger of logic in everyday language.”120

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4.3.4 Set-theoretical worldview (= logical worldview)
4.3.4.1 Set-theoretical worldview
Aristotle’s syllogism (Socrates’ death) in Sec. 4.3.3 such that(G1 ) Since Socrates is a human being, and every human being is mortal, thenSocrates is mortal
is very instructive. Translating this into the language of set theory, we get the followingih(G2 ) (=(F))[Socrates ∈ H]
∧ ∀x[x ∈ H → x ∈ M ]→ [Socrates ∈ M ](Socrates is a human being)(every human being is mortal)(Socrates is mortal)

Thus, what appeared to be a syllogism in (G1 ) was actually a variant of modus ponensin (G1 ). Thus, the ”logic” of everyday language is insuﬃcient,(G3 ) The meaning of ”logic” becomes clear only after (G1 ), in which the everyday language (G1 ) is described under the set-theoretic worldview.
Thus, the set-theoretic worldview (= the style of describing phenomena in the languageof set theory (= the language using ”symbol ∈” and symbolic logic)) is a very, very goodworldview.
Now, let us consider the following.
Exercise 4.6. Is Descartes’ cogito proposition ”I think therefore I am” a proposition?
[Proof?] Deﬁne T as the set of all thinking organisms. Let E be the set of all that exist.
Then, it is clear that T ⊆ E. Put I=”I”.
Then, we see that I ∈ T and T ⊆ E, therefore, I ∈ E
However, I do not think that the above argument is related to the cogito proposition ”I
think therefore I am”. Thus, the set-theoretic worldview may impotent on the cogito
proposition.
4.3.4.2 From a Questioning of the Set-theoretic Worldview to a Quantum Mechanical Worldview
Even so, there is room to consider whether a set-theoretic worldview is the best way. For
example, the set-theoretic worldview cannot answer the following questions(H1 ) In the ﬁrst place, is there such a scientiﬁc concept as ”the set of all humans”? (Thisis a problem called ”Hempel’s raven Problem”(cf. Sec. 12.7). It was Hempel who
ﬁrst questioned the set-theoretic worldview.
(H2 ) Is the ﬁrst proposition of philosophy ”I think, therefore I am” a proposition?
(H3 ) Is there a relationship between the set-theoretic worldview and Descartes-Kant’s
epistemology?
(H4 ) Does the syllogism hold in quantum systems?
(H5 ) Isn’t symbolic logic useful only for mathematics (=set theory)?
Thus, various problems emerge. The most important of these problems is the following,
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4.3 Practical logic; Why does logic hold in this world?
(I) Why does logic work in this world?
Wittgenstein tried to solve this problem in his philosophical book TPL (i.e., ref. [125]:”Tractatus Logico-Philosophicus”, but he was theoretically unsuccessful. However,(J) In philosophy, it is sometimes more important to pose a problem (I) than to solveit
In this sense, TPL is a good book.
Our interest in this text is the following quantum mechanical worldview.
We then claim the following.
(K1 ) quantum language is the language of science
And in [74], we already showed that
(K2 ) Statistics is one aspect of quantum language.
Therefore, to answer the question ”Why does logic work in this world? we can show
the following.
(K3 ) By deﬁning ”propositions” in the quantum language, we show that the logic is valid.
This will be discussed in Chap.12.
After all, Figure 0.1 in Preface asserts the following.
(K4 ) Statistics, quantum mechanics, scholastic philosophy, Cartesian-Kantianphilosophy, and analytic philosophy are each an aspect of quantum language (or an immature state of quantum mechanics).

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Chap. 4 The Big Three in Greek Philosophy (Aristotle)

4.3.5 Appendix: Syllogism does not always hold in quantum systems?
We have the following theorems:
Theorem 4.7. (L1 ) Syllogism does not necessarily hold in quantum systems(L2 ) Syllogism always holds in both classical and quantum systems////
Since (L1 ) and (L2 ) have diﬀerent premises, (L1 ) and (L2 ) are not contradictory (cf.
Remark 12.9 later). The proof of (L2 ) is given in Corollary 12.8.
Proof 4.8. The proof of Theorem 4.7 (G1 ) due to the following:• [79] Ishikawa, S. (2020) Wittgenstein’s picture theory in the quantum mechanical worldview Journal of quantum information science, Vol. 10, No.4, 104-125,
DOI:10.4236/jqis.2020.104007(https://www.scirp.org/journal/paperabs.aspx?paperid=106233)(M) : [Syllogism does not hold in quantum systems]:
i.e. the following does not always hold in quantum language:• if P1 −→ P2 and P2 −→ P3 , then it holds P1 −→ P3 .
Let us prove it as follows. A quantum two particles system S is formulated in a tensor
Hilbert space H = H1 ⊗ H1 = L2 (Rq1 ) ⊗ L2 (Rq2 ) = L2 (R2(q1 ,q2 ) ). The state u0 ( ∈ H =
H1 ⊗ H1 = L2 (R2(q1 ,q2 ) )) or precisely, ρ0 = |u0 ihu0 | of the system S is assumed to ber
u0 (q1 , q2 ) =1 − 12 (q1 −q2 +a)2 − 12 (q1 +q2 )2
8σ
e 8ϵ
2πϵσ

(4.1)

where a 6= 0, a positive number ϵ is suﬃciently small, and a positive number σ is suﬃciently
large. Thus, we see thatr
1 − 12 (p1 −p2 )2 − 12 (p1 +p2 )28ϵ|c
u0 (p1 , q2 )| = |
e 8σ
|
(4.2)
2πϵσ
where uc0 is the Fourier transform of u0

q2 6

p2 6
q1 − q2 + a = 0

a

a

O

q1

O

p1
p 1 + p2 = 0

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4.3 Practical logic; Why does logic hold in this world?
2
For each k = 1, 2, deﬁne the self-adjoint operators Qk : L2 (R(q) → L2 (R2(q1 ,q2 ) ) and
1 ,q2 )
Pk : L2 (R2(q1 ,q2 ) ) → L2 (R2(q1 ,q2 ) ) by

Q 1 = q1 ,Q2 = q2 ,

ℏ∂
i∂q1
ℏ∂
P2 =
i∂q2

P1 =

(4.3)

(♯1 ) Let O1 = (R3 , BR3 , F1 ) be the observable representation of the self-adjoint
operator (Q1 ⊗ P2 ) × (I ⊗ P2 ). And consider the measurement MB(H) (O1 =(R3 , BR3 , F1 ), S[|u0 ⟩⟨u0 |] ). Assume that the measured value (q10 , p02 , p02 )(∈ R3 ). Thatis,(q10 , p02 )

p02

=⇒

the momentum of A2(the position of A1 , the momentum of A2 )(♯2 ) Let O2 = (R2 , BR2 , F2 ) be the observable representation of (I ⊗ P2 ) × (P1 ⊗ I). And
consider the measurement MB(H) (O2 = (R2 , BR2 , F2 ), S[|u0 ⟩⟨u0 |] ). Assume that themeasured value (p02 , −p02 )(∈ R3 ). That is,p02=⇒

the momentum of A2−p02
the momentum of A1(♯3 ) Therefore, if syllogism holds, we may conclude that(q10 , p02 )(the position of A1 , the momentum of A2 )



−p02

=⇒

the momentum of A1

that is, the momentum of A1 is equal to −p02

But, the above argument (particularly, “syllogism”) is not true. That is because(♯4 ) (Q1 ⊗ P2 ) × (I ⊗ P2 ) and (I ⊗ P2 ) × (P1 ⊗ I) ( therefore, O1 and O2 ) do notcommute, and thus, the simultaneous observable does not exist.
Thus, we can not test the (♯3 ) experimentally.
Remark 4.9. Some arguments diﬀer from the above. In Corollary 12.8 (= Remark 12.9)
later, we present the diﬀerent result, i.e.
(♯) syllogism always holds in classical and quantum systems.
which is more formal. the above (♯) does not contradict the result of this section, since
the deﬁnitions of ”proposition” are diﬀerent.

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Chapter 5

Around Alexandria; Hellenistic period
Wisdom of pyramid building for thousands of years was accumulated by Egypt (Alexandria). Bright people studied in Egypt from each place of the Mediterranean Sea coast tolearn it. For example,
Euclid· · · geometry

Aristarchus· · · Heliocentrism

Archimedes· · · buoyancy, lever

Eratosthenes· · · the measurement of the earth

Ptolemaios· · · Geocentrism

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5.1 Around Alexandria; Hellenistic period

5.1 Around Alexandria; Hellenistic period
Inﬂuenced by the tradition of pyramid construction engineering, the studies of the Alexandrian school are solid and scientiﬁc. Put diﬀerently, it could be said that there was no
philosophical appeal that transcended mathematical logic. The Alexandrians must have
known of Plato’s work. However, they had little inﬂuence from Plato. I guess that• they didn’t think that Plato’s philosophy would survive more than 2,000 years later
Hellenistic period is located as follows:poem

science

The arche is−−−−→

Socrates−−−−−→

student

Aristotle −−−−→ Hellenistic periodxstudent
Plato

−−−−→

···

−−−−→ Scholasticism
Middle Ages

vs. Sophists

Many Greek philosophers were not willing to put their theories to work. But, in Hellenistic
period, Practical research was respected. In Section 5.3, I will explain Heliocentrism ofAristarchus a little in detail as the preparation of Chap. 7, in which we say that• ”Geocentrism vs. Heliocentrism” is a metaphysical problem that cannot be put onblack-and-white in the experiment.

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Chap. 5 Around Alexandria; Hellenistic period

5.2 Euclid(BC.330 - BC.275)
5.2.1 Euclid geometry - Parallel postulate

Three great pyramids in the Egyptian Giza desert (deceased person is Khufu, Khafre,
Menkaure) erecting time of is the 2500 BC. Since then more than 2,000 years later, Euclid(BC.330 - BC.275) was born. Euclid is referred to as the ”father of geometry” who was
active in Alexandria (the mouth of the Nile). His book ”Elements” is one of the mostinﬂuential works in human cultural history. It has been estimated to be second only to the
Bible in the number of editions published since the ﬁrst printing in 1482 AD. In ”elements”(Volume 1), 23 deﬁnitions, 5 postulates, and 9 axioms are listed at the beginning, and 48
theorems (such as Pythagoras’ theorem) are proved on the basis of them. In short, what
Euclid did was(A1 ) After determining the obvious (?) axioms (or axioms), complex theorems (e.g.
Pythagorean theorem) were proved or deduced from them.
This can be seen as a mathematical version of [worldviewism (cf. Sec.1.1)].
Here, ﬁve postukates are illustrated as follows.

Euclide geomety: Postulates 1∼ 5

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5.2 Euclid(BC.330 - BC.275)

This is explained below. We focus on the postulate 5 (i.e., the parallel postulate).
• If a line segment intersects two lines on the same side whose sum of two interior
angles is less than two right angles, then extending the two lines inﬁnitely, the sumof the angles (= α + β) intersects the line on the side whose sum of the angles isless than two right angles.

A + B < 180◦=⇒ the intersection point • exists

A
B•

Therefore,(A2 ) Euclid was a mathematician who proposed the axiomatization of geometry, considered the parallel postulate, and intuited that the notion of ”obviousness” isnon-trivial.
Despite Euclid’s meticulous care, the following false beliefs were formed by ”Elements”.
(B) It is best to start from the obvious.
Many philosophers (Descartes, Spinoza, etc.) have fallen prey to this false belief. It isalready well known that Descartes’ cogito proposition ”I think, therefore I am” is far fromself-evident. In other words, nothing is self-evident (i.e., there is no unquestionable truth).
However, many philosophers have emulated the ”Elements” as follows.

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Chap. 5 Around Alexandria; Hellenistic period

introduction part (unquestionable truth)
z
}|
{

Plato



Idea theory








early Scholasticism
Plato+Christianity








late Scholasticism

Aristotle+Christianity 






I think, therefore I am 







Kant




thing-in-itself








Analytic philosophy
mathematical logic

commonTherefore−−−−−−−→ ethics, moral

Descartes

It should be noted here that the worldviews on the left side above are all propositions
and theories that are impossible to deny. In short, they are all starting from ”pseudoscientiﬁc propositions” in the sense of Popper’s disprovability .
♠Note 5.1. (i): In ”Elements”, geometry is not only written but also algebra. For example,
it is shown that prime numbers are inﬁnite. The proof is as follows.
(♯) Assume that the set of prime numbers is ﬁnite, that is, {2, 3, 5, 7, ..., n}. Put
N = (2 × 3 × 5 × 7 × ... × n) + 1
Then, N is a prime number or it can be divided by the larger prime number than n.
In each case, it contradicts the assumption that n is the largest prime number.
(ii): Euclid’s axiomatic system is now known not to provide a complete logical foundation
for the modern world, and now seems to have only historical value. Birkhoﬀ’s axiomatic
system is a well-known modern version, but there are many others, of which the author is
not familiar.

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5.2 Euclid(BC.330 - BC.275)

5.2.2 non-Euclidean revolution

Discovery of non-Euclid geometry (due to Gauss(1777 - 1855), etc.) defeated the wrong
belief (B) and asserted(C) Start from ”productive” than ”self-evident”!
that is, ”all is well that ends well”. In this text, the (C) is called the non-Euclidean
revolution, that is,(D) non-Euclidean revolution (B): self-evident−−−−−−−−−−−−−−−−→(C): productive
non-Euclidean revolution

It can’t be said that the non-Euclidean revolution is still generally also recognized sufﬁciently in today. There is no successful theory which starts from ”self-evident things”.
For example, Newtonian mechanics, the theory of relativity, quantum mechanics, etc. do
not start from ”self-evident things”. Paradoxically saying, we see that(E) The question: ”What is ’self-evidence’ ?” is not self-evident.
Axiom of choice of mathematics is not self-evident, where axiom of choice is as follows.
• Given any set X of pairwise disjoint non-empty sets, there exists at least one set C
that contains exactly one element in common with each of the sets in X.
(For example, consider a set X = {{a, b}, {c, d, e}, {g}, {h, i, j, k}}. Then, we canconstruct a set C = {a, c, g, j})

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This is not self-evident (i.e., trivial). For instance, Banach-Tarski theorem says that(F) If we adopt axiom of choice, we must admit the following• A ball B is resolved into parts of several ﬁnite numbers, and we assume thatit’s put together again. Then, we can get the same two balls which are also thesame as the ball B.

resolve into parts of several ﬁnite numbers−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→and put together again

Then, we want to doubt axiom of choice, but a description of the mathematics
largely decreases when I do not accept axiom of choice. Hence, we usually accept
axiom of choice.

♠Note 5.2. There was also tradition of pyramid construction, and Egypt was an advanced
country of mathematics. Pythagoras and Archimedes also learned geometry in Egypt.
Then Alexandria was an academic city as there was Alexandria library having 700,000
collection of books. After Euclid, we know that
• Eratosthenes (BC.275 - BC.194) : He was determined to 46250km the whole circumference of the earth. Cf. Sec. 5.5.
• Cleopatra(BC.69 - BC.30): The most beautiful woman in human history.
• Ptolemaios (AD.83 - 168): Geocentrism

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5.3 Aristarchus (BC.310 - BC.230)

5.3 Aristarchus (BC.310 - BC.230)
5.3.1 the diameter of the moon: the diameter of the sun

Aristarchus (BC.310 - BC.230) was an ancient Greek astronomer and mathematician who
presented Heliocentrism. He calculated as follows.
Proposition 5.1.
(A1 ) the diameter of the moon: the diameter of the earth ≈1:3 (Recent result says that
1 : 3.669), where a : b = c : d means a/b = c/d.
(A2 ) the diameter of the moon: the diameter of the sun≈1:19(A3 ) Thus, the diameter of the earth : the diameter of the sun ≈1:6.333 (Recent result
says that 1 : 109)
(A4 ) Since each volume is proportional to [diameter]3 , the sun is much larger than theearth.

The answer to (A1 ) Look at the lower left ﬁgure (lunar eclipse). Since the sun is very
far, it suﬃces to consider thatthe diameter of the earth ≈ the diameter of the earth’s shadow
Hence, measuring by eye, we see (A1 ).
The answer to (A2 ): Look at the lower left ﬁgure (the ﬁrst quarter moon). Note thatcos 87◦ ≈ 1/19. And using the fact that the sun and the moon are seen as the same size,
we can calculate:the diameter of the moon
the distance between the moon and the earth
1= cos 87◦ ≈=the diameter of the sun
the distance between the sun and the earth
19

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5.3.2 Ancient Heliocentrism
Aristarchus considered as follows:(B1 ) The sun is overwhelmingly larger than the Earth. If so, it is wrong that the big
sun goes around the small earth. It is sure that the small earth goes around the
big sun.
That is,(B2 ) Aristarchus proposed Heliocentrism

His argument is almost complete since the diﬀerence between ”the volume” and ”themass” is trivial.
Therefore, I agree that

Aristarchus was the ﬁrst proponent of the heliocentric theory
Next problem is as follows.
(C) measuring the diameter of the earth
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5.3 Aristarchus (BC.310 - BC.230)
This was solved by Eratosthenes (cf. Sec.5.5).

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5.4 Archimedes (BC.287 - BC.212)
Archimedes was born in Syracuse on the island of Sicily in the Mediterranean. Archimedes
studied in Alexandria that was a center of the study and engaged in the study of ”Elements” with pupils of Euclid afterwards. He returned to Syracuse later and spent life in
Syracuse.

5.4.1 Buoyancy (Archimedes’ principle)
Archimedes’ principle on buoyancy is as follows.
(A) Any object, wholly or partially immersed in a ﬂuid, is buoyed up by a force equal
to the weight of the ﬂuid displaced by the object. If some want to avoid the term
”force”, then
the weight of the matter in water= the weight of the matter − [buoyancy]/g(5.1)
where (g: gravitational constant)[buoyancy] = Sum of the water pressure from the bottom of the object
− Sum of the water pressure from the top of the object= (f1 + f2 )S = (|f2 | − |f1 |)S = hSg = V g

-

f1
? ? ?

⇑

h
buoyancyS(the density of water is ρ(= 1).)

S: bottom area, h: height
V = hS; volume
fk : water pressure, k = 1, 2







6 6 6
f2
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5.4 Archimedes (BC.287 - BC.212)♠Note 5.3. A famous anecdote of the golden crown is the delicate anecdote that there isnot connected with Archimedes’ principle. In like there is a relationship, I try to write
this in what follows.

• The King of Syracuse asked Archimedes ”Can you check whether silver is not mixed by
the crown without breaking the crown”. Archimedes notices next answer (♯) duringbathing: while shouting with joy too much ”Heureka!”(=”I have found it!”), wasrunning around the streets naked without even wearing clothes.
He found:(♯) Prepare a gold nugget of the same weight as the crown. Next, determine the volumeof the gold nugget and the volume of the crown as shown in the ﬁgure below. This
allows us to determine the speciﬁc gravity of the gold nugget and the crown. Thisway, we do not need to use Archimedes’ principle of buoyancy.

This way, Archimedes’ principle of buoyancy would not have to be used. However,136

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Chap. 5 Around Alexandria; Hellenistic period
there is no way that Archimedes would jump out of the bath and run around thecity because of something as simple as this. It would be more natural to assume thatArchimedes knew the above before he took a bath and discovered the ”Archimedes’
principle of buoyancy” when he took a bath.

♠Note 5.4. For each great discovery, an anecdote (or, a catchphrase, stage eﬀect) is left as
follows.
(♯1 ) Archimedes· · · · · · golden crown, heureka! (cf. Sec.5.4)(♯2 ) Galileo· · · · · · Leaning Tower of Pisa, ”And Yet It Moves” (cf. Sec.7.3.4)(♯3 ) Newton· · · · · · ⃝:Newton’s
apple, ”Geocentrism vs. Heliocentrism”(cf. Note 7.8)
1(♯4 ) Descartes· · · · · · ⃝:ﬂyon the ceiling (cf. Note 8.2), ⃝:1
2 I think, therefore I am, (cf.
Sec.8.2)(♯5 ) Kant· · · · · · clock (cf. Note 10.2), dogmatic slumber (cf. Note 10.5)(♯6 ) Wittgenstein· · · · · · primary school teacher, Gardener, Guardian: Russell (cf.
Sec.12.1.1)(♯7 ) Einstein· · · · · · Elevator(♯8 ) quantum mechanics· · · · · · Heisenberg’s uncertainty principle (cf. ref. [38]).

I have the following opinion about (♯8 ):(♭) There is a big gap between Heisenberg’s uncertainty relation (refs. [28, 38]) and Kennard=Robertson uncertainty relation (refs [90, 111]). Therefore, just as the ”Newton’s apple” was used as a symbol for Newton mechanics, ”Heisenberg’s uncertainty
relation” was used as a symbol for quantum mechanics (cf. Note 4.1 of ref. [74]).

5.4.2 The tomb of Archimedes
Consider the ball B of radius r. Archimedes showed the followings:
The volume of the ball B =

4πr3
,
3

The surface area of the ball B = 4πr2

If you are a mathematical genius, you may ﬁnd the proof by seeing the lower illustration(’the cylinder which is circumscribed to a ball’ called ”the tomb of Archimedes”). If you
are not genius, you can calculate it by using the diﬀerential and integral calculus.

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5.4 Archimedes (BC.287 - BC.212)

Thus,”Volume of Sphere” = ”Volume of Cylinder” − 2 × ”Volume of Cone”= 2πr3 − 2 ×

πr3
4πr3
=
3
3

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5.4.3 Principle of leverage
Archimedes found” principle of a lever” and did more various invention with a lever.

Principle of leverage: ml = M L

M

m
L

l

Archimedes said ”Give me a lever long enough and a fulcrum on which to place it, and I
shall move the world”. In spite that he referred Aristarchus’ Heliocentrism in his book:”The Sand Reckoner”, he supported Aristotle’s Geocentric model. However, Archimedes,
found ” principle of a lever”, have to restate Aristarchus’ Heliocentrism((B 1 ) in Sec.5.3)
as(B) Because the sun is so much bigger than the earth. The center of gravity of the
combined Earth and Sun is extremely close to the Sun. Hence, the Sun and theEarth revolve around the gravity of both the earth and the Sun.
If Archimedes said so, science history would be history which is completely diﬀerent fromnow.
♠Note 5.5. Archimedes’ arguments were so clear that even elementary school students could
understand them, and he did not say ambiguous and unintelligible things like philosophy(Plato, etc.). This clarity is a factor in Archimedes’ popularity. As I have mentioned
before, his words pierce our hearts,• Heureka!”(=”I have found it!”)• Give me a lever long enough and a fulcrum on which to place it, and I shall move theworld
and so on. And the ”last words” was• Do not disturb my circles!

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5.4 Archimedes (BC.287 - BC.212)
The city of Syracuse, where Archimedes lived, was a battleground between Carthage(Hannibal the General) and Rome. The Roman army knew that Archimedes was a famousscientist, so they instructed him not to do any harm. However, when Archimedes was
thinking about writing a ﬁgure on the sand, he was almost taken away by the Romansoldiers, who refused to do so, saying ”Do not disturb my circles! And thus he was killed.
It can be said that he was the ”greatest star of the ancient scientists” until the end of his
life.

♠Note 5.6. Note that
• Archimedes did not speak ambiguous things like Plato’s philosophy.
Therefore the work of Archimedes is quantitative, clear and easy to understand. Since political power could interpret the vague philosophy conveniently, philosophy could inﬂuence
to maintain harmony with religion or politics. In fact, philosophy survived in the middle
ages as a maid of theology. On the other hand, Archimedes’ work was almost forgotten.

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Chap. 5 Around Alexandria; Hellenistic period

5.5 Eratosthenes (BC.275 - BC.194)

5.5.1 The biggest ancient observer
Pythagoras believed that the earth must be a beautiful shape and believed that the earth
was a sphere. Aristotle deduced that the lunar eclipse was the shadow of the Earth andbelieved that the Earth was a sphere. When you look at the ocean in the distance, it
looks like an arc, so there were probably people who believed that the earth was a spheresince long ago. However, if we were to mention the two certain discoverers, it would
be the scientist Eratosthenes (BC.275 - BC.194) and the explorer Magellan (AD.1480 AD.1521).

• Syene (=Aswan) is on the tropic of cancer, thus, the sun is seen in right above atnoon on the summer solstice.
• Syene (=Aswan) is located just south of Alexandria. The distance =AS=925km.
Eratosthenes measured the whole circumference of the earth as follows.
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5.5 Eratosthenes (BC.275 - BC.194)
NP

A

Sun

θ

θ
S

sun light

SP

NP:North pole, SP:South pole, A:Alexandria, S:Syene(=Aswan)Hence,
the whole circumference of the earth = 2 × 3.14 × [the radius of the Earth] = 360AS/θ=360 × 925/7.2 = 46250km
As the recent result:40009km, it may be surprising.
♠Note 5.7. Since Aristarchus discovered

 
 

the diameter of the moon : the diameter of the earth : the diameter of the sun= 1 : 3 : 19then,
 by Eratosthenes’s result, we know that the diameter of the moon , the diameter of the earth , the diameter of the sun .

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5.6 Claudius Ptolemaios (AD.83 - AD.168)
5.6.1 A culmination of Ancient Astronomy
Ptolemaic Dynasty is ruined by the death of Cleopatra, Rome became the heyday of theFive Good Emperors era. At this time, Ptolemaios (AD.83 - 168) played an active partin Alexandria. In his book ”Almagest”, he adopted Aristotle’s Geocentrism (i.e., the sungoes around the earth). Ptolemaios explained the retrogression seen at a planet in Marssuch as Mars revolves around the earth while drawing a small circle as ”epicycle”.

Ptolemaios compiled the latest theory in those days and concluded the Geocentrism (=Ptolemaic system) under the enormous measured data.
(A) Ptolemaios followed Aristotle, Archimedes, etc.
And it is sure(B) Ptolemaios is a top-notch researchers.
He was the scientist who gave the most importance to observation among the ancientscientists.
Although, approximately 1500 years later (at Galileo’s trial (1633)), his Ptolemaic system was replaced by the Copernican system, he was surely one of scientists who thought
observation and experiment as important most.
♠Note 5.8. S. Weinberg (1933 -2021), a physicist at the University of Texas, Austin, won
a Nobel Prize in 1979 for work that became a cornerstone of particle physics, said of his
book [123] ”To explain the world; The discovery of modern science” as follows:• [in Chapter 8]: In one respect the work on this theory described in the Almagest is
strikingly modern in its methods. Mathematical models are proposed for planetary
motions containing various free numerical parameters, which are then found by constraining the predictions of the models to agree with observation.
Decades ago, when I saw the planetarium when I was a kid, a commentator explainedthat ”Ptolemaios asserted a foolish Geocentrism, and this was corrected by Copernicus.”

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5.6 Claudius Ptolemaios (AD.83 - AD.168)
However, when I went to the planetarium recently, a commentator praised Ptolemaios,
saying, ”His epicycle model is groundbreaking at the time.” It was great to hear a similar
opinion to Weinberg’s at a planetarium in the Far East island nation.

♠Note 5.9. Recall the following ﬁgure:poem
The arche is ⃝⃝

−
−−−−
→

Socrates−−−−−→

sciencestudent

Aristotle −−−−−→ Hellenistic periodxstudent
Plato

−
−−−−
→

···

−
−−−−
→ Scholasticism
Middle Ages

vs. Sophists

That is, Aristotle bridged the gap between this pseudoscience (≈ the arche is ⃝⃝) andthe base for the foundation that would become science. Archimedes is the discovererof ”principle of buoyancy”, which belongs the realistic worldview. Ptolemaic system isbased on the realistic motion function method.

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Chapter 6

The Middle Ages
It is often said that(⋆) the Middle Ages is characterized as ”the time of the thought stop for about 1500
years” Thus, it is called ”Philosophy is a maidservant of theology”
However, I think that this is due to our modern low philosophical background.
Thus the (⋆) is completely wrong. Of all the philosophers to date, I believe it is Anselmus,
Thomas Aquinas (and Descartes) who have understood Platonic philosophy the best.

In this chapter, we discuss:(♯1 ) Augustinus(354 - 430): Christianity became the state religion of the Roman Empire.
His time theory (i.e., subjective time theory) should be reconsidered in the Copenhagen interpretation.
(♯2 ) Anselmus(1033 - 1110): the father of Scholasticism, Arguments for the existence ofGod, ”Realismus (i.e., dualism)” in Problem of universals(♯3 ) Abelard, (1079 - 1142): Argued for monism due to Aristotle’s inﬂuence(♯4 ) Thomas Aquinas(1225 - 1274): Completion of the Scholasticism (Summa Theologica):
He deepen Anselmus’s theory in dualism despite Aristotle’s inﬂuence(♯5 ) Ockham(1285 - 1347): Ockham’s razor, ”Nominalismus (i.e., monism)” in Problem ofuniversals
And therefore, I assert that they are located as shown in the next ﬁgure.

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the realistic worldview (monism, physics)

Aristotle + Hellenism+Islam

Plato + Augustinus















⇒

Abelard, Ockham
Scholaticism(A.D.1000∼1500)
Anselmus,Thomas

⇒



















Newton → theory of relativity

Descartes·Kant
Wittgenstein→quantum
languagethe idealistic worldview (dualism, no experiment)

Abelard and Ockham were followers of Aristotle’s philosophy. Notable above is [Anselmus,
Thomas]. They adopted Aristotle’s method to deepen Plato’s philosophy. In the historyof Western philosophy (from Plato to analytic philosophy), these two are the only two
who have attempted to take the theory of ideas head-on and rationally. Spinoza, Leibniz,
and Wittgenstein also tried and failed. I think they (Anselmus, Thomas) are too geniuses
to be properly appreciated.

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Chap. 6 The Middle Ages

6.1 Augustinus(AD. 354 - AD.430)
6.1.1 The benefactor who saved ”Idea theory”

One of the largest events in the Western history isAD.380: Christianity became the state religion of the Roman Empire
A wonder of Western philosophy(≈Plato philosophy) is:(A) Western philosophy met with a dying crisis many times. Each time, Western philosophy was rescued by a hand of someone’s help (such as a hand of help of a
god).
Augustinus (354-430), the greatest Catholic priest, adopted Plato’s Idea theory to reinforce Christianity such as
(B)

Plato

:

real world


y

Idea


y

Idea world


y

:

city of this world

churchCity of GodyAugustinus

Here, it should be noted that ”church” is a bridge between ”city of this world” and ”Cityof God”. Augustinus’ theory is too visible since this is a theory for Christians, written
in a way that everyone can understand. However, Augustinus’ dualism is rather religious(i.e., good and evil dualism), not scientiﬁc. All proceeded as Augustinus’ plan.
♠Note 6.1. If we emulate Remark 3.6 and forcefully apply the three key words of dualism
to Augustine, we get the following

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6.1 Augustinus(AD. 354 - AD.430)[B](Mediating of A and C)

dualism \ three key-words[A](= mind)⃝:
3 Plato: sunRemark 3.6
Augustinus; Christianity

actual world
earthly city

Idea ( Sun)

quantum language[measured value][observable](body)

church[C](= matter)
Idea world
the city of God
system[state]This idea of ”church” alone cannot be said to have advanced Idea theory, but in the sensethat it saved Idea theory from dying out, it should be noted as follows.
(♯) Plato −−−−−→ Augustinus −−−−−−→ · · · −−−−−−→ Quantum language
regress

progress

progressAlso, as stated in the next section, Augustinus is closely related to the Copenhagen interpretation. Thus I assert (♯).

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Chap. 6 The Middle Ages

6.1.2* ”Confessions” by Augustinus: Only the present exists

This section is written in the following reference:(C) [76]:Ishikawa, S: Leibniz-Clarke correspondence, Brain in a vat, Five-minute hypothesis, McTaggart’s paradox, etc. are clariﬁed in quantum language Open Journal of philosophy, Vol. 8, No.5, 466-480, 2018, DOI: 10.4236/ojpp.2018.85032(https://www.scirp.org/Journal/PaperInformation.aspx?PaperID=87862)[Revised version] (https://philpapers.org/rec/ISHLCB)(http://www.math.keio.ac.jp/academic/research_pdf/report/2018/18001.
pdf)• [85] S. Ishikawa, K. Kikuchi, (2021) Quantum Fuzzy Logic and Time, Journal ofApplied mathematics and physics, Vo.9 No.11 2021, 2609-2622(https://www.scirp.org/journal/paperinformation.aspx?paperid=112972)
We want to know:
1 How should we live?
○
2 How is the world made?
○Augustine (354-430), the greatest Catholic priest, used Plato’s philosophy as the ”God’s
intellect = Idea” and armed Christianity. And everything proceeded according to Augustine’s plan and intention. What we ordinary people want to know most is, how should
we live? Christian Fathers, as God’s spokespersons, preached this to the people as theteachings of Christ. Therefore, Christian fathers, like God, had to be able to answer any
questions immediately. Among them, the question that puzzled them were, ”how is theworld made?”, ”if this world was made by God, what was it like before God made it?”etc. Bible says:(D1 ) This world was made by God.
If so, people may have a question:(D2 ) How about before God made it?
However, if (D1 ) is believed, then the priest would respond as follows.
(D3 ) Time was also made at the same time as the world.
Therefore,
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6.1 Augustinus(AD. 354 - AD.430)(D4 ) The sentence ”before God made it” is nonsense.
If we are told by fathers of Christianity so, We are convinced, ”I see, I understand.”It should be noted that people want such a short story, and not scientiﬁc arguments.
That is, note that it is not an understanding of the world for the sake of truth, but anunderstanding of the world for the sake of deepening our faith.

Augustinus asserted the following in his book ”Confessions”.
(E):Augustinus’ theory of time(E) Only present exists, and neither future nor past exist.

It is well known that St. Augustinus said that(F) the past does not exist because of its being already gone, that the future does notexist because of its not coming yet, and that the present really exists.
Here, consider(G) ”Only the present exists”
Note that this proposition (G) is related to ”tense”. Here, according to the Copenhagen
interpretation (E2 ) in Sec. 1.3.3, there is no tense in quantum languages. Therefore, this(G) is not a statement of a quantum language. Therefore, this (G) is not scientiﬁc, i.e.,there is no experiment to verify (or deny) it. This means that (G) is a very convenient
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Chap. 6 The Middle Ages
statement for religious people.
This is the beginning of subjective time. This ”time” cannot be used in science, but
can be used in philosophy as follows.
Only the present exists

Live happily in the moment!
therefore

idealistic worldview −−−−−→
introduction·preface·ﬁction

Ethics · morals
main subject

The Copenhagen Interpretation is ”a rule that draws a line between scientiﬁc and pseudoscientiﬁc propositions,” but it does not mean that pseudoscientiﬁc propositions are useless.
For example, the pseudoscientiﬁc proposition ”only the present exists” is a proposition
that cannot be denied by experiment (it does not have Popper’s disprovability). If so,”Only the present exists” could be used as an ”absolute truth” in non-scientiﬁc ﬁelds.
Rather, the use of ”pseudoscientiﬁc absolute truth” as a catchphrase is a staple ofWestern philosophy, as shown below. For example
introduction part (unquestionable truth)
}|
{
z




AugustinusOnly present exists








DescartescommonI think, therefore I am 
Therefore−−−−−−−→ ethics, moral



Kant





thing-in-itself







Analytic philosophy
mathematical logic

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6.1 Augustinus(AD. 354 - AD.430)

6.1.3 ”Subjective time” is a magic word which excites our delusion
Augustine’s problems such as ”subjective time,” ”tense,” and ”observer’s time” did notenter into the realm of science, but they continued to attract the interest of philosophers.
For example, Bergson, a philosopher of ”subjective time”, tried to challenge Einstein of”theory of relativity” to an argument. But he was rejected by Einstein, saying ”I don’tknow the time of philosophers”. Even now, some are still misled by this ”observer’s time”.
In quantum mechanics, for example, observer’s time is often assumed. For example, someresearchers may accept ”So-called Copenhagen interpretation” such as• at the moment when an observer measures it, a wave function collapses.
In order to explain ”At the moment when observer measured it”, von Neumann made a
non-scientiﬁc word ”abstract ego”, and said• ”At the moment when observer measured it” is ”at the moment when a signal reach
abstract ego”
which is of course prohibited by the linguistic Copenhagen interpretation (cf. (E2 ) inSec.1.3.3 earlier).

For the quantum linguistic understanding of ”wave function collapse”, see:ref. [65] S. Ishikawa, Linguistic interpretation of quantum mechanics; Projection
Postulate, JQIS, Vol. 5, No.4, 150-155, 2015, DOI: 10.4236/jqis.2015.54017(http://www.scirp.org/Journal/PaperInformation.aspx?PaperID=62464)

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6.1.3.1 Summing up
I think Augustine knew that word play was a useful tool for the spread of Christianity.
And I think Augustine also knew that subjective time is a treasure trove of wordplay.
Many philosophers were interested in ”subjective time”. For example,(♯1 ) McTaggart’s paradox: ”(♯2 ) Russell’s ”Five-minute hypothesis”(♯3 ) Bergson’s subjective timeetc.
However, the linguistic Copenhagen interpretation says that(H) Observer’s space-time does not exist. That is, the subjective space-time ( = observer’s space-time) is non-sense

Therefore, QL does not bother with ”subjective time.
However, it was Augustine who ﬁrst noticed the aﬃnity between Platonic philosophy
and the Copenhagen interpretation.
♠Note 6.2. ”What is the subjective time?” This is a problem of brain science. It is surethat cats and dogs etc. have clock gene or biological clock, thus they surely feel thesubjective time. This is a scientiﬁc problem. Also, measuring the time with a clock isalso a measurement. However, when you measure time with your brain clock, it is not ameasurement. A measurement that only you can make is not a measurement. In science,”I”, ”now” and ”here” are forbidden. Thus, ’Now I am here’ is not a scientiﬁc proposition(i.e., a proposition in quantum language). Enjoyment of wordplay is an important partof a successful philosophy, e.g., ”I know I know nothing”, ”Only present exists”, ”I think,therefore I am”, etc. However, it should be noted that these are not statements in quantum
language, i.e. these violate the linguistic Copenhagen interpretation.
♠Note 6.3. After all, I think Augustinus’s subjective time theory deepened our understanding of the Copenhagen interpretation. .Therefore, we insist as follows.
⃝1
Plato −−−−−−→
progresssubjective time

Augustinus
linguistic Copenhagen interpretation

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6.1 Augustinus(AD. 354 - AD.430)

The key question is, ”To whom do they tell their philosophy? ”.
Augustine· · · believers, general public
Anselmus, Thomas · · · theological controversy, thus, professional priests
Descartes,...,Kant· · · by Gutenberg’s typography, the general publicWittgenstein - - - · · · professional philosophers and general public*1
In short, when it is aimed at the laypersons, the Copenhagen Interpretation is chosen. For
professionals, ”Axiomatic theory” tends to be chosen.

*1 The part of the book that tried to be written to professional philosophers fell apart, but it had astrong message and helped popularize analytic philosophy.

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Chap. 6 The Middle Ages

6.2 Scholasticism –Bamboo (=Aristotle) cannot be grafted to atree (=Plato) –6.2.1 Aristotle’s philosophy spread to the Islamic world
Plato’s philosophy survived with the backing of Christianity (Augustine). The philosophyof Aristotle spread to Islam. I don’t know the details of the reason why,(A) Plato’s philosophy spread to the Christian world and Aristotle’s philosophy to the
Muslim world.
Probably, there were various conﬂicts in Christianity, and the winners stayed in Rome andsupported Plato’s philosophy. The losers were driven to the Muslim world, and Aristotle’s
philosophy must have spread to the Muslim world by such a process. Eastern Islam wascentered in Baghdad which was famous on the Arabian Nights. Western Islamic culture
developed around Cordoba in the Andalusian region of southern Spain and became thelargest city in the world in the 10th century. At that time, the Islamic world learned a lotof wisdom from the books of ancient Greeks and Romans and developed its own thought
and technology. The Islamic culture was at the forefront of the world under Aristotle’s
philosophy.
♠Note 6.4. In this text we adopt the story such as (A). Actually, it may not be such asimple story.

6.2.2 Crusade expedition and Inﬂow of Islamic culture

In the era of crusade expedition (1096 - 1270), the Western countries were in a downturn.
Such public opinion had been drifting.
• The achievements of the Crusades do not rise by Plato’s way. Thus, let’s study
Aristotle at the tip of the Islamic culture!
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6.2 Scholasticism –Bamboo (=Aristotle) cannot be grafted to a tree (=Plato) –• in every age and every place in the world, the human resources required are

the humanities in peacetime, and the sciences in wartime.
As a byproduct of the pilgrimage to the Holy Land of Jerusalem and the crusade torecapture it, interaction with Islamic culture was facilitated. Aristotle’s philosophy ﬂowed
into the West, merged with Plato’s philosophy, and settled in as Scholastic philosophy.
That is, Scholasticism was born. As the typical persons of Scholasticism, we list up as
follows.
(B1 ) Anselmus (1033 - 1109) ”The father of Scholasticism”, ”Realismus(≈dualistic idealism)”(B2 ) Abelard (1079-1142) ”Nominalismus(≈monistic realism)”(B3 ) Thomas Aquinas (1225 - 1274) ”Realismus”, ”Summa Theologica”, Greatest theologian in Scholasticism(B4 ) Ockham (1285 - 1347) ”Ockham’s razor”, ”Nominalismus”
After all,• in the beginning, Plato’s dualistic idealism was the most popular, but graduallyAristotle’s inﬂuence increased. And it has become like a product of the fusion of
Platonic and Aristotelian philosophies.
Of course, it is impossible to succeed this ”fusion”. That is because Plato philosophy andAristotle philosophy are ”oil (dualistic idealism) and water (monistic realism)”, and theseare diﬀerent categories (cf. Figure 0.1 [ the history of worldview] in Preface). However,in this text, we prepare the story such as• In the process of fusing Platonic and Aristotelian philosophy, Idealism was deepened
by Anselmus, Thomas Aquinas, etc.

the realistic worldview (physics)
Aristotle +Hellenism+Islam

Plato + Augustinus



















⇒

Abelard, Ockham
Scholasticism(A.D.1000∼1500)
Anselmus,Thomas

⇒











Newton → theory of relativity










Descartes·Kant
Wittgenstein→quantum language

the idealistic worldview (no experiment)

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Chap. 6 The Middle Ages

6.3 The discovery of zero

6.3.1

Positional notation (= the discovery of zero): Arabic numerals

As mentioned in the previous section,(A) Plato was introduced to the Christian world and Aristotle to the Muslim world.
Although many people must have suﬀered miserably during the Crusade expedition, themerits for Christian culture is that Aristotle philosophy and the positional notation
ﬂowed into Europe from Islam.
♠Note 6.5. The phrase ”the discovery of zero” can mean many things. In this text, ”discovery of zero” = ” positional notation ”.

The positional notation is how to write numbers to learn in an elementary school.
That is,• The next number after 9 is written as 10
For example,+5040302, −15, +39.045, −81.5, +0.3333 · · ·−1000, +0.009876, +3.1415 · · · , 0,and so on. That is, By 13 symbols ”0, 1, 2, 3, 4, 5, 6, 7, 8, 9, +, −, . (radix point)”, we
can express all real numbers by the positional notation . Hence, we may say•

the discovery of the positional notation (= Arabic numerals)= the discovery of real numbers.

(the radix point was discovered in Europa of 16 century AD.) Of course, the discovery ofzero is(B) the discovery of how to use zero called the positional notation

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6.3 The discovery of zero

6.3.2 Arabic numerals and Roman numerals
Roman numerals are often used on the clock face such as
1=I, 2=II, 3=III, 4=IV, 5=V,...,10=X, 11=XI,
However, it is too hard to represent large numbers such as
495 = CDXCV, 1888 = MDCCCLXXXVIII, 3999 = MMMCMXCIX

6.3.3 The explosion of mathematics
European mathematics originally had the high potential of Euclidean geometry. With
the introduction of the positional notation (the discovery of zeros), computation became
easier. The word ”ALGEBRA (algebra)” is originally an Arabic word. The formula forthe solution of the quadratic equation: ax2 + bx + c = 0:√−b ± b2 − 4acx=2a
was also understood by the Arabic mathematician al-Khwarizmi (about 790 - about 850)
Mathematician Gauss(1777 - 1855) said(C) ”If genius Archimedes invented the positional notation , I am certain that themathematics must have progressed drastically.”
The positional notation triggered the ”math explosion” such that(D) Solution of algebraic equations, complex numbers, the function concept, betting of
problem (probability), analytic geometry (Descartes coordinates), calculus, diﬀerential equations, linear algebra, number theory, etc.

♠Note 6.6. There may be several opinions about the three big discoveries of mathematics.
We think as follows.
1 the discovery of the plane (geometry)○
2 the discovery of zero ( positional notation )○3 the discovery of sets○0
Of course, it is needless to say that the biggest discovery is ” ○:thediscovery of natural
4
numbers”. And ”○:theinvention of computer” would be almost certain.

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Chap. 6 The Middle Ages

6.4 Anselmus; the problem of universals and the proof of theexistence of God
Anselmus is the founder of Scholastic philosophy, famous for ”the problem of universals”
and his ”Proof of the Existence of God”.

6.4.1 Aristotelian Philosophy’s Inﬂow from Islam
The ﬁgure below shows the location of Anselmus.
positional notation
logic, proof

Islam
Aristotle −−−−−→ syllogism −−−−−→monismPlato
dualism−−−−−→ Ockham
via Islamyinﬂow
God
−
−−−−
→ Augustinus −
−−−−
→
Idea

• the existence of God• the problem of universals−−−−−→ Thomas

AnselmusAs can be seen from above, Anselmus is located at the conﬂuence of the ”PlatonicAugustinian line of faith” and the ”Aristotelian line of reason via Islam.” Thus, Anselmus’
next two jobs make sense.
−→ the existence of God
 (A1 ): Considering God rationally(A)(A2 ): Considering Idea theory rationally −→ the problem of universals
Now anyone can see that (A1 ) failed. Opinions may diﬀer on (A2 ), but this text considers(A2 ) to be the greatest advance in Plato’s philosophy. If it were not for the work ofAnselmus, the Idea Theory would never have gone beyond the realm of a fairy tale.

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6.4 Anselmus; the problem of universals and the proof of the existence of God

6.4.2 Review: the worldviewism
In fact, the two issues in (A) are closely related to worldviewism. Let’s review worldviewism here .
(B): Worldviewism (cf. Sec.1.1)
The worldviewism is as follows.
conclusion

world is so(B)thereforeworldview −−−−−→ discussions, calculation, logic, properties
premisesubject

That is,(B1 ) Worldviewism is the spirit of ”Let’s start with worldview!”
To put it more clearly.
(B2 ) Without a worldview, there’s nothing to say.
We have seen many paradoxes (e.g., Zeno’s paradox, etc.) that arise from ignoring theworldview.

In this text, we have mainly discussed the quantum mechanical worldview (= quantum
language), but a worldview based on relativity is also acceptable, and we do not deny thatthe worldview may change depending on the problem to be solved.

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Chap. 6 The Middle Ages

6.4.3 The proof of the existence of God
Anselmus is the founder of Scholastic philosophy, famous for his ”Proof of the Existenceof God”. His proof is nothing more than a play on words that we use in our daily lives.
Its premises are not clearly stated (i.e., ”under which worldview is the proof?” is not
explicitly stated). Also, the meaning of ”existence” is not clear. Therefore, there is noneed to read the proofs seriously.

Proof 6.1. Anselmus: the proof of God’s Existencea○:
God is a being than which none greater can be imagined . That is, the greatest
possible being that can be imagined.
b○:
It is obvious that God exists as an idea in the mind.
c○:
A being that exists as an idea in the mind and in reality is greater than a beingthat exists only as an idea in the mind.
d○: Thus, if God exists only as an idea in the mind, then we can imagine somethingthat is greater than God. But we cannot imagine something that is greater than
God.
e
○:
Therefore, God exists in reality.

My opinion on this proof is as follows.
(C) Before examining the above proofs in detail, it is necessary to ascertain”under what worldview is it being discussed? As we will soon see, this isa ∼ ○e is not worth
not an argument from a certain worldview. Thus, ○verifying. This is not a proof but a play on words.
♠Note 6.7. (i):The ”proof of God’s existence” has been challenged by many philosophers.
However, we believe that the only credible proof is the following.
(♯) What is described in the Bible is the Christian worldview. And the Bible says that
God exists. Therefore, God exists . ([Existence] = [keyword]. cf.Note 1.2)(ii):All scientists are interested in God. What are the neural circuits of the brain regarding
faith? What is subjective time? (= What about the biological clock?) These are one of
the most interesting questions in neuroscience.

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6.4 Anselmus; the problem of universals and the proof of the existence of God

6.4.4 Overview of the problem of universals; see next section for details.
The problem of universals is the following controversy.

Problem of universals(D) It is clear that individuals such as Jack and Betty exist. So the question is, ”Does
the human race exist?”
Realism claims that it ”exists”, and Anselm is the representative.
Nominalism asserts that “it does not exist,” and Occam is the representative.

Anyone would ask, ”Why did the great priests spend 250 years in such a play on words?”
But if this is a profound question, we must marvel at the level of scholastic philosophy.
The problem of universals is the greatest theme not only in Schola philosophy but alsoin Western philosophy. This shows the high potential of Scholastic philosophy (i.e., thea ○g are alldepth of its understanding of Plato’s philosophy). In fact. The following ○∼problems of the same kind as the universal argument.
Table 6.1 (=Table 4.1) : monistic realism (Aristotle) vs. dualistic idealism (Plato)
dispute ⧹ [R] vs. [L]⃝:a motion⃝:Ancient
Greeceb⃝:c Problem of universals⃝:
d space-time⃝:
e quantum theory⃝:philosophyof sciencef⃝:g fuzzy sets

Realistic worldview

Idealistic worldview(monism, realism, no measurement)(dualism, idealism, measurement)

Hērakleitos
Aristotle”Nominalismus”(Ockham)
Newton ( Clarke)
Einstein
CarnapKalmanParmenides
Plato”Realismus”(Anselmus)
Leibniz
Bohr
Quine
Zadeh

For the precise argument, see Table 4.1.

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Chap. 6 The Middle Agesc The Problem of universals was a 250-year debate between the Aristotelians andthe Platonists The Problem of universals was a debate between the Aristotelians
and the Platonists in Scholasticism, with the Platonists gaining the upper hand
at ﬁrst, and then the Aristotelians gradually gaining the upper hand. Although it
was not a quantitative scientiﬁc debate, the problem of universals universal shows
the high philosophical competence of Scholastic philosophy (both Aristotelian and
Platonist).
This will be discussed in the following section ( Sec. 6.5).

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6.5

Scholasticism; Problem of universals

6.5.1

What is the problem of universals?

This section is due to the following paper:• ref.[83],S. Ishikawa, (2022) The problem of universals from the scientiﬁc point ofview: Thomas Aquinas should be more appreciated Open Journal of Philosophy, Vol.
12, No. 1, 86-104,(https://www.scirp.org/journal/paperinformation.aspx?paperid=115252)

In the ﬁgure below, let us review again the historical position of Anselmus.
positional notation
logic, proof

Islam
Aristotle −−−−−→ syllogism −−−−−→monismPlato
dualism−−−−−→ Ockham
via Islamyinﬂow
God
−
−−−−
→ Augustinus −
−−−−
→
Idea

• the existence of God• the problem of universals−−−−−→ Thomas

AnselmusAs can be seen from above, Anselmus is located at the conﬂuence of the ”PlatonicAugustinian line of faith” and the ”Aristotelian line of reason via Islam.” Thus, Anselmus’
next two jobs make sense.
−→ the existence of God
 (A1 ): Considering God rationally(A)(A2 ): Considering Idea theory rationally −→ the problem of universals
Now anyone can see that (A1 ) failed. The (A2 ) is a controversy that arises when Plato’s
Idea Theory enters the realm of science, and readers will be surprised at Anselmus’ deep
knowledge of Plato’s philosophy.

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(B): What is ”Problem of universals”?
Problem 6.2. ”Problem of universals” is as follows.
(♯) It is certain that Mx. Smith, Mx. White, Mx. Brown, etc. exist as matters(i.e., individuals). Then, we have the following problem:
Do ”universals” (i.e., ”Honesty”, ”Optimism”, ”Kindness”, ”Female”,”Intelligence”,etc.) exist?
If ”Yes”, then, ”Realismus”. If ”No”, then, ”Nominalismus”.
The following ﬁgure will promote the reader’s understanding

Many readers may ask the following question?
(B′ ) Why did the great Fathers argue so fervently on such a matter of wordplay?
When studying the problem of universals scientiﬁcally, we must ﬁrst answer two questions.
(B1 ) Under what worldview should we discuss the problem of universals ?
(B2 ) What is ”universal”?
In fact, however, ”universal” is a concept that can be clearly understood for the ﬁrst time
under a scientiﬁc worldview.
Therefore, in the next section, review the scientiﬁc worldviews and then answer questions (B1 ) and (B2 )165

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6.5.2 Review of Sec 1.4 (Appendix: The mathematical foundations of quantum
language)
6.5.2.1 Axioms 1 and 2
Now we review Axioms 1 and 2 of QL in Sec. 1.4 as follows.
Axiom 1 [State, Measurement]•[General (i.e., quantum and classical ) case]:(C1 ) [State]: With any system S, a C ∗ -algebra A(⊆ B(H)) can be associated. A
state of the system S is represented by an element ρ(∈ Sp (A∗ )).
(C2 ) [ Measurement ]: An observable is represented by an observable
O =(X, P(X), G) in A. And, the measurement of the observable O for thesystem S with the state ρ is denoted by MA (O, S[ρ] ) or more precisely,
MA (O =(X, P(X), G), S[ρ] ) . An observer can obtain a measured value x(∈ X) by the measurement M(O, S[ρ] ). The probability that a measured
value x (∈ X) obtained by the measurement MA (O =(X, P(X), G), S[ρ] ) is
given by ρ(G({x}))(≡ A∗ hρ, G({x})iA ).
•[Classical case, i.e. the case that A = C(Ω)]: The above (C1 ) and (C2 ) can besummarized for the classical system as follows(Cc1 ) [Classical state]: A classical state of the system S is represented by anelement ω(∈ Ω).
c(C2 ) [Classical measurement ]: When an observer take a measurement of an observable O =(X, P(X), G) for a system with a state ω(∈ Ω), the probability
that a measured value x (∈ X) is obtained is given by [G({x})](ω).
Note that it suﬃces to identify ”observable” with ”measuring instrument”.

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(D): Axiom 2 [ Causality ]Let (T ≡ {t0 , t1 , ..., tn }, ≤) be a ( ﬁnite) tree-like semi-ordered set.
(D1 ): [General (i.e., quantum and classical ) case]
Causality is represented by a causal relation { Φt2 ,t1 : At2 → At1 , t1 , t2 ∈ T(t1 ≤ t2 ) }.
(D2 ): [Classical case]Classical causality is represented by a classical causal relation { ϕt2 ,t1 : Ωt1 → Ωt2 ,
t1 , t2 ∈ T (t1 ≤ t2 ) }.

6.5.2.2 Three worldviews
Here we focus on three worldviews
Remark 6.3. (= Remark 1.11)(E):quantum language (=QL)
QL =



+ causality + Copenhagen interpretation(Axiom 1; (i) and (ii))
statemeasurement(Axiom 2)(how to use Axioms 1 and 2)(F): classical quantum language (= classical QL)
classical QL =statemeasurement++ Copenhagen interpretation

causality(Axiom 1[Classical case (i), (ii)]) (Axiom 2[Classical case])(how to use Axioms 1 and 2)

Also, we deﬁne ”abstract classical mechanics” as follows.
(G): abstract classical mechanics
abstract classical mechanics =

state

+

(Axiom 1 [Classical case] (i))

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6.5.3 Scientiﬁc deﬁnition of ”universal”
As mentioned in the previous section, ”universal” is a concept that can be clearly understood for the ﬁrst time under a scientiﬁc worldview. And, when studying the problem ofuniversals scientiﬁcally, we must ﬁrst answer two questions.
• (=(B1 )) Under what worldview should we discuss the problem of universals ?
• (=(B2 )) What is ”universal”?
Then, we can assert the deﬁnition:(H): Deﬁnition of ”universal” in scientiﬁc worldview
Deﬁnition 6.4. [Universal]
Assume a certain scientiﬁc worldview (worldview expressed in mathematics).
(H1 ) The axiom system of the assumed world view has several keywords. Among
those keywords, those with mathematical expressions (= parameter expressions) are called ”universal”.
[Note]: For non-scientiﬁc worldviews (worldviews not expressed in mathematics),”universal” cannot be deﬁned. However, I dare say that ”universal” = ”essential keywords”. This is because it is an axiom that quantitatively expresses therelationship between several essential keywords.

♠Note 6.8. Thus, to use Pythagorean language, we can express it as follows(♯1 ) Number is the within of all universals.

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6.5.4 Examples: universals in Classical QL and Abstract Classical Mechanics

6.5.4.1 Example: universals in Classical QL ( or, QL)
Explained in classical QL (same for QL), but for simplicity I will explain in classical QL.
(I): The classical QL is formulated as (F)classical QL =statemeasurement++ Copenhagen interpretation

causality(Axiom 1[Classical case (i), (ii)]) (Axiom 2[Classical case])(how to use Axioms 1 and 2)

Here, the key-words in Axiom 1 [Classical case (i), (ii)] and Axiom 2 [Classical case] areas follows.
(J) ”state”, ”system”, ”observable”, ”measurement”, ”measured value”, ”probability”,”causality”
Among the above, those that have mathematical expressions are as follows.
(K) ”state” (ω ∈ Ω),”observable” (O =(X, P(X), G))”measured value”(x ∈ X),”causality” ({ ϕt2 ,t1 : Ωt1 → Ωt2 )(Here, ”measurement” MC(Ω) (O =(X, P(X), G), S[ω] )) and ”probability” [G({x})](ω) arenot simple, thus, the two are omitted. )Therefore, the universals in classical QL are as follows.
•”state”, ”observable”, ”measured value”, ”causality”

6.5.4.2 Example: universals in abstract classical mechanics
The same is true for Newtonian mechanics, but for the sake of simplicity I will explain it
in abstract classical mechanics.
(L): abstract classical mechanics is formulated as follows ( = (G)):
abstract classical mechanics =

state

+

(Axiom 1 [Classical case] (i))

causality(Axiom 2 [Classical case])

Therefore, key-words are as follows:”state”, ”causality”
Among the above, those that have mathematical expressions are as follows.
”state” (ω ∈ Ω),”causality” ({ ϕt2 ,t1 : Ωt1 → Ωt2 )Therefore, abstract classiical mechanics has the following universals:•”state”, ”causality”
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6.5.5 Detailed discussion: Problem of Universals in science[[I]: The problem of universals in classical QL]:
Let Ω be the compact state space, in which every human’s state is assumed to be represented. Let mH : Ω → [0, 1] [resp. mO : Ω → [0, 1] ] be the membership function of”Honesty” [ resp. ”Optimism”] (see Figure below) (cf. ref. [129], 1965).

1

mH

mO

•
ω

Ω

Figure 6.1: [Membership functions mH (resp. mO ) of ”Honesty” ( resp. ”Optimism”)]
Here, assume that the value mH (ω) (∀ω ∈ Ω) is determined as follows:(M) [The probabilistic interpretation of QL]: Suppose a group of 100 respondents are
asked the question ”Is this person (with the state ω ∈ Ω) honest or not? and100mH (ω) of the respondents answered ”yes”. Then, we can deﬁne the membershipfunction mH : Ω → [0, 1] concerning ”Honesty”. Similarly, mO : Ω → [0, 1] isdeﬁned.
Further, deﬁne the observables OH = ({y, n}, 2{y,n} , GH ) and OJ = ({y, n}, 2{y,n} , GO ) (where ”y”=”yes”, ”n”=”no”) such that
[GH ({y})](ω) = mH (ω),[GH ({n})](ω) = 1 − mH (ω)[Go ({y})](ω) = mo (ω),[Go ({n})](ω) = 1 − mo (ω)

for all ω ∈ Ω.
And thus, we have the following identiﬁcations:OH ≈ m H ,O O ≈ mO

Therefore, we see that(N) the probability that a measured value y (∈ X = {y, n}) obtained by the measurement MC(Ω) (OH =({y, n}, P({y, n}), GH ), S[ω] ) is given by [GH ({y})](ω)(=
mH (ω)).
That is, choosing one at random from a group of 100 people, you ask to him/her thequestion ”Is this person (with state ω) honest or not?. Then he/she surely answers, withthe probability mH (ω), ”This person is honest”.
Thus, in the framework of Axioms 0 and 1, we can say that the measurement
MC(Ω) (OH =({y, n}, P({y, n}), GH ), S[ω] ) is meaningful, that is, three important
key-words (i.e., three universals ) concerning ”Honesty” etc. exist as follows.
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(O1 ) measured values {y, n},(O2 ) observable OH ( equivalently, membership function mH : Ω → [0, 1]),
(O3 ) state ωThis is our conclusion in scientiﬁc dualism (i.e., QL).
Now, the original universal was as shown in the ﬁgure below.

If so,• observableOH is called ”primary universal”or simply, ”universal”
Also,• state ω(∈ Ω) is called ”secondary universal”• measured value (i.e., the element of {y, n}) is called ”tertiary universal”[[II]:Universal in Abstract classical mechanics]
Now, in the case of abstract classical mechanics,(P) state ω(∈ Ω)is meaningful (≈ exists). Therefore, as in the case of [I], we see• state ω(∈ Ω) is ”secondary universal”.
For example, assume that some say that(Q) A person with the state ω is honest (i.e. mH (ω) = 1)”
However, abstract classical mechanics does not have an observable that represents ”honesty (i.e., primary universal)”. Thus, the statement (Q) is only sound.

[[III]: Summary]
Summary 6.5. Putting the above together, we get the following

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6.5 Scholasticism; Problem of universals/
monismNewtonian mechanics
abstract classical mechanics
dualismquantum mechanics
QL, classical QL

tertiary
universal

primary universal
or simply, universal

secondaryuniversal

/

/

state

measured value

observable

state♠Memo 6.1. Instead of Summary 6.5, there is a reason to deﬁne as follows.
(♭) ”state”, ”observed value” and ”measured value” are respectively ”primary universal”
and ”secondary universal” ”tertiary universal”
In fact, in ref. [82], this (♭) was adopted. However, in this text, Summary 6.5 is adopted
because it is close to the spirit of Anselmus and Thomas Aquinas.

Remark 6.6. It is generally believed that the problem of universals is diﬃcult to understand. The reason for this is that the problem of universals can only be fully understoodwithin the context of scientiﬁc dualism. And, this is because the discovery of scientiﬁcdualism (that is, the discovery of quantum language (or quantum mechanics)) is only recent. It is amazing to see the genius of Anselmus, who took Plato’s ideas one step further
in the direction of science.

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6.5.6 Primary universals in incomplete worldview
Now we have the following problem:(R) Can we discuss the problem of universals in the case that the worldview is nonscientiﬁc?
It is a matter of course that in an incomplete worldview, only an incomplete argument
can be made. However, we think that(S) If we can assume that
incomplete worldview (S) −−−−−→ Newtonian mechanics
progress

then, the incomplete worldview (S) does not have the (primary) universal.
(T) If we can assume that
incomplete worldview (T) −−−−−→ Quantum language
progress

then, the incomplete worldview (T) have the (primary) universal.
If so, we see that following progress of ”(primary) universal” in the history of philosophy:(U) The progress of (primary) universals in history of philosophy
Idea

linguistic version−−−−−−−−−−−→

signiﬁer, signiﬁed

Plato’s idea theoryChristianyversion

Saussure’s linguistic theory(primary) universal

membership function

Anselmus, Thomas

Zadeh’s fuzzy theoryquantiﬁcationycognitiveyversion

secondary qualityyidentiﬁcation

linguistic version−−−−−−−−−−−→

observable
quantum language

John Locke♠Note 6.9. The arguments in (S) and (T) above are valid for the history of philosophy (U),but generally inadequate. For example. we think(♯) statistics −−−−−−→ Quantum language
progress(cf. see Note 1.3)

But, I don’t want to say that statistics has ”universals (= primary universals)”.

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When I was in high school, I studied Locke’s ”secondary quality”. The simplicity of”secondary quality” left a lasting impression on me, but I never dreamed that this wasa central concept in philosophy. However, the fact that it was mentioned in high school
textbooks must mean that ordinary philosophers knew that it was a central concept inphilosophy.

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6.6 The problem of universals in history of western philosophy
6.6.1 Problem of universals in Scholasticism; Anselmus, Thomas Aquinas, etc.

6.6.1.1 Anselmus (1033-1109); dualism, Realismus
First, Anselmus of England, the father of Scholasticism, argued that(A) universals are real and exist before individuals,
and preached realism (i.e., conceptual realism, in this text it is also called dualistic idealism
in spite of the risk of confusing you). We appreciate (A) from a scientiﬁc point of view.
Because Plato is fairy tale and Augustine is too religious.
6.6.1.2 Abelard (1079-1142) ; monism, Nominalismus
Since ”eidos” is usually regarded as ”Aristotle’s Idea”, it is natural to consider that
(B1 ) universals are in individuals
Therefore, it is reasonable that Roscellinus (1050-1125), Abelard (1079-1142) and others
proposed an argument known as nominalism argued in what follows:(B2 ) universals (in the sense of Anselmus) are merely nominal terms or ”sound”
6.6.1.3 Thomas Aquinas (1225-1274); dualism, Realismus

The Catholic priest: Thomas Aquinas (1225 - 1274) wrote ”Summa Theologica” as thesumming-up of Scholasticism. He was the most important at the intermediate time of
Scholasticism (or, Problem of universals), that is,
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6.6 The problem of universals in history of western philosophy
(B3 )




















(BC.384−322)(BC.427−347)(1285−1347)(1079−1142)

Aristotle −→ · · · Isalam · · · −→Abelard(monism)(Nominalismus)(354−430)(1033−1109)−→(1595−1650)

Ockham −→ Newton(Nominalismus)(mechanics)(1225−1274)(1595−1650)

Plato−→ Augustinus −→ Anselmus −→ Thomas Aquinas −→Descartes(dualism)(Realismus)(nind-body dualism)(Platonism)(Realismus)

He developed Realismus (due to Anselmus) as follows.
It is usually said that Thomas Aquinas (1225-1274) proposed to integrate conceptual
realism (= dualistic idealism) and nominalism (= monistic realism) such as

Theory of Heaven( by Thomas )(C1 ) :universals exist’after individuals’ in the human intellect .
universalia post rem as actual world
QL: (i.e., measured value is sensed by the brain (i.e., mind)).
(C2 ) :universals exist ’before
 individuals’ in the divine intellect, .
universalia ante res as Plato’s Idea
QL: (i.e., observable (or, measuring instrument) could be in the heavens, like theNorth Star (cf. Note 1.7))(C3 ) :universals exist ’in individuals’
in the world, .

universalia in rebus as Aristotle’s eidos
QL: (i.e., state exists in matter)♠Note 6.10. In Sec. 8.3.3, I introduce Descartes mind-body dualism, which is also called”Theory of Earth” in this text.

I would like to think, however, that Thomas pursued true dualism and not a Platonic/Aristotelian compromise.
I think that(D) The closest theories to quantum language in the history of philosophy are Thomas’s[(C1 )∼(C3 )] above and Descartes problem (Sec. 8.3.2).

6.6.1.4 Ockham (1285-1347); Monism, Nominalismus
In the 14th century, however, William Ockham and others revived nominalism (≈ monistic
worldview), which led to the emergence of modern thought that abolished conceptual
thinking and sought truth through observation and experimentation.
Nominalism was very helpful in enlightening Aristotelian philosophy. As I’ve said manytimes so far, I think that monism and dualism are not mutually exclusive, but coexist
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such as the above (B3 ).
Under the trend that the correct answer to the problem of universals is Newtonianmechanics, nominalism has somehow become dominant, and this continues to this day.
On the other hand, because no theory succeeding Thomas has emerged, the philosophicalmeaning of the problem of universals has remained unclear until the present day.
Table 6.2 : Summing up: universals in worldview (monism and dualism)/

monismdualism/
Aristotle
Abelard(1079-1142)
Ockham(1285-1347)
Newton(complete form)
Plato (Allegoryof the sun)Augustinus(354-430)
Anselmus(1033-1109)
Thomas Aquinas(1225-1274)
Descartes(1596-1650)
Quantum language

A(= mind)[tertiary
universal(A)]B(= body; medium)[(primary)
universal (B)]

————

————

————

————

————

————

————

————

C(= matter)[secondary
universal(C)]hyle[eidos]individual[universal]
individual[universal]
particle[state]actual world

Idea

idea world

city of this world

churchCity of God[/][universal]

human intellectdivine intellectindividual[/]
individual[universale post rem]

[universale ante rem]

[universal in re ]

mind

body

matterobservermeasuring instrument

matter[measured value][observable][state ]I think philosophy is to answer ”What is a dualism?”, or equivalently, ”What is Plato’s Idea?”
In what follows, the reader will see that the meaning of ”universal” is clear only in Newtonianmechanics (= scientiﬁc monism) and quantum language (= scientiﬁc dualism), and that in allother cases the meaning of ”universal” is ambiguous.

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6.7 Ockham’s razor and Plato’s beard

6.7 Ockham’s razor and Plato’s beard
William of Ockham (1285 - 1347), a Scholastic philosopher or theologian born in Ockham in England, who revived nominalism (≈ monistic worldview), which led to the emergence of modern
thought that abolished conceptual thinking and sought truth through observation and experimentation. Further he is known as an advocate of Ockham’s razor(=the law of parsimony) in
philosophy and science.
(A):Ockham’s razor(=the law of parsimony)
Ockham’s razor is as follows:(A) Shave unnecessary assumptions with a razor!
(A’) All things being equal, the simplest solution tends to be the best one.

However, this may be a self-evident truth. For example,(B) Assume that you were a student of Plato and Plato asked you• ”The sun goes around the earth? or the earth goes around the sun?”Then, which did you answer to Plato?
Probably, you, by Ockham’s razor, answer that the sun goes around the earth. In fact Aristotle
did so. Ockham’s razor is dependent on the environment around. If the surrounding environmentchanges, so will the way of thinking. Thus I have a question:(C) Is there a case as which Ockham’s razor is functioning eﬀectively (besides the mathematicaltheorems) ?
We see that
(D1 ) Relying on Ockham’s razor, Ptolemy claimed the theory of celestial motion from theobservational data of the time.
And(D2 ) relying on Ockham’s razor, Galileo claimed for a geodynamic theory from the observational
data of the time.
Ockham’s Razor is an explanation added later and thus, powerless. But then again...
(E) Is there a case (other than mathematics) in which ”Ockham’s Razor” is working eﬀectively?
This makes me skeptical of Ockham’s razor. Even in the Middle Ages, it is diﬃcult to imagine
that people would have been impressed by such a statement if it had been made without any
background. The question is, ”In what kind of background and context did Ockham say it? I

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Chap. 6 The Middle Ages
think so. The author is a layman and does not know much about it. Ockham, however, is theleading edge of the Aristotelian sect. Therefore, I agree the following opinion:(F) What Ockham wanted to shave with a razor is Plato’s beard ( = the theory of Ideas)

It makes sense, then. However, it would be too short-sighted to think that it must be better toeliminate such hypotheses as the existence of ”something ideal in the heavens. Plato must havethought that the idea theory was a fairy tale. It would be immature and foolish to cut oﬀ thefairy tale. As mentioned above, Ockham’s razor is dependent on the environment around. Under
Newtonian mechanics, Idea theory, purpose (= ﬁnal cause), Geocentric model must be cut oﬀ.
And thus, we think that(G) under QL, ideationalism need not be truncatedBecause, I would like to argue the following story:

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6.7 Ockham’s razor and Plato’s beard

6.7.1 Statistics or quantum language: which will Occam’s razor cut oﬀ?How
sharp is Ockham’s razor?
Figure 0.1 ( Preface ) is restated as follows.
Figure 0.1: The location of QL in the history of western philosophy(realistic) monism (i.e., without measurement axiom)
relativity
4 −→
theory−−−−−−−−−−→○(monism)(unsolved)3○
−−→ Newton ○6 TOE:theory of(realism)−−→ everything
quantum
5 −→ mechanics −−−−−−−−→○(quantum phys.)

Socrates
Aristotle
0○:Greek

7○

Schola-8○−−−−−−−−−−−−−−−−−→

8
○



quantization
7○9−−−−−−−−−→○
−−−−−−−−−→ statistics

dualism
applied math.
system theoryclassical mechanical worldview


(=MT)



(mind-body dualism)(Copernican revolution)) ○QL:quantum
15
linguistic−−→
language
Descartes
Kant scientiﬁc
10
○

11 
−−→ Spinoza, Locke,... −−→−−−−−−→○
 quantum mechanicalturnepistemological worldview
worldview

12
○



linguistic turn



?

(logic in philo.))(logic, set theory)Wittgensteinmechanical
Saussure −−−−−−−→○

Boole, Frege,
14
13
○
turn
Russell, Cantor −−→ Zadeh

philosophy ○21−−−−−→○
sticism
Parmenides, Zeno
Platolinguistic turnmath.

logical worldview(idealistic) dualism (i.e., with measurement axiom)
Figure 0.1 (in Preface): The history of the world-descriptions
Philosophy (≈ dualistic idealism) has progressed toward QL
0
1
2
10
12
14
15 )
(i.e., ○→
○→
○→
○→
○→
○→
○
7 and ○](i.e.,
8
As will be discussed in Sec. 11.7, the Copernican revolution [○from realism to idealism) from Newtonian mechanics and quantum mechanics respectively produce
statistics and quantum mechanics.

copy theory
phenomena −→ QM
quantum mechanics (=QM)
xa○quantumization
copy theory
phenomena −→ CMclassical mechanics (=CM)

8 Copernican revolution○−−−−−−−−−−−−−−−−−−−−−→
in quantum mechanics

7 Copernican revolution○−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
in classical mechanics (due to Fisher, etc.)

constructionism
QL −→ phenomena
quantum language (=QL)
c ○yclassicization
constructionism
statistics −→ phenomena
statistics

However, I believe that the true form of statistics is the ”simpliﬁcation (or, classicization
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Chap. 6 The Middle Agesc in the above Figure) of quantum language”.
○

Scope of application of statistics (=dynamical system theory) ⊂ Scope of application of QL
Why doesn’t Occam’s razor work?

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183

Chapter 7

Early modern – Scientiﬁc revolution:
From Geocentrism to Heliocentrism
We assume that the three greatest paradigm shifts are as follows(♯1 ) Aristotelian worldview (purpose) −→ Newtonian worldview (causal relation)(♯2 ) Ptolemaic system: Geocentrism −→ Copernican system: Heliocentrism(♯3 ) Christianity: Adam and Eve −→ Darwin: evolution theory
In this chapter, we are concerned with (♯1 ) and (♯2 ), and conclude that• (♯2 ) is a metaphysical dispute, which cannot be made clear by experiments. And it
was clariﬁed by (♯1 ). In this sense, (♯2 ) is included in (♯1 ).
• The (♯3 ) is peculiar to the West.

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Chap. 7 Early modern – Scientiﬁc revolution: From Geocentrism to Heliocentrism

7.1 Paradigm shift
Eastern Roman Empire was made to be ruined by Ottoman Turkey in 1453.
• 1453:The Eastern Roman Empire extinction (Constantinople surrender)
The inﬂuence on Christ cultural area of this great event is immeasurable.
Traﬃc of ”Silk Road” became inconvenient. And thus,
Age of Discovery had begun
Also, engineers, artists, cultural people, etc. (of Eastern Roman Empire) had ﬂowed into
Western Europe as refugees. And hence,
Renaissance rose suddenly.
The time series is as follows.
Table 7.1. Scientiﬁc revolution; Chronological table
Before Galileo: The era of observation and experiment
inventor• 1450: Gutenberg’s printing press• 1453:The Eastern Roman Empire extinction (Constantinople surrender)• 1492: Columbus, discovery of the American Continental
navigator• 1498: Vasco da Gama, discovery of the sea route to India
navigator• 1500s: Leonardo da Vinci, ”Mona Lisa’s smile”
artist• 1510: Copernicus, Heliocentrism
scientist• 1510: Raﬀaello, ”The School of Athens”, Admiration to ancient Greece
artist• 1517: Luther, Protestant Reformation
religionist• 1519 - 20: Magellan, the ﬁrst circumnavigation of the Earth• 1540s: Michelangelo, ”The Last Judgment”
artist• 1600: G. Bruno, who supported heliocentrism along with Galileo, was burned at
the stake by the Vaticanphilosopher• 1609∼1619: Kepler’s laws of planetary motion
scientist• 1610: Galileo, A telescope was made and moons of Jupiter were found scientist• 1620: F. Bacon, ”knowledge is power”, the father of British Empiricism illuminator• 1633 Galileo’s trial ”And yet it moves”
scientist

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7.1 Paradigm shift

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Chap. 7 Early modern – Scientiﬁc revolution: From Geocentrism to Heliocentrism

After Galileo: The era of thought• 1637: ”Discourse on the Method”, Rene Descartes(1596 - 1650),
the father of modern philosophy, Cogito proposition philosopher

Table 7.2.

• 1670: Pascal, ”Pensèes”
enlightener• 1685 - 1750: Bachartist• 1687: Newton, ”Principia”
scientist• 1688: Glorious Revolution• 1690: John Locke, the father of British Empiricism, ”An Essay Concerning
Human Understanding”, tabula-rasa, the secondary quality
philosopher• around 1700: Jakob Bernoulli, the law of large numbers
mathematician
philosopher• 1703: Leibniz, ”New Essays on Human Understanding”• 1715 - 16: Leibniz-Clarke correspondence (cf. Sec. 9.2.2)• 1739: Hume, ”A Treatise of Human Nature”
philosopher• 1781: Kant, ”Critique of Pure Reason”
philosopher• 1788: Lagrange, Mécanique analytiquescientists• 1789: Lavoisier, Elementary Treatise on Chemistryscientists
scientists• 1799-1825: Laplace, Traité de mécanique céleste♠Note 7.1. The law of large numbers, discovered by J. Bernoulli(1654 - 1705), is as follows.
(♯) Suppose a fair coin (with a probability of heads equal to 1/2) is tossed N times. Thenit holds that
Number of times the coin comes up Heads
lim
= 1/2
N →∞
N
Number of times the coin comes up Tails
lim
= 1/2
N →∞
N

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7.1 Paradigm shift
I think that Bernoulli’s achievement equals Galileo’s achievement. That is,Scientiﬁc pioneer in the realistic worldview · · · Galileo
Scientiﬁc pioneer in the idealistic worldview · · · J. Bernoulli
It is diﬃcult to identify the founder of the probability theory to one person. But, I thinkthat J. Bernoulli is one of the most important founders (e.g., P.S. Laplace (1749-1827), A.
Kolmogorov (1903-1987), etc.).

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Chap. 7 Early modern – Scientiﬁc revolution: From Geocentrism to Heliocentrism

7.2 Francis Bacon (1561 - 1626): The father of empiricism,
Inductive reasoning

7.2.1 How to create science: The exclusion of idols (=prejudice, preconception)
F. Bacon has been called the father of empiricism. He was the greatest enlightener of”scientiﬁc revolution” as follows.
the birth of worldview•

Movement(Hērakleitos, Parmenides, Zeno,
Aristotle, Ptolemaios)

the birth of modern science”purpose” (and no experiment)−−−−−−−−−−−−−−−−−−−−−−−−→ Causality (and experiment)
Aristotle’s spirit :(About 1500 years)( Galileo, Bacon, Descartes, Newton)

In 1620, he proposed ”how to create science” ( called inductive reasoning, or induction
principle) in his book ”Novum organum”.
(A):Induction principle (by bad idols), how to create scienceHis proposal is as follows.
1(A) ○:Exclusionof bad idols
23−→○:data, collection by observation and experiment−→○:scientiﬁctheory,principle

Let us explain this in what follows.
1 : Firstly, we have to exclude bad idols (=prejudice, preconception) Here, idols○
is as follows.
Idols of the Tribe: prejudice due to sense organs
Idols of the Cave: prejudice due to custom, the education
Idols of the Market: prejudice due to language
Idols of the Theatre: prejudice due to thought, theory
2 : Next, we have to collect data by observation, experiments,○
3 : Lastly, ﬁnd the essence from the data, and build science theory.
○23Here, ”○+○”
is called ”induction”.

Recall that Greek philosophy starts from ”principle”, e.g., ”the arche (= the ﬁrst principle of all things) is”, or, Euclid advocated geometric axiomatization, and a lot oftheorems were derived from Euclid axioms, that is,
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7.2 Francis Bacon (1561 - 1626): The father of empiricism, Inductive reasoning(B1 ) Mathematics:

deduction
Euclid axioms −−−−−−−−→ theorems

On the other hand, Bacon emphasized the importance of observation (or experiments),
that is,(B2 ) Science:induction
data, collections −−−−−−−→ principle

which is the scientiﬁc method proposed by Bacon (who was called the father of empiricism).

7.2.1.1 Isaac Newton (the exclusion of bad idols) (1642-1727)
Newton said:”I frame no hypotheses”
And he practiced Bacon’s induction principle, and proposed Newtonian mechanics asfollows.
12(C): ○exclude
bad idols (i.e., Aristotle’s purpose, Geocentrism) −→ ○Datacollection
3(due to Tycho Brahe, Kepler, Galileo) −→ ○Science
theory (Newtonian mechanics)♠Note 7.2. This may be said on the philosophy side. The next section (good idols) must
be added.

7.2.1.2 Isaac Newton (good idols) (1642-1727)
Bacon’s induction principle is not simple. there is another way (by good idols) such as(D): Induction principle (by good idols)
Induction principle (by good idols) is as follows.
123(D) ○believe
good idols −→ ○Datacollection −→ ○Science
theoryNewton said:”I frame no hypotheses”
And he practiced Bacon’s induction principle (good idols), and proposed Newtonian mechanics as follows.
12(E): ○believe
good idols (i.e., Causal relation) −→ ○Datacollection (due to Tycho
3
Brahe, Kepler, Galileo) −→ ○Science
theory (Newtonian mechanics)♠Note 7.3. Although ironically,(♯) Bacon, who proposed the exclusion of idols, was also one of discoverers of ”good idols”
called ”causal relation”.
If so, what
 Bacon wanted  to say really may bebad idols = dogmatism in Scholasticism (or, Aristotle’s purpose)190

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Chap. 7 Early modern – Scientiﬁc revolution: From Geocentrism to Heliocentrism♠Note 7.4. S. Weinberg (1933 -2021), a physicist at the University of Texas, Austin, won
a Nobel Prize in 1979 for work that became a cornerstone of particle physics, said in his
book [123] ”To explain the world; The discovery of modern science” as follows:(♯1 ) [in Chapter 13]: They (i.e., Bacon and Descartes) are, in my opinion, the two individuals whose importance in the scientiﬁc revolution is most overrated. .... Scientists
in the seventeenth and eighteenth centuries would invoke Bacon as a counterweightto Plato and Aristotle, .... It is not clear to me that anyone’s scientiﬁc work was
actually changed for the better by Bacon’s writing. Galileo did not need Bacon to tell
him to do experiments, and neither I think did Boyle or Newton.
From the pure scientiﬁc point of view, I almost agree to the above opinion. However, it
should be noted that the scientiﬁc revolution was not only achieved by scientists, but by
the collective power of philosophers, enlighteners, astronomers, mathematicians, adventurers, artists, educators, politicians, religionists, and the general public. The diﬃcultand important thing is to get public support for the scientiﬁc revolution. I think Bacon
did this work well. He enlightened the method to create science (i.e., the importance of
causality and experiment) to the general public, not the scientists. As I will show in thenext chapter, I consider as follows:(♯2 ) the greatest task of modern philosophy (Descartes, Kant, analytic philosophy) is toenlighten the general public about science.
Even so, I respect Dr. Weinberg’s honest opinion.

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7.3 From Geocentrism to Heliocentrism

7.3 From Geocentrism to Heliocentrism

7.3.1 What is ”Geocentrism vs. Heliocentrism”?
As mentioned in Chapter 5,• Heliocentrism due to Aristarchus (BC.310 - BC.230) is based on the arguments:
The big sun cannot go around the small earth.
I think his Heliocentrism to have reached the scientiﬁc level. (cf. Sec. 5.3).
• Geocentrism due to Ptolemaios (AD.83 - AD.168) can explain the motion on planets
by epicycle theory. Thus, I also think his Geocentrism to have reached the scientiﬁclevel at the time. (cf. Sec. 5.6).
However, Heliocentrism due to Copernicus (1473 - 1543) might be controversial whether
it had reached the scientiﬁc level. For example, there is an opinion that
• At the time, Europe is in the cold period, the masses were hungry for ”the sun”.
The public was hungry for the sun central principle. Therefore, there is a foundationthat allows the germination of Heliocentrism.
In spite of the above, I want to assert that Copernicus is a great scientists. The reason I
think so is written in the following Note 7.5.
♠Note 7.5. S. Weinberg (1933 -2021), An American physicist, won a Nobel Prize of physics
in 1979, said in his book [123] ”To explain the world; The discovery of modern science”
as follows:(♯) [Chapter 11] Copernicus could not claim in the Commentariolus that his scheme ﬁtted
observation better than that of Ptolemy. For one thing, it didn’t. Indeed, it couldn’t,
since for the most part Copernicus based his theory on data he inferred from Ptolemy’s
Almagest, rather than on his own observations. Instead of appealing to new
observations, Copernicus pointed out a number of his theory’s advantages.
It was very interesting to me that Copernicus, the ﬂag bearer of the scientiﬁc revolution, valued beauty over observation. In fact, my conclusion in section 7.4 of this text,

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Chap. 7 Early modern – Scientiﬁc revolution: From Geocentrism to Heliocentrism”Heliocentrism vs. Geocentrism,” is also ”Heliocentrism was chosen because it is more
beautiful.” I’ve grown to love Copernicus. Thus I would like to regard him as the greatscientist.

The following question is the main theme of this chapter.
(A): What is ”Heliocentrism vs. Geocentrism”?
Now,(A1 ) Note that motion is relative. Thus, if the earth is assumed to be at center, thesun goes around the earth (i.e., Geocentrism). Also, if the sun is assumed to
be at center, the earth goes around the sun (i.e., Heliocentrism). Hence,
The diﬀerence between Heliocentrism and Geocentrism
may be only a diﬀerence of how to take the coordinate system.
Or,erael(A2 ) Under what worldview should ”Heliocentrism vs. Geocentrism” be considered?

7.3.2

Somehow ”from Geocentrism to Heliocentrism”

In what follows, I will arrange the history of ”Heliocentrism vs. Geocentrism”.

• 1510: Copernicus, Heliocentrism in the Commentariolus• 1600: G. Bruno, who supported heliocentrism along with Galileo, was burned at
the stake by the Vatican
Analyzing the enormous data obtained by Tycho Brahe’s steady astronomical observation, Kepler found the following laws:• Kepler’s laws of planetary motion:
1609:(♯1 ) The ﬁrst law of elliptical orbits,
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7.3 From Geocentrism to Heliocentrism(♯2 ) The second law of areal velocity,1619:(♯3 ) The third law of Periods:

And
1610: Galileo found the moons of the Jupiter by his telescope of the self-madeAnd further,
1633: Galileo said ”And yet, it moves” in the Trial of Galileo♠Note 7.6. In the next year of 1642 when Galileo died, Isaac Newton was born in the
British country.

1687: Newton ”Principia”
In this way, we think:(B) Somehow the air ”to Heliocentrism” has been formed.
Still, I am worried about this problem (A)”What is ’Geocentrism vs. Heliocentrism’ ?”That is,(C1 ) Did both Galileo and the church understand the essence of ’Geocentrism vs. Heliocentrism’ ?
which is equivalent to(C2 ) In order to win the deﬁnitive victory, what should they (Galileo or the church) havedone?
In order to answer to this question, we ﬁrst have to clarify the meaning of ”Heliocentrism
vs. Geocentrism”.

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Chap. 7 Early modern – Scientiﬁc revolution: From Geocentrism to Heliocentrism

7.3.3 ”Geocentrism vs. Heliocentrism” is the problem of the worldview
As mentioned in the previous section, how to decide ”Geocentrism vs. Heliocentrism” issomewhat diﬃcult. That is because(D) Thus, if the earth is assumed to be at center, the sun goes around the earth (i.e.,
Heliocentrism). Also, if the sun is assumed to be at center, the earth goes aroundthe sun (i.e., Geocentrism). Hence, The diﬀerence between Heliocentrism and Geocentrism is only a diﬀerence of how to take the coordinate system.
In the same sense, we say that(E) No matter how much there are exact observation data, we cannot decide”Geocentrism or Heliocentrism”
In the famous trial of Galileo, he said”And Yet It Moves”
However, I wonder if Galileo knew the (E)?
♠Note 7.7. No matter how much there are exact observation data, we cannot decide ”Geocentrism vs. Heliocentrism” we have to need the worldview. Namely,(♯1 ) it is a matter of course that there is no science without measurement
However, we believe that(♯2 ) there is no science without worldview
Thus, as seen later, we cannot decide ”Geocentrism vs. Heliocentrism” without worldview

7.3.4 The Galileo legend; Leaning Tower of Pisa, Trial of Galileo
The worldview of Aristotle has kept its position for 1500 years such as (cf. Sec. 4.2.1):
the birth of worldview•

Movement(Hērakleitos, Parmenides, Zeno,
Aristotle, Ptolemaios)

the birth of modern science”purpose” (and no experiment)−−−−−−−−−−−−−−−−−−−−−−−−→ Causality (and experiment)
Aristotle’s spirit :(About 1500 years)( Galileo, Bacon, Descartes, Newton)

Thus, Aristotle’s worldview is not a so bad worldview.
But, this worldview was a little inconvenient to organize the data, obtained by technological innovation (e.g., telescope, navigation, etc.). The history of the increase of the195

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7.3 From Geocentrism to Heliocentrism
inconvenience is as follow:







Copernicus ⇒ Kepler ⇒ Galileo

However, these are not suﬃcient to decide ”Geocentrism vs. Heliocentrism”. It is a matterof course that there were excellent persons in the church. And they might think:• if they insisted that motion is relative, they did not lose the dispute, at least, they
could make ”Geocentrism vs. Heliocentrism” endless dispute.
Galileo legend Galileo (1564 - 1642) was an active leader of the overthrow of theworldview of Aristotle, and his targets were the following (F1 ) and (F2 ):(F1 ) Ptolemaic Geocentrism(F2 ) Aristotelian purpose such as ”Heavy objects fall faster”
Concerning the two, We have two episodes called ”Galileo legend” as follows.
For (F1 ), ”And Yet it moves” in trial of Galileo
For (F2 ), Leaning Tower of Pisa

Thus, Galileo might think it was a matter of measurement data, but the church might
think it was a matter of worldview. Therefore,• at the time (1633) of the trial of Galileo, the Church was convinced that it would
not lose its argument with Galileo and, at worst, could bring it into an endless
dispute.
An endless dispute implies the win of the church. The church is not so stupid.
No way, the church did not think that Newton would appear
No one would have predicted the appearance of Newton.

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After all, Galileo was the active leader of the overthrow of the worldview of Aristotle,
but he could not propose the new worldview. In that sense, the Galileo legend is just the
beginning of Newton’s appearance

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7.4 Principia; Newtonian worldview

7.4 Principia; Newtonian worldview
7.4.1 Principia (1687)”PhilosophiæNaturalis Principia Mathematica” (in short, ”Principia”) , written by Newton (1687), is the most famous and important book in science. Three laws of Kepler were
derived from three laws of dynamics and the law of universal gravitation. Principia waswritten based on elementary geometry and not the diﬀerential and integral calculus. Why
did Newton (= advocate of diﬀerential and integral calculus) not write Principia basedon diﬀerential and integral calculus? Although there may be several opinions for thisquestion, The work (based on diﬀerential and integral calculus) was succeeded by Leibniz,
J. Bernoulli, Euler, d’Alembert, Lagrange and Laplace, etc. and was completed.

7.4.2* What is ”Geocentrism vs. Heliocentrism”? After all, the worldviewism
The following biggest paradigm shift in the history of science is as follows.
(A) Motion  Motion function method: (Parmenides, Zeno, Aristotle, Ptolemaios) cf.
Sec. 2.4−−−−−−−−−→
paradigm shift

Causal relation



Kinetic diﬀerential equation method: (Newton)198

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Chap. 7 Early modern – Scientiﬁc revolution: From Geocentrism to Heliocentrism
That is, we see:(B): Paradigm shift [Newtonian mechanical worldview ]Now,(B) Aristotle’s worldview(purpose)−−−−−−−−−→
paradigm shift

Newtonian worldview(causal relation)

Here, Newtonian worldview means Newtonian mechanics, that is,
Newton’s kinetic equation (i.e., the chain of causal relations)+ the law of universal gravitation
Recall that the main theme of this text is the worldviewism(cf. Sec. 1.1). As mentioned
frequently up to this point,(C1 ) The argument in ordinary language (or, in the motion function method (cf.
Sec.2.3.4)) is fuzzy, and thus, ”Geocentrism vs. Heliocentrism” cannot be decided.
Thus, we need a new worldview.
In Principia, Newton proposed Newtonian mechanics (i.e., Newtonian worldview) andshowed that
(C2 ) When the motion of the sun and the planets is studied, the calculation becomes easy under the assumption that the planets go aroundthe sun.
Therefore, even the deﬁnitions ”center” and ”go around” depend on the worldview. Afterall, we conclude that(D) ”Geocentrism or Heliocentrism” is not an issue that can be settled, no matter how
accurate the observations are. That is, ”Geocentrism vs. Heliocentrism” is not the
problem of measurements, but the problem of the worldview.

♠Note 7.8. (i): There are many ways to understand the ”Geocentrism vs. Heliocentrism”.
Since the main theme of this text is ”Nothing can be said without a worldview,” I wrote
a ﬁction based on this theme.
(ii): Next is said to be the three major discoveries of modern science(♯1 ) Aristotelian worldview (purpose) −→ Newtonian worldview (causal relation)

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7.4 Principia; Newtonian worldview(♯2 ) Ptolemaic system: Geocentrism −→ Copernican system: Heliocentrism(♯3 ) Christianity: Adam and Eve −→ Darwin: evolution theory

However, it should be noted that (♯2 ) is a consequence of (♯1 ). There may be a reason toconsider that (♯2 ) is a great episode of the birth of (♯1 ).

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Chap. 7 Early modern – Scientiﬁc revolution: From Geocentrism to Heliocentrism

7.5 Appendix; About ”Dialogues concerning two new sciences”
by Galileo Galilei

The following issues are discussed• If you drop a heavy iron ball A and a light iron ball B at the same time, which one
will fall to the ground faster?
The only way to settle this issue would be to conduct an experiment. In fact, Galileo
conducted this experiment on the Leaning Tower of Pisa, and proved that ”two iron balls
fall at the same time”. This is the famous ”Leaning Tower of Pisa legend”. So far, so
good.

But, in ”Dialogues concerning two new sciences”, Galileo must have wanted to say something like this:• It is not necessary to experiment to know that two iron balls fall at the same time.
You don’t have to go to the Leaning Tower of Pisa to conclude that ”two iron balls
fall at the same time” if you think about it while lying in bed in your own home.
I will explain Galileo’s idea in the following.

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7.5 Appendix; About ”Dialogues concerning two new sciences” by Galileo Galilei(A):”Dialogues concerning two new sciences” by Galileo Galilei ref. [24]Here, assume that
(A1 ) The heavier iron ball A falls faster than the lighter ball B.
If so, at what speed will the iron ball [A+B], which is made up of a heavy iron ball
A and a light iron ball B connected by a string, fall?

Common sense tells me that
(A2 ) The iron ball [A+B] will fall at a speed halfway between the heavy iron ball
A and the light iron ball B
However, this contradicts (A1 ). Since the iron ball [A+B] is heavier than theheavier iron ball A, according to the assumption (A1 ), the iron ball [A+B] should
fall faster than the heavier iron ball A. The assumption (A1 ) is wrong because it isstrange that the falling speed increases only by connecting with a string. Assumingthat the lighter iron ball falls faster than the heavier one at (A1 ) is a contradiction
as well. Therefore,(A3 ) without experiment ( and without Newtonian mechanics) , we can conclude a heavy iron ball and a light iron ball will fall at the same time.
If it is true, was the ”Leaning Tower of Pisa” experiment unnecessary?
The above discussion seems both strange and obvious. I would like the reader to consider
this exercise.
♠Note 7.9. For convenience, consider two kinds of propositions such as(B1 ) analytic propositions: propositions grounded in meanings, independent of matters offact.
(B2 ) synthetic propositions: propositions grounded in fact.
If so, we consider:(C1 ) Is the proposition ”A heavy ball and a light ball fall at the same speed” an analytic
proposition or a synthetic proposition?
(C2 ) Does (A3 ) imply that the problem (C1 ) is a analytic proposition without Newtonina
mechanics?
I don’t have a deﬁnitive answer to this question, but I can say the following in terms ofquantum language:(D) all proposition in QL are synthetic (Sec. 12.3).

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Chap. 7 Early modern – Scientiﬁc revolution: From Geocentrism to Heliocentrism
In addition, the problem (A) raises the following issue(E) There may be a worldview of common sense (or the worldview called common sense).
(Sec. 12.2.3: Wittgenstein’s paradox)203

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205

Chapter 8Modern philosophy: Continental
rationalism: Descartes, Spinoza, Leibniz
The genealogy of dualistic idealism is as follows

Scholastic philosophy was ”philosophy for the priests.” Descartes opened the door tomodern philosophy by proposing a ”philosophy for the general public. He raised two
problems: the mind-body problem and the subjectivity problem (the so-called Descartes
problem (=Cartesian problem)).

Descartes problem = mind-body problem and subjectivity
problemHere,(♯1 ) mind-body problem: How are the mind and body connected?
(♯2 ) subjectivity problem: Is the world I see the same world you see?

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In summary:(♯3 )mind-body problem (Continental rationalism)(classical) Descartes problem
(Establish mind-body dualism!)  subjecyive problem· · · How are the mind and body related?
· · · Is the world I see the same world you see?

(British empiricism)

Some readers may ask, ”Why is the Cartesian problem important?” or ”Is it a philosophical problem?”. Therefore, to state the conclusion ﬁrst, the Cartesian problem is thesame as the following ”(modern) Cartesian problem”.
(♯4 )
mind-body problem (Continental rationalism)(modern) Descartes problem
(Establish mind-body dualism!)  subjectivity problem(British Empiricism)(Establish QL!)···

Find Axioms including
the key-words (mind, body, matter)· · · Find the interpretation
concerning this Axioms”Why are (♯3 ) and (♯4 ) the same?” will be explained in this section.

Modern philosophy (Descartes-Kantian philosophy) enjoyed 300 years of prosperity. I think
the reason is as follows.
(♭1 ) The Cartesian problem (♯3 ) was presented using plain language [(♯1 ) and (♯2 )] that thegeneral public could understand(♭2 ) As a matter of consequence, Plato’s Idea is a magic word such as[Idea]−−−−−−→ [universal]−−−−−−→ [body]progress
progress−−−−−−→ [secondary quality] −−−−−−→[observable]
progress
progress(♭3 ) The general public expected Descartes-Kant philosophy to be a counterweight (i.e., rival)
to Newtonian mechanics
The often stated opinion that ”epistemology could have been done without” is an opinion worth
listening to, but we ﬁrmly disagree with it.
In this chapter, I will explain Descartes philosophy (i.e., mind-body dualism) and present mysolution to Descartes problem. In addition, I will discuss continental rationalism (i.e., Spinoza
and Leibniz).

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Chap. 8 Modern philosophy: Continental rationalism: Descartes, Spinoza, Leibniz

8.1 Popularization of philosophy; Descartes=Kant philosophy
8.1.1 Background of Descartes=Kant philosophy
The historical background of Descartes=Kant philosophy is as follows.

Table 8.1. Scientiﬁc revolution; Chronological table
Historical Background of ”Discourse on the Method”: R. Descartes• Scholastic philosophy, Thomas Aquinas (1225 - 1274), Ockham (1285 - 1347) ), The
Black Death pandemic that struck Europe between 1347 and 1351 was the worst in
history, claiming the lives of one-third of the European population.
• 1450: Gutenberg’s printing press
inventor• 1453:The Eastern Roman Empire extinction (Constantinople surrender)• 1492: Columbus, discovery of the American Continental
navigator• 1498: Vasco da Gama, discovery of the sea route to India
navigator
artist• 1500s: Leonardo da Vinci, ”Mona Lisa’s smile”• 1510: Copernicus, Heliocentrism
scientist• 1510: Raﬀaello, ”The School of Athens”, Admiration to ancient Greece
artist• 1517: Luther, Protestant Reformation
religionist• 1519 - 20: Magellan, the ﬁrst circumnavigation of the Earth• 1540s: Michelangelo, ”The Last Judgment”
artist• 1600: G. Bruno, who supported heliocentrism along with Galileo, was burned at the
stake by the Vaticanphilosopher• 1609∼1619: Kepler’s laws of planetary motion
scientist• 1610: Galileo, A telescope was made and moons of Jupiter were foundscientist
illuminator• 1620: F. Bacon, ”knowledge is power”,The father of British Empiricism• 1633 Galileo’s trial ”And yet it moves”
scientist• 1637:”Discourse on the Method”, Rene Descartes (1596 -

1650),
the father of modern philosophy, Cogito proposition
philosopher• 1670: Pascal, ”Pensèes”
enlightener• 1677: Spinoza, ”Ethica”
enlightener• 1685 - 1750: Bachartist• 1687: Newton, ”Principia”
scientist• 1688: Glorious Revolution• 1690: John Locke, the father of British Empiricism, ”An Essay Concerning
Human Understanding”, tabula-rasa, the secondary quality
philosopher• around 1700: Jakob Bernoulli, the law of large numbers
mathematician
philosopher• 1703: Leibniz, ”New Essays on Human Understanding”• 1715 - 16: Leibniz-Clarke correspondence (cf. Sec. 9.2.2)
philosopher• 1739: Hume, ”A Treatise of Human Nature”• 1781: Kant, ”Critique of Pure Reason”
philosopher• 1788: Lagrange, Mécanique analytiquescientists• 1789: Lavoisier, Elementary Treatise on Chemistryscientists• 1799-1825: Laplace, Traité de mécanique célestescientists

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8.1 Popularization of philosophy; Descartes=Kant philosophy

8.1.2 Philosophy for the general public (≈ literary philosophy≈ science without
formulas≈ Copenhagen interpretation)
The persons Scholastic philosophers talk about philosophy are priests. However, in modern
ages (i.e., since the invention of the Gutenberg printing press), the persons philosophers
talk about philosophy are the general public. In this sense, modern philosophy may be
a kind of literature or enlightenment philosophy. Descartes was the ﬁrst to talk about
philosophy to the general public.
Descartes’s ”Discourse on the Method” was the ﬁrst philosophical book written in
French, whereas philosophical works at the time were usually written in Latin. It was
written in plain language for the general public. Thus I think that
(A1 ) Descartes-Kant philosophy was a philosophy for the general publicThat is, I think that Descartes-Kant philosophy is a type of enlightenment philosophy.
As can be seen in Table 8.1, at the beginning of the scientiﬁc revolution, Descartes andKant philosophy was born to educate the general public on how to deal with science.
On the other hand, in Chap. 6, I asserted the following.
(A2 ) Scholastic philosophy was a philosophy for professionals.

As shown throughout this text, the diﬀerence between ”philosophy for the general public” and ”philosophy for professionals” often caused confusion in Western philosophy. For
example, the philosophy of science was originally supposed to be a ”philosophy for professionals (i.e., scientists),” but historically it has been presented as a ”philosophy for thegeneral public.” Because of this, the philosophy of science was not accepted by scientists.
( See ref. [78], in which ”philosophy of science for scientists” is written.)

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Chap. 8 Modern philosophy: Continental rationalism: Descartes, Spinoza, Leibniz
It is well known that quantum mechanics is formulated as follows.

(B)

axiomatic theory
empirical theory
Axiom 1 (measurement)
quantum mechanics
Copenhagen=
Axiom 2 (causality)(dualism)+ interpretation
narrative stylemath styleThat is, there are two main components: the axioms (laws) and the Copenhagen Interpretation (a manual on how to use the axioms).
Therefore, if Descartes-Kant philosophy and quantum mechanics are related, and if
Descartes-Kant philosophy is philosophy for the general public, then,• Descartes-Kant philosophy must resemble the Copenhagen interpretation(since Axioms 1 and 2 are too mathematical).
Also, recall thataxiomatic theory
Axiom (causality)
Newtonian mechanics(C)(monism)= Newtonian equation
math styleempirical theory+

No interpretation(or obvious interpretaion)
narrative styleThus, Newtonian mechanics has no ”interpretation” (or if it does, it is an obvious interpretation). This suggests that a philosophy founded on Newtonian mechanics is not very
promising. In fact, nominalism (by Abelard, Ockham, and others) has not produced afruitful philosophy. Statistics (= dynamical system theory), which is a mathematical generalization of Newtonian mechanics, has been regarded as a kind of applied mathematics
rather than philosophy. In other words, a ”theory without interpretation,” such as (C)above, is hardly a philosophy for the general public. That is,(D) Monism is hardly a philosophy.
The above can be summarized as follows.
Philosophy for Professionals· · · Axiomatic theoryThe problem of universals, Continental Rationalism, Statistics,





Philosophy of Science (Hempel, etc.), QL (Axiomatic theory)








(A great scientiﬁc success has been statistics, but it is not generally
regarded as a philosophy since it does not belong to dualism)
(E)




Philosophy for the Masses· · · Empirical theoryDescartes-Kant
Epistemology,
Philosophy of Science (Popper, etc.)





Philosophy of Mind, QL (Copenhagen Interpretation)







(Gaining the support of the masses tends to be considered a success)
The caveat above is that the diﬀerence between professionals and amateurs is not the levelof research. One can think of it as ”professional=mathematics required (especially Axioms
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8.1 Popularization of philosophy; Descartes=Kant philosophy
1 and 2)” and ”amateur=explanation of rules that cannot be expressed in mathematicalformulas (Copenhagen Interpretation)”. For example, although not mentioned much inthis text, various examples discussed in the philosophy of mind (e.g., the qualia problem
and the brain in a vat (Sec. 9.4)) are very useful for forming Copenhagen interpretations.
♠Note 8.1. Many readers will ask the following question.
(♯1 ) While quantum mechanics has an ”interpretation (i.e., the Copenhagen interpretation),” classical mechanics and the theory of relativity do not have ”interpretations”.
Why is this?
(♯2 ) If the Copenhagen Interpretation is a ”manual on how to use axioms”, is there sucha thing as a complete manual? (cf. Sec. 12.2.3 Wittgenstein’s paradox).
These questions are one of the motivations for my interest in the study of the Copenhagen
interpretation. I do not yet know the exact answer to these questions, but so far I believe
that(♭1 ) All theories, except mathematics, possess some ”interpretation” of some sort. Thus,
both classical mechanics and relativity have ”interpretation”. However, since they are”common sense and self-evident interpretation,” there was no particular inconveniencein not being aware of the ”interpretation”.
(♭2 ) I believe that there is no perfect interpretation. I believe that all theories, not only
quantum mechanics (except mathematics), cannot be perfect (cf. Sec. 12.2.3 Wittgenstein’s paradox).
I don’t quite understand the ”Wittgenstein paradox”, but this paradox is related to theabove.

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Chap. 8 Modern philosophy: Continental rationalism: Descartes, Spinoza, Leibniz

8.2 The cogito proposition ”I think, therefore I am” is acatchphrase of modern philosophy
8.2.1 Discourse on the Method (1637)

Descartes(1596-1650) was a French philosopher, mathematician, and scientist. He is also
widely regarded as one of the founders of modern philosophy.
♠Note 8.2. Descartes’ well-known mathematical achievement is the discovery of Cartesian
coordinates or analytic geometry. There is even a plausible legend that Descartes, who wassleeping in his bed, came up with the system in order to track the position of a ﬂy movingaround the ceiling. However, Cartesian coordinates have nothing to do with Cartesianphilosophy.

y 6(2, 1)

1O

2x[Cartesian coordinates; parameterization ]Let us study the most famous book in philosophy: ”Discourse on the Method (1637)”
by René Descartes, which is the abbreviation of ”The method of rightly conducting one’s
reason and of seeking truth in the sciences”. Probably after Descartes read Bacon’s”Novum organum”, he decided ”Start from the unquestionable truth”. And he found themost famous philosophical proposition (= cogito proposition):”I think, therefore I am”
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8.2 The cogito proposition ”I think, therefore I am” is a catchphrase of modern
philosophy
That is, Descartes think:I think that ’I think, therefore I am’
In spite that this is a most doubtful proposition (and thus, it belongs to pseudoscienceand not in quantum language), Descartes believed that it was the unquestionable truth.
And further, bearing in mind Euclid’s success in geometry, Descartes was convinced
that(A) Every statement derived from the cogito proposition are absolutely trusted (despite
the fact that there is nothing that can be derived from the cogito proposition)
This is just ”The method of rightly conducting one’s reason and of seeking truth in thesciences”. As I have said many times, Descartes issued his philosophy to the generalpublic, and I believe he used forced logic as a technique to communicate his philosophy.
Summing up, we see:
Proclaim 8.2. The ﬁrst principle (= cogito proposition) in philosophy
Descartes doubted everything. And he arrived in the cogito proposition which has no
doubted room. That is, he arrived in(B1 ) ”I think, therefore I am.”And, he proclaimed that the cogito proposition (B1 ) is the ﬁrst principle in philosophy,
and asserted:(B2 ) ”Everything deduced from the cogito proposition is absolutely correct.”♠Memo 8.1. This was enough because Descartes aimed for a philosophy for the generalpublic. If Descartes aimed at a philosophy for philosophers, he had to start with thequestion ”What is a proposition?” ( See Chap. 11). And since the cogito proposition is
not a (scientiﬁc) proposition, (B1 ) and (B2 ) are absurd. However, as we will see in thenext section (Sec. 8.3), a miracle occurs. It is not clear whether all of this is Descartes’
calculation or not, but I think that Descartes was convinced that ”mind-body dualism”
was the way forward for philosophy.

8.2.2* Regrettably, the cogito proposition is not a (scientiﬁc) proposition.
What is described in this section has been mentioned many times in the references[46]∼[76].
Recall the linguistic Copenhagen interpretation:”No observer can measure himself.”

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Chap. 8 Modern philosophy: Continental rationalism: Descartes, Spinoza, Leibniz

We will show two (scientiﬁc or non-scientiﬁc) interpretations of the cogito proposition: ”I
think, therefore I am”.
Lemma 8.3. [(i): Scientiﬁc interpretation of the cogito proposition]:
Let start from the scientiﬁc interpretation of the cogito proposition.
For the sake of convenience, put ”I”=”Tom”. Thus,• ”I think, therefore I am”= ”Tom thinks, therefore Tom is”
If this is a scientiﬁc proposition, it must be experimentally veriﬁable. However, it iseasy. That is because it is usual that
(C1 ) a doctor says ”Tom’s brainwaves are normal, so he’s alive.”
which is of course a statement in quantum language, since”observer = doctor”,”matter = Tom”.

However, the interpretation (C1 ) is a trivial one, and this would not be Descartes’
intention.
Proposition 8.4. [(ii): Non-scientiﬁc interpretation (i.e., Descartes’ intention)of the cogito proposition]:
Similarly, put ”I”=”Tom”. Put(♭) ”I think, therefore I am”= ”Tom thinks, therefore Tom is”
Thus, we consider the situation (which is the same as Descartes’s intension) :(C2 ) Tom thinks ”Tom thinks, therefore Tom is”
which means that”observer = Tom”,”matter = Tom”,

That is, this violates the linguistic Copenhagen interpretation: ”No observer can measure
himself.” Therefore, the (C2 ) is not in quantum language, that is, it is just a play onwords.
Thus, the (♭) is a non-scientiﬁc proposition (i.e., an experimentally unveriﬁable proposition), but this play on words had captured the interest of philosophy enthusiasts. Thatis,213

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8.2 The cogito proposition ”I think, therefore I am” is a catchphrase of modern
philosophy
This play on words opened the door to modern philosophy.

♠Note 8.3. As concluded in chapter 12, ”I think, therefore I am” is not a proposition ( precisely, a proposition in QL). in spite that it is called ”the ﬁrst proposition of philosophy”.
Using Wittgenstein’s words, we says that
”I think, therefore I am”=”non-proposition” = ”what we cannot speak about”
Therefore, Wittgenstein’s famous words (i.e., ”What we cannot speak about we must passover in silence”) may have been invoked in the Descartes-Kant philosophy.

The catchphrase ”I think, therefore I am” may be unscientiﬁc, but it does not mean that
Descartes’ philosophy is unscientiﬁc. On the contrary, Descartes’ philosophy is a great
achievement (comparable to Newtonian mechanics in physics (monism)).
Descartes opened the door to modern philosophy.
We will discuss this below.

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Chap. 8 Modern philosophy: Continental rationalism: Descartes, Spinoza, Leibniz

8.3 Descartes’ genius strategy
8.3.1 From the cogito proposition to mind-body dualism
The most important key-word in Descartes’ philosophy is ”I”. However, Descartes thought
that
Nobody pays attention even if Descartes appeals for the importance of ”I” aloud.
Thus,(A) Descartes used the advertising slogan (i.e., the cogito proposition): ”I think, therefore I am”
The cogito proposition is nonsensical, but it is very impressive. Thus, this proposition
could be a perfect advertising slogan.
non-scientiﬁc”exist”=keyword

derivation

Note 1.2

cogito proposition −−−−−−−−−→ the existence of ”I” −−−−−−−−−−−→ ”I” as keyword
Thus, by the cogito proposition, what Descartes wanted to say was(B) ”I” is the most important key-word in Descartes philosophy.
His strategy succeeded wonderfully. If ”I” is accepted, the existence of ”matter” (whichis perceived by ”I”) is accepted. And further, the medium of ”I” and ”matter” is automatically accepted as ”body (= sensory organ)”. Therefore, the key-words of Descartes
philosophy (= mind-body dualism = mind-body·matter dualism) is(C)”I”(=”mind”, ”brain”, ”observer”) ,|{z}
spiritual dynamics”body”(=”sensory organ”) ,|{z
physical dynamics”matter”}

in this sense, we usually assume that ”mind-body dualism” = ”mind-body·matter dualism” =”mind-matter dualism”.
Again, it should be noted that this is not a consequence of the cogito proposition. Thatis the cogito proposition is the reason added afterwards. The cogito proposition is nothing
more than a catchphrase. This is the greatest catchphrase in human history.
Hence, I think215
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8.3 Descartes’ genius strategy• ”cogito proposition” does not matter as long as Descartes can insist on ”mind-body
dualism”.
♠Note 8.4. (i): Descartes derived ”existence of I” from ”cogito proposition” as the ﬁrst
proposition of philosophy, and further derived• ”existence of things”, ”existence of body”, ”existence of causality” and ”existence ofGod”.
It can only be said that it is a ”miracle of Western philosophy” that such an unreasonable
story led to quantum language.
(ii): The next gap is quite big.
[I] −→ [observer], [body] −→ [sensory organ] [mind-body dualism] −→ [epistemology]
This may be due primarily to John Locke (in Sec 9.1)
(iii): Note that
• For example, assume that an apple is put on food scale. We are not sure if he istrying to weigh an apple or make sure the food scale isn’t broken (see Figure below).
Thus, the diﬀerence between ”body” and ”matter” is arbitrary. This is determinedby the observer. Both ”body” and ”matter” obey the laws of physics. However, thelaws governing ”mind”(i,e., spiritual dynamics) are assumed to be unknown.

////

Principle 8.5. The principle of Descartes’ philosophy is(D) Think of everything in terms of mind-body dualism!
In other words, Descartes proposed a worldview of mind-body dualism that takes thethree key words ”mind, body, and matter” as its starting point.

This text claims that ”the scientiﬁc realization of Descartes mind-body dualism is quantum language.

(♯)

axiomatic theory
empirical theory
Axiom 1 (measurement)
quantum language
Copenhagen+ interpretation(dualism)=
Axiom 2 (causality)
narrative stylemath style216

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Chap. 8 Modern philosophy: Continental rationalism: Descartes, Spinoza, LeibnizzAxiom 1 [measurement]}|”I”(=”mind”, ”brain”, ”observer”) ,|{z}
spiritual dynamics{”body”(=”sensory organ”) ,”matter”|{z}physical dynamics (Axiom 2 [causality])217

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8.3 Descartes’ genius strategy

8.3.2 Descartes problem (= mind-body problem + subjectivity problem
And, Descartes proposed the following two problems;
Problem 8.6. [Descartes problem (mind-body problem, subjectivity

problem)]Raising the following two issues is Descartes’s greatest task in philosophy.
(E1 ) mind-body problem:
How are ”mind” and ”body” connected?

(E2 ) subjectivity problem (= qualia problem):Is the world I perceive the same as the world you perceive?
Or more precisely,
How do we use mind-body dualism to solve the subjectivity problem (and problems
mentioned in List (D1 ) in Preface)?

In summary, the following is a summary.
(E3 )mind-body problem (Continental rationalism)(classical) Descartes problem
(Establish mind-body dualism!)  subjective problem− How are the mind and body related?

− Is the world I see the same world you see?

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Chap. 8 Modern philosophy: Continental rationalism: Descartes, Spinoza, Leibniz♠Note 8.5. The above two problems have been touted as the two most diﬃcult problems inmodern philosophy. Without raising Descartes’ issue, the prosperity of modern philosophy
might not have been realized. In fact, Kant, Husserl, and others challenged these issues,and their results were supported by the general public. Although their approach is rather
pseudo-psychological (i.e., pseudo-scientiﬁc), they are respected as great philosophers.
However, no scientist thinks that their achievements have ”guaranteed the objectivityof natural science.” (However, as will be mentioned in Chap. 10, Kant’s transcendental
idealism is the gold standard of modern philosophy.)

No scientist would interpret the (original) Descartes problem in the literal sense. Of
course, another way to describe the Descartes problem is as follows.

(E4 ) Descartes problem is as follows.
(♯) Establish mind-body dualism so that the mind-body andsubjectivity problems can be solved!
Answer 8.7. [The solutions of the above Descartes’ problems]: Now,
if we agree with (E4 ), then Descartes problem has already been solved (in terms of thistext) and we have(E5 )
mind-body problem (Continental rationalism)(modern) Descartes problem
(Establish mind-body dualism!)  subjectivity problem(British Empiricism)(Establish QL!)···

Find Axioms including
the key-words (mind, body, matter)· · · Propose the interpretation
concerning this Axioms

In short, with the discovery of the ”quantum language” the (modern) Descartes
problem (E5 ) is solved. A few additional explanations are provided below.
That is,(F1 ) mind-body problem:[How do mind, body, and matter relate?]
This answer has already been answered in Axiom 1. Because(♯1 ) Axiom 1 explains the quantitative relation among measured value(≈mind),observable(≈body), state(≈matter)
Also, see (♭) in Principle 8.5). The mind-body problem is said to be the biggest
problem in philosophy, and although it may be persistent, Sec. 12.8 will also
discuss it in detail.
(F2 ) subjectivity problem (= qualia problem):[Is the world as I perceive it the same as the world as you perceive it?]This problem is called the Qualia problem, but it is not the only one, and ourgoal was to create a ”Copenhagen Interpretation” so that the various unsolved
problems described in the ”Foreword to (D1 )” could be solved all at once.
(♯2 ) Our goal was to ”create a Copenhagen interpretation,” and we answered thisin Sec. 1.3 and throughout this text.
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8.3 Descartes’ genius strategy
Therefore, from the quantum language standpoint, (F2 ) has already been solved.
Putting together (F1 ) and (F2 ), we see;quantum language(dualism)=axiomatic theory
Axiom 1 (measurement)
Axiom 2 (causality)

empirical theory
linguistic Copenhagen+
interpretation
narrative
style;
subjectivity problem
math style; mind-body problem

Thus, we say that

The solution to the Cartesian problem was made possible byquantum language
Throughout this text, the above answers will be explained precisely.
♠Note 8.6. (i): I believe that no matter how lofty a theory may be, it is meaningless if it
does not solve a variety of problems. What I have done is to rewrite the ”Copenhagen
Interpretation” to solve the problems mentioned in List (D 1 ) in Preface.
(ii): Many philosophers may not be immediately convinced because our approach is not
traditional. However, I am sure that our approach is truly philosophical and scientiﬁc.
That is, I think that philosophers should not regard Problem 8.6 as the problems in psychology and brain science. I believe that Kant, Husserl, and others have become immersedin pseudoscience. Even so, their considerations have facilitated our understanding of the”Copenhagen Interpretation.”(iii):As mentioned in Note 1.6, in Mermin’s book [101], he proposed the excellent and interesting explanation of ”Copenhagen interpretation” as follows.
(♯1 ) If I were forced to sum up in one sentence what the Copenhagen interpretation says
to me, it would be “Shut up and calculate”(♯2 ) Stop being bothered!
If it were a quantum mechanics course at a university, these (♯1 ) and (♯2 ) would be appropriate advice. However, in quantum language, I think [The linguistic Copenhagen
interpretation (E0 )-(E7 ) in Sec. 1.3] is necessary. I believe that the linguistic Copenhagen
interpretation [(E0 ),... ,(E7 )] is the ”true Copenhagen Interpretation”.

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Chap. 8 Modern philosophy: Continental rationalism: Descartes, Spinoza, Leibniz

8.3.3 Two Geniuses: Thomas (Theory of Heaven) to Descartes (Theory of
Earth)
As noted in Sec. 6.6.1, Thomas Aquinas asserted ”Theory of Heaven” as follows.

Theory of Heaven( by Thomas )(C1 ) :universals exist’after individuals’ in the human intellect .
universalia post rem as actual world
QL: (i.e., measured value is sensed by the brain (i.e., mind)).
(C2 ) :universals exist ’before
 individuals’ in the divine intellect, .
universalia ante res as Plato’s Idea
QL: (i.e., observable (or, measuring instrument) could be in the heavens, like theNorth Star (cf. Note 1.7))(C3 ) :universals exist ’in individuals’
in the world, .

universalia in rebus as Aristotle’s eidos
QL: (i.e., state exists in matter)

It was Descartes who rewrote this to ”Theory of Earth”.

Theory of Earth (by Descartes)
Solve the following Cartesian problem in mind-body dualism, which consists of three
key words:mind, body, and matter!
(E3 ) in Sec. 8.3.2

mind-body problem (Continental rationalism)(classical) Descartes problem
(Establish mind-body dualism!)  subjective problem− How are the mind and body related?

− Is the world I see the same world you see?

(British empiricism)In the same sense:(E5 ) in Sec. 8.3.2
mind-body problem (Continental rationalism)(modern) Descartes problem
(Establish mind-body dualism!)  subjectivity problem(British Empiricism)(Establish QL!)···

Find Axioms including
the key-words (mind, body, matter)· · · Propose the interpretation
concerning this Axioms

These two might be greater than Kant!

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8.4 Continental Rationalism; Descartes, Spinoza, Leibniz

8.4 Continental Rationalism; Descartes, Spinoza, Leibniz
Inspired by Descartes’s mind-body dualism, several philosophers (Spinoza, Leibniz, etc.)
sought to further develop mind-body dualism. Their philosophy was called Continental
Rationalism. Therefore, they devoted themselves to the mind-body problem:(A) How are ”mind” and ”body” connected?

That is,(B) They wanted to ﬁnd a principle for the relationship between ”mind” and ”body(matter)”just as Newton discovered a principle for the motion of ”matter”.
However, as one can easily surmise, their challenge was not successful as a science.
A scientiﬁc failure does not mean a philosophical failure, but from the standpoint of
this text, we cannot devote a page to it.
However, their attempt to break out of the shell of dualism left a mark on the historyof Western philosophy, so I will mention a few things.

8.4.1 Spinoza: Pantheism

Baruch Spinoza (1632-1677) was a Dutch philosopher, who was one of the foremost
exponents of 17th-century Rationalism. In his work ”Ethica”, he claimed to have improvedon Descartes’ mind-body dualism as follows.









monism−−−−−−−−−−−−−−−→ God(Spinoza’s synthesis)










(due to Descartes) 

mind

(C)

(due to Newton)

body
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Chap. 8 Modern philosophy: Continental rationalism: Descartes, Spinoza, Leibniz
That is, Spinoza combined the two entities (”mind” and ”body/matter”) into one entity
called ”God”. Namely, he proposed ”God monism” (i.e., pantheism).
Since the interest of this text is the ”scientiﬁc part of Western philosophy,” it must be
an unfavorable assessment for Spinoza.
However, the spirit with which he challenged the great attempt to unify the ”mind-bodyproblem,” ”God,” and ”ethics” was favorably received by many philosophical enthusiasts.
♠Note 8.7. This text claimed the following(♯1 ) The logic is determined by the worldview. For example, mathematics has a mathematical logic, quantum language has a quantum language logic (cf. Sec. 12.2).
Now, if it is a philosophy that is more akin to literature or religion, ”Is it logical or not?”is not often pursued. However, since ”Ethica” was written by Spinoza with the intentionof ”logical reﬂection,” the question of ”logical or not? is often debated. And it is often
pointed out that it is ”absurdly logical” from a modern point of view. I think this pointis probably right on the mark. And I think the main way to read ”Ethica” is to enjoy its”absurd logic” from a literary point of view. Of course, this kind of literary enjoyment
is acceptable in philosophy. However, from the standpoint of this text, I would like toconsider the following possibilities.
(♯2 ) Pantheism has the logic of pantheism.
In other words, it would be interesting if we could discover a ”logic of pantheism” diﬀerent
from the ”logic of mathematics” or the ”logic of quantum language”. Quantum language isa quantiﬁcation and verbalization of Cartesian philosophy (cf. Answer 8.7), so pantheism(and monads) may also be quantiﬁable and verbalizable. Descartes philosophy and pantheism are equally dubious from a scientiﬁc point of view as monads are from a scientiﬁcpoint of view. And yet, since quantum language emerged from Cartesian philosophy, I
expect something scientiﬁc to emerge from both pantheism and monads. I hope that arival to quantum language will emerge.

8.4.2 Leibniz: Monadology

Gottfried Wilhelm Leibniz (1646-1716) was one of the great thinkers of the seventeenth
and eighteenth centuries and is known as the last ”universal genius”. He made deep andimportant contributions to the ﬁelds of metaphysics, epistemology, logic, philosophy of religion, as well as mathematics, physics, geology, jurisprudence, and history. In particular,
he discovered Calculus at the same time as Newton.
Leibniz claimed to have improved on Descartes’ mind-body dualism as follows.
(D) The true elements that make up the world are the monads, which are mutuallyindependent, indivisible entities, and the source of all vital activity. Every monadis diﬀerent from each other.
In this sense, it is pluralism. I believe that the success of Newtonian mechanics inspired
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8.4 Continental Rationalism; Descartes, Spinoza, Leibniz
Leibniz. I think Leibniz aimed at the summit of both the material and spiritual worlds.
The genius Leibniz must have thought so. However, since the interest of this text is ”thescientiﬁc part of Western philosophy as seen from modern science (quantum language)”,I have to give Leibniz an unfavorable evaluation. An evaluation from the perspective of
science 100 years later might be the opposite of my opinion.
However, Leibniz’s space-time theory in the Leibniz-Clarke correspondence (in Sec.
9.2) is one of the most important achievements of the Copenhagen interpretation, and his
genius is on full display.
♠Note 8.8. In the book ”The astonishing hypothesis” (by F. (the most noted for beinga co-discoverer of the structure of the DNA molecule in 1953 with James Watson)), Dr.
Crick said that(♯1 ) You, your joys and your sorrows, your memories and your ambitions, your sense of
personal identity and free will, are in fact no more than the behavior of a vast assemblyof nerve cells and their associated molecules.
That is, he believed the monistic realism such as(♯2 ) the movement of the human spirit is also a kind of physical phenomenon.
I agree to his opinion. And I believe that with the development of brain science, evenconsciousness can be understood in the future. However, it should be noted that Crick’s
claim and my claim (H) are not contradictory. Because Crick’s interest is brain science,on the other hand, our interest is philosophy (more precisely, the language of science). For
further discussion, see Sec. 12.8.

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225

Chapter 9Modern philosophy: British empiricism:
Locke, Leibnitz, Berkeley, Hume
Kant proposed a ”Kantian synthesis”*1 to answer the following question.
(♯1 ) Why are there two schools of thought, Continental rationalism and British empiricism?
(♯2 ) Are these two schools of thought to be merged?
However, ”Kantian synthesis” is pseudoscientiﬁc and cannot be adopted from the standpoint of
this text. Our assertion is ”QL synthesis (integrated by quantum language)” as follows. *2







Develop Descartes theory Leibniz
Spinoza(=mind-body dualism)Monadology
Pantheism



, •
•



Solve






the mind-body problem
• −−−−−−−−−−−−−−−−−−−−−→

Axiomatic theory

QL





WittgensteinContinental
rationalism


Axiom
1

Descartes theory





TLP


•
,
Axiom
2



mind-body dualism 
Derived from cogito















How
to
use
Locke,
Berkley,
QLDescartes theory
epistemology
linguistic




Solve
the
subjectivity(Copernican
Copenhagen
problem,
etc.





• −−−−−−−−−−−−−−−−−→
revolution)≈ interpretationEmpirical theory
Hume, Kant

British empiricism




QL

Figure 9.1: QL synthesis
The main purpose of chapters 9 and 10 is to explain this.

*1 See Sec. 9.1.3*2 This was, for the ﬁrst time, proposed in ref.

[84]: S. Ishikawa, (2022) Empiricism, Rationalism,
and Kantian synthesis in quantum linguistic point of view OJPP, Vol. 12, No. 2, 182-198,(https://www.scirp.org/journal/paperinformation.aspx?paperid=117114)

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In the previous chapter, we concentrated on Continental Rationalism (= Descartes, Spinoza,
Leibniz), especially Descartes’ mind-body dualism. Then, I showed that the Cartesian problem(i.e., the mind-body problem and the subjectivity problem) can be solved if we assume that
Descartes’ mind-body dualism evolves and moves toward quantum language. We also showedthat the works of Spinoza and Leibniz are not scientiﬁcally successful (i.e., their theories do notresemble Axiom 1 (measurement) at all).
In this and the next chapters, we will discuss British empiricism (Locke, Berkeley, Hume, andKant) and show that their theories (Descartes-Kant epistemology) were scientiﬁcally successful
and had a great inﬂuence on the Copenhagen interpretation of quantum language. The following
ﬁgure summarizes the results.

(However, attempts at Axiomatic theory were too diﬃcult at the time.)

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Chap. 9 Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, Hume9.1 Locke (1632 - 1704): The father of British Empiricism
9.1.1 ”An Essay Concerning Human Understanding” by Locke (1689)
In England, there has been a tradition of empiricism and inductive reasoning since F. Bacon(founder of empiricism, 1561-1626). John Locke (1632 - 1704) attempted to view Descartes’ work
(mind-body dualism) empirically and inductively.

John Locke was called the father of British empiricism and systematized empirical epistemology
in his main book, An Essay Concerning Humanity.
(A) :”An Essay Concerning Human Understanding” by Locke (1689)
Locke is the heir to Descartes philosophy. He clariﬁed the most diﬃcult to understand”body” in Descartes mind-body dualism. Since Isaac Newton published his ”Principia(=
Mathematical Principles of Natural Philosophy (1686))”, John Locke humbly wrote in his”Essay Concerning Human Understanding (1689)” that.
(B) ... and in an age that produces such masters as the great Huygenius andthe incomparable Mr. Newton , with some others of that strain, it is ambition
enough to be employed as an under-labourer in clearing the ground a little, ...

♠Note 9.1. I think that ”the incomparable Mr. Newton” is the most important key-wordin the whole of Descartes-Kant philosophy (i.e., in British empiricism as well as continental rationalism). I think almost all philosophers (or, the general public) at the time
wanted someone to tell them that philosophy is as great as Newtonian mechanics (or, morehonestly, philosophy is greater than Newtonian mechanics). And it was Kant who said
it. It is through this process that the general public learned to deal with science. In thissense, Cartesian Kantian philosophy is an Enlightenment philosophy that communicatesto the general public the coming of the scientiﬁc revolution. Thus,(C1 ) even if Descartes-Kant philosophy is a pseudoscience about cognition, its importanceremains constant.
(C2 ) also note that the scientiﬁc revolution that began 500 years ago is still ongoing. (cf.
Note 3.4).
With the advent of Newtonian mechanics, the role of the scientiﬁc part of philosophy
threatened to disappear. For example, I think the nominalists of Scholastic philosophy(Ockham’s successors) could have been more inﬂuential than Descartes. However, while

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9.1 Locke (1632 - 1704): The father of British Empiricism
there may have been a rivalry of ”dualism” against ”monism,” British empiricism (Locke,
Berkeley, Hume, Kant) found a ﬁeld that, in hindsight, corresponds to the Copenhagen
interpretation of quantum mechanics. This insight is so great that no amount of praise
can be too much. I think it is the ”sense of balance” of the people of that time. And thus,
I think(C3 ) the emergence of a monistic giant, Newtonian mechanics, resulted in the strengtheningof dualistic idealism and the ﬂourishing of modern philosophy

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Chap. 9 Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, Hume9.1.2 Tabula·Rasa (British empiricism) vs Nativism (Continental rationalism)
The main purpose in Chap. 9(=this chapter)a and Chap. 10 is as follows.
(A) To assert the superiority of ” quantum linguistic synthesis (= QL-synthesis)” over”Kant’s synthesis.”
Therefore, in this section, we will give an outline of ”Kant’s synthesis” and ” QL-synthesis.
9.1.2.1 Kantian synthesis
Philosophy enthusiasts like the story that Descartes’ philosophy developed in Englandand on the Continent as empiricism and rationalism, respectively, and that Kant united
the two about a hundred years later. Let me explain this theory as follows.

The Flower of Modern Philosophy ( Kantian Synthesis)
Descartes’ work gave rise to two philosophies: British empiricism and continental rationalism.
British empiricism “tabula rasa(= blank paper )“An Essay Concerning Human Understanding (by Locke, 1690)(B) He eliminated the possibility of innate knowledge before experience. Human being
is born as the blank state ( “tabula rasa” ) . In other words, the ”brain circuit”
is initially a blank slate, but concepts (i.e., complex brain circuits) are createdthrough seeing and hearing various things. (Locke, Berkeley, Hume, · · · )Continental rationalism nativism“New Essays on Human Understanding” (by Leibniz, 1703) says that(C) Nativism: Human ”brain circuits” already contain ideas and thoughts when weare born. (Spinoza, Leibniz· · · )That is, the opposition structure:

British empiricism vs.

Continental rationalism(nativism)(tabula·rasa)

was born.
Then, about 100 years later, after many twists and turns, Kant came up with [Critique of Pure Reason: 1781];(D) to synthesize “tabula·rasa and nativism”
Kant, however, has taken both sides of the issue into account, and concluded that• while innate qualities are important (innate theory), experience, training, and229

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9.1 Locke (1632 - 1704): The father of British Empiricism
education are also important (empiricism).

This is illustrated in the following ﬁgure.

(E) Kantian synthesis
Deduction (due to Descartes) 








Continental Rationalism 
mind-body problem






•
(nativism)Descartes








[Spinoza,Leibniz](the father of−−−−−−−−→ Kant
modern phil. −→(synthesis)Induction(due
toBacon)

subjectivity problem






British Empiricism










•
(tabula
rasa)






[Locke, Berkeley, Hume]♠Note 9.2. Recall the theory of Plato’s anamnesis (Sec.3.3). Let’s recall:
Plato presented the concept of ”recollection” (=anamnesis) as a mediator between the realworld and the ideal world. i.e.
• We saw the Idea before we were born. But when we are born we have forgotten it.
Therefore, to know an Idea is to recall it. In other words, to learn is nothing but torecall (=anamnesis).
In the history of Western philosophy, the theory of anamnesis is placed as follows.

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Chap. 9 Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, Hume[modern philosophy]

Descartes
Plato[anamnesis]

−→












[tabula rasa, experience]

British Empiricism[Locke, Berkeley, Hume][a priori, nativism]





Continental
Rationalism[Leibniz]





















[”Critique of Pure Reason”]−→*3

Kant philosophy[summing-up, compromise]

I believe that “tabula rasa vs. nativism” is based on the tradition of Plato’s philosophy.
It is a pseudo-scientiﬁc argument in , and I think that sticking to it risks losing sightof the essence of modern philosophy. Therefore, I think Kant’s synthesis needs to bereconsidered.

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9.1 Locke (1632 - 1704): The father of British Empiricism

9.1.2.2 QL-synthesis
I think the above ”Synthesis of Kant” is too pseudoscientiﬁc. Write my opinion in thenote below.
♠Note 9.3. Thinking in terms of ”language” instead of ”recognition”(♯1 ) ”Everyday language” is similar to tabla-rasa. When a baby is born, it does not know
any ordinary language at all (= tabula rasa). The baby acquires everyday language
through repeated trial and error.
(♯2 ) Mathematics is similar to Continental Rationalism. All mathematical theorems canbe proved (in principle) if only the axioms of set theory are determined.
(♯3 ) The quantum language is similar to the compromise between (♯1 ) and (♯2 ) above as
follows.
rationalism

Quantum language =(empiricism)

Axiom 1 ( measurement)
Axiom 2 ( causality)

linguistic+ Copenhagen interpretation////

Taking this Note 9.3 as a hint, I consider the following. The main purpose of thistext (chapters 8, 9, and 10) is to propose a rewriting of the history of modern Western
philosophy (i.e., from [(E): Kantian synthesis] to [(F): QL synthesis]).

(F) QL-synthesis)
This will be explainedin chapters 8, 9, and 10, but only a ﬁgure
is given here.

Deduction (due to Descartes)
WittgenContinental Rationalismmind-body problem 


•
→
stein
(axiomatictheory)Descartes[Spinoza,Leibniz](the father ofmodern phil. →Induction (due to Bacon)
subjectivity problem British Empiricism
(empirical theory) • Copenhagen interpretation
[Locke, Berkeley, Hume, Kant]

232

















Axioms 1, 2−−−−−−−−→QL
(synthesis)Copenhagen






interpretation











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Chap. 9 Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, Hume

(However, attempts at Axiomatic theory were too diﬃcult at the time.)

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9.1 Locke (1632 - 1704): The father of British Empiricism

9.1.3 Secondary quality ( and primary quality )According to Locke, we see(G1 ) primary quality (i.e., inherent nature (=primary quality)) · · · weight, temperature, length, etc.
(G2 ) secondary quality (i.e., quantities perceived by sensory organs)· · · sweet, red,hot, salty, etc.

In terms of quantum language, we say:primary quality=⇒state,secondary quality=⇒observable
Here,(H) Locke represents the most important concept in dualistic idealism as the term ”secondary quality”. The terms such as Idea, body, etc. may not be epistemological.
However,• ”secondary quality” is a word we can understand. Hence, Locke’s achievement
should be honored.
Again, note that ”secondary quality” is a word that forms the foundation of mindbody dualism.
Of course, it was Locke who steered Cartesian philosophy in the direction of ”recognition
and measurement”. Thus, he argues the following.
(Idea)[(primary) universal](body)

progress

progressmodel-change

model-change

Plato −−−−−−−−→ Anselmus, Thomas −−−−−−−−→ Descartes(sensory organ), [secondary quality]
progress−−−−−−−−→
model-change(measuring instrument), [observable]
progress−−−−−−−−→

Locke

model-change

epistemology

quantum language

British empiricist philosophers spoke philosophy to ordinary philosophy enthusiasts. Thus,
they always spoke philosophy in plain language. The term ”secondary quality” can beunderstood by anyone.
Corollary 9.1. (Cf. (W) in Sec. 6.5.4) Of these three (”measured value,” ”observed
value,” and ”state”), ”observable” is the most diﬃcult to understand. So, the history of”observable” can be summarized as follows.
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Chap. 9 Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, Hume
(I)
Idea

linguistic version−−−−−−−−−−−→

signiﬁer, [signiﬁed]

Plato’s idea theoryascholasticyversion ⃝Saussure’s linguistic theory[(primary) universal][membership function]

Anselmus, Thomas

Zadeh’s fuzzy theoryquantiﬁcationycognitiveyversion ⃝b[secondary quality]yidentiﬁcation

linguistic version ⃝c−−−−−−−−−−−−→

John Locke[observable]
quantum language(The history from Plato Idea to observables)

Through scholastic philosophy a , Idea became the scientiﬁc concept of ”universal. Afterb by Locke, Idea became a concept that everyone can understand [secondary properties].
After that, it only becomes a [computable form] called an ”observable” by c .
♠Note 9.4. (i):The concept of ”measuring instrument” (≈ body) is not easy. For example,
telescopes, glasses and eyes are a type of measuring instrument. A directional magnet is,of course, a measuring instrument. If so, then the polar star may be also regarded as a
type of measuring instrument.

(ii): By the way, Merleau-Ponty (1908 - 1961) might think in the following manner.
• I shake hands with my right hand and the left hand. In this case, if I regard theright hand as the measuring instrument, I feel the existence of my left hand. On thecontrary, if I regard the left hand as the measuring instrument, I feel the existence of
my right hand.

(iii): I have mentioned this before, but I will write it again. For example, assume thatan apple is put on food scale. We are not sure if he is trying to weigh an apple or makesure the food scale isn’t broken. Thus, the diﬀerence between ”body” and ”matter” isarbitrary. This is determined by the observer. Both ”body” and ”matter” obey the lawsof physics. However, the laws governing ”mind” are assumed to be unknown. Maybe there
is no such thing.

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9.1 Locke (1632 - 1704): The father of British Empiricism(v): As readers are already aware:• the magic word ”Idea” is what made Western philosophy ﬂourish!
A.N.Whitehead (1861 - 1947) said as follows.
(♯) Western philosophy is characterized as a series of footnotes to Plato
In short, the word ”Idea” is a profound word with many aspects, as described in (G), andit has kept philosophy enthusiasts occupied for 2,500 years.
////

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Chap. 9 Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, Hume9.2* Leibniz-Clarke Correspondence: What is space-time?

The contents of this section are published in the followings:• ref. [76]: S. Ishikawa; Leibniz-Clarke correspondence, Brain in a vat, Five-minutehypothesis, McTaggart’s paradox, etc. are clariﬁed in quantum language
Open Journal of philosophy, Vol. 8, No.5, 466-480, 2018, DOI: 10.4236/ojpp.2018.85032(https://www.scirp.org/Journal/PaperInformation.aspx?PaperID=87862)• ref. [77]; S. Ishikawa; Leibniz-Clarke correspondence, Brain in a vat, Fiveminute hypothesis, McTaggart’s paradox, etc. are clariﬁed in quantum language; [Revised version] ; Keio Research report; 2018; KSTS/RR-18/001, 1-15
(https://philpapers.org/rec/ISHLCB)(http://www.math.keio.ac.jp/academic/research_pdf/report/2018/18001.
pdf)• [85] S. Ishikawa, K. Kikuchi, (2021) Quantum Fuzzy Logic and Time, Journal ofApplied mathematics and physics, Vo.9 No.11 2021, 2609-2622(https://www.scirp.org/journal/paperinformation.aspx?paperid=112972)

The problems (“What is space?” and “What is time?”) are the most important inmodern science as well as the traditional philosophies. In this section, we give the quantum
linguistic answer to these problems. As seen later, our answer is similar to Leibniz’s
relationalism concerning space-time. In this sense, we consider that Leibniz is one of the
discoverers of the linguistic Copenhagen interpretation. No one could understand Leibniz’s
monad in science (cf. Sec. 8.4), but everyone can understand his excellent ”space-timetheory” in science.

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9.2
* Leibniz-Clarke Correspondence: What is space-time?

9.2.1 “What is space?” and “What is time?”

9.2.1.1 Space in quantum language(How to describe “space” in quantum language)(Cf. the C ∗ -algebraic formulation in Sec 1.4) is used)
In what follows, let us explain “space” in measurement theory (= quantum language).
For example, consider the simplest case, that is,(A)“space”=Rq (one dimensional space)

Since classical system and quantum system must be considered, we see (B1 ): a classical particle in the one dimensional space Rq(B)(B2 ): a quantum particle in the one dimensional space Rq
In the classical case, we start from the following state:(q, p) = (“position”, “momentum”) ∈ Rq × Rp
Thus, we have the classical basic structure:(C1 )[C0 (Rq × Rp )*4 ⊆ B(L2 (Rq × Rp )]Also, concerning quantum system, we have the quantum basic structure:(C2 )[C(L2 (Rq ) ⊆ B(L2 (Rq )]Summing up, we have the basic structure (C1 ): classical [C0 (Rq × Rp ) ⊆ B(L2 (Rq × Rp )](C) [A ⊆ A ⊆ B(H)](C2 ): quantum [C(L2 (Rq ) ⊆ B(L2 (Rq )]Since we always start from a basic structure in quantum language, we consider that

*4 See Note 1.11

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Chap. 9 Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, HumeHow to describe “space” in quantum language⇔ How to describe [(A):space] by [(C):basic structure]

This is done in the following steps.
Assertion 9.2. [The linguistic Copenhagen interpretation concerning ”space”]
How to describe “space” in quantum language(D1 ) Begin with the basic structure:[A ⊆ B(H)](D2 ) Next, consider a certain commutative C ∗ -algebra A0 (= C0 (Ω)) such that
A0 ⊆ B(H)(D3 ) Lastly, the spectrum Ω (≈ Sp (A∗0 )) is used to represent “space”.
Therefore, in quantum language, we see• space is a kind of state of a ”thing”.
[Note]: Therefore,(D4 ) Space can be thought of as a kind of state space (a space representing the stateof ”things”).
Therefore, if there is no ”thing”, there is no basic structure,and therefore there is no space.
For example,(E1 ) in the classical case (C1 ):[C0 (Rq × Rp ) ⊆ B(L2 (Rq × Rp ))]we have the commutative C0 (Rq ) such that
C0 (Rq ) ⊆ B(L2 (Rq × Rp ))And thus, we get the space Rq as mentioned in (A)(E2 ) in the quantum case (C2 ):[C(L2 (Rq ) ⊆ B(L2 (Rq ))]we have the commutative C0 (Rq ) such that
C0 (Rq ) ⊆ B(L2 (Rq ))And thus, we get the space Rq as mentioned in (A)239

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9.2
* Leibniz-Clarke Correspondence: What is space-time?

9.2.1.2 Time in quantum language: How to describe time in quantum language?
Axiom 2 (i.e., causality) should be restated (For details, see ref. [74]).
Axiom 2 [ Causality ] in the C ∗ -algebraic formulation•[General case (i.e., quantum case (Ati = Bc (Hti )) and classical case (Ati =C(Ωti ))) ]:Assume [t1 ≤ t2 ].
(♯1 ): Causality is represented by a causal operator Φt1 ,t2 : At2 → At1
In the same sense,(♯2 ): Causality is represented by a dual causal operator Φ∗t1 ,t2 : A∗t1 → A∗t2•[Classical case (Ati = C(Ωti )]Assume [t1 ≤ t2 ]:(♭1 ): Classical causality is represented by a causal operator Φt1 ,t2 : C(Ωt2 ) →C(Ωt1 )In the same sense,(♭2 ): Classical causality is represented by a dual causal operator Φ∗t1 ,t2 : C(Ωt1 )∗→ C(Ωt2 )∗Also,(♭3 ): Classical causality is represented by a continuous map { ϕt1 ,t2 : Ωt1 → Ωt2 ,(t1 ≤ t2 ) }♠Note 9.5. In the above, t1 , t2 , ... does not always imply time. For example, when formulating regression analysis in quantum language, use the parallel structure shown in thelower left ﬁgure (cf. ref. [74]). If it is semi-ordered, it is acceptable.

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Chap. 9 Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, Hume

ϕ0,1

Ω0

 Ω1
1 Ω2
ϕ0,2
Φ0,1

Φ1,2

Φ2,3

C(Ω0 ) ←−−−− C(Ω1 ) ←−−−− C(Ω2 ) ←−−−− C(Ω3 )

ϕ0,3
q Ω3
s
ϕ0,4 Ω4

Assertion 9.3. [Copenhagen Interpretation of ”Time”]
How to describe ”time” in quantum language
Let [A ⊆ B(H)] be a C ∗ -basic structure. Let T be the real line R. for any t1 , t2 ∈
T (t1 ≤ t2 ), deﬁne a causal operator Φt1 ,t2 : A → A such that
Φt1 ,t2 Φt2 ,t3 = Φt1 ,t3(∀t1 , ∀t2 , ∀t3 ∈ T such that t1 ≤ t2 ≤ t3 )(9.1)

Also, let Φ∗t1 ,t2 : A∗ → A∗ be the dual causal operator of Φt1 ,t2 : A → A. Let ρ0 (∈ S(A∗ )be an initial state. Then, the time evolution is repretented by Φ∗0,t ρ0 (0 ≤ t < ∞).
[Note](F) In the above, time T is a parameter that to describe the change in state of the”thing”
Therefore, without ”things”, there is no basic structure, and time is meaningless.
♠Note 9.6. According to the Copenhagen interpretation, the thing (state) does not move.
the above (F) formally states that(F’) time T is a parameter used to describe the change of an observable
The Schrödinger picture (change of state) is more intuitive than the Heisenberg picture(change of observables), so I wrote it as above.

For example, consider the following speciﬁcs. Note that T (∈ T ) is treated as a parameter, and without the ”thing”, there is no basic structure and time T is meaningless.
[I]: For quantum systems(the case of C ∗ -basic structure:[Bc (H) ⊆ B(H)]): Consider
b as follows:
the Schrödinger equation with Hamiltonian H
iℏ

d
b ti
|ψt i = H|ψdt( in H) (t ∈ T, ψ ∈ H, ||ψ||H = 1)b∗∗
Clearly the solution is equal to exp[ Htiℏ ]|ψ0 i. Using the dual causal operator Φt2 ,t1 : At1→ A∗t2 (i.e., Sp (Bc (H)∗ → Sp (Bc (H)∗ ), the causality is written by

b
b
Ht
Ht
]|ψ0 ihψ0 | exp[
]
iℏ
iℏ
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Φ∗t,0 (|ψ0 ihψ0 ) = exp[

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9.2
* Leibniz-Clarke Correspondence: What is space-time?
[II]: For classical case(C ∗ -basic structure:[C(Ω), B(H)]): Consider the dynamical system { ϕt1 ,t2 : Ω → Ω, (t1 , t2 ∈ T such that t1 ≤ t2 ) }, which is equivalento Newtonian
equation. Clearly, we see
ϕt3 ,t2 ϕt2 .t1 = ϕt3 ,t1(0 ≤ t1 ≤ t2 ≤ t3 < ∞)Assume the initial state is ω0 (∈ Ω). Then we seeωt = ϕ0,t (ω0 ) (t ≥ 0)

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Chap. 9 Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, Hume9.2.2 Leibniz-Clarke Correspondence

Read the following while looking at the above ﬁgure.
The above argument urges us to recall Leibniz-Clarke Correspondence (1715–1716: cf.
[1]), which is important to know both Leibniz’s and Clarke’s (=Newton’s) ideas concerningspace and time.
(G) [The realistic space-time]

Newton’s absolutism says that the space-time should be regarded as a receptacle of a “thing.” Therefore, even if “thing” does not exits, the space-timeexists.
On the other hand,(H) [The metaphysical space-time]

Leibniz’s relationalism says that
(H1 ) Space is a kind of state of “thing”.
(H2 ) Time T is also a kind of state space R (or, Z), which represents an order ofoccurring in succession which changes one after another.
Thus,(I) Without ”things”, there would be no space-time. (cf. (D4 ) to (F4 ) )Therefore, I regard this correspondence as
Newton (≈ Clarke)←→vs.

(realistic view)

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Leibniz(linguistic and idealistic view)

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9.2
* Leibniz-Clarke Correspondence: What is space-time?
which should be compared toAristotle(realistic view)

←→
vs.

Plato
(linguistic and idealistic view)(also, recall Note 4.4).
In the end, it is the diﬀerence between existential and linguistic worldviews that can besummarized as follows
(space-time in physics)

1 : Newton, Clarke○· · · realistic space-time(successors: Einstein, etc.)(realistic worldview)“What is space-time?”






2 :
Leibniz ○(linguistic worldview)(space-time in measurement theory)···

linguistic space-time(i.e., spectrum, tree)“How should space-time be represented?”

Therefore, both Clarke (≈Newton) and Leibniz are correct, but the decisive diﬀerence is
as follows.
(J1 ) Newton created the language of Newtonian mechanics, under which he discussed
space-time.
(J2 ) Leibniz discussed space-time without (quantum) language, only by his genius intuition.
Because of this diﬀerence, Leibniz’s theory of space-time relationalism has been diﬃcult
to understand until today.

I would also ask you to review the following table again
Table 9.1 (=Table 4.1) : realistic worldview (Aristotle) vs. idealistic worldview (Plato)
dispute ⧹ [R] vs. [L]a motion○:b○:Ancient
Greecec Problem of universals○:
d space-time○:
e quantum theory○:f○:philosophyof scienceg fuzzy sets○:

Realistic worldview

Idealistic worldview(monism, realism, no measurement)(dualism, idealism, measurement)

Hērakleitos
Aristotle
OckhamNewton ( Clarke)
Einstein
CarnapKalmanParmenides
Plato
AnselmusLeibniz
Bohr
Quine
Zadeh

For the precise argument, see Table 4.1.
♠Note 9.7. Leibniz is an undisputed genius. I think Leibniz is the only one who can discuss
space-time on equal terms with Newton. Leibniz space-time theory is almost the sameas the space-time theory in the Copenhagen interpretation. Thus, Leibniz was one of theleading contributors to the creation of the Copenhagen interpretation. Therefore, I would
like to consider that Leibniz also belongs to British empiricism.
I cannot say anything about this because I do not understand Leibniz’s monadology.
However, I would have hoped that the genius Leibniz would have discussed space-time244

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Chap. 9 Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, Humeunder the worldview of monadology and put this up against Newton.

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9.2
* Leibniz-Clarke Correspondence: What is space-time?

9.2.3 Appendix: motion function method ( in classical system )The ” motion function method (Sec. 2.3)” discussed in the Zeno’s paradox is supplemented
here with its fundamentals.
9.2.3.1 Schrödinger and Heisenberg pictures are equivalent in the classical system
According to Leibniz, ”time” is just a ”parameter” that can be conveniently created. Let’s
introduce ”Parallel time” and ”Series time*5 .
Consider a classical dynamical system (Ω, ϕt1 ,t2 ). Here, Ω is compact space, t1 , t2 ∈T = [0, 1] such that 0 ≤ t1 ≤ t2 ≤ 1, a map ϕt1 ,t2 (·) : Ω → Ω is bi-continuous and satisﬁes
the following condition:(♯1 ) limt2 →t1 ϕt1 ,t2 (ω) = ω (ω ∈ Ω)(♯2 ) [ϕt2 ,t3 ◦ ϕt1 ,t2 ](ω) = ϕt2 ,t3 (ϕt1 ,t2 (ω)) = ϕt1 ,t3 (ω) (ω ∈ Ω, 0 ≤ t1 ≤ t2 ≤ t3 ≤ 1)
As mentioned before(K) there exists a homomorphism Φt1 ,t2 : C(Ω) → C(Ω) such that[Φt1 ,t2 (gt2 )](ωt1 ) = gt2 (ϕt1 ,t2 (ωt1 ))(∀ωt1 ∈ Ω, ∀gt2 ∈ C(Ω)),

Consider the following time series (i.e., the case that N = 3, Ωi = Ω, i = 0, 1, 2, 3):(L)
O=(X,P(X),F )O=(X,P(X),F )Φ0,1C(Ω0 )←−−−−
state spaceyC(Ω1 )←−−−−
state space
y

ϕ0,1

−−−−→

Ω0

O=(X,P(X),F )
Φ1,2

Ω1

O=(X,P(X),F )Φ2,3

C(Ω2 )←−−−−
state space
y

ϕ1,2

−−−−→

Ω2

ϕ2,3

−−−−→

C(Ω3 )
state space
y
Ω3

[ω0 ]

[(i): Schrödinger pictures ( a state moves) :Parallel time)] of (L):
Figure ( the case that N = 3; Ω = Ωi , i = 0, 1, 2, 3)

ϕ0,0



Ω0 : O = (X, P(X), F )ϕ0,0 : identity mapΩ0ω01 Ω1 : O = (X, P(X), F )ϕ0,1ϕ0,2 = ϕ1,2 ◦ ϕ0,1
ϕ0,3 = ϕ2,3 ◦ ϕ1,2 ◦ ϕ0,1

ϕ0,2
q Ω2 : O = (X, P(X), F )sϕ0,3 Ω3 : O = (X, P(X), F )*5 See [85]; Parallel time represents the time lapse of a dice throw or the law of large numbers, etc.

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Chap. 9 Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, HumeAssume that the state ω0 (∈ Ω) at time t0 (= 0) evolves in time to become ϕ0,tk (ω0 )(k = 0, 1, ..., N ) as follows:(♯3 ) state ϕ0,t0 (ω0 ) = ω0 at time t0 = 0/n(= 0)state ϕ0,t1 (ω0 ) at time t1 = 1/nstate ϕ0,t2 (ω0 ) at time t2 = 2/n···state ϕ0,tk (ω0 ) at time tk = k/n···state ϕ0,tn (ω0 ) at time tn = n/n = 1
And assume:(M) At each time t0 (= 0), t1 (= 1/n), ..., tk (= k/n), ..., tn (= 1), measurement MC(Ω) (O =(X, P(X), F ), S[ϕ0,tk (ω0 )] ) is taken.
That is, putting Tn = {t0 (= 0), t1 (= 1/n), · · · , tn (= 1)}, we take the tensor exact
measurement:O
MC(Ω) (O = (X, P(X), F ), S[(ϕ0,tk (ω0 ))] )tk ∈TnO=MC(ΩTn ) (OEΩ = (ΩTn , B(ΩTn ),tk ∈TnO
tk ∈TnF ), S[(ϕ0,tk (ω0 ))tk ∈Tn ] )Then, we see that, for any Ξk ⊆ X (k = 1, 2, ..., n),(N) the probability that the measued value belongs to ×ki=0 Ξk is given by
×nk=0 [F (Ξk )](ϕ0,tk (ω0 ))[(ii): Heisenberg picture ( observable moves: (Series time)]:Figure ( the case that N = 3; Ω = Ωi , i = 0, 1, 2, 3)
O=(X,P(X),F )C(Ω0 )

O=(X,P(X),F )
Φ0,1

←−−−−

C(Ω1 )

O=(X,P(X),F )
Φ1,2

←−−−−

C(Ω2 )

O=(X,P(X),F )
Φ2,3

←−−−−

C(Ω3 )[ω0 ]As mentioned in the above, assume that the state ω0 (∈ Ω) at time t0 (= 0), and Tn ={t0 (= 0), t1 (= 1/n), · · · , tn−1 (= (n − 1)/n), tn (= 1)}.
Assume that, at each t0 (= 0), t1 (= 1/n), · · · , tn−1 (= (n − 1)/n), tn (= 1), an observable
O = (X, P(X), F ) is set.
(♭4 ) the observable O(= (X, P(X), F )) at time tn (= 1) is identiﬁed with the observableΦtn−1 ,tn O(= (X, P(X), Φtn−1 ,tn F )) at time tn−1 . At time tn−1 , we originally havean observable O, and the product of this O and Φtn−1 ,tn O gives the observable attime tn−1 :O × (Φtn−1 ,tn O)= (X 2 , P(X 2 ), Fbn−1 )Similarly, the observable it time tn−2 is represented by
O × (Φtn−2 ,tn−1 (O × (Φtn−1 ,tn O)))
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9.2
* Leibniz-Clarke Correspondence: What is space-time?
Further, the observable at time tn−3 is represented by,
O × (Φtn−3 ,tn−2 (O × (Φtn−2 ,tn−1 (O × (Φtn−1 ,tn O))))= (X 4 , P(X 4 ), Fbn−3 )Iteratively, after all, the observable at time t0 is represented by,
b t = O × (Φt ,t (· · · (O × (Φt ,t (O × (Φt ,t (O × (Φt ,t O))))) · · · ))O0
0 1
n−3 n−2n−2 n−1n−1 n= (X n+1 , P(X n+1 ), Fb0 )b t , S[ω ] ) at time t = 0. Therefore, putting
Thus, we get the measurement MC(Ω) (O00Ξk ⊆ X (k = 1, 2, ..., n), we see that(O) the probability that its measured value belongs to ×ki=1 Ξk is given by [Fb0 (Ξ0 × Ξ1 ×
· · · × Ξn )](ω0 )Here, we see[Fb0 (Ξ0 × Ξ1 × · · · × Ξn )](ω0 )=[F (Ξ0 )](ω0 ) × Φtn−1 ,tn−2 [Fb1 (Ξ1 × · · · × Ξn )](ω0 )=[F (Ξ0 )](ω0 ) × [Fb1 (Ξ1 × · · · × Ξn )](ω1 )···n= × [F (Ξk )](ϕ0,tk (ω0 ))k=0Here, note that (N)=(O) holds. Thus, we can conclude tha(P) Schrödinger and Heisenberg pictures are equivalent in the classical system
9.2.3.2 Derivation of the motion function method from (classical) quantum language
In classical system, we see that the Schrödinger picture (N) and Heisenberg picture (O)are equivalent, we use Schrödinger picture (N) from here. Put T = [0, 1]. Consider thecase that O = (X, P(X), F ) is the precise observable OE (cf Sec. 1.4.5), namely, O =(X, P(X), F ) = OE = (Ω, B(Ω), Eω ). That is, we consider the case that X = Ω, F = Eω ,namely,precise measurement MC(Ω) (OE = (Ω, B(Ω), Eω ), S[ϕ0,t (ω0 )] ) (t ∈ T = [0, 1])
And further, consider the tensor product precise measurement
O
MC(Ω) (OE = (Ω, B(Ω), EΩ ), S[ϕ0,t (ω0 )] )t∈T=MC(ΩT ) (O

OE = (ΩT , B(ΩT ),t∈TOEΩ ), S[(ϕ0,t (ω0 ))t∈T ] )t∈TProposition 1.20 says:
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Chap. 9 Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, Hume(Q) (=(F2 ) in
N Proposition 1.20): When the tensor product precise measurement
MC(ΩT ) ( t∈T OEΩ = (ΩT , B(ΩT ), EΩT ), S[(ϕ0,t (ωω0 )t∈T ] ) is take, the probability
that the measured value (xt )t∈T (∈ ΩT ) belongs to any open set which includes(ωt )t∈T (∈ ΩT ) is 1. In the same sense, the measured value (xt )t∈T (∈ ΩT ) is surely
equal to (ϕ0,t (ω0 ))t∈T
X(= Ω) 6

xt = ϕ0,t (ω0 )- T (= [0, 1])t

In general, deﬁne the position map P ′ : Ω(= X) → X ′ such thatP′Ω(= X) 3 [state]=[position, momentum] 7→ [position](= X ′ )Then, the motion function m : T → X ′ can be written as follows.
m(t) = P ′ (ϕ0,t (ω0 ))249(∀t ∈ T )For further information, see my homepage

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9.3 Subjective idealism: Berkeley, ”To be is to be perceived”

9.3 Subjective idealism: Berkeley, ”To be is to be perceived”

9.3.1 Priest: Berkeley
Berkeley (1685 -1753) is famous as follows.
(A1 ) Berkeley is a priest, and he interpreted Locke’s primary quality as the state ofthings that come from a supernatural power such as a god. Thus his philosophy iscalled subjective idealism.

Let’s check the next table to see where Berkley is located.
Table 9.2: The key-words of mind-body dualism

mind-body dualism
Plato(Allegory of the sun)Thomas(1225-1274)
Descartes(1596-1650)
Locke(1632-1704)
Berkeley(1685 -1753)
Kant(1724-1804)
quantum language

A(= mind)[tertiary universal]

B(body:between A and B[(primary) universal]

C(= matter)[secondary universal]

actual world
human intellect[tertiary universal]

Idea
divine intellect[(primary) universal]

idea world
individual[secondary universal]

I, mind, brain

body

mattermatter[primary quality]mind

body (≈ sensory organ)[secondary quality]
body (≈ sensory organ)[secondary quality]

phenomenon

cognition

thing-in-itself

observer[measured value]measuring instrument

system[state]mind[observable]God

If we were to evaluate this Table description alone, we would have to say the following.
 Berkeley
Locke −−−−−→
regress Kant
Berkley was too extreme and Kant said too many pseudo-scientiﬁc nitpicks. Of course,
they hit an inverted home run.
Berkeley was an important ﬁgure in the following sense.
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Chap. 9 Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, Hume(x)”
(A2 ) Berkeley indicated that Newton’s deﬁnition of diﬀerentiation ”lim h→0 f (x+h)−fh
is not complete(A3 ) He said
• To be is to be perceived,• If a tree falls in a forest and no one is around to hear it, does it make a sound?
which represented the essential spirit of the mind-body dualism

Summing up, Berkeley was always the standpoint of anti-Newton (= anti-realism).
• If we think that modern philosophy (from Descartes to Kant) has signiﬁcance asa buﬀer zone of Christianity with Newtonian mechanics, we can conclude thatBerkeley was honest.
♠Note 9.8. The mathematical deﬁnition of lim 00 (i.e., (ϵ, δ)-deﬁnition of limit) was more
important than Newton thought, and it was discovered one hundred and tens of years
later (by Cauchy (1789-1857), Weierstrass (1815-1897), etc.).

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9.3 Subjective idealism: Berkeley, ”To be is to be perceived”

9.3.2 (A3 ): To be is to be perceived
Berkeley is too extreme and many may disagree next:(♯)

Locke −−−−−−→ Berkeley
progress

However, his saying ”To be is to be perceived” is the essence of the Copenhagen interpretation. Thus, I agree to the (♯).
Consider the following saying:(B1 ) There is no science without measurement(≈ [To be is to be perceived])
Everyone may believe that this saying (B1 ) is absolutely true. In fact, the importance of”measurement” is emphasized as follows (cf. Sec. 1.1).
(=quantum language)
measurement theory[Axiom 2][Axiom 1]=

measurement + causal relation+

linguistic Copenhagen interpretation[the manual to use Axioms 1 and 2](9.1)
But, it is Genius Newton (and Einstein) that neglect this absolute truth (B 1 ). In fact,
Newtonian mechanics is formulated as follows.
[Newtonian kinetic equation]

Newtonian mechanics = No measurement +causal relation
No interpretation

+

(9.2)

[the manual to use Newtonian mechanics]

Here, note that Newton removed ”measurement” from (9.2) in spite of the maxim that
there is no science without measurement. Perhaps he intuited that adding ”measurement”
would be tricky. If so, the insightfulness of Newton is surprising. A genius isn’t confusedby ”the absolute maxim (B1 )”. The following is my ﬁction:(B2 ) ”Exclusion of measurement” is the conclusion reached by the deep consideration of
Newton. However, Berkeley, an excellent controversialist of anti-Newton, consideredthat the exclusion was a weak point of Newtonian mechanics. And he said
To be is to be perceived
This is the golden rule of anti-Newtonianism (i.e., anti-physicalism, idealism). The opposing structure: [Newton vs. Berkeley] continues to [Einstein vs. Bohr] as mentioned innext section.
♠Note 9.9. ”Why does Newtonian mechanics (9.2) have no” interpretation ”? I don’tknow the answer. Perhaps, the way to interpret Newtonian mechanics is so simple that I
don’t bother to mention it. I believe that there is always an ”interpretation” of any nonmathematical theory (cf. ref. [84]). I also believe that this is related to the ”Wittgensteinparadox” (cf. Sec. 12.2.3).

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Chap. 9 Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, HumeI think that a theory that contains something that cannot be explained in one word requires”interpretation”. And I think that is the ” idea (or observable quantity, or measurement)”.
This thinking explains the perpetuation of Western philosophy, with Plato as its source.

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9.3 Subjective idealism: Berkeley, ”To be is to be perceived”

9.3.3 ”Einstein-Tagore Meeting” and ”Bohr-Einstein debates”
Concerning ”realistic worldview vs. idealistic worldview”, Einstein-Tagore (poet, thinker
in India) meeting in 1930 is famous, in which they asserted as• Tagore: Truth is always limited by human perception.
• Einstein: Truth is independent of our consciousness, For instance, if nobody is in
this house, yet that table remains where it is*6 .

In the above, Tagore’s assertion is similar to Berkeley’s ”To be is to be perceived”, which
belongs to the situation of dualistic idealism(=idealistic worldview).

On the other hand, Einstein’s saying:(C1 ) if nobody is in this house, yet that table remains where it is (= Does the moon
disappear when I’m not looking at it?)is the same as(C2 ) Truth is independent of us (= realistic worldview)Thus, Einstein and Newton are similar, in the sense that
Truth is independent of human being (i.e., physics holds without measurement)
Therefore, it should be noted that the formula (9.2) is signiﬁcant. In this text, we are
not concerned with Bohr-Einstein debates in quantum mechanics (in order to solve thisproblem, I proposed quantum language), (cf. ref. [74])). However, Bohr-Einstein debates
is similar to the above. Thus, summing up, we see:

*6 Einstein often said this kind of statement at various places, for example, ”Does the moon disappear

when I’m not looking at it?”254

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Chap. 9 Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, Hume
Table 9.3 realistic worldview vs. idealistic worldview
Realistic worldview

Idealistic worldview[monism, realism, no measurement ][dualism, idealism, measurement ]NewtonNewton (and Clarke)
Einstein
Einstein

Berkeley
Leibniz
TagoreBohr

Concerning Bohr-Einstein debates, The impression that Einstein lost now has been left,
but the author does not think so (cf. ref. [74]).
♠Note 9.10. Omitting ”Newton vs. Berkeley” and ”Einstein vs. Tagore” in the abovetable, I repeatedly mention the following table (cf. Assertion 4.2):

Table 9.4 realistic worldview vs. idealistic worldview (cf. Table 4.1 in Sec. 4.1.3)
dispute ⧹ [R] vs. [L]

Realistic worldview

Idealistic worldview(monism, no measurement)(dualism, measurement)

Hērakleitos⃝:
a motion
Greece⃝:AncientAristotleb⃝:c Problem of universals ”Nominalismus”(Ockham)⃝:
Newtond space-time⃝:
Einstein
e quantum theory⃝:philosophyof science
Carnapf

Parmenides
Plato”Realismus”(Anselmus)
Leibniz
Bohr
Quine

For the precise argument, see Table 4.1.
⃝
a is my ﬁction, ⃝c is a confusion (Aristotle vs. Plato). ⃝d is the Leibniz=Clarkecorrespondence (cf. Sec. 9.2.2), ⃝e is Bohr-Einstein debates. Quantum language is
proposed as one of answers to Bohr-Einstein debates(cf. ref. [74]). Quantum language
is proposed as one of answers to Bohr-Einstein debates(cf. ref. [74]). ⃝:f Quineunderstood the spirit of the linguistic Copenhagen interpretation (i.e., ”If you don’t
measure it, you don’t know anything”) in the Carnap=Quine debate (cf. Sec. 12.3).
♠Note 9.11. In Japan, I learned the dualistic proposition such as ”If a tree falls in aforest and no one is around to hear it, does it make a sound?” in a Zen dialogue (i.e., aquestion-and-answer exchange between Zen priests and their followers). Zen is one schoolof Buddhism. In modern Japan, most people may think that Zen monologue is a kind of
wordplay.

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9.3 Subjective idealism: Berkeley, ”To be is to be perceived”

9.3.4 Bohr-Einstein debates: Do the laws of physics require measurement?

For Bohr-Einstein debates, I discussed in ref. [74]. However, I would like to write
something in this section. Almost people agree to the following maxim:• There is no science without measurement
However, genius Newton neglected this maxim, and proposed Newtonian mechanics asfollow: Newtonian mechanics and quantum mechanics are formulated as follows:(♯1 ) Newtonian mechanics =Nothing+

causality(Newtonian equation)

On the other hand, quantum mechanics is formulated as follows.
(=quantum language)(♯2 ) measurement theory[Axiom 2][Axiom 1]=

measurement++

causality

linguistic Copenhagen interpretation[the manual to use Axioms 1 and 2]

N. Bohr, the leader of the Copenhagen school, agreed to the (♯2 ), on the other hand,
A. Einstein asserted that measurement is not needed for physics since he believed in(♯3 ) The moon is there whether one looks at it or not.
in Einstein and Tagore’s conversation. So far, many experimental results support Bohr.
However, if Einstein says the following (♯4 ), everyone has no choice but to shut up.
(♯4 ) Then, did the laws of physics not work before the birth of mankind (e.g., the ageof the dinosaurs)?
Thus, I think that Bohr-Einstein debates is not settled yet. All we can do is to wait foran Einsteinian genius to emerge.

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9.4 Qualia problem and Brain in a vat argument

9.4 Qualia problem and Brain in a vat argument
This section was written with reference to the following.
• [76]:Ishikawa, S: Leibniz-Clarke correspondence, Brain in a vat, Five-minute hypothesis, McTaggart’s paradox, etc. are clariﬁed in quantum language Open Journal of philosophy, Vol. 8, No.5, 466-480, 2018, DOI: 10.4236/ojpp.2018.85032(https://www.scirp.org/Journal/PaperInformation.aspx?PaperID=87862)[Revised version] (https://philpapers.org/rec/ISHLCB)(http://www.math.keio.ac.jp/academic/research_pdf/report/2018/18001.
pdf)• [85] S. Ishikawa, K. Kikuchi, (2021) Quantum Fuzzy Logic and Time, Journal ofApplied mathematics and physics, Vo.9 No.11 2021, 2609-2622(https://www.scirp.org/journal/paperinformation.aspx?paperid=112972)

9.4.1 * The problem of qualia (= the subjectivity problem)
Jack and Betty were looking at the pink ﬂowers. Betty had the following question (i.e.,qualia problem) .
(♯) Is the ”pink” that I felt the same as the ”pink” that Jack felt?

This question can be answered in the following way.
[(A): Scientiﬁc answer (To measure is to believe)]:
With Jack and Betty as test subjects, you (i.e., scientist) can perform various tests (colorblindness test, EEG measurement, electroretinogram, etc.). And if no diﬀerence is foundin any of the tests, we can conclude that each ”pink” felt by Jack and Betty is the same.
[(B): Non-scientiﬁc argument (To think is not to measure)]:However, Betty may say:(♭1 ) I like the ”pink” so much that I has pink walls in my room too. Jack loves blue
and always wears blue. Therefore, I ﬁnd ”pink” to be more beautiful than Jack.
However, the scientist may say that258

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Chap. 9 Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, Hume(♭2 ) According to the linguistic Copenhagen interpretation (cf. (E1 ) in Sec. 1.2.1),
you (= Betty) are not qualiﬁed to be an observer. Even if you are qualiﬁed to
be an observer, you should perform the same tests as in [(A):Scientiﬁc answer]
for Jack and yourself as test subjects, that is, various tests (colorblindness test,
EEG measurement, electroretinogram, etc.). However, if so, it is the same as thesituation [(A):Scientiﬁc answer].

Then, Betty says:(♭3 ) OK. I (=Betty) can understand. In short, the measurements I can only makeagainst myself are not scientiﬁc measurements. Or, a measurement that only I canmake is not a measurement. Is it OK?
Then, the scientist says to Betty:(♭4 ) That’s right.

♠Note 9.12. In the book ”The astonishing hypothesis” (by F. Crick (the most noted forbeing a co-discoverer of the structure of the DNA molecule in 1953 with James Watson)),Dr. Crick said that(♯1 ) You, your joys and your sorrows, your memories and your ambitions, your sense of
personal identity and free will, are in fact no more than the behavior of a vast assemblyof nerve cells and their associated molecules.
That is, he believed the monistic realism such as(♯2 ) the movement of the human spirit is also a kind of physical phenomenon.
I agree to his opinion. And I believe that with the development of brain science, evenconsciousness can be measured in the future. However, no matter how much brain sciencedevelops, I believe that there is no solution to the qualia problem other than the one

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9.4 Qualia problem and Brain in a vat argument
described above.

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Chap. 9 Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, Hume9.4.2* Brain in a vat argument
Let us introduce a famous problem ”Brain in a vat” due to H. Putnam (cf. ref. [108]) as
follows:

Figure 9.1: [Brain in a vat ]There is a possibility of the following.
(C1 ) a mad scientist has removed your brain, and placed it into a vat of liquid to keep it
alive and active. The scientist has also connected your brain to a powerful computer,which sends neurological signals to the brain in the way the brain normally receives
them. Thus, the computer is able to send your the data to you brain to fool you
into believing that you are still walking around a forest.
Here, you may say as follows.
(C2 ) Am I a brain in a tank? Or can you conﬁrm that I am not?
And you may say(C3 ) ”I cannot decide if I am a brain vat or not”.
That is, ”I cannot decide if I am in Case[1] or Case[2]?
Therefore, you may say:(C4 ) ”I cannot know if I have a limb or not.”Then, we have the following problem:
Problem 9.4. [The problem concerning ”Brain in a vat”]• Are the above arguments correct?
[Ansewr]:
One might think that (C3 ) and (C4 ) are obviously correct since (C1 ) is assumed, but
this is not the case from the quantum language standpoint. Because.you ask someone,”Am I walking through the woods?” or ”Do I have a limb?” Then, they reply, ”yes,” so
you can be sure that you are walking through the woods and have a limb. In short.
• You just have to trust the measurement results.
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9.4 Qualia problem and Brain in a vat argument
After all, we can believe in the following ”spirit of quantum language (Copenhagen
interpretation)”.
• To be is to be perceived♠Note 9.13. I believe that ”the above discussion has solved the ’brain in a vat,’” but one
might disagree with this. In fact, some of my friends are not convinced by the above
explanation. If you are one of them, I would like you to try the following problem.
(♯) What rules should be added to the Copenhagen Interpretation (Sec. 1.3.3) to put an
end to the ”brain in a vat”? The question is: What rules should be added to the
Copenhagen interpretation (Sec. 1.3.3)?
As I have said many times, the author believes that there is no perfect Copenhagen
Interpretation. What we can do is to make the Copenhagen interpretation as close toperfect as possible. The most interesting part of my position is to collect problems thatcannot be solved by any means, and to sort out the laws that they have in common.

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Chap. 9 Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, Hume9.5 Hume; skeptic who didn’t measure, ”A Treatise of Human
Nature”
9.5.1 Hume’s straying Brain pseudo-science (≈Less than brain science) ;Hume’s wordplay
Descartes takes the ”Cogito proposition [I think therefore I am]” as the ﬁrst propositionof philosophy, and from this he derives the ”existence of I”. And further, he asserted that
he can derive the followings:(A) ”Existence of Matter”, ”Existence of Body”, ”Existence of Causality”, ..., ”Existence of God”
Given modern common sense, the cogito proposition is not a scientiﬁc proposition, andeverything Descartes claimed to have derived from it is suspect.
However, these were just Descartes’ tricks and what Descartes really wanted to say wasto propose a mind-body dualism (consisting of ”mind”, ”body” and ”matter”).
But Descartes philosophy is a philosophy that risks entering into pseudoscience. In fact,
Hume fell into the zone of pseudoscience.
In ”A Treatise of Human Nature” (1739), Hume doubted (A) above. Probably, Humeassumed something like the ”brain in a vat” described in the previous section.

Figure 9.1 in Sec. 9.4.2: [Brain in a vat ]And he concluded that263

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9.5 Hume; skeptic who didn’t measure, ”A Treatise of Human Nature”(B) the existence of ”bundle of perceptions (= brain circuit)” is certain.
• [matter], [body], [causality] etc. are kinds of states of brain circuit

It is said that Hume’s philosophy is the achievement of British empiricism (i.e., nofurther progress can be expected).
I think the reason they say so is as follows.
(C1 ) Once we fall into the trap of the ”aquarium brain,” we cannot easily get out of it.
As shown in the previous section, the ”brain in a vat” can be solved only when themeaning of ”measurement” is clariﬁed.
(C2 ) The study of perceptual bundles (= brain circuits) is a matter of brain science,
which is impossible in Hume’s time.
In either of these two cases, Hume was 300 years too early.
If so, I am forced to conclude as follows:
regress

Descartes −−−−−→ Locke −−−−→ Humeprogress

This is because Descartes and Locke found three key words [mind, body, matter], which
are the essence of mind-body dualism, but Hume placed [body, matter] under [mind].
However, Hume is revived by Kant (see Kant: Copernican revolution in the next chapter). There, we see that
Descartes −−−−−→ Locke −−−−−→ Hume −−−−−→ Kant
progress

progress

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Chap. 9 Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, Hume9.5.2 Hume; The causality problem
It is a matter of course that the understanding of ”causal relation” is the most important
theme in worldview. In Newtonian mechanics, the causality is represented by Newtonian
kinetic equation. In Descartes=Kant philosophy, the representation of ”causal relation”
is as follows.
(♯1 ) [Cognitive causality]: David Hume, Immanuel Kant, etc. thought as follows. :We cannot say that ”causality” actually exists in the world, or that it does
not exist in the world. And when we think that ”something” in the world is”causality”, we should just believe that the it has ”causality”. Hume argues
that scientiﬁc wisdom is a product of ”habitual thinking” , not objective truth.

Most readers may regard this as ”a kind of rhetoric”, however, several readers may beconvinced in ”Now that you say that, it may be so.” Surely, since you are looking throughthe prejudice ”causality”, you may look such. This is Kant’s famous ”Copernican revolution” (i.e., ”Kant was awakened from his dogmatic slumber by Hume’s idea and came up
with the Copernican revolution”, this will be discussed in Sec. 10.2 [What is causality?]),that is,”cognition constitutes the world.”
which is considered that the cognition circuit of causality is installed in the brain, andwhen it is stimulated by ”something” and reacts, ”there is causal relationship.” Probably,
many readers doubt about the substantial inﬂuence which this (♯) had on the science after
it. However, Hume served as a bridge between Locke and Kant. Thus we think that

This will be discussed again in the next chapter (cf. Sec. 10.2 What is causality?).
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9.5 Hume; skeptic who didn’t measure, ”A Treatise of Human Nature”♠Note 9.14. (i): Note that
habitual thinking ( in Hume philosophy) ←−−−−−−−−−−→ Copenhagen interpretation (in QL)
correspondence(ii): In the book ”The astonishing hypothesis” (by F. Crick (the most noted for being a
co-discoverer of the structure of the DNA molecule in 1953 with James Watson)),Dr. Crick said that(♯1 ) You, your joys and your sorrows, your memories and your ambitions, your sense of
personal identity and free will, are in fact no more than the behavior of a vast assemblyof nerve cells and their associated molecules.
That is, he (as well as we) believed the monistic realism such as(♯2 ) the movement of the human spirit is also a kind of physical phenomenon.
However, Crick did not reject the mind-body dualism. Because I think that(♯3 ) British empiricism and brain science have nothing to do with each other.
which is also what Kant emphasized (though Hume may have confused pseudoscience withphilosophy).

Throughout this text, I argue that(♯4 ) it is the ”Copenhagen interpretation (i.e., how to use Axioms 1 and 2)”, not the ”brain
science”, that is related to British empiricism (≈ Descartes=Kant epistemology).

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Chap. 9 Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, Hume9.6 Hume’s problem of induction in the quantum mechanicalworldview
This section was written with reference to the following.
• [78]:Ishikawa, S: Philosophy of science for scientists; The probabilistic interpretationof science Journal of quantum information science, Vol. 9, No.3, 140-154,
DOI: 10.4236/jqis.2019.93007(https://www.scirp.org/Journal/paperinformation.aspx?paperid=95447)

9.6.1 If discussions are not based on a certain worldview, the discussions willturn into chit-chat, and will not be scientiﬁc
For example, consider the following inferences (= inductive inferences):(A1 ) Until now, the sun has always risen in the east. So, tomorrow the sun will rise in
the east again.
(A2 ) When a coin is thrown ten times, we get (H,H,T,H,H,H,H,H,H,T) (where
H=”head”, T=”tail”). Thus, we infer that ”H” will be obtained with probability
8/10 by the next coin-tossing(A3 ) White birds are swimming in a row. I can’t see the last bird very well, but it must
be a white bird.

Here, our current problem (= Hume’s problem of induction) is(B) Can we justify the above enumerative induction (i.e., inductive reasoning)?
If this is a problem of science, this is one of the most important problems in science.
Now, the position of this text is always the same(C) If discussions are not based on a certain worldview, the discussions will turn into
chit-chat, and will not be scientiﬁc
For example, when I discussed ”Zeno’s paradox (cf. Sec.2.4), I explained this spirit (C)to the point of endless exposition. Therefore, what we should do is(D) Within a quantum language (i,e., under a quantum mechanical worldview), justifying inductive inferences such as (A1 )∼(A3 ) above♠Note 9.15. It is important to doubt a certain worldview, but you need to be careful
about how you doubt it. For example, some may be interested in the justiﬁcations of thefollowings:

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9.6 Hume’s problem of induction in the quantum mechanical worldview(♯1 ) Newtonian mechanics has been right to this day. So, will the Newtonian mechanics
be right tomorrow?
(♯2 ) Quantum language have been very useful to this day. So, will quantum language be
useful tomorrow?
The justiﬁcation problem of these may be called Hume’s problem of induction, though
David Hume suspected the justiﬁcation of induction. It cannot be said that these aremeaningless. In fact, the following (♯3 ) was exactly Einstein’s interest.
(♯3 ) Newtonian mechanics has been right to this day. However, Is Newtonian mechanics
correct when particles move at very high speeds?
However, Einstein was praised for advocating the theory of relativity. He would not havebeen evaluated if he had only doubted as in (♯3 ). It is important to doubt, but if there is
no result, it is a word-play. In the end, we consider that there are two kinds of ”inductive
inference” as follows.
(♭1 ) Inductive reasoning under a certain worldview(♭2 ) Inductive reasoning for the validity of a certain worldview
Our interest is in (♭1 ), that is,
The Problem of Hume’s Induction in the Framework of Quantum Language
You are free to be skeptical about (♭2 ), but it will be accepted as a science only when it
produces results like Einstein.

9.6.2 The solution of Hume’s problem of induction; Without a worldview, nothing can be said
If Hume’s induction problem has a scientiﬁc aspect, then there must be scientists who
have clariﬁed this scientiﬁc aspect without knowing Hume’s induction problem. Thus,
I think it is natural to expect that the scientiﬁc part of Hume’s problem of inductionis almost identical to already well-known scientiﬁc results. That is, I consider Hume’s
problem of induction to be essentially equivalent to the law of large numbers.
♠Note 9.16. In Note 7.1, we say that Bernoulli’s achievement (i.e., the discovery of thelaw of large numbers) equals Galileo’s achievement. That is, I think that Bernoulli’s
achievement equals Galileo’s achievement. That is,Scientiﬁc pioneer in the realistic worldview · · · Galileo
Scientiﬁc pioneer in the idealistic worldview · · · J. Bernoulli

The reason for this is that Galileo-Newtonian mechanics can predict the future in serial
time, and Bernoulli’s law of large numbers has the power to predict the future in paralleltime. (cf. Sec. 9.2.3, or ref. [85]). Furthermore, in an essential sense, I think

Only these two theories have the power to predict the
future.
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Chap. 9 Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, Hume
Hume’s problem of inductive inference is a problem of predicting the future (in paralleltime), I think that Hume’s problem of inductive inference and Bernoulli’s law of large
numbers are closely related.

In this section, we show that the validity of Hume’s problem of induction is easily
resolved in the quantum mechanical worldview. First, ”Hume’s principle of the uniformityof nature” is the following assumption(E) [Hume’s principle of the uniformity of nature] Events that occur normallyin this world do not occur completely randomly, but there is some kind of order. If
it is conﬁrmed that the same phenomenon occurs over and over again under similar
conditions, the same phenomenon will occur again under similar conditions.
Of particular importance in (E) above is the part about ”Events that occur normallyin this world”. Therefore, I think that ”the problem of Hume’s induction” is as follows.
(F) Quantum language is a world view that describes events that occur normally in thisworld, so we can justify inductive reasoning by deﬁning the “uniformity principleof nature” under quantum language.
The theme of this text is the quantum mechanical worldview, and I assert:[To do science]=[To speak in quantum language]
Thus, we assume a quantum mechanical worldview as a worldview that describes theevents that normally occur in this world. Therefore, we must start from Deﬁnition 9.5 [The uniformity principle of nature ] in quantum language.
Deﬁnition 9.5. [The uniformity principle of nature] Quantum system is also acceptable, but for simplicity, the C ∗ -basic structure of the classical system is used. Consider[C(Ω) ⊆ B(L2 (Ω, ν))]. A family of measurements {MC(Ω) (Oi := (X, P(X), Fi ), S[ωi ] ) | i =−n, −n + 1, ..., −1, 0, 1, 2, ..., N } is said to be satisﬁed with the condition ”the uniformity
principle of nature”, if it satisﬁes that(♯) there exists a probability space (X, P(X), µ) that satisﬁes[Fi (Ξ)](ωi ) = µ(Ξ)∀Ξ ∈ P(X), ∀i = −n, −n + 1, ..., −1, 0, 1, 2, ..., N////

Under the above deﬁnition, we assert the following theorem (essentially the same as thelaw of large numbers).
Theorem 9.6. [Inductive inference(= inductive reasoning); (The solution of Hume’s
problem of induction] Consider a classical C ∗ -basic structure [C(Ω) ⊆ B(L2 (Ω, ν))].
Assume that A family of measurements{MC(Ω) (Oi := (X, P(X), Fi ), S[ωi ] ) | i =−n, −n + 1, ..., −1, 0, 1, 2, ..., N } satisﬁes the uniformity principle of natur Let
N(x−n , x−n+1 , ..., x−1 , x0 , x1 , ..., xN ) ∈ ×i=−n X be a measured value of the parallel269

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9.6 Hume’s problem of induction in the quantum mechanical worldview
measurementNN

i=−n MC(Ω) (Oi := (X, P(X), Fi ), S[ωi ] ). Then, it holds that♯{k | xk ∈ Ξ, k = −n, −n + 1, ..., −1}≈ µ(Ξ)(= [Fi (Ξ)](ωi ))nΞ ∈ P(X), i = −n, −n + 1, ..., −1, 0, 1, 2, ..., N(9.3)

where n is assumed to be a suﬃciently large number. Also, ♯[Θ] is the number of theelements of a set Θ.

Proof: The proof is simple, but just to be sure, it is described below.
. Let Ξi ∈ P(X) (i = −n, −n + 1, ..., −1, 0, 1, ..., N ). According to Axiom 1 [measurement](in Section 1.2.1), we say(♯) the probability that a measured value (x−n , x−n+1 , ..., x−1 , x0 , x1 , ..., xN ) of a parNNN
allel measurement i=−n MC(Ω) (Oi := (X, P(X), Fi ), S[ωi ] ) belongs to ×i=−n Ξiis equal to ×i=−n [Fi (Ξi )](ωi ) = ×i=−n µ(Ξi )N

N

Therefor, a sequence {xi }Ni=−n can be regarded as a sequence which is independently
selected from the probability space (X, P(X), µ). Therefore, by the law of large numbers,
we get (9.3).
After all, we may say

inductive inference≈ the quantum linguistic representation of the law of large numbers
NN♠Note 9.17. (i): For the idea that a parallel measurement
i=−n MC(Ω) (Oi :=(X, P(X), Fi ), S[ωi ] ) can be regarded as the time series of the family of measurements{MC(Ω) (Oi := (X, P(X), Fi ), S[ωi ] ) | i = −n, −n + 1, ..., −1, 0, 1, 2, ..., N }, see ref. [85].
(ii): The reader may wonder why philosophers have failed to formulate inductive reasoning? Because they did not ﬁrst declare a worldview and then formulate induction
under that worldview. We cannot say anything without a worldview ( such as quantum
language). Again, note that
• No formulation is possible without assuming a worldview.
• If you don’t assume a worldview, it can only be a chat270

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Chap. 9 Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, Hume9.6.3 Examples (Coin toss, Emerald)
Example 9.7. [Coin tossing]. [Coin tossing]. Let us discuss the unfair coin tossing as
the most understandable example of Theorem 9.6 [Inductive reasoning]. Consider a basic
structure [C(Ω) ⊆ B(L2 (Ω, ν))]. Here, Ω is the state space, ω(∈ Ω) is a state of ”how tothrow a coin”. Let {ωi }Ni=−n be a sequence in Ω, where ωi is the state of i-th coin tossing(i = −n, −n + 1, ..., 0, 1, 2, 3, ..., N ). Let O = (X, 2X , F ) be an observable in C(Ω) suchthat
X = {H, T }, (where H: head, U : tail) ,[F ({H})](ωi ) = µ({H}) = 2/3,

[F ({T })](ωi ) = µ({T }) = 1/3(∀i = −n, −n + 1, ..., −1, 0, 1, 2, ..., N )(9.4)

Therefore, a family of measurements{MC(Ω) (O := (X, 2X , F ), S[ωi ] ) | i = −n, −n +1, ..., −1, 0, 1, 2, ..., N } satisﬁes ”the uniformity principle of nature”.
N
Let (x−n , x−n+1 , ..., x−1 , x0 , x1 , ..., xN ) ∈ ×i=−n X be a measured value of the parallelNNmeasuremment i=−n MC(Ω) (O := (X, 2X , F ), S[ωi ] ).
Theorem 9.6 [Inductive reasoning] says that it is natural to assume that, for suﬃciently
large n,(x−n , x−n+1 , ..., x−1 , x0 ) = (T| H H T H H H{zT T ..... T H H})
n+1

where

♯[{i = −n, −n + 1, ..., 0 | xi = H}]/(n + 1) ≈ 2/3♯[{i = −n, −n + 1, ..., 0 | xi = F }]/(n + 1) ≈ 1/3
Therefore,xi = Hxi = T(probability 2/3)
(probability 1/3)

(i = 1, 2, ..., N )

(9.5)

(9.6)

Here, it should be noted that(♯) Even without knowing the condition (9.4)(i.e., µ({H}) = 2/3), we can know the
future (9.6) from the past data(9.5). future (9.6). However, if we have no information about the future, we cannot estimate the future. Note that we assume that
we know the information that both the past and the future will have the same
skewness.
////
Example 9.8. [Emerald]. In order to prepare for the next section, we will divide thediscussion into two parts: ”[I]: Solutions without quantum language” and ”[II]: Solutions
with quantum language”. All of these are too easy, but they are exercises in preparation
for the next section.
[I]: Solution without quantum language
Suppose there is an emerald mine M1 . Assume the following.
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9.6 Hume’s problem of induction in the quantum mechanical worldview(G) Assume that the probability that an emerald excavated at mine M1 is green isp(0  p ≤ 1). (However, the speciﬁc value of p is unknown.)
Unearth emeralds in the mine M1 .
Consider times −n, −n + 1, ..., −1, 0(= t0 ), 1, 2, ..., N . For simplicity, put n = 10, N =9. For example, the result of the emerald excavation is ”green at time −10 (=Y = Ygr )”and ”not green at time −9 (=N = Ngr )”, .... Suppose that the following table is obtained.
Table 9.5: Mine M1 ; time; -10, -9, -8,...,-1
time−10 −9 −8 −7 −6 −5 −4‘Green’, or ‘not Green’
Y
N
Y
Y
Y
Y
N

−3
Y

−2
Y

−1
Y

Then, from Table 9.5 we can infer,p ≈ 0.8
If so, we can infer, by the hypothesis (G), that at time t = 0, 1, 2, ..., 9, ..., the color of
the emerald that will be found in mine M1 is as follows.
Green probability p≈ 8/10Non-green probability (1 − p) ≈ 2/10
Thus, it is not surprising if we obtain the following table.
Table 9.6: Mine M1 ; time; 0,1,...,9time
0 1 2 3 4 5 6‘Green’, or ‘not Green’ Y Y Y N Y Y Y

7
Y

8
N

9
Y

[II]: Solution that uses quantum language
As above, consider assumption (G) for the excavation of emeralds in the mine M1 .
(G′ ) (=(G)) Assume that the probability that an emerald excavated at mine M1 is green
is p(0 ≤ p ≤ 1). (However, the speciﬁc value of p is unknown.)
Let us translate the above meaning into quantum language.
Consider a C ∗ NN
basic structure[C(Ω) ⊆ B(L2 (Ω, ν))].
a parallel measurement
i=−n MC(Ω)(Oi := (X, P(X), Fi ), S[ωi ] ). Here,Ω = {ω | ω is a state of an emerald},ΩM1 = {ω ∈ Ω | ω is a state of an emerald in mine M1 }Let ωi (∈ ΩM1 ) be an state of an emerald (discovered in mine M1 ) at time i(= −n, −n +1, ..., 0, 1, 2, ..., N ). Also, the observable O = (X = {Y, N}, 2X , F) in C(Ω) is satisﬁed with(H) [F({Y})](ω) = p[F({N})](ω) = 1 − p(∀ω ∈ ΩM1 )Assume thatNN• by the parallel measurement
i=−n MC(Ω) (Oi := (X, P(X), Fi ), S[ωi ] ), we get”green (= Ygr = Y ) at time −10”, ”not green (= Ngr = N ) at time −9”,,,,
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Chap. 9 Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, Humeand thus we get the following table:
Table 9.7 (≈ Table 9.5): Mine 1 ; time; -10, -9, -8,...,-1
time−10 −9 −8 −7 −6 −5 −4 −3‘Green’, or ‘not Green’
Y
N
Y
N
Y
Y
N
Y

−2
Y

−1
Y

···
···

The, we see, by the assumption (G′ ), thatp ≈ 0.8
It wouldn’t be surprising if this part [· · · ] in the above table looked like the table below.
Table 9.8 (= Table 9.6): Mine M1 time; 0,1,...,9time
0 1 2 3 4 5 6 7 8‘Green’, or ‘not Green’ Y Y Y N Y Y Y Y N

9
Y
////

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9.7
* Grue paradox cannot be represented in quantum language

9.7* Grue paradox cannot be represented in quantum language♠Note 9.18. I’ve read several books that explain Goodman’s arguments concerning ”grueParadox” (cf. ref. [25]), but I’ve never been impressed with them. Rather, I think thatthe grue paradox is a strange paradox that no one understands, yet everyone knows its
importance ( If this paradox wasn’t so important, it wouldn’t be so famous.) I think thatgrue Paradox evokes the inevitability of the quantum mechanical worldview. That is, grueparadox can not be understood without quantum language.

This section improves upon the following literature• [78]:Ishikawa, S: Philosophy of science for scientists; The probabilistic interpretationof science, Journal of quantum information science, Vol. 9, No.3, 140-154,
DOI: 10.4236/jqis.2019.93007(https://www.scirp.org/Journal/paperinformation.aspx?paperid=95447)

9.7.1 Grue paradox (explanation without quantum language)
Let us extend the discussion in Example 9.8 of the previous section a little. Suppose thatmine M2 is located near mine M1 . Assume the following.
(I1 ) In mine M1 , we assume that the probability that the excavated emerald is green isp(0  p ≤ 1).
Further,(I2 ) Suppose that the probability that an emerald excavated in mine M2 is blue is p.
This is a somewhat unnatural assumption, but not impossible. For example, if we consider
that a part of the original mine M1 became mine M2 due to tectonic movement, and thegreen emerald turned blue, the above condition is also possible.

First, let’s excavate emeralds at the mine M1 . Let’s take t0 (= 0) as a reference point.
For example, ”green (= Ygreen = Y ) at time −10”, ”not green (= Ngreen = N )at time−9”, .... , we obtain the following table.
Table 9.9: Mine M1 ; time; -10,-9,...,-1
time−10 −9 −8 −7 −6 −5 −4‘Green’, or ‘not Green’
Y
N
Y
Y
Y
Y
N

−3
Y

−2
Y

−1
Y

If so, by the assumption (I1 ), it is natural to considerp ≈ 0.8274

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Chap. 9 Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, HumeAnd further, if so, by the assumption (I2 ), we can know that the probability p in (I2 )is also equal to 0.8.
Thus, when excavating emeralds in mine M2 , we get, for example, ”blue (= Yblue = Y )at time 0, ... , not blue (= Nblue = N ) at time 3,...”, and so, we get the following table.
This is of course, natual.
Table 9.10 : Mine M2 ; time; 0,1,...,9time
0 1 2 3 4 5 6 7‘Blue’, or ‘not Blue’ Y Y Y N Y Y Y Y

8
N

9
Y

So far, the above makes some sense, even if there are some unnaturalness of (I1 ) and (I2 ).
However, the following opinion is natural.
(J) The above argument asserts the following• Since most of them were green before time t = 0, most of them are blue after
time t = 0

Is this really ”inductive reasoning” under ”the uniformity principle of nature” ?
Now rewrite ”the uniformity principle of nature (E)” in Sec. 9.6:
(E) (in Sec. 9.6) [Hume’s principle of the uniformity of nature] Events that
occur normally in this world do not occur completely randomly, but there is somekind of order. If it is conﬁrmed that the same phenomenon occurs over and overagain under similar conditions, the same phenomenon will occur again under similar
conditions.
This must have been the starting point (Hume’s original intention). The (J) above seems
to be far from the original intention.
Here, Goodman asserted (cf. ref. [25]) that(K) the (J) is surely ”inductive reasoning”
He argued for (K) with the following very original idea.
(L1 ) There is a dialect ”grue” in the area of mine M1 and mine M2 , which means ”green”
in the area of mine M1 and ”blue” in the area of mine M2 .
Put it this way.
(L2 ) (J) above can be rewritten as• If most of them are grue before time t = 0, most of them will be grue after
time t = 0 as wellThis would be in accordance with ”Hume’s original intention”. Nevertheless, is Goodman’s(K) correct? (proceed to next section).
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9.7
* Grue paradox cannot be represented in quantum language

9.7.2 Grue paradox (explanation with quantum language)
There are pros and cons to Goodman’s arguments [(L1 ) and (L2 )]. The reason there arepros and cons is that we do not yet understand ”inductive reasoning” very well. In sucha case, we can describe it in quantum language. Then we can know whether it is right orwrong.
Suppose that mine M2 is located near mine M1 . Assume the following as in the previous(I1 ) and (I2 ).
(I′1 ) In mine M1 , we assume that the probability that the excavated emerald is green isp(0  p ≤ 1).
Further,(I′2 ) Suppose that the probability that an emerald excavated in mine M2 is blue is p.
Then,Ω1 = {ω1 | ω1 is a state of an emerald in mine M1 }
Ω2 = {ω2 | ω2 is a state of an emerald in mine M2 }T
where, Ω1 Ω2 = ∅ (Strictly speaking, no two diﬀerent emeralds are in the same state)
Here, deﬁne the observable Ogreen= {{Y, N }, 2{Y,N } , F1green ) in C(Ω1 ) as
1
[F1green ({Y })](ω1 ) = p(∀ω1 ∈ Ω1 )Thus, we see:(M1 ) The probability that a measured value ”Y ” is obtained by the measurement
MC(Ω1 ) (Ogreen= {{Y, N }, 2{Y,N } , F1green ), S[ω1 ] ) is
1
[F1green ({Y })](ω1 ) = p(∀ω1 ∈ Ω1 )Similarly, deﬁne the observable Oblue= {{Y, N }, 2{Y,N } , F2blue ) in C(Ω2 ) as
2
[F2blue ({Y })](ω1 ) = p(∀ω2 ∈ Ω2 )Thus, we see:(M2 ) The probability that a measured value ”Y ” is obtained by the measurement
MC(Ω2 ) (Oblue= {{Y, N }, 2{Y,N } , F2blue ), S[ω2 ] ) is
2

Here, putting Ω = Ω1
C(Ω) by[F grue ({Y })](ω) =S[F2blue ({Y })](ω2 ) = p(∀ω2 ∈ Ω2 )

Ω2 , deﬁne the observable Ogrue = {{Y, N }, 2{Y,N } , F grue ) ingreen({Y })](ω) (ω ∈ Ω1 ) [F1[F2blue ({Y })](ω)(ω ∈ Ω2 )and F grue ({N }) = 1 − F grue ({Y }).
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Chap. 9 Modern philosophy: British empiricism: Locke, Leibnitz, Berkeley, Hume
Let’s take t0 (= 0) as a reference point. At time t = −n, −n + 1, ..., −1, 0, 1, ..., N ameasurement MC(Ω1 S Ω2 ) (Ogrue , S[ωi ] ) is taken. (Here, ωi ∈ Ω1 (i = −n, −n + 1, ..., −1),ωi ∈ Ω2 (i = 0, 1, ..., N )).
N
That is, a parallel measurement i=−n,−n+1,...,0,...,N MC(Ω1 S Ω2 ) (Ogrue , S[ωi ] ) is taken.
For example, assuming n = 10, N = 9, we get ”grue (= Ygrue = Y ) at time −10, notgrue(= Ngrue = N ) at time −9, ....” Then, we get the following table.
Table 9.11 (≈ Table 9.7): Mine 1 ; time; -10, -9, -8,...,-1, 0,1,...,9time−10 −9 −8 −7 −6 −5 −4 −3 −2‘Grue’, or ‘not Grue’
Y
N
Y
N
Y
Y
N
Y
Y

−1
Y

···
···

If so, from the assumption (I′1 ), it is reasonable to infer the following .
p ≈ 0.8
Furthermore, if so, from the assumption (I′2 ), it is reasonable to infer that the part [· · · ]in the above table is as follows.
Table 9.12 (≈ Table 9.8): Mine M2 time; 0,1,...,9time
0 1 2 3 4 5 6 7 8‘Grue’, or ‘not Grue’ Y Y Y N Y Y Y Y N

9YThus we can conclude that(N) Goodman’s arguments [(L1 ) and (L2 )] is true

The readers here may not need any more ”seediness”, but I will mention it here in casethey are interested. To restate Deﬁnition 9.5 [The principle of the uniformity of nature],
we see(O) Consider [C(Ω) ⊆ B(L2 (Ω, ν))]. A family of measurements {MC(Ω) (Oi :=(X, P(X), Fi ), S[ωi ] ) | i = −n, −n + 1, ..., −1, 0, 1, 2, ..., N } is said to be satisﬁedwith the condition ”the uniformity principle of nature”, if it satisﬁes that(♯) there exists a probability space (X, P(X), µ) that satisﬁes[Fi (Ξ)](ωi ) = µ(Ξ)∀Ξ ∈ P(X), ∀i = −n, −n + 1, ..., −1, 0, 1, 2, ..., NThis says• the measurable space (X, P(X)) must be common, but Fi and ωi are arbitrary
but it must satisfy the condition (♯).
In short, if we rely on the quantum language Deﬁnition 9.5 (uniformity principle ofnature) instead of the vague ”Hume’s uniformity principle of nature (E),” we knowthat Goodman’s argument is correct.
♠Note 9.19. In the above discussion, the reader should have easily understood how Grue’s
paradox works and how interesting it is. The author believes that it is not easy to understand the workings and interest of Grue’s paradox without using the quantum language
formulation.

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279

Chapter 10

Kant: Copernican revolution[I]: The following is ”Kantian synthesis”, which is called the ﬂower of modern philosophy:(♯1)


Deduction (due to Descartes)Continental Rationalism
mind-body problem




•
(nativism)Descartes[Spinoza,Leibniz](the father ofmodern phil. −→Induction (due to Bacon)subjectivity problem

British Empiricism




•
(tabula rasa)
[Locke, Berkeley, Hume]















−−−−−−−→ Kant
(synthesis)













I think that Kant dreamed of a good relationship between philosophy and natural science.
Speciﬁcally, he dreamed that philosophy would be the foundation of Newtonian mechanics. In
other words, his dream was to create the philosophical foundation of Newtonian mechanics, andhe tried to realize it by ”the Copernican revolution (regarding cognition)”. He wanted to write
about this dream in ”Critique of Pure Reason”. In other words
Knowledge gained from various experiences
(British empiricism)
Kantian−−−−−−−→ Critique of Pure Reasonsynthesis
Innate reason integrating miscellaneous knowledge (Continental rationalism)

Although his challenge was not successful theoretically, his work was very successful as a kind ofenlightening book that promoted a good relationship between philosophy and natural science.
In other words, I would like to think that his work had a great inﬂuence on the acceptance ofKSTS/RR-22/001
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natural science by the general public.
However, the above ” Synthesis of Continental Rationalism and British Empiricism” will be
realized in the following ”QL Synthesis”.
From a scientiﬁc point of view, I believe that this (♯1 ) should be replaced by the following QL
synthesis (♯2 ) (=Quantum Linguistic Synthesis).*1(♯2) QL-synthesis






Develop Descartes theory Leibniz
Spinoza(=mind-body
dualism)
Monadology• Pantheism , •



Solve






the mind-body problem
• −−−−−−−−−−−−−−−−−−−−−→

Axiomatic theory

QL






 WittgensteinContinental
rationalism


Axiom
1

Descartes theory




•
TLP

,
Axiom
2



mind-body dualism 





Derived from cogito















How
to
use
Locke,
Berkley,
QLDescartes theory
epistemology
linguisticSolve the subjectivity(Copernican
Copenhagen problem, etc.







revolution)≈interpretation
•
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
→



Empirical
theoryHume, Kant

British empiricism




QL

Figure 9.1: QL-synthesis(However, attempts at Axiomatic theory were too diﬃcult at the time)*1 This was, for the ﬁrst time, proposed in ref.

[84]: S. Ishikawa, (2022) Empiricism, Rationalism,
and Kantian synthesis in quantum linguistic point of view OJPP, Vol. 12, No. 2, 182-198,(https://www.scirp.org/journal/paperinformation.aspx?paperid=117114)

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Chap. 10 Kant: Copernican revolution[[II]: But miracles do happen. This was not to say that Kant’s ”Copernican revolution” was abad idea.
• With the ”Copernican revolution,” Kant proposed scientiﬁc

idealism (the so-called ”transcendental idealism”).
That is, Descartes, Locke, and Hume strictly speaking belong to ”pseudo-scientiﬁc realism” andnot scientiﬁc idealism. Kant discovered another conceptual science, which is diﬀerent from therealistic science (=physics). Quantum language is one of the most successful examples of theCopernican revolution (= scientiﬁc idealism). (cf. Sec. 11.7 in Chapter 11).
Therefore, if Kant said this, we would agree.

Philosophy (idealism) is as useful and great as physics, even if
limited to the scientiﬁc point of view.

Summing up,• Kant’s synthesis is enlightenment, but the Copernican revolution is scientiﬁc.

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10.1 Critique of Pure Reason

10.1 Critique of Pure Reason
10.1.1 Three Critiques

Immanuel Kant (1724 - 1804), a professor of at the University of Königsberg, was oneof the most inﬂuential philosophers in the history of Western philosophy. His main workis ”Critique of Pure Reason (1781)”, ”Critique of Practical Reason (1788)”, ”Critique ofJudgment(1790)”, whose theme is respectively ”truth” (i.e., ”pseudo- truth” in the senseof this text), ”virtue”, ”beauty”.
Kant, who is at the top of the three ﬁelds, may look like a superhuman (for example,
just as one person simultaneously wins three Nobel Prizes (physics, chemistry, physiology
or medicine), however, in the case of philosophy, the three (”truth, goodness, and beauty”)
are one. As I said before, concerning ”good” and ”beauty”, most philosophers have similararguments. Therefore, they appeal a clear diﬀerence in the part of ”true”. It is the wisdomof Western philosophy (not found in Eastern philosophy) to keep fresh by model change.
That is, he followed the formulation of an idealistic worldview (cf. Sec.1.1) as follows.
”Critique of Practical Reason (1788)””Critique of Judgment(1790)””Critique of Pure Reason (1781)”(A1 )ﬁctional worldview (literary truth)
preface, introduction, (ﬁctional)premise, expedient

therefore−−−−−→

ethics, morals, aesthetics
main subject

Of course, the interest of this text is concentrated to the worldview (i.e., ”Critique ofPure Reason” ). Have said many times in this text, philosophy of the worldview is a”preface” of (A1 ). Thus, it must be built with as ”meticulous logic” as a detective novel.
In fact,(A2 ) The Critique of Pure Reason succeeds as literature (or, enlightenment philosophy)
in impressing upon its readers that it has succeeded in laying the foundation for
Newtonian mechanics.
Considering the state of the world at that time, it is not surprising that it was favorably
accepted by the general public. What kind of arguments Kant used to draw this conclusion
is a testament to his skill. In fact, as a philosophy of the Enlightenment, it was a great
success.
However, even the meaning of such things as ”the foundation of Newtonian mechanics”
is unclear, and there was no chance whatsoever that the ”Critique of Pure Reason” would
succeed scientiﬁcally.
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Chap. 10 Kant: Copernican revolution(A3 ) The Critique of Pure Reason (whose purpose is the ”Enlightenment of Science”)
and Quantum Language (whose purpose is the ”Language of Science”) have little
in common.
♠Note 10.1. As mentioned frequently up to this point, the idealistic worldview is really”asserted ﬁction”, however, we must pretend not to regard the ﬁctional worldview as”non-logical”, or we must accept it as ”logical in a wide sense”. That is because it
must be prohibited that the diﬀerence between philosophy and religion becomes fuzzy.
Therefore, philosophers must use the terms such as ”logic”, ”reason”, etc. a lot in theidealistic worldview (e.g., ”Critique of Pure Reason”, ”Tractatus Logico philosophicus(=TLP)”, etc.). I think that the formulation of an idealistic worldview (cf. Sec.1.1) isa survival strategy to prevent Western philosophy from being swallowed by Christianity.
Philosophers have to say ”logical” even if it is a lie. Of course, philosophy enthusiasts
know this and enjoy reading it, so it may not be a ”lie”.
♠Note 10.2. It is often held that Kant lived a very strict and disciplined life, leading to
an oft-repeated story that neighbors would set their clocks by his daily walks. In fact, he
may have been a cheerful and sociable man. However, the image of Kant as strict andhonest promotes an understanding of Kant’s philosophy. For the philosopher, his image
itself is a part of his work. Therefore, if you have seen a painting of Kant chatting withfriends at a luncheon, it is better to pretend you did not see it and forget about it. To
have a strong impact on his word ”Copernican Revolution,” Kant must be a modest andnon-joking philosopher.

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10.1 Critique of Pure Reason

10.1.2 The purpose of ”Critique of Pure Reason” (1781)

The following is a story that philosophy buﬀs will love.
(B1 ) with the advent of Newton, the natural sciences have grown exponentially. People
believe in that systematic knowledge as ”objective truth” and come to have great
expectations. It was Hume who made a bitter blow to it. He argues that(♯) scientiﬁc wisdom is a product of ”habitual thinking” , not objective truth.
Kant is shocked by Hume’s opinion. One of the central issues of ”Critique of Pure
Reason” is to rebuild the objectivity and reliability of the natural sciences that wereshattered by Hume.

Therefore,(B2 ) Kant’s purpose is to answer the following• Why does Newtonian mechanics hold?
• Why can Newtonian mechanics be understood?
There is no way that philosophy can answer such a question, but Kant has come up with asolution that satisﬁes philosophy enthusiasts. Of course, it does not have to be a solutionthat satisﬁes physicists. Because, remember, modern philosophy is philosophy for thegeneral public.
♠Note 10.3. (i): If we replace ”habitual thinking” with ”Copenhagen Interpretation” in(♯) in (B1 ) above, i.e.,(♯) scientiﬁc wisdom is a product of ”Copenhagen Interpretation” , not objective truth,
this is true for the explaination of quantum language. Thus, it is tempting to think that
Kant envisioned a quantum language, not Newtonian mechanics.
(ii): Next is Einstein’s words:• The most incomprehensible thing about the world is that it is comprehensible.
Or
One may say ”the eternal mystery of the world is its comprehensibility.” It is oneof the great realizations of Immanuel Kant that the postulation of a real external
world would be senseless without this comprehensibility. (on page 292 of ref. [18]:284

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Chap. 10 Kant: Copernican revolution
Ideas and Opinions, Crown Publishers, Inc . . New York 1954)

Note that the above (B2 ) is essentially the same as the following:• Why is the world comprehensible?
With Einstein’s endorsement above, it seems certain that Kant is the discoverer of ”themost incomprehensible problem in the history of science” even if his theory is scientiﬁcallyinadequate.
I don’t think any researcher would challenge this today, because everyone would thinkthat such a problem is impossible to solve. This is a problem to enjoy not knowing, not a
problem to solve.
(iii): To mount against Newtonian mechanics, Kant may have proposed the following
question:• Why does Newtonian mechanics hold? Why can we understand Newtonian mechanics?
Even if the answer is uncertain, it is better to just discuss it. We may have the feelingthat ”philosophy is better than physics.” If so, it is a success for the philosophy of the
Enlightenment.
(iv): As seen in Chap. 12 later, I assert that the purpose of Wittgenstein’s TPL (i.e., ref.
[125]: ”Tractatus Logico-Philosophicus”) is to explain• Why does logic work in our usual world?
My solution to this problem is not enlightening.

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10.1.3 Thing-in-itself, Copernican revolution; from copy theory to constitution
theory, Kantian synthesis
In order to solve the problem (B), Kant thought that(C) We can understand the ”world” only through the human perception. Also, cats canunderstand the ”world” only through the cat perception. Thus,• There is ”cat’s world” for cats. and further, there is ”butterﬂy’s world” forbutterﬂies.

If there are aliens who have better cognitive abilities than we humans do, their worldis diﬀerent from ours. Although the diﬀerence of the worlds is made by that of thecognitive ability, it is sure there exists something, which is called ”thing-in-itself”
by Kant.
That is, Kant thought as follows.
(D) we do not perceive the world as copy, but we perceive the world as it is constituted
by cognition ability.

That is, Kant proposed so called Copernican revolution such that”copy theory” −−−−−−−−−−−−−−−−−→ ”constitution theory”
realism
Copernican revolution (transcendental) idealism
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Chap. 10 Kant: Copernican revolution
namely,(E1 ) It’s not ”the world ﬁrst, cognition later”, but ”cognition ﬁrst, the world later”.
or(E2 ) Cognition is not about painting a photorealistic picture, but it is similar to painting
an abstract picture.
Then our problem is as follows.
Problem 10.1.
[Kant’s metaphysical foundation for Newtonian mechanics (natural science)]:
Assume the above Copernican revolution is true. Then, study the following problems:(F1 ) Why can we make and understand Newtonian mechanics?
(F2 ) Is there a question we can’t answer?

[A rough sketch of Kant’s Answer for (F1 )]:In order to understand Newtonian mechanics, we need the scientiﬁc ability, that is,(G1 ) Sensibility: The cognitive ability to organize the various sensations received
through the sense organs within the framework of ”space-time”(G2 ) Understanding: The ability to judge and understand material obtained throughsensibility based on concepts such as quantity and cause-eﬀect relationships.
(G3 ) These abilities (sensitivity and understanding) are a priori, i.e., innate, and should
be common to all human beings.
And,(G4 ) With experience and training, these abilities will increase.

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Hence, our habitual thinking is also common to all human beings at its core, and in thissense, it is universal. Therefore, we can understand Newton mechanics ( though
it depends on our sensibility and understanding).

[A rough sketch of Kant’s Answer for (F2 )]:Kant describes four examples of unanswerable problems.
(H1 ) Is the universe ﬁnite or inﬁnite?
(H2 ) Does the atom exist?
(H3 ) Does free will exist?
(H4 ) Does God exist?
Kant proves these by deriving antinomies, but this is a play on words rather thana proof. If a proof is to be given, it must be preceded by the declaration of a premise(or worldview), but since no premise (or worldview) has been declared, it is not a proof
but a play on words.
The above proof is not a scientiﬁc one. However, it is certain that many philosophicalfans agreed with Kant’s explanation (G) in Problem 10.1. In particular(G3 ) · · · Sensibility, Understanding (nativism)(G4 ) · · · experience and training (experiment)This is usually called ”Kantian synthesis” such as(I) Descartes[mind-body dualism]−→[ tabula rasa , experience ]






British Empiricism
 [Locke, Berkeley, Hume][ nativism , a priori ]







 Continental Rationalism[Spinoza, Leibniz]














Kantian[”Critique of Pure Reason”]−−−−−−−→
Kant philosophy
synthesis[summing-up,
compromise]









In this sense, it is usually said that Kant integrated British empiricism and continental
rationalism.

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Chap. 10 Kant: Copernican revolution

10.1.4 Summary
As we can see from the above discussion,(J) the purpose of Kant’s Critique of Pure Reason (= CPR) was to provide a foundation
for Newtonian mechanics from a philosophical perspective.
To begin with, this objective was a scientiﬁcally unintelligible objective. However, byadding ”common human sensitivity and understanding” to subjectivism such as Hume’sbundle of perceptions, it was concluded that humans possess the intelligence to understand
Newtonian mechanics. This is an argument ”with a conclusion at the beginning,” and ”thecommon sensitivity and understanding of mankind” is only an afterthought. Nevertheless,”CPR” was a great success because it was the ”philosophy of the Enlightenment. It isbecause the general public was eager to know ”the position of Newtonian mechanics” and”CPR” proposed ”the position of Newtonian mechanics” as the general public wished.
After all,Get along with science!
To begin with, philosophy for the general public would have to be Enlightenment philosophy to be a hit.

No one would dispute that Kant’s philosophy was a great success as an Enlightenment philosophy. However, it is not easy to evaluate Kant’s philosophy from a scientiﬁc
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10.1 Critique of Pure Reason
viewpoint.
(K) Is the following true?
Locke −−−−−−−−−→ Kant *2
progress (QL)

The Critique of Pure Reason is pseudo-scientiﬁc, and it is a bit of a stretch to agree withit.
However, if we think in terms of ”language” rather than ”recognition,” it was as simple
as shown in Note 9.3. That is,
Repost of [Note 9.3]
Thinking in terms of ”language” instead of ”recognition”(♯1 ) ”Everyday language” is similar to tabla-rasa. When a baby is born, it does not know any
ordinary language at all (= tabula rasa). The baby acquires everyday language throughrepeated trial and error.
(♯2 ) Mathematics is similar to Continental Rationalism. All mathematical theorems can beproved (in principle) if only the axioms of set theory are determined.
(♯3 ) The quantum language is similar to the compromise between (♯1 ) and (♯2 ) above as
follows.
rationalism

Quantum language =Axiom 1 ( measurement)
Axiom 2 ( causality)(empiricism)linguistic+ Copenhagen interpretation

Also, as noted in (♯) of (B1 ) (or, (i) in Note 10.3), the following is a central claim ofquantum language.
(♯) scientiﬁc wisdom is a product of ”Copenhagen Interpretation” , not objective
truth,
Therefore, if we consider that Kant wrote the spirit of quantum language (i.e., the theoryof the reach of dualistic idealism) in the ”Critique of Pure Reason,” Kant’s reputation
rises to the highest level. If philosophy fans also read ”Critique of Pure Reason” as a bookof dualistic idealism, we can understand why they were so enthusiastic about it. If we takethis into account and judge the overall situation, we can agree with (k) above. In addition,
this chapter does not elaborate on Kant’s achievement, the Copernican revolution.
After the discussion on the Copernican turn, we have to answer (K), and the ﬁnal
answer is given in Answer 11.22 of Sec. 11.7.

*2 ”X −−−−−−−−−−→ Y ” means ””Y” is closer to quantum language than ”X”.

progress(QL)

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Chap. 10 Kant: Copernican revolution

10.2 * Kant’s Copernican revolution: What is causality?
Sec. 11.7 will discuss the details of the Copernican revolution [II] in detail, but in thissection, I will give a light introduction to the Copernican revolution in connection with”causality”.
As mention in Sec 4.2, Aristotle considered the cause of the movement to be the ”purpose” of the movement. Although this was what should be praised, it was not able to besaid that ”the purpose was to the point.” For human beings to discover that the essenceof movement and change is ”causal relationship”, we had to wait for the appearance ofGalileo, Bacon, Descartes, Newton, etc.
Revolution to ”Causality” from ”Purpose”
is the greatest paradigm shift in the history of science. It is not an overstatement even if
we call it ”birth of modern science”.
the birth of worldview

Movement

the birth of modern science”purpose”−−−−−−−−−−−−−−−−−−−→(Hērakleitos, Parmenides, Zeno) Aristotle :(About 1500 years)

Causality( Galileo, Bacon, Descartes, Newton)♠Note 10.4. I cannot emphasize too much the importance of the discovery of the term:”causality”. That is,(♯1 ) Science is the discipline about phenomena can be represented by the term ”causality”.
Thus, I consider that the discovery of ”causality” is equal to that of science. However, my
opinion is not the above (♯1 ) but(♯2 ) Science is a discipline that describes the world in terms of ”measurement” and ”causality”.

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10.2* Kant’s Copernican revolution: What is causality?

10.2.1 Four answers to ”what is causality?”
As mentioned above, about ”what is an essence of movement and change?”, it wasonce settled with the word ”causality.” However, not all were solved now. We do not yet
understand ”causality” fully. In fact,
Problem 10.2. Problem:”What is causality?”is the most important outstanding problems in modern science.
Answer this problem!

There may be some readers who are surprised with saying like this, although it is theoutstanding problems in the present. Below, I arrange the history of the answer to thisproblem.
(A) [Realistic causality]: Newton advocated the realistic describing method of Newtonian mechanics as a ﬁnal settlement of accounts of ideas, such as Galileo, Bacon,and Descartes, and he thought as follows. :”Causality” actually exists in the world. Newtonian equation described faithfully this ”causality”. That is, Newtonian equation is the equation of a causal
chain.

This realistic causality may be a very natural idea, and you may think that you cannot
think in addition to this. In fact, probably, we may say that the current of the realistic
causal relationship which continues like
”Newtonian mechanics−→ Electricity and magnetism−→ Theory of relativity−→ · · · ”
is the mainstream of science.
However, there are also other ideas, i.e. three ”non-realistic causalities” as follows.
(B) [Cognitive causality]: David Hume, Immanuel Kant, etc. thought as follows. :We can not say that ”Causality” actually exists in the world, or that it does
not exist in the world. And when we think that ”something” in the world is”causality”, we should just believe that it has ”causality”.
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Chap. 10 Kant: Copernican revolution

Most readers may regard this as ”a kind of rhetoric”, however, some readers may believe
it. It may look like that, because you are looking through the prejudice of ”causality.” This
is Kant’s famous ”Copernican revolution” (i.e., ”Kant was awakened from his dogmatic
slumber by Hume’s idea and came up with the Copernican revolution”), that is,”cognition constitutes the world.”
which is considered that the cognition circuit of causality is installed in the brain, andwhen it is stimulated by ”something” and reacts, ”there is causal relationship.”♠Note 10.5. About his discovery of ”the Copernican revolution”, Kant says in his book
”Prolegomena” (1783):(♯) I freely admit that it was the remembrance of David Hume which, many years ago, ﬁrst
interrupted my dogmatic slumber and gave my investigations in the ﬁeld of speculative
philosophy a completely diﬀerent direction.
Readers may ask, ”Why did Kant, an honest and humble man, make such an exaggerated
statement?” It is a matter of course that Kant had great conﬁdence such that it was thegreatest discovery in the history of philosophy. I agree to his opinion. For additional
explanation about this, see Problem 10.5 later. Also, see Section 11.5.2.

(C) [Mathematical causality(Dynamical system theory)]:293

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10.2* Kant’s Copernican revolution: What is causality?
Since dynamical system theory has developed as the mathematical technique inengineering, they have not investigated ”What is causality?” thoroughly. However,In dynamical system theory, we start from the state equation (i.e., simultaneous
ordinary diﬀerential equation of the ﬁrst order) such that dω1dt (t) = v1 (ω1 (t), ω2 (t), . . . , ωn (t), t) dω
2dt (t) = v2 (ω1 (t), ω2 (t), . . . , ωn (t), t)······ dωndt (t) = vn (ω1 (t), ω2 (t), . . . , ωn (t), t)(10.1)

and, we think that(♯) the phenomenon described by the state equation has ”causality.”This is the spirit of dynamical system theory (= statistics). Although this is proposed
under the confusion of mathematics and worldview, it is quite useful. In this sense, I thinkthat (C) should be evaluated more.
(D) [Linguistic causal relationship (Measurement Theory)]:

The causal relationship of measurement theory is decided by the Axiom 2 (causality;
Sec. 1.1) of Chap. 1. If I say in detail,:• Although measurement theory (= quantum language) consists of the two Axioms 1 and 2, it is the Axiom 2 that is concerned with causal relationship. Whendescribing something in quantum language and using Axiom 2 (causality; Sec.
1.1) , we think that it has causality.

Summary 10.3. The above is summarized as follows.
(A) World is ﬁrst
(B) Recognition is ﬁrst
(C) Mathematics(buried into ordinary language) is ﬁrst
(D) Language (= quantum language) is ﬁrst

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Chap. 10 Kant: Copernican revolution
Now, in measurement theory, we assert the next as said repeatedly:
Quantum language is a basic language which describes various sciences.
Supposing this is recognized, we can assert the next. Namely,

In science, causality is just as mentioned in the above (D).

This is my answer to ”What is causality?”.

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10.3 From Kantian synthesis to quantum linguistic synthesis

10.3 From Kantian synthesis to quantum linguistic synthesis
In Preface, I said that
Without seeing the ﬁnished product, it is impossible to know exactly what it is.
A typical example of this is Kant’s synthesis. Let us discuss this below.
It is usually said that there were two trends in modern philosophy. That is,Continental rationalismDescartes, Spinoza, Leibniz, ...
•British empiricism

Bacon, Descartes, Locke, Berkley, Hume, ...
And the following is the established theory.
(A1 ) ”Continental rationalism vs. British empiricism” is due to ”nativism vs. tabularasa”. And Kant synthesized the two.
That is, The following is called the ﬂower of modern philosophy:(A2 ) : [Establish theory]: Kantian synthesis
Deduction (due to Descartes) 









Continental
Rationalism
mind-body problem






•
(nativism)Descartes








[Spinoza,Leibniz](the father of (nativism) +(experience)−−−−−−−−−−−−−−−−−−−→ Kant
modern phil. −→(Kantian synthesis)Induction(due
toBacon)

subjectivity problem






British Empiricism










•
(tabularasa,
experience)






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Chap. 10 Kant: Copernican revolution
More precisely,

I think that Kant dreamed of a good relationship between philosophy and natural science.
Speciﬁcally, he dreamed that philosophy would be the foundation of Newtonian mechanics. In other words, his dream was to create the philosophical foundation of Newtonianmechanics, and he tried to realize it by ”the Copernican revolution (regarding cognition)”.
He probably wanted to write about his dream in the Critique of Pure Reason. That is,
The innate human reason to integrate
miscellaneous knowledge(Continental rationalism)

Miscellaneous knowledge gained
from various experiences(British empiricism)












−−−−−−−−−−−−−−→ Critique of Pure Reason
Kantian synthesis









Although his challenge was not scientiﬁcally successful, his work was very successful asa kind of enlightening book that promoted a good relationship between philosophy andnatural science. In other words, I would like to think that this work had a great inﬂuenceon the acceptance of natural science by the general public.

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10.3 From Kantian synthesis to quantum linguistic synthesis
However, the oppositional structure [nativism vs. tabula rasa] seems to be pseudoscientiﬁc. Thus, in this text I think that
(B1 ) *3

QL
Axiomatic theory

Continental rationalism −−−−−−−−−−−−−−→

Axioms 1 and 2

Empirical theory

British empiricism −−−−−−−−−−−−−−→

Copenhagen interpretation

Here, recall that
Axiom 1 ( measurement)
QL
linguistic(B2 ) (quantum language) =Axiom 2 ( causality)+ Copenhagen interpretation
Thus, (B1 ) implies that

(C): QL synthesis






Develop Descartes theory Leibniz
Spinoza(=mind-body dualism)Monadology
Pantheism



, •
•



Solve






the mind-body problem
• −−−−−−−−−−−−−−−−−−−−−→

Axiomatic theory

QL





WittgensteinContinental
rationalism
Axiom 1Descartes theory





TLP


, Axiom 2•
mind-body dualism 





Derived from cogito















How
to
use
Locke,
Berkley,
QLDescartes theory
epistemology
linguistic




the
subjectivity
Solve




(Copernican
Copenhagen
problem,
etc.





revolution)• −−−−−−−−−−−−−−−−−→≈ interpretationEmpirical theory


Hume,
Kant


British empiricism




QL

Figure 10.1(=Figure 9.1): QL-synthesis
We have not yet discussed Wittgenstein, but will do so in chapters 11 and 12.

*3 This was, for the ﬁrst time, proposed in ref.

[84]: S. Ishikawa, (2022) Empiricism, Rationalism,
and Kantian synthesis in quantum linguistic point of view OJPP, Vol. 12, No. 2, 182-198,(https://www.scirp.org/journal/paperinformation.aspx?paperid=117114)

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Chap. 10 Kant: Copernican revolution
That is,Since the diﬃculty of axiom 1 (measurement) of QL is almost the same as the diﬃcultyof quantum mechanics, they (Spinoza, Leibniz, Wittgenstein) tried something that was
too diﬃcult for the level of science at the time, and of course they did not achieve any
scientiﬁc results. However, many philosophy enthusiasts were moved by the way theysincerely challenged the most important and diﬃcult problem. Their philosophies are
scientiﬁcally understood to anyone (perhaps even to themselves), yet they are celebrated
by many philosophy enthusiasts, a strange thing that continues to this day.
British empiricism philosophers, on the other hand, have had some success. The taskof creating a Copenhagen Interpretation (= manual for the use of Axiom 1) without
knowing Axiom 1 may seem impossible, but their results were not so far oﬀ the mark. I
don’t know exactly why such a miracle occurred, but I ﬁnd it very interesting. It is very
interesting that it was possible to imagine dualistic idealism without knowing the axioms(i.e., principle).
With the above discussion, I reject the Kantian synthesis and adopt the QL synthesis
as follows.

However, it was Kant who ﬁrst realized that continental rationalism and British empiricism were to be integrated.
QL synthesis explains the following question:(D) Why does modern philosophy consist of two theories (i.e., continental rationalism
and British empiricism)?
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10.3 From Kantian synthesis to quantum linguistic synthesis
That is,(E) the two theories (i.e., continental rationalism and British empiricism) are actually
one theory such as (B).
Remark 10.4. Summing up, in chapters 8,9,10, we see that*4(♯1 ) Descartes: mind-body dualism (mind-body problem and subjectivity problem are
the most important questions in modern philosophy)(nevertheless, cogito proposition is a kind of wordplay)(♯2 ) Locke: secondary quality (≈ universal by Anselmus) is the most important in dualistic idealism(nevertheless, ”tabula rasa” is nonsense)(♯3 ) Leibniz: relationalism of space-time is a kind of Copenhagen interpretation(nevertheless, nativism is nonsense, and monadology is incomprehensible)(♯4 ) Berkeley: ”To be is to be perceived” is the most important spirit of QL(nevertheless, his subjective idealism is rather religion)(♯5 ) Hume: the Copenhagen interpretation is a kind of ”habitual thinking”(nevertheless, ”a bundle of perceptions” is a pseudoscientiﬁc ﬁction)(♯6 ) Kant: transcendental idealism is the greatest discovery in modern philosophy (thiswill be discussed in Se, 11.7)(nevertheless, ”thing-in-itself” is a kind of wordplay, also, Kantian synthesis is
his pseudoscientiﬁc ﬁction)

These achievements are the essence of the linguistic Copenhagen interpretation (= how
to use Axiom 1 (measurement) and Axiom 2 ( causality). What follows is astonishing.
(♭) they arrived at the essence of the Copenhagen interpretation without
knowing Axioms 1 and 2.
On the other hand, it was too diﬃcult to propose an axiom theory to solve the mind-bodyproblem. Even with geniuses such as Spinoza, Leibniz, and Wittgenstein, they could not
move a millimeter forward.
However, the following problems still remain.
Problem 10.5. If Kantian synthesis is Kant’s pseudoscientiﬁc ﬁction, is the next
”progress” believable?

*4 When I was young, I was taught by my teacher that researcher’s achievement should be evaluatedby their ”max”.

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Chap. 10 Kant: Copernican revolution
Descartes, Locke, Berkeley, Hume −−−−−→ Kant
progress

Or is Kant really great in the ﬁrst place?
This will be answered in Answer 11.22

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303

Chapter 11

Linguistic philosophy (Before TLP)
12
In this chapter, we discuss analytic philosophy and its surroundings (the [gray] and ○,13 ○14 ) in Figure 0.1 (in Preface) below (= Figure 0.1 in the Introduction).
○,

Figure 0.1 (in Preface): The location of QL in the history of western philosophy(realistic) monism (i.e., without measurement axiom)
relativity
4 −→ theory−−−−−−−−−−→○(monism)(unsolved)3○
−−→ Newton ○6 TOE:theory of(realism)−−→ everything
quantum
5 −→ mechanics −−−−−−−−→○(quantum phys.)

Socrates
Aristotle
0○:Greek

7○

Schola-8○−−−−−−−−−−−−−−−−−→

8
○



quantization

7
○

9
○
−
−
−
−
−
−
−
−
→
−

statistics−−−−−−−−−→

dualism
applied math.
system theoryclassical mechanical worldview


(=MT)



(mind-body dualism)(Copernican revolution)) ○QL:quantum
15
linguistic−−→
language
Descartes
Kant scientiﬁc
10
○

11 
−−→ Spinoza, Locke,... −−→−−−−−−→○
quantummechanicalturn
epistemological worldview
worldview

12
○



linguistic turn



?

(logic in philo.))(logic, set theory)Wittgensteinmechanical
Saussure −−−−−−−→○

Boole, Frege,
14
13
○
turn
Zadeh
Russell, Cantor −−→philosophy ○21−−−−−→○
sticism
Parmenides, Zeno
Platolinguistic turnmath.

logical worldview(idealistic) dualism (i.e., with measurement axiom)
Figure 0.1 (in Preface): The history of the world-descriptions
Philosophy (≈ dualistic idealism) has progressed toward QL
0
1
2
10
12
14
15 )
(i.e., ○→
○→
○→
○→
○→
○→
○

It is usually said that there is a gap between Kant philosophy and analytic philosophy. In
12 above is not theoretically connected. But this is a superﬁcial conclusion.
other words, ○From a quantum linguistic point of view, studying Kant philosophy and analytic philosophy reveals that the two are closely related. This will be studied in this and next chapters

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(in particular, Sec. 12.1 in next chapter). That is, the above ﬁgure asserts ”philosophy
is one”, just as physics is one.
To put it provocatively,
12 is more essential than ○.
13
• ○

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Chap. 11 Linguistic philosophy (Before TLP)

11.1 Elementary knowledge of symbolic logic in mathematics
11.1.1 Propositional logic and predicate logic
Let’s review the basics of logic. That’s enough to read this text.
Postulate 11.1. [ Symbolic logic (i.e., Propositional logic and predicate logic: G. Boole,
G. Frege)]
For any proposition P , the truth function ϕ(P ) is determined such that
ϕ(P ) =

1 (or, ”T”, ”Yes”) (if P is true)
0 (or, ”F”, ”No”) (if P is false (i.e., not true))

Propostional logic(A1 ) Assume that P, P1 , P2 are propositions. Then, ¬P , P1 ∧ P2 , P1 ∨ P2 , P1 → P2 arepropositions.
And it holds that

1 (if ϕ(¬P ) = 0)0 (if ϕ(¬P ) = 1)
1 (if ϕ(P1 ) = ϕ(P2 ) = 1)
ϕ(P1 ∧ P2 ) =0 (otherwise)
1 (if (ϕ(P1 ), ϕ(P2 )) = (1, 1), (1, 0), (0, 1))ϕ(P1 ∨ P2 ) = ϕ(¬(¬P1 ∧ ¬P2 )) =0 (otherwise)
1 (if (ϕ(P1 ), ϕ(P2 )) = (1, 1), (0, 1), (0, 0))ϕ(P1 → P2 ) = ϕ(¬P1 ∨ P2 )) =0 (otherwise)ϕ(¬P ) =where ∧, ∨, ¬, → respectively is called ”and”, ”or”, ”not”, ”implies”.
Also, assume that Pθ is a proposition (θ ∈ Θ ≡ {1, 2, ..., n}), then it holds(i): P1 ∧ P2 ∧ ... ∧ Pn (denoted by ∧ Pθ , ) is a propositionθ∈Θ(ii): P1 ∨ P2 ∨ ... ∨ Pn (denoted by ∨ Pθ , ) is a proposition.
θ∈ΘHere, ϕ(P1 ∧ P2 ∧ ... ∧ Pn ) = mini=1,...,n ϕ(Pi ), ϕ(P1 ∨ P2 ∨ ... ∨ Pn ) = maxi=1,...,n ϕ(Pi ).
(A2 ) The above ﬁnite set Θ(≡ {1, 2, ..., n}) can be extended to an inﬁnite set Θ.
Predicate logic(i): ∧ Pθ is denoted by ∀θ[[θ ∈ Θ] → [Pθ ]]θ∈Θ(ii): ¬[∀θ[¬[Qθ ]]] is deﬁned by ∃θ[Qθ ].
Therefore, we see∨ Pθ = ¬[ ∧ ¬Pθ ] = ¬[∀θ[[θ ∈ Θ] → [¬Pθ ]] = ∃θ[[θ ∈ Θ] ∧ [Pθ ]]

θ∈Θ

θ∈Θ

Remark 11.2. The point to note above is the sudden appearance of the word
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11.1 Elementary knowledge of symbolic logic in mathematics”Proposition P ”. The end result is that the ”proposition” is not deﬁned until the end.
This is supplemented in (i)-(iv) below.
(i):In mathematics, predicate logic is common knowledge. However, in the ﬁelds ofphilosophy and science, predicate logic is not yet understood by anyone. This fact wasﬁrst pointed out in ”Hempel’s raven paradox (cf. Sec. 12.7). Thus, depending on one’s
point of view, the logic of philosophy and science is more diﬃcult than the logic ofmathematics. Relying on the ”ZFC axiom system (1921)” proposed by Zermelo and
Frenkel, we already have the answer to ”What is a (mathematical) proposition?”. On theother hand, in philosophy and science, the answer to ”What is a proposition? is unknownin philosophy and science.
(ii):Descartes considered the unintelligible proposition ”I think, therefore I am” to be theﬁrst proposition of philosophy, so he may not have seriously considered the question ”Whatis a proposition? Since Kant discussed the distinction between ”analytic propositions” and”synthesis propositions,” he may not have realized the importance of the question, ”Whatis a proposition? The problem of ”analytic propositions” and ”synthesis propositions”
raised by Kant is still an unresolved issue today as the Carnap-Quine controversy (cf.
Sec. 12.3).
(iii): The history of the question ”What is a (scientiﬁc) proposition?” in science is not very
old. In Newtonian mechanics, the question is ”Is it a proposition of Newtonian mechanics,
or not?” was often self-evident and did not become a serious problem (personally, I am
interested in [Galileo’s free fall] in section 7.5). In quantum mechanics, however, things
have changed. Quantum mechanics assumes a ”observer (i.e., mind-body dualism),” which
complicates the story. Therefore, it became necessary to have rules for judging ”is it aproposition of quantum mechanics or not?”. This is what has been described so far in thistext. Or rather, my motivation for entering this ﬁeld was ”Why does quantum mechanics
have a (Copenhagen) interpretation?” because I wondered.
(iv): As readers may have already noticed, I think• the most important question in philosophy is ”What is a proposition in this world?”
which might be, for the ﬁrst time, founed by Wittgenstein. Of course, there is not always
one answer, but this is the main theme of this text.
(v): Thus, in this text (chapters 11 and 12) we show the following(♭1 ) What is a proposition (of this world)?
(♭2 ) Why does propositional logic hold in this world?
(As for predicate logic, it is strictly incomplete.)
Exercise 11.3. The truth table below is well known. Wittgenstein was one of the contributors to the popularization of truth tables.

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Chap. 11 Linguistic philosophy (Before TLP)(i): The proof of syllogism: [(p → q) ∧ (q → r)] → (p → r)
p
1
1
1
1
0
0
0
0

q
1
1
0
0
1
1
0
0

r
1
0
1
0
1
0
1
0

p→q
1
1
0
0
1
1
1
1

q→r
1
0
1
1
1
0
1
1

p→r
1
0
1
0
1
1
1
1

Truth Table(p → q) ∧ (q → r)
1
0
0
0
1
0
1
1

(p → q) ∧ (q → r) → (p → r)
1
1
1
1
1
1
1
1

Thus, syllogism: [(p → q) ∧ (q → r)] → (p → r) is always true.
(ii):Modus ponens ((p → q) ∧ p) → q is always true, but ((p → q) ∧ q) → p is not always
true.
Proof of (ii)p q p → q (p → q) ∧ p (p → q) ∧ q ((p → q) ∧ p) → q ((p → q) ∧ q) → p
1 1
1
1
1
1
1
1 0
0
0
0
1
1
0 1
1
0
1
1
0
0 0
1
0
0
1
1
For example, put p := ”it rains”, q := ”the ground is wet”. Modus ponens says that[[”it rains” → ”the ground is wet”]∧ ”it rains”]→ ”the ground is wet”.

(iii):[The elementary mathematical problem concerning predicate logic]”limn→∞ an = a (i.e., ”A real-valued sequence {an }∞n=1 converges to a”) is deﬁned by∀ϵ(> 0)∃N (natural number)∀n(natural number)[[n > N ] → |an − a| < ϵ]This was due to the great mathematicians Cauchy(1789-1857) and Weierstrass(1815-1897)
etc.
(iv):The following is not a syllogism.(cf. Exercise 4.5 in Sec. 4.3.2).
(♯) Socrates is a man, and all men are mortal, so Socrates dies.

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11.1 Elementary knowledge of symbolic logic in mathematics

11.1.2 Which do you trust more, logic or statistics?
In a word, logic is used in ﬁelds where quantitative discussion is diﬃcult, and statistics isused in ﬁelds where detailed data is available as follows.

In motto terms, this ﬁgure says the following.
(B) ”Logic for the humanities, statistics for the sciences”
which is common sense that everyone knows.
Thus we have three kinds of ”logics”(C1 ) logic for the humanities: everyday logic (QL logic (=fuzzy logic))(C2 ) logic for the sciences: statistics (a kind of many-valued logic)(C3 ) logic for mathematicians: mathematical logic
Frege and Russell may have deliberately used the confusion of the two logics of (C 1 )and (C3 ) to promote analytic philosophy. In this and the next chapters, (C1 ) and (C2 )will be explained. For this,(D) It suﬃces to derive statistics (C2 ) and everyday logic (≈ fuzzy logic)(C1 ) from QL.

Recall the following part of Figure 0.1 in Preface, that is,308

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Chap. 11 Linguistic philosophy (Before TLP)

(E)


8
○
quantum mechanics −−−−−−−−−→ linguistic turn 






 ○
9
○
15
statistics −−−−−→−−→ QLdualism 





math.


14
○
13
○

Boole, Frege −−−−−−−→ Wittgenstein, Hempel, etc. −−−−−→ philosophy

dualism9 ○)15 of statistics from QL is described in Note 1.3 (or, generally,
The simple derivation (○←ref.[74]), so in this text (chapters 11, 12, especially, in Sec, 12.1) we devote ourselves to
14
15 of everyday logic (fuzzy logic) from QL.
the derivation (○←○)♠Note 11.1. I am in the department of mathematics and know many brilliant mathematicians, many of whom do not know much about mathematical logic. I was surprised thatphilosophers know diﬃcult theorems in mathematical logic (e.g., Gödel’s incompletenesstheorem, Löwenheim-Skolem theorem, Banach-Tarski’s Theorem, etc.). This fact wasstrange to me because I did not think that mathematical logic and philosophy are closely
related. There can be a philosophy of mathematics, but it should not be the center ofphilosophy.

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11.2 Probabilistic symbolic logic in mathematics

11.2 Probabilistic symbolic logic in mathematics
This section consists of excerpts from the following paper:(A) Ref. [81]: Ishikawa, S. (2021) Fuzzy Logic in the Quantum MechanicalWorldview ; Related to Zadeh, Wittgenstein, Moore, Saussure, Quine, Lewis
Carroll, etc. JAMP, Vol. 9, No.3, 140-154, (https://www.scirp.org/journal/
paperinformation.aspx?paperid=110830)

Now let us introduce probabilistic symbolic logic, which is a slight generalization ofmathematical symbol logic mentioned in the previous section. Probabilistic symbolic logic
plays an essential role in QL logic (= fuzzy logic) ( in Sec. 12.1). Since I am convincedthat this fuzzy logic is what Wittgenstein had wanted to insist, I encourage readers toread this section. This section is written in such a way that it can be read without
knowledge of quantum language. It’s very easy and I think it’s readable by normal highschool students. However, it should be noted that this probabilistic symbolic logic, likemathematical symbolic logic, is not suﬃcient from a philosophical perspective in the sensethat the ”proposition” is deﬁned only in mathematics (or, equivalently in the sense that
it is not discussed in a certain worldview).

11.2.1 Easy example
Let us start from the following easy example.
Example 11.4. For example, consider a proposition P1 such that(B)

¬P1 = ”this tomato is not red”,

P1 = ”this tomato is red”,

And suppose that there are 100 respondents, and furthermore, the following question is
asked to them.
(C) Is this tomato red? (i.e., is the proposition P1 true or not?)Assume that the results of the responses are as follows.

70 respondents say ”Yes, this tomato is red” (i.e., the proposition P1 is true, i.e. ”T”)
(D)
30 respondents say ”No, this tomato is not red” (i.e., the proposition P1 is false, i.e. ”F”)
It is possible to consider that T = ”Yes” = 1 and F = ”No” = 0.

This can be probabilistically interpreted as follows.
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Chap. 11 Linguistic philosophy (Before TLP)
(E) When any respondent is randomly selected out of 100, the probability that this
respondent will answer ”yes” to question (C) is p1 (= 0.7). Or simply, the probability
that the proposition P1 is true is p1 . In symbolic form,
Prob[P1 ; {T }] = p1 (= 0.7)
Then we generally denote that
Prob[P1 ; {T }] = p1 ,Prob[P1 ; {F }] = 1 − p1

(where 0 ≤ p1 ≤ 1)

Also, note that Prob[¬P1 ; {T }] = Prob[P1 ; {F }] = 1 − p1 .
Remark 11.5. (i):We do not plan to do anything unusual. Propositions” are the sameas what we have usually called ”propositions. However, it might have been better to use
a more ambiguous proposition such as ”cats are cute” or ”the rainbow has seven colors”
as an example.

(ii): Zadeh often emphasized that fuzziness and probability are diﬀerent concepts. However, I believe that fuzziness without a probabilistic interpretation cannot be a scientiﬁc
concept. In this sense, (e) above is essential

11.2.2 Quasi-product probability
In addition, assume another proposition P2 such that Prob[P2 ; {T }] = p2 , Prob[P2 ; {F }] =1 − p2 . Thus, we have two probability spaces ({T, F }, 2{T,F } , µi ), (i = 1, 2), such that
µ1 ({T }) = p1 ,µ1 ({F }) = 1 − p1 ,µ2 ({T }) = p2 ,µ2 ({F }) = 1 − p2

And consider the quasi-product probability space ({T, F }2 , P({T, F }2 ), µ1 ×qp µ2 ), which
satisﬁes the following marginal conditions:(µ1 ×qp µ2 )({T } × {T, F }) = µ1 ({T }),(µ1 ×qp µ2 )({F } × {T, F }) = µ1 ({F }),and(µ1 ×qp µ2 )({T, F } × {T }) = µ2 ({T }),(µ1 ×qp µ2 )({T, F } × {F }) = µ2 ({F }),311

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11.2 Probabilistic symbolic logic in mathematics
Putting the above together, we get the following table (i.e., Table 11.1).

×qp

Table 11.1: quasi-product probability measure µ1µ2µ1 ({T })µ1 ({F })\µ2 ({T })µ2 ({F })(µ1 ×qp µ2 )({(T, T )})(≡ α)(µ1 ×qp µ2 )({(T, F )})(≡ µ1 ({F }) − α)(µ1 ×qp µ2 )({(F, T )})(≡ µ2 ({F }) − α)qp(µ1 × µ2 )({(F, F )})(≡ 1 − µ1 ({F }) − µ2 ({F }) + α)

wheremax{µ1 ({T }) + µ2 ({T }) − 1, 0} ≤ α ≤ min{µ1 ({T }), µ2 ({T })}Thus, the quasi-product probability is not unique in general.
Remark 11.6. The followings (i) and (ii) are typical. Assume that P1 = P2 . Thus,µ1 = µ2 .
(i):[Product probability space ]Assume that each respondent randomly chooses ”T” [resp. ”F”] with probability 0.7 [resp.
qp0.3] in the same way for the two questions P1 and P2 . Then, the µ1 × µ2 is consideredas the ordinary product probability µ1 × µ2 such thatqp(µ1 × µ2 )({(x1 , x2 )}) = µ1 ({x1 }) × µ1 ({x2 })(∀(x1 , x2 ) ∈ {T, F }2 )If we write the above in a table, we get the following table (i.e., Table 11.2).
Table 11.2: product probability measure µ1µ1 ({T })\µ1 ({T })µ1 ({F })µ1 ({T }) × µ1 ({T })(≡ α)µ1 ({T }) × µ1 ({F })(≡ µ1 ({F }) − α)×qp µ1 (= µ1 × µ1 )µ1 ({F })µ1 ({F }) × µ1 ({T })(≡ µ1 ({F }) − α)µ1 ({F }) × µ1 ({F })(≡ 1 − µ1 ({F }) − µ1 ({F }) + α)(ii):[Standard situation when P1 = P2 ] It is natural to think that the respondent selectedin (C) will give the same answer to the same question (i.e., P1 and P1 ). In this case we
qp
qp
qp
see µ1 × µ1 such that (µ1 × µ1 )({(T, T )}) = p1 and (µ1 × µ1 )({(F, F )}) = 1 − p1 .
Thus,
( if (x1 , x2 ) = (T, T ) p1
qp
1 − p1 ( if (x1 , x2 ) = (F, F )( × µ)({(x1 , x2 )}) =

i=1,2
0
( if (x1 , x2 ) = (T, F ), (F, T )If we write the above in a table, we get the following table (i.e., Table 11.3).
Table 11.3: quasi-product probability measure µ1 × µ1\
µ1 ({T })µ1 ({F })qpµ1 ({T })µ1 ({F })(µ1 × µ1 )({(T, T )}) = p1
qp(µ1 × µ1 )({(T, F )}) = 0
qp(µ1 × µ1 )({(F, T )}) = 0
qp(µ1 × µ1 )({(F, F )}) = 1 − p1
qp

In this text, unless otherwise stated, this quasi-product probability measure µ1 × µ1 will
be used. However, if so, it is not necessary to ask the same question twice; we only needqp
to ask it once. For further discussion, see Example 11.8 later. Also, this ×i=1,2 µi (= µ)qp
is easily extended to the case that ×i=1,2,...,n µi (where µi = µ, i = 1, 2, ..., n). such that
qp


 p1
n
1 − p1( × µ)({(xi )i=1 }) =

i=1,2,...,n
0
qp

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Chap. 11 Linguistic philosophy (Before TLP)

11.2.3 Logic symbols and logical operations
For the sake of convenience, we will deﬁne as follows (also, see (G) for the formal deﬁnition):(F) P1 ∧ P2 = ”P1 and P2 ”,¬P = not P

P1 ∨ P2 = ”P1 or P2 ”,

P1 → P2 = ”¬P1 or P2 ”,

Also, P1 ∧ P2 , ¬P1 ∧ P2 , P1 ∧ ¬P2 , ¬P1 ∧ ¬P2 ) are located in the following table (i.e.,
Table 11.4).

×

qp

Table 11.4: quasi-product probability measure µ1µ2 concerning P1 ∧ P2 , ¬P1 ∧ P2 , P1 ∧ ¬P2 , ¬P1 ∧ ¬P2\
P1 ; Prob[P1 ; {T }]¬P1 ; Prob[¬P1 ; {T }]\p1 = µ1 ({T })p1̄ = µ1 ({F })P2 ; Prob[P2 ; {T }]P1 ∧ P2 ; Prob[P1 ∧ P2 ; {T }]¬P1 ∧ P2 ; Prob[¬P1 ∧ P2 ; {T }]p2 = µ2 ({T })p12 (= (µ1 ×qp µ2 )({(T, T )}) ≡ α)p1̄2 (= (µ1 ×qp µ2 )({(F, T )}) ≡ p2 − α)P1 ∧ ¬P2 ; Prob[P1 ∧ ¬P2 ; {T }]¬P1 ∧ ¬P2 ; Prob[¬P1 ∧ ¬P2 ; {T }]¬P2 ; Prob[¬P2 ; {T }]p12̄ (= (µ1 ×qp µ2 )({(T, F )})p1̄2̄ (= (µ1 ×qp µ2 )({(F, F )})≡ p1 − α)≡ 1 − p1 − p2 + α)p2̄ = µ2 ({F })where max{p1 + p2 − 1, 0} ≤ α ≤ min{p1 , p2 }.
Now we can explain the following example:
Example 11.7. [Simple probabilistic truth table] The following table (i.e., Table 11.5:
Simple probabilistic truth table) is the same as a well-known truth table, except for the”probability column”.
Table 11.5: Simple Probabilistic Truth ( Elementary propositions P1 , P2 )qp
P1 P2 probability p = ×i=1,2 µi ¬P1 P1 → P2 P1 ∧ P2 P1 ∨ P2
qp
TT
p12 = ×i=1,2 µi ({(T, T )})
F
T
T
T
qp
T
F
p12̄ = ×i=1,2 µi ({(T, F )})
F
F
F
T
qp
F
T
p1̄2 = ×i=1,2 µi ({(F, T )})
T
T
F
T
qp
F F
p1̄2̄ = ×i=1,2 µi ({(F, F )})T
T
F
F
Thus, it will be enough to explain hoe to use the ”probability column” as follows. For
example, consider the above proposition ¬P1 , which can be regarded as the map from{T, F }2 to {T, F } such that
T if (x1 , x2 ) = (F, T ), or (F, F )[¬P1 ](x1 , x2 ) =F if (x1 , x2 ) = (T, T ), or (T, F )(or ¬P1 : {T, F } → {T, F } such that [¬P1 ](T ) = F , [¬P1 ](F ) = T ).
Thus, we see that Prob[¬P1 ; {T }] (i.e., the probability that ¬P1 is true) is equal toqp(µ1 × µ2 )([¬P1 ]−1 ({T })) = p1̄2 + p1̄2̄
Next, consider [P1 → P2 ] (= [¬P1 ∨ P2 ]), which is regarded as the map from {T, F }2 to{T, F } such that
T if (x1 , x2 ) = (T, T ) or (F, T ) or (F, F )[P1 → P2 ](x1 , x2 ) =F if (x1 , x2 ) = (T, F )313

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11.2 Probabilistic symbolic logic in mathematics
Thus, we see that Prob[P1 → P2 ; {T }] (i.e., the probability that [P1 → P2 ] is true) is
equal toqp(µ1 × µ2 )([P1 → P2 ]−1 ({T })) = p12 + p1̄2 + p1̄2̄
Similarly we see
Prob[P1 ∧ P2 ; {T }] = p12 ,Prob[P1 ∨ P2 ; {T }] = p12 + p1̄2 + p12̄Furthermore, we must note that(G) ¬P1 , [P1 ∧ P2 ], [P1 ∨ P2 ] and [P1 → P2 ] can be respectively regarded as maps from{T, F }2 to {T, F } as shown in the above Table 11.5. Rather than (F), this map isthe formal deﬁnition of logic symbols (i.e., ¬, ∧, ∨, →).

Example 11.8. [Probabilistic truth table ((i): P1 6= P3 , (ii): P1 = P3 )](i) (Simplest case: P1 6= P3 ): The following table (i.e., Table 11.6) is the preparation of
the next (ii) (i.e., Tables 11.7, 11.8, 11.9). Consider the truth table of [P1 → P2 ] ∧ P3 asfollows.
Table 11.6: Probabilistic Truth Table ( Elementary propositions P1 , P2 , P3 )qp
P1 P2 P3
probability: p = ×i=1,2,3 µi[P1 → P2 ] ∧ P3
qp
T
T
T
p123 = ×i=1,2,3 µi ({(T, T, T )})
T
qp
T
T
F
p123̄ = ×i=1,2,3 µi ({(T, T, F )})
F
qp
T
F
T
p12̄3 = ×i=1,2,3 µi ({(T, F, T )})
F
qp
T
F F
p1̄2̄3̄ = ×i=1,2,3 µi ({(T, F, F )})
F
qp
F
T
T
p1̄23 = ×i=1,2,3 µi ({(F, T, T )})
T
qp
F
T
F
p1̄23̄ = ×i=1,2,3 µi ({(F, T, F )})
F
qp
F F
T
p1̄2̄3 = ×i=1,2,3 µi ({(F, F, T )})
T
qp
F F F
p1̄2̄3̄ = ×i=1,2,3 µi ({(F, F, F )})FThis says that the proposition [P1 → P2 ] ∧ P3 is the map from{T, F }3 → {T, F } such that[[P1 → P2 ] ∧ P3 ](x1 , x2 , x3 )T if (x1 , x2 , x3 ) = (T, T, T ), (F, T, T ), (F, F, T )=F if (x1 , x2 , x3 ) = (T, T, F ), (T, F, T ), (T, F, F ), (F, T, F ), (F, F, F )
Thus,
qp

Prob[[[P1 → P2 ] ∧ P3 ]; {T }] = (× µi )([[P1 → P2 ] ∧ P3 ]−1 ({T })) = p123 + p1̄2̄3 + p12̄3̄

i=1,2,3(ii) ( Case; P1 = P3 in the above (i)): Furthermore, assume that P1 = P3 in the above.
Then, recalling Remark 11.6 (ii), we usually assume that ×qp
i=1,2,3 µi ({(x1 , x2 , x3 )}) = 0
qp
qp(if x1 6= x3 ). Thus, putting ×i=1,2 µi ({(x1 , x2 )}) = ×i=1,2,3 µi ({(x1 , x2 )} × {T.F }), we see

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Chap. 11 Linguistic philosophy (Before TLP)
the following: ( Note that Table 11.6= Table 11.7 except the probability column).

P1
T
T
T
T
F
F
F
F

P2
T
T
F
F
T
T
F
F

Table 11.7: Probabilistic Truth Table( Overlapping elementary propositions P1 , P2 , P3 (= P1 ))qp
P3 (= P1 )probability: p = ×i=1,2 µi[P1 → P2 ] ∧ P3 (= P1 )qp
Tp123 (= p12 ) = ×i=1,2 µi ({(T, T )})T
F
p123̄ (= 0)F
qpT
p12̄3 (= p12̄ ) = ×i=1,2 µi ({(T, F )})F
F
p12̄3̄ (= 0)FT
p1̄23 (= 0)T
qpF
p1̄23̄ (= p1̄2 ) = ×i=1,2 µi ({(F, T )})F
T
p1̄2̄3 (= 0)T
qpF
p1̄2̄3̄ (= p1̄2̄ ) = ×i=1,2 µi ({(F, F )})F

Since the case of the probability 0 (i.e., p123̄ = p1̄23 = p12̄3̄ = p1̄2̄3 = 0 ) can be omitted,
we have the following table:

P1
T
T
F
F

P2
T
F
T
F

Table 11.8: Probabilistic Truth Table(Overlapping elementary propositions P1 , P2 , P3 (= P1 ))qp
P3 (= P1 )probability: p = ×i=1,2 µi[P1 → P2 ] ∧ P3 (= P1 )qp
Tp123 (= p12 ) = ×i=1,2 µi ({(T, T )})T
qp
T
p12̄3 (= p12̄ ) = ×i=1,2 µi ({(T, F )})F
qp
F
p1̄23̄ (= p1̄2 ) = ×i=1,2 µi ({(F, T )})F
qp
F
p1̄2̄3̄ (= p1̄2̄ ) = ×i=1,2 µi ({(F, F )})F

Therefore, we see that Prob[[P1 → P2 ] ∧ P3 (= P1 ); {T }] (i.e., the probability that [P1 →F2 ] ∧ P1 is true) is equal toqp(µ1 × µ2 )([[P1 → P2 ] ∧ P3 (= P1 )]−1 ({T })) = p12
Note that this is essentially the same as the following table.
Table 11.9: Probabilistic Truth Table(Non-overlapping elementary propositions P1 , P2 )qp
P1 P2 probability: p = ×i=1,2 µi [P1 → P2 ] ∧ P1
qp
TT
p12 = ×i=1,2 µi ({(T, T )})T
qp
TF
p12̄ = ×i=1,2 µi ({(T, F )})F
qp
F
T
p1̄2 = ×i=1,2 µi ({(F, T )})F
qp
F F
p1̄2̄ = ×i=1,2 µi ({(F, F )})F
Hence, we conclude that Tables 11.7, 11.8 and 11.9 are essentially the same. And therefore,
we see;(H) the calculation will be more concise if we start with non-overlapping propositions(such as Table 11.9). (For example, if P1 , P2 , P3 = P4 = P5 = P6 , P7 , then it suﬃces
to consider P1 , P2 , P3 .P7 )315

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11.2 Probabilistic symbolic logic in mathematics
Example 11.9. [Modus ponens in probabilistic truth table] The following table (i.e.,
Table 11.10: Probabilistic truth table) is the same as a well-known truth table, except forthe ”probability column”.
Table 11.10: Probabilistic Truth Table ( Elementary propositions P1 , P2 )qp
P1 P2
probability: p =[P1 → P2 ] ∧ P1
i=1,2 µi
T

T

T

F

F

T

F

F

×
p12̄ = ×p1̄2 = ×p1̄2̄ = ×
p12 =

×

qp
i=1,2 µi ({(T, T )})qp
i=1,2 µi ({(T, F )})qp
i=1,2 µi ({(F, T )})qp
i=1,2 µi ({(F, F )})[P1 → P2 ] ∧ P1 → P2

T

T

F

T

F

T

F

T

For example, consider the proposition [P1 → P2 ] ∧ P1 , which is regarded as the map
from {T, F }2 to {T, F } such that
T if (x1 , x2 ) = (T, T )[[P1 → P2 ] ∧ P1 ](x1 , x2 ) =F if (x1 , x2 ) = (T, F ), (F, T ), (F, F )Thus, we see that Prob[[[P1 → P2 ] ∧ P1 ]; {T }] (i.e., the probability that [[P1 → P2 ] ∧ P1 ]is true) is equal toqp(µ1 × µ2 )([[P1 → P2 ] ∧ P1 ]−1 ({T })) = p12
By the same way, we can calculate as follows.
(µ1 ×qp µ2 )([((P1 → P2 ) ∧ P1 ) → P2 ]−1 ({T })) = p12 + p12̄ + p1̄2 + p1̄2̄ ) = 1
That is,Prob[((P1 → P2 ) ∧ P1 ) → P2 ]; {T }] = p12 + p12̄ + p1̄2 + p1̄2̄ = 1
Thus, modus ponens is always true even in probabilistic logic.
For example, put P1 :=”it rains”, P2 :=”the ground is wet”. Modus ponens says that[[”it rains” → ”the ground is wet”]∧ ”it rains”]→”the ground is wet”.

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Chap. 11 Linguistic philosophy (Before TLP)The statement (H) in Examples 11.8 says that it suﬃces to consider the case of nonoverlapping propositions. Thus we have the following theorem.
Theorem 11.10. [Logical sample space] Let P1 , P2 , ..., Pi , ..., Pn be non-overlapping
propositions (i.e., Pi 6= Pj (∀i, j such that i 6= j, cf. (H) above), and consider the probability space ({T, F }, 2{T,F } , µi ) such that µi ({T }) = Prob[Pi ; {T }], µi ({F }) = Prob[Pi ; {F }].
qp
Consider a quasi-product probability space ({T, F }n , P({T, F }n ), ×i=1,2,...,n µi ) such
that
qp

(

×

i=1,2,...,nµi )({T, F }k−1 ×{x}×{T, F }n−k ) = Prob[Pk ; {x}](x ∈ {T, F },k = 1, 2, ..., n)qpThen, the pair {P1 , P2 , ..., Pn }; ({T, F }n , P({T, F }n ), ×i=1,2,...,n µi ) is called the logicalsample space. Let P be a proposition which is constructed by P1 , ..., Pn . Note that P isregarded as the map from {T, F }n → {T, F }. Then, we see thatqp

Prob[P ; {T }] = (× µi )(P −1 ({T }))),
i=1,2,...,nqp

Prob[P ; {F }] = (× µi )(P −1 ({F })))
i=1,2,...,nAlso, it is clear that the above Theorem 11.10 implies the following corollary.
Corollary 11.11. [Tautology in probabilistic logic] Let P be a proposition constructedfrom non-overlapping propositions P1 , P2 , ..., Pn . Then, the followings are equivalent:(i) P is a tautology in the sense of usual logic.
(ii) Prob[P ; {T }] = 1.
That is, tautology always holds even in probabilistic logic. For example,(♭) syllogism (i.e., [[P1 → P2 ] ∧ [P2 → P3 ]] → [P1 → P3 ] always holds.

Remark 11.12. It is usually said that the following is the typical example of logic.
(♯) Since Socrates is human being, and human being is mortal, it follows that Socrates
is mortal.

However, so far, we do not know the deﬁnition of ”proposition” other than ”mathematical
proposition”. For example, as seen in Sec. 8.2, ”I think” and ”I am” are not propositions
in QL. Strictly speaking, there is no guarantee that ”Socrates is a man” is a proposition.
A discussion of mathematics alone, without a worldview, is powerless.
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11.2 Probabilistic symbolic logic in mathematics

Therefore, the above Corollary 11.11 does not guarantee that (♯) holds. This will be
discussed in Sec. 12.1.

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Chap. 11 Linguistic philosophy (Before TLP)

11.3 George Boole, Gottlob Frege and Bertrand Russell:
Mathematics or philosophy?
George Boole (1815-1864) was a mathematician, philosopher and logician in England.
Particularly, it’s said that the propositional logic (i.e., Boolean algebra) was proposed byhim. Also, Gottlob Frege (1848-1925) was a German philosopher, logician, and mathematician. He is called the father of analytic philosophy , concentrating on the philosophyof language, logic, and mathematics. Particularly, it’s said that the predicate logic wasproposed by him. Bertrand Russel (1872-1970) was mathematician, philosopher, logician,social critic in England. With A. N. Whitehead he wrote Principia Mathematica, anattempt to create a logical basis for mathematics.

Many times we have discussed the following idealism:z

introduction part
}|
Plato

Idea theory
Descartes(A)

I think, therefore I am
Kant

thing-in-itself
Kant’s philosophy provided thefoundation for Newtonian mechanics

{












common Therefore−−−−−−−−→ spirit, ethics, moraltrust me













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11.3 George Boole, Gottlob Frege and Bertrand Russell: Mathematics or philosophy?

That is, it was the method of philosophy (i.e., Plato’s way of telling philosophy) to ﬁnd
something impossible to doubt, and then declare that they had ”deduced” various thingsfrom it.
Following the example of Kant, B. Frege and B. Russell must have thought the following(or they must have thought that the general public would support them if they said thefollowing):(B1 ) Mathematical logic provides the foundation for mathematics. Mathematics is justone area of mathematical logic. Mathematical logic is so powerful that there is no
doubt about it. Thus, mathematical logic is greatest. Therefore , we have to be”logical”.
That is,Frege, Russell(B2 )mathematical logic
Logic is so powerful that it is the foundation of mathematics

Therefore−−−−−−−−−−−−−−−−→ ”Think everything logically!”
trust logic!

B. Russell wrote in ”The Basic Writings of Bertrand Russell, 1903-1959”, (p.608, Psychology Press) such as(C) Ordinary language is totally unsuited for expressing what physics really asserts,
since the words of everyday life are not suﬃciently abstract. Only mathematics andmathematical logic can say as little as the physicist means to say.
I think Russell was sociable and interacted with many scientists in diﬀerent ﬁelds, so ofcourse he knew that ”Einstein was not familiar with mathematical logic”. So why did hemake the above statement? My guess is as follows. He knew the method of philosophy(i.e., Plato’s way of telling philosophy), that is,320

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• Philosophy is ”asserted ﬁction” that is, to say ”Therefore” in the (B), even if it’sa lie.
Moreover, Russell was a man of integrity, so I think he wanted to say somewhere that
(D1 ) mathematics (mathematical logic) alone cannot be a worldview,
because he could not lie. At the end of Wittgenstein’s Tractatus Logico-philosophicus(TLP: 1921), he wrote, ”What we cannot speak about we must pass over in silence. )”.
(D2 ) Don’t talk about anything other than what we can talk about (= scientiﬁc propositions)!
Therefore. I think Wittgenstein was saying on Russell’s behalf what Russell wanted toargue. If so, I can understand Russell’s tolerance for Wittgenstein.
Russell must have been concerned about scientiﬁc propositions. Otherwise, Russell’s
generosity to Wittgenstein cannot be understood.
♠Note 11.2. The rule of philosophy (≈ idealism) is as follows:(♯1 ) Only discussion, no experimentation.
(♯2 ) Winners are determined by the popularity of the general public (or philosophical enthusiasts).

Therefore, philosophers are not necessarily in pursuit of truth. Sometimes it is necessary
to bluﬀ a little to gain a lot of support. Russell did not mean to say that logic is greaterthan mathematics. However, I think Russell succeeded in making the average philosophy
enthusiasts feel that ”logic is greater than mathematics.” Gaining the support of thegeneral public is not an easy task. Even if you write about meaningless things in anesoteric and lofty way, the general public will not follow you unless there is something inyour writing that attracts them. Wittgenstein had a gift for writing philosophy.

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11.4 Peirce (1839-1914): Abduction

11.4 Peirce (1839-1914): Abduction
This section was written with reference to the following.
• [78]:Ishikawa, S: Philosophy of science for scientists; The probabilistic interpretationof science Journal of quantum information science, Vol. 9, No.3, 140-154,
DOI: 10.4236/jqis.2019.93007(https://www.scirp.org/Journal/paperinformation.aspx?paperid=95447)

Charles Sanders Peirce (1839-1914) was an American philosopher, logician, mathematician, and scientist who is sometimes known as ”the father of pragmatism”. This paper
will not touch on pragmatism, but will speak of his ”abduction” (also called abductive
reasoning, abductive inference, or retroduction). In ref. [104] (Vol. II p.375), as the
typical example of ”abduction”, Peirce mentioned as follows.
(♯1 ) Fossils are found; say, remains like those of ﬁshes, but far in the interior of thecountry. To explain the phenomenon, we suppose the sea once washed over thisland.
This kind of reasoning was called ”abduction” by Peirce. As most readers will immediately
realize, abduction is essentially the same as the qualitative representation of Fisher’s
maximum likelihood method. Recall the following ﬁgure in Sec. 1.4.4:(♯2 ) A person in the distance is standing. I am not sure of the person’s gender, but theperson is wearing a skirt. Therefore, I presume the person to be a woman.
Such an estimation ((♯1 ), (♯2 )) is called ”abduction” by Peirse. As many readers will
immediately notice, abduction is essentially the same as the qualitative expression of322

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Chap. 11 Linguistic philosophy (Before TLP)
Fisher’s maximum likelihood method.
The purpose of this section is to explain the following table:
Logical worldview

Quantum mechanical worldview[ no measurement (logical approach))][ measurement (probabilistic approach)]

where
(♯)

deduction (cf. Sec. 11.4.1.1 (i))←→

measurement

abduction (cf. Sec. 11.4.1.1 (ii))←→

inference (maximum likelihood method)

induction (cf. Sec. 11.4.1.1 (iii))←→

the law of large numbers the logical approach(rough, handy)

the probabilistic approach (exact, calculation required)♠Note 11.3. In the table above (or Figure 0.1 in Preface), I used the term ”logical worldview,” but it is quite diﬃcult to explain what this means. In a nutshell, think of it asfollows.
• Arguments that use many [and, or, not, ⇒] to advance the argument.
Or, think of it as the same as the ”set-theoretic worldview” of Sec. 4.3.4. I know it may
seem lax, but I think• In dualistic idealism, only the quantum mechanical worldview (quantum language) is
solid

Recall the ﬁgure in Sec. 11.1.2

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11.4 Peirce (1839-1914): Abduction♠Note 11.4. Peirce’s work was done before 1900. The pioneering nature of his work can
be clearly seen in the following time series(i) Born’s discovery “the probabilistic interpretation of quantum mechanics” in ref. [6](1926)
which is essentially the same as Axiom 1 (measurement) of quantum language(ii) Fisher’s great book “Statistical Methods for Research Workers” in ref. [22] (1925)(in which Fisher’s maximum likelihood method was introduced)
These two are among the major works of the top ten most inﬂuential writings of the 20th
century. Recall that Axiom 1 (measurement) of quantum language and Fisher’s maximum
likelihood method are essentially the same (cf. Sec. 1.4.4). Therefore, I believe that
Peirce’s work was ahead of its time.

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Chap. 11 Linguistic philosophy (Before TLP)

11.4.1 * What is Peirce’s abduction?
11.4.1.1 Deduction, abduction and induction in ”logic”
According to Peirce, three kinds of inferences (i.e., deduction, abduction, induction) are
important. Let us explain deduction, abduction and induction as follows.
[(i):Deduction]b ′ ) are often
A typical example of deduction is as follows: In the following, (A′1 ) and (A1omitted.
(A1 ) All the beans in this bag B1 are white: [bag B1 −→”w”(≈ white)](A′1 ) All the beans in that bag B2 are white or black ﬁfty-ﬁfty (or generally, the ratio of
white beans to black beans is p/(1 − p) where 0 < p < 1):[bag B2 −→”w”(≈
white) or ”b”(≈ black)](A2 ) This bean is from this bag B1 : [bag B1 ]
(A3 ) Therefore, this bean is white: [”w”(≈ white)]

It is, of course, obvious and ordinary.
♠Note 11.5. It is clear that the following is a tautology:


(♯1 )
[B1 −→ w] ∧ [B2 −→ [w ∨ b]] ∧ B1 −→ [w]Thus, the above conclusion (A3 ) can be understood as a consequence of this tautology.

[(ii): Abduction]
C.S, Peirce (cf. ref. [104]) proposed abduction as follows.. The example of abduction is
as follows:b 1 ) All the beans in this bag B1 are white: [bag B1 −→”w”(≈ white)](Ab ′ ) All the beans in that bag B2 are white and black ﬁfty-ﬁfty (or generally, the ratio(A
1of white beans to black beans is p/(1 − p)): [bag B2 −→”w”(≈ white) or ”b”(≈
black)]b(A2 ) This bean (from B1 or B2 (i.e., it is not known whether it is B1 or B2 ) ) is
white: [”w”(≈ white)]b(A3 ) Therefore, this bean is from this bag B1 : [bag B1 ]325

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11.4 Peirce (1839-1914): Abduction[(iii): Induction]
Further, induction (inductive reasoning) is as follows.
e 1 ) 1000p white beans and 1000(1 − p) black beans are mixed well in this bag B3 (here,(A
0 < p < 1). Assume that we do not know the value p (0 < p < 1).
e(A2 ) When we took 20 beans out of this bag B3 , every bean was white.
e 3 ) Therefore, the bean picked out from this bag B3 next can be presumed to be white.
(A

This will be again discussed in the following section.

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Chap. 11 Linguistic philosophy (Before TLP)

11.4.1.2 Deduction, abduction and induction in quantum language (i.e., the quantum mechanical worldview)
In our worldview (i.e., the quantum mechanical worldview ≈ the quantum linguistic worldview), the relation among deduction, abduction and abduction is characterized as follows.
b 1 )-(Ab 3 )] can be justiﬁed in quantum language.
First, we will show that the abduction [(AConsider the state space Θ = {θ1 , θ2 } with the discrete topology, and the classical basic
structure [C(Θ) ⊆ B(L2 (Θ, ν))], where ν({θ1 }) = ν({θ2 }) = 1/2. Assume that
θ1 ≈ the state of the bag B1 ,

θ2 ≈ the state of the bag B2 ,Assume that 1000 white beans belong to bag B1 , and further, 1000p white beans and1000(1 − p) black beans belong to the bag B2 (where 0 < p < 1). Thus we have theobservable O = ({w, b}, 2{w,b} , F ) in C(Θ) such that
[F ({w})](θ1 ) = 1[F ({b})](θ1 ) = 0[F ({w})](θ2 ) = p[F ({b})](θ2 ) = 1 − p(0 < p < 1)

where ”w” and ”b” means ”white” and ”black” respectively.
Thus, we have the measurement MC(Θ) (O := ({w, b}, 2{w,b} , F ), S[θi ] ), i = 1, 2.
[(i): Measurement ≈ Deduction]
Axiom 1 [measurement] (in Section 1.1) says thatb 1 ) [measurement]: The probability that the measured value w is obtained by(B
MC(Θ) (O := ({w, b}, 2{w,b} , F ), S[θ1 ] ) is equal to 1
This is the same as the deduction (i.e., (A1 )–(A3 )).
[(ii): Inference ≈ Abduction]
Next, under the circumstance that bags B1 and B2 cannot be distinguished, we consider
the following inference problem:b 2 ) [inference problem]: When the measured value w is obtained by the measurement(B
MC(Θ) (O := ({w, b}, 2{w,b} , F ), S[∗] ), which do you infer, [∗] = θ1 or [∗] = θ2 ?
Fisher’s maximum likelihood method Theorem 1.15 [Fisher’s maximum likelihood
method] says that [∗] = θ1 , sincemax{F ({w})](θ1 ), F ({w})](θ2 )} = max{1, p} = 1 = [F ({w})](θ1 )b 3 ).
This implies (Ab 2 ) is the quantum linguistic representation of abduction (i.e.,
Therefore, the above (Bb 1 )–(Ab 3 )).
(A[(iii): Induction ≈ the law of large numbers]This has already been discussed in Sec, 9.6, ”Hume’s problem of induction,” so we omit
it here.

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11.4 Peirce (1839-1914): Abduction

11.4.1.3 Summary; Logic-like understanding (by Peirce) vs. quantum linguistic understanding
Generally speaking,(C) Quantum linguistic method (in Sec. 11.4.1.2) is natual, and logic-like method (≈pseudo-logical method in Sec. )(by Peirce) is incomplete.
However, the world is not only about ”theory”. In many cases, quantitative data can
not be obtained, thus, logic-like method (by Peirce) is often used.
• In fact, Sherlock Holmes, the famous detective, must have been a master in the useof Peirce’s abduction.
If Holmes had used the maximum likelihood method, his detective novels would nothave become bestsellers.

The slogan is ”logic for rough arguments, statistics for precise arguments,” to put itroughly,
Logic for the humanities, statistics for the sciences
When quantitative data are available, the ”quantum mechanical worldview” is superior tothe ”logical worldview. However, the ”logical worldview” has an advantage that it can beused without calculation. Therefore, in daily life, the ”logical worldview” is often superior
to the ”quantum mechanical worldview”.
In summary, it is as follows.
Correspondence 11.13.
(B1 ): measurement−−−−−−−−−→ [(A1 ),(A′1 ),(A2 ) −→(A3 )]simpliﬁed form
deduction
Axiom 1 [measurement] (in Section 1.2.1)

yreverse
yreverseb 2 ): inference(B(Fisher’s maximum likelihood method)b 1 ),(Ab ′ ),(Ab 2 ) −→(Ab 3 )]−−−−−−−−−→ [(A1simpliﬁed form
abduction(11.1)

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Chap. 11 Linguistic philosophy (Before TLP)
Logical worldview

Quantum mechanical worldview[ no measurement (logical approach))][ measurement (probabilistic approach)]deduction (cf. Sec. 11.4.1.1 (i))←→

measurement

abduction (cf. Sec. 11.4.1.1 (ii))←→

inference (maximum likelihood method)

induction (cf. Sec. 11.4.1.1 (iii))←→

the law of large numbers

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11.5 Bertrand Russell: ﬁve-minute hypothesis, McTaggart’s paradox, Moore’s paradox

11.5 Bertrand Russell: ﬁve-minute hypothesis, McTaggart’s
paradox, Moore’s paradox
This section consists of excerpts from the following paper:• Ref. [81]: Ishikawa, S. (2021) Fuzzy Logic in the Quantum Mechanical Worldview; Related to Zadeh, Wittgenstein, Moore, Saussure, Quine, Lewis Carroll, etc.
JAMP, Vol. 9, No.3, 140-154,(https://www.scirp.org/journal/paperinformation.aspx?paperid=110830)
Bertrand Russell (1872-1970) was a great intellectual and one of the founders of analytic
philosophy along with Gottlob Frege (and student Ludwig Wittgenstein). Russell is amulti-talented philosopher who has produced many achievements. However,In this section, I mention ”Russell’s paradox”, ”Five-minute hypothesis”, and ”McTaggart’s paradox”, Moore’s paradox. And we see that these are closely related to the
linguistic Copenhagen interpretation (or equivalently, to the problem ”What is a ’proposition’ ?”). Thus, it can be said that• the three of them prepared the ground for the appearance of Wittgenstein at Cambridge University.

11.5.1 Russell’s paradox in set theory
As mentioned frequently in this text, I think that mathematical logic is one of ﬁelds ofmathematics and thus, Russell’s study of logic has little to do with traditional Western
philosophy (i.e., Descartes-Kant philosophy).
However, some things should be mentioned below. The following is famous as Russell’sparadox (or, Russell-Zermelo paradox).
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Chap. 11 Linguistic philosophy (Before TLP)The naive set theory (i.e., Cantor’s set theory) involves contradictions. For example,
Russell showed Russell’s paradox such that(A) if it is assumed that {U | U ∈/ U } is a set, then it leads contradiction (1902).*1
To avoid such a paradox, Alfred North Whitehead and Bertrand Russell proposed ”typetheory” (i.e., a kind of axiomatic set theory), which was published as the Principia Mathematica on the foundations of mathematics in 1910–1913. Also, the axiomatic system of
set theory was developed by Zermelo and others. Most modern mathematicians study
mathematics developed under a system of axioms called ZFC (1921), which consists ofeight axioms by Zermelo and Fraenkel plus an axiom called the Axiom of Choice.
Remark 11.14. The above is closely related to the problem: ”What is a ’mathematical
proposition’ ?”. That is because this problem is essentially the same as the problem:”What is a ’set’ ?”. That is, from the theoretical point of view, we see
What is a ’mathematical proposition’ ? ⇐⇒ What is a ’set’ ?
equiv.

Thus, the discovery of ”something that like a set but is not a set” ( such as Russell
discovered) promoted the birth of axiomatic set theory (e.g., Zermelo-Fraenkel set theory).
It is believed that the problem:”What is a ’mathematical proposition’ ?” was solved by
Zermelo and Fraenkel.
Therefore, the next biggest and only remaining philosophical question is• ”What is a ’non-mathematical proposition’ ?”This, of course, was the problem Wittgenstein pursued in ref. [125]: ”TLP (=Tractatus
Logico-Philosophicus)”.
////
Summing up, we see that(B) Russell’s paradox (i.e., the discovery of something like a set but not a set) prompted
the birth of axiomatic set theory (e.g., Zermelo-Fraenkel set theory), in which”mathematical proposition” can be completely deﬁned.

This (B) should be compared with the following (C).

*1 If it is assumed that ”the set of all sets” is a set, it leads contradiction.

This is called Cantor’s

paradox.

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11.5 Bertrand Russell: ﬁve-minute hypothesis, McTaggart’s paradox, Moore’s paradox(C) Similarly, the discovery of something like a proposition but not a proposition (e.g., ”I think, therefore I am”, Moore’s paradox, etc.) prompted the developmentof the linguistic Copenhagen interpretation ( which serves to distinguish between”scientiﬁc proposition” and ”non-scientiﬁc proposition”).

Recall the ﬁnal chapter of Wittgenstein’s TLP (i.e., ref. [125]: ”Tractatus LogicoPhilosophicus”) in which only ”What we cannot speak about we must pass over in silence”
is written. This implies that Wittgenstein’s central theme in his TLP is to draw a line
between what we can speak about (= scientiﬁc propositions ) and what we cannot speak
about (= non-scientiﬁc propositions), which is the same as the theme of this text, that is,
the linguistic Copenhagen interpretation=the rule of how to use axioms 1 (measurement) and 2 (causality)=the rule of how to draw a line between scientiﬁc propositions and=non-scientiﬁc propositions♠Note 11.6. Note that
Russell’s paradox(1902) −−−−−−−−−−−−−→Zermelo-Fraenkel set theory
about 20 years later

Descartes’ ”I think therefore I am”(1637) −−−−−−−−−−−−−−→linguistic Copenhagen
about 350 years laterinterpretation
This is due to the fact:• There was a strong belief in mathematics that paradox should be avoided at all costs.
On the other hand, philosophers have enjoyed and exploited paradoxes.

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11.5.2 * Five-minute hypothesis
In summary, it is as follows.
(♯1 ) Russell discovered the Russell-Zermelo paradox. This discovery led to the biggest
question in the history of mathematics: ”What is a mathematical proposition?”.
(♯2 ) He solved this problem with Whitehead.
(♯3 ) Next, Russell challenged the question ”What is a scientiﬁc proposition?” It was only
a proposal of a ”Five-minute hypothesis,” and no noteworthy results were obtained.
(♯3 ) Therefore, he entrusted the problem to his disciple, Wittgenstein. However,
Wittgenstein could not solve this problem either.
(♯4 ) But Wittgenstein’s words had power. He was notably gifted as a poet. As a result,
he became a symbolic philosopher in the propagation of analytic philosophy.

Bertrand Russell was a very eloquent philosopher who communicated many thoughtprovoking things to the general public. For a layman like me, he is a very helpful philosopher. If the following quote had not come from Russell, I don’t think it would have been
as famous as it is.
• There is no logical impossibility in the hypothesis that the world sprang into being
ﬁve minutes ago, exactly as it then was, with a population that ”remembered” awholly unreal past. There is no logically necessary connection between events at
diﬀerent times; therefore nothing that is happening now or will happen in the future
can disprove the hypothesis that the world began ﬁve minutes ago.
Ref. [114] ”The Analysis of Mind”, p.223, Bertrand Russell
This section was written with reference to the following.
• [76]:Ishikawa, S: Leibniz-Clarke correspondence, Brain in a vat, Five-minute hypothesis, McTaggart’s paradox, etc. are clariﬁed in quantum language, Open Journal of philosophy, Vol. 8, No.5, 466-480, 2018, DOI: 10.4236/ojpp.2018.85032(https://www.scirp.org/Journal/PaperInformation.aspx?PaperID=87862)[Revised version] (https://philpapers.org/rec/ISHLCB)(http://www.math.keio.ac.jp/academic/research_pdf/report/2018/18001.
pdf)• [85] S. Ishikawa, K. Kikuchi, (2021) Quantum Fuzzy Logic and Time, Journal ofApplied mathematics and physics, Vo.9 No.11 2021, 2609-2622(https://www.scirp.org/journal/paperinformation.aspx?paperid=112972)
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11.5 Bertrand Russell: ﬁve-minute hypothesis, McTaggart’s paradox, Moore’s paradox
The ﬁve-minute hypothesis is a skeptical hypothesis put forth by the philosopher
Bertrand Russell. However, as seen later, I do not think that this hypothesis is not
related to skepticism though my understanding to skepticism may be insuﬃcient.
The ﬁve-minute hypothesis, proposed by B. Russell (cf. ref. [114]), is as follows.
(A1 ) the universe was created ﬁve minutes ago. Or equivalently, the universe was createdten years ago.
Now we show that this (A1 ) is not the statement in quantum language as follows (i.e.,
The ﬁrst answer (i) and the second answer (ii))

Answer:Recall the linguistic Copenhagen interpretation:Thus, observer’s time ( e.g., ”tense”, ”now”,...) can not used in quantum language.
Note that this hypothesis (A1 ) is related to ”tense”. Thus, the linguistic Copenhagen
interpretation (E2 ) in Sec. 1.2.1 says that this (A1 ) is not a statement in quantum
language. Thus, the (A1 ) is not scientiﬁc, that is, there is no experiment to verify thestatement (A1 ).
Therefore, we can conclude that
(A2 ) ”the universe was created ﬁve minutes ago” is not a proposition in QL.

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Some may want to relate this hypothesis to skepticism (cf. ref. [114]), However we donot think that this direction is productive.
Remark 11.15. (i): Also, the above (A2 ) should be compared to the following (A3 )(A3 ) The universe was created in A.D. 2010. (Or equivalently, now is A.D. 2020, andthe universe was created ten years ago.)
This (A3 ) can be denied by experiment, that is, it is diﬀerent from the fact. Thus, this is
a proposition in quantum language.
(ii): The following is a trick question;
In the diagram below, which is less scientiﬁc, Russell or Wittgenstein?

In the above, we say that(♯1 ) Russell’s statement is scientiﬁc, but it is wrong.
(♯2 ) Wittgenstein is making a logical error. That is, he misunderstands Russell’s scientiﬁc statements as unscientiﬁc.

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11.5 Bertrand Russell: ﬁve-minute hypothesis, McTaggart’s paradox, Moore’s paradox

11.5.3 * McTaggart’s paradox
This section was written with reference to the following.
(A) [76]:Ishikawa, S: Leibniz-Clarke correspondence, Brain in a vat, Five-minute hypothesis, McTaggart’s paradox, etc. are clariﬁed in quantum language Open Journal of philosophy, Vol. 8, No.5, 466-480, 2018, DOI: 10.4236/ojpp.2018.85032(https://www.scirp.org/Journal/PaperInformation.aspx?PaperID=87862)[Revised version] (https://philpapers.org/rec/ISHLCB)(http://www.math.keio.ac.jp/academic/research_pdf/report/2018/18001.
pdf)• [85] S. Ishikawa, K. Kikuchi, (2021) Quantum Fuzzy Logic and Time, Journal ofApplied mathematics and physics, Vo.9 No.11 2021, 2609-2622(https://www.scirp.org/journal/paperinformation.aspx?paperid=112972)
J.M.E. McTaggart (1866–1921) was an English philosopher.

He was a member of the Cambridge Apostles, along with B. Russell (1877-1970). In ref.
[98], McTaggart asserted ”the Unreality of Time” as follows.
The sketch of McTaggart’s proof(B1 ) Assume that there are two kinds of times. i.e. ”observer’s time (A-series)” and”objective time (B-series)”. (Note that this assumption is against the linguistic
Copenhagen interpretation (E2 ) in Sec. 1.2.1.)(B2 ) · · · · · ·(B3 ) After all, the contradiction is obtained
Therefore, by the reduction to the absurd (i.e., the proof by contradiction), we get;(B4 ) A-series does not exist (in science).
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Recall the linguistic Copenhagen interpretation:

This implies that the above McTaggart’s proof is non-sense.
For completeness, we add the following. About this proof, there are various opinions
also among philosophers. Although I cannot understand the above part (B2 ) (since theproperties of A-series are not clear), I agree to him if his assertion is (B4 ) (cf. ref. [49]).
That is, I agree that McTaggart noticed ﬁrst that observer’s time is not scientiﬁc. Recall
the linguistic Copenhagen interpretation (E2 ) in Sec. 1.2.1:• While ”matter” is in the space-time, the observer is not.
Thus, I agree to the opinion that McTaggart is one of discoverers of the linguistic Copenhagen interpretation. I think, from the quantum linguistic point of view, that he shouldbe estimated more highly.
Therefore, we can say that(C) McTaggart was also related to the linguistic Copenhagen interpretation(≈ the problem such that ”What is a ’proposition’ ?”)337

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11.5 Bertrand Russell: ﬁve-minute hypothesis, McTaggart’s paradox, Moore’s paradox

11.5.4 Moore’s paradox: ”It is raining, but I do not believe it is raining”
11.5.4.1 ”It is raining, but I do not believe it is raining” and Wittgenstein
G.E. Moore (1873-1958) was an English philosopher. He was, with Bertrand Russell,
Ludwig Wittgenstein, and Gottlob Frege, one of the founders of analytic philosophy.

As his quote, the following is famous:(A) ”It is raining, but I do not believe it is raining”
That is, Moore raised the following question (i.e., Moore’s paradox).
(B) Is the (A) a proposition?

That is, can we deﬁne the truth value of (A)?

This problem is famous because Wittgenstein was very interested in it. I don’t remember
where it was written, but I heard that Wittgenstein said that the discovery of (A) was
the greatest of all Moore’s achievements. The reason why Wittgenstein was interested in(A) is clear as follows.
(C1 ) His purpose at TLP was to answer ”What is a proposition?” And at the end of
his TLP, he wrote, ”For something that is not a proposition, you must be silent.”
However, TLP was not theoretically successful, and ”what is a proposition?” wasthe philosophical theme of his life.
And Wittgenstein thought:(C2 ) The ﬁrst thing to do in order to answer ”what is a proposition?” is to study apseudo-proposition such as ”It is raining, but I do not believe it is raining”. It
would be great if we could make a theory of pseudo-propositions (and pseudo-truth
values).
Now that I think about it, there are various detailed themes in this direction (ordinary
language school), which are practically interesting (relationship between everyday language and AI, etc.), but there was not much possibility that a big theory would be born.
Of course, the future is still undecided, and it’s too early to say for sure.

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11.5.4.2 ”It is raining, but I do not believe it is raining” in the linguistic Copenhagen interpretation
Remember the linguistic Copenhagen interpretation (cf. Sec. 1.1) as follows.
(D1 ) Audience should not go on stage(D2 ) The measurement is not dependent on the observer(D3 ) the observer cannot measure the observer himself(D4 ) Don’t use terms ”I”, here”, ”now”.

”It is raining, but I do not believe it is raining” can be interpreted in many ways, buthere I will explain the following scientiﬁc interpretation.
First, let us assume that
(E1 ) ”I” in ”It is raining, but I do not believe it is raining” is ”Mr. X”.
(E2 ) ”(D1 ):Audience should not go on stage” implies that the observable is not Mr X.
That is, it implies that(F) The sentence ”It is raining, but I do not believe it is raining” is regarded as themeasuered value of [Weather conditions, Mr. X’s state of mind]
If so, this is scientiﬁc. But, it is somewhat unnatural.
Therefore, it is natural to assume that(G) ”It is raining, but I do not believe it is raining” is not considered as a scientiﬁcproposition, but as a literary art that enjoys playing with words.
11.5.4.3 Wittgenstein was interested in Moore’s paradox
However, the purpose here is not to examine Moore’s paradox in detail. What I havefocused on is the following(H) when Wittgenstein ﬁrst heard this paradox one evening (which Moore had earlier
stated in a lecture), he rushed round to Moore’s lodgings, got him out of bed andinsisted that Moore repeat the entire lecture to him.
If this anecdote is true, it is because it clearly shows that the central theme of Wittgenstein’s research is ”What is a scientiﬁc proposition?”(I) ”It is raining, but I do not believe it is raining” is ”what we cannot speak about”.
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11.5 Bertrand Russell: ﬁve-minute hypothesis, McTaggart’s paradox, Moore’s paradox
Furthermore, he must have thought Descartes’ cogito proposition ”I think, therefore Iam” ( which is called the ﬁrst principle of philosophy) was not also a proposition (cf. thelinguistic Copenhagen interpretation (E1 ) in Sec. 1.3.3).
At the time, Russell, Moore, and McTaggart were all professors in the Department of
Philosophy at Cambridge University. Moore’s paradox, Russell’s ”ﬁve-minute hypothesis,”and McTaggart’s ”non-existence of time” were all based on questions such as ”What isa scientiﬁc proposition? (Equivalently, ”What is the Copenhagen Interpretation? ) .
This was the environment in which analytic philosophy (symbolically, ”Wittgenstein”)
was born.
In other words, they showed an unusual interest in ”nonsense phrases”. Since the
Copenhagen Interpretation is the principle to eliminate these ”nonsense phrases” fromscience, we can say that their interest was in the Copenhagen Interpretation. However,I would like to think that Wittgenstein’s real purpose was ”axiomatic theory”. Axioms
1 (measurement) and 2 (causality) in quantum language are diﬃcult to discover without
knowing quantum mechanics, so it does not matter whether Wittgenstein was a scientiﬁc
success or failure. Therefore, I would like to argue the following diagram






Develop Descartes theory Leibniz
Spinoza(=mind-body
dualism)
Monadology• Pantheism , •



Solve






the mind-body problem
• −−−−−−−−−−−−−−−−−−−−−→

Axiomatic theory

QL




 WittgensteinContinental rationalism 


Axiom
1

Descartes theory





TLP
•
,
Axiom
2



mind-body dualism 





Derived from cogito















How
to
use
Locke,
Berkley,
QLDescartes theory
epistemology
linguistic




Solve
the
subjectivity(Copernican
Copenhagen
problem, etc.





−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
→
•
revolution)≈interpretationEmpirical
theoryHume, Kant

British empiricism




QL

Figure 9.1: QL-synthesis

That is, the above ﬁgure explains the following(J) The reason why the studies on the Copenhagen interpretation in analytic philosophy(Russell with the ”world ﬁve minutes before” hypothesis, McTaggart’s paradox, andMoore’s paradox) were not connected with TLP (see the discussion) is that TLP
belonged to the axiomatic approach.

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Chap. 11 Linguistic philosophy (Before TLP)Up to this point, the ”pre-Wittgenstein” period has ended, and after Wittgenstein,philosophy of science is still considered to play the leading role.

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11.6 Saussure: Copernican revolution in language

11.6 Saussure: Copernican revolution in language

Ferdinand de Saussure (1857-1913) was a Swiss linguist. He is widely considered one of
the founders of 20th-century linguistics.

11.6.1 Saussure’s linguistics: What comes ﬁrst, things or words?
Let’s think a little more about the implications of Saussureian linguistics. We tend to
think that there are things at the beginning, and that we give each thing a name, just like
we put a label on it. However, that is not the case. Rather, Saussureian linguistics saysthat we understand the order of things by the act of giving them names. I think that his
theory is the linguistic version of the Copernican revolution of Kant. That is,copy theory (realism)

Saussure’s revolution
things −→ words −−−−−−−−−−−−−−−−−−−−−−−→
Copernican revolution in language

constructionism (transcendental idealism)words −→ things
Saussure’s linguistics(11.2)For example,(A1 ) I, who live in Japan, an island nation, know the names of many ﬁsh, but rarely
distinguish between ”cow, bull, ox, calf,” etc.

(A2 ) Also, in Japan, the rainbow has seven colors (i.e., red, orange, yellow, green, blue,
indigo, violet(=purple)), and I was surprised when I ﬁrst heard that there is acountry where the rainbow does not have seven colors.
(A3 ) Quantum language changes the way we see the world. In other words, quantum
language is like a kind of worldview. That is, we see the world through the coloredglasses of quantum language.

In the above, I’d like to assume that (A1 )∼(A3 ) are about the same.
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11.6.2 * The quantitative expressions of ”signiﬁed”
Now let us explain the terms ”signiﬁer” and ”signiﬁed”, which were introduced by
Saussure.
Deﬁnition 11.16. [”signiﬁer” and ”signiﬁed” in ordinary language]
For example, we explain ”signiﬁer” and ”signiﬁed” concerning ”dog”(B1 ) The ”dog” in front of you is itself a physical being, isn’t it? The image of the dog
you have in mind, the barking, or the image of the dog in your head, is ”signiﬁed”.
(B2 ) When this becomes a word (letter/sound) such as ”dog”, it is called ”signiﬁer”.

Although Saussure’s proposal (i.e., the above deﬁnition) is very signiﬁcant, his ideas are
not quantitative, so they are not very useful from a scientiﬁc standpoint. As emphasized
frequently throughout the paper, I consider(C) Without a quantitative worldview, we can’t say anything solid from a scientiﬁcstandpoint.
In fact, Saussure’s idea does not play an important role in science.

Thus, I will propose Deﬁnition 11.17, in which Saussure’s idea is realized in quantum
language as follows.
Deﬁnition 11.17. [Membership function (= Fuzzy set, cf. [41, 42, 43, 44, 129])] Let Ω
be a state space. A continuous function m : Ω → [0, 1] (i.e., the closed interval in the realR) is called a membership function. Assume that the state (i.e., quantitative property)of any animal can be expressed by a point in the state space Ω. Deﬁne the membershipfunctions mD : Ω → [0, 1] of dogs as follows. Suppose that there are 100 zoologists andthe following question is made them.
(D) Is this animal with the sate ω1 (∈ Ω) a dog or not?
The answer is as follows.

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11.6 Saussure: Copernican revolution in language(E)

70 zoologists say that this bird is a dog.
30 zoologists say that this animal is not a dog.

Then the value of mD (ω1 ) is deﬁned by 0.7. For many animals with the state ωi (i =2, 3, ...N ), repeating the experiment in the same way, the value of mD (ωi ) (i = 2, 3, ...N )is determined. And the membership function mD : Ω → [0, 1] of dogs is deﬁned by theinterpolation method (which may be rather subjective). Put ΩD = {ω ∈ Ω | mD (ω) = 1},
which is called the dog state class.
mD

1

ΩD
[membership function mD : Ω → [0, 1], and ΩD ]Ω

Thus we have the dog-like observable OD = ({T, F }, 2{T,F } , GD ) in C(Ω) with the membership function GD ({T }) (= mD ) (cf. Deﬁnition 12.3).

With the above preparation, we get the following deﬁnitions.
Deﬁnition 11.18. [”signiﬁer” and ”signiﬁed” in quantum language](F1 ) ”signiﬁer”—–”dog”(F2 ) ”signiﬁed”—– membership function mD : Ω → [0, 1] (or roughly, ΩD )The above will play an essential role in Hempel’s raven problem in Sec. 12.7.

♠Note 11.7. (i); The above deﬁnition is essential to the solution of Hempel raven problem.
See Section 12.7(ii): From the quantum linguistic point of view, Saussure’s theory is located in the follow-

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Chap. 11 Linguistic philosophy (Before TLP)
ing.
•
Idea

linguistic version−−−−−−−−−−−→

signiﬁer, signiﬁed

Plato’s idea theoryChristianyversion

Saussure’s linguistic theory

universal

membership function

Anselmus, Thomas

Zadeh’s fuzzy theoryquantiﬁcationycognitiveyversion

secondary qualityyidentiﬁcation

linguistic version−−−−−−−−−−−→

observable
quantum language

John Locke

Some readers may disagree with me about Plato’s Idea, as I’ve made a rather dogmatic
assumption about it. However, I like the diagram above because it makes me feel like I
understand the history of Western philosophy as the history of dualistic idealism.

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11.7 Several Copernican revolutions [II]11.7 Several Copernican revolutions [II]Continued from [Copernican revolution [I] (cf.] Sec. 10.2)].

11.7.1 Copernican revolution as a method of making idealism from realism
The ﬁgure below is the same as Figure 0.1 in Preface.
The ”Copernican revolution” was discussed in section 10.2, but let us consider thefollowing problem again.
10 below is Kant’s famous Copernican revolution. However, there
Problem 11.19. The ○10 in this ﬁgure. Find them. The answer isare ”two other Copernican revolutions than ○”given in Answer 11.20.
Figure 0.1: The location of QL in the history of western philosophy(realistic) monism (i.e., without measurement axiom)
relativity
4 −→ theory−−−−−−−−−−→○(monism)(unsolved)3○
−−→ Newton ○6 TOE:theory of(realism)−−→ everything
quantum
5 −→ mechanics −−−−−−−−→○(quantum phys.)

Socrates
Aristotle
0○:Greek

7○

Schola-8○−−−−−−−−−−−−−−−−−→

8
○



quantization

7
○

9
−
−
−
−
−
−
−
−
−
→
○

statistics−−−−−−−−−→

dualism
applied math.
system theoryclassical mechanical worldview


(=MT)



(mind-body dualism)(Copernican revolution)) ○QL:quantum
15
linguistic−−→
language
Descartes
Kant scientiﬁc
10
○

11 
−−−−−−→○−−→ Spinoza, Locke,... −−→ quantum mechanicalturnepistemological worldview
worldview

12
○



linguistic turn



?

(logic in philo.))(logic, set theory)Wittgensteinmechanical
Saussure −−−−−−−→○

Boole, Frege,
14
13
○
turn
Zadeh
Russell, Cantor −−→philosophy ○21−−−−−→○
sticism
Parmenides, Zeno
Platolinguistic turnmath.

logical worldview(idealistic) dualism (i.e., with measurement axiom)
Figure 0.1 (in Preface): The history of the world-descriptions
Philosophy (≈ dualistic idealism) has progressed toward QL
0
1
2
10
12
14
15 )
(i.e., ○→
○→
○→
○→
○→
○→
○

Answer: The answer is written in Answer 11.20. Since there are three Copernican revolutions in the above ﬁgure, you can understand how great a discovery Kant’s
Copernican revolution is.

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♠Note 11.8. The original Copernican revolution is of course as follows (cf. Chap. 7):

Thus, the original Copernican revolution means
Developing science

the original Copernican revolution−−−−−−−−−−−−−−−−−−−−−−−−−→

realism
Ptolemaic systemscientiﬁc progress

More advanced science

realism
Copernican system////

However, Kant’s original Copernican revolution implies the following
copy theory [The world ﬁrst, cognition later]

Hume’s pseudoscience, pseudo realism
constructionism [Cognition ﬁrst, the world later]
Kantian Copernican revolution−−−−−−−−−−−−−−−−−−−−−→
scientiﬁc progress(transcendental) idealism

In this text, the term is used to describe the shift from ”realism (the world comesﬁrst and then we make a theory of it)” to ”idealism (the theory comes ﬁrst and then we
understand the world).
copy theory [World First, Theory Second]

realism
realistic painting
constructionism [Theory ﬁrst, world second]
as a way of making idealism from realism.

−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
Copernican revolution

transcendental idealism
abstract painting

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11.7 Several Copernican revolutions [II]In short,(A) the Copernican revolution is a method of turning

realism into idealism.
♠Note 11.9. In this text, the idealism generated by the ”Copernican turn” is called ”transcendental idealism. I believe that scientiﬁc idealism is always transcendental idealism.
(♯) The discovery of scientiﬁc idealism that is not transcendental idealism would be thegreatest discovery in the history of philosophy.
Previously, I thought that probability theory belongs to scientiﬁc idealism and not transcendental idealism. However, since probability theory is a part of QL. Thus, probability
theory may be transcendental idealism. I think it is worth challenging (♯), because there
might be a position that mathematics or evolutionary theory can be regarded as scientiﬁc
idealism that is not transcendentalist idealism.

Answer 11.20. Answer of Problem 11.19:
As discussed above, in this text, we understand the “Copernican revolution” as follows.
copy theory [World First, Theory Second]

realism
realistic painting
constructionism [Theory ﬁrst, world second]
as a way of making idealism from realism.

−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
Copernican revolution

transcendental idealism
abstract painting

7 ○,
8 ○”.
10
Thus, the answer is the following three ”○,
7○:
copy theory

constructionism−−−−−−−−−−−−−−−−→ transcendental idealism

realism
Newtonian mechanics

7
Copernican revolution○

statistics

8○:
copy theory

realismquantum language

constructionism−−−−−−−−−−−−−−−−→ transcendental idealism
8
Copernican revolution○

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10○:
copy theory

constructionism−−−−−−−−−−−−−−−−→ transcendental idealism

pseudorealism

Copernican revolution10○

Kant idealism

Hume’s pseudoscience

////

7 ○,
8 ○
10 and Saussure’sAssertion 11.21. Four Copernican revolutions: the above ○,revolution
Summing up as follows;

copy theory
the world −→ cognition

10○:Copernican
revolution [I]−−−−−−−−−−−−−−−−−−−−−−−→
in cognition (due to Kant)

constructionism
cognition −→ the worldKant
12 ylinguistic turn○

Descartes=Locke

Copernican revolution [II]copy theory
things −→ words
language(= Saussure revolution )−−−−−−−−−−−−−−−−−−−−−−−→
in language (due to Saussure)

Saussure, Wittgenstein
14 ymechanical turn○

copy theory
phenomena −→ QM
quantum mechanics (=QM)
xa○quantumization
copy theory
phenomena −→ CMclassical mechanics (=CM)

constructionism
words −→ things

8 Copernican revolution [III]○−−−−−−−−−−−−−−−−−−−−−−−−→
in quantum mechanics

constructionism
QL −→ phenomena
quantum language (=QL)
c ○yclassicization

7 Copernican revolution [IV]○−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
in classical mechanics (Fisher, etc.)

constructionism
statistics −→ phenomena
statistics, system theory

Thus, we say:(B) There are various types of idealism, such as spiritual, religious, and anti-scientiﬁc,and it is true that all of them are “useful in some sense.” However, there are
ideologies such as [II]∼[IV] above (especially statistics) that can become the basic
theories of the ongoing scientiﬁc revolution. Moreover, it should be noted that
their prototype is Kant’s Copernican revolution [I] (cf. Sec. 10.2).

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11.7 Several Copernican revolutions [II]11.7.2 From Descartes to Kant: It is a progress?
Now that you understand the meaning of the Copernican turn, I will give the answer toProblem 10.5 ”Is Descartes to Kant a progress?”
Answer 11.22. [ The answer of Problem 10.5: In the ﬁgure below, is Descartes to Kanta progress?]

The immature state of a discipline at the time of its birth is sometimes regarded asphilosophy. For example, in ancient Greece, mathematics, astronomy, and atomic theory
were philosophies. It is also reasonable to consider the epistemology of Locke and Humeas the birth of brain science. Kant, however, rejected this trend as follows.
(♯1 ) Greatness of Kant is to have prevented that epistemology faces the direction of
the brain science by the showy name called ”Copernican revolution”.
That is, Kant understand that, even if the epistemology is clariﬁed by the brain science,
this is non-sense from the philosophical point of view. We think that(♯2 ) using the term ”Copernican revolution”, Kant prevented that epistemology enters into the zone of brain science.

Naming of ”Copernican revolution” does not mean that self-congratulation of Kant. I’dlike to believe that350

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• strong intention of Kant which says ”Epistemology is not experimental science,but philosophy” (or, ”philosophy should not aim at experimental sciences”), is
included in the term: ”Copernican revolution”.
Therefore, I think that• ”The Copernican revolution (due to Kant)” is the greatest discovery
in the history of philosophy.
That is,the Copernican revolution = the discovery of ”scientiﬁc idealism”

So, in conclusion, I think the following is obvious.
Descartes, Locke, Berkeley, Hume −−−−−−−−→ Kant *2
progress(QL)

since the term ”Copernican revolution” prevented epistemology from entering the realmof brain science.
♠Note 11.10. Kant’s pseudo-scientiﬁc work (”thing-in-itself”, reason, understanding,antinomy, etc.) is considered more Enlightenment than scientiﬁc. Here, we did notcount anything other than the ”Copernican turn” as either positive or negative. However, the analytical and comprehensive propositions are highly evaluated (cf. Sec.12.3).

Remark 11.23. Someone may want to consider the following progress:
Descartes=Kant −−−−−→ Husserl
progress

This is because it is widely accepted that Husserl overcame the heights of Kant. In termsof literary enjoyment, I think the above is correct. If so, then Husserl’s position is asfollows*2 The symbol ”X −−−−−−−−−−→ Y ” is deﬁned by [”Y” is closer to quantum language than ”X”]

progress (QL)351

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11.7 Several Copernican revolutions [II]Thus I conclude that
Descartes=Kant −−−−−−−−→ Husserl∗2

regress (QL)

(For Kant −−−−−−−−−→ Wittgenstein , see the following Chapter.)progress (QL)

I think Husserl developed the pseudo-scientiﬁc aspects of Kant. I think he enjoyed theterm ”rigor” from a literary point of view. Husserl’s phenomenology is one of the great
achievements of Western philosophy, but it has little to do with science. If you see it
as a kind of literature, it is ﬁne if there are philosophy lovers who prefer this style ofphilosophy.

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353

Chapter 12

Linguistic philosophy ( After TLP)
I agree with the following Dummett’s opinion: (cf. ref. [15]: Origins of analytical philosophy):(♯1 ) If we identify the linguistic turn as the starting-point of analytical philosophy proper,
there can be no doubt that, to however great an extent Frege, Moore and Russell prepared the ground, the crucial step was taken by Wittgenstein in the Tractatus Logicophilosophicus (= TLP) of 1922 ( 127 page)

Since I can not understand the non-mathematical work of Frege and Russell, I have notjudged them as comprehensively as Dummett, but I consider the following.
Frege, Russell(♯2 )mathematical logic
Logic is so powerful that
it is the foundation of mathematics

Wittgenstein−−−−−−−−−→
philosophy

TLP
What we cannot speak about
we must pass over in silence

Many philosophers may not agree with the above opinion (♯1 ) (=(♯2 )). Because Wittgenstein failed to fulﬁll the purpose of TLP, i.e. to answer the following questions:(♯3 ) What is the deﬁnition of ”what we can speak about” (=”proposition”)?
(♯4 ) Why does logic work in our daily lives?*1*1 About the question ”Why does statistics work in our daily lives?”, we already answered in refs.

[42], [44], [50],...,etc. These are summarized in [74]

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Although he failed in this task, his aspirations were enthusiastically praised by the generalpublic, and he laid the foundation for today’s analytic philosophy.
I agree with opinion (♯1 ) because I think that the philosopher’s job is not only to solve
problems, but also to ”present problems and create new trends in philosophy *2 .

In Sec. 12.2, I answer the above questions (♯3 ) and (♯4 ) in QL. And thus, the reader canbe sure that Wittgenstein’s direction was the right one.
Just as ”physics is one,” I like to think, ”philosophy is one”. For this, I assert that Wittgenstein must be located in the mainstream of philosophy (i.e., the mind-body dualism) as
follows.

the mainstream of philosophy (i.e., the mind-body dualism)z}|{
Descartes → Locke → Berkeley → Hume → Kant → Wittgenstein → QL

*2 Wittgenstein may have only agitated and did nothing substantial. However, we consider ”agitating

and involving the general public” to be of special importance as an achievement of philosophy. In
this text, most of ”what Wittgenstein did in science” is ”what the author thinks ’what Wittgensteinwanted to do in science’”.

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12.1 QL-logic (i.e., Fuzzy logic in QL); My scientiﬁcunderstanding of TLP
This section consists of excerpts from the following paper:• Ref. [81]: Ishikawa, S., (2021) Fuzzy Logic in the Quantum Mechanical Worldview; Related to Zadeh, Wittgenstein, Moore, Saussure, Quine, Lewis Carroll, etc.
JAMP, Vol. 9, No.3, 140-154,(https://www.scirp.org/journal/paperinformation.aspx?paperid=110830)

12.1.1 Wittgenstein: Tractatus Logico Philosophicus (=TLP)
In ref. [125] (i.e., Wittgenstein’s ”Tractatus Logico Philosophicus” (abbreviated as
TLP)), which is one of the most famous philosophy books of the 20th century, Wittgenstein
studied ”logic” in philosophy (and not in mathematics). However, in hindsight, he may
have written literary work on the subject of ”logic” (cf. refs. [13], [93]). It is a fact,
however, that TLP was accepted by much of the general public. I think the general publicfelt that Boole and Frege’s ”symbolic logic” was just mathematics, and that this was not
suﬃcient as philosophy. In other words, they expected TLP to answer the question, suchthat(♯1 ) Why does logic work in our world?
and(♯2 ) What is the deﬁnition of non-mathematical proposition?

Wittgenstein’s motivation for ”TLP” should have been to answer these questions ((♯1 )and (♯2 )), but he was too much a poet and a conﬁdent man. Moreover, ”TLP” waspublished in 1921, and Wittgenstein did not know the discovery of quantum mechanics(around 1925). Therefore, Wittgenstein’s chances of scientiﬁc success were very small.

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12.1.2 Zadeh and Kalman; The problem of universals in 20th century

In 1965, L.A. Zadeh proposed an engineering concept called ”fuzzy sets”( cf. ref. [129]),
which was enthusiastically supported by some engineers. However, R.E. Kalman did notrecognize ”fuzzy sets” as a scientiﬁc concept, and argued as follows (1972):”Let me say quite categorically that there is no such thing as a fuzzy concept...
We do talk about fuzzy things but they are not scientiﬁc concepts. Some people in
the past have discovered certain interesting things, formulated their ﬁndings in a
non-fuzzy way, and therefore we have progressed in science” (cf. ref. [130]).

I consider this the ”problem of universals of the 20th century” .
Even now, more than 50 years later, I don’t think Kalman’s claim can be denied. In
fact, the concept of ”fuzzy sets” has not yet acquired the status of more than a convenientengineering method. Kalman might have thought that the basic philosophy of engineering
is a mechanical worldview, and thus, ”scientiﬁc concepts”=”concepts in the (classical)
mechanical worldview”. Note that dynamical system theory (which is essentially thesame as statistics ) is considered as the mathematical representation of the (classical)
worldview. That is,(classical mechanical worldview)(measured value, state)(A1 ) Dynamical system theory = state equation + measurement equation(= statistics )( causality)(probability)(See 7 in Figure 0.1 (in Preface) )356

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Chap. 12 Linguistic philosophy ( After TLP)On the other hand, quantum language is characterized as follows:(quantum mechanical worldview)(A2 )Quantum language(= measurement theory)(measured value, observable, state)= Axiom 2 +Axiom 1

(causality)(measurement)+ linguistic ( Copenhagen ) interpretation(how to use Axioms 1 and 2)

8 in Figure 0.1 (in Preface) )(See ○

which is a mathematical representation of quantum mechanical worldview (cf. refs. [46],[74], [69]).
In fact, ”fuzzy sets” correspond to ”observables” in QL. If so, it is understandable that
Kalman did not recognize ”fuzzy sets”. Zadeh’s ref.[129] is one of the most cited papersof the 20th century. This means that many engineers felt that ”mathematical sets” were
not suﬃcient and expected his ”fuzzy sets”.
I think Wittgenstein and Zadeh were in a very similar position in the sense that they
were enthusiastically supported by many ordinary people (or engineers in the ﬁeld). I
would rather believe in the sense of the general public than in the sense of a few outstanding experts. But what Wittgenstein and Zadeh lacked, I think, was a world view.
Therefore, their arguments are vague and diﬃcult to understand, but if we understandthem under the quantum mechanical worldview, they seem to be arguing almost the samething. In this section, I will explain them. In other words, we explain the validity of
Wittgenstein’s TLP and Zadeh’s fuzzy set theory at the same time.

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12.1.3 QL proposition (= fuzzy proposition in QL = TF-measurement) in classical system
Now let us explain ”Fuzzy logic in QL” (i.e., my understanding of TLP). Let us start fromeasy example as follows.
Although our theory is valid for quantum systems as well as classical systems, in this Sec.
12.1 we explain our idea in classical systems (i.e., the commutative case that A = C(Ω)).
Again let us start from the following example (= Example 11.4).
Example 12.1. [= Example 11.4]: For example, consider a proposition P1 such that(B) P1 = ”this tomato is red”,¬P1 = ”this tomato is not red”,

And suppose that there are 100 respondents, and furthermore, the following question is
asked to them.
(C) Is this tomato red? (i.e., is the proposition P1 true or not?)Assume that the results of the responses are as follows.
(D) 70 respondents say ”Yes, this tomato is red” (i.e., the proposition P1 is true, i.e.”T”)


30 respondents say ”No, this tomato is not red” (i.e., the proposition P1 is false, i.e.”F”)

For simplicity, we consider T = ”Yes” = 1 and F = ”No” = 0.

This can be probabilistically interpreted as follows.
(E) When any respondent is randomly selected out of 100, the probability that this
respondent will answer ”yes” to question (C) is p1 (= 0.7). Or simply, the probability
that the proposition P1 is true is p1 . In symbolic form,
Prob[P1 ; {T }] = p1 (= 0.7)
Then we generally denote that(F)

Prob[P1 ; {T }] = p1 ,Prob[P1 ; {F }] = 1 − p1

(where 0 ≤ p1 ≤ 1)

Also, note that Prob[¬P1 ; {T }] = Prob[P1 ; {F }] = 1 − p1 .
////

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Chap. 12 Linguistic philosophy ( After TLP)The above will be formulated in terms of QL as follows. Let Ω a state space, which is, forsimplicity, assumed to be a compact set Ω ⊆ RN (= N -dimensional real space), where N
is suﬃciently large natural number. Consider many tomatoes, that is, roughly speaking,consider T as the set of all tomatoes. Assume that any tomato t(∈ T ) is represented bya state ω, which is an element of the state space Ω. Thus, we have the map ωb : T → Ω.
That is, the quantitative property of a tomato t is represented by ωb (t). For example, it
suﬃces to consider Ω such that(G) ω = ω (1) (= weight), ω (2) (= diameter), ω (3) (= color value),

(4)
(5)
(N )
ω (= calorie), ω (= sugar content), ..., ω (= ...) ∈Ω ⊆ RN

Consider a binary observable (or, {T, F }-valued observable ) O1 = ({T, F }, 2{T,F } , G1 )in C(Ω). The measurement MC(Ω) (O1 , S[δω ] ) is called a (T F )-measurement, which is also
called a fuzzy proposition. Axiom 1 says that(H) the probability that measured value T is obtained by the (T F )-measurement
MC(Ω) (O1 , S[δω ] ) is given by δω (G1 )(= C(Ω)∗ hδω , G1 iC(Ω) = G1 (ω))
This is the quantum linguistic representation of the above (E). That is, we identify aproposition P1 with a (TF)-measurement MC(Ω) (O1 , S[δω ] ).
Remark 12.2. (i): Someone might say that the term ”the set of all tomatoes” is asambiguous as ”the set of all dinosaurs”. However, for the sake of convenience, here weuse the term ”the set of all tomatoes”. This problem is the same as that of the Hempel’raven paradox (cf. Sec. 12.7, i.e. ”the set of all ravens” leads to contradiction).
(ii): If we want to consider another proposition P2 (=MC(Ω) (O2 , S[δω′ ] ) ), we must deﬁneMC(Ω) (O1 , S[δω ] ) ∧ MC(Ω) (O2 , S[δω′ ] ) (ω 6= ω ′ ). This will be explained in Sec. 12.1.7(iii): If we want to consider both tomato’s world Ω1 and apple’s world Ω2 , it suﬃces tostart from the tensor space C(Ω1 ) ⊗ C(Ω2 ) (=C(Ω1 × Ω2 )). This will be also explainedin Sec. 12.1.7.

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12.1.4 Fuzzy logic in QL
Let’s start with the following deﬁnition.
Deﬁnition 12.3. [(TF)-measurement (=Fuzzy proposition), Fuzzy set (= Membership function)]
Let O = ({T, F }, 2{T,F } , G) be a binary observable (or, (TF)-observable, {T, F }valued observable ) in a C ∗ -algebra A. A measurement MA (O, S[ρ] ) is called a(TF)-measurement, which is also called a fuzzy proposition.
Since Axiom 1 says that the probability that a measured value T is obtained by(TF)-measurement MA (O, S[ρ] ) is given by ρ(G({T })), we say that
(I1 ) a (TF)-measurement MA (O, S[ρ] ) is true with probability ρ(G({T }))Or,(I2 ) Prob[MA (O, S[ρ] ); {T }] = ρ(G({T })) (= A∗ hρ, G({T })iA )Also, G({T })(∈ A) is called the membership function of O (cf. Deﬁnition 11.17).

♠Note 12.1. (i): We sometimes use a binary observable O = ({y, n}, 2{y,n} , G) be a binary observable (or, (yes-no)-observable, {y, n}-valued observable ) instead of a binary
observable O = ({T, F }, 2{T,F } , G) (or, (TF)-observable, {T, F }-valued observable ).
(ii): Let O = ({T, F }, 2{T,F } , G) be an (TF)-observable in C ∗ -algebra A. The observable
O = ({T, F }, 2{T,F } , G)(or, in classical case, a membership function (= fuzzy set) G({T })itself is not related to ”probability”, as Zadeh frequently emphasized. However, when we
take a measurement MA (O, S[ρ] ); {T }, a probability occurs. (See (E) above.)(iii): More generally, I think that(♯) ”Proposition in QL”= ”measurement” (i.e., Axiom 1)”

though it is essentially the same as a proposition in statistics. As is repeatedly emphasizedin this text,• Use statistics for rigorous considerations and logic for coarse considerations!

Our interest is to show that
Kant philosophy −−−−−−−−−−→ Analytic philosophy ,progress(QL)
and thus,• the main concept ”proposition” in analytic philosophy must be formulated in theframework of dualistic idealism. This is realized in the above deﬁnition.
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That is, Wittgenstein took over the baton of ”dualistic idealism (i.e., the mainstream ofphilosophy)” from Kant (cf. ref. [81], or Sec. 12.1). In other words, I would like to thinkthat ”philosophy is one” just as ”physics is one”.

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12.1.5 Calculations in QL-proposition (= TF-measurement )In general, we must consider many propositions {Pi = MAi (Oi , S[ρi ] ) : i = 1, 2, ..., n}. In
this section, we devote ourselves to the following simple case:(J1 ) A is ﬁxed, i.e. A1 = A2 = ... = An
(J2 ) O1 , O2 , ..., On commute,(J3 ) a state ρ is only one, i.e. ρ1 = ρ2 = ... = ρn
For the general case, we discuss in next section (i.e., Sec.12.1.7). That is, in this section, wedevote ourselves to {Pi = MA (Oi = ({T, F }, 2{T,F } , Gi ), S[ρ] ) : i = 1, 2, ..., n}. However,it should be noted that the above simple case (N) is essential, that is, the general case isan easy consequence of the simple case as seen in the next section.
Deﬁnition 12.4. [Fuzzy logic symbols ( ¬, ∧, ∨, →) )] Let Oi = ({T, F }, 2{T,F } , Gi ) be
binary observables (or, {T, F }-valued observable) in a C ∗ -algebra A. (i = 1, 2). Assume
that Oi (i = 1, 2) commute.
1G2 ({T })G1 ({T })Ω
Figure 12.1a: [G1 ({T }) and G2 ({T }) in the classical case such that A = C(Ω) ]*3
Fix the quasi-product observable O1 ×qp O2 = ({T, F }2 , P({T, F }2 ), G1 ×qp G2 ). Consider (TF)-measurement MA (Oi = ({T, F }, 2{T,F } , Gi ), S[ρ] ) (which is abbreviated as Pi) in a C ∗ -algebra A. Put µi (Ξ) = ρ(Gi (Ξ)) (Ξ ∈ {T, F }, i = 0, 1, 2), and (×qp
i=1,2 µi )(Ξ1 ×qp
Ξ2 ) = (ρ(G1 × G2 )(Ξ1 × Ξ2 )) (Ξ1 , Ξ2 ⊆ {T, F }).

Deﬁnition:
below!

[(i): NOT:¬], [(ii): AND:∧], [(iii): OR:∨], [(iv): implication:→](i): [NOT: ¬]Put i = 1, 2. Deﬁne ¬MA (Oi , S[ρ] ) such that
¬MA (Oi , S[ρ] ) = MA (π ¬ Oi , S[ρ] ) = MA (π ¬ Oi = ({T, F }, 2{T,F } , π ¬ Gi ), S[ρ] )where the map π ¬ : {T, F } → {T, F } is deﬁned by π ¬ (T ) = F, π ¬ (F ) = T . And, π ¬ may
also be used as follows: π ¬ Gi ({T }) = Gi ({F }), π ¬ Gi ({F }) = Gi ({T }). Clearly it holds
that Prob[¬MA (Oi , S[ρ] ); {T }] = ρ(Gi ({{F })). = µi ({F }).

*3 For simplicity, we assume that {ω ∈ Ω | [G1 ({T })](ω)}

362T{ω ∈ Ω | [G2 ({T })](ω)} = ∅.
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G1 (π ¬ {T })G1 ({T })1

Figure 12.1b: [G1 (π ¬ {T }) in the classical case such that A = C(Ω) ]Ω(ii): [AND: ∧]Deﬁne MA (O1 , S[ρ] ) ∧ MA (O2 , S[ρ] ) such that
MA (O1 , S[ρ] ) ∧ MA (O2 , S[ρ] ) =MA (π ∧ (O1 ×qp O2 ), S[ρ] )=MA (({T, F }, 2{T,F } , π ∧ (G1 ×qp G2 )), S[ρ] )where π ∧ : {T, F }2 → {T, F } is deﬁned by π ∧ (T, T ) = T, π ∧ (T, F ) = π ∧ (F, T ) =π ∧ (T, F ) = F .
It holds that Prob[MA (O1 , S[ρ] ) ∧ MA (O2 , S[ρ] ); {T }] = ρ(G1 ×qp G2 )((π ∧ )−1 ({T })) =(µ1 ×qp µ2 )({(T, T )}).
(G1 ×qp G2 )((π ∧ )−1 ({T }))1G2 ({T })G1 ({T })Ω
Figure 12.1c: [(G1 ×qp G2 )((π ∧ )−1 ({T })) in the classical case such that A = C(Ω) ](iii): [OR: ∨]Deﬁne MA (O1 , S[ρ] ) ∨ MA (O2 , S[ρ] ) such that
MA (O1 , S[ρ] ) ∨ MA (O2 , S[ρ] ) = MA (π ∨ (O1 ×qp O2 ), S[ρ] )where π ∨ : {T, F }2 → {T, F } is deﬁned by π ∨ (T, T ) = π ∨ (T, F ) = π ∨ (F, T ) =T, π ∨ (F, F ) = F .
It holds that Prob[MA (O1 , S[ρ] ) ∨ MA (O2 , S[ρ] ); {T }] = ρ(G1 ×qp G2 )((π ∨ )−1 ({T }))= (µ1 ×qp µ2 )({(T, T ), (T, F ), (F, T )}).

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12.1 QL-logic (i.e., Fuzzy logic in QL); My scientiﬁc understanding of TLP(G1 ×qp G2 )((π ∨ )−1 ({T }))1G2 ({T })G1 ({T })Ω
Figure 12.1d: [(G1 ×qp G2 )((π ∨ )−1 ({T })) in the classical case such that A = C(Ω) ](iv): [implication: →]Deﬁne MA (O1 , S[ρ] ) → MA (O2 , S[ρ] ) such that
MA (O1 , S[ρ] ) → MA (O2 , S[ρ] ) = MA (π → (O1 ×qp O2 ), S[ρ] )where π → : {T, F }2 → {T, F } is deﬁned by π → (T, T ) = π → (F, T ) = π → (F, F ) =1, π → (T, F ) = F .
It holds that Prob[MA (O1 , S[ρ] ) → MA (O2 , S[ρ] ); {T }] = ρ(G1 ×qp G2 )((π → )−1 ({T }))= (µ1 ×qp µ2 )({(T, T ), (F, F ), (F, T )}).

(G1 ×qp G2 )((π → )−1 ({T }))1G2 ({T })G1 ({T })Ω
Figure 12.1e: [(G1 ×qp G2 )((π → )−1 ({T })) in the classical case such that A = C(Ω) ]364

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12.1.6 Fundamental theorem in QL logic
With the above preparation (+ Theorem 11.10 [Logical sample space]), we obtain thefollowing theorem.
Theorem 12.5. [Fundamental theorem in Fuzzy logic] Let Oi = ({T, F }, 2{T,F } , Gi ) be
binary observables (i.e., {T, F }-valued observable) in a C ∗ -algebra A. (i = 1, 2, ..., n).
Assume that Oi 6= Oj (∀i, j such that i 6= j) (i.e., non-overlapping condition in
Theorem 11.10) and Oi (i = 1, 2, ..., n) commute. Fix the quasi-product observqp
nnable ×qp
i=1,...,n Oi = ({T, F } , P({T, F } ), ×i=1,...,n Gi ). Consider (TF)-measurement
MA (Oi = ({T, F }, 2{T,F } , Gi ), S[ρ] ) (which is abbreviated as Pi ) in a C ∗ -algebra A).
qp
And consider the quasi-product measurement ×i=1,...,n MA (Oi = ({T, F }, 2{T,F } , Gi ),qp
qp
S[ρ] ) = MA (×i=1,...,n Oi = ({T, F }n , P({T, F }n ), ×i=1,...,n Gi ), S[ρ] ). Put•µi ({xi }) = ρ(Gi ({xi }))

qp

•

×

i=1,...,n(xi ∈ {T, F }, i = 1, 2, .., n)µi ({(x1 , x2 , ..., xn )}) = ρ (×qp
G
)({(x
,
x
,
...,
x
)}
i
1
2
n
i=1,...,n= (×qpi=1,...,n µi )({(x1 , x2 , ..., xn )})(∀(x1 , x2 , ..., xn ) ∈ {T, F }n )qpHere, the pair {P1 , P2 , ..., Pn }; ({T, F }n , P({T, F }n ), ×i=1,2,...,n µi ) is a logical sample
space.
Then, by Theorem 11.10 [Logical sample space], we see the following:• Let P be a proposition which is constructed by P1 , ..., Pn . Note that P is regarded
as the map from {T, F }n → {T, F }. Then, we see thatqp

Prob[P ; {T }] = (×

i=1,2,...,nµi )(P −1 ({T })))

Remark 12.6. Since the linguistic Copenhagen interpretation says that ”only one
qp
measurement is permitted”, we only take the measurement: ×i=1,...,n MA (Oi =qpqp({T, F }, 2{T,F } , Gi ), S[ρ] ) = MA (×i=1,...,n Oi = ({T, F }n , P({T, F }n ), ×i=1,...,n Gi ),S[ρ] ). Therefore, the measurements (i.e., MA (π ∧ (O1 ×qp O2 ), S[ρ] ) (= MA (O1 , S[ρ] ) ∧MA (O2 , S[ρ] )) etc. in Deﬁnition 12.4) are not actually done. To be precise, theseqpmeasurements are included in the quasi-product measurement ×i=1,...,n MA (Oi =({T, F }, 2{T,F } , Gi ), S[ρ] ).
Example 12.7. [The QL version of Table 11.10]. Replacing P1 and P2 with MA (O1 , S[ρ] )365
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12.1 QL-logic (i.e., Fuzzy logic in QL); My scientiﬁc understanding of TLP
and MA (O2 , S[ρ] ), we get the following Table 12.1, i.e. the QL version of Table 11.10.
Table 12.1: Probabilistic Truth Table ( Elementary propositions MA (O1 , S[ρ] )(= P1 ), MA (O2 , S[ρ] )(= P2 ))probability:
MA (O1 , S[ρ] )MA (O2 , S[ρ] )(= P1 )(= P2 )

p=

×
×
p =×
p =×

×

qp

T

T

T

F

p12̄ =

p12 =

F

T

1̄2

F

F

1̄2̄

qp

i=1,2 µi

i=1,2 µi ({(T, T )})qp
i=1,2 µi ({(T, F )})qp
i=1,2 µi ({(F, T )})qp
i=1,2 µi ({(F, F )})[P1 → P2 ] ∧ P1

[P1 → P2 ] ∧ P1 → P2

T

T

F

T

F

T

F

T

Thus, we see that
Prob[[P1 → P2 ] ∧ P1 ; {T }] = p12
Similarly, we see the modus pones:
Prob[[[P1 → P2 ] ∧ P1 ] → P2 ; {T }] = 1
For example, put P1 :=”it rains”, P2 :=”the ground is wet”. Modus ponens says that[[”it rains” → ”the ground is wet”]∧ ”it rains”]→”the ground is wet”.

The following is the quantum linguistic version of Corollary 11.11.
Corollary 12.8. [Tautology in fuzzy logic] Let P (= MA (O, S[ρ] )) be a proposition constructed from elementary propositions P1 (= MA (O1 .S[ρ] )), P2 (= MA (O2 , S[ρ] )), ..., Pn (=MA (On , S[ρ] )). Then, the followings are equivalent:(i) P is a tautology in the sense of crisp logic.
(ii) Prob[P ; {T }] = 1.
That is, tautology always holds even in practical fuzzy logic. For example,(K) syllogism (i.e., [[P1 → P2 ] ∧ [P2 → P3 ]] → [P1 → P3 ] always holds.

Remark 12.9. We have two results such that(i) in ref. [79], I showed that syllogism does not always hold in quantum system (cf.
Sec.4.3.3).
(ii) in Corollary 12.8, I showed that syllogism always holds in classical and quantum
systems.
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Thus, readers may think that (i) and (ii) are contradictory. However these are not contradictory since Corollary 12.8 requires that O1 , O2 , O3 commute. On the other hand, inref. [79], the commutativity of O1 and O3 is not required. The most important one, ofcourse, is Corollary 12.8.
Remark 12.10. [ Cogito proposition is not a proposition in QL; cf. Sec. 8.2, also, seerefs. [46], [70]]: Examine the cogito proposition ”I think, therefore I am”, in which it isnatural to consider that”observer” = ”I” = ”measurement object (=system)”This is against the linguistic Copenhagen interpretation (E1 ) ”No observer can measureitself” in Sec. 1.2.1. Therefore, the cogito proposition is not a proposition in QL. The
fact that the ﬁrst proposition of philosophy is not a proposition is interesting.

Thus we see the following non-sense table:”I think”(= P1′ )?
?
?
?

Table 12.1′ : Probabilistic Truth Table( non-propositions ”I think”(= P1′ ), ”I am”(= P2′ ))probability:”I am”
qp′[P1′ → P2′ ] ∧ P1′ [P1′ → P2′ ] ∧ P1′ → P2′(= P2 )
p = ×i=1,2 µi?
p12 =?
?
?
?
p12̄ =?
?
?
?
p1̄2 =?
?
?
?
p1̄2̄ =?
?
?

Remark 12.11. The linguistic Copenhagen interpretation (E0 ) in Sec.1.2.1 says that(♯1 ) Don’t talk about what you can’t measure!
Thus, as shown in ref. [65], we see:367

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Also, the (♯1 ) is equivalent to Berkely’s saying(♯2 ) To be is to be perceived:

This implies the following ﬁgure:368

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12.1.7 General caseIn previous section, we devote ourselves to {MA (Oi = ({T, F }, 2{T,F } , Gi ), S[ρ] ) : i =1, 2, ..., n} under the conditions (N1 )∼ (N3 ), which are not so wide as mentioned in Remark 12.2 (ii) and (iii), In this section, we consider the general case: {Pi = MAi (Oi =({T, F }, 2{T,F } , Gi ), S[ρi ] ) : i = 1, 2, ..., n}. Put
b i = ({T, F }, 2{T,F } , Gb i ), S[⊗n ρ ] )Pbi = MNnj=1 Aj (O
j=1 j
where
b i ](Ξi ) = (
[G

i−1
O

I) ⊗ Gi (Ξi ) ⊗ (

j=1

n
O

I) (∀Ξi ∈ P({T, F }), i = 1, 2, ..., n)j=i+1

b i = ({T, F }, 2{T,F } , Gb i ), S[⊗n ρ ] ) : i = 1, 2, ..., n}Here, note that {Pbi = MNnj=1 Aj (Oj=1 jsatisﬁes thatb = Nn Ai is ﬁxed,(L1 ) A
j=1
b 1, Ob 2 , ..., Ob n commute,(L2 ) On(L3 ) a state ⊗j=1 ρi is only one.
Therefore, the general case: {Pi = MAi (Oi = ({T, F }, 2{T,F } , Gi ), S[ρi ] ) : i = 1, 2, ..., n}can be understood.
Remark 12.12. As a special case of measurement, (QL) logic and statistics are derived
as follows.
measurement
MA (O = (X, P(X), G), S[ρ] )binary valued logicMA (O = ({T, F }, P({T, F }), G), S[ρ] )−−−−−−−−→
derivation distribution with a parameter ω(R, B(R), [G(·)](ω))ω∈R· · · QL logic cf. this section· · · statistics cf. ref.[74]Thus, (QL) logic can be seen as a two-valued logic and statistics as a multi-valued logic.
Thus, the following motto has a point.
• The broad arguments of the humanities are discussed in ”logic,”• the quantitative arguments of the sciences are discussed in ”statistics,”
The actual segregation is shown in the following ﬁgure

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Problem 12.13. [Any sweet tomato is red]Consider (T F )-valued observables OSW = ({T, F }, 2{T,F } , GSW ) and ORD =({T, F }, 2{T,F } , GRD ) in C(Ω), where OSW and ORD is respectively called the sweetobservable and the red observable. It is natural to consider that ”Any sweet tomato isred” is deﬁned by(M)

SW ⊆ RD370

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Chap. 12 Linguistic philosophy ( After TLP)
where SW = {ω ∈ Ω|GSW (ω) = 1} and RD = {ω ∈ Ω|GRD (ω) = 1}.

GRD

1
GSW

Ω

RD

Ω

SW

In order to examine (M) (i.e., to answer the problem: ”Is the (M) a proposition?”), it
suﬃces to check the following:
Prob[MC(Ω) (ORD , S[δω ] ); {T }] = 1(∀ω ∈ SW )Thus we have the following problem:(N) Problem: Show that [SW ⊆ RD ] is a QL proposition!
Now, here is the next division of cases.
 (I): SW is ﬁnite•(II): SW is inﬁnite

hiCase (I) SW is ﬁnite
Assume that SW = {ω1 , ω2 , ..., ωn }. Recall the linguistic Copenhagen interpretation
Nn (L1 )such that only one measurement is permitted. Thus, consider the tensor space i=1 C(Ω)= C(Ωn ), and the tensor product measurement
MC(Ωn ) (nO

ORD = ({T, F }n , P{T, F }n ,

i=1

n
O

GRD ), S[δ(ω1 ,ω2 ,...,ωn ) ] )

i=1

where
(

n
O
i=1

GRD )({(x1 , x2 , ..., xn )}) =nO

GRD ({xi })(∀(x1 , x2 , ..., xn ) ∈ {T, F }n )i=1

Assume that the measured value x = (x1 , x2 , ..., xn ) belongs to {T }n . Then, we canconclude that (M) is true. Also, as an analogy of (ii) in Deﬁnition 12.4, we may consider
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as follows. Let π ∧ : {T, F }n → {T, F } be a map such thatT
if x1 = x2 = ... = xn = T∧
π (x1 , x2 , ..., xn ) =F (otherwise)
Then, (M) is equivalent toProb[MC(Ωn ) (π ∧ (⊗ni=1 ORD ), S[δ(ω1 ,ω2 ,...,ωn ) ] ); {T }] = 1
Thus,(M) isi a QL proposition.
hCase (II) SW is inﬁnite
This tis the same as the Problem of Hempel’s rven problem, which will be answered inSec. 12.7.

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12.2 Wittgenstein; the biggest star of analytic philosophy.
12.2.1 TLP attempted to answer the question, ”What is a proposition?”

Ludwig Wittgenstein (1889 - 1951), who was the student of B. Russel, was the most
famous philosopher (in analytic philosophy) in the 20th century.
Wittgenstein wrote two books, i.e., TLP and PI,
Tractatus Logico-Philosophicus (=TLP)(A)

the former Wittgenstein
1921

Philosophisch Investigations (=PI)
teacher−−−−−−→
gardenerthe later Wittgenstein
1953

The former is concerned with world description, while the latter is literary. Thus, I thinkthat the above is similar to the following:”Critique of Practical Reason (1788)””Critique of Judgment(1790)””Critique of Pure Reason (=CPR)(1781)”(B)

ﬁctional worldview (literary truth)
preface, introduction, (ﬁctional)premise, expedient−→ethics, morals, aesthetics
main subject

Thus we devote ourselves to TLP in this text.
It has already been more than 100 years since analytic philosophy was born. Moreover,today, many philosophers specialize in analytic philosophy. However, strangely enough,I don’t think that the evaluation of TLP has been settled yet. I think that TLP istheoretically insuﬃcient. In fact, in the last 100 years, no philosopher has been able toread TLP in a theoretical way*4 .
In the preface of his book ”Tractatus Logico-Philosophicus”, L. Wittgenstein said that(C) This book will perhaps only be understood by those who have themselves alreadythought the thoughts which are expressed in it — or similar thoughts.

*4 This is no mystery in philosophy. No philosopher has ever been able to read Descartes’ ”Discourse

on Method” or Kant’s ”Critique of Pure Reason” scientiﬁcally. We do not even know if they are
scientiﬁcally decipherable in the ﬁrst place. That is why I am writing this text!

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12.2 Wittgenstein; the biggest star of analytic philosophy.
Here, what is ”the thoughts which are expressed in it ”? I think that his interest is not”mathematical logic”, but ”non-mathematical logic (i.e., practical logic)”.

Thus I believe:(D) What Wittgenstein wanted to insist on in TLP is to answer the following questions:(D1 ) What is ”proposition”? (i.e., what is the deﬁnition of ”(practical)
proposition”?)(D2 ) Why does logic work in our usual world?

If so, we can consider (D) (= (D1 ) + (D2 )) directly, without reading the TLP. The
reason is that we have many powerful weapons (statistics, quantum mechanics, etc.)that Wittgenstein did not know about. It should be noted that
• The 20th century was the century of statistics and quantum mechanics.
though the 21th century may be the century of (quantum) computer.
♠Note 12.2. I believe that (D) is due to Wittgenstein. However, even if it were not, it iscertain that (D1 ) and (D2 ) are the most important in philosophy.

If (C) above is true, I would be in a very favorable situation. Because I already know
the answer to this problem.
(E) Some twenty years ago, in the papers on the subject ”Why does logic work inthis world?” (cf. refs. [41, 42, 43, 44]). To be precise, I proved that• both logic and statistics work in this world because quantum language
works in this world.

*4 It was a reﬂective paper in the sense that it did not get much response. I should have written it interms that people who were passionate about ”fuzzy” at the time could understand. I also did not
realize that this issue is closely related to philosophy.

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Chap. 12 Linguistic philosophy ( After TLP)At the time, I was not aware that ”Why logic does work in our world?” was a questionof philosophy. After reading (C) in TLP, I immediately rewrote the following papers.
(F1 ) Ref. [79]: Ishikawa, S., (2020) Wittgenstein’s picture theory in the quantum
mechanical worldview, Journal of quantum information science, Vol. 10, No.4, 104125, DOI:10.4236/jqis.2020.104007(https://www.scirp.org/journal/paperabs.aspx?paperid=106233)(F2 ) Ref. [81]: Ishikawa, S., (2021) Fuzzy Logic in the Quantum Mechanical Worldview; Related to Zadeh, Wittgenstein, Moore, Saussure, Quine, Lewis Carroll, etc.
JAMP, Vol. 9, No.3, 140-154,(https://www.scirp.org/journal/paperinformation.aspx?paperid=110830)
This is essentially the same as Sec. 12.1 in this text.
(F3 ) Ref. [85]S. Ishikawa, K. Kikuchi, (2021) Quantum Fuzzy Logic and Time,
JAMP, Vo.9 No.11 2021, 2609-2622 (https://www.scirp.org/journal/
paperinformation.aspx?paperid=112972)

Thus, I can assert the following:375

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12.2 Wittgenstein; the biggest star of analytic philosophy.
Answers to the above (G)(=(D))(G1 ) (=(D1 )) What is ”proposition”? (i.e., what is the deﬁnition of”proposition”?)(G2 ) (=(D2 )) Why does logic work in our usual world?

Answer 12.14. The above problems (G) are already solved in the previous section.
That is;(♯1 ) Problem (G1 ) is already answered in Deﬁnition 12.3.
That is, ”QL proposition” = ”measurement (=Axiom 1)”.

(cf. Deﬁnition 12.3, Note 12.1).
Also,(♯2 ) Problem (G2 ) is solved in Theorem 12.5.
Summing up, we say•”To speak what we can speak about” = ”To speak QL”♠Note 12.3. There may be a certain number of people in the world who have a special
personality that feels status in reading diﬃcult books that are impossible to understand.
I would like to believe that Wittgenstein did not write TLP for those people. I like thefollowing Einstein quote;• ”If you can’t explain it to a six year old, you don’t understand it yourself”

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Therefore, I thought Wittgenstein himself did not understand TLP at all. Reading his
conﬁdent explanation, I thought he was convinced that he understood TLP. For example,
Wittgenstein conﬁdently concluded(H1 ) ”What we cannot speak about we must pass over in silence”
in TLP. If so,• The rules for drawing the line between ”what can be said” and ”what cannot besaid” (i.e., the deﬁnition of ”proposition”) should be the main theme of TLP.
However, the deﬁnition is nowhere to be found, and without the deﬁnition, anyone cansay (H1 ), etc., which is the worst. However, I was relieved when I read the following (§38
in [126])(H2 ) ”The basic of Russell’s logic, as also of mine in the TLP, is that what a propositionis illustrated by a few commonplace examples, and then pre-supposed as understoodin full generality”
When I read this, I knew I could trust Wittgenstein. So, the following §6.54 in TLP meansthat Wittgenstein knew that TLP was like a ladder that could be thrown down.
(H3 ) §6.54: ”He must so to speak throw away the ladder, after he has climbed up on it”
Also, despite the hot love call from logical positivism, Wittgenstein set himself apart fromlogical positivism because he had an intuition that TPL was incomplete and powerless toscience.
It does not help that his TLP is incomplete. No matter how much of a genius Wittgenstein was, it would have been impossible for him to discover ”Fuzzy logic in QL”. The
environment in 1920 was as follows.
(i) Born’s discovery “the probabilistic interpretation of quantum mechanics” in ref. [6](1926)(ii) Tractatus Logico-Philosophicus by L. Wittgenstein in ref. [125] (1921)(iii) Fisher’s great book “Statistical Methods for Research Workers” in ref. [22] (1925)
The integration of these three is QL, and I don’t think any genius at the time could havereached QL. I think that von Neumann’s [121] and Davies’ [14] are indispensable for QL.
If von Neumann had taken an interest in TLP, though, the history of analytic philosophy
might have been diﬀerent.
However, the most important task of the philosopher isto ”create a larger stream of philosophy”.
In this sense, it is natural that he is in the position shown in the diagram below.

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Even Descartes only raised the issue (i.e., Cartesian problem). Wittgenstein also raised
important issues ((G1 ) and (G2 )), so it is only natural that he should be depicted in thisﬁgure. Only quantum language has the power to solve the various unsolved problems ofdualistic idealism.

♠Note 12.4. (i):Even if TLP is a poem whose logic is broken, if it raises issues (D 1 ) and(D2 ), TLP is still a top-notch philosophy book. In philosophy, raising a question may be
more important than answering. In fact, TLP has changed the mainstream of philosophy.
I believe that the correctness of the direction pointed by TLP has been proven by this
text.
(ii): As mentioned in the previous section ( Sec. 12.1), I believe that QL is the central
theory of analytic philosophy, which is also obvious if you look at List (D 1 ) [Wittgenstein’s
dream came true] in Preface. Of course, Kripke’s possible worlds semantics is one ofgreatest theories in analytic philosophy. However, I have the same opinion ( for Kripke’s
possible worlds semantics ) as Dummett as follows(♯) I have been struck by the enormous inﬂuence of fashion in philosophy: possible-worlds
semantics is an excellent example. Such a fashion seizes almost everyone at a particular moment and they all go haring oﬀ after it. I don’t think that the vague forpossible-worlds semantics was just a mistake. It is occurred because Kripke succeededin using that apparatus to make some strong points that struck everybody forcibly;
most then got themselves into a state of mind in which they could hardly think except
in those terms. ( 188-189 pages; M. Dummett. Origins of Analytical philosophy)

I think Dummett’s assessment is harsh because he expected analytic philosophy to be
more than statistics and quantum mechanics.

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12.2.2 The power of Wittgenstein’s word: Linguistic turn
However, in this text, I want to assert that Wittgenstein is one of the greatest philosophers(Plato, Descartes, Kant, etc.). That is because he said the following sayings (I 1 ) - (I3 ):(I1 ) ”The limits of my language mean the limits of my world.”(however, note that Wittgenstein did not propose ”my language”).

(I2 ) ”What we cannot speak about we must pass over in silence”(however, note that Wittgenstein did not deﬁne ”what we cannot speak about (=proposition)”).
(I3 ) ”Language-game”(however, his ”language game” was about everyday language ( as mentioned in(I1 ), he did not propose his language. The rule of the language game of quantum
language is the Copenhagen Interpretation.)).
The above is just the spirit of quantum language. Instead of my poor explanation of thespirit of quantum language, I prefer to saying(J) ”The spirit of quantum language is represented by the above (I1 ) - (I3 )”Seeing the above (I1 )∼ (I3 ), I can understand ”Why did Russell support Wittgenstein
as his guardian?”. Russell must have thought ”Without Wittgenstein, we (i.e., Frege
and Russell) cannot spread analytic philosophy alone.” That is, I think Russell expected
Wittgenstein to be an enlightener of the philosophy of language.
♠Note 12.5. For each great discovery, an anecdote (or, a catchphrase, stage eﬀect) is left
as follows.
(♯1 ) Archimedes· · · · · · golden crown, heureka! (cf. Sec.5.4)(♯2 ) Galileo· · · · · · Leaning Tower of Pisa, ”And Yet It Moves” (cf. Sec.7.3.4)(♯3 ) Newton· · · · · · :Newton’s apple, ”Geocentrism vs. Heliocentrism”(cf. Note 7.8)(♯4 ) Descartes· · · · · · :ﬂy on the ceiling (cf. Note 8.2), : I think, therefore I am, (cf. Sec.8.2)(♯5 ) Kant· · · · · · clock (cf. Note 10.2), dogmatic slumber (cf. Note 10.5)(♯6 ) Wittgenstein· · · · · · Son of a steel magnate, Hitler’s schoolmate, Primary school
teacher, Gardener, Architect, Guardian: Russell (cf. Sec.12.1.1)(♯7 ) Einstein· · · · · · Elevator(♯8 ) quantum mechanics· · · · · · Heisenberg’s uncertainty principle (cf. ref. [38]).

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I have the following opinion about (♯8 ):(♭) There is a big gap between Heisenberg’s uncertainty relation (refs. [28, 38]) and Kennard=Robertson uncertainty relation (refs [90, 111]). Therefore, just as the ”Newton’s apple” was used as a symbol for Newton mechanics, ”Heisenberg’s uncertainty
relation” was used as a symbol for quantum mechanics (cf. Note 4.1 of ref. [74]).

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12.2.3 Philosophical Investigations (1953) and Wittgenstein’s paradox
12.2.3.1 the later Wittgenstein
Wittgenstein wrote two books, i.e.
Tractatus Logico-Philosophicus (=TLP)(K)

the former Wittgenstein
1921

Philosophisch Investigations (=PI)
teacher−−−−−−→
gardenerthe later Wittgenstein
1953

Despite concluding in TLP that ”What we cannot speak about we must pass over insilence,” Wittgenstein spoke of ”what cannot be spoken about” in PI. This is not to saythat he had a ”change of heart,” since science is not the only target of philosophy.
The outline of the PI (the part 1) is as follows.
(i) Sec.1-88; language game(ii) Sec.89-133; logic and philosophy(iii) Sec.134-242; Wittgenstein’s paradox(iv) Sec.242-315; private language(v) Sec.315-693; psychology
Although PI (the part 2) is famous for rabbit-duck illusion, we are not concerned withpsychology and brain science. Psychology (or, cognitive science) is an important discipline,
but it is a discipline in which experimentation and observation are essential, and it is notreally Wittgenstein’s cup of tea. I have an opinion that the ”philosophy of mathematics”
should be left to mathematicians. That is, I believe that philosophers should concentrateon ”world description (= world view)” in the ﬁrst sense.

All of them (i.e., parts 1 and 2) may be of interest to Wittgenstein fans. From a QL pointof view, I think the above (i), (iii), and (iv) are important. (i) has already been mentioned
in Section 11.8.2.
12.2.3.2 (iii): Wittgenstein’s paradox;
I tried several times to understand this paradox (= Wittgenstein’s paradox), but I could
not understand it interestingly. About Wittgenstein’s paradox, S. Kripke says in ref. [94]S. Kripke says the following about Wittgenstein’s paradox(L) ( Wittgenstein has invented a new form of skepticism. Personally I am inclined toregard it as the most radical and original skeptical problem that philosophy has seen
to date, one that only a highly unusual cast of mind could have produced. )381

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12.2 Wittgenstein; the biggest star of analytic philosophy.
If Wittgenstein’s paradox is something that Kripke admires in this way, I would certainly
like to understand it, and I tried several more times to understand it, but could not.
However, the following (M), which may or may not be related to Wittgenstein’s paradox,
but which has been mentioned many times in this text, is interesting.
(L): Wittgenstein’s paradox (I like to think so though I could not understand his intentions)
I guess to think that Wittgenstein wanted to say the following in his paradox:(L1 ): [Worldviewism] we must start from a worldview(L2 ): however, no worldview can be accurately described.

For skepticism, the most important question is: ”What should we be skeptical about?” I
think. The object of skepticism is neither ”chess game”, ”quus calculation” nor ”private
language”. That is, we must skeptical about QL (particularly, the linguistic Copenhagen
interpretation), i.e.
(= dualistic idealism)

Quantum language = measurement + causality(= measurement theory)(Axiom 1)(Axiom 2)+ linguistic (Copenhagen) interpretation(how to use Axioms 1 and 2)

since Axioms 1 and 2 are kinds of spells (i.e., incantation, magic words, metaphysicalstatements) as mentioned in Sec. 1.2.1. Also, recall the following sprit of ”Copenhagen
interpretation” due to Mermin (cf. Note 1.6):(♯1 ) If I were forced to sum up in one sentence what the Copenhagen interpretation says
to me, it would be “Shut up and calculate!”
It is true that many researchers have won Nobel prizes in this spirit. However, I think
it is time to become skeptical about QL. In other words, I think it is time to seriously
pursue the next as an unsolved problem.
(♯2 ) What is ”Copenhagen interpretation”?
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Thus I believe that the following problems about language game are important:[Question 1 about language game]:(♭1 ) Why is ”interpretation” needed in QL (or, quantum language)?
Why does Newtonian mechanics need no ”interpretation”?
Or, really, Newtonian mechanics needs no ”interpretation”?
(monistic realism)
Newtonian mechanics = Newton’s Three Laws of Motion + No interpretation[My answer]: I think that Newtonian mechanics also has an ”interpretation”. However, I think that Newtonian mechanics does not bother to emphasize ”interpretation”
because Newtonian mechanics does not have the ”observable”. One of the arguments ofthis text was that the diﬃculty in understanding the concept of ”observable (≈Idea)” hasperpetuated the history of Western philosophy for 2,500 years, as shown below.
(History of progress from Ideas to observables)

Idea

linguistic version−−−−−−−−−−−→

signiﬁer, [signiﬁed]

Plato’s idea theoryascholasticyversion ⃝Saussure’s linguistic theory[(primary) universal][membership function]

Anselmus, Thomas

Zadeh’s fuzzy theoryquantiﬁcationycognitiveyversion ⃝b[secondary quality]yidentiﬁcation

linguistic version ⃝c−−−−−−−−−−−−→

John Locke[observable]
quantum language////[Question 2 about language game]:(♭2 ) If the Copenhagen Interpretation is a ”manual for the successful use of theaxioms,” isn’t it impossible to create a complete manual, no matter how
detailed the rules?
[My answer]: I believe that there is no perfect interpretation. No matter how detailed
the ”rules” are, you cannot make a perfect manual. In the end, we have to use common
sense to ﬁll in the gaps in the manual. Then, a new problem arises: What is ’common
sense’ ? This is a reminder of ”Hume’s habitual thinking.
////
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12.2 Wittgenstein; the biggest star of analytic philosophy.
[Question 2 about language game]:(♭3 ) How are the rules of QL learned?

[My answer]: Note that QL is a kind of idealism (or, a kind of metaphysics ). I
myself made quantum language as an analogy after learning quantum mechanics. Therefore, I initially thought that it would be diﬃcult to learn quantum language until after
I had mastered quantum mechanics. However, in my graduate laboratory, students withno knowledge of quantum mechanics have easily learned quantum languages. This fact
surprised me, and I still wonder about it. Just as a baby gradually learns an everyday
language, a quantum language can be gradually learned by solving exercises. It is possible
to gradually expand the range of objects that can be expressed in a quantum language. I
can’t help but marvel at the human capacity for language.
////♠Note 12.6. Mathematics requires only axioms and does not require interpretation. Therefore, AI (= artiﬁcial intelligence) that understands mathematics better than humans will
be a reality in 20 to 30 years. However, I think the emergence of AI that understands QL
will be delayed a bit longer. Of course, with a quantum computer, the diﬀerence between
math and QL may be trivial.

12.2.3.3 (iv): Private language
As mentioned in Sec. 4.3.1, if the ecology of various animals is observed, it will be clearthat the base of language was due to intimidation, solidarity, reproduction. Language wasone of the strongest arms for the survival and breeding. Such a time have continued formillions of years. Of course, the biggest events in the ”history of language” happened one
after the other. For example,(N) ”rhythm and song”, ”logical structure”, ”quantity concept”, ”grammar”, ”tense”,
etc.
Among these, ”logical structure” and ”quantity concepts” are relevant to the language ofstating science. However, this is a characteristic of only a small portion of language as awhole.
Now, a private language is a language that expresses inner experiences such as senses,
emotions, wills, and thoughts. . I feel like it has more aﬃnity with AI than philosophy,
but I don’t know.

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12.3 Quine’s analytic-synthetic distinction in QL

12.3

Quine’s analytic-synthetic distinction in QL

From a scientiﬁc point of view, the late Wittgenstein seems to have been a disappointment.
However, It is not that analytic philosophy since TPL has been sterile. From this section
onward, we describe the fruitful results within the analytic philosophy.
This section consists of excerpts from the following paper:• Ref. [81]: Ishikawa, S., (2021) Fuzzy Logic in the Quantum Mechanical Worldview; Related to Zadeh, Wittgenstein, Moore, Saussure, Quine, Lewis Carroll, etc.
JAMP, Vol. 9, No.3, 140-154,(https://www.scirp.org/journal/paperinformation.aspx?paperid=110830)

12.3.1 Analytic and Synthetic Propositions in QL
Rudolf Carnap (1891-1971) was a philosopher active in Europe before 1935 and in
the United States after that. He was a central member of the Vienna School of logicalpositivism. The claim of logical positivism is diﬃcult to understand, but in the author’s
style, it is as follows.
(A) Taking Wittgenstein’s ”Treatise on the Philosophy of Logic” as a starting point,
the aim was to develop it. Thus, they aimed to create ”my language” in ”Thelimits of my language mean the limits of my world”.
If so, their goal is the same as the author’s (i.e., quantum language), but logical positivism
was exceptionally strong in its commitment to mathematics and logic. The diﬀerence may
just be that I know that the last 100 years have been the era of quantum mechanics (andnot mathematical logic).
Quine (1908-2000) was an American-born philosopher who taught at Harvard University. He was originally a follower of R. Carnap’s logical positivism. Later, however, hebecame the sharpest critic of Carnap’s philosophy (and logical positivism). Thus, heformed his own philosophy.
The distinction between ”analytic propositions” and ”synthetic propositions” begins
with Kant. Here,(B) The classiﬁcation of ”a thing is true or not true” is of interest to any scientist.
Moreover, the meaning of ” true” diﬀers between analytic propositions and synthetic386

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Chap. 12 Linguistic philosophy ( After TLP)
propositions. This classiﬁcation is considered one of the core elements of logicalpositivism.
”All bachelors are not married.” ”Horses are animals” ”1+3=5”(C) A ” synthetic proposition” is a proposition whose truth or falsity depends on what
the facts about the world are. ”The cat is on the mat.” ”Horses are fast”, ”Jack isa bachelor”
The classiﬁcation of ”a thing is true or not true” is of interest to any scientist. Moreover,the meaning of ” true” diﬀers between analytic propositions and synthetic propositions.
This classiﬁcation is considered one of the core elements of logical positivism.

Now, in his 1951 article, ”Two Dogmas of Empiricism” ref. [109], Quine rejects thedistinction between analytic and synthetic propositions and he asserted(D) Every proposition is synthetic.
When I learned of Quine’s claim (D) some years ago, I remember thinking, ”That’s a very
bold claim,” and was surprised. However, I did not quite understand the basis on which
Quine made this claim (D).
12.3.1.1 My answer
Many philosophers have entered into this issue, but they are all too obscure for the author
to be able to give an adequate explanation. As I have said many times, I believe that:(E1 ) If we don’t discuss under a worldview, we are just chatting.
(E1 ) If there is no deﬁnition, there is no discussion.
That is, recalling the following:(F) ”QL-proposition”=”(TF)-measurement (= experimental veriﬁcation)”(Or, more generally, ”proposition in QL” = ”Axiom 1”)(cf. Deﬁnition 12.3, Note 12.1).
we can naturally conclude that all propositions in QL are synthetic .
Of course, only this deﬁnition (F) is important. ”Synthesis or analysis?” is not so important since someone may consider (F) analytic.

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12.3 Quine’s analytic-synthetic distinction in QL♠Note 12.7. What I mean by (D) is that the worldview and the ”deﬁnition of propositions”
under it are of paramount importance. Without deﬁning propositions, it is meaningless to
discuss ”whether or not analytic propositions can be distinguished from synthetic propositions”.

Of course, diﬀerent worldviews (=theoretical systems) have diﬀerent ways of thinking,and my proposal is not the only one. For example, in the theoretical system of mathematics, it is obvious that all propositions are analytic. I hope that many readers will
oﬀer various suggestions. I think that philosophers of analytic philosophy should discussproblems after presenting their worldview. Logic without a worldview is, I think, merelya branch of mathematical logic.
The ”Carnap-Quine Controversy” is a dispute over the distinction between analytic andsynthetic propositions. Quine argued that all propositions are synthetic.

Since in quantum language ”every proposition is a total proposition,” QL agrees with
Quine. The spirit of the Copenhagen Interpretation is ” without measurement we cannot
know if it is true or false”.

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12.4 Popper’s falsiﬁability in the Copenhagen interpretation
From the quantum linguistic point of view, as asserted in Assertion 1.5, we think thatto do science =to describe by QL=to study in the quantum mechanical worldview
However, when it comes to ”what is science?”, I can’t help but mention Popper.
let me mention a little about Karl Popper (1902-1994).

Thus

12.4.1 Popper’s falsiﬁability

This may be common to many philosophers of science, if not Popper, but their work is”enlightening” and it is diﬃcult for us in science to understand its true meaning. Popper’s
books on probability theory and quantum mechanics are also a little far from the interestsof science, so I do not fully understand them.
Popper is famous for ”falsiﬁability” as follows. Falsiﬁability is the following view ofscience (cf. ref. [105]).
(A):Popper’s falsiﬁability(A) In order to guarantee the objectivity of a scientiﬁc theory, there must be apossibility that the hypothesis will be disproved by experiment or observation. In other words, truth must always be subjected to experiments thatnegate it. And if the denying experiment is conﬁrmed, the truth must be
denied.
Many scientists would take this claim for granted. For example,(B1 ) it is a common belief that mammoths are extinct, but if someone discovers a mammoth near the North Pole, this belief will be debunked.
This is a matter of course. Moreover, this is not particularly interesting.
How about the following? ((B2 ) ”I think, therefore I am” is against the falsiﬁability because there is no experiment
to deny it. Therefore, it is not a truth.
There may be some philosophy buﬀs who ﬁnd this interesting. Popper’s falsiﬁability may
not be a statement for scientists, but for philosophy enthusiasts.
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12.4 Popper’s falsiﬁability in the Copenhagen interpretation

12.4.2 Falsiﬁability for worldviews
It should be obvious that
(C1 ) Newtonian mechanics has falsiﬁability. The power of this is remarkable such as
Therefore
Newtonian mechanics −−−−−−−→ heliocentrism
Now, as I have pointed out many times, many good philosophers have followed Plato (or,Euclid) in adopting the following worldview principle.
(C2 )introduction part (unquestionable truth)z}|{DescartesI think, therefore I am 







Kant
thing-in-itself
Analytic philosophy

mathematical logic












commonTherefore−−−−−−−→ ethics, moral

In the above, all of the statements on the left (introduction part) are contrary to falsiﬁcationism. In other words, philosophers tried to start from ”unquestionable truths” forfear of being pointed out in error. In fact, no scientiﬁcally productive philosophy was everborn.
If so we want to say(D) The biggest risk is not taking any risk. Take a risk as Newton!
But, it is not so.

Now let us consider quantum language. For example, let us restate Axiom 1 (in Sec.
1.4.2).
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Chap. 12 Linguistic philosophy ( After TLP)
Axiom 1 [State, Measurement] in the C ∗ -algebraic formulation•[General (i.e., quantum and classical ) case]:(i) [State]: With any system S, a C ∗ -algebra A(⊆ B(H)) can be associated. A
state of the system S is represented by an element ρ(∈ Sp (A∗ )).
(ii) [ Measurement ]: The probability that a measured value x (∈ X) obtainedby the measurement MA (O =(X, P(X), G), S[ρ] ) is given by ρ(G({x}))(≡A∗ hρ, G({x})iA ).

Again and again, as I have said, Axiom 1 is a nonsensical incantation. This is not thelevel of satisfying or not satisfying falsiﬁability. Since it is unintelligible, we can only saythat it does not satisfy it.
Therefore,(E1 ) Newtonian mechanics satisﬁes falsiﬁability. However Quantum language
does not satisfy falsiﬁability.
In other words, quantum language has more power than statistics, yet does not satisfy
the same falsiﬁability as statistics.
Summing up the above (i.e., (C1 ), (C2 ), and (E1 ), we have(E2 ) Falsiﬁability is the rule that draws the line between realism and idealism
This is not a rule for drawing the line between science and pseudoscience.)
This may be a noble claim, but it is not a particularly gratifying claim for scientists.

12.4.3 Falsiﬁability in the Copenhagen interpretation
In conclusion, we would like to argue the following(F) Falsiﬁablity (A) should be regarded as one of the Copenhagen interpretations! (cf.
(E0 ) in Sec. 1.1.2.2)
Because,Falsiﬁablity (A) ≈ Reinforcement of ”To be is to be perceived”

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12.4 Popper’s falsiﬁability in the Copenhagen interpretation

Let us explain it as follows.
(G) The conclusion of Wittgenstein’s TLP was ”what we cannot speak about we must
pass over in silence”. In order to do so, we need a rule to distinguish between”what we cannot speak about (=pseudo-science)” and ”what we can speak about(=science). Otherwise, the argument is meaningless. It is with this in mind that
Popper formulated falsiﬁability.
Recall that QL has the following form.
[Axiom 2](=quantum language)[Axiom 1]
measurement theory = measurement + causal relation+

linguistic Copenhagen interpretation[the manual to use Axioms 1 and 2](*)

Above, Axiom 1 and Axiom 2 alone lead to a messed up conclusion. For example,(H1 ) [Schrödinger’s cat]: There is a half-dead cat that we don’t know whether it is alive
or dead, and the moment we see it, it is determined whether the cat is alive or
dead*5 .

(H2 ) An incomprehensible non-proposition such as ”I think, therefore I am” is regarded
as ”ﬁrst proposition of philosophy.”*5 This was solved in ref. [65]392

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It is the ”Copenhagen interpretation” that contains such pseudoscientiﬁc propositions
and keeps them out of the open. Therefore, we have two deﬁnitions of ”Copenhagen
interpretation” such as(I1 ) Copenhagen Interpretation is a manual on how to use Axioms 1 and 2(I2 ) Copenhagen interpretation draws a line between scientiﬁc and pseudoscientiﬁcpropositions.
That is, we may consider as follows.
(J) (=(F)) Falsiﬁablity (A) should be regarded as one of the Copenhagen interpretations! (cf. (E0 ) in Sec. 1.1.2.2)
In this light, no scientist would feel uncomfortable with ”falsiﬁability.”♠Note 12.8. This is a story about a philosophical study group led by Wittgenstein. Popper
was also a participant in that seminar. For Wittgenstein, there were no philosophicalproblems, only trivial puzzles. All the while, Wittgenstein, seated by the ﬁreplace, had,says Popper, ”been nervously playing with the poker”. After an exchange of views on
ethics, Wittgenstein asked Popper to give an example of an ethical rule. Popper replied,”Not to threaten visiting lecturers with pokers.” Wittgenstein then threw away the poker
and stormed oﬀ.

This episode implies an adversarial relationship between Wittgenstein and Popper. But
what Popper did was to do Wittgenstein’s homework (What is ”what we cannot speak
about”?) as follows.
• using falsiﬁability, he drew a line between scientiﬁc propositions (= what we can speak
about) and pseudoscientiﬁc propositions (= what we cannot speak about).
However, from the standpoint of quantum language, falsiﬁability is only one part of the
Copenhagen interpretation. Furthermore, a scientiﬁc proposition is a QL-proposition (Sec.
12.1).

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12.4 Popper’s falsiﬁability in the Copenhagen interpretation
If we consider falsiﬁability as one of the rules of Copenhagen interpretation, Popper’s
place is determined as follows.

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Chap. 12 Linguistic philosophy ( After TLP)

12.5

Lewis Carroll’s logical paradox

This section is quoted from reference [81], i.e.
• Ishikawa, S., (2021) Fuzzy Logic in the Quantum Mechanical Worldview ; Related
to Zadeh, Wittgenstein, Moore, Saussure, Quine, Lewis Carroll, etc. Journal of
applied mathematics and physics, Vol. 9, No.7, 1583-1610,(https://www.scirp.org/journal/paperinformation.aspx?paperid=110830)
Lewis Carroll (1832–1898) was an English writer of children’s ﬁction, notably ”Alice’s
Adventures in Wonderland” and its sequel ”Through the Looking-Glass.” He was also amathematician, photographer, inventor, and Anglican deacon.

In ref. [10] ”What the Tortoise said to Achilles” (1895), Lewis Carroll raised the following
question.
• Is logic logical?
According to [10], let us explain it as follows.
Achilles says: ”Can you understand the following modus ponens?”a (P1 → P2 )Premise ○:b P1Premise ○:z P2Conclusion: ○a and ○.
b However, why do ○a and ○
b imply ○?”z
the Tortoise says: ”I can understand ○Achilles says: ”I see, then, the following OK?”a (P1 → P2 )Premise ○:b P1Premise ○:c ○a and ○b are ”true”, therefore ○z is ”true”
Premise ○:z
Conclusion: ○ P2a ○b and ○.
c However, why do ○,a ○b
the Tortoise says: ”I can understand Premises ○,c conclude ○?”zand ○Achilles says: ”I see, then, the following OK?”a (P1 → P2 )Premise ○:b P1Premise ○:c ○a and ○b are ”true”, therefore ○z is ”true”
Premise ○:395

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12.5 Lewis Carroll’s logical paradoxd ○,a ○b and ○
c are ”true”, therefore ○z is ”true”
Premise ○:z P2Conclusion: ○a ○,b ○c and ○.
d However, why do ○,athe Tortoise says: ”I can understand Premises ○,b ○c and ○d conclude ○?”z○,... (Inﬁnite regress)
This is Lewis Carroll’s Paradox
Several philosophers have tried to resolve Carroll’s paradox. For example, Bertrand
Russell discussed the paradox brieﬂy in ref. [115]. His opinion is as follows.
a ○b and ○z are propositions, on the other hand, ○c and ○d are(A1 ) The above ○,inference rules. And, therefore, both the Tortoise and Achilles are confusing this.
I agree to his opinion. I think that their confusion is due to the lack of a deﬁnition of
proposition.
♠Note 12.9. In this paradox, the following is of essential importance(♯) Is Lewis Carroll’s paradox a mathematical problem or not?
If it is a mathematical problem, it is not a paradox. Because ”proposition” and ”inferencerule” can never be confused since mathematics is formulated under axiomatic set theory.
(Lewis Carroll’s paradox was proposed in [10] (1895). The axiomatic set theory (e.g.,
Zermelo-Fraenkel set theory) was proposed was proposed in the early 20th century. Therefore, we can say that, in 1895, Lewis Carroll’s paradox was also a paradox within mathematics. )On the other hand, if Lewis Carroll’s paradox is not a mathematical problem, it is noteasy. That is because we have no deﬁnition of ”proposition” in general. However, as
indicated before, we already know the deﬁnition of ”proposition” within QL. Thus, in QL
we can argue as follows.

Let us prove that Lewis Carroll’s paradox is not a paradox in QL as follows. Note that
QL says that
(A2 ) ”proposition P ”=”(TF)-measurement MA (O, S[ρ] )”. And the essence of the modus
ponens is all described in the following Table 12.2 (= Table 12.1). Therefore, logic
is a simple calculation of elementary arithmetic.
Table 12.2: Probabilistic Truth Table ( Elementary propositions MA (O1 , S[ρ] )(= P1 ), MA (O2 , S[ρ] )(= P2 ))probability:
MA (O1 , S[ρ] )MA (O2 , S[ρ] )(= P1 )(= P2 )

T

T

T

F

F

T

F

F

×
p12̄ = ×p1̄2 = ×p1̄2̄ = ×
p12 =

×

qp
i=1,2 µiqp
i=1,2 µi ({(T, T )})qp
i=1,2 µi ({(T, F )})qp
i=1,2 µi ({(F, T )})qp
i=1,2 µi ({(F, F )})p=[P1 → P2 ] ∧ P1

[P1 → P2 ] ∧ P1 → P2

T

T

F

T

F

T

F

T

For example, put P1 :=”it rains”, P2 :=”the ground is wet”. Modus ponens says that396
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Chap. 12 Linguistic philosophy ( After TLP)[[”it rains” → ”the ground is wet”]∧ ”it rains”]→”the ground is wet”.

Maybe the Tortoise thought of logic as a noble discipline and tried to understand it
rigorously. However, all we have to do is only what is written in the table above. Notec and ○
d are not written in this table.
that the above ○As mentioned in Note 12.9, in the case of mathematics, the deﬁnition of ”proposition”
is so clear that I think it is rare for mathematicians to take Carroll’s logical paradox
seriously. When I asked the mathematicians around me, they all just said, ”What are you
saying stupid things?” or ”Isn’t that the same as saying, I don’t understand ’1+1=2’ ?”.
Since the deﬁnition of a proposition is not clear in the case of philosophy, we tend to
confuse propositions with inference rules. We know the deﬁnition of a proposition in QL
so clearly that we could avoid any confusion.
I don’t know Carroll’s own intentions, but I think the reason this paradox has been ofinterest to people for over 100 years is that it is closely related to the question ”what isa scientiﬁc proposition?”. Carroll’s logical paradox is not the childish problem that mostmathematicians think it is.

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12.6 Flagpole problem: What is a scientiﬁc explanation?

12.6 Flagpole problem: What is a scientiﬁc explanation?
This section was written with reference to the following.
• [78]:Ishikawa, S: Philosophy of science for scientists; The probabilistic interpretationof science Journal of quantum information science, Vol. 9, No.3, 140-154,
DOI: 10.4236/jqis.2019.93007(https://www.scirp.org/Journal/paperinformation.aspx?paperid=95447)
Remark 12.15. Recall that QL logic (= fuzzy logic) (in Sec. 12.1) is practical logic within
the quantum mechanical world view. As I have said many times in this text, philosopherslike ”logic” (or, the word ”logic”) too much. They always want to ”be logical”. I feel asfollows.
• Philosophers are more fond of ”logic” (or, the word ”logic”) than mathematicians
That is, I assert that• the scientiﬁc explanation should not be described in the logical worldview (=the logical spirit=the spirit of ”Think logically!”), but in the quantum mechanical worldview.
The ﬂagpole problem below is caused by philosophers’ excessive love of logic.

12.6.1 * The quantum linguistic solution of Flagpole problem
Carl Gustav Hempel (1905-1997) was a German philosopher. He was a major ﬁgure
in logical empiricism, a 20th-century movement in the philosophy of science. His studies
of induction, explanation, and rationality in science exerted a profound inﬂuence upon ayoung generation of philosophers of science. He is also known for the raven paradox (alsoknown as ”Hempel’s paradox”).
We think that most modern scientists know Newtonian mechanics, but not mathematical logic. Thus most scientists today would feel that(A) the basis of science is not ”logic” but a ”mechanical worldview”.
However, for early 20th century philosophers, however, (A) was not common knowledge.
It was Hempel who ﬁrst noticed the incompatibility of logic (or set theory) and science.
12.6.1.1 Flagpole problem in Hempel’s model (= the deductive-nomological model (DNmodel))
D-M explanation (deductive-nomological explanation) is one of the ”explanatory methodsof science” and has the following forms

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D-M explanation (deductive-nomological explanation)
Deﬁnition 12.16. [D-M explanation (deductive-nomological explanation)] D-M explanation is the scientiﬁc explanation such as[fol]1[fol]2···where[fol]k means ”fact” or ”law”[fol]n−1[fol]nC(Conclusion)

•

Sun
−− 1
ω10
Flag

α◦
ω20

|
2

Figure 12.2: [Flagpole problem ]Let us explain the ﬂagpole problem as follows. Suppose that the sun is at an elevation
angle α◦ in the sky. Assume that tan α◦ = 1/2. There is a ﬂagpole which is ω10 meterstall (0 ≤ ω10 ≤ 1. The ﬂagpole casts a shadow ω20 meters long. Suppose that we want
to explain the length of the ﬂagpole’s shadow. On Hempel’s model, we see the following
explanation.
(B1 ) 1. [fact]: The sun is at an elevation angle α◦ in the sky.
2. [law]: Light propagates linearly.
3. [fact]: The ﬂagpole is ω10 meters high.
Then,
4. [conclusion]: The length of the shadow is ω20 = ω10 / tan α◦ = 2ω10 meters
This is a good explanation of ”Why is that shadow 2ω10 meters long?”
Similarly, we may consider as follows.
(B2 ) 1. [fact]: The sun is at an elevation angle α◦ in the sky.
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12.6 Flagpole problem: What is a scientiﬁc explanation?
2. [law]: Light propagates linearly.
3. [fact]: The length of the shadow is ω20 metersThen,
4. [conclusion]: The ﬂagpole is ω10 (= (tan α◦ )ω20 = ω20 /2) meters tall.
This is somewhat disconcerting (since the cause and eﬀect are reversed) as an explanation
for the question ”Why is the ﬂagpole height ω20 /2 meters?”• Hempel did not miss this minute discomfort.
Thus we have the ﬂagpole problem as follows:
Problem 12.17. [Hempel’s ﬂagpole problem](B3 ) [ Flagpole problem]
Why do we feel that the solution (B2 ) is unnatural?
Is there a deﬁciency in the D-M description?
Therefore, in what follows, I will present the quantum linguistic explanation.
My opinion is as follows.
(C) the above explanations (B1 ) and (B2 ) rely on DN model (i.e., deductive-nomologicalmodel), which is due to a kind of the logical worldview (=the logical spirit=the spiritof ”Think logically!”) (i.e., the spirit that science should be written logically). And
thus, the most important concepts ”measurement” and ”causality” are not used in(B1 ) and (B2 ).

Therefore, in what follows, I will present the quantum linguistic explanation.
12.6.1.2 Flagpole problem by the quantum linguistic explanation
Consider two state spaces Ω1 = [0, 1] and Ω2 = [0, 2]. Here, assume that a state ω1 (∈ Ω1 )means the height of the ﬂagpole. And an state ω2 (∈ Ω2 ) means the lenght of the shadowof the ﬂagpole. The relation between ω1 and ω2 is represented by the following map:
Ω1 (= [0, 1]) −−→ Ω2 (= [0, 2])
ϕ1,2

For simplicity, the speed of light is assumed to be ∞. It would also be easier to use
precision measurements, but we will not use them. Since the sun is located at an elevation
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angle α◦ in the sky, the causal map ϕ1,2 : Ω1 → Ω2 is determined as follows
ω2 = ϕ1,2 (ω1 ) = ω1 / tan(α◦ ) = 2ω1 (∀ω1 ∈ Ω1 (= [0, 1]))Thus, the causal operator Φ1,2 : C(Ω2 ) → C(Ω1 ) is deﬁned by(Φ1,2 f2 )(ω1 ) = f2 (ϕ(ω1 )) (∀f2 ∈ C(Ω2 ), ω1 ∈ Ω1 (= [0, 1]))
Let N be suﬃciently large number. Put XN = {0(= N0 ), N1 , N2 , ..., 2NN }. The observable(2)(2)
ON = (XN , P(XN ), FN ) in C([0, 2]) is deﬁned by0(−∞ ≤ ω ≤ (k − 1)/N )((k − 1)/N ≤ ω ≤ k/N )Nω + 1(2)[FN ({ Nk })](ω2 ) =−Nω+
1(k/N ≤ ω ≤ (k + 1)/N )0((k + 1)/N ≤ ω < ∞)

1
(2)

k
})](ω2 )[FN ({ N(k = 0, 1, 2, ..., 2N ), ω2 ∈ Ω2 (= [0, 2])(k − 1)/N

k/N(k + 1)/N

Here, we consider it in terms of the Heisenberg picture. That is,
(2)

(2)

(1)

Φ1,2 ON = (XN , P(XN ), Φ1,2 FN ) = (XN , P(XN ), FN )
where
(1)

[FN ({

k
k
k
k
(2)
(2)
(2)
})](ω1 ) = [FN ({ })](ϕ1,2 (ω1 )) = [FN ({ })](2ω1 ) = [FN ({})](ω1 )N
NN
2N(∀k = 0, 1, ..., 2N, ∀ω1 ∈ Ω1 (= [0, 1]))
(1)

(1)

Thus, we get the measurement MC(Ω1 ) (Φ1,2 ON = (XN , P(XN ), FN ), S[ω10 ] ) with thestate ω10 (∈ Ω1 ).

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12.6 Flagpole problem: What is a scientiﬁc explanation?

(2)

(2)

k
FN ({ 2N})|
ω1 = 0k−1
2NFN ({ k+1
2N })

↓0

k
2N

ω1

k+1
2N

(2)

FN ({ k+2
2N })|
ω1 = 1

k+2
2N

Now, from Axiom 1 we can say the following.
(1)

(1)

(D1 ) [Measurement]; We take a measurement MC(Ω1 ) (Φ1,2 ON = (XN , P(XN ), FN ), S[ω10 ] ).
k
If it holds that 2N≤ ω10 ≤ k+1
2N , then we see thatk
the probability that a measured value 2N(∈ X) is obtained is given by



(2)
k
0

[FN ({ 2N })](ω1 )the probability that a measured value k+1
2N (∈ X) is obtained is given by

(2)
(2)
k+1
k
[FN ({ 2N })](ω10 ) (=1 − [FN ({ 2N})](ω10 ))This is the quantum linguistic representation of (B1 ).
Also,k
is obtained by a(D2 ) [Inference]; (cf. Sec. 1.4.4): Assume that the measued value 2N
(1)

(1)

measurement MC(Ω1 ) (Φ1,2 ON = (XN , P(XN ), FN ), S[∗] ). Then, Fisher’s maximal
likelihood method says that(2)max [FN ({ω1 ∈Ω1kkk(2)})](ω1 ) = 1 = [FN ({
})](
)
2N
2N
2N

k
and thus we can infer that unknown state [∗] is equal to 2N.

This is the quantum linguistic representation of (B2 ).
Therefore, we asset that(E) ”Scientiﬁc explanation” is an explanation in quantum lan-guage

Of course, since the claim of this text is that ”quantum language is the language ofscience,” it would be troubling if it were not (E).

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12.7 * Hempel’s raven paradox: A milestone in the philosophyof science
As we have repeatedly stated, Wittgenstein’s interest was to clarify the problem: ”whatis a scientiﬁc proposition?”. For example, the next are doubtful.
• ”I think, therefore I am”, ”thing-in-itself”, ”1 + 1 = 2”, etc.
But the most accomplished masterpiece of them all is Hempel’s raven paradox, i.e.

”All Ravens Are Black”.
That is, ”All Ravens Are Black” is not a scientiﬁc proposition.
This section was written with reference to the following.
• [78]:Ishikawa, S: Philosophy of science for scientists; The probabilistic interpretationof science Journal of quantum information science, Vol. 9, No.3, 140-154,
DOI: 10.4236/jqis.2019.93007(https://www.scirp.org/Journal/paperinformation.aspx?paperid=95447)”Hempel’s raven” is a paradox that arouses skepticism toward the ”set-theoretic worldview. I would like to pay tribute to Hempel’s research attitude for pointing out something(”ﬂagpole problem” or ”Hempel’s raven”) that is easy to miss if one does not know the”quantum mechanical worldview.” Philosophy of science is a broad ﬁeld, but I think thedirection that Hempel pointed out was the direction that philosophy of science should go.
He said that
• The propositions of mathematics are devoid of all factual content; they convey no
information whatever on any empirical subject matter.

I think this quote from Hempel suggests that he was exploring something like quantum
language.

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12.7 * Hempel’s raven paradox: A milestone in the philosophy of science

12.7.1 Is ”the set of all tyrannosaurus” meaningful? : the set theoreticalworldview
Let us explain ”the set theoretical worldview”, (which is a kind of spirit of ”Think logically!”, and thus should not be regarded as a worldiew). Logic and set theory are similar(i.e., ”logic + set” = ”set theory” =”mathematics”), and thus, it usually believed that
set theory as well as logic are considered reliable.
As mentioned Sec. 11.1, the following is the greatest history of mathematics (i.e., thebeginning of modern mathematics):
naive set theory
axiomatic set theory
Cantor−−−−−→
ZFC-Axiom−−−−−→ modern math.
(1891)(1921)

However, the diﬀerence between naive set theory and axiomatic set theory is negligible
for most mathematicians (except mathematicians specializing in foundations of mathematics). Therefore, for simplicity, let’s assume that ”set theory” = ”naive set theory” inthis section (cf. Note 1.1). Therefore, we assume that ”set” is deﬁned by ”a collection ofthings”. This is the same as the use of ”set” in everyday language.
General people (including philosophers) may think(A1 ) Set theory is very reliable because it is used to lay the foundation for mathematics.
Therefore, if we use sets to describe the concept of this world, we will not fall intoa mistake.
In this text, this is called the set theoretical worldview. However, I am skeptical of theset theoretical worldview (♯1 ). That is, I think that
(A2 ) It is true that set theory is a very reliable and solid discipline. However, we must
be cautious in using sets to describe the world.
For example,(B) Is ”the set of all tyrannosaurus” meaningful? Or, is ”the set of all raven” meaningful? If it is not meaningless, how do we represent ”All ravens are black”?
This (B) is Hempel’s raven problem.

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Remark 12.18. TLP begins with the following statement:(♯) The world is the totality of facts (i.e., the totality of states (= state space)), not ofthings (i.e., the set of things ).

Note that Hempel doubted the ”set of things” (i.e., ”the set of ravens”). I think that
Hempel was one of the philosophers who understood TLP best.

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12.7 * Hempel’s raven paradox: A milestone in the philosophy of science

12.7.2 Hempel’s raven problem in the set theoretical worldview; What is the
problem?

First, let us review the traditional arguments concerning Hempel’s raven problem (cf.
refs. [31, 32]). Thus, we start from the followings:(C) Let U be the set of all birds. Let B(⊆ U ) be a set of all black birds. Let R(⊆ U )be a set of all ravens.
Although these should be doubtful (since these are as ambiguous as ”the set of all tyrannosaurs” ),we advance towards the next argument. The statement: ”Every raven is black” is logicallydenoted by(D1 ) ”Every raven is black” : (∀x)[x ∈ R −→ x ∈ B]

i.e.R ⊆ B ⊆ U,

U
R

B

Figure 12.3:The set-theoretical worldview (Sec. 4.3.4)[U : the set of all birds, B: the set of all black birds, R: the set of ravens]

Also, this is logically equivalent to the following contraposition:(D2 ) ”Every non-black bird is a nonraven” : (∀x)[x ∈ U \ B −→ x ∈ U \ R]i.e. U \ B ⊆ U \ R
However, if these are equivalent, then we have the following problems (i.e., raven problem):(E1 ) Why is the actual veriﬁcation of (A2 ) much more diﬃcult than the actual veriﬁcationof (A1 )?
(E2 ) Why can the truth of ”(A1 ): any raven is black” be known by (A2 ), i.e. without
seeing a raven also at once?
(E3 ) Is it possible to experimentally verify ”Every raven is black”?
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These are so called Hempel’s raven paradox. However, there is a reason to consider that”the set of all ravens” is as ambiguous as ”the set of all tyrannosaurs”. If so, that is, if theabove (C) is ambiguous, all other (D1 )-(E3 ) are also ambiguous, and thus non-scientiﬁc.
Now we think that the most essential problem concerning Hempel’s raven problem is
as follows:(F) What is the scientiﬁc meaning of ”Every raven is black”?
In order to study this problem, we must prepare the quantum linguistic formulation ofornithology, under which the meaning of ”Every raven is black” will be clariﬁed in thissection.
Remark 12.19. (i): I am impressed by Hempel’s powers of observation that he did not
overlook ”discomfort” that would normally be overlooked. Before I wrote paper [78], I
thought the above paradox [(E1 )∼(E3 )] was ”mere sophistry.”(ii): Just to be sure, in this text we assume that the followings are the same:”any raven is black” = ”every raven is black” = ”all ravens are black”.
This is the same as the usage in mathematics (i.e., ”any” = ”every” = ”all” = ”∀”).

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12.7 * Hempel’s raven paradox: A milestone in the philosophy of science

12.7.3 Hempel’s raven problem in the quantum mechanical worldview
In this section we slightly improve our result in• Ref. [78]: Ishikawa, S., (2019) Philosophy of science for scientists; The probabilistic
interpretation of science, Journal of quantum information science, Vol. 9, No.3, 140154,(https://www.scirp.org/Journal/paperinformation.aspx?paperid=95447)
We think that Hempel’s raven problem raises the problem of ”What is the scientiﬁcmeaning of ‘Every raven is black’?”. In order to answer this problem, we must prepare
the quantum linguistic formulation of ornithology
The arguments below are essentially the same as Saussure’s ”signiﬁer” and ”signiﬁed”
in Sec. 11.6.
12.7.3.1 Quantum linguistic solution of Hempel’s raven paradox
Now, we will discuss the continuation of Problem 12.13[All sweet apples are red]. However,since it will be virtually the same as ”Hempel’s raven problem,” we will rewrite ”All sweet
apples are red” to ”All ravens are black” for the discussion.
Problem 12.20. (=Problem 12.13) [All ravens are black]

Consider two (T F )-observables OR = ({T, F }, 2{T,F } , GR ) and OB = ({T, F }, 2{T,F } , GB )in C(Ω), where OR and OB is respectively called ”raven observable” and ”black birdobservable”.
mB (= GB ({T }))1mR (= GR ({T }))
Ω

ΩB

Ω

ΩR

Figure 12.4: [All ravens are black: (ΩR ⊆ ΩB ) ]408

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Now, it is natural to deﬁne the proposition ”all ravens are black” as follows(G) ΩR ⊆ ΩB
where mR = GR ({T }), mB = GB ({T }), ΩR = {ω ∈ Ω|mR (ω) = 1}, ΩB = {ω ∈Ω|mB (ω) = 1}.
Now we have the folowing problem:

Problem[(Q): Is [ΩR ⊆ ΩB ] a QL-propositin?.

(H)

I will answer this below.
Answer 12.21. Answer of Problem 12.20: Note that ΩR (= {ω ∈ Ω | [GR ({T })](ω) = 1})
is compact. Further, it is natural to assume that ΩR is separable (i.e., there exists a
countable dense set Ω′R (= {ω1 , ω2 , ...} ⊆ ΩR ). The next is obvious.
[ΩR ⊆ ΩB ] ⇐⇒ [Ω′R ⊆ ΩB ]Using the notation in ”Sec. 1.4.6: Inﬁnite tensor product observable”, deﬁne the following:∞

P0 ( × {T, F })
n=1

n ∞
=



× Ξn Ξn ⊆ {T, F } (n = 1, 2, ...),

n=1

n(= 1, 2, ...) | Ξn 6= {T, F } is ﬁniteoAlso, let n0 be any natural number. And put
∞

0
PT,n
( × {T, F })
0

n ∞
=

n=1

× Ξn Ξk = {T } (k = 1, 2, ..., n0 ), Ξk = {T, F } (k = n0 + 1, n0 + 2, ...)

n=1

Consider the inﬁnite tensor measurement:∞OMC(Ω) OB , S[δωn ] = MC(×∞n=1 Ω)

n=1=MC(×∞
n=1 Ω)

∞
O
n=1

N∞

n=1 MC(Ω)

∞
O

o


OB , S[δωn ] . That is,OB , S[⊗∞n=1 δωn ]

n=1
∞

∞

n=1

n=1

OB = ( × {T, F }, P0 ( × {T, F }),

∞
O



GB ) , S[⊗∞n=1 δωn ]n=1

Assume that, by the measurement, we get the measured value x = (xn )∞n=1 (∈∞
×n=1 {T, F }). Note thatT,n0
• the probability that x = (xn )∞(×n=1 {T, F }) is quaal to 1.
n=1 belongs to P0∞

That is, we see thatωk ∈ ΩB (= {ω ∈ Ω : mB (ω) = 1})(k = 1, 2, ..., n0 )Here, since n0 is arbitrary, we get:
ω k ∈ ΩB

(k = 1, 2, ...)

thus,Ω′R ⊆ ΩB

Therefore, we the that the followings are equivalent:409

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12.7 * Hempel’s raven paradox: A milestone in the philosophy of science(I1 ) Nthe measured value x = (xn )∞n=1 by the inﬁnite tensor product measurement
∞
M
O
,
S
satisﬁew
that xn = T (∀n = 1, 2, ...).
B
C(Ω)[δωn ]
n=1(I2 ) ΩR ⊆ ΩB
This completes the proof.

♠Note 12.10. (i):The condition of separability may not be necessary, but I did so because
it is natural to assume separability.
(ii): In fact, it is almost self-evident if we consider the following. Let me reiterate theﬁgure.
mB (= GB ({T }))1mR (= GR ({T }))

Ω

ΩB

Ω

ΩR

Figure 12.5, which is the same as Figure 12.4: [All ravens are black: (ΩR ⊆ ΩB ) ]This ﬁgure was created by measurement in the ﬁrst place. Then it is natural that”ΩR ⊆ ΩB ” is also obtained by measurement. The above proof is nothing more thana mathematical expression. Conversely, if you want to regard ”Ω R ⊆ ΩB ” as a proposition, you must set up an axiom (Axiom’ 1) so that you can show it with a mathematicalformula.

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12.7.3.2

Popper’s falsiﬁcationism in measurement theory♠Note 12.11. As mentioned in Sec. 12.4.1, Karl Popper (1902-1994) was one of the 20th
century’s most inﬂuential philosophers of science. Popper claims that, in order for something to be considered science, it must be falsiﬁable. If it is false, it can be shown throughobservation and experiment to be false. However, I think that the meaning of ”observationand experiment” is ambiguous. That is, it must be ”observation and experiment that is
described by quantum language”. This will be done below.

In the previous section, we discussed ”ΩSB ⊆ ΩB ” (i.e., Every small black bird isblack). Since this is a priori statement, we can accept this statement without veriﬁcation
by experiment.
In this section we will discuss the statement ”ΩR ⊆ ΩB ” (i.e., Every raven is black),
which is not a priori proposition but a posteriori proposition.
Hence, our problem is as follows:(J) How can we be sure of ΩR ⊆ ΩB (i.e., ”Every raven is black” )?
i.e. What should we do to be sure of ”ΩR ⊆ ΩB ”?.
In order to do it, we obey Popper’s falsiﬁcationism (cf. ref. [105]) such that(K) ”ΩR ⊆ ΩB ” should be accepted, if many experiments which deny ”ΩR ⊆ ΩB ” areconducted and ”ΩR ⊆ ΩB ” still cannot be denied.
For instance, we mention the following two tests ( [Test I] and [Test II])
[Test I]: In order to deny ”ΩR ⊆ ΩB ”,(L) we try to ﬁnd a bird with the state ω0 such that ω0 ∈ ΩR \ {ω | mB (ω) = 0} (SeeFigure 11.5 below)
This test is quite natural, and thus, we should try this ﬁrst.
mB

1
mR
ω0

↓

↔

Ω

≈0; negligible

Ω
ΩR
• ω0

ΩB

Figure 12.6: [ω0 ∈ ΩR \ {ω | mB (ω) = 0}, ρ0 ({ω ∈ Ω | 0 < mB (ω) < 1}negligible. ]411

TΩR }) ≈ 0, i.e.

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12.7 * Hempel’s raven paradox: A milestone in the philosophy of science[Test II: statistical hypothesis testing]: In order to deny ”ΩR ⊆ ΩB ”,(M0 ) we try to conﬁrm the hypothesis that there are non-black ravens by 3 percentages in 100 ravens. That is, we take the parallel mixed measurement (cf. [74,
N100
{y,n}
78])
, FB ), S[∗] (ρ0 )), where a mixed state ρ0 (∈i=1 MC(Ω) (OB :=({y, n}, 2m∗S (C(Ω) )) satisﬁes ρ0 (ΩR ) = 1 and ρ0 (ΩTR \ ΩB ) = 0.03. Here, we, for simplicity,
assume that ρ0 ({ω ∈ Ω | 0 < mB (ω) < 1} ΩR }) ≈ 0, i.e. negligible. (See Figure11.5 above.)
And assume that
(M1 ) as the result of the (M0 ), we get that one hundred ravens were black continuously
which is written in terms of quantum language as follows:
N100(M2 ) By the parallel mixed measurement (cf. [74, 78]) i=1 MC(Ω) (OB :=({y, n}, 2{y,n} , FB ),S[∗] (ρ0 )), a measured value (y, y, y, ..., y ) is obtained.
| {z }
100

Then, we calculate:(M3 ) the probability that a measured value (y, y, y, ..., y ) is obtained by the parallel| {z }
100
N100
mixed measurement
M
(O
:=
({y,
n}, 2{y,n} , FB ), S[∗] (ρ0 )) is given by
B
C(Ω)
i=1
(97/100)100 (< 0.048). That is, the probability that (M2 ) is realized (i.e., we meet
one hundred black ravens continuously) is less than 0.048 (> (97/100)100 )).
Thus, if we believe (M0 ), a very rare thing (i.e., (M3 )) happened since probability 0.048
is quite rare. Therefore, we should doubt the hypothesis (M0 ). That is, we couldn’t deny”ΩR ⊆ ΩB (i.e., any raven is black)”. When we can’t do such test many times and still
deny ”ΩR ⊆ ΩB (i.e., any raven is black)”, according to Popper’s falsiﬁcationism, we will
believe this.
Remark 12.22. [Any small black bird is black]
Consider the membership function mB : Ω → [0, 1] of ”black bird” and the membershipfunction mSB : Ω → [0, 1] of ”small black bird” as follows:412

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mB

1
mSB

Ω

ΩB

Ω

ΩSB

Figure 12.7: [All ravens are black: (ΩR ⊆ ΩB ) ]Put ΩB = {ω ∈ Ω|mB (ω) = 1}, ΩSB = {ω ∈ Ω|mSB (ω) = 1}. Then, it is natural toconsider ”ΩSB ⊆ ΩB ” is self-evident without any need for measurement.
However, as mentioned in Sec.12.3, a proposition in quantum language is a ”measurement”. Again, see Sec. 4.1.3 f : Carnap-Quine controversy.
Remark 12.23. I don’t think anyone can read (Wittgenstein’s) TLP scientiﬁcally. But
Wittgenstein showed us the direction in which philosophy should go. This is, of course, themost important thing a philosopher can do. Logical positivists such as Carnap, Quine,
Hempel, and others, through the problems of analytic and synthetic propositions, theﬂagpole problem, and Hempel’s crow paradox, called for the necessity of a ”languageof science”. And they must have been convinced that the ”language of science” wassomething diﬀerent from mathematical logic and set theory. However, there was too much
distance between their area of expertise and quantum mechanics. I had always wondered
how the ”philosophy of mind” came about. However, if the ”philosophy of mind” can beused as a tool to examine the Copenhagen Interpretation (cf. Sec. 9.4.2: Brain in vat),then I can understand its importance.

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12.8 Three approaches to the mind-body problem

12.8 Three approaches to the mind-body problem
The mind-body problem (i.e., to clarify the relation between ”mind” and ”body”) is themost famous problem in Descartes philosophy. There are three standing positions onwhether we see this as a problem of science or a problem of philosophy.
(A) [Science; Sec. 12.8.2: the ﬁrst approach]: If we are in the position of existential
monism (i.e., the scientiﬁc position), i.e. if we expect a scientiﬁc solution, then themind-body problem is a problem of brain science, AI, and cognitive science.
(B) [Wordplay in a literary sense; Sec. 12.8.3: the second approach][Illusory problem]:
The position that one should enjoy wordplay in a literary sense has historically been
most favored by philosophy fans.
(C) [Philosophy; Sec. 12.8.4: the second approach]: if we consider the mind-bodyproblem as a philosophical problem (i.e., dualistic idealism), it is a question of
proposing a worldview with ”mind” and ”body” as the key words.

Of course, our interest is focused on (C). This section was written with reference to thefollowing.
• [70] Ishikawa,S., A Final solution to mind-body problem by quantum language,
Journal of quantum information science, Vol. 7, No.2, 48-56, 2017, DOI:
10.4236/jqis.2017.72005 (http://www.scirp.org/Journal/PaperInformation.
aspx?PaperID=76391)
If quantum language is the only scientiﬁcally successful theory in dualistic idealism, it isnatural to study the mind-body problem in quantum language. This will be discussed in
Section 12.8.4 [The third approach].

12.8.1 The mind-body problem
Now let us introduce the mind-body problem, which is said to be the greatest unsolved
problem in dualistic idealism.
In spite that the cogito proposition ”I think, therefore I am” is non-sense (cf. Sec. 8.2),
Descartes used it in order to propose Descartes philosophy (i.e., mind-body dualism).
That is, his argument is as follows.
(D) If the existence of ”I” is deduced from the cogito proposition, the existence of414
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Chap. 12 Linguistic philosophy ( After TLP)
”matter” (which is perceived by ”I”) is accepted. And further, the medium of ”I”
and ”matter” is automatically accepted as ”body (= sensory organ)”.
Therefore, the key-words of Descartes philosophy (= mind-body dualism) is(E)”I”(=”mind”), ”body”(=”sensory organ”), ”matter”

Here, we have the following problem:(F): Descartes problem (i.e., the mind-body problem) in dualistic idealism
It is natural to consider that ”body” and ”matter” are ruled in physics.
Then, we have the following problem:
How are ”mind” and ”body” connected?
(or more generally, how are ”mind”, ”body” and ”matter” connected?)

This is generally considered to be the most important problem in Descartes philosophy.

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12.8 Three approaches to the mind-body problem

12.8.2 The ﬁrst approach; Cognitive scientiﬁc approach
As mentioned in Note 9.12, Dr. Crick (the most noted for being a co-discoverer of thestructure of the DNA molecule in 1953 with James Watson) said in his book ”The astonishing hypothesis” [12]) as follows.
(G1 ) You, your joys and your sorrows, your memories and your ambitions, your senseof personal identity and free will, are in fact no more than the behavior of a vastassembly of nerve cells and their associated molecules.

From the scientiﬁc point of view, I agree to his opinion (G1 ). (i.e., the denial of thesubstance dualism ). Therefore, I believe that the following will be realized.
(G2 ) Descartes(Kant,Husserl)−−−−−→
progress

Solution

Philosophy of mind−−−−−→
progress

Brain science−−−−−→
progress(based on cognitive science).

(the mind-body problem)

This means the following.
(G3 ) The mind-body problem is solved scientiﬁcally (as a matter of quantum mechanics(≈ quantum language))That is,But this is what almost all modern scientists believe.
That is because quantum language says:(H) Describe any monistic phenomenon (such as (G1 ) ) by dualistic language (=quantum language) !
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Chap. 12 Linguistic philosophy ( After TLP)

12.8.3 The second approach; Illusory problem?
It should be noted that
(I1 ) the term ‘mind’ and ‘body’ in the mind-body problem (F) is ambiguous in
Descartes=Kant epistemology.
Thus, there may be a reason to consider that the mind-body problem is just ”what wecannot speak about”. Therefore, according to Wittgenstein’s famous saying ”What wecannot speak about we must pass over in silence” (in [125]), some may conclude that wemust speak nothing about the problem (F). That is, the mind-body problem is an illusory
problem. As mentioned before, I think that the Wittgenstein’s next assertion is spot on.
(I2 ) philosophical problems arise from insuﬃcient attention to the variety of natural
language use.

12.8.4 * The third approach; Quantum linguistic solution to the mind-bodyproblem
The solution to Descartes’ psychosomatic problem with quantum language has alreadybeen discussed in Sec.8.3.2, but let us discuss it again here. The mind-body problem was
I have already described the solution to Descartes’ mind-body problem using quantum
language in section ? The mind-body problem was(J) Describe (in quantum language) the relationship between ”mind,” ”body (senseorgans),” and ”objects”!
I have already mentioned this many times. I have mentioned this many times, but let us
note the following.

mind
body(sensory organ)quantum language
quantum languageyymeasured value

observable

matterquantum languageystate

Illustrated,

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12.8 Three approaches to the mind-body problemz

Axiom 1 [measurement]}|”I”(=”mind”, ”brain”) ,|{z}
spiritual dynamics{”matter””body”(=”sensory organ”) ,|{z}
physical dynamics (Axiom 2 [causality])Thus, problem (J) can be interpreted as the following problem (K).
(K) Describe in quantum language the relation among ”measured value”, ”observable”
and ”state”!
Of course, this answer is just Axiom 1 (measurement in Sec. 1.2.1) as follows.
Axiom 1 = The solution to the mind-body problem• With any system S, a C ∗ -algebraic basic structure [A ⊆ B(H)] can be associated in which measurement theory of that system can be formulated. Whenthe observer takes a measurement of an observable (or, by a measuring instrument) O=(X, P(X), F ) for a system S[ρ] i.e. a system S with a state ρ), theprobability that a measured value x (∈ X) obtained by the measurement isgiven by ρ(F ({x}))(≡ A∗ (ρ, F ({x}))A ).
Obviously, this Axiom explains the following three relationshipsmindquantum languageymeasured valuex(∈ X)

Body (sensory organs)quantum languageyobservable O = (X, P(X), F )matterquantum language
ystate ρThis, plus Axiom 2 (causality), is the ”solution to the mind-body problem”. That is,(♯) (= (E5 ) of Answer 8.7 in Sec. 8.3.2)
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Chap. 12 Linguistic philosophy ( After TLP)
mind-body problem (Continental rationalism)(modern) Descartes problem
(Establish mind-body dualism!)  subjectivity problem(British Empiricism)(Establish QL!)

419

···

Find Axioms including
the key-words (mind, body, matter)· · · Propose the interpretation
concerning this Axioms

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421

Chapter 13

Postscript: Philosophy (of worldviews)
has progressed towards quantum
language : Philosophy is one
13.1 What is the core of philosophy of science?
Comparing Western and Eastern philosophy, I have the impression that there is something consistently
scientiﬁc in Western philosophy. The purpose of this text was to clarify something scientiﬁc. That is, I
concentrated on the question, ”What is the core of philosophy of science? I proposed the following Figure
0.1 (in Preface):
Assertion 13.1. (= Figure 0.1 in Preface): The location of QL in the history of western
philosophy(realistic) monism (i.e., without measurement axiom)
relativity
4 −→ theory−−−−−−−−−−→○(monism)(unsolved)3○
−−→ Newton ○6 TOE:theory of(realism)−−→ everything
quantum
5 −→ mechanics −−−−−−−−→○(quantum phys.)

Socrates
Aristotle
0○:Greek

7○

Schola-8○


linguistic turn−−−−−−−−−−−−−−−−−→

8
○



quantization

7
○

9
−
−
−
−
−
−
−
−
−
→
○

−−−−−−−−−→ statistics

dualism
applied math.
system theoryclassical mechanical worldview


(=MT)



(mind-body dualism)(Copernican revolution)) ○QL:quantum
15
linguistic−−→
language
Descartes
Kant
10○
scientiﬁc
11 −−→ Spinoza, Locke,... −−→−−−−−−→○
quantummechanicalturn
epistemological worldview
worldview

12
○



linguistic turn



?

(logic in philo.))(logic, set theory)Wittgensteinmechanical
Saussure
Boole, Frege,
14
−
−
−
−
−
−
−
→
○
13
○
turn
Zadeh
Russell, Cantor −−→philosophy ○21−−−−−→○
sticism
Parmenides, Zeno
Plato

math.

logical worldview(idealistic) dualism (i.e., with measurement axiom)
Figure 0.1 (in Preface): The history of the world-descriptions
Philosophy (≈ dualistic idealism) has progressed toward QL
0
1
2
10
12
14
15 )
(i.e., ○→
○→
○→
○→
○→
○→
○

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13.1 What is the core of philosophy of science?
Writing a history of Western philosophy, a discipline that has produced the greatest
number of geniuses, was a rather courageous undertaking for me, a non-specialist in philosophy. Many readers are probably more familiar with the history of Western philosophy
than I am. My only advantage is that I am somewhat familiar with quantum language,
which is probably the scientiﬁc ﬁnality of dualistic idealism. If so, I thought that even asa layman I might be able to have some say in the history of Western philosophy.

0
1
2
10
12
14
15 )
”Does the series (i.e., ○→
○→
○→
○→
○→
○→
○
in the above ﬁgure
mean progress?” is the biggest question in the History of Western philosophy. Quantum
language gives an aﬃrmative answer to this question, if the following deﬁnition is acceped:(B) ”X −−−−−−−−→ Y ” is deﬁned that ”Y is more like QL than X”
progress(QL)

Therefore, in this text, I examined this (B) to conﬁrm the progress of the above series *1 .

*1 ”History is an unending dialogue between the present and the past.” (cf. ref. [10]422

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Chap. 13 Postscript: Philosophy (of worldviews) has progressed towards quantum
language : Philosophy is one

13.2 Reviews from Plato to Wittgenstein

A summary of their scientiﬁc accomplishments follows.

13.2.1 (C1 ): Plato
Plato left us with the word ”Idea”. This was, in fact, the ”magic word”.
(History of progress from Idea to observables)

Idea

linguistic version−−−−−−−−−−−→

signiﬁer, [signiﬁed]

Plato’s idea theoryascholasticyversion ⃝Saussure’s linguistic theory[(primary) universal][membership function]

Anselmus, Thomas

Zadeh’s fuzzy theoryquantiﬁcationycognitiveyversion ⃝b[secondary quality]yidentiﬁcation

linguistic version ⃝c−−−−−−−−−−−−→

John Locke[observable]
quantum language

By scholastic philosophers a Anselmus (and Thomas), Idea was understood as ascientiﬁc concept of ”universal”. And by Locke b , Idea became a concept thateveryone can understand as ”the secondary quality”, and in QL c , Idea became acomputable concept as observables.

13.2.2 (C2 ): Augustines
St. Augustine oﬀered the hand of God (the hand of salvation) to the dying Platonic
philosophy. I think that Parmenides is the founder of ”Copenhagen interpretation (cf.
Sec. 2.3). However, it was Augustine who introduced the Copenhagen Interpretation to
Plato’s philosophy, arguing that ”Only the present exists” as follows:423

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13.2 Reviews from Plato to Wittgenstein

Idea

Plato →



























mind-body dualism

universal
Axiomatic−−−−−−−→
theoryCopenhagen−−−−−−−−−→
interpretation

Anselms, Thomas −−−−→

Descates
subjectivityyproblem

problem

secodary quality

Only the present exists

Augustinus

mind-body−−−−−−−→ · · ·

−−−−→

Locke

epistemology−−−−−−−−→ · · ·13.2.3 (C3 ): Anselmus and Thomas
Anselmus and Thomas pursued the scientiﬁc meaning of ”Idea”.
(C′3 )

Theory of Heaven( by Thomas )(C13 ) :universals exist’after individuals’ in the human intellect .
universalia post rem as actual world
QL: (i.e., measured value is sensed by the brain (i.e., mind)).
(C23 ) :universals exist ’before
 individuals’ in the divine intellect, .
universalia ante res as Plato’s Idea
QL: (i.e., observable (or, measuring instrument) could be in the heavens, like theNorth Star (cf. Note 1.7))(C33 ) :universals exist ’in individuals’
in the world, .

universalia in rebus as Aristotle’s eidos
QL: (i.e., state exists in matter)

13.2.4 (C4 ): Descartes
Descartes proposed Descartes problem of mind-body dualism (i.e., the mind-body problem
and the subjectivity problem)
That is,(♭1 ) The mind-body problem: How are mind and body related?
(♭2 ) The subjectivity problem: Is the world as I see the same as the world as you see it?

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Chap. 13 Postscript: Philosophy (of worldviews) has progressed towards quantum
language : Philosophy is one

This can be stated in relation to quantum language as follows.
(C′4 )

Theory of Earth ( by Descartes )mind-body problem (Continental rationalism)(modern) Descartes problem
(Establish mind-body dualism!)  subjectivity problem(British Empiricism)(Establish QL!)···

Find Axioms including
the key-words (mind, body, matter)· · · Propose the interpretation
concerning this Axioms

13.2.5 (C5 ): Kant
In a Copernican revolution, Kant discovered how to create scientiﬁc idealism (transcendental idealism). It is quite powerful, as follows. Especially 8 is the most important
copy theory
the world −→ cognition

10○:Copernican
revolution [I]−−−−−−−−−−−−−−−−−−−−−−−→
in cognition (due to Kant)

constructionism
cognition −→ the worldKant
12 ylinguistic turn○

Descartes=Locke

Copernican revolution [II]copy theory
things −→ words
language(= Saussure revolution )−−−−−−−−−−−−−−−−−−−−−−−→
in language (due to Saussure)

Saussure, Wittgenstein
14 ymechanical turn○

copy theory
phenomena −→ QM
quantum mechanics (=QM)
xa○quantumization
copy theory
phenomena −→ CMclassical mechanics (=CM)

constructionism
words −→ things

8 Copernican revolution [III]○−−−−−−−−−−−−−−−−−−−−−−−−→
in quantum mechanics

constructionism
QL −→ phenomena
quantum language (=QL)
c ○yclassicization

7 Copernican revolution [IV]○−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
in classical mechanics (Fisher, etc.)

constructionism
statistics −→ phenomena
statistics, system theory

However, Kant’s synthesis can be rewritten as QL-synthesis as follows.

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13.2 Reviews from Plato to Wittgenstein






Develop Descartes theory Leibniz
Spinoza(=mind-body dualism)Monadology• Pantheism , •



Solve






the mind-body problem
• −−−−−−−−−−−−−−−−−−−−−→

Axiomatic theory

QL





WittgensteinContinental
rationalism
Axiom 1Descartes theory





TLP

, Axiom 2•
mind-body dualism 





Derived from cogito















How to use
Locke,
Berkley,
QLDescartes theory
epistemology
linguistic




Solve
the
subjectivity(Copernican
Copenhagen
problem,etc.
 • −−−−−−−−−−−−−−−−−→
revolution)≈ interpretationEmpirical theory


Hume,
Kant


British empiricism




QL

QL-synthesis

13.2.6 (C6 ): Wittgenstein
I could not read Wittgenstein’s work TLP (=”Tractatus Logico-Philosophicus”) as a scientiﬁc academic book. However, as can be seen from the ﬁgure below, the inﬂuence of
TLP was enormous.

Looking at this ﬁgure, I think he insisted, ”Make up the language of science!” in TLP.
Though he said ”What we cannot speak about we must pass over in silence” in TPL, he
could not deﬁne ”scientiﬁc proposition”. However, as shown in the above ﬁgure, Wittgenstein changed the trend of philosophy. I think this is what should be most appreciated inphilosophy. I believe Wittgenstein’s dream was made possible by quantum language (cf.
Sec. 12.1).

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Chap. 13 Postscript: Philosophy (of worldviews) has progressed towards quantum
language : Philosophy is one

13.3 Almost all open questions of dualistic idealism have beensolved; Wittgenstein’s dream came true
I thought I would continue the discussion in the previous section a bit further, but thereader may ﬁnd the following(B′ ) ”X −−−−−−−−→ Y ” is deﬁned that ”Y is more like QL than X”
progress(QL)

Aren’t you just making up your own arbitrary yardstick like the one above and dogmatically claiming that ”quantum language” is the goal, etc.? Enough with the nonsense, if
you claim that quantum language is the ﬁnal form of dualistic idealism, then solve themultitude of unsolved problems!
”Proof is better than argument!”
I think you are right. Thus I showed the solutions of unsolved problems ((D 1 ) and (D2 ))as follows:(D1 ): solutions of unsolved problems in this text(D2 ): solutions of unsolved problems in ref. [74](D1 ) List of unsolved problems in Western philosophy for which

solutions are given in this text

See (D1 ) in Preface
In other words, this text’s argument was the following.
Anselmus,Thomas Aquinas

Plato →|

Christianity
Augustinus

Spinoza, Leibnitz

Wittgenstein→ Descartes=Kant → Analytic phil.
Locke, Berkeley, HumePhil. of science{z}
Era of search for dualistic idealism (various unresolved issues arose)y
QL
Goal (Solutions in this text)

In this text, this was realized. Thus I believe that

QL is the scientiﬁc ﬁnal form of dualistic idealism.

(D2 ) The list of my answers for scientiﬁc unsolved problems in(D2 )ref. [74]; Linguistic Copenhagen interpretation of quantum mechanics; Quantum Language [Ver 5], Research Report, Dept. Math. Keio University,
KSTS/RR-19/003 (2019); 473 p (http://www.math.keio.ac.jp/academic/research_
pdf/report/2019/19003.pdf)

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13.3 Almost all open questions of dualistic idealism have been solved; Wittgenstein’s
dream came true
• Kolmogorov’s extension theorem in quantum language (Sec.4.1 in ref. [74])(Sec.4.1 in ref. [74])• The law of large numbers in quantum language (Sec.4.2 in ref. [74])• The discovery of Heisenberg’s uncertainty relation (cf. ref. [38], or, Sec. 4.3 inref. [74])Has Heisenberg’s uncertainty relation ever been used eﬀectively in physics? (ref.
[38], or, Sec. 4.3 in ref. [74])• Bell’s inequality holds in both classical and quantum systems (Sec. 4.5.2 in ref.
[74])
• Measurement theoretical formulation of measurement, inference, control (Sec. 5.2
in ref. [74])• Monty-Hall problem in quantum language (non-Bayesian approach) (ref. [51],Sec.5.5 in ref. [74])• Two envelope problem in quantum language (non-Bayesian approach) (ref. [61],Sec.5.6 in ref. [74])• Conﬁdence interval and statistical hypothesis test (Chapter 6 in ref. [74])• Analysis of variance (Chapter 7 in ref. [74])• Syllogism holds in classical systems, but not in quantum systems (Sec.8.6 andSec.8.7 in ref. [74])• Mixed measurement theory (Bayesian measurement theory) (Chap. 9 in ref. [74])• The measurement theoretical characterization of the wave-function collapse (=projection pustulate) (cf. ref. [65], or Sec.11.2 in ref. [74])(von Neumann-Lüders Projection Postulate can be justiﬁed in QL: ref. [65])• The measurement theoretical characterizations of de Broglie’s paradox, quantum
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Chap. 13 Postscript: Philosophy (of worldviews) has progressed towards quantum
language : Philosophy is one
Zeno eﬀect, Schrödinger cat, Wigner’s friend, Wheeler’s delayed choice experiment, Hardy Paradox, quantum eraser (Sec.11.3∼Sec.11.8 in ref. [74])• The measurement theoretical characterizations of double-slit experiment, Wilson
cloud chamber (Sec.12.2, Sec.12.3 in ref. [74])• The measurement theoretical characterizations of regression analysis (Sec.13.2 inref. [74])• The measurement theoretical characterizations of Brownian motion, Zeno’s paradox (Sec.14.2, Sec.14.4 in ref. [74], also, see [47])• The measurement theoretical characterizations of least-squares method (Chap.15
in ref. [74])• The measurement theoretical characterizations of Kalman ﬁlter (Chap.16 in ref.
[74])
• The measurement theoretical characterizations of equilibrium statistical mechanics (ref. [52] or, Chap.17 in ref. [74])• The measurement theoretical characterizations of psychological tests (Chap.18 inref. [74])• The measurement theoretical characterizations of belief (Chap.19 in ref. [74])• The mathematical foundation of science (Hempel’s raven paradox) (Chap.20 inref. [74])We believe that the primary goal of the philosophy of science is to propose a ”language ofscience.” Looking at the above list (D1 ) and (D2 ), many readers may then agree followings:(E) As a candidate for the language of science, statistics is currently the most promising
candidate, but there is a good chance that quantum language will be more promisingin the near future429

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13.4 Philosophy of science is the scientiﬁc part of Western philosophy

13.4

Philosophy of science is the scientiﬁc part of Western
philosophy

13.4.1 Dr. Hawking
Dr. Hawking said, in his best seller ”A Brief History of Time”, as follows.

• However, in the nineteenth and twentieth centuries, science became too technical andmathematical for the philosophers, or anyone else except a few specialists. Philosophers reduced the scope of their inquiries so much that Wittgenstein the most famous
philosopher this century, said ”The sole remaining task for philosophy is the analysis of language.” What a comedown from the great tradition of philosophy from
Aristotle to Kant!
I agree to his opinion. Three notable jobs of the 1920s are the following.
(i) Tractatus Logico-Philosophicus by L. Wittgenstein in ref. [125] (1921)(ii) Fisher’s historical masterpiece “Statistical Methods for Research Workers” in ref.
[22] (1925)(iii) Born’s discovery “the probabilistic interpretation of quantum mechanics” (cf. ref.
[6] (1926)
Since then, quantum mechanics and statistics have made great developments. Comparedto this, the development of analytic philosophy has been slow. No wonder it is about timethat a theory integrating the three has emerged.

13.4.2 The purpose of philosophy of science is to create a language of science
My specialty is nonlinear analysis and mathematical physics (e.g. refs. [36, 37, 38, 39]).
Heisenberg’s uncertainty principle (cf ref. [28]; 1927), derived from his γ ray microscope
thought experiment, is expressed as follows:δq · δp ≈ ℏ(:= Plank constant)(HU)

The author proposed the mathematical deﬁnitions ∆q and ∆p , which are considered as
the mathematical representions of δq and δp . And I proved the following (mathematically,430

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Chap. 13 Postscript: Philosophy (of worldviews) has progressed towards quantum
language : Philosophy is one
not by thought experiment) (ref. [38]).
∆q · ∆p ≥ ℏ/2(MU)

which is the mathematical representation of (HU).
I immediately shifted my research to the application side to propose a revolutionary
measuring instrument based on this principle, but without remarkable results. I still ﬁnd
it curious that the most famous principle in quantum mechanics has no applications,despite the fact that the formula for the calculation of errors has been clearly deﬁned *2 .
It was at this time that I became interested in the Copenhagen interpretation, thinking
that it was due to the author’s lack of understanding of the Copenhagen interpretation.
This is how I got hooked on the Copenhagen Interpretation, a collection of aphorisms
on how to use quantum mechanics.
And I proposed: (F1 ): [mathematical generalization of quantum mechanics]
3*(F) : [quantum language] =(F2 ):[(true) Copenhagen interpretation]
That is, the true Copenhagen interpretation does not belong to quantum mechanics but
quantum language.
I put the (F) against the ”fuzzy sets” that were popular at the time and claimed thescientiﬁc validity of ”fuzzy logic” (refs. [41, 42, 43, 44]; 1997-2000, and ref. [46]; 2006)
Adding ref.[81]; 2021, the whole picture of quantum language looked like the follows:
8○quantum mechanics −−−−−−−−−→ linguistic turn 




 ○
15
9
○
−−→ QL(G) (=(E) in Sec. 11.1.2)
statistics −−−−−→
dualism 






14
○

Wittgenstein −−−−−→ dualism

Since quantum mechanics, statistics, and logic were the fundamental theories that inﬂuenced ordinary scientists in the 20th century, it may be a promising ﬁeld if all three canbe combined in a single theory called quantum language.
Around this time, I learned that there was a discipline called ”philosophy of science. At
ﬁrst, the sense of philosophy of science was very humanistic, and there were many aspectsof it that I did not understand.
However, the author realized that philosophy of science could be read interestingly fromthe viewpoint of ”quantum language. He thought that the purpose of philosophy of sciencewas ”to create a language to talk about science. If so, I thought that this purpose hadalready been realized in ”quantum language.

*2 If I am allowed to joke, I would like to add ”Heisenberg’s uncertainty principle is of no practicaluse” to the Copenhagen interpretation.
*3 Although not mentioned in this document, (F1 ) and (F2 ) may be independent.

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13.4 Philosophy of science is the scientiﬁc part of Western philosophy
One book on the philosophy of science said, ”The scientiﬁc part of Western philosophy
is the philosophy of science,” so I read some philosophy books, but I did not feel that
I understood. However, I did understand that the history of Western philosophy is atreasure trove of unsolved problems concerning dualistic idealism. I thought that thescience part of the history of Western philosophy is the history of ”creating a language
to talk about science. I concentrated on this when I realized that most of the unsolved
problems of philosophy could be easily solved by quantum language. The results of thiswork are described in this text.
The author has taken a very shortcut when it comes to ”creating a language that speaks
science”. He has merely taken advantage of the labor of geniuses (Bohr, Heisenberg,
Schrödinger, Born, von Neumann, etc. in quantum mechanics; Fischer, Kolmogorov,
Kalman, etc. in statistics; Frege, Cantor, Zadeh, etc. with regard to logic and sets).

13.4.3 QL wants rivals!
Philosophy is a kind of literature that can be enjoyed by the general public. Moreover,even if we limit ourselves to the scientiﬁc aspects of philosophy, the general public canenjoy it. This is one of the attractions of philosophy. I have often emphasized in this textthat this appeal is due to the following distinction (cf. (E) in Sec. 8.1.2 ).

(H1 ): Philosophy for Professionals· · · Axiomatic theory




The
problemofuniversals,Continental Rationalism, Statistics,
Philosophy of Science (Hempel, etc.), QL (Axiomatic theory)









(A great scientiﬁc success has been statistics, but it is not generally
regarded as a philosophy since it does not belong to dualism)
(H)




(H2 ): Philosophy for the Masses· · · Empirical theoryDescartes-Kant
Epistemology,
Philosophy of Science (Popper, etc.)





Philosophy of Mind, QL (Copenhagen Interpretation)







(Gaining the support of the masses tends to be considered a success)

The caveat above is that the diﬀerence between professionals and amateurs is not the levelof research. One can think of it as ”professional=mathematics required (especially Axioms
1 and 2)” and ”amateur=explanation of rules that cannot be expressed in mathematicalformulas (Copenhagen Interpretation)”. For example, although not mentioned much inthis text, various examples discussed in the philosophy of mind (e.g., the qualia problem
and the brain in a vat (Sec. 9.4)) are very useful for forming Copenhagen interpretations.

This text was written for the general public, focusing on the Copenhagen interpretation(Ref. [74] is for professional use). Since this text is of type (H2 ), there should be a deep
discussion of the Copenhagen interpretation, even if the mathematics is treated lightly.
Therefore, I have stuck to a simple C ∗ -algebraic formulation of the quantum language.
Moreover, in the measurable space (X, P(X)), X was assumed to be a ﬁnite set, which
made it easier to understand, but also prevented deeper discussion. The W ∗ -algebraic
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433
formulation of the quantum language was also appealing, but to make it clear that thistext is intended for the general public, we stuck to the C ∗ -algebraic formulation.
A more fundamental problem is that ” ’Why is quantum language possible?’ is notknown”. In physics, we can proceed by comparing the phenomena/experimental results
to conﬁrm their truth or falsity.
However, Axioms 1 and 2 are mere spells, and in the (linguistic) Copenhagen interpretation, there are no clear criteria for ”adopting or rejecting a statement.
In extreme terms, it is just a decision based on my preference.
Thus, I feel we need a theory on the criteria for the Copenhagen interpretation, butthat may not be possible.
”Is the Copenhagen interpretation uniquely determined in the ﬁrst place?” and ”It may
be impossible to create a perfect Copenhagen interpretation (Wittgenstein’s paradox; Sec.
12.2.3),” and there are many other things I cannot answer when theorists prod me.
In the end, all I can say now is the following(I) We can solve a large number of unsolved problems listed in (D1 ) in Preface and(D2 ) in Sec. 13.3, so let’s not get too theoretical and keep solving problems.
Quantum language is a mysterious discipline. Quantum language is a proposed discipline
as one attempt to integrate 2,500 years of Western philosophical history, quantum mechanics, and statistics. Therefore, the world the author sees now is a very small part ofquantum language. And I believe that QL is the ﬁnal form of dualistic idealism. Evenso, the most desirable thing for quantum language is for rivals to emerge. I believe thatcomparing with strong rivals will clarify the various issues of quantum language.
I hope that the wise readers will explore the many aspects of quantum language.
December 2022
Shiro ISHIKAWA*4
E-mail: ishikawa@math.keio.ac.jp*4 For the further information concerning quantum language (e.g.

an improved version of this text
may be published), see home page: https://ishikawa.math.keio.ac.jp/indexe.html

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Index
abduction, 322, 325
Abelard (1079-1142), 175
abstract ego (von Neumann), 152
Aesop(BC.620 - BC.510), 79
allegory of the cave, 81
allegory of the sun, 85
ambiguity of ordinary language
Geocentrism and Heliocentrism, 199
Zeno’s paradoxes, 63
analytic proposition, 386
Anselmus(1033 - 1109), 106, 162, 255
antinomy, 63
Archimedes(BC287 - BC212), 135
Aristarchus(BC310 - BC230), 132
Aristotle(BC384 - BC322), 100
asserted ﬁction, 76
Augustinus(354-430), 150
axiom of choice, 131
Axiom 1 [measurement], 7
Axiom 2[causality], 8Bacon(1561 - 1626), 112, 292
Banach-Tarski theorem, 131
Bell’s inequality, 255
Berkeley, George(1685-1753) , 250
J. Bernoulli(1654 - 1705), 188, 268
Bohr(1885 - 1962), 106, 162, 255
Bohr-Einstein debates, 254–256
Boole, George(1815-1864), 319
Born(1882-1970), 324, 377, 430
brain in a vat, 261
British Empiricism, 227
buoyancy, 135
C ∗ -algebraic formulation, 23
Cantor’s paradox, 331
Carnap, Rudolf (1891-1971) , 386
Carroll, Lewis(1838-1898), 395
Carroll’s logical paradox, 395
Cartesian problem=Descartes problem, 211
causal relation, 6, 190, 199
cogito proposition, 128, 211, 214Continental Rationalism, 222, 229
Copenhagen interpretation (=linguistic
Copenhagen interpretation), 10
Copernican revolution, 265, 293
F. Crick (1916 – 2004), 224, 259
Critique of Pure Reason (Kant), 282
Descartes(1596-1650), 11, 13, 211
Descartes ﬁgure, 11, 13
Descartes problem=Cartesian problem, 211

Dialogues concerning two new sciences, 201
Discourse on the Method (Descartes), 211
discovery of zero, 157
dogmatic slumber, 265, 293
DST-formula, 66
Dummett, David (1925-2011) , 353
eidos(Aristotle), 103
Einstein(1879 - 1955), 106, 162, 255
Einstein-Tagore meeting, 254
Eratosthenes(BC275 - BC194), 141
ethics · morals, 72
Euclid(BC330 - BC275) , 127
Eudoxus(BC.400 BC.347), 49
falsiﬁability, 17, 129, 389, 411
Fisher(1890-1962), 324, 377, 430
Five-minute hypothesis, 333
ﬂagpole problem, 398
Frege, G.(1848-1925), 319
Fuzzy logic in QL, 360
fuzzy logic(=QL-logic), 355
Galileo(1564 - 1642), 112, 188, 268, 292
Gauss(1777 - 1855), 158
Geocentrism(Ptolemaios), 143
Goodman, Nelson(1906-1998), 274
grue paradox, 274
habitual thinking (= interpretation), 62
Harari.Y.N(1976 -), 79
Hawking(1942-1998), 430
Heisenberg’s uncertainty principle, 137, 379, 430
Hezenberg picture, 30
Heliocentrism, 192
Heliocentrism(Aristarchus), 132
Hempel, C.G(1905-1997), 398
Hērakleitos(BC.540 -BC.480), 54, 110
Hume, David (1711-1776) , 263, 267
Hume’s problem of induction, 267
Husserl (1859-1938) , 351
hyle(Aristotle), 103
I think, therefore I am., 211
Idea(Plato), 250
I know that I know nothing, 74
induction, 189
inductive reasoning, 189
Kalman(1930 - 2016), 356
Kant(1724-1804), 265, 293
Kantian synthesis, 296

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J, Kepler(1571 - 1630), 190, 194
Kripke’s quus, 382
law of large numbers, 187, 268
leaning Tower of Pisa, 195
Leibniz(1632-1677), 222
Leibniz(1646-1716), 223, 238
Leibniz’s relationalism, 238
Leibniz-Clarke Correspondence, 238
linguistic interpretation (=linguistic Copenhagen
interpretation), 10
linguistic philosophy, 373
List of the scientiﬁc problems that have beensolved, 427
J. Locke (1632 - 1704), 227
logical worldview (= set-theoretical worldview),
121
maximum likelihood method, 34
McTaggart. J.M.E.(1866–1921), 336
McTaggart’s paradox, 336
measured value, observable, state, 6
measurement, 7Merleau-Ponty(1908-1961), 15, 235
mind-body dualism=mind-matter dualism, 215
mind-body problem, 218, 414
model-change, 97Monty-Hall problem, 428
Moore, E.J. (1873-1958) , 338
Moore’s paradox, 338
motion function method, 60, 246
nativism, 229
necessity of dualism, 91
necessity of idealism, 91
Newton(1643-1727), 112, 244, 292
non-Euclidean revolution, 130
observable, state, measured value, 6
Ockham(1285 - 1347), 106, 162, 255
Ockham’s razor, 178
Only the present exists, 150
paradigm shift, 185parallel postulate, 128
Parmenides(born around BC. 515), 110
Peirce, C.S.(1839-1914), 322
PI(= Philosophisch Investigations), 373
Plato’s beard, 179
Popper(1902-1994), 17, 389, 411
predicate logic, 305
primary quality, secondary quality, 250
Principia, 198, 227
Principia Mathematica, 319
Problem of universals, 106, 162, 255
product observable, 32
projection postulate, 19, 152
Proof of the existence of God, 161
propositional logic, 305
Protagoras(BC490 - BC420), 72
Ptolemaios(AD.83 - AD.168), 143

pure Pythagoreanism, 52
purpose, 110
Pythagoras(BC582 - BC496), 46
QL(=quantum language), v
QL-logic(= fuzzy logic), 355
QL-synthesis, 225
qualia (=subjectivity) problem, 218, 258
quantum linguistic synthesis, 296
quasi-product observable, 32
Quine, W.V.O. (1908-2000) , 386
raven paradox, 404
Russell. B.(1872 - 1970), 88, 103, 330
Russell’s paradox, 330
Saussure, F.(1857-1913), 342
Scholasticism, 155
Schrödinger picture, 30
secondary quality, primary quality, 250
self-reference
subjective time, 153
set-theoretical worldview (= logical worldview),
121
signiﬁed, 343
signiﬁer, 343
Socrates(BC469 - BC399), 72
spectrum, 239
spin observable, 27
Spinoza, Baruch(1632-1677), 222
state, observable, measured value, 6state equation, 294
statistics from QL, 7
subjective time, 152
subjectivity (= qualia) problem, 218
subjectivity problem, 218, 220syllogism(Aristotle), 115
syllogism (in QL), 366
synsetic proposition, 386
Tagore, Rabindranath(1861 - 1941), 254
tensor quasi-product observable, 32
Thalēs(BC640 - BC546), 41
time theory (Augustinus), 150
TLP(=Tractatus Logico-Philosophicus), 353
trial of Galileo, 194
two envelope problem, 428
universal, 164, 165, 175
von Neumann(1903- 1957) , 152
von Neumann-Lüders Projection Postulate, 428
W ∗ -algebraic formulation, 23
wave function collapse, 19, 152
Weinberg(1933 -2021), 42, 79, 100, 112, 143, 191,
192
What is space-time?, 61
Whitehead(1861 - 1947), 43, 71, 236
worldviewism, 2, 39

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Zadeh, A. L.(1921 -2017), 357
Zadeh’s fuzzy set, 93, 356
Zeno(BC490 - BC430), 63
Zeno’s paradoxes , 63

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Department of Mathematics
Faculty of Science and Technology
Keio University
Research Report
2020[20/001] Shiro Ishikawa,
History of Western Philosophy from the quantum theoretical point of view; [Ver. 3 ],KSTS/RR-20/001, November 15, 2020
2021
[21/001] Shiro Ishikawa,
History of Western Philosophy from the quantum theoretical point of view; [Ver. 4 ],KSTS/RR-21/001, August 23, 2021
2022
[22/001] Shiro Ishikawa,
History of Western Philosophy from the quantum theoretical point of view; [Ver. 5 ],KSTS/RR-22/001, December 20, 2022