Research Report

KSTS/RR-21/001
August 23, 2021

History of Western Philosophy fromthe quantum theoretical point of view; [Ver. 4]

byShiro Ishikawa

Shiro Ishikawa
Department of Mathematics
Keio University

Department of Mathematics
Faculty of Science and Technology
Keio University
c
⃝2021
KSTS
3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522 Japan

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History of Western Philosophy from the quantum theoreticalpoint of view; [Ver. 4]
Shiro ISHIKAWA*1 (Mail: ishikawa@math.keio.ac.jp)

Preface:

Toward post-analytic philosophy; Philosophy has madeprogress towards QL
Quantum language (i.e., QL (=MT = measurement theory), the linguistic Copenhagen interpretation of quantum mechanics, the quantum mechanical worldview) proposed by myself is a scientiﬁclanguage that is inspired by the Copenhagen interpretation of 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. The contents of my lecture are roughly as follows.
Assertion 0.1 ( in Preface ):the realistic worldview (physics): monism
relativity−→ theory−−−−−−−−−−→ 3 (monism)(unsolved)2
−−→ Newton
 5 TOE:theory of(realism)−−→ everything
quantum(quantum phys.)−→ mechanics −−−−−−−−→ 4 OO

Socrates
Aristotle

O:Greek0philosophy

Parmenides, Zeno
Plato

Schola-O−−−−−→ 1
sticismOO−−−−−−−−−−−−−−→ 7 





dualism


8
−
−
−
−
−
−
−
−
→
−−−−−−−−−→ 6 statistics
quantization
applied math.
system theoryclassical mechanical worldview


(=MT)



(mind-matter dualism)(Copernican revolution)) 14 QL:quantum
Descartes−−→ language
Kant
9mechanical−−→ Locke,...
−−−−−−−→ 10
−−→quantum mechanical


turn

ﬁctional worldview
worldview
ﬁctional worldview


11



linguistic
turn




?

(logic in philo.))(logic, set theory)
Wittgensteindualism
Boole, Frege, 12 Saussurre−−−−−−−−→ 13
measurement
Zadeh
Russell,Cantor −−→
language

O

O

O

O
O

O

O

O

O

logical worldview

math.

the idealistic worldview (no experiment, metaphysics): dualism
Figure 0: The history of the world-descriptions
Philosophy (≈ dualistic idealism) has progressed toward QL( i.e., 0 → 1 → 9 → 11 → 13 → 14 )O O O O O O

The main part of quantum language
already studied in the following:



⃝
2 - ⃝
7 - ⃝
14





and statistics ⃝in the above were
6 - ⃝8 ⃝14*1 Department of mathematics, Faculty of science and Technology, Keio University, 3-14-1, Hiyoshi,

Kouhokuku, in Yokohama, 223-8522, Japan,

HP:http://www.math.keio.ac.jp/~ishikawa/indexe.html

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(A1 ) ref. [71]: 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)

Therefore, in this text I devote myself to the part  ⃝
0 - ⃝
1 - ⃝
9 - ⃝11 ⃝13 ⃝
14 , which is almostequal to the history of western philosophy. This part ⃝was not detailed
0 - ⃝
1 - ⃝
9 - ⃝11 ⃝13 ⃝14
in my lecture, but it will be a good preparation to read (A1 ) (since (A1 ) may be too hard). Thatis, this paper is written as the supplementary reader in my lecture, and thus it can be read without
the mathematical preparation.
Thus, the above ﬁgure says that
(A2 ) statistics, quantum mechanics, Descartes=Kant philosophy and analytic philosophy are each one aspect of quantum language.
Also, I think that the following is one of the most fundamental problems of western philosophy:
The progress problem (B) :(B) Do the time series

Has philosophy made progress?

⃝
0 - ⃝
1 - ⃝
9 - ⃝11 ⃝13 ⃝14mean the progress of philos-

ophy?

In this paper I conclude that
(C): My answer; the progress of philosopy (≈ dualistic idealism)(C1 ) deep If ”to make progress” is deﬁned by ”to come near quantum language” (i.e., ”becoming more and more
0  like quantum language” we can say that the time series ⃝
⃝
⃝
⃝⃝⃝
can
be
regardedasprogress,
that
is,
1
9
11 13 14This will be proved throughout this text.
Note0.1:Wittgenstein could not discover ”scientiﬁc logic”, but I believe that he is the biggestii

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contributor.
Here, in this paper, it suﬃces to consider that ”idealism”≈”linguistic theory”≈”metaphysics” (cf.
Deﬁnition 1.8). Thus, we conclude that
(C2 ) a scientiﬁc perfection of dualistic idealism is realized by quantum language
which is surely one of the most important assertions in western philosophy.
quantum language is intended as a language of science. Therefore,

Also recall that

(C3 ) the purpose of philosophy concerning dualistic idealism was to make the languageof science.
That is, I think that
(C4 ) none of the philosophies of dualistic idealism (other than quantum language) has beensuccessful.
It is sure:• If anyone believes that he has found a theory that goesbeyond ⃝,13 he will certainly want totalk about the landscape ⃝
⃝
⃝
⃝⃝⃝
seen
from
the
theory.
01
9
11 13 14Hence I wanted to do that too. This is my motive of writing this paper.
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 to otherimmature philosophies), I can expect that most unsolved problems in dualistic idealism canbe solved in the framework of quantum language.

In this text, I will show that the following unsolved philosophical problems are easily solved ascorollaries of the (C2 ):It is said that
• Wittgenstein carried out one of the most ambitious attempts in the history of philosophy
to solve all the problems of philosophy at once in [117]: ”Tractatus Logico-Philosophicus(abbreviated as TLP)”.
However, 100 years later, we can’t say that he succeeded in his challenge.
following list shows, QL realized his dream.

However, as thefamous unsolved problems concerning dualistic
idealism can be solved in QL(D) Almost• What is probability (or, measurement, causality) ? cf. Sec. 1.1.1)• Zeno paradox (Flying arrow, Achilles and a tortoise), (cf. ref. [45], or Sec. 2.4)

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( ”to solve Zeno paradox” = ”to understand DST-formula” (cf. Sec. 2.4.3 )ZenoDST• the measurement theoretical understanding of Plato’s allegory of the sum , (cf. Sec. 3.3.2)• Plato’s Idea theory≈Zadeh’s fuzzy theory≈Sausuure’s linguistic theory (cf. Sec. 3.5.3)• 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)• Only the present exists (cf. Sec. 6.1.2)• What is the problem of universals? (cf. Sec. 6.5.1)• What is Geocentrism vs. Heliocentrism? After all, the worldviewism (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. 9.3)
• The problem of qualia (cf. Sec. 9.5.1)• Brain in a vat argument (cf. Sec. 9.5.2)• The solution of Hume’s problem of induction (cf. Sec. 9.7.1)• grue paradox cannot be represented in quantum language (cf. Sec. 9.8)
• What is causality? (cf. Sec. 10.2)• 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)• My scientiﬁc understanding of ”Tractatus Logico-Philosophicus (=TLP)” and ”Zadeh’s fuzzysets” (cf. ref. [78], or Sec. 12.1)(i.e., the mainstream ofThat is, Wittgenstein took over the baton of ”dualistic idealism
philosophy)” from Kant. This is essentially important, since Wittgenstein’s picture theory
must belong to dualistic idealism in order to assert (C) ( or, Assertion 0.1)
• Lewis Carroll’s paradox (cf. Sec. 12.4)• Flagpole problem, (cf. Sec. 12.5)• Hempel’s raven paradox (cf. Sec. 12.6)• the mind-body problem (i.e., How are mind and body connected?), (cf. Sec. 12.7)
Also, for the solutions of unsolved problems in quantum mechanics, statistical mechanics, statistics
and probability theory, see ref. [71]), that is,(A1 ) ref. [71]: 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)
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The list will be written is in (D2 ) in Chap. 13: ”Postscript.”
After all, I will clarify that(E) the strangeness of philosophy (i.e., dualistic idealism) is due to thestrangeness of the Copenhagen interpretation of quantum mechanics

Lastly I should add the following:• My specialty is not philosophy. Perhaps most readers are more familiar with the history ofWestern philosophy. The only advantage I have is that I am familiar with quantum language( which may be the scientiﬁc ﬁnal goal of dualistic idealism). Thus,
my purpose of this text is to ﬁnd the great power of quantum language (i.e., the quantum mechanical worldview) everywhere in the history of western
philosophy.
• This text is written as the supplementary reader, and thus, it can be read without the knowledge of quantum theory. I used a lot of illustrations to explain things since I’m not conﬁdent in
my English. The mathematical arguments were left to ref. [71] mentioned in (A): KSTS/RR19/003 (2019); 473 p(http://www.math.keio.ac.jp/academic/research_pdf/report/2019/19003.pdf)• this text is the revised version of ref. [65]: History of Western Philosophy from the quantum
theoretical point of view [Ver. 3], Research Report (Department of mathematics, Keio university, Yokohama), (KSTS-RR-20/001, 2020, 296 pages)(http://www.math.keio.ac.jp/academic/research_pdf/report/2017/17004.pdf)
Roughly speaking, I say that(∗)
this [Ver. 4] = ”some corrections of [Ver. 3]” + ref. [78] + ”illustrations”
I believe that philosophy, like literature, should be written for the general public. I think it should
be mostly a playful part rather than an academic part. I hope many readers will enjoy it. *2 .
August 2021
Shiro Ishikawa*2 For the further information concerning quantum language, see my home page:~ishikawa/indexe.html

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vii

Contents
The sections containing solutions of famous unsolved problems are marked with the red *xi

Chapter 1
1.1

1
2

1.2

1.3

1.4

1.5

Chapter 2
2.1

2.2

2.3

2.4

Chapter 3
3.1

3.2

The outline of quantum language (=measurement theory)
The overview of quantum language(=measurement theory) . . . . . . . . . . . . . . . .
1.1.1 * Axiom 1 (What is probability (or, measurement) ?) and Axiom 2 (What is
causality ?) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Linguistic Copenhagen interpretation (or in short, Linguistic interpretation) . .
The history of worldview and our purpose . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Quantum language in the history of worldviews . . . . . . . . . . . . . . . . . . .
1.2.2 Our purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Realistic worldview and idealistic worldview . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 The worldviewism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 The realistic worldview(physics) (♭1 ) . . . . . . . . . . . . . . . . . . . . . . . .
1.3.3 The idealistic worldview (♭2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Preview of the assertions in this paper . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Realistic worldview?, (ﬁctional, mechanical) worldview? . . . . . . . . . . . . . .
1.4.2 Keywords: Monism and dualism . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix: The mathematical foundations of quantum language . . . . . . . . . . . . .
1.5.1 Mathematical preparations (the C ∗ -algebraic formulation of QL) . . . . . . . .
1.5.2 Axiom 1 [Measurement] and Axiom 2 [Causality] . . . . . . . . . . . . . . . . .
1.5.3 Inference; Fisher’s maximum likelihood method) . . . . . . . . . . . . . . . . . .

2
5
13
13
14
17
17
21
21
26
26
27
30
30
31
32

Ancient Greek philosophy (before Socrates)
Thalēs (BC.640 - BC.546) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Thalēs: the ﬁrst philosopher: ” the arche (= the ﬁrst thing of all things) 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 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4 Appendix: The discussion about Zeno’s paradoxes (e.g., Achilles and a tortoise])in the motion function method . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35
36
36
39
41
41
46
49
49
50
52
54
54

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 – . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Allegory of the cave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61
62
62
63
65
65
69

56
57
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3.3

3.4
3.5

3.6

Chapter 4
4.1

4.2

4.3

Chapter 5
5.1
5.2

5.3

5.4

5.5
5.6

Chapter 6
6.1

6.2

6.3

6.4

3.2.3 The theory of anamnesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The allegory of the sun: Measurement theoretical aspect of Idea theory . . . . . . . . .
3.3.1 The allegory of the sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2* The measurement theoretical understanding of the allegory of the sun . . . . .
Plato: The ﬁctional worldview (=Plato’s way of telling philosophy) . . . . . . . . . . .
3.4.1 The necessity of idealism and dualism . . . . . . . . . . . . . . . . . . . . . . . .
Key words of dualism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Three key-words of dualism . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2 Is Idea theory related to measurement (i.e., dualism) ? . . . . . . . . . . . . . .
3.5.3* Plato’s Idea theory ≈ Locke’s secondary quality ≈Sausuure’s linguistic theory ≈Zadeh’s fuzzy theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary: Plato’s way of telling philosophy . . . . . . . . . . . . . . . . . . . . . . . .
3.6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70
72
72
74
75
75
78
78
78

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Why does the motion happen? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 From purpose to causality: Modern science started from the discovery of ”causality” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Practical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Practical logic; Aristotle’s syllogism in ordinary language . . . . . . . . . . . . .
4.3.2 Mathematical logic; due to Boole, Frege . . . . . . . . . . . . . . . . . . . . . .
4.3.3* Syllogism does not always hold in quantum systems . . . . . . . . . . . . . . . .

87
88
88
88
91

81
83
83

91
94
94
96
97

Around Alexndria; Hellenistic period
101
Around Alexandria; Hellenistic period . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Euclid(BC.330 - BC.275) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2.1 Euclid geometry - Parallel postulate . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2.2 non-Euclidean revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Aristarchus (BC.310 - BC.230) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.3.1 the diameter of the moon : the diameter of the sun . . . . . . . . . . . . . . . . 107
5.3.2 Ancient Heliocentrism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Archimedes (BC.287 - BC.212) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.4.1 Buoyancy (Archimedes’ principle) . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.4.2 The tomb of Archimedes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.4.3 Principle of leverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Eratosthenes (BC.275 - BC.194) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.5.1 The biggest ancient observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Claudius Ptolemaeus (AD.83 - AD.168) . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.6.1 The ancient scientiﬁc collected studies . . . . . . . . . . . . . . . . . . . . . . . 116
The Middle Ages
119
Augustinus(AD. 354 - AD.430) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.1.1 Philosophy is a maidservant of theology . . . . . . . . . . . . . . . . . . . . . . 120
6.1.2* ”Confessions” by Augustinus: Only the present exists . . . . . . . . . . . . . . . 122
6.1.3 ”Subjective time” is a magic word which excites our delusion . . . . . . . . . . 123
Scholasticism –Graft Bamboo (=Aristotle) to a tree (=Plato) – . . . . . . . . . . . . . 126
6.2.1 Aristotle’s philosophy spread to the Islamic world . . . . . . . . . . . . . . . . . 126
6.2.2 Crusade expedition and Inﬂow of Islamic culture . . . . . . . . . . . . . . . . . 126
The discovery of zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.3.1 Positional notation (= the discovery of zero): Arabic numerals . . . . . . . . . 128
6.3.2 Arabic numerals and Roman numerals . . . . . . . . . . . . . . . . . . . . . . . 128
6.3.3 The explosion of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
The proof of the existence of God . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.4.1 Anselmus (1033 - 1109) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.4.2 Review: the worldviewism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

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Contents

6.5
6.6
6.7
Chapter 7
7.1
7.2
7.3

7.4

7.5
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
9.8
Chapter 10
10.1

6.4.3 The inﬂow of Aristotle philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Scholasticism; Problem of universals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.5.1* What is the problem of universals? . . . . . . . . . . . . . . . . . . . . . . . . . 133
Thomas Aquinas (1225 - 1274); Scholasticism . . . . . . . . . . . . . . . . . . . . . . . 136
6.6.1 Grafting Bamboo (Aristotlelian science) to a Tree (Plato’s literature) . . . . . . 136
Ockham’s razor and Plato’s beard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Early modern – Scientiﬁc revolution: From Geocentrism to Heliocentrism
141
Paradigm shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Francis Bacon (1561 - 1626): The father of empiricism, Inductive reasoning . . . . . . . 145
7.2.1 How to create science: The exclusion of idols (=prejudice, preconception) . . . 145
From Geocentrism to Heliocentrism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.3.1 What is ”Geocentrism vs. Heliocentrism”? . . . . . . . . . . . . . . . . . . . . . 148
7.3.2 Somehow ”from Geocentrism to Heliocentrism” . . . . . . . . . . . . . . . . . . 149
7.3.3 ”Geocentrism vs. Heliocentrism” is the problem of the worldview . . . . . . . . 150
7.3.4 The Galileo legend; Leaning Tower of Pisa, Trial of Galileo . . . . . . . . . . . . 150
Principia; Newtonian worldview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.4.1 Principia (1687) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.4.2* What is ”Geocentrism vs. Heliocentrism”? After all, the worldviewism . . . . 152
Appendix; About ”Dialogues concerning two new sciences” by Galileo Galilei . . . . . 155
Modern philosophy (Dualism): Descartes
157
The theory of Ideas has expired . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158”I think, therefore I am” is meaningless from the scientiﬁc point of view . . . . . . . . 160
8.2.1 Discourse on the Method (1637) . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8.2.2* Two (scientiﬁc or non-scientiﬁc) interpretations of ”I think, therefore I am” . . 161
Descartes’ strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
8.3.1 From cogito to dualistic idealism . . . . . . . . . . . . . . . . . . . . . . . . . . 163
The correspondence of key-words in Descartes philosophy and quantum language . . . 166
Modern philosophy(Locke, Leibnitz, Berkeley, Hume)
169
Locke (1632 - 1704): The father of British Empiricism . . . . . . . . . . . . . . . . . . 170
9.1.1 ”An Essay Concerning Human Understanding” by Locke (1689) . . . . . . . . . 170
9.1.2 ”Tabula rasa”, primary quality and secondary quality . . . . . . . . . . . . . . . 171
Dramatic presentation of ”British Empiricism vs. Continental Rationalism” . . . . . . 173
9.2.1 Would Leibniz be serious for this argument (i.e., nativism) ? . . . . . . . . . . . 174* Leibniz-Clarke Correspondence: What is space-time?
. . . . . . . . . . . . . . . . . . 176
9.3.1 “What is space?” and “What is time?” . . . . . . . . . . . . . . . . . . . . . . . 177
9.3.2 Leibniz-Clarke Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
Subjective idealism: Berkeley, ”To be is to be perceived” . . . . . . . . . . . . . . . . . 181
9.4.1 Priest: Berkeley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
9.4.2 (A3 ): To be is to be perceived . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
9.4.3 ”Einstein-Tagore Meeting” and ”Bohr-Einstein debates” . . . . . . . . . . . . . 182
9.4.4 Bohr-Einstein debates: Do the laws of physics require measurement? . . . . . . 184
Qualia problem and Brain in a vat argument . . . . . . . . . . . . . . . . . . . . . . . . 185
9.5.1* The problem of qualia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
9.5.2* Brain in a vat argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Hume; skeptic who didn’t measure, ”A Treatise of Human Nature” . . . . . . . . . . . 189
9.6.1 The review of Descartes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

9.6.2 Hume’s straying Less than brain science ; Hume’s wordplay . . . . . . . . . . . 190
9.6.3 Hume; The causality problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
Hume’s problem of induction in the quantum mechanical worldview . . . . . . . . . . . 193
9.7.1* The solution of Hume’s problem of induction . . . . . . . . . . . . . . . . . . . 193* grue paradox cannot be represented in quantum language . . . . . . . . . . . . . . . . . 197
Kant: Copernican revolution
201
Critique of Pure Reason . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
10.1.1 Three Critiques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

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10.2
10.3

10.4

Chapter 11
11.1

11.2

11.3
11.4
11.5

11.6

Chapter 12
12.1

12.2

12.3

12.4
12.5
12.6

12.7

10.1.2 The purpose of ”Critique of Pure Reason” (1781) . . . . . . . . . . . . . . . . . 203
10.1.3 Thing-in-itself, Copernican revolution; from copy theory to constitution theory
204* Summary: What is causality? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
10.2.1 Four answers to ”what is causality?” . . . . . . . . . . . . . . . . . . . . . . . . 208
Summary; Descartes=Kant philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
10.3.1 Hume might be more scientiﬁc than Kant . . . . . . . . . . . . . . . . . . . . . 212
10.3.2 Why was Kant so successful? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
Is Kant a progress from Descartes ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
10.4.1 The inevitability of Kant’s appearance . . . . . . . . . . . . . . . . . . . . . . . 215
10.4.2 Is Kant a progress from Descartes ? . . . . . . . . . . . . . . . . . . . . . . . . . 216
Linguistic philosophy ( Before TLP)
221
Elementary knowledge of symbolic logic in mathematics . . . . . . . . . . . . . . . . . 223
11.1.1 Propositional logic and predicate logic . . . . . . . . . . . . . . . . . . . . . . . 223
11.1.2 Which do you trust more, logic or statistics? . . . . . . . . . . . . . . . . . . . . 225
Probabilistic symbolic logic in mathematics . . . . . . . . . . . . . . . . . . . . . . . . 226
11.2.1 Easy example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
11.2.2 Quasi-product probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
11.2.3 Logic symbols and logical operations . . . . . . . . . . . . . . . . . . . . . . . . 228
George Boole (1815-1864), Gottlob Frege (1848-1925) and Bertrand Russell (1872-1970) 232
Peirce (1839-1914): Abduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
11.4.1 * What is Peirce’s abduction? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
Bertrand Russell: ﬁve-minute hypothesis, McTaggart’s paradox, Moore’s paradox . . . 241
11.5.1 Russell’s paradox in set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
11.5.2 * Five-minute hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
11.5.3 * McTaggart’s paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
11.5.4 Moore’s paradox: ”It is raining, but I do not believe it is raining” . . . . . . . . 247
Saussure: Copernican revolution in language . . . . . . . . . . . . . . . . . . . . . . . . 249
11.6.1 Saussure’s linguistics: What comes ﬁrst, things or words? . . . . . . . . . . . . 249
11.6.2 Several Copernican revolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
11.6.3 * The quantitative expressions of ”signiﬁed” . . . . . . . . . . . . . . . . . . . . 252
Linguistic philosophy ( After TLP)255
* Fuzzy logic in QL; My scientiﬁc understanding of ”Tractatus Logico-Philosophicus(=TLP)”. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
12.1.1 Wittgenstein and Zadeh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
12.1.2 Easy example; classical system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
12.1.3 Fuzzy logic in QL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
12.1.4 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Wittgenstein; the biggest star of analytic philosophy. . . . . . . . . . . . . . . . . . . . 265
12.2.1 What did he wanted to insist on in TLP? . . . . . . . . . . . . . . . . . . . . . 265
12.2.2 The power of Wittgenstein’s word: Linguistic turn . . . . . . . . . . . . . . . . . 268
12.2.3 Philosophical Investigations (1953) and Wittgenstein’s paradox . . . . . . . . . 270
Quine’s analytic-synthetic distinction and Popper’s falsiﬁability in QL; . . . . . . . . . 272
12.3.1 Quine’s analytic-synthetic distinction in QL; . . . . . . . . . . . . . . . . . . . . 272
12.3.2 Popper’s falsiﬁability in QL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
Lewis Carroll’s logical paradox in fuzzy logic . . . . . . . . . . . . . . . . . . . . . . . 277
Flagpole problem in the quantum mechanical worldview . . . . . . . . . . . . . . . . . 279
12.5.1 * The quantum linguistic solution of Flagpole problem . . . . . . . . . . . . . . . 279* Hempel’s raven problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
12.6.1 Is ”the set of all tyrannosaurus” meaningful? : the set theoretical worldview . . 282
12.6.2 Hempel’s raven problem in the set theoretical worldview . . . . . . . . . . . . . 283
12.6.3 Hempel’s raven problem in the quantum mechanical worldview . . . . . . . . . 283
12.6.4 A priori proposition: ”Any small black bird is black” . . . . . . . . . . . . . . . 286
12.6.5 A posteriori proposition: ”Every raven is black” . . . . . . . . . . . . . . . . . . 287
Three approaches to the mind-body problem . . . . . . . . . . . . . . . . . . . . . . . . 289
12.7.1 The mind-body problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
12.7.2 The ﬁrst approach; Cognitive scientiﬁc approach . . . . . . . . . . . . . . . . . . 289

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Contents
12.7.3 The second approach; Illusory problem? . . . . . . . . . . . . . . . . . . . . . . 290
12.7.4 * The third approach; Quantum linguistic solution to the mind-body problem . 290
Chapter 13
13.1
13.2

13.3

Postscript: Can QL be post-analytic philosophy?
293
Philosophy (of worldviews) has progressed towards quantum language . . . . . . . . . 293
If the above (A) and (B) are true, almost all unsolved problems concerning dualistic
idealism have to be solved in QL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
13.2.1 The list of my answers of unsolved problems . . . . . . . . . . . . . . . . . . . . 294
Can QL be post-analytic philosophy? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

References

301

Index

305

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1

Chapter 1

The outline of quantum language(=measurement theory)
Abstract:
Recently I proposed ”quantum language” (or, ”measurement theory”, ”the linguistic
Copenhagen interpretation of quantum mechanics”, ”the quantum mechanical worldview”), which
is not only characterized as the metaphysical and the linguistic turn of quantum mechanics but
also as the linguistic turn of Descartes=Kant epistemology. And further, I believe that quantum
language is the only scientiﬁcally successful theory in dualistic idealism. Hence, this turn encourages
us to understand Western philosophies (Parmenides, Plato, Descartes, Locke, Berkeley, Hume, Kant,
Wittgenstein, etc.) within the framework of quantum language. This is done in this paper. Thatis, I show that most unsolved philosophical problems in dualism are solved within the framework ofquantum language.
Readers don’t need any prior knowledge to read this paper. It is suﬃcient to read the overview
of quantum language (and the linguistic Copenhagen interpretation) described in section 1.1 of thischapter. However, if you are a graduate student, I recommend you to read this paper and ”quantum
language (ref. [71])” together.
In Section 1.2, I remark that(♯1 ) roughly speaking, western philosophy and the linguistic Copenhagen interpretation are similar.
Further, in Section 1.3, I explain the worldview (= world description), which is classiﬁed by thefollowing four,(♯2 ) • the realistic worldview(≈ physics),• the ﬁctional worldview (≈ western philosophy),• the logical worldview (=the logical spirit=the spirit of ”Think logically!”) (≈ analytic, linguistic, scientiﬁc philosophy),• the mechanical worldview (≈ statistics, quantum language).
(cf. Assertion 0.1 in Preface).

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1.1 The overview of quantum language(=measurement theory)

1.1 The overview of quantum language(=measurement theory)
1.1.1 *

Axiom 1 (What is probability (or, measurement) ?) and Axiom 2 (What is
causality ?)The idea of quantum language (also known as ”measurement theory”) is inspired and structured
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 noted that quantum language is not almighty, forexample, the theory of relativity is beyond the description of quantum language.
The framework of measurement theory(=quantum language) is simple, that is, it is composedwith two axioms (Axiom 1 concerning measurement and Axiom 2 concerning causal relation) andthe manual to use Axioms 1 and 2 (called the linguistic Copenhagen interpretation). That is, (cf.
refs [46, 71] ),(=quantum language)[Axiom 2][Axiom 1]

measurement theory = measurement + causal relation+

linguistic Copenhagen interpretation(1.1)[the manual to use Axioms 1 and 2]

That is,Although you do not need to fully understand Axiom 1 and Axiom 2 to read this text, I, forcompleteness, mention them as follows. (For the mathematical foundations of Axioms 1 and 2, see
Section 1.5 (Appendix: the mathematical foundations of quantum language ) later. The following
axiom can be regarded as the mathematical generalization of the probabilistic interpretation ofquantum 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 1.1)
(This will be explained more precisely in Section 1.5 )With any system S, a basic structure [A ⊆ N ⊆ B(H)] can be associated in which measurement theory of that system can be formulated. When the observer takes a measurement of anobservable (or, by a measuring instrument) O=(X, F, F ) for a system S[ρ] i.e., a system S witha state ρ), the probability that a measured value x (∈ X) obtained by the measurement belongsto Ξ (∈ F) is given by ρ(F (Ξ))(≡ A∗ (ρ, F (Ξ))N ).

And, the following is a mathematical generalization of Heisenberg’s kinetic equation (≈Schrödinger equation).
Axiom 2 [causality] (in Section 1.1)
(This will be explained more precisely in Section 1.5 )Let T be a tree (i.e., semi-ordered tree structure). For each t(∈ T ), a basic structure [At ⊆Nt ⊆ B(Ht )] is associated. Then, the causal chain is represented by a sequential causal
operator {Φt1 ,t2 : Nt2 → Nt1 }(t1 ,t2 )∈T 2 .
≦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 1.1. If metaphysics did something wrong in the history of science, it is because metaphysics

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1.1 The overview of quantum language(=measurement theory)
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 alanguage to deal with the keywords. So we replace (♣1 ) by(♣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 historyof 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 Newtonian mechanics”. Recall that Newton
mechanics does not answer the question ”What are mass, force, acceleration?”, but ”How should theybe used?” or equivalently, ”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 ) making physics called Newtonian mechanics.
(♭2 ) proposing the worldview called Newtonian mechanics.
(♭3 ) making 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 the idealistic pointof view,(♯1 ) making a philosophy called □ □(♯2 ) proposing a worldview called □ □(♯3 ) making 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 1.1, the question: ”What is measurement, probability, causality?” should beinterpreted as the question: ”How should the terms: measurement, probability, causality be used?”.
Since measurement, probability, causality are key-words of quantum language (i.e., Axioms 1 and2), what is important is to use Axioms 1 and 2 appropriately. To this end, I recommend readers
to read ref. [71] and this text carefully.
♠Note 1.2. The above problem is the most diﬃcult unsolved problem in the history of philosophy. 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 success of Newtonian mechanics. Again, recall
that Newtonian mechanics never answer to ”What are forces, accelerations, and masses?”. What
Newtonian mechanics answers to is ”How are used the terms: forces, accelerations, and masses?”

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

Linguistic Copenhagen interpretation (or in short, Linguistic interpretation)

1.1.2

1.1.2.1 Descartes ﬁgure (the linguistic Copenhagen interpretation)
In the previous section, we introduced the outline of quantum language. Here, note that
(C1 ) the axiom is a kind of spells (i.e., incantation, magic words, metaphysical statement), so that,
it is impossible to verify it experimentally.
And thus, we say:(C2 ) Since quantum language is a language, it may be diﬃcult to use it well at ﬁrst. We need tomake practice, and will master it only by trial and error.
♠Note 1.3. In Mermin’s book [93], he said
• If I were forced to sum up in one sentence what the Copenhagen interpretation says to me, itwould be “Shut up and calculate”• Stop being bothered!
This means that ”What is the Copenhagen interpretation? is unsolved problem. Thus, I assert thatthe linguistic Copenhagen interpretation is the true Copenhagen interpretation (cf. ref. [71] ). Thatis, we assert that the Copenhagen interpretation is justiﬁed in philosophy (i.e., language) and not inphysics.

However,(C3 ) if we want to master quantum language as quick as possible, we will need a good manual to use theaxioms.
Here, we think that(C4 )the linguistic Copenhagen interpretation= manual to use spells (Axioms 1 and 2)

Since Axiom 1 includes the term ”measurement”, the concept of ”measurement” should be, at ﬁrst,
understood in dualism (i.e., ”observer” and ”measuring object”) as illustrated in Figure 1.2.

OO

Figure 1.2. [Descartes Figure]: Image of ”measurement(= a + b )” in dualism

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1.1 The overview of quantum language(=measurement theory)

In the ﬁgure, ”measurement” is characterized as interaction between ”observer” and ”system”(matter orobject to be measured, measuring object), composed of(D1 ) a projection of light onto the object (i.e., someone, not necessarily an observer, shines the light.)
b perception of the reaction of the object (i.e., the observer receives the reaction.)O
O

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, and that 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 paper.
♠Note 1.4. (i): The following argument, called the Bohr-Einstein debates, is extremely profound (cf.
Sec.9.4.3 Bohr-Einstein debates).
• For example, if the voltmeter needle points to 5V, but the observer doesn’t see it, it’s just aphysical phenomenon. Therefore, Axiom 1 (measurement) is not required. However, in the
dualistic position of introducing measurement into science, Axiom 1 is essential.

Einstein is the former position (i.e., monism) and Bohr is the latter (i.e., dualism). It’s the biggest
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 Assertion 0.1 in Preface,
they should be compatible.
(ii): 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 is also a type of measuring instrument.

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

1.1.2.2 The linguistic Copenhagen interpretation [ (E0 )-(E7 ) ]The linguistic Copenhagen interpretation is ”the manual to use Axiom 1 and 2”. Thus, there are various
explanations for the linguistic Copenhagen interpretations. However, it is usual to consider that the linguistic
Copenhagen interpretation is characterized by statements in Panel (E), among which the most important is(E4 ) Only one measurement is permitted.
(E):The linguistic Copenhagen interpretation

With Descartes ﬁgure 1.2, 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 be perceived.”(ii); There in no scientiﬁc truth without experiment (= measurement). [Popper’s Falsiﬁability; ref. [97]] In order to guarantee the objectivity of a scientiﬁc theory, there must be apossibility that the hypothesis will be disproved by experiment or observation. That is, truth
must always be subjected to experiments that deny its truth. And if the denying experiment7
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1.1 The overview of quantum language(=measurement theory)is conﬁrmed, then the truth must be rejected.
(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.

To be more speciﬁc, the words ”I”, ”Here”, ”Now” are forbidden . Hence, ”I think, therefore I am” is non-sense.
♠Note 1.5. 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 as a 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.
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 suchas ”now,” ”here,” and ”I” should not be used in a scientiﬁc proposition. If you are goingto use 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. McTaggart’s paradox,
Russell’s ﬁve-minute hypothesis).
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Chap. 1 The outline of quantum language (=measurement theory)(E3 ) In measurement theory, ”observable(=measuring instrument≈body)” is the most important
than ”measured value(≈mind)” and ”state(≈matter)” in (D3 ).
(E4 ) Only one measurement is permitted.

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

For the virtual wave function collapse, see:ref. [63] 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)♠Note 1.6. The Schrödinger cat is the most famous paradox in quantum mechanics. However,
we are not bothered by this paradox since the state after measurement is not described inquantum language.

(E5 ) There is no probability without measurement. Also, the measurement cannot be measured.

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1.1 The overview of quantum language(=measurement theory)
There is no logic without measurement. See Assertion 0.1 in Preface such as
⃝
13Analytic philosophy ( logic) −−−−−−−−−−−→ (See Chapter 11)  ⃝14−−→ QL (= Measurement theory)


⃝
8

statistics ( probability) −−−−−−−−−−−→(E6 ) State never moves. Thus, we always use the Heisenberg’s picture (and not the Schrödinger
picture)and so on.
Since it is believed thatO

quantum language is the ﬁnal goal of dualistic idealism(cf. 10 in Figure 0.1 in Assertion 0.1 (in Preface), it is deduced that
(E7 ) most maxims of the philosophers (particularly, the dualistic idealism) can be regarded as
expressions in linguistic Copenhagen interpretation.

For example, Locke’s ”Secondary quality”, Berkeley’s ”To be is to be perceived”, Kant’s ”Copernican revolution”, Wittgenstein’s ”The limits of my language mean the limits of my world”, ”What
we cannot speak about we must pass over in silence”, etc. teach us the spirit of quantum language.

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Chap. 1 The outline of quantum language (=measurement theory)
Some people may wonder (E7 ). However, note(F1 ) Descartes=Kant philosophy and quantum language have the same purpose of pursuing 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.
We assert the following.

Assertion 1.3. [descriptive power of quantum language](cf. ref. [71])Roughly speaking,• quantum language has a great descriptive power more than statistics.
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 mostsciences (e.g., economics, psychology, engineering, etc.). Thus, as a modiﬁed version of ”statistics is thelanguage 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, the theory of relativity
is beyond the description of quantum language.

♠Note 1.7. It is one of the roles of the linguistic Copenhagen interpretation to exclude a scientiﬁcallynonsense propositions from quantum language. For example,(♯1 ) Cogito proposition ”I think, therefore I am” is not the proposition within quantum language.
In cogito proposition, we see that ”observer”=”I” and ”object to be measured”=”I”, which is inconsistent with that the above (E1 ). Thus, cogito proposition is not a proposition in quantum language.
This will be discussed again in Proposition 8.3.
Next,(♯2 ) The hypothesis that the world was created ﬁve minutes ago (due to B. Russell in Sec.11.4)

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1.1 The overview of quantum language(=measurement theory)is not within quantum language.
That is because this hypothesis is considered under the premise such that the observer’s time, which
is prohibited by (E2 ).

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

1.2 The history of worldview and our purpose
1.2.1 Quantum language in the history of worldviews
We assert that, in the history of worldview, quantum language is located as follows.

Assertion 1.4. (= Assertion 0.1 in Preface) [The location of quantum language in the history of worlddescription (cf. ref.[47, 71]) ]the realistic worldview (physics): monism−−→ Newton O(monism)O

relativity−→ theory−−−−−−−−−−→ 32(realism)

Socrates
Aristotle

O:Greek0

Schola-O−−−−−→ 1

philosophy
Parmenides, Zeno
Plato

sticismOquantum−→ mechanics −−−−−−−−→ 4









O
5

−−→

O

O
O

O

OO

(unsolved)

TOE:theory of
everything(quantum phys.)language−−−−−−−−−−−−−−−−−
→7 




dualism


8
−
−
−
−
−
−
−
−
→

−−−−−−−−−→ 6 statistics

quantization
applied math.
system theoryclassical mechanical worldview



(=MT)
(mind-matter dualism) (Copernican revolution)) 14 QL:quantum
DescartesKant
9mechanicallanguage
10
Locke,...
−−→−−→−−−−−−−→  −−→turn
mechanicalquantum
ﬁctional
worldview

ﬁctional worldview
worldview


11



turn
linguistic





?
(logic in philo.))(logic, set theory)Wittgensteindualism
Saussurre
Boole, Frege, 12−−−−−−−−−→ 13
measurement
Russell,Cantor −−→ Zadeh

O

O

O

math.

logical worldview

the idealistic worldview (no experiment, metaphysics): dualism
Figure 0: The history of the world-descriptions
Philosophy (≈ dualistic idealism) has progressed toward QL( i.e., 0 → 1 → 9 → 11 → 13 → 14 )O O O O O O

Therefore, quantum language has the following four aspects:
The four aspects of quantum language
7 : the linguistic turn of quantum mechanics
(i.e., the true color of the Copenhagen interpretation)

8 : the dualistic reconstruction of statistics=
10 : the ﬁnal goal of dualistic idealism
(i.e., the linguistic and mechanical turn of the Descartes=Kant epistemology)
13 : the measurement theoretical understanding of Wittgenstein’s picture theory

O
O
O
O

13

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1.2 The history of worldview and our purpose♠Note 1.8. I cannot answer the question: ”What is quantum mechanics in physics?”, since this isrelated to ⃝5 in the above ﬁgure. However, I am convinced that our interpretation (i.e., the linguistic
Copenhagen interpretation ) is the true color of the Copenhagen interpretation.

1.2.2 Our purpose
Our purpose is to answer the problem ”Has western philosophy made progress?”, and to conclude thefollowing Assertion 1.5.
In this paper I assert thatHas western philosophy (particularly, dualistic idealism) made progress?
(A) If ”to make progress” is deﬁned by ”toAssertion 1.5.
 come near quantum language”, we
can say that the time series ⃝→
⃝→
⃝→
⃝→
⃝⃝
0
1
9
11
13 14 can be regarded as progress, or more
precisely,
Cogito turn (dualism)

Plato −−−−−−−−−−−−−−−−→
progressmind-matter dualism
transcendental idealism
Copernican revolution
Kant
Descartes=Locke −−−−−−−−−−−−−−−−→
progresslinguistic philosophy
linguistic turn−−−−−−−−−−→
progress

Wittgensteinmechanical turn (scientiﬁc turn)−−−−−−−−−−−−−−−−−−−−−−−−→
progress

the quantum mechanical worldview
Quantum language
quantitative argument is possible(1.3)

The name of ”scientiﬁc turn” is due to the fact that quantitative argument is possible in quantum
language and that it is only quantum language which scientists need in above-mentioned. Thus, weconclude that(B) a scientiﬁc perfection of dualism and idealisma is realized by quantum language.

a Throughout this paper, it suﬃces to consider that ”idealistic”≈”linguistic”≈”metaphysical” (cf. Deﬁ-nition 1.8)

It is sure:(C) if someonebelieves that he/she ﬁnds
13 he/she certainly wants to talk about the
 the theory beyond ⃝,
⃝
⃝
⃝
⃝
seen
from
the
theory.
landscape ⃝
0
1
9
11
13
This will be done in this paper.
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 (B)[a scientiﬁc perfection of dualism andidealism is realized by quantum language] is true, we can expect that most unsolved problems can be solvedin the framework of quantum language (cf. (D) in Preface). In fact, in this paper we show the solutions of

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Chap. 1 The outline of quantum language (=measurement theory)the following problems
List 1.6. It is said that
• Wittgenstein carried out one of the most ambitious attempts in the history of philosophy to solve
all the problems of philosophy at once in [117]: ”Tractatus Logico-Philosophicus (abbreviated as
TLP)”.
However, 100 years later, we can’t say that he succeeded in his challenge. However, as the following listshows, QL realized his dream.
(D1 )Almost famous unsolved problems concerning dualistic
idealism can be solved in QL• What is probability (or, measurement, causality) ? cf. Sec. 1.1.1)• The solution of Zeno paradox (Flying arrow, Achilles and a tortoise), (cf. ref. [45], or, Sec. 2.4.2,
Sec. 2.4.3)( ”to solve Zeno paradox” = ”to understand DST-formula” (cf. Sec. 2.4.3 )• the measurement theoretical understanding of Plato’s allegory of the sum , (cf. Sec. 3.3.2)• Plato’s Idea theory≈Zadeh’s fuzzy theory≈Sausuure’s linguistic theory (cf. Sec. 3.5.3)• 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)• Only the present exists (cf. Sec. 6.1.2)• What is the problem of universals? (cf. Sec. 6.5.1)• What is Geocentrism vs. Heliocentrism? After all, the worldviewism (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. 9.3)
• The problem of qualia (cf. Sec. 9.5.1)• Brain in a vat argument (cf. Sec. 9.5.2)• The solution of Hume’s problem of induction (cf. Sec. 9.7.1)• grue paradox cannot be represented in quantum language (cf. Sec. 9.8)
• What is causality? (cf. Sec. 10.2)• 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)• My scientiﬁc understanding of ”Tractatus Logico-Philosophicus (=TLP)” and ”Zadeh’s fuzzy sets”(cf. ref. [78], 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 picture theory must belong to

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1.2 The history of worldview and our purpose
dualistic idealism in order to assert 1.4 (= Assertion 1.5)
• Lewis Carroll’s paradox (cf. Sec. 12.4)• Flagpole problem, (cf. Sec. 12.5)• Hempel’s raven paradox (cf. Sec. 12.6)• the mind-body problem (i.e., How are mind and body connected?), (cf. Sec. 12.7)
Also, for the solutions of unsolved problems in quantum mechanics, statistical mechanics, statistics andprobability theory, see ref. [71]), that is,(A1 ) ref. [71]: 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)
The list will be written is in (D2 ) in Chap. 13: ”Postscript.”
Since most unsolved problems are easily solved in quantum language, we can be convinced thatquantum language is the ﬁnal goal of the dualistic idealism[⃝
0 - ⃝
1 - ⃝
9 - ⃝
10 ]
♠Note 1.9. It should be noted that Einstein’s success is due to the proposal of ”the language called
the theory of relativity”. On the other hand, I think that(♯4 ) philosophers failed because they did not propose a suitable language.
Talking cynically, we say that(♯5 ) Philosophers has investigated ”linguistic Copenhagen interpretation” (=”how to use Axioms 1
and 2”) without language (i.e., Axiom 1 (measurement) and Axiom 2 (causality)).
Therefore, in most cases many philosophers wander. However, great philosophers rarely miss thepoint. For example, Wittgenstein could only incompletely propose his language, but he left the
maxim such that(♯2 ) ”The limits of my language mean the limits of my world”
which is just the spirit of quantum language (as mentioned in Chapters 11 and 12).

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

1.3 Realistic worldview and idealistic worldview
1.3.1 The worldviewism
The worldviewism isthe spirit that ”Start from worldview!”.
That is,(A): The worldviewism
The worldviewism has the following form:
world is so(A1 ) worldview
premiseconclusion
therefore−−−−−−→ practical logic (≈ inference, how to think, calculation), discussions, etc.
subject

That is,(A2 ) Since the world is like this, think like this!
That is, the worldviewism is the spirit such that ”Start from worldview!”. I think that
(A3 ) It’s not an exaggeration to say ”philosophy = worldviewism”.
[Remark] This is not trivial. That is because the above (A) says that the worldview is greater thanlogic. That is,”logic” (precisely, ”practical logic”) is created by a worldview.
(cf. Section 2.4.1 [ Zeno’s paradoxes], Section 4.3.2 [Aristotle’s syllogism] etc.).
Arguing repeatedly in this paper, we consider that the ignoring of the worldviewism (or, relying ona bad worldview ) causes that philosophy falls into a blind alley. Slogan-wise, we say(A4 ) Without a worldview, there is no logic (i.e., no calculation)(A5 ) Without a worldview, we can’t have any discussion.
(A6 ) Paradoxes arise from a lack of worldview.
• ······♠Note 1.10. Without a worldview, we have paradoxes as follows.
• Zeno’s paradoxes ( cf. Sec. 2.4.2)• The solution of Hume’s problem of induction (cf. Sec. 9.7.1)• grue paradox cannot be represented in quantum language (cf. Sec. 9.8)
• Lewis Carroll’s paradox (cf. Sec. 12.4)• Wittgenstein’s paradox (cf. Sec. 12.2.3)
As mentioned in Chapter 11 later, I think what makes analytic philosophy useless is that analytic
philosophy does not follow worldviewism, but uses ”logic”.
////
The following questions often appear in philosophy:
Does time exist?

Does I exist?

Does position exist?

Does velocity exist?

Does beauty exist?

Does mathematics exist?

Does a particle exist?

Does motion exist?

Does God exist?
However, these statements are ambiguous since the term ”exist” (or ”to be”) is not determined. Thus I
think as follows.
Deﬁnition 1.7. [”existence” (i.e., ”to be”]
Fix a worldview. Then, we say ”[X] exists”, if [X] is a keyword in the worldview (or more generally, if [X]17

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1.3 Realistic worldview and idealistic worldview
is a thing that can be expressed using these keywords)
Remark: For example, in Newtonian mechanics, ”velocity”, ”acceleration”, ”force”, etc. exist. Also, if
we see Christianity as a kind of worldview, we say that God exists. In the cases of QL, the followings aremost essential:

measured value,observable,

state,

causal map

We begin with the following deﬁnition:
Deﬁnition 1.8.
[”realism”, ”idealism”],[ ”monism” , ”dualism”](B) realism := a worldview (or, theory) that requires experimental veriﬁcation.
idealism := a worldview (or, theory) that does not necessarily require experimental veriﬁcation.
(”idealism” is also called ”linguistic theory”, ”non-realism” or ”metaphysics” in this paper)(C) monism := a worldview about objects
dualism := a worldview about humans and objects, (i.e., a theory with ”observers” and ”measuring
objects”)
Though the above is quite rough, it is suﬃcient to read this paper. I think that there is no strict deﬁnitionin philosophy (not mathematics). And the more rigorous we try to say it, the more imprecise it becomes.
♠Note 1.11. The rule of philosophy (≈ idealism) is as follows:(♯1 ) Only discussion, no experimentation.
(♯2 ) the winner will be determined by popularity vote of the general public.

As seen in Assertion 1.4, we have the following classiﬁcation of philosophies:
Classiﬁcation 1.9. [(D): Classiﬁcation of worldviews]
 (♭1 ) : the realistic worldview : Aristotle, Archimedes, Galileo, Newton, Einstein(monism)









(♭21 ): the ﬁctional worldview (≈ western philosophy)








Plato, Descartes, Locke, Kant









(♭
):
the
logical worldview (=the logical spirit =the spirit

22




of
”Thinklogically!”) (≈ analytic, scientiﬁc phil.)
(♭
)
:
the
idealistic
worldview2
(dualism)Frege,
Saussure, Russell, Wittgenstein...








(♭
):
the
mechanicalworldview

23




statistics, quantum language
Throughout this paper, we explain that
cogitoCopernican(♭21 ): Plato −−−−→ (♭21 ): Descartes −−−−−−−−→ (♭21 ): Kant



turn
revolution






linguistic
mechanical−−−−−−→ (♭22 ): Wittgenstein −−−−−−−→ (♭23 ): QL



turn
turn







(♭3 )Others (Thinking Tip, etc.) : Darwin’s theory of evolution, Hegel’s dialectic, etc.






(♭4 )mathematics
Particularly, we devote ourselves to (♭2 ), i.e.,

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Chap. 1 The outline of quantum language (=measurement theory)////
I think mathematical logic in (♭22 ) should be classiﬁed in (♭4 ) , but for the sake of convenience, in thistext I sometimes think that mathematical logic (or, set theory) belongs to (♭22 ). This is not generally
a mistake. This is because, for example, when we use the word ”set” in everyday language (or, in thenaive set theory (cf. [29])), it is naturally interpreted as ”a collection of things”. It can be diﬃcult todistinguish between non-mathematical concepts and mathematical concepts when mathematical concepts
are naturally embedded in natural language, as in ”two apples and three apples together make ﬁve apples.”This must always be kept in mind.
♠Note 1.12. Lord Kelvin (1824–1907) said(♯) Mathematics is the only good metaphysics.
The (♭2 ), (♭3 ) and (♭4 ) are metaphysics, as they does not require experimental veriﬁcation. For
example, note that Darwin’s theory of evolution has not been settled by experimental veriﬁcation.
Thus, if we believe the above (♯), we must consider (♭2 ), (♭3 ) and (♭4 ) as bad theories. However, I
think that at least quantum language (which is not mathematics) is a good metaphysics. Hence, I
think Kelvin is wrong.

♠Note 1.13. For example, it is certain that(♯1 ) there is no physics without the world
However,(♯2 ) mathematics itself is not related to the world. In an extreme case, there may exist mathematics
without the world
Therefore, it is natural to consider that mathematics is not a kind of worldview. Truth be told, I
think(♯3 ) mathematical theory (including logic, set theory, category theory) has nothing to do with worldview.
However, for example, consider the term: ”set”, which is the term in ordinary language as well as
mathematics. Thus, some may comes up with the idea (which may be called the set theoreticalworldview) of describing the world through the naive set theory (and not the axiomatic set theory).
This set theoretical worldview is rather general in science. In Chap. 12, we will see that thisworldview triggers the ﬂagpole problem and Hemple’s raven paradox.
I also think of symbolic logic as a tool for constructing mathematics, but some people may have the idea

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1.3 Realistic worldview and idealistic worldview(which may be called the logical worldview (=the logical spirit =the spirit of ”Think logically!”)) thatmodal logic can be used to describe the world. Thus, the relationship between logic and philosophy
varies somewhat from researcher to researcher. Also, it is sure that some philosophers consider that
logic is one of ﬁeld of philosophy. Therefore, in spite that I believe in (♯3 ), in this paper,(♯4 ) logic (or, set theory) is regarded as a worldview called the logical worldview (=thelogical spirit=the spirit of ”Think logically!”).
However, I think the chances of the logical worldview succeeding are very small. In fact, in Chapters
11 and 12. we will see that the logical worldview (=the logical spirit=the spirit of ”Think logically!”)
was not very successful.
////More precisely, let us examine the above (D):(♭1 ) the realistic worldview (= physics)
The world is ruled by Newtonian mechanics

worldview

Heliocentrism is true
therefore−−−−−−→ discussions, calculation, practical logic, properties

premise,law
therefore

discussion

where ”−−−−−−→” is reliableand(♭21 ) the ﬁctional worldview (≈ western philosophy)
don’t steal a glance

invisible man exists
thereforeworldview (literary truth, pseudo- truth) −−−−−−→ ethics, practical logic, discussions
main subject(ﬁctional)premise
therefore

where ”−−−−−−→” is not necessarily reliable
and(♭22 ) the logical worldview (=the logical spirit=the spirit of ”Think logically!”) (≈ Analytic
philosophy)
Describe the world by symbolic logic, set theory

worldview (symbol logic, set theory)

therefore

A → B, B → C, therefore, A → C−−−−−−→ calculation, discussions, properties
theorem

Axiomatic logic
therefore

where ”−−−−−−→” is not necessarily reliable
and(♭23 ) the mechanical worldview (= the classical and quantum mechanical worldview)
Describe the world by (mechanical) language

If an apple falls, it has a cause.
therefore−−−−−−→ calculation, practical logic,properties

worldview
language inspired by mechanics
therefore

discussions

where ”−−−−−−→” is reliable20

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

1.3.2 The realistic worldview(physics) (♭1 )(E): The realistic worldview (physics) (♭1 )The realistic worldview has the following form:
Such phenomena happen

world is sotherefore

realistic worldview −−−−−−→ scientiﬁc properties · calculation

(E0 )

law
therefore

discussion

where ”−−−−−−→” is reliable
Hence, (E1 ) world is so is a law, truth, (which should be veriﬁed by experiment),(E2 ) such phenomena happen is a consequence

As examples of the realistic worldview, we see that
Newtonian mechanics, electromagnetism, theory of relativity, etc.
For example,
Heliocentrism is true (cf. Chap. 7)

Newtonian mechanics(F)

worldview
premise · law

therefore−−−−−−→ discussions, calculation, practical logic, properties
discussion

Note that the realistic worldview (=physics) is the most authorized.

1.3.3 The idealistic worldview (♭2 )In what follows, consider the idealistic worldview (♭2 ) in

1.3.3.1 The ﬁctional worldview (Wester philosophy) (♭21 )Our main theme of the preprint is the following ﬁctional worldview:

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1.3 Realistic worldview and idealistic worldview(G): The ﬁctional worldview (western philosophy) (♭21 )The ﬁctional worldview ( in Plato’s way of telling philosophy) is as follows.
you should do soworld is so(G0 )ﬁctional worldview (literary truth)

therefore−−−−−−→ practical logic, ethics, morals
main subject

preface, introduction, (ﬁctional)premise, expedient
therefore

where ”−−−−−−→” is not necessarily reliable
Therefore,(G1 ) world is so is secondary,(G2 ) you should do so is main theme

For example,
don’t steal a glance

invisible man exists
therefore(H1 ) ﬁctional worldview (literary truth, pseudo- truth) −−−−−−→ discussions, properties
main subject(ﬁctional)premiseandultimate beauty exists

she is an embodiment of beauty.
therefore(H2 ) ﬁctional worldview (literary truth, pseudo- truth) −−−−−−→ ethics, morals, aesthetic sense
main assertion

premise,ﬁction

The ﬁctional worldview forms the main current of western philosophy such as
Plato, Descartes, Locke, Hume, Kant,
etc.

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

1.3.3.2 the logical worldview (=the logical spirit=the spirit of ”Think logically!”) (analytic philosophy)(♭22 )(I1 ): the logical worldview (=the spirit of ”Think logically!”) (axiomatic logic) (♭22 )As mentioned in Note 1.13, I think:(I1 ) the logical worldview (=the logical spirit=the spirit of ”Think logically!”) (or, set
theoretical worldview)
Describe the world by symbolic logic, set theory

worldview (symbol logic, set theory)

understandable propositions (cf. Remark 1.10)
therefore−−−−−−→

scientiﬁc discussions
science, science philosophy

Axiomatic logic
therefore

where ”−−−−−−→” is not necessarily reliable
Main philosophers (mentioned in this paper) are as follows.
Boole, Frege, Peirce, Russell, Wittgenstein, Hempel, Quine, Popper, Zadeh,...
However, concerning Boole, Frege, Russell, I can only understand their mathematics, but not theirphilosophy, since I think that the logical worldview (=the logical spirit=the spirit of ”Think logically!”)
is not true worldview.

In ref. [117]: ”Tractatus Logico-philosophicus”, Wittgenstein opposed the straightforward introductionof mathematical logic into philosophy, but it was incomplete. It is my opinion that his spirit has cometo fruition as fuzzy logic in QL (cf. Sec. 12.1).

Remark 1.10. The above (I1 ) is motivated by the following:
Describe mathematics by logic, set theory(I2 )axiomatic logic, set theory

therefore−−−−−−→ mathematics
theorem

Axiomstherefore

where ”−−−−−−→” is completely reliableThis is the greatest achievement of mankind, which was completed by Cantor, Zermelo=Fraenkel, etc. Thus,
many philosophers might consider that(J) (I2 ) is the greatest, hence, (I1 ) is also promising.
This optimistic outlook, I believe, prompted the birth of analytic philosophy (Frege, Russell, etc.). To answer”Why does logic work in our world?”, The proposal of fuzzy logic in QL was indispensable (cf. Sec. 12.1).
♠Note 1.14.

Dr. Hawking said in his best seller book [27]:• However, in the nineteenth and twentieth centuries, science became too technical and mathemat-

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1.3 Realistic worldview and idealistic worldview
ical for the philosophers, or anyone else except a few specialists. Philosophers reduced the scopeof their inquiries so much that Wittgenstein the most famous philosopher this century, said ”Thesole 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 him. And thus, I may not appreciate analytic philosophy. In this paper, I think that oneof the main purposes of western philosophy (from Plato to Kant) is to clarify dualistic idealism. To
be honest, I have no clear answer on ”What is the purpose of analytic philosophy?” However analytic
philosophy (or, linguistic philosophy) is important since it bridges between the ﬁctional worldview
and the quantum mechanical worldview. That is,linguistic philosophy
the ﬁctional worldview
linguistic turn
Kant: (dualism)−−−−−−−−−→ Saussure, Wittgensteintranscendental idealism
the quantum mechanical worldview
mechanical turn−−−−−−−−−−−→ Quantum language: (dualism)
linguistic idealism(1.4)♠Note1.15. Readers should not confuse two logics (i.e., axiomatic logic and practical logic) such that
(K1 ): axiomatic logic (i.e., mathematical logic) in (I 1 ) or (J1 )
propositional logic, predicate logic, modal logic, temporal logic,
intuitionistic logic, quantum logic, etc.
(K)(K2 ): practical logic in (A) (i.e., logic generated from worldview, or, inference)
e.g., the logic of Newton mechanics, statistical inference in statistics, etc.
Recall the worldviewism (A), i.e.,
conclusion

world is so(A1 ) (=(A))

thereforeworldview −−−−−−→ discussions, calculation (= practical logic), properties
premisesubject

That is, ”logic” (precisely, ”practical logic”) is created by a worldview. As mentioned in(A), we see:
• Newtonian mechanics has the logic speciﬁc to Newtonian mechanics.
• Statistics has the logic speciﬁc to statistics (i.e., statistical inference).
• quantum language has the logic speciﬁc to quantum language.
• ······
Slogan-wise, we say• Without a worldview, there is no logic• Without a worldview, there is no discussion• ······
I have an opinion that philosophers should be more interested in practical logic than in mathematical
logic. Also,(L) I don’t think the mathematical achievements of the logical worldview (=the logical spirit=thespirit of ”Think logically!”) (e.g., modal logic, temporal logic, etc.) were very successful asphilosophy.
That is because these were powerless in solving unsolved philosophical problems, compared to quantum
language (cf. List 1.6).

1.3.3.3 The mechanical worldview (statistics, quantum language) (♭23 )In the era after Kant, disciplines (i.e., mathematics, physics, medicine, chemistry, biology, economics, psychology, etc.) were born one after another, and the weight of philosophy became smaller in the wholediscipline. Therefore, philosophy is no longer regarded as the king of academia. But that doesn’t mean thatphilosophy has run out of material. I think there is still a lot of work to be done on the scientiﬁcation of”dualistic idealism.”24

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Chap. 1 The outline of quantum language (=measurement theory)(M): The mechanical worldview (Statistics, quantum language) (♭23 )The mechanical worldview is as follows.
language is proposed like mechanics(M)

describe phenomena by the language
therefore(M1 ): mechanical worldview −−−−−−→ (M2 ): calculation, scientiﬁc properties
language, words
therefore

discussions

where ”−−−−−−→” is reliable
Therefore,(M1 ) language is proposed , that is, ”establish the verbal system”(M2 ) describe phenomena by the mechanical language
The key words are ”measurement” and ”causality” as follows.

For example, it suﬃces to consider statistics (and moreover, quantum language). Also, recall that(M3 ) statistics (= classical mechanical worldview)
statistical arguments
therefore−−−−−−→ practical logic (i.e., calculation, inference), control
discussions

It was such a huge success that it has been said:(M4 ) statistics is the language of scienceAlso,(M5 ) quantum language (= quantum mechanical worldview)
statistical arguments
therefore−−−−−−→ practical logic (i.e., calculation, inference), control
discussions
therefore−−−−−−→ Almost philosophical problems (in dualistic idealism) can be solved(M6 ) QL can explain ”Why does logic work in our world?” (cf. Sec. 12.1)
Assertion 1.3 says that, for example,• Economics is to describe economical phenomena in terms of quantum language (or statistics)• psychology is to describe psychological phenomena in terms of quantum language (or statistics)

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1.4 Preview of the assertions in this paper

1.4 Preview of the assertions in this paper
1.4.1 Realistic worldview?, (ﬁctional, mechanical) worldview?
Classiﬁcation 1.11. [= Classiﬁcation 1.9: the realistic worldview, the ﬁctional worldview,the logical worldview (=the logical spirit=the spirit of ”Think logically!”), the mechanicalworldview] ](♭1 ) : the realistic worldview : Aristotle, Archimedes, Galileo, Newton, Einstein(monism)









(♭21 ): the ﬁctional worldview (≈ western philosophy)








Plato, Descartes, Locke, Kant







 (♭22 ): the logical worldview (=the logical spirit=the spirit of






”Think logically!”) (≈ analytic, scientiﬁc phil.)(♭2 ) : the idealistic worldview(dualism)Frege, Saussure, Russell, Wittgenstein...








(♭
):
the
mechanical worldview

23




statistics, quantum language
Throughout this paper, we explain that
cogitoCopernican(♭21 ): Plato −−−−→ (♭21 ): Descartes −−−−−−−−→ (♭21 ): Kant



turn
revolution






linguistic
mechanical−−−−−−→ (♭22 ): Wittgenstein −−−−−−−→ (♭23 ): QL



turn
turn







(♭3 )Others (Thinking Tip, etc.) : Darwin’s theory of evolution, Hegel’s dialectic, etc.






(♭4 )mathematics
Particularly, we devote ourselves to (♭2 ), i.e.,////
I think mathematical logic in (♭22 ) should be classiﬁed in (♭4 ) , but for the sake of convenience, in thistext I may think of a ﬁeld of mathematics as a kind of worldview. That is because, compared to people inscience, philosophers like the word ”logic”.

Assertion 1.12. [Dispute: realistic worldview vs. idealistic worldview]”realistic worldview? idealistic worldview?” is the biggest dispute in the history of philosophy. However,from our view-point, the two have to exist together as follows.
Table 1.1 : realistic worldview vs. idealistic worldview

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Chap. 1 The outline of quantum language (=measurement theory)
dispute ⧹ [R] vs. [L]

Realistic worldview

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

Hērakleitos
Aristotle”Nominalismus”(Ockham)
NewtonEinstein
CarnapParmenides
Plato”Realismus”(Anselmus)
Leibniz
Bohr
Quine

⃝:
a motion⃝:Ancient
Greeceb⃝:c Problem of universals⃝:
d space-time⃝:
e quantum theory⃝:philosophyof sciencef⃝a is my ﬁction, ⃝c is a confusion. ⃝d is the Leibniz=Clarke correspondence (cf. Sec. 9.3.2), ⃝e is BohrEinstein debates. Quantum language is proposed as one of answers to Bohr-Einstein debates(cf. ref.
[71]). ⃝:f Quine understood 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. 13.3).

1.4.2 Keywords: Monism and dualism
Recall Figure 1.2 (in Section 1.1.2: The linguistic Copenhagen interpretation) below:

OO

Figure 1.13. (Figure 1.2 in Section 1.1.2) [Descartes Figure]: Image of ”measurement(= a + b )” in
dualism:
anddualismmonismdualismmonismmeasuring instrument

observer(I(=mind))•(body, eye, etc.)[observable][measured value]
a project lightO
Ob perceive reaction

system, hyle, point mass
(matter, measuring object)[state]- [eidos]

Assertion 1.14. [The correspondence of key-words]
It is a matter of course that each worldview has the corresponding key-words. If Western philosophy (i.e.,
worldview ) makes progress, its key-words are naturally reﬁned and clariﬁed. The key-word’s progress of
the realistic worldview [resp. idealistic worldview] is written as follows.
[The key-words of the dualism] The idealistic worldview is the mind-matter dualism,
which is composed of three key-words, that is, [A](= mind), [C](= matter) and [B](=body: something connecting [A:mind] and [C:matter]). Thus, we see that:27

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1.4 Preview of the assertions in this paper\[A](= mind)[B](Mediating of A and C)

Platoactual world

Idea

Thomas Aquinas

universale post rem

universale ante rem

DescartesI, mind, brain

body

Locke

mind

Berkeley
Kant
Saussure
Zadeh/[Idea world]/[universale in re]/[matter]primary quality[matter]/[God]/[thing-in-itself]

secondary quality
secondary quality

mind
phenomenon

cognition/

signiﬁed [signiﬁer]
fuzzy set (= membership function)

Wittgenstein

truth value

statistics

sample ( space )quantum mechanics

measured value

quantum language

measured value[C](= matter)/logical space[object]
parameter[population]
state[particle]
state[system][proposition]/[ trial]observable[ measurement]
observable[ measurement][The key-words of the monism] The realistic worldview is monism, and its completed version is realized
as Newtonian mechanics, whose key-words are only ”point mass” and ”state”. Thus, we see:\[A](= mind)[B](Mediating of A and C)

Aristotle

nothing

nothing

Newtonnothing

nothing[C](= matter)eidos[hyle]
state[point mass]

That is, we consider the following progress:
quantiﬁcation[eidos] −−−−−−−−−→[state],[hyle] −−−−−→[point mass]

progress

progress♠Note 1.16. (i): It is somewhat unreasonable to regard Plato’s Idea theory as a type of measurement
theory. Therefore, Plato and Aquinas above are doubtful, but I dare to write (cf. Sec. 3.3).
(ii): In mind-matter dualism, [B: medium] is the most important (cf. the linguistic Copenhagen
interpretation (E3 ) in Sec. 1.1.2). Thus, we consider that Plato’s Idea theory is dualism. On theother hand, statistics lacks [B: medium]. Thus, statistics is not usually regarded as dualism but
mathematical theory. However, in this paper, statistics is listed up as ”incomplete dualism” in theabove table.
(iii): The readers may wonder that ”actual world≈mind(=human)” in Plato. However, it should
be understood under the maxim: ”Man is the measure of all things”. Similarly, we think that”measured value≈mind(=human)” in quantum language. That is because there is no ”measured
value” without ”mind(=brain)”.
(iv): In [C: matter], the terms ”state” and ”system” in quantum language are always used as the form
”the system with the state” (cf. Axiom 1 in Sec. 1.1). In the history of western philosophy, ”state”
and ”system” were often confused, and in most cases, only one of the two has been found. Also,seeing statistics in the above table, the reader may ﬁnd that understanding the diﬀerence between”parameter” and ”population” is surprisingly diﬃcult. And thus, in this paper, the diﬀerence is not

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Chap. 1 The outline of quantum language (=measurement theory)
emphasized.
Corollary 1.15. Of the three (i.e., ”measured value”, ”observable”, ”state”), the ”observable” is the mostdiﬃcult to understand. Thus, Let me summarize the history of ”observable” as follows.

•
linguistic version−−−−−−−−−−−→

Idea
Plato’s idea theory

signiﬁer, signﬁedSaussure’s linguistic theory

body, secondary quality

membership function

Descartes, J. Locke

Zadeh’s fuzzy theoryquantiﬁcationycognitiveyversionj
yidentiﬁcation

observable (= measuring instrument)

quantiﬁcation

quantum language

Of course, Plato’s Idea can be interpreted in a variety of ways. Therefore, the above ﬁgure may be mydesire.

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1.5 Appendix: The mathematical foundations of quantum language

1.5

Appendix: The mathematical foundations of quantum language

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 the mathematical part of this text. For the
precise explanation, see [71]. If you want to read this book as literary as that refs. [23, 108], it might berecommended to skip this section. Literary enjoyment is also one of the charms of philosophy. 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). We have two formulations of QL such that(A1 ): the C ∗ -algebraic formulation ( see Sec. 1.5.1)
(A) the formulation of QL(A2 ): the W ∗ -algebraic formulation( see, for example, Section 2 in [76], or precisely [71])(A1 ) is simple, On the other hand, (A2 ) is rather mathematical. In this text, we usually use the C ∗ -algebraic
formulation (A1 ). But (A2 ) will be used in the case that ”exact measurement” is needed.

1.5.1 Mathematical preparations (the C ∗ -algebraic formulation of QL)
Following refs. [44, 46, 47, 71] (all our results until present are summarized in ref. [71]), we will review
quantum language, which has the following form:
Quantum language = measurement + causality + linguistic (Copenhagen) interpretation(= measurement theory)(Axiom 1)(Axiom 2)

(how to use Axioms 1 and 2)


(1.5)

which asserts that ”measurement” and ”causality” are the most important concepts in science.
Consider an operator algebra B(H) (i.e., an operator algebra composed of all bounded linear operators on a
Hilbert space H with the norm ∥G∥B(H) = sup∥u∥H =1 ∥Gu∥H . Let A(⊆ B(H)) is a C ∗ -algebra, (i.e., normclosed subalgebra of B(H)) (cf. refs. [113], [110], [120]). Our purpose of this section is not to explain QL ingeneral situation but to explain QL in an understandable setting. Thus, from here, we devote ourselves tothe 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
Let A ⊆ B(H)) be a C ∗ -algebra, and let A∗ be the dual Banach space of A.
That is, A∗ = {ρ | ρ is∗a continuous linear functional on A }, and the norm ∥ρ∥A is deﬁned by sup{|ρ(G)(= A∗ ⟨ρ, G⟩A )| | G ∈A such that ∥G∥A (= ∥G∥B(H) ) ≤ 1}. Deﬁne the mixed state ρ (∈ A∗ ) such that ∥ρ∥A∗ = 1 and ρ(L) ≥ 0 for
all L ∈ A such that 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 if it satisﬁes that “ρ = θρ1 + (1 − θ)ρ2 for some ρ1 , ρ2 ∈30

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Chap. 1 The outline of quantum language (=measurement theory)
Sm (A∗ ) and 0 < θ < 1” implies “ρ = ρ1 = ρ2 ”. Put
Sp (A∗ )={ρ ∈ Sm (A∗ ) | ρ is a pure state},
which is
It is well known (cf. ref. [110]) thatcalled a state space.
p
n ∗
[Case
(C
)];
S
(B(C
) ) = {ρ = |u⟩⟨u| (i.e., the Dirac notation) | ∥u∥Cn = 1}

1




[Case (C2 )]; Sp (C(Ω)∗ ) = {ρ = δω0 | δω0 is a point measure at ω0 ∈ Ω} ≈ Ω.
(C)
Under the identiﬁcation: Sp (C(Ω)∗ ) ∈ δω ↔ ω ∈ Ω, ω and Ω is also called a stateand state space respectively.
Deﬁnition 1.16. [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 nonnegative element in A such that 0 ≤ G(Ξ) ≤ I, (b): G(∅) = 0 and G(X) = I, where 0 and I is the0-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. I will devote myself to binary ( i.e.,{T, F }-valued ) observables in most of the cases in this paper. 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).

1.5.2

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) ofthat 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 interpretationof quantum mechanics. And thus, it is a statement without reality.
Now we can present Axiom 1 as follows.
Axiom 1 [ Measurement ]The probability that a measublack value x (∈ X) obtained by the measurement MA (O =(X, P(X), G),S[ρ] ) is given by ρ(G({x}))(≡ A∗ ⟨ρ, G({x})⟩A ).

Next let us explain Axiom 2 ( which is not used in this paper). Let A1 (⊆ B(H1 )) and A2 (⊆ B(H2 )) be
A continuous linear operator Φ1,2 : A2 → A1 is called a Markov operator, if Φ1,2 (L) ≥ 0 (∀L ∈ A2 such thatL ≥ 0 and ϕ1,2 (I2 ) = I1 , where I1 and I2 is identity maps in A1 and A2 respectively.
Now we can propose Axiom 2 (i.e., causality). (For details, see ref. [71].)
Axiom 2 [ Causality ]The causality is represented by a Markov operator Φ1,2 : A2 → A1 .

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1.5 Appendix: The mathematical foundations of quantum language
Deﬁnition 1.17. [(i): 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 observablennqp( i=1 Xi , P( i=1 Xi ),i=1,2,...,n Gi ) such that

×

×

×

×qp
i=1,2,...,n Oi =qp[

× Gi ](X1 × X2 × ... × Xj−1 × Ξj × Xj+1 × ... × Xn ) = Gj (Ξj ) (∀Ξj ∈ P(Xj ), j = 1, 2, ..., n)i=1,2,...,n
×qp

×n

×n

×qp

Also, MA (i=1,2,...,n Oi = (
i=1 Xi , P(
i=1 Xi ),
i=1,2,...,n Gi ), S[ρ] ) is called a quasi-product measurement of MA ( Oi = (Xi , P(Xi ), Gi ) , S[ρ] ) (i = 1, 2, ..., n).
[(ii): Tensor C ∗ -algebra, tensor quasi-product observable, tensor quasi-product measurement]: Let
Oi = (Xi , P(Xi ) , Gi ) be observables in Ai , (i = 1, 2, ..., n). Deﬁne a tensor quasi-product observable
Nqp
Nqp
Nn
n
n
∗
i=1 Xi , P(
i=1 Xi ),
i=1,2,...,n Oi = (
i=1,2,...,n Gi ) in the tensor C -algebra
i=1 Ai such that

×

[

qp
O

×

Gi ](X1 × X2 × ... × Xj−1 × Ξj × Xj+1 × ... × Xn ) = (

i=1,2,...,n

j−1
O
i=1

I) ⊗ Gj (Ξj ) ⊗ (

n
O

I)

i=j+1

(∀Ξj ∈ P(Xj ), j = 1, 2, ..., n)
N
Nqp
n
n
Nn
Also, MNni=1 Ai ( qpi=1 Xi , P(
i=1 Xi ),
i=1,2,...,n Gi ), S[ i=1 ρi ] ) is called a tensor quasii=1,2,...,n Oi = (product measurement of MAi ( Oi = (Xi , P(Xi ), Gi ) , S[ρi ] ) (i = 1, 2, ..., n).
.

××

1.5.3 Inference; Fisher’s maximum likelihood method)

We begin with the following notation:
Notation 1.18. [ML∞ (Ω,ν) (O, S[∗] )]: Consider a measurement ML∞ (Ω,ν) (O:=(X, F, F ), S[ω0 ] ) formulated
in the basic structure [C(Ω) ⊆ L∞ (Ω, ν) ⊆ B(L2 (Ω, ν))]. Here, note that
(E1 ) in most cases that the measurement ML∞ (Ω,ν) (O:=(X, F, F ), S[ω0 ] ) is taken, it is usual to think thatthe state ω0 (∈ Ω) is unknown.
That is because(E2 ) the measurement ML∞ (Ω,ν) (O, S[ω0 ] ) may be taken in order to know the state ω0 .
Therefore, when we want to stress that we do not know the state ω0 , the measurement ML∞ (Ω,ν)(O:=(X, F, F ), S[ω0 ] ) is often denoted by ML∞ (Ω,ν) (O:=(X, F, F ), S[∗] )Theorem 1.19. [Inference; Fisher’s maximum likelihood method (cf. ref. [42] or §5.2 in ref. [71]] For simplicity, assume that X is ﬁnite set. Assume that the measured value x(∈ X) is obtained by the measurement
ML∞ (Ω,ν) (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

Proof. It is an easy consequence of Axiom 1 (cf. §5.2 in ref. [71]).
Remark 1.20. [ Inference and Control cf. §5.2 in ref. [71]] The inference problem is characterized as thereverse problem of measurements.

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

That is, we consider thatML∞ (Ω,ν) (O:=(X, 2X , F ), S[ω0 ] )(E1 ) (state ω0 , observable O) −−−−−−−−−−−−−−−−−−−−−−−−−−→ measured value x0
measurement (Axiom 1)
On the other handML∞ (Ω,ν) (O:=(X, 2X , F ), S[∗] )(E2 ) (measured value x0 , observable O) −−−−−−−−−−−−−−−−−−−−−−−−−→ state ω0
inference (reverse Axiom 1)
Thus, (E1 ) and (E2 ) are in reverse problem.
Also, we note, from the mathematical point of view, that inference problem (E 3 ) and control problem (E4 )are essentially the same as follows.
(E3 ) [Inference problem; statistics]: when measured value x0 is obtained, infer the unknown state ω0and(E4 ) [Control problem; dynamical system theory]: Settle the state ω0 such that measured value x0 will beobtained!
Thus, we think, from the theoretical point of view, that statistics and dynamical system theory are essentiallythe same. Thus, we consider that statistics (= dynamical system theory) is the mathematical representationof classical mechanical worldview. On the other hand, quantum language is regarded as the mathematicalrepresentation of quantum mechanical worldview.

1.5.3.1 Exercise: Fisher’s maximum likelihood method
Problem 1.21. You can’t tell if it’s a man or a woman, but you can see someone in the distance. He/she
is wearing a skirt. Estimate whether this person is a man or a woman.

Answer:Consider the state space Ω = {ωM , ωW }, where ωM [ resp. ωW ] is considered as the state of ”man” [ resp.
”woman”]. Deﬁne the observable O = ({y, n}, 2{y,n} , F ) in C(Ω) such that
Number of men wearing skirts< 0.01,
Number of men
Number of women wearing skirts[F ({y})](ωW ) => 0.4
Number of women[F ({y})](ωM ) =[F ({n})](ωM ) = 1 − [F ({y})](ωM )[F ({n})](ωW ) = 1 − [F ({y})](ωW )Fisher’s maximum likelihood method says that• ﬁnd ω ∈ Ω such that
[F ({y})](ω) = max{[F ({y})](ωM ), [F ({y})](ωW )}33

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1.5 Appendix: The mathematical foundations of quantum language
Thus, we see that ω = ωW .
That is, we can infer that the person is a woman.

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35

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 phase ”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

Firstly we examine Pythagoras’ saying ” Number is within all things”. But unfortunately, this golden
rule has never been followed in the history of philosophy (≈ dualistic idealism).
• The reason that quantum language has a strong descriptive power is due to the fact 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. 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 ofquantum language. It is not a mathematical problem 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 that without worldview, there is no logic.

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2.1 Thalēs (BC.640 - BC.546)

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 ofGod. Myths have been handed down from generation to generation from our ancestors by oral andwritten word. There is a great deal of maritime trade, and many ethnic groups come together tointeract, 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 havemerged 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 ”thealphabet 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 a common
letter between diﬀerent ethnic groups. Egypt had an advanced civilization (such as 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 thequestion ”why does an earthquake occur?” Most myths will say ”because God is angry”. Thalēs
appeared in Miletus in ancient Greece and said:(B) The arche (= the ﬁrst principle of all things) is water. Therefore, the earthquake is caused by
the vibration of the waterThis may be childish, but is an explanation that does not brought out the ”God” . This is the reason
why Aristotle said Thalēs was ”the ﬁrst philosopher.”36

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Chap. 2 Ancient Greek philosophy (before Socrates)♠Note 2.1. S. Weinberg (1933 -2021), An American physicist, won a Nobel Prize of physics in 1979,
said in his book [115] ”To explain the word; The discovery of modern science” as follows:(♯) It seems to me that to understand these early Greeks, it is better to think of them not as 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, Descartes, Kant, etc. all should be regarded as
poets or ﬁction writers. As we will discuss later, I believe that the purpose of Kant’s Critique of Pure
Reason is not the pursuit of truth, but a proposal for a symbiosis with science. That is because therule of philosophy is as follows:(♯1 ) Only discussion, no experimentation.
(♯2 ) the winner will be determined by popularity vote of the general public.

♠Note 2.2. Who is the ﬁrst philosopher? Of course there may be a lot of opinions for this question.
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

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2.1 Thalēs (BC.640 - BC.546)
philosopher”, which is the same as the spirit of this paper mentioned later.

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Chap. 2 Ancient Greek philosophy (before Socrates)

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 ability of 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 inelementary geometry is neglected in middle and high schools, only about 60
However, if the next theorem is the discovery of Thales, we can trust Thales’ mathematical ability.
Theorem 2.2.
b=d

• The vertical angles are equal. That is, in the ﬁgure below, it holds that a = c,

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

(2.1)

thus,
a = 180° − b = c////
However, this proof has changed from ”trivial” to ”trivial”, and I can’t feel like proof. 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 of ancient
Greek philosophy may have given rise to Euclid’s Elements.

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2.1 Thalēs (BC.640 - BC.546)♠Note 2.3. (i): When Thales visited Egypt, there is a story that the king of Egypt was impressedby 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 inthose days (2000 years before Thalēs) in Egypt. If so, Thalēs’ theorem should be doubt whether it isdue to Thalēs. However, even as a true prover was unknown, the ability of mathematics at the time(i.e., the discovery of the concept of ”proof”) should be surprising. This led to Euclid’s ”Elements”(due to Euclid (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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Chap. 2 Ancient Greek philosophy (before Socrates)

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. Modern scholars
disagree regarding Pythagoras’s education and inﬂuences, but they do agree that, around 530 BC, he
travelled to Croton in southern Italy, where he founded a school for mathematics. Pythagoras wasa 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 of a regularpentagonand so on.
Theorem 2.3. Finding of irrational members, e.g.,√

2 is an irrational number.

Theorem 2.4. (Pythagorean theorem): In △ABC, the followings are equivalent:
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2.2 Pythagoras (BC.582 - BC.496)(♯1 ) ∠A = 90◦(♯2 ) AB2 + CA2 = BC2
Proof. The proof of Pythagoras himself is unknown, but the next proof will be learned in juniorhigh 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 con√structed as follows (the ﬁgure below (right)). Thus, we easily get 1+2 5 .

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Chap. 2 Ancient Greek philosophy (before Socrates)
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 (ofPythagorean theorem). Supplement the two proofs as follows:.

(b)(a)
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 regularpentagon for beginners. Even university students in the department of mathematics sometimes don’tknow this.
♠Note 2.6. The above two (the discovery of irrational numbers and the Pythagorean theorem) are oneof the most important discoveries in mathematics. If the following episodes are true, we can trust hismathematical ability.
(♯1 ) Pythagoras was killing the disciple, who found the irrational number, in order to hide the existenceof irrational numbers.
(♯2 ) When Pythagoras discovered the Pythagorean theorem, he celebrated it, oﬀered the sacriﬁce ofthe 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 with an essential
meaning. Nevertheless, their importance had been recognized in Pythagorean organization. Thissuggests 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 thefollowing astronomical system:

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2.2 Pythagoras (BC.582 - BC.496)

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

the earth is round(♭1 )Pythagoras−→

BC582-496

Eudoxusgeocentrism 2−→

BC400-347

geocentrism 3

Aristotle−→ Ptolemaeus

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 the greatestof ancient Greek mathematicians), a student of Plato. He proposed a kind of geocentrism and themethod 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 Earth was ﬂatand 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 ofexhaustion−→

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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Chap. 2 Ancient Greek philosophy (before Socrates)

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2.2 Pythagoras (BC.582 - BC.496)

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 is number”.
Now let us explain the following principle (called Pythagoreanism in this paper):(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”
I believe that this is the most important claim throughout Western philosophy.

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.
♠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.

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Chap. 2 Ancient Greek philosophy (before Socrates)(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. [117]: 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, hiswork 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 the most beautiful shape wasa 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 (though religious 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 to the present day and is inhibiting the healthy
development of our worldview (e.g., see ”The theory of probability” in [71], or ”Analytic philosophy”
in Chap. 12).

Hence we have the following problem (i.e., the problem of worldview), which is the main problem inthis paper:
Problem 2.6. The problem of worldview is as follows.

 A scientiﬁc worldview has the form: ”scientiﬁc worldview= mathematics + α”. If 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 thispaper, we devote ourselves to ”measurement (Axiom 1) and causality (Axiom 2) (cf. Sec.1.1).
♠Note 2.10. Possibly in this time, distinction of mathematics and science was not clear. What ismathematics? We had to wait for Cantor’s set theory and axiomatic set theory (by Zermelo andFraenkel) (circa 1900 A.D.) before we would know a deﬁnite answer to this question.

Pythagoreanism blossomed for the ﬁrst time in the early modern times (i.e., the scientiﬁc revolution
due to Galileo=Newton mechanics and Bernoulli’s law of large numbers, etc.cf. Chapter 7 later). In
this paper, we consider the following as a genealogy of dualism:• Plato → Descartes → Kant → Wittgenstein
However, it can be called only mysteriousness that Pythagoreanism has not blossomed in the mainstream of western philosophy. As emphasized later, our assertion ismeasurement, causality + mathematics −→ Quantum languagethat is, we assert that
• Pythagoreanism blossoms as quantum language in western philosophy
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2.2 Pythagoras (BC.582 - BC.496)
if the following is accepted:

Pythagoreanism• Plato → Descartes → Kant → Wittgenstein −−−−−−−−−−−→ Quantum language
quantiﬁcation

Note that, from Plato to Wittgenstein, Pythagoreanism was ignored. Finally, the dream of Pythagoras was realized in quantum language.

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Chap. 2 Ancient Greek philosophy (before Socrates)

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 paper, we interpret ”Everything ﬂows” asfollows.
(B) ”motion” is the most fundamental key-word in science.
If so, we can relate the (B) to Parmenides.
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2.3 Hērakleitos and Parmenides

2.3.2 Parmenides; The ancestor of the Copenhagen interpretation
In the same period of Hērakleitos, Parmenides said the exact opposite of words of Hērakleitos. 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 think logically withreason. 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 linguistic Copenhagen interpretation(cf. Sec. 1.1.2), i.e.,”only one measurement is permitted”, ”State does not move”, etc.
Also, in case of quantum mechanics, its object is too small, is not seen. Thus, we cannot relyon the sense, but only calculation. We can completely consent to Parmenides’ assertion (C 2 ) incase of quantum mechanics.

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

However, I may have an opinion such that50

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Chap. 2 Ancient Greek philosophy (before Socrates)(♯) It is not too much to say that Parmenides and Kolmogorov are the founders of the linguistic
Copenhagen interpretation of quantum mechanics.
( where Kolmogorov is the founder of the modern theory of probability (cf. ref. [83]). It is known
that Kolmogorov’s extension theorem is the most fundamental in the theory of probability.) 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 ).

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’ assertionis 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 and Parmenides:(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).
Therefore, as mentioned in Assertion 1.12, I consider that
Realistic worldview vs. idealistic worldview
dispute ⧹ [R] vs. [L]⃝:a motion
Greece⃝:Ancientb⃝:c Problem of universals⃝:
d space-time⃝:
e quantum theory⃝:philosophyof sciencefRealistic worldview

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

Hērakleitos
Aristotle”Nominalismus”(Ockham)
NewtonEinstein
CarnapParmenides
Plato”Realismus”(Anselmus)
Leibniz
Bohr
Quine

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2.3 Hērakleitos and Parmenides

2.3.3 Motion function method as a worldview
As mentioned in the previous section, Pythagoras said ”The arche (= the ﬁrst principle of all things)
is number”, Hence,(F) Mathematics (or, parameterization) is indispensable to describe the world. However, we needwords (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) thought of ”motion”
As one of the mechanical worldviews, we introduce the following ”motion function method”, whichis assumed to be due to Parmenides in this paper 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 a motionfunction.
Then, the motion function method (in the mechanical worldview) is proposed as follows:(H1 ) ”motion” should be described by the motion function f : T → X.
X 6

f (t)

- T

♠Note 2.13. In the above, we should note thatt(♯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 wecannot 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 can be derived from
Newton mechanics. However, motion functions are not limited to those derived from Newtonian mechanics. Thus, in this paper we usually consider that it belongs to the linguistic
worldview.
(♯3 ) For the derivation of motion function method from quantum language, see ref. [45], or
Chap. 14 in ref.[71].

The motion function method is easy, and it is usually studied in elementary school as follows.
Problem 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?

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Chap. 2 Ancient Greek philosophy (before Socrates)[Answer] Amy’s motion function fA : R(time axis) → R(one dimensional space axis) is deﬁned byfA (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 not consistent with
Parmenides’ saying: there exists only ”one” and not ”many”(cf. Sec.1.1.2: 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.
[43, 44, 71]). Namely,
quantum language −−−−−−−−→ dynamical system theory −−−−−−−−→ motion function method
derivation
derivation
That is,(♯) the motion function method is one aspects of the quantum mechanical worldview(i.e., quantum language)
Although I do not know, from the historical point of view, the discoverer of the motion function
method, I want to assume that Parmenides is the main character, since he was a teacher of Zeno (cf.
next section). Strictly speaking, the discovery might not be in Ancient 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 understood by Pythagoras, Aristotle, Archimedes, etc.
(b): Of course, the above ”motion function method” is incomplete and temporary. The motionfunction f : T (time) → X(space) is not suﬃcient without the answers to the questions ”What istime?” and ”What is space?” (the Leibniz=Clarke correspondence (cf. Sec. 9.3.2). For the quantum
linguistic answers to this questions, see ref. [71](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” (≈ ”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 important is”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, the mechanical worldview is discovered by plural persons. For example, the discoverer of the theory 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 paper (cf. ref. [71]), we want to regard
quantum theory, discovered by Heisenberg, Schrödinger and Born, as a kind of mechanicalworldview (and not realistic worldview).

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2.4 Zeno(BC490 - BC430): The Motion Paradox

2.4 Zeno(BC490 - BC430): The Motion Paradox
In this section, from the quantum linguistic point of view, we study Zeno’s paradoxes, the oldest
paradox in science. Pay attention to the following.
• All the arguments in this section are my opinion, not the common.
Therefore, I invite readers to read this section from a critical perspective.

2.4.1 What is Zeno’s paradoxes?

Without a worldview, we cannot say anything

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 and a tortoise”, ”dichotomy”,”stadium”, etc.), I think that these are essentially the same problem. 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 taste the literary pleasure.

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 composed of instants, thenmotion 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 ”to exit there” and ”notto 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’ ?”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 froma worldview and not logic”.
since Paradox 2.9 shows that ”antinomy” happens in the logic in ordinary language. If so, we haveto obey the worldviewism in Sec. 1.3.1. that is,54

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Chap. 2 Ancient Greek philosophy (before Socrates)• First declare the worldview, and discuss the world in the worldview. Namely,
world is soconclusion
thereforeworldview −−−−−→ discussions, calculation, practical logic, properties
premisesubject

Seeing the literature-like proof in Paradox 2.9, we naturally have the question:• Under what kind of worldview is the above literature-like proof presented ?
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) canbe discussed.
[Answer]; This is answered in Answer2.11 below.
Slogan-wise, we say(B) ”Without a worldview, there is no logic (or precisely, practical logic)”.
Again see ”the worldviewism (A)” in Sec. 1.3.1.
♠Note 2.17. The above is not to be taken for granted. As mentioned in Chaps. 11 and 12 later, thespirit of analytic philosophy is ”Start from logic” or ”Think logically !”, though the great traditionof philosophy from Plato to Kant was always based on the worldviewism (cf. Chaps. 3-10). Thus, Iam somewhat skeptical of analytic philosophy since I believe that the above (B) is the fundamental
spirit of philosophy.

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2.4 Zeno(BC490 - BC430): The Motion Paradox

2.4.2* The solution⋆ about Zeno’s paradoxes (e.g., Flying arrow) in the motion function
methodIf we obey the motion function method (in the mechanical worldview), 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 motion function
method (cf. Section 2.3.3) in the mechanical worldview, 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 iscorresponded. It is obvious that• ”for each time t, the position x(t) of the arrow is corresponded” do 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 acertain worldview is determined, Zeno’s paradoxes can be solved. 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 or the theory of relativity.
♠Note 2.19. As mentioned in Section 2.3.2, Parmenides’ assertion is similar to the linguistic Copenhagen interpretation. Therefore, we think that the ﬁrst step of the worldview was formed by Parmenides who belongs to the (quantum) mechanical worldview. Also, Hērakleitos is located in Paramenides’ opposite side. Thus we see the following. (cf. Classiﬁcation 1.11 [the classiﬁcation ofphilosophers]):
We consider
the following classiﬁcation of philosophers.
(♭1 ): the realistic worldview (physics)
Hērakleitos, Aristotle, Archimedes, Galileo, Newton, Einstein, · · ·








(♭21 ): the ﬁctional worldview (Western philosophy)Plato, Scholasticism, Descartes, Locke, Leibniz, Berkeley, Hume, Kant, Husserl

(♭)


(♭22 ): the logical worldview (=the logical spirit=the spirit of ”Think logically!”)Boole, Frege, Peirce, Saussure, Russell, Wittgenstein, Hempel, Quine, Popper









(♭ ): the mechanical worldview (statistics, quantum language) 23
Parmenides, Zeno, J. Bernoulli, statistics (e.g., Fisher), quantum language
Since Zeno is a student of Parmenides, it is natural to consider that Parmenides and Zeno are in the56

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Chap. 2 Ancient Greek philosophy (before Socrates)same position. Also, Pythagoras has no position in the above since we decide that mathematics is
not a kind of worldview.

2.4.3 DST formula

In this section, we deﬁned and discussed the ”motion function method”, which was a metaphysicalmethod. However, it is not diﬃcult, and it is similar to the ”DST formula” learned in elementary
school. To be precise, the ”motion function method” is the premise ﬁrst, and the DST formula hasmeaning under it. 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 school mathematics.
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 not available when the author was inelementary school, seems to be quite popular nowadays.

This ﬁgure is quite well known even though it does not seem to be a very good memory method. It
is not written in the math textbooks of elementary school students, so they must have learned it ata cram school. When I asked the fourth-year students in my laboratory, about 50% of them knew
about it. I was surprised at this prevalence. To be sure, let me repeat,• The DST formula (♣) is one of the most diﬃcult formulas in elementary arithmetic. Rather,I think it is one of the most diﬃcult formulas taught in junior high school, high school, anduniversity.
The Pythagorean Theorem is a mathematical formula. The law of conservation of momentum is aformula of Newtonian mechanics. Now let’s ask,
(A) In which ﬁeld is the DST formula (♣) a formula?
This problem is equivalent to the following problem:(B) Solve Zeno’s paradoxes!
That is, as the answer for (A), 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 junior high school,
high school, and university.
Zeno’s paradox attracted the interest of many philosophers because, in the author’s opinion, the57
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2.4 Zeno(BC490 - BC430): The Motion Paradox
kinetic function method is a metaphysical method. However, Zeno’s paradox is mentioned by many
philosophers, including Aristotle, Spinoza, Bertrand Russell, and Bergson. But I always think, ”Evena wise philosopher looks like a fool when talking about Zeno’s paradox.” Therefore, if the reader
does not consider me stupid, I am lucky.

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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 theessence of Zeno’s paradox, because Parmenides and Zeno’s interest should be a 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. [45], or Chap. 14 inref. [71]). Readers should also taste the literary 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. Even if Achilles continues to do this inﬁnite times, he will not be able to keep
up with the turtle.

[The scientiﬁc answer to Zeno’s paradox (Achilles and a tortoise) by the motion function
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 the position of thequickest [resp. slowest] runner at time t = 0 is equal to 0 [resp. a (> 0)]. Thus, we can assume thatthe 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 motion function:
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)):[(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, ...)
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2.4 Zeno(BC490 - BC430): The Motion Paradoxk)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)

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

q1 , q26

a
1−γ

..
.

..
..
.

(1−γ 3 )a1−γ

q2 (t) = γvt + a?

2(1−γ )a1−γ······

6aq1 (t) = vt0(= t0 )

av(= t1 )

(1−γ 2 )a (1−γ 3 )a(1−γ)v(1−γ)v(= t2 )(= t3 )···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 2.20. (♭1 ): Note that the above (ii) [= the solution using a geometric series] is another solutionof (i). Of course, there was no need to use geometric series. The point is that there is a diﬀerence inthe position of whether one considers the Achilles and a turtle paradox to be a mathematical problem
or a philosophical problem. Many philosophers have gotten into dead ends by confusing the two
positions. Philosophers should have considered it as a problem of world description.
(♭2 ): 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 it easy for solvers to fall into it.
In fact, for 2,500 years, most solvers have fallen for this trick.

♠Note 2.21. As mentioned in Preface, the purpose of this paper is to understand the history 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. Then the problem
should be entrusted to physicists. However, in this paper, we assume that Zeno’s paradoxes belong
to philosophy and not physics. Also, for the answer to ”What is space-time in quantum language?”,see Sec. 9.3.2: (Leibniz=Clarke correspondence).

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61

Chapter 3

The Big Three in Greek Philosophy(Socrates, Plato)
By the appearance of The Big Three in Greek Philosophy (Socrates, Plato, Aristotle), the origin ofwestern philosophy was formed as follows.
(♭1 ) : the realistic worldview(Aristotle → Newton)









(♭21 ) : the ﬁctional worldview





 the main current of western philosophy










(Socrates, Plato, Descartes, Kant, etc.)





(♭)



(♭22 ) : the mechanical worldview(♭2 ) : the idealistic worldview









 (Parmenides → statistics, quantum language)














(♭ ) : the linguistic philosophy



 23
(Aristotle → Frege, Saussure, Wittgenstein)though I think philosophers are divided on whether Aristotle belongs to the (♭23 ) or not.

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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 theme ofphilosophy. In the following dispute:”relativism (rational sophists) ” vs. ”absolutism (a man of faith: Socrates) ”
Socrates has 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 the sameproblem (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.
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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)
It is a matter of course that
(D2 ) If these words mentioned in the above (C) didn’t spread, the human race might have beenruined. 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 paper, 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. Society of ants has”ethics and morals of ants”, and apes must have ”ethics and morals of apes”. The ethical philosophyof the Neanderthals must have been quite diﬀerent from ours. Therefore, our ancestors must havedestroyed the Neanderthals. Communication and contact 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 late Dr. Hawking had emphasized the same opinion.

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, ”I know 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 varietyof opinions on the following issues.
• Which is playing with sophistry, Socrates or sophists?
I may feel that Socrates uses more sophistry (since Socrates uses taboo statement: I know I know
nothing).

In physics, you can draw conclusions with experiments, in mathematics you can prove them, and inengineering you just need to employ something useful. However, ethical philosophy does not solve
the problem in that way. So Socrates (= Plato) came up with the magic word ”I know that I
know nothing” (self-referential proposition, anti-Copenhagen interpretive proposition) as away to end the discussion.
(E) Since Socrates is the main character of Plato’s novel, he must always be undefeated.
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3.1 Protagoras and Socrates
Recall Figure 1.2 (in Section 1.1.2: The linguistic Copenhagen interpretation) below:

OO

Figure 3.1. (= Figure 1.2 in Section 1.1.2) [Descartes Figure]: Image of ”measurement(= a + b )”in dualism

Note that(F) the self-referential propositions (e.g., ”I know that I know nothing”, ”I think, therefore I am”,
etc. ) are out of quantum language (= dualistic language).
That is because the linguistic Copenhagen interpretation says that ”observer” and ”measuring object” must be completely separated. However, it holds that ”observer” = ”I” = ”measuring object” inthe above self-referential propositions. Thus, the self-referential propositions are taboo statements
in quantum language (= scientiﬁc language).
However, it is sure that the wordplay of self-referential propositions (”I know that I know nothing”,”I think, therefore I am”, etc.) is an important part of philosophy. In the art of argumentation,taboo propositions are often valid. Taboo propositions are so unclear in their meaning that listeners(i.e., readers ) often misinterpret them as profound propositions. This is the reason that Socrates
and Descartes succeeded. They were excellent sophists. A good sophist doesn’t let his readers knowthat he is a sophist. This is natural. Because philosophy is a kind of literature.

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

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 inorder 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 the insurance company does nothold. 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 we live?”) fromworldview (i.e., ”How is the world ?”), that is,worldview−−−−−−→ practical logic, ethics

How is the world? therefore”How should we live?

This method (= the form of philosophy) has been accepted as the standard form of ”how to tellphilosophy” in the history of two thousands hundreds of years.
(C): The ﬁction called ”the theory of Ideas”
Theory 3.2. [Theory of Ideas]:
The theory of Ideas is as follows• It cannot be said that love always overcomes money in the real world. However, there
exists another world (i.e., the world of Idea), where ”love always overcomes money” isbelieved as the objective truth. That is, there exists Idea (= the true form) in heavens.
A thing existing on the ground is only the shadow.
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.
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3.2 Plato(BC.427 - BC.347)
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” and has always been themainstream of Western philosophy, despite the twists and turns that followed. The theory of ideas,
which was supposed to be a logistic support of ethical philosophy (Socrates), became the mainstreamof the philosophy of worldview.
(E) Our human DNA prefers logic (or reasoning) as if the philosophy of ethics were derived fromthe philosophy of worldview. That is,”the world is so”, therefore ”we should live so”
namely,world is so

we should live so
thereforeﬁctional worldview −−−−−→ practical logic, ethics·morals(F1 ) (=(B))

introduction·preface·ﬁction

main subject

Also, for example,
don’t steal a glance

invisible man exists•

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

andultimate beauty exists•

she is an embodiment of beauty.
thereforeﬁctional 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 ”How should welive?” 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 sensuous common soil. Logic alone is notenough, 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 of telling philosophy (F1 )may not be reckless. That is because the (F1 ) means
world is so(F2 )we should live so
thereforeﬁctional worldview −−−−−→ practical logic, Ethics · morals
introduction·preface·ﬁctioncommon sensory soil

main subject

That is, ”logic” depends on ”worldview”. Slogan-wise, we say(G)”Without a worldview, there is no logic (= practical logic)”.

Recall that this slogan plays an essential role in the solution of Zeno’s paradox (cf. Sec.2.4 (B)).
Also, again see ”the worldviewism (A)” in Sec. 1.3.1 and Note 1.14.
♠Note 3.3. Here, ”logic in the common soil of the senses” has a similar meaning to ”logic under acertain description of the world”. In a broader sense, this ”logic” is the logic usually used by ordinary
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 of the various kinds of logic is thelogic of mathematical logic (= symbolic logic), which is so universal that even aliens may know it.
However,(♯) this does not mean that mathematical logic experts can master the logic of other disciplines well.
Naturally, a good understanding of economics means that one can use the logic of the common soil of66

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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)
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) (cf.
Note1.15), i.e.,
symbolic logic (= axiomatic logic =mathematical logic)•
practical logic (= non-mathematical logic)
Practical logic plays an important role in worldviews. On the other hand, symbolic logic plays animportant role in mathematics. I am skeptical of analytic philosophy, in which symbolic logic andpractical logic may be confused. I feel that the above spirit (♯) is scarce in analytic philosophy.
♠Note 3.4. (i): Aesop (BC.620 - BC.510): Idea theory is similar to Aesop’s fable in some ways. It isnatural 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 well known before the late 5th century
BC. Of course, Plato must have been aware of the persuasive 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.

(ii): In the book ”Sapiens: A Brief History of Humankind” (cf. [30], 2016), Harai, the author,
describes three revolutions which brought about dramatic changes to human societies.
 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

O

O
O
O

The above 1 implies that
• we can rule the world because we are the only animals that can create and believe in ﬁctions like
God, the state, money and human rights.
Did Plato know about the cognitive revolution?
////

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 truth that we seek. Theremay 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 Prizein 1979 for work that became a cornerstone of particle physics, said in his book [115] ”To explain theword; 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 of them attempted to verify oreven (aside perhaps from Zeno) seriously to justify their speculations.
Plato’s philosophy is a representative of idealism (cf. for the deﬁnition of ”idealism”, see Deﬁnition
1.8), 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.

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3.2 Plato(BC.427 - BC.347)
Bacon (i.e., the father of British Empiricism) of the scientiﬁc revolution (the 17th century) that theimportance of observation was, for the ﬁrst time, emphasized. See Section 7.2
Recall the rule of philosophy (≈ idealism) is as follows:(♯1 ) Only discussion, no experimentation.
(♯2 ) the winner will be determined by popularity vote of the general public.

♠Note 3.6. In order to avoid eternal argument, Socrates invented ”Socratic method”, that is, the magic
sentence ”I know that I know nothing”. Plato also invented the theory of Ideas.

For completeness, we add the following:(F) 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 the theory of Ideas
is sophistry, and there is no truth in ethics.
Even so, there may be a reason to consider that(G) something such as the sense of ethics of the human commonness is printed in the arrangementof 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)

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

Allegory 3.3. [Allegory of the cave]

A group of prisoners are looking at the shadows on the back wall of a cave. The shadows justare 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 loverof wisdom. Let’s assume he is Socrates. He turns around and the ﬁrst thing he sees is objects ofstone and wood made to resemble the shapes of 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 the entrance/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 thinkthat 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 the sunlight. The returning prisoner,
whose eyes have become accustomed to the sunlight, would be blind when he re-enters the cave, justas he was when he was ﬁrst exposed to the 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
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3.2 Plato(BC.427 - BC.347)
similar journey. Hence, there is a possibility that the prisoners would therefore reach out and killanyone who attempted to drag them out of the cave. In fact Socrates was killed.
Remark 3.4. Thus, Allegory of the cave says that three key-words in Plato philosophy:[A](mind)←− [B] −→[C](matter)(medium)

correspond to as follows:\[A](= mind)[B](Mediating of A and C)[C](= matter)

Plato (cave)actual world

sunlight(See Review 3.10 later)quantum language

measured value

observable/[Idea world]state[system]

3.2.3 The theory of anamnesis
As a mediator between the real world and the idea world, Plato presents the concept of ”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 anamnes 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”]−→

Kant philosophy[summing-up, compromise]

Remark 3.5. If we believe in the theory of anamnesis, three key-words in Plato philosophy:[A](mind)←− [B] −→[C](matter)(medium)

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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)
correspond to as follows:\[A](= mind)[B](Mediating of A and C)[C](= matter)

Plato (anamnesis)

actual world

anamnesis(See Review 3.10 later)quantum language

measured value

observable/[Idea world]state[system]71

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3.3 The allegory of the sun: Measurement theoretical aspect of Idea theory

3.3 The allegory of the sun: Measurement theoretical aspect of Idea
theoryThe 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 child of goodness”. For the answer
to ”Why a child?”, see Note 3.9.

3.3.1 The allegory of the sun
Note that the North Star can be also regarded as a measuring instrument for orientation. With thisin mind, I hope you will read the following.
Allegory 3.6. [The Allegory of the Sun]: The Allegory of the Sun explains what the ”Idea ofGood” 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 yourvision 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 (”thebeholder,” ”the sun (the mediating thing),” ”the thing to be seen”) are clear and very easy tounderstand.
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 can make the72

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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)
object ”visible”. In the same way, the mastery of the measuring instrument, which is called the”idea of Good,” makes clear virtue and courage.
Remark 3.7. The polar star can be regarded as a measuring instrument such as a kind of compass(cf. Note 1.4 (ii)). Thus, it is reasonable to regard the sum (Idea) as a measure of ”Good”.

That is, I want to think:”Idea” = ”instrument to make Idea world visible”.
though this may not be standard.
Therefore, three key-words in Plato philosophy:[A](mind)←− [B] −→[C](matter)(medium)

correspond to as follows:\[A](= mind)[B](Mediating of A and C)[C](= matter)

Plato (Sun)actual world

Idea(See Review 3.10 later)quantum language

measured value

observable/[Idea world]state[system]

It should be noted that this response is somewhat unreasonable. That is because Idea Theory was
not proposed with the intention of a theory of measurement (or epistemology).

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3.3 The allegory of the sun: Measurement theoretical aspect of Idea theory

3.3.2* The measurement theoretical understanding of the allegory of the sun

OO

Recall Figure 1.2 [ Descartes ﬁgure] in Sec. 1.1.2, namely,
Figure 1.2 ; [Descartes Figure]: Image of ”measurement(= a + b )” in dualism

Here, it should be noted that this is essentially the same as the ﬁgure [Allegory of the sun].
Summary 3.8. We see:Figure B [Idea of good]in Allegory 3.6xDeﬁnegoodness

science→======←philosophy

equivalent

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

Descartes Figure
in Figure 1.2

in Allegory 3.6xIllustration

Plato philosophy (Theory 3.2)

science

Theory of IdeasAllegoryy

Axiom 1 [measurement]

Allegory of the cave(Allegory 3.3)+ AnamnesisyAnamnesis(Sec. 3.2.3)

sciencephilosophy
We consider that• the understanding of the ﬁrst line above, namely,
Figure B [Idea of good]in Allegory 3.6

science→======←philosophy

equivalent

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

Descartes Figure
in Figure 1.2

in Allegory 3.6

is equal to the measurement theoretical understanding of Allegory 3.6 [Allegory of the sun].
Thus, ”Descartes ﬁgure” should have been called ”Plato ﬁgure”.
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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)

3.4 Plato: The ﬁctional worldview (=Plato’s way of telling philosophy)
3.4.1 The necessity of idealism and dualism
Let us review Plato’s way of telling philosophy (=the ﬁctional worldview, cf. Sec. 1.3.3.1).
(A): The ﬁctional worldview (=Plato’s way of telling philosophy)
Plato’s way of telling philosophy (in the main current of western philosophy) is as follows.
world is so(A)(♯): ﬁctional worldview (literary truth, pseudo-truth)
preface, introduction, (ﬁctional)premise, expedient
you should do sotherefore−−−−−→ (♭): ethics, morals, etc.
main subject

Therefore,(A1 ) world is so is secondary,(A2 ) you should do so is main theme
In Plato philosophy, the theory of Ideas (=the ﬁctional worldview) is only the ﬁction (= parable,
fable). That is,
the theory of Ideas is only a prologue for Plato’s ethics.
Plato’s way of telling philosophy is common to all philosophies in the genealogy of the dualistic
idealism as follows:(B) Plato(the theory of Ideas)−→Augustinus−→Thomas Aquinas −→Descartes−→Kant(epistemology)
If so, we may hesitate to reply ”Yes” for the following question:• Does the philosophy of worldview proposed by them merit serious and scientiﬁc discussion?
As mentioned throughout this paper, we consider that• every worldview in the genealogy (B) of the dualistic idealism is an allegory as similar as thetheory of Ideas.
♠Note 3.7. (i): Kant is highly regarded on today. As seen later (Chapter 10: Kant), the reason is that
Kant understood Plato’s intention perfectly.
”Critique of Practical Reason (1788)””Critique of Judgment(1790)””Critique of Pure Reason (1781)”•

therefore−−−−−−→

ﬁctional worldview

ethics, morals, aesthetics

preface, introduction, (ﬁctional)premise, expedient

main subject

where it should be noted that Critique of Pure Reason is not scientiﬁc.
(ii): In Plato’s way of telling philosophy:
world is so(♯): worldview

therefore−−−−−−→ (♭), etc.

it should be noted that (♭) is not necessarily deduced from (♯). In most cases, (♯) and (♭) are nottheoretically related. In this text, we are concerned with the following four:75

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3.4 Plato: The ﬁctional worldview (=Plato’s way of telling philosophy)

In fact, as seen in the above, ”Critique of Pure Reason” and ”Critique of Practical Reason”(or,”Critique of Judgment”) ) are not related. Also, recall that
• Descartes philosophy:
therefore(♯): I think, therefore I am −−−−−−→ (♭), God exists• analytic philosophy is generally considered as follows.
therefore(♯): mathematical logic −−−−−−→( Frege, Russell etc.

(♭), philosophy of language, philosophy of mind, analytic aesthetics(as I have said throughout this text, I do not believe that mathematics can be a worldview).
That is,

Say!, ”therefore,” even if it’s a lie.
That is because (♭) will not be never experimentally veriﬁed.

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

we should live so
thereforeﬁctional worldview −−−−−→ Ethics · morals, etc.
introduction·preface·ﬁctioncommon sensory soil

main subject

3.4.1.2 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 some medication.
Philosophers are not to be trusted.
Thus, philosophers have to assume an unrealistic world (metaphysical world). This is because if thesettings can be veriﬁed by experiment, mistakes may be pointed out. Therefore,(E) 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

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

3.4.1.3 The necessity of dualism
Also,(F) since the goal is ethics and morality, a world that reﬂects human beings in some way ispreferable. In other words, the dualism of ”things” and ”people” is preferable.
For example,• We don’t usually take moral lessons from monism, such as Newtonian mechanics.
This is the reason to adopt ”mind-matter dualism” in philosophy.
After all, we conclude that(G) in the philosophy of worldview, dualistic idealism is desirable.
Remark 3.9. Recall that our purpose of this paper is to show that
Plato −−−−−→ Descartes, Locke −−−−−→ Kant −−−−−→ Wittgenstein −−−−−→ QL
progress

progress

progress

progress(3.1)
where −−−−−→ means ”getting closer to quantum language”.
progress

However, some ask the following question:• Is the series (3.1) inevitable?
Concerning this question, Allegory 3.6 [ the Sun] is suggestive. Note that the following facts:(♯1 ) Plato proposed Idea theory, which was an fairy tale as dualistic idealism.
(♯2 ) Plato attempted to explain Idea theory in a variety of ways, but the most scientiﬁc andrational explanation was Allegory 3.6 [ the Sun]. (Of course, ”scientiﬁc and rational” does notnecessarily mean ”good” since Idea theory is not scientiﬁc and rational.)(♯3 ) Allegory 3.6 [ the Sun] is similar to quantum language (= measurement theory).
Further note that(♭)[ History of Western philosophy]= [dualistic idealism] + [ the spirit of being scientiﬁc and rational]= [Find the scientiﬁc destination of dualistic idealism]

If so (i.e. if (♯1 ), (♯2 ), (♯3 ) and (♭) are true), we want to consider that the series (3.1) is inevitable.

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3.5 Key words of dualism

3.5 Key words of dualism
3.5.1 Three key-words of dualism
The key-words of dualism are simple. That is because• Since dualism is a theory concerning measurement, [A](observer) and [C](matter) are needed.
However, if the two are not related, this implies only that there are two monism. Therefore,
there must be [B](medium(=device that mediate [A] and [C]))Recall Figure 1.2 (in Section 1.1.2: The linguistic Copenhagen interpretation) below:

OO

Figure 1.2 (in Section 1.1.2); [Descartes Figure]: Image of ”measurement(= a + b )” in dualism[A](mind) ←− [B] −→ [C](matter)(medium)(A)measured value ,(observer, brain, mind)(B)
observable (= measuring instrument) ,

(C)
state

(thermometer, eye, ear, body, polar star)(matter, measuring object).

If so,• The structure of mind-matter dualism is as follows[A](mind)←− [B] −→[C](matter)(medium)

That is, it is composed of [A](mind),[B](medium),[C](matter).
In history, there are incomplete dualism that does not include three [A](mind), [B](medium),[C](matter).
• We consider that [B](medium) is the most important than the other two (cf. Linguistic
Copenhagen interpretation (E3 ) in Sec.1.1.2 ). Therefore, if the theory includes [B](medium),
it is called ”dualism”.

3.5.2

Is Idea theory related to measurement (i.e., dualism) ?

Now we have a question:• Where should these key-words (i.e., Idea world, actual world, Idea, anamnesis) be assigned

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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)
next?
(A)
measured value ,(observer, brain, mind)(B)
observable (= measuring instrument) ,

(C)
state

(thermometer, eye, ear, body, polar star)(matter, measuring object).

This is answered in what follows.
Review 3.10. Recall that Remark 3.4 (Allegory of the cave), Remark 3.5 (Anamnesis), Remark
3.7 (Allegory of the cave) in the following table:\
cave⃝:Plato:1
Remark 3.4⃝:Plato:anamnesis
2
Remark 3.5⃝:Plato:sun
3Remark 3.7
quantum language[A](= mind)[B](Mediating of A and C)

actual world

sunlight

actual world

anamnesis

actual world

Idea

measured value

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

Plato’s theory of ideas was not created for science, so what follows may be too forceful. However,In this text, I consider the following. Here,(♯) I don’t know Plato’s intention, but I want to choose Remark 3.7(Allegory of thesun), which is more scientiﬁc than the other two.
That is, I assert that\

[A](= mind)[B](Mediating of A and C)⃝:
3 Plato: sunRemark 3.7

actual world

Idea

measured value

observable

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

The reason the I think that the above is somewhat persuasive is as follows.
• Since the Sophists said ”Man is the measure of all things”, Socrates-Plato must have thought
the exact opposite claim, ”Idea is the measure of all things”. Namely,(♭) There exists the absolute standard (= idea) of ”love”, ”beauty”, ”goodness”, etc.
Thus it is not unnatural to regard Idea as the meter standard, or the touchstone (i.e., a kindof measuring instrument).
♠Note 3.8. I am not conﬁdent in the correspondence ⃝3 in Review 3.10 because the theory of ideas
was not created with the intention of measurement. However, I don’t care if the above claim is wrong.
That is because our purpose of this paper is to show that(♯) Plato(the theory of Ideas) −−−−−−→Descartes −−−−−−→Kant −−−−−−→Quantum language
progress

progress

progress

if approaching quantum language is called ”progress”. It is easy to see that the above (♯) holds in allcases of ⃝,1 ⃝,2 and ⃝.
3
♠Note 3.9. However, I think that
the Allegory of the Sun is not a proper example for Idea theory,
since it is too scientiﬁc and thus too easy to understand. Probably, I think Plato gave too muchscientiﬁc explanation in an attempt to make readers understand Idea theory. I think Plato was notsatisﬁed with the parable in ”the Allegory of the Sun”, so he referred to ”the Allegory of the Sun” as”the child of goodness”. The book : ”History of western philosophy” due to B. Russell says that (cf.
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3.5 Key words of dualism• 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.
In the same sense, Idea theory and ”the Allegory of the Sun” do not mix easily. Thus, this allegory
may hinder our understanding of Idea theory. However, it is interesting that the scientiﬁc explanationof Idea theory is similar to measurement theory. Thus, in this paper, I choose ⃝3 as mentioned in
Review 3.10.
♠Note 3.10. (i): Literature has various genres. As an example, it is a love story, a detective story, SF(science ﬁction), poetry, nonﬁction. In the same sense, the philosophy of worldview is a kind ofliterature.
Therefore, I think that the emphasis on the relationship to mathematics hinders our understandingof ideation theory. The following is famous:(♯) It was written in the gate of the entrance of the school (Platonic Academy) which Plato established, ”The person who does not know the geometry should not pass through thisgate”

If this is to be taken seriously, Plato’s researchers must be well trained in mathematics. But inreality, that has not happened. In science, it is more experimental veriﬁcation than logic. Thus, theimportance of logic is not so emphasized in science. That is, the proof of the pudding is in the eating.
The ﬁeld where importance of the logic is emphasized is the ﬁeld where it is hard to do logical andquantitative arguments. For example, the importance of logic is emphasized in courts. There must
be many unknowns in many cases. Still, the judge must judge the person. Therefore, the impressionof ”logically judged” is very important.
However, ”mathematical, logical, reasonable” must be emphasized in philosophy. That is because thediﬀerence between philosophy and religion becomes obscure without the emphasis. The reason why
the importance of logic is not emphasized in science is that I think it’s because if we emphasize theimportance of experimentation, it won’t be confused with religion.
(ii): The above discussion reminds me of modern ”analytic philosophy”, in which the importance of
logic is emphasized.
Humble people do not promote humility. Also, for mathematicians, logic is like air. Therefore,
mathematicians do not emphasize the importance of logic.
Thus, I don’t know if analytic philosophers are logical or not.

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3.5.3* Plato’s Idea theory ≈ Locke’s secondary quality ≈Sausuure’s linguistic theory ≈Zadeh’s fuzzy theory
If we assume that(D) Plato’s Idea theory, Locke’s secondary quality, Sausuure’s linguistic theory and Zadeh’s fuzzy
theory are one of aspects of quantum language
then, we can discuss as follows.
⃝:Idea−−−−−−→⃝:
1
2 secondary qualities −−−−−−−−→⃝:
3 observable (= measuring instrument)
cognitive

quantitative

where,⃝
1 Idea”the meter standard of beauty, the meter standard of goodness, · · · ”=⇒”device that make beauty visible”, ”device that make the good visible”⃝
2 secondary qualities(=sensations of inherent nature (=primary qualities))”sweet, pungent”, ”hot, cold”, ”beautiful, ugly” · · ·⃝3 observable(=measuring instrument)saccharimeter, thermometer, · · ·In this real world, ”Man is the measure of all things” may be almost true. On the other hand, in
the Idea world, the absolute goodness, the absolute beauty. are always believed. That is,• There exists the absolute standard of ”love”, ”beauty”, ”goodness”, etc.
Example 3.11. [The argument under the assumption that Idea theory is regarded as measurement
theory]Let Ω be a compact state space. Here, any tree can be represented by a certain state ω(∈ Ω).
Thus, we have the basic structure [C(Ω) ⊆ L∞ (Ω, ν) ⊆ B(L2 (Ω, ν))]. Let OB = ({y, n}, 2{y,n} , B)
be the continuous observable, which is assumed to be measurement instrument of beauty. Let T0
be a tree with the state ω0 (∈ Ω). Put ω0 = ωe (T0 ). Then, we have the measurement ML∞ (Ω,ν)(OB = ({y, n}, 2{y,n} , B), S[eω(T0 )] ), where ”y” [resp. ”n” ] means ”yes” [ resp. ”no”].
B({y})1•[B({y})](ω0 )
•
ω0

Ω

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3.5 Key words of dualism
Axiom 1 (in Section 1.5.2) implies that• the probability that a measured value y is obtained by the measurement ML∞ (Ω,ν) (OB =({y, n}, 2{y,n} , B), S[eω(T0 )] ) is given by [B({y})](ω0 ).
That is,• the probability that the tree T0 is beautiful is given by [B({y})](ω0 ).
Remark 3.12. [Plato’s Idea theory ≈ Locke’s secondary quality ≈Sausuure’s linguistic theory ≈Zadeh’s fuzzy theory] Readers may ﬁnd this example trivial. However, it should be noted that thisexample is essentially the same as Deﬁnition 11.20 (”signiﬁer” and ”signiﬁed”). That is, we see,from the quantum linguistic point of view, the following three are similar:
Plato’s Idea theory
Locke’s secondary quality (cf. Sec. 9.1)
Saussure’s linguistic theory (cf. Sec. 11.5)
Zadeh’s Fuzzy theory (cf. refs. [39, 40, 41, 42])
Or more precisely,

(D)
Idea

linguistic version−−−−−−−−−−−→

signiﬁer, signﬁed

Plato’s idea theorycognitiveyversion

Saussure’s linguistic theory

secondary quality

membership function

John Locke

Zadeh’s fuzzy theoryquantiﬁcationyyidentiﬁcation

quantiﬁcation

j observable

quantum language

For Zadeh’s fuzzy theory, just imagine the following diagram:82

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

3.6 Summary: Plato’s way of telling philosophy
3.6.1 Summary(A):
Plato’s way of telling philosophy (i.e., the ﬁctional worldview)
In Plato’s way of telling philosophy,(A1 ) a ﬁctional worldview is characterized as the premise (or, introduction, preface, ﬁction) of themain theme (i.e., ethics, moral, etc.).
In other words, consider the following ﬁgure, i.e., Plato’s way of telling philosophy:
world is so(A2 )ﬁctional worldview (literary truth, pseudo- truth)
preface, introduction, (ﬁctional)premise, expedient, prelude
you should do sotherefore−−−−−→ practical logic, ethics, morals
main subject

Literature has various genres. As an example, it is a love story, a detective story, SF (science
ﬁction), poetry, nonﬁction. In the same sense, the above ﬁctional worldview (as the supportof the main assertion [ethics. moral]) is a kind of literature.
(Notice)therefore(A3 ) Some consider that the term: ”−−−−−→” implies that the ﬁctional worldview should be”logical”. Here, it should be noted that the ”logical” is similar to the ”logical” of detective
story.
Remark 3.13. For example we see,z

introduction part
}|

{




Idea theory








early Scholasticism
Plato+Christianity








late Scholasticism

Aristotle+Christianity 

Plato






I think, therefore I am 







Kant




thing-in-itself








Analytic philosophy
mathematical logic

commonTherefore−−−−−−−→ spirit, ethics, moral

Descartes

Here it should be noted that all of the worldviews in the left-side of the above are theories that are
impossible to deny (thus, non-scientiﬁc theories).
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3.6 Summary: Plato’s way of telling philosophy

If we think about it simply, the ”introduction (= ﬁctional worldview)” part is unnecessary.
In fact, Socrates, Confucius, etc. focused only on ethics and morality. However, the ”introduction”
part had a variety of beneﬁts. In what follows, let us explain some beneﬁts.
Remark 3.14. Whitehead(1861 - 1947) said:(B) Western philosophy is characterized as a series of footnotes to Plato
Although we do not know his true intention, our understanding is as follows.
(C) The various fashions of the Buddhism, Confucianism and the Taoism Confucius jumbled up
at the Orient. And those continued for 2500 years by a subtle strained relation. On the other
hand, in the Western, Christianity was too strong. Thus, the main theme (i.e., ethics, morals)
is mostly due to ”Christianity + Socrates”, and thus, the various fashions were not born.
However, according to Plato’s way of telling philosophy, the introduction part (i.e., the ﬁctional
worldview) is changeable. Therefore, the progress of Western philosophy (which was notrealized in the East) can be realized as follows.
• Plato−→Augustinus−→Thomas Aquinas −→Descartes−→Locke−→...−→Kant
That is,Plato’s way of telling philosophy could keep freshness.
If the example is say, we think that there has been an eﬀect, such as the model changeof car. The Plato’s way of telling philosophy is almost always the main current of Western
philosophy. This device (i.e., the model change) brought the prosperity of Western philosophy.
If we do not consider so, we cannot explain the fact that useless world-description (in western
philosophy) lasted for 2500 years.
♠Note 3.11. We consider that(♯) Philosophy of ethics is common to mankind and is the world standard.
Or, we want to consider so. Otherwise, world peace cannot be achieved. There is a point to thetheory that ethics is a rule that keeps society alive. Depending on what kind of society we envision,
ethics will be somewhat diﬀerent. However, we want to consider (♯). For example, ”Don’t lie” or”Don’t kill people”, etc., are common in the world. Also, the Golden Rule ”Do unto others what you
would have them do unto you” has been chanted by many philosophers and religious ﬁgures (Christ,
Confucius , Muhammad, etc.). In this sense, I think that ethics is logical.
However, worldview (= Plato’s ﬁctional worldview in the way he tells his philosophy) is not universal.
In other words, it is one of the local philosophies that has evolved in its own way. We might say”Galapagos philosophy”. For example, consider(♯1 ) Idea theory −−−−−−−−−→ epistemology −−−−−−−−−−−→ analytic philosophy
cogito turnlinguistic turnIf there was no epistemology, it is no wonder. In fact, there is no epistemology in the East or theUnited States. In this sense, the world description is not logical. On the other hand,(♯2 ) the realistic worldview is the world standard (moreover, the universe standard)
Aristotle−→Archimedes−→Newton−→ · · ·(♯3 ) the logical worldview (=the logical spirit=the spirit of ”Think logically!”) can’t be called theworld standard since it is quite inﬂuenced by Descartes=Kant philosophy.

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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)
Aristotle −→ Frege −→ Wittgenstein−→ · · ·I think that philosophers need not be more logical than scientists. Thus, I can’t understandwhy philosophers like logic. Kant philosophy was too literary, and it may be the reaction. I
highly estimate Wittgenstein’s ”Tractatus Logico-Philosophicus (widely abbreviated and cited
as TLP)”, in which he attempted to explore a diﬀerent logic than the ”logic of mathematics”.
(cf. Section 11.6 later) )(♯4 ) the mechanical worldview is also the world standard:
Parmenides·Zeno−−−−−−−−−−−−−−−−−−→
Bernoulli→Fisher(statistics)(motion function method)

About two thousand years (the law of large numbers)And, it is a matter of course that the above two (the realistic worldview and the mechanical worldview) are useful. It is remarkable that the philosophy of worldview (=the ﬁctional worldview) isn’t usefulat all but it continued for 2500 years. That is, we think that ”Galapagos” ⇔ ”scientiﬁcally useless”.
This may be due to the shadow supporter (i.e., Christianity), i.e., as mentioned in the above remark,Christianity is too strong.
In spite of the above arguments(i.e., above, (♯1 ) is not the only inevitability), in this note I show thefollowing:This is a miracle.

♠Note 3.12. Now we have the following classiﬁcation of philosophers. (cf. Classiﬁcation 1.11 [theclassiﬁcation of philosophers]): Recall the ﬁgure (Allegory of the cave) in Allegory 3.3, which isclearly related to measurement (cf. Deﬁnition 1.8 of ”idealism”). Thus, Plato’s Idea theory belongs
to the ﬁctional worldview (Western philosophy):(♭1 ): the realistic worldview (physics)
Hērakleitos, Aristotle, Archimedes, Galileo, Newton, Einstein, · · ·








(♭21 ): the ﬁctional worldview (Western philosophy)Plato, Scholasticism, Descartes, Locke, Leibniz, Berkeley, Hume, Kant, Husserl

(♭)


 (♭22 ): the logical worldview (=the logical spirit=the spirit of ”Think logically!”)Boole, Frege, Peirce, Saussure, Russell, Wittgenstein, Hempel, Quine, Popper









(♭ ): the mechanical worldview (statistics, quantum language) 23
Parmenides, Zeno, J. Bernoulli, statistics (e.g., Fisher), quantum language
Spirit of Pythagoras is inherited, and Parmenides and Zeno have argued establishment of the worldview as science sincerely. However, Plato used the ﬁctional worldview as a means of the protection of
Socrates(ethic philosophy) and has dwarfed the worldview in non-scientiﬁc way. But, as Whiteheadsaid ”Plato’s footnote”, the ﬁctional worldview, that is,(♯1 ) Plato→Augustinus→Thomas Aquinas −→Descartes−→Locke→...→Kant−→Husserl
has continued to be supported over a long time of more than 2000 years. On the other hand, thescientiﬁc idealistic worldview was established by Fisher , etc. as follows:
Tunnel of more than 1700 years(♯2 ) Parmenides·Zeno →−−−−−−−−−−−−−−−−−−−−−−−→→

the classical mechanical worldview

Early modern period
J.Bernoulli,Bayes,Laplace,etc.

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3.6 Summary: Plato’s way of telling philosophy→ Fisher(statistics)−→quantum language

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Chapter 4

The Big Three in Greek Philosophy(Aristotle)
Although Aristotle was a student of Plato, he proposed the realistic worldview, which was completely
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, wesee:

the realistic worldview (monism, physics)

Aristotle

Plato















→ Scholasticism →















Newton → theory of relativity

Descartes·Kant → statistics·quantum language

the idealistic worldview (dualism, no experiment)
Therefore, it’s best to assume that the two men have very diﬀerent areas of expertise.

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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 ofthe realistic worldview). He could not accept Plato’s theory of Ideas(= asserted ﬁction ). Namely,• Philosopher Plato preferred asserted ﬁction (without experiment) to truth (withexperiment)
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 Prizein 1979 for work that became a cornerstone of particle physics, said in his book [115] ”To explain theword; 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 ethical philosophy.
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 not tedious.
On the other hand, Aristotle might be aim for science. Thus, from the modern point of view, 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 hasn’t
faded after 2,000 years. However, our purpose of this text is to show that
Plato −−−−−→ Descartes −−−−−→ Kant −−−−−→ Wittgenstein −−−−−→ Quantum language
progress

progress

progress

progress

4.1.2 Edios and Hyle
Aristotle proposed the concepts such as ”eidos” and ”hyle” as follows.

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

OO

Figure 1.2 (in Section 1.1.2); [Descartes Figure]: Image of ”measurement(= a + b )” in
dualism:
dualism and monism (= Aristotle’s worldview)
dualismmonismmeasuring instrument

observer(I(=mind))•(body, eye, etc.)[observable][measured value]
a project lightO
Ob perceive reaction

system, hyle, point mass
(matter, measuring object)[state]- [eidos]

Assertion 4.1. (= Assertion 1.14 )[The key-words of the realistic worldview] The realistic worldview is monism, and its completed version is realized as Newtonian mechanics, whose key-words are ”point mass” and ”state”.
Thus, we see:\[A](= mind)[B](Mediating of A and C)

Aristotle

————

————

Newton————————[C](state)[matter]eidos[hyle]
state[point mass]

That is, we consider the following progress:[eidos] −−−−−→[state]progress[hyle] −−−−−→[point mass]
progress

We can easily use Newtonian mechanics as follows.
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4.1 Aristotle (BC.384 - BC.322)• point mass(=particle with the mass m ) with the state(=(position, momentum)= (x, p)∈ R2 (= state space))Thus, it is a matter of course that we conclude that• Aristotle is the founder of the realistic worldview (= physics)[Note]: The above table should be compared to the following table in Review 3.10 [ i.e., Plato’s
Idea theory ]:\[A](= mind)[B](Mediating of A and C)⃝:
3 Plato: Sun
Remark 3.7

actual world

Idea

measured value

observable

quantum language[C](= matter)/[Idea world]state[system]♠Note 4.2. The book : ”History of western philosophy” due to B. Russell says that (cf. ref. [108]),• 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 of expertise.
Aristotle proposed a realist philosophy that is quite diﬀerent from the Platonic philosophy of idealism.
As the following diagram shows, Plato and Aristotle are water and oil. In fact, Scholasticism (= acompromise between Plato and Aristotle) was not successful because Plato and Aristotle do not mix.

the realistic worldview (physics)
Aristotle + Hellenism+Islam

Plato + Augustinus















A.D.1000∼1500⇒ Scholasticism ⇒compromise















Newton → theory of relativity

Descartes·Kant → statistics·quantum language

the idealistic worldview (no experiment)”Realistic worldview (= realism) or idealistic worldview (= metaphysics) ?” is the biggest disputein the history of philosophy as shown in Assertion 1.12, that is,realistic worldview vs. idealistic worldview (cf. Assertion 1.12)
dispute ⧹ [R] vs. [L]⃝:a motion⃝:Ancient
Greeceb⃝:c Problem of universals⃝:
d space-time⃝:
e quantum theoryof science⃝:philosophyfRealistic worldview

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

Hērakleitos
Aristotle”Nominalismus”(Ockham)
NewtonEinstein
CarnapParmenides
Plato”Realismus”(Anselmus)
Leibniz
Bohr
Quine

⃝
c is a confusion. ⃝d is the Leibniz=Clarke correspondence (cf. Sec. 9.3.2), ⃝ea is my ﬁction, ⃝is Bohr-Einstein debates. Quantum language is proposed as one of answers to Bohr-Einstein debates(cf. ref. [71]). Quantum language is proposed as one of answers to Bohr-Einstein debates(cf.
ref. [71]). ⃝:f Quine understood 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. 13.3).
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Chap. 4 The Big Three in Greek Philosophy (Aristotle)

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 are Hērakleitos andParmenides:• 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.

However, Aristotle (BC384–BC322) pursued an even more fundamental problem:(C) What is the essence of movement and change?
and concluded as follows.

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4.2 Why does the motion happen?
(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, and smoke 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 From purpose to causality
Under the inﬂuence of Aristotle, ”Purpose” had remained as a mainstream idea of ”Movement” fora long period of 1500 years or more.
We were freed from the spell of ”Purpose”, only after Galileo, Bacon, Descartes, and Newton etal. 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 theshift ”birth of modern science”.
the birth of worldview(E)

Movement

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

ZXC

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 the realistic
worldview, Newtonian kinetic equation (i.e., the equation of the chain of causality) was ﬁnal in asense. 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 appearance of quantum language (Axiom
2 (causal relation) in Sec.1.1.1, also, see ref. [71]).
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Chap. 4 The Big Three in Greek Philosophy (Aristotle)
Summary 4.2. [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 1.1) in quantum language(Continued to Sec. 10.2: What is causality?).

♠Note 4.3. S. Weinberg (1933 -2021), a physicist at the University of Texas, Austin, won a Nobel Prizein 1979 for work that became a cornerstone of particle physics, said in his book [115] ”To explain theword; The discovery of modern science” as follows:• [ in Chapter 3]:We can agree with the classical scholar R. J. Hankinson that ”we must not lose
sight of the fact that Aristotle was a man of his time and for that time he was extraordinarily perspicacious, acute, and advanced.” Nevertheless, there were principles running all throughAristotle’s thought that had to be unlearned in the discovery of modern science. For one thing,Aristotle’s work was suﬀused with teleology: 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) understandthe meaning of ”science”.
Aristotle is also the father of biology. I think the reason Aristotle failed to discover causality isthat 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 is 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 born out of [(i) geodynamics, (ii) causality (Newtonian mechanics), and(iii) the importance of experimentation (i.e., British empiricism)]. That is,(i) Geocentrism(ii) causality (Newtonian mechanics)=⇒ scientiﬁc revolution(iii) the importance of experimentation

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4.3 Practical logic

4.3 Practical logic
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 proposition inmathematical logic is usually considered to be a mathematical proposition. Also, A proposition inpractical 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 was due tointimidation, solidarity, reproduction. Language was one of the strongest arms for the survival andbreeding. Such a time have continued for millions of years. Of course, the biggest 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 logicalstructure is passed down as ”Aristotle’s syllogism,” Namely,
Aristotle’s syllogism in ordinary language(B2 ) Since Socrates is human being, and human being is mortal, it follows that Socrates ismortal.

Although this is quite famous, the reader may have several questions concerning this as follows,(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 was endorsed by Aristotle andremained authoritative for almost 2,000 years.
In fact, Immanuel Kant said that there was nothing else to invent after the work of Aristotle.
♠Note 4.4. Here,(♯1 ) Who is the discoverer of the syllogism?

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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,(♯1 ) Who is the discoverer of the motion function method?
As mentioned in Note 2.16, it is not possible to identify the names of the discoverer. However, thesyllogism and motion function method in QL were respectively discovered in refs. [76, 78] and [45]95

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4.3 Practical logic

4.3.2 Mathematical logic; due to Boole, Frege
Let us review symbolic logic as follows.
Postulate 4.3. (= Postulate 11.1)[ 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))(A1 ) Assume that P1 , P2 are propositions. Then, P1 ∧ P2 , P1 ∨ P2 , ¬P1 , P1 → P2 are propositions.
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 ).
(A2 ) The above ﬁnite set Θ(≡ {1, 2, ..., n}) can be extended to an inﬁnite set Θ.
Exercise 4.4. (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.
The above is elementary. However, it should be noted that(E) We have no deﬁnition of ”non-mathematical proposition”.
For example, we can not answer to the following questions:(F1 ) Is the ﬁrst proposition of philosophy ”I think, therefore I am” a proposition?
(F2 ) Is the statement ”” in Moore’s paradox a proposition?
(F3 ) Is true syllogism in quantum systems?
If so, symbolic logic is powerless without mathematics. R. Wittgenstein thought so, he challenged
to clarify logic in philosophy. However, he failed his challenge.
In spite of his failures, analytic philosophy ﬂourished in the following ways
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Chap. 4 The Big Three in Greek Philosophy (Aristotle)

propositional logic

G. Boole(1847)−→

predicate logic

Type theory

Gottlob Frege −→B. Russell(1879)|{z(1903)}

Spirit of ”Think logically!”

Analytic phil.

L. Wittgenstein
Prosperity of analytic philosophy−−−−−−−−→ (Tractatus Logico-Philosophicus −→philosophy(1921)♠Note 4.5. To compare Review 3.10 and Assertion 4.1, we can guess that Aristotle cannot understand Plato’s theory of Ideas. That is, Aristotle tried to propose the other theory (i.e., the realistic
worldview) than Plato’s theory (i.e., the ﬁctional worldview). And he discovered ”eidos” and ”hyle”,
which are the most basic concepts in mechanics. Thus, we conclude that Aristotle is the founder ofthe realistic worldview. As mentioned before, I am skeptical of the logical worldview (=the logicalspirit=the spirit of ”Think logically!”), but the following diagram is usual in philosophers:
Arithtotle −−→ Boole −−→ Frege −−→ Russell −−→ ...
Aristotle’ssyllogism may not belong to symbolic logic, thus, I do not add Aristotle to (♭22 ).
(♭):
 1 the realistic worldview (physics)
Hērakleitos, Aristotle, Archimedes, Galileo, Newton, Einstein, · · ·








(♭21 ): the ﬁctional worldview (Western philosophy)Plato, Scholasticism, Descartes, Locke, Leibniz, Berkeley, Hume, Kant, Husserl

(♭)


(♭22 ): the logical worldview (=the logical spirit=the spirit of ”Think logically!”)Boole, Frege, Peirce, Saussure, Russell, Wittgenstein, Hempel, Quine, Popper









(♭ ): the mechanical worldview (statistics, quantum language) 23
Parmenides, Zeno, J. Bernoulli, statistics (e.g., Fisher), quantum language
In science, quantitative discussion and computation become important. In this sense, we consider thatphysics (or, mechanics) is located in the center of science. On the other hand, the logical worldview(=the logical spirit=the spirit of ”Think logically!”) is rather qualitative, and therefore, the logicalworldview (=the logical spirit=the spirit of ”Think logically!”) is rather inﬂuential in the ﬁeld of
humanities. For example, trials are an area where the importance of logic is most emphasized. Also,
I think that philosophers are people who like logic most.

4.3.3* Syllogism does not always hold in quantum systems
We have the following theorem
Theorem 4.5. (G1 ) Syllogism does not necessarily hold in quantum systems(G2 ) Syllogism always holds in both classical and quantum systems////
Since (G1 ) and (G2 ) have diﬀerent premises, (G1 ) and (G2 ) are not contradictory (cf. Remark 12.9 later).
The proof of (G2 ) is given in Corollary 12.8.
Proof 4.6. The proof of Theorem 4.5 (G1 ) due to the following:• [76] Ishikawa, S., (2020) Wittgenstein’s picture theory in the quantum mechanical worldview Journalof quantum information science, Vol. 10, No.4 , 104-125, DOI:10.4236/jqis.2020.104007(https://www.scirp.org/journal/paperabs.aspx?paperid=106233)(H) : [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 .

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4.3 Practical logic
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 ⟩⟨u0 | 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 ̸= 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 )2
8ϵ
|
(4.2)
|c
u0 (p1 , q2 )| = |
e 8σ
2πϵσ
where uc0 is the Fourier transform of u0

p2 6

q2 6
q1 − q2 + a = 0

a

a

O

O

q1

p1
p 1 + p2 = 0

For each k = 1, 2, deﬁne the self-adjoint operators Qk : L2 (R2(q1 ,q2 ) ) → L2 (R2(q1 ,q2 ) ) and Pk : L2 (R2(q1 ,q2 ) ) →L2 (R2(q1 ,q2 ) ) by
Q 1 = q1 ,Q 2 = q2 ,

ℏ∂
i∂q1
ℏ∂
P2 =
i∂q2

P1 =

(4.3)

(♯01 ) 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 ). That is,(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 themeasurement MB(H) (O2 = (R2 , BR2 , F2 ), S[|u0 ⟩⟨u0 |] ). Assume that the measured 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 momentum of A1(the position of A1 , the momentum of A2 )



−p02

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Chap. 4 The Big Three in Greek Philosophy (Aristotle)
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 not commute, and thus,
the simultaneous observable does not exist.
Thus, we can not test the (♯3 ) experimentally.
Remark 4.7. 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.

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101

Chapter 5

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

Aristarchus· · · Heliocentrism

Archimedes· · · buoyancy, lever

Eratosthenes· · · the measurement of the earth

Ptolemaeus· · · 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 aresolid 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, Practicalresearch was respected. In Section 5.3, I will explain Heliocentrism of Aristarchus 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 on black-and-whitein the experiment. That is,(♯) ”Geocentrism vs. Heliocentrism” is a philosophical problem (i.e., the problem of worldview), andnot the problem of truth or falsehood.

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Chap. 5 Around Alexndria; 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 most inﬂuential works in the history of mathematics. It has been estimated to besecond only to the Bible in the number of editions published since the ﬁrst printing in 1482 AD. When I
think from now,(A) Euclid advocated geometric axiomatization and considered the parallel postulate, and was the mathematician who intuited that the concept of ”self-evident” isn’t self-evident. Here, the parallel postulate
is as follows:• If a line segment intersects two straight lines forming two interior angles on the same side thatsum to less than two right angles, then the two lines, if extended indeﬁnitely, meet on that side
on which the angles sum to less than two right angles.

α + β < 180◦=⇒ the intersection point• exsts

β
α

•

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5.2 Euclid(BC.330 - BC.275)

In spite of close attention of Euclid, the next wrong belief has been formed by ”Element”.
(B) It is the best method to start from a self-evident thing.
Much philosophers (Descartes and Spinoza, etc.) have fallen into this wrong belief. It is well known by now
that Descartes’ cogito proposition ”I think, therefore I am” is a proposition that is far from self-evident.
That is, there is nothing self-evident (= there is not unquestionable truth ).
As mentioned in Remark 3.13, many excellent philosophers follow Euclid as follows.
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

Here it should be noted that all of the worldviews in the left-side of the above are theories that are impossible
to deny.
♠Note 5.1. In ”Elements”, geometry is not only written but also algebra. For example, it is shownthat 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 eachcase, it contradicts the assumption that n is the largest prime number.

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5.2.2 non-Euclidean revolution

Discovery of non-Euclid geometry (due to Gauss(1777 - 1855), etc.) defeated the wrong belief (B) andasserted(C) Start from ”productive” than ”self-evident”!
that is, ”all is well that ends well”.
that is, In this paper, the (C) is called the non-Euclidean revolution,(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”. Paradoxicallysaying, 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 can construct a set
C = {a, c, g, j})This is not self-evident (i.e., trivial). For instance, Banach-Tarski theorem says that(F) If we adopt axiom of choice, we have to admit the following• A ball B is resolved into parts of several ﬁnite numbers, and we assume that it’s put together
again. Then, we can get the same two balls which are also the same as the ball B.

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

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5.2 Euclid(BC.330 - BC.275)

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.
Remark 5.1. I have an opinion that the main stream of Western philosophy is

Plato




Find the axiomatic approach to dualism
dualism−−−−−−−−−−−−−→ ?
Descartes-Kant philosophy=⇒

not yet completed
Euclidaxiomatic approach 
Euclid’s ”Elements” and the Bible are cultures that we non-Westerners ﬁnd astonishing.
♠Note 5.2. There was also tradition of pyramid construction, and Egypt was an advanced country ofmathematics. 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, weknow that• Eratosthenes (BC.275 - BC.194) : He was determined to 46250km the whole circumference ofthe earth. Cf. Sec. 5.5.
• Cleopatra(BC.69 - BC.30): The most beautiful woman in human history.
• Ptolemaeus (AD.83 - 168): Geocentrism

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Chap. 5 Around Alexndria; Hellenistic period

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 5.2.
(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 the earth.

The answer to (A1 ) Look at the lower left ﬁgure (lunar eclipse). Since the sun is very far, it suﬃces toconsider 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 that cos 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 Aristarchus (BC.310 - BC.230)

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 thesmall 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 ”the mass” is trivial.
Therefore, I agrre that

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

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Chap. 5 Around Alexndria; Hellenistic period

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,the shape of this matter is assumed a cone.)[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(the density of water is ρ(= 1).)

S: bottom area, h: height
f1

-

? ? ?

⇑ h

buoyancyS

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 is not connectedwith 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 thecrown without breaking the crown”. Archimedes notices next answer (♯) during bathing: while
shouting with joy too much ”Heureka!”(=”I have found it!”), was running around the streets
naked without even wearing clothes.
He found:(♯) Preparing the gold bullion of the weight same as the crown, compare the weight the gold bullion
and the weight is the crown in water. Then, we can, by the (5.1), compare the volume of thegold bullion and the volume of the crown.
♠Note 5.4. For each great discovery, an anecdote (or, a catch copy, 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 7.9)
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 10.6)(♯6 ) Wttgenstein· · · · · · primary school teacher, Gardener, Guardian: Russell (cf. Sec.12.1.1)(♯7 ) Einstein· · · · · · Elevator(♯8 ) quantum mechanics· · · · · · Heisenberg’s uncertainty principle (cf. ref. [36], or, Note 4.1 of ref.
[71])
Here, the (♯8 ) is my opinion (Has Heisenberg’s uncertainty relation ever been used eﬀectively inphysics? (Sec. 4.3 in ref. [71])).

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

4πr 3
,
3

The surface area of the ball B = 4πr 2

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

Thus,”Volume of Sphere” = ”Volume of Cylinder” − 2 × ”Volume of Cone”= 2πr 3 − 2 ×

4πr 3
πr 3
=
3
3

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5.4 Archimedes (BC.287 - BC.212)

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 theworld”. 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((B1 ) 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 the Earth revolve around the gravity
of both the earth and the Sun.
If Archimedes said so, science history would be history which is completely diﬀerent from now.
♠Note 5.5. Archimedes’ arguments were so clear that even elementary school students could understandthem, 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 the worldand so on. And the ”last words” was• Do not disturb my circles!
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 famous scientist, so they instructed him not to do any harm. However, when Archimedes was thinking about writing a ﬁgureon the sand, he was almost taken away by the Roman soldiers, 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 ancientscientists” until the end of his life.

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Chap. 5 Around Alexndria; Hellenistic period♠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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5.5 Eratosthenes (BC.275 - BC.194)

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 and believed that the Earth was asphere. 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 sphere since 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 at noon 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.

NP

A
θ
S

Sun

θ

sun light114
SP

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Chap. 5 Around Alexndria; Hellenistic period
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 : 19
then, 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 Ptolemaeus (AD.83 - AD.168)

5.6 Claudius Ptolemaeus (AD.83 - AD.168)
5.6.1 The ancient scientiﬁc collected studies
Ptolemaic Dynasty is ruined by the death of Cleopatra, Rome became the heyday of the Five Good Emperors
era. At this time, Ptolemaeus (AD.83 - 168) played an active part in Alexandria. In his book ”Almagest”, headopted Aristotle’s Geocentrism (i.e., the sun goes around the earth). Ptolemaeus explained the retrogression
seen at a planet in Mars such as Mars revolves around the earth while drawing a small circle as ”epicycle”.

Ptolemaeus compiled the latest theory in those days and concluded the Geocentrism (= Ptolemaic system)
under the enormous measured data.
(A) Ptolemaeus followed Aristotle, Archimedes, etc.
And it is sure(B) Ptolemaeus is a top-notch researchers.
He was the scientist who gave the most importance to observation among the ancient scientists.
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 Prizein 1979 for work that became a cornerstone of particle physics, said of his book [115] ”To explain theword; 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

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Chap. 5 Around Alexndria; Hellenistic periodvarious free numerical parameters, which are then found by constraining the predictions of themodels to agree with observation.
Decades ago, when I saw the planetarium when I was a kid, a commentator explained that ”Ptolemaeus
asserted a foolish Geocentrism, and this was corrected by Copernicus.” However, when I went to theplanetarium a few months ago, a commentator praised Ptolemaeus, saying, ”His epicycle model isgroundbreaking at the time.” It was great to hear a similar opinion to Weinberg’s at a planetariumin the Far East island nation.

♠Note 5.9. Recall the following ﬁgure:poem

science

The arche is ⃝⃝

−−−−
→
−

Socrates−−−−−→

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

student

Plato

−
−−−−
→

···

−
−−−−
→ Scholasticism
Middle Ages

vs. Sophists

That is, Aristotle bridged the gap between this pseudoscience (≈ the arche is ⃝⃝) and the base forthe foundation that would become science. Archimedes is the discoverer of ”principle of buoyancy”,
which belongs the realistic worldview. Ptolemaic system is based on the realistic motion function
method. Thus, we can get as follows (cf. Classiﬁcation 1.11 [the classiﬁcation of philosophers]).
(♭1 ): the realistic worldview (physics)
Hērakleitos, Aristotle, Aristarchus, Archimedes, Eratosthenes, Ptolemaeus,Galileo, Newton, Einstein, · · ·(Although mathematics is not a worldview,Pythagoras, Eudoxus, Euclid)








 (♭21 ): the ﬁctional worldview (Western philosophy)
Plato, Scholasticism, Descartes, Locke, Leibniz, Berkeley, Hume, Kant, Husserl
(♭)






(♭22 ): the logical worldview (=the logical spirit=the spirit of ”Think logically!”)





Boole, Frege, Peirce, Saussure, Russell, Wittgenstein, Hempel, Quine, Popper








 (♭23 ): the mechanical worldview (statistics, quantum language)
Parmenides, Zeno, J. Bernoulli, statistics (e.g., Fisher), quantum language
In the above, the following two are greatest (cf. Note 2.7):
geocentrism 1

the earth is round(C1 ) [astronomy]:

Pythagoras−→

Pythagoras
BC582-496

geocentrism 2−→

Aristotle

geocentrism 3−→ Ptolemaeus

BC400-347
BC384-322
method ofexhaustion
Elements

BC582-496
Pythagorean theorem(C2 ) [math]:

Eudoxus−→EudoxusBC400-347

117−→

EuclidBC330-275

AD83-168
quadrature−→ Archimedes
BC287-212

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Chapter 6

The Middle Ages
The Middle Ages may be characterized as ”the time of the thought stop for about 1500 years” Thus,
it is called”Philosophy is a maidservant of theology”
In this chapter, we discuss:(♯1 ) Augustinus(354 - 430): Christianity became the state religion of the Roman Empire. Subjective
time theory(♯2 ) Anselmus(1033 - 1110): the father of Scholasticism, Arguments for the existence of God, ”Realismus” in Problem of universals(♯3 ) Thomas Aquinas(1225 - 1274): Completion of the scholasticism (Summa Theologica): Compromise between Plato philosophy and Aristotle philosophy(♯4 ) Ockham(1285 - 1347): Ockham’s razor, ”Nominalismus” in Problem of universals

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6.1 Augustinus(AD. 354 - AD.430)

6.1 Augustinus(AD. 354 - AD.430)
6.1.1 Philosophy is a maidservant of theology

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 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 (AD. 354 - 430) is one who extended a helping hand to dying Plato philosophy. Catholic father
Augustinus used Plato philosophy to reinforce a theoretical backbone of Christianity. For this,(B) It’s desirable that God (in Christianity) and Idea (in Plato philosophy) have the similar nature.
The opinion of the sophist (”Man is the measure of all things”) is contrary to the opinion of Socrates-Plato,
and also to the opinion of Christianity (”God is the measure of all things”). Thus, Socrates-Plato andChristianity have an aﬃnity. In other words, there are reasons to think as follows.
Plato’s Idea = (A device that make the absolute goodness visible)≈ (A device that make the intellect of ”God” visible) = church
Augustinus might think so. Of course, they cannot accept Aristotle’s Idea (i.e., the Idea that came down toearth) . Because it would be more valuable if the Idea were in heaven. It could be said that ”Christianity
was Idea theory arranged for the people (Nietzsche),” and conversely, it could be said that ”Idea theory wascelebrated as Christianity for the intellectuals.”
The Plato philosophy got the strongest supporter (i.e., Christianity).
(C) Philosophy won a help from Christianity. But this implied ”Philosophy is a maidservant of theology”.
And philosophy fell into a thought stop, but, at least, Philosophy survived.
All proceeded as Augustinus’ plan.
♠Note 6.1. If we emulate Review 3.10 and forcefully apply the three key words of dualism to Augustine,
we get the following
dualism \ three key-words[A](= mind)[B](Mediating of A and C)[C](= matter)⃝:
3 Plato: sunRemark 3.7

actual world

Idea

Augustinus; Christianity

earthly city

churchquantum language

measured value

observable/[Idea world]/[the city of God]state[system]If so, we have to acknowledge the following,• Plato −−−−−→ Augustinus −−−−−−→ Descartes −−−−−−→ · −−−−−−→ Quantum language
regress

progress

progress

progressbut we do not stick to this since Christianity is not related to a measurement theory.

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Chap. 6 The Middle Ages
It is unclear whether the Christian fathers (such as Augustine) took Plato’s philosophy seriously ornot. However, the famous philosopher Anselmus barely appeared 600 years after Augustine, so I don’t
think they were serious.
♠Note 6.2. Readers may have the following question:(♯1 ) Why is paranormal theory always the mainstream of Western philosophy?
My opinion is as follows. Recall the Platonic method of telling philosophy:
world is so(♯2 )Live sothereforeﬁctional worldview (literary truth, pseudo- truth) −−−−−−→ Ethics · morals
main subject

introduction·preface·ﬁction

Therefore,• world is so is secondary, subsidiary,• you should do so is main theme
Since the ethics and morality of the main subject is fully backed by Christianity, the ”description of
the world” in the introduction part can be any fairy tale or ﬁction. However, as I explained in Plato’s
section,(♯3 )) The description of the world in philosophy requires a ”dualistic idealism”.
Also, note that dualistic idealism is paranormal. This is the answer to the question (♯1 ). However,assume that there is a scientiﬁc theory in dualistic idealism. Then, we can expect that(♯4 )) while fancy theories wander around in dualistic idealism, they gradually evolve and converge intoa solid theory.
It is the purpose of this paper to show this. Speciﬁcally, we show that
• Plato → Descartes → Locke → Kant → Wittgenstein → Quantum language
scientiﬁc theory

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6.1 Augustinus(AD. 354 - AD.430)

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

This section is written in the following reference:• [73]: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)
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 weordinary people want to know most is, how should we live? Christian Fathers, as God’s spokespersons,preached this to the people as the teachings 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 the world
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 we believe in (D1 ), then we consider that
(D3 ) Time was also made at the same time as the world.
Therefore,(D4 ) The sentence ”before God made it” is nonsense.
If we are told by fathers of Christianity so, we think that
(D5 ) Well, I didn’t read the Bible very well myself. I had a boring question ((D 2 )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 an understanding of the world for the sake ofdeepening our faith.

O
O

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Chap. 6 The Middle Ages
Augustinus asserted the following in his book ”Confessions”.
(E):Augustinus’ theory of time as a short story(E) Only present exists, and neither future nor past exist.

In fact,(F) the future is in ”prediction”, the past is in ”memory”. There is what we can realize ”only now”.
This is the beginning of the subjective time (which may be a main theme in philosophy). Although this”time” cannot be used in science, this time can be used in the Platonic method of telling philosophy as
follows.
Only present exists

Live joyfully now!
therefore−−−−−−→ Ethics · morals

ﬁctional worldview
introduction·preface·ﬁction

main subject

It is well known that St. Augustinus said that• the past does not exist because of its being already gone, that the future does not exist 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”. Thus, the linguistic Copenhagen interpretation (E 2 ) in
Sec. 1.1.2 says that this (K) is not a statement in quantum language. Thus, the (G) is not scientiﬁc, thatis, there is no experiment to verify the (G). Now,
• Augustinus’ tense (past, present, future) is a kind of sermon. But it may be interesting in comparison
with the linguistic Copenhagen interpretation (cf. (E2 ) in Sec.1.1.2), i.e.,
There is no tense in science.
Thus, it is prohibited that Augustinus’ tense (i.e., the subjective time) is discussed in science. However, we
can appreciate literary pleasure from the philosophical discussions.

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 not enter into the realmof 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’t know 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, some researchers
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 ) in Sec.1.1.2 earlier).

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6.1 Augustinus(AD. 354 - AD.430)

For the quantum linguistic understanding of ”wave function collapse”, see:ref. [63] 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)

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 interestedin ”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(♭) 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.
♠Note 6.3. ”What is the subjective time?” This is a problem of brain science. It is sure that catsand dogs etc. have clock gene or biological clock, thus they surely feel the subjective time. This isa scientiﬁc problem. Also, measuring the time with a clock is also a measurement. However, when
you measure time with your brain clock, it is not a measurement. A measurement that only you canmake 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 part of 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.

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Chap. 6 The Middle Ages
♠Note 6.4. For completeness, let us rewrite as follows.
1 How should we live?
2 How is the world made?
Here,• ” 1 : the problems of life” and ” 2 : the problem of world” are diﬀerent things
In spite of the diﬀerence, we prefer to Platonic method of telling philosophy:• the ”logic” which is dressed so that 1 may be derived from 2 .
In this sense, the 2 is a reason added later. We might be, by common sense, convinced that ”theworldview was to describe the world plainly and with no ﬁction”. However, Plato and Augustinus
consider that• the worldview is to create the world that it is convenient for faith or doctrine.

O
O

O

O

O

O

O

This is a replacement of the problem. However, this succeeds in science as well as philosophy. As
seen later (Kant’s Copernican revolution, Wittgenstein’ words ”The limits of my language mean thelimits of my world”, and ﬁnally, quantum language),• the worldview is not to describe the world plainly and with no ﬁction, but to create the worldthat it is convenient for faith or doctrine.
That is, ”More abstract painting than realistic painting”. Concretely saying, for example,• When there is a kind of the paint only in ”red” and ”green”, We draw as much as possible itseems realistic picture in this two colors
This is not only the philosophical case but also the scientiﬁc case (i.e., quantum language). Quantum
language is prohibited from using anything other than two axioms (Axiom 1 and Axiom 2 in Sec.1.1),
which is the same as the paint example above. Thus I think that
• It is not an exaggeration to say that, since Plato, the Copernican revolution has been the normof philosophy.

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6.2 Scholasticism –Graft Bamboo (=Aristotle) to a tree (=Plato) –6.2 Scholasticism –Graft Bamboo (=Aristotle) to a tree (=Plato) –6.2.1 Aristotle’s philosophy spread to the Islamic world
Plato’s philosophy survived with the backing of Christianity (Augustine). The philosophy of Aristotle spreadto Islam. I don’t know the details of the reason why,(A) Plato’s philosophy must have 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 and supported 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 was centered in Baghdad which was famous on the ArabianNights. Western Islamic culture developed around Cordoba in the Andalusian region of southern Spain andbecame the largest city in the world in the 10th century. At that time, the Islamic world learned a lot of
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 followership.
♠Note 6.5. In this paper we adopt the story such as (A). Actually, it may not be such a simple 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 tipof the Islamic culture!
I think it is true that• 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 to recapture it, interaction with Islamic culture was facilitated. Aristotle’s philosophy ﬂowed into the West, merged with Plato’s
philosophy, and settled in as Schola’s philosophy. That is, Scholasticism was born. As the typical persons ofScholasticism, we list up as follows.
(B1 ) Anselmus (1033 - 1109) ”The father of Scholasticism”, ”Realismus”(B2 ) Thomas Aquinas (1225 - 1274) ”Summa Theologica”, Greatest man in Scholasticism(B3 ) Ockham (1285 - 1347) ”Ockham’s razor”, ”Nominalismus”
After all,• in the beginning, Plato’s dualistic idealism was the most popular , but gradually Aristotle’s inﬂuenceincreased. It has become so ”science-like” that it has abandoned dualistic idealism. 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 and Aristotle philosophy
are ”oil (idealistic dualism) and water (realistic monism)”, and these are diﬀerent categories (cf. Assertion126

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Chap. 6 The Middle Ages
1.4 [ the history of worldview], Assertion 1.14). However, in this paper, we prepare the story such as• in the process of fusion of Plato philosophy and Aristotle philosophy, disadvantages of the theory of
Ideas became clear, which led to Descartes.
Figure 6.1 ( Scholasticism is a compromise between Plato and Aristotle).
the realistic worldview (physics)
Aristotle + Hellenism+Islam

Plato + Augustinus















A.D.1000∼1500⇒ Scholasticism ⇒















Newton → theory of relativity

Descartes·Kant → statistics·quantum language

the idealistic worldview (no experiment)
Also, by-product of crusade expedition, we have to note• ”Positional notation (= the discovery of zero)” of the origin in India
which will be mentioned in what follows.

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6.3 The discovery of zero

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, the merits for Christianculture is that Aristotle philosophy and the positional notation ﬂowed into Europe from Islam.
♠Note 6.6. Which was inﬂuential, Aristotle philosophy or the positional notation ? If this questionis the same as ”Which was indispensable for the proposal of Newtonian mechanics?”, we may choose
the positional notation , because Newton was a calculation maniac.
The positional notation is how to write numbers to learn in an elementary school. For example,+5040302, −15, +39.045, −81.5, +3.1415−1000, +0.009876, +0.3333 · · · , 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 all real numbers.
(the radix point was discovered in Europa of 16 century AD.) Of course, the discovery of zero is(B) the discovery of how to use zero called the positional notation

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 thepositional notation (the discovery of zeros), computation became easier. The word ”ALGEBRA (algebra)”
is originally an Arabic word. The formula for the solution of the quadratic equation : ax2 + bx + c = 0:√−b ± b2 − 4ac
x=
2a

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Chap. 6 The Middle Ages
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 the mathematics musthave 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 6.7. 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
Of course, it is needless to say that the biggest discovery is ” 0 :the discovery of natural numbers”.

O
O
O

O

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6.4 The proof of the existence of God

6.4 The proof of the existence of God
6.4.1 Anselmus (1033 - 1109)

Anselmus is the founder of Scholastic philosophy, famous for his ”Proof of the Existence of God”. His proof
is nothing more than a play on words that we use on a daily basis. The premise is not clearly stated (i.e., it
is not proven under any one worldview). In particular, the meaning of ”existence” is not clear. Therefore,
there is no need to read the proofs seriously.

O
O
O
O
O

Proof 6.2. Anselmus: the proof of God’s Existence
1 : God is a being than which none greater can be imagined . That is, the greatest possible being thatcan be imagined.
2 : It is obvious that God exists as an idea in the mind.
3 : A being that exists as an idea in the mind and in reality is greater than a being that exists only as
an idea in the mind.
4 : Thus, if God exists only as an idea in the mind, then we can imagine something that is greater thanGod. But we cannot imagine something that is greater than God.
5 : Therefore, God exists in reality.
The above proof is not worth validating since it is not discussed under a certain worldview.
This is a play on words.
♠Note 6.8. (i): I don’t understand the above proof. However, I associate the above argument with ”aset of all sets”, which is related to Russell’s paradox. Thus, the above may be a kind of self-referential
word play.
(ii): If the meaning of ”existence” is deﬁned in the sense of Deﬁnition 1.7, the existence of ”God” isobvious. Because ”God” is the most important key-word in the Bible.

6.4.2 Review: the worldviewism
Let us review the worldviewism.
(A): Worldviewism (cf. Sec.1.3.1)
The worldviewism is as follows.
world is so(A)conclusion
thereforeworldview −−−−−−→ discussions, calculation, logic, properties
premisesubject

That is,(B) The worldviewism is the spirit ”Start from the worldview” or ”Start from the ﬁrm premise!”.

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Thus, from our standing-point (i.e., worldviewism ), Proof 6.2 is not trusted. So far, we have seen many
paradoxes caused by arguments that ignore worldviewism. For example,(C) The list of our answers for philosophical unsolved problems (cf. (D) in Sec. 1.2.2)• What is probability (or, measurement, causality) ? cf. Sec. 1.1.1)• The solution of Zeno paradox (Flying arrow, Achilles and a tortoise), (cf. ref. [45], or, Sec. 2.4.2,
Sec. 2.4.3)( ”to solve Zeno paradox” = ”to understand DST-formula” (cf. Sec. 2.4.3 )• the measurement theoretical understanding of Plato’s allegory of the sum , (cf. Sec. 3.3.2)• Plato’s Idea theory≈Zadeh’s fuzzy theory≈Sausuure’s linguistic theory (cf. Sec. 3.5.3)• Syllogism holds in classical systems, but not in quantum systems (cf. Sec. 4.3.2)• Only the present exists (cf. Sec. 6.1.2)
which are a part of List 1.6.

6.4.3 The inﬂow of Aristotle philosophy
Although we cannot understand Proof 6.2, we think that Anselmus did not say much more than the following:(D1 ) ”Aristotle philosophy ﬂowed in via Islam, and I underwent the inﬂuence”.
That is,Aristotle −−−−−→ syllogism −−−−−→ logic −
−−−−
→
proof


via Islamyinﬂow
Plato

−
−−−−
→ Augustinus −−−−−→ God −−−−−→ The existence of God

In other words, a priest who had studied Aristotle’s syllogism felt like he had ”proved the existence of God”
by using the syllogism. We can see that ”Proof of the Existence of God” is a product of Schola’s philosophy(a reckless attempt to fuse and compromise Plato and Aristotle). If one were to choose Aristotle’s ”syllogism”
over Christ’s ”Bible, we think that
(D2 ) the expiry date of Plato’s myth ”God’s intellect = Idea” by Augustine has expired, and Aristotle’s
inﬂuence has increased.

Anselmus’ argument itself is nonsense. However, to exaggerate, the Schola philosophy, which is based on
Anselmus, is thought as(E) It is a revolution in Christianity that allows not only ”faith” but also ”reason”.
Of course, even if the theme of thinking is limited, there would be no objection if it is ”proof of God’s
existence”. The reason for Anselmus’ fame is the above (E), namely,(F) It is the discovery of the magic word ”proof of the existence of God” to be freed from the mind controlof ”stop thinking”.
After all, we see,(G) the history of Schola’s philosophy is that the ”eyes to see reality” of Aristotle’s style gradually matured
from the blind state of faith alone, and Descartes received the baton and opened the curtain of themodern era.
Nevertheless, great philosophers in Europe have repeatedly challenged the ”proof of God’s existence”, but I(especially non-Christian Japanese) would have no right to speak to the signiﬁcance and motivation of thischallenge.
♠Note 6.9. All scientists are interested to ”God”. ”What is God ?( = How about neuronal circuit

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6.4 The proof of the existence of God
concerning belief?)” and ”What is subjective time? (= How about biological clock?) are one of themost interesting problems in brain science.

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Chap. 6 The Middle Ages

6.5

Scholasticism; Problem of universals

6.5.1* What is the problem of universals?
The following is a review. Augustine (354-430), the greatest Catholic priest, adopted Plato’s Idea theory
to reinforce Christianity. Fortunately, the fundamental keywords (=existence) of both God (Christianity)and Idea (Plato’s philosophy) are quite similar, since both of them have the mood of heavenly existence,
so Augustine’s plan succeeded and the honeymoon era between Christianity and Plato’s philosophy lasted
for more than 500 years. There, Aristotle’s philosophy ﬂowed in via Islam. Plato’s philosophy, which is adualistic and idealistic description of the world, and Aristotle’s philosophy, which is a monistic and realistic
description of the world, are like water and oil (or, grafting a bamboo to a tree), and thus , naturally it
was confusing. As shown in the previous section, the ﬁrst confusion was the ”proof of God’s existence” ofAnselmus. The second confusion is the ”Problem of universals” in this section. The problem of universals isthe biggest dispute in Scholasticism. This problem is as follows.
(A): What is ”Problem of universals”?
Problem 6.3. ”Problem of universals” is as follows.
(A1 ) It is certain that Mr. Smith, Mr. White, Mr. Brown, etc. exist. Then, we have the following
problem:
Do ”optimism”, ”American”, ”Japanese”, ”honesty”, ”intelligence”,etc. exist?
If ”Yes”, then, ”Realismus”. If ”No”, then, ”Nominalismus”.
Most people may have the following question:• Why did great Fathers argue eagerly in a problem like such word game?
Thus, our present problem is ”What is ’Problem of universals’ ?”. In what follows, this will beanswered from the quantum mechanical worldview ?
The following diagram will promote the reader’s understandingLet Ω be a compact space, in which every human’s state is assumed to be represented. Let mH : Ω → [0, 1][resp. mJ : Ω → [0, 1] ] be the membership function of ”honesty” [ resp. ”Japanese”] (see the picture below).

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6.5 Scholasticism; Problem of universals

mH

1

mJ

Ω
Here, deﬁne the observables OH = ({y, n}, 2{y,n} , FH ) and OJ = ({y, n}, 2{y,n} , FJ ) ( where ”y”=”yes”,”n”=”no”) such that
[FH ({y})](ω) = mH (ω),[FH ({n})](ω) = 1 − mH (ω)[FJ ({y})](ω) = mJ (ω),[FJ ({n})](ω) = 1 − mJ (ω)

for all ω ∈ Ω.
And consuder the following identiﬁcation:
OH ≈ m H ,OJ ≈ m J

Therefore, in the framework of QL, we can say that• ”Japanese” and ”honesty” exist.
in the sense of Deﬁnition 1.7.
Also, recall Example 3.11 (and Remark 3.12), that is,(A2 )linguistic version−−−−−−−−−−−→

Idea
Plato’s idea theory

signiﬁer, signﬁedSaussure’s linguistic theory

secondary quality

membership function

John Locke

Zadeh’s fuzzy theoryquantiﬁcationycognitiveyversion


yidentiﬁcation

quantiﬁcationjobservable

quantum language

That is,(A3 ) ”Idea of honesty”(= ”Degree of honesty”)−−−−−−−−−→ ”Membership function of honesty mH : Ω → [0, 1]”
quantiﬁcation−−−−−−−−−−−−−−−−−−→ ”the observable concerning honestyOmH = ({y, n}, 2{y,n} .FmH )”probabilistic interpretation

Also, note that\
Plato
idealism (no experiment)
Aristotle
realism (experiment)
quantum language[A](= mind)[B](Mediating of A and C)

actual world

Idea

————

————

measured value

observable

134[C](= matter)/[Idea world]
eidos[hyle]
state[system]

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Chap. 6 The Middle Ages
Thus, we see:
Answer 6.4. (A3 ) says that
(B1 ) ”Honesty” is an observable
Also, recall Deﬁnition 1.7, that is,(i): in quantum language (i.e., idealistic dualism), ”observable”, ”measured value” and ”state” exist(ii): in Aristotle standing point (i.e., realistic monism), ”state” exists.
Therefore, we say that(B2 ) Plato school agree to ”Realismus”: that is, ”honesty” exists.
e.g., Anselmus (1033 - 1109)(B3 ) Aristotle school agree to ”Nominalismus”: that is, ”honesty” does not exist.
e.g., William of Ockham (1285 - 1347)

After all, we think• This is the problem such that ”(since Augustinus) Plato school (”Realismus”) vs. (via Islam)Aristotle
school (”Nominalismus”). The power in the church gradually shifted to the Aristotle group. In thissense, it may be called ”confusion” than ”dispute”.
As seen in the following table, the problem ”realistic worldview(monism) vs. idealistic worldview(dualism)”
is the biggest dispute in philosophy and science. ”Nominalismus” (Ockham) in Problem of universals is alittle irrational since religion is not realistic.
Table 1.1 : realistic worldview vs. idealistic worldview
dispute ⧹ [R] vs. [L]⃝:a motion⃝:Ancient
Greeceb⃝:c Problem of universals⃝:
d space-time⃝:
e quantum theoryof science⃝:philosophyfRealistic worldview

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

Hērakleitos
Aristotle”Nominalismus”(Ockham)
NewtonEinstein
CarnapParmenides
Plato”Realismun”(Anselmus)
Leibniz
Bohr
Quine

⃝
c is a confusion. ⃝d is the Leibniz=Clarke correspondence (cf. Sec. 9.3.2), ⃝e is Bohra is my ﬁction, ⃝Einstein debates. Quantum language is proposed as one of answers to Bohr-Einstein debates(cf. ref.
[71]). Quantum language is proposed as one of answers to Bohr-Einstein debates(cf. ref. [71]). ⃝:f
Quine understood 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. 13.3).

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6.6 Thomas Aquinas (1225 - 1274); Scholasticism

6.6 Thomas Aquinas (1225 - 1274); Scholasticism
6.6.1 Grafting Bamboo (Aristotlelian science) to a Tree (Plato’s literature)

The Catholic priest: Thomas Aquinas (1225 - 1274) wrote ”Summa Theologica” as the summing-up of
Scholasticism. He was the most important at the intermediate time of Scholasticism (or, Problem of universals), that is,(1033−1109)(1285−1347)(1225−1274)−→
OckhamAnselmus −→Thomas Aquinas(”Realismus”)(”Nominalismus”)(compromise or fusion)(6.1)

His proposal is the compromise of ”Realismus” (due to Plato) and Ockham (due to Aristotle), thus, histheory has three key-words: (A1 ) : universalia ante res as Plato’s Idea (i.e., measuring instrument in quantum language)(A2 ) : universalia in rebus as Aristotle’s eidos (i.e., state in quantum language)(A3 ) : universalia post rem as actual world (i.e., measured value in quantum language).
Since Plato philosophy and Aristotle philosophy are ”oil and water”, and these are diﬀerent categories (cf.
Assertion 1.4[ the history of worldview]), it is a matter of course that Aquinas’ idea is irrational. However,as seen in Table 5.2, we say:(B) in the process of fusion of Plato philosophy and Aristotle philosophy, deﬁciencies in the theory of
Ideas is turned to reveal, this led to Descartes.
that is,Table 5.2 : Key-words in each worldview
Platoidealism (no experiment)
Aristotle
realism (experiment)

actual world

Idea

————

————

Thomas Aquinas

universale post rem

universale ante rem

DescartesI, mind, brainbodyquantum language

measured value

measuring instrument/[Idea world]
eidos[hyle][universale in re]//[matter]state[system]♠Note 6.10. As mentioned in Sec.3.5, I am not conﬁdent in the above table, because the theory ofideas and Scholasticism were not created with the intention of measurement. However, I will proceedwith this policy of discussion in this paper.

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Review 6.5. In the above table, the meaning of the correspondence of key-words is as follow.
(actual world,Idea world)

Plato(post rem,ante rem)(mind,body)compromise+ Aristotle −−−−−−−−−−−−−→ Thomas Aquinas −−−−−−−−−−−−→ Descartes(eidos,hyle)

Impossible attempt(in re, /) or (/, in re)

Descartes’ genius(/,matter)(measured value,measuring instrument)−−−−−→
progressquantum language(state,system)

Also, the formula 6.1(=the history of Scholasticism ) is rewritten as follows.
Plato (+ Aristotle)
Plato + Aristotle
Aristotle
Plato
Augustinus −→Anselmus−→Thomas Aquinas−−−−−→
Ockham
(6.2)
overstep(”Realismus”)(”Nominalismus”)(”Realismus”)(compromise or fusion)
Thus, I guess that(C) Ockham wanted to shave Plato’s Idea theory with a razor. (cf. (C3 ) in Sec. 6.7).
♠Note 6.11. It’s said that the problem of universals is incomprehensible. This is due to the fact:• The fusion of Plato and Aristotle is an unreasonable trial,
That is,(♯1 ) Aristotle (as well as Newton) do not ﬁt in Christianity.
(♯2 ) Although the key-words of Thomas Aquinas philosophy and those of Descartes philosophy are
similar (i.e., those have three key-words as seen in Table 5.2), this may be accidental. (cf.
Review6.5).
(♯3 ) The formula (6.2) implies the confusion in Scholasticism. Therefore, the ”−→ in formula (6.2)”
does not mean ”progress”.
♠Note 6.12. A seen in the above, the problem of universals is in confusion. However, we thinkthat Scholasticism belongs to the linguistic worldview. Thus, we have (cf. Classiﬁcation 1.11 [theclassiﬁcation of philosophers]).
 (♭1 ): the realistic worldview (physics)
Hērakleitos, Aristotle, Aristarchus, Archimedes, Eratosthenes, Ptolemaeus,Galileo, Newton, Einstein, · · ·(Although mathematics is not a worldview,Pythagoras, Eudoxus, Euclid)








 (♭21 ): the ﬁctional worldview (Western philosophy)
Plato, Scholasticism, Descartes, Locke, Leibniz, Berkeley, Hume, Kant, Husserl
(♭)






(♭22 ): the logical worldview (=the logical spirit=the spirit of ”Think logically!”)





Boole, Frege, Peirce, Saussure, Russell, Wittgenstein, Hempel, Quine, Popper








(♭ ): the mechanical worldview (statistics, quantum language) 23
Parmenides, Zeno, J. Bernoulli, statistics (e.g., Fisher), quantum language

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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, isknown 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!

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. Thus I have a question:(C) Is there a case as which Ockham’s razor is functioning eﬀectively (besides the mathematical theorems)?
We see that
(D1 ) Relying on Ockham’s razor, Ptolemy claimed the theory of celestial motion from the observational
data of the time.
And(D2 ) relying on Ockham’s razor, Galileo claimed for a geodynamic theory from the observational data ofthe 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 think so. The author is a layman anddoes not know much about it. Ockham, however, is the leading 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)(cf. (C) in Sec. 6.6).

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Chap. 6 The Middle Ages

It makes sense, then. However, it would be too short-sighted to think that it must be better to eliminate
such hypotheses as the existence of ”something ideal in the heavens. Plato must have thought that the idea
theory was a fairy tale. It would be immature and foolish to cut oﬀ the fairy tale. As mentioned above,
Ockham’s razor is dependent on the environment around. Under Newtonian mechanics, Geocentric model
must be cut oﬀ. And thus, we think that(G) under QL, the theory of Idea must not be cut oﬀ
since our proposal in this text is the following:♠Note 6.13. Realism and idealism are based on diﬀerent principles. In realism (e.g., physics), experimental veriﬁcation is everything. On the other hand, the principles of idealism (e.g., metaphysics) arevarious, such as ”beauty”, ”usefulness”, and ”Ockham’s razor”, but in the end, idealism must be supported by many people. Mathematics is the most successful metaphysics and meets all of ”beauty”,”usefulness”, ”Ockham’s razor” and so on. In order for quantum language to gain the support ofmany people, it is necessary to appeal its ”usefulness”. However, in this book, we adopted a method
to deepen the understanding of quantum language by appealing its similarity to Western philosophy.

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141

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 clariﬁedby (♯1 ). In this sense, (♯2 ) is included in (♯1 ).

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7.1 Paradigm shift

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 timeline 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
artist• 1510: Raﬀaello, ”The School of Athens”, Admiration to ancient Greece
religionist• 1517: Luther, Protestant Reformation• 1519 - 20: Magellan, the ﬁrst circumnavigation of the Earthartist• 1540s: Michelangelo, ”The Last Judgment”• 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• 1620: F. Bacon, ”knowledge is power”,The father of British Empiricism
illuminatorscientist• 1633 Galileo’s trial ”And yet it moves”

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

After Galileo: The era of thought
Table 7.2.

• 1637: ”Discourse on the Method”, Rene Descartes (1596 - 1650),

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7.1 Paradigm shift
the father of modern philosophy, Cogito proposition• 1670: Pascal, ”Pensèes”• 1685 - 1750: Bach• 1687: Newton, ”Principia”• 1688: Glorious Revolution• 1690: John Locke, the father of British Empiricism, ”An Essay Concerning
Human Understanding”, tabula-rasa, the secondary quality• around 1700: Jakob Bernoulli, the law of large numbers• 1703: Leibniz, ”New Essays on Human Understanding”• 1715 - 16: Leibniz-Clarke correspondence (cf. Sec. 9.3.2)• 1739: Hume, ”A Treatise of Human Nature”• 1781: Kant, ”Critique of Pure Reason”

philosopher
enlightenerartistscientist

philosopher
mathematician
philosopher
philosopher
philosopher♠Note 7.1. The law of large numbers, discovered by J. Bernoulli(1654 - 1705), is as follows.
(♯) If a fair coin (one with probability of heads equal to 1/2) is ﬂipped a large number of times, theproportion of heads will tend to get closer to 1/2 as the number of tosses increases.

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

Causality (and experiment)( Galileo, Bacon, Descartes, Newton)

the realistic worldview (physics)

Aristotle· · ·Plato. . .

British empiricism
−−−−−−−−−−−−−−→ Newton → theory of relativity
Bacon;
Induction
A.D.1000∼1500 ⇒ Scholasticism ⇒












Continental rationalism −−−−−−−−−−−−−−−−−→ Descartes,Locke,Leibniz,...,Kant → QL
Descartes: Deduction








the linguistic worldview (no experiment)
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 scienceOO

His proposal is as follows.
(A) 1 :Exclusion of bad idols−→ 2 :data, collection by observation and experiment−→ 3 :scientiﬁc theory, 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.
Here, ” 2 + 3 ” is called ”induction”.

O
O
O

O

OO

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 of theorems were derived from Euclid
axioms, that is,145

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7.2 Francis Bacon (1561 - 1626): The father of empiricism, Inductive reasoning
deduction
Euclid axioms −−−−−−−−→ theorems(B1 ) Mathematics:
On the other hand, Bacon emphasized the importance of observation (or experiments), that is,
induction(B2 ) Science:
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”

OO

And he practiced Bacon’s induction principle, and proposed Newtonian mechanics as follows.
(C): 1 exclude bad idols (i.e., Aristotle’s purpose, Geocentrism) −→ 2 Data collection (due to Tycho
Brahe, Kepler, Galileo) −→ 3 Science theory (Newtonian mechanics)

O♠Note 7.2. This may be say in a philosophy side. We must add the next section (good idols).

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)

O

O

O

Induction principle (by good idols) is as follows.
(D) 1 believe good idols −→ 2 Data collection −→ 3 Science theory
Newton said:”I frame no hypotheses”

OO

And he practiced Bacon’s induction principle (good idols), and proposed Newtonian mechanics as follows.
(E): 1 believe good idols (i.e., Causal relation) −→ 2 Data collection (due to Tycho Brahe, Kepler,
Galileo) −→ 3 Science theory (Newtonian mechanics)

O♠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)♠Note 7.4. S. Weinberg (1933 -2021), a physicist at the University of Texas, Austin, won a Nobel Prizein 1979 for work that became a cornerstone of particle physics, said in his book [115] ”To explain theword; 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 andeighteenth centuries would invoke Bacon as a counterweight to Plato and Aristotle, .... It is notclear 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 ofphilosophers, enlighteners, astronomers, mathematicians, adventurers, artists, educators, politicians,
religionists, and the general public. The diﬃcult and important thing is to get public support for thescientiﬁc 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 willshow in the next chapter, I consider as follows:146

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

the realistic worldview (physics)
Aristotle· · ·Plato. . .

British empiricism
−−−−−−−−−−−−−−→ Newton → theory of relativity Bacon; Induction

A.D.1000∼1500 ⇒ Scholasticism ⇒












Continental rationalism −−−−−−−−−−−−−−−−−→ Descartes,Locke,Leibniz,...,Kant → QL
Descartes: Deduction








the linguistic worldview (no experiment)

And, the part:
Continental rationalism(♯1 ) ”−−−−−−−−−−−−−−−−→ Descartes,Locke,Leibniz,...,Kant ”
Descartesis usually rewritten by the following (♯2 ), which is called ”the ﬂower of modern philosophy”
Induction (due to Bacon)








BritishEmpiricism








Deduction
[Locke, Berkeley, Hume](♯2 )Descartes−→−→ Kant philosophy[The father of modern philosophy]
Deduction(due
toDescartes)[summing-up]








Continental
Rationalism[Leibniz](Note that, in spite of ”British Empiricism”, Locke, Berkeley, Hume were not concerned with experiment. Thus, I assume that they did not belong to the realistic worldview.)

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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 Ptolemaeus (AD.83 - AD.168) can explain the motion on planets by epicycle
theory. Thus, I also think his Geocentrism to have reached the scientiﬁc level at the time. (cf. Sec.
5.6).
However, Heliocentrism due to Copernicus (1473 - 1543) might be controversial whether it had reached thescientiﬁc 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 washungry for the sun central principle. Therefore, there is a foundation that 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 [115] ”To explain the word; 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 aesthetic 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 paper, ”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 great scientist.
In this paper, I discuss the next.

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Chap. 7 Early modern – Scientiﬁc revolution: From Geocentrism to Heliocentrism(A): What is ”Heliocentrism vs. Geocentrism”?
Now,(A) Note that motion is relative. Thus, if the earth is assumed to be at center, the sun goes aroundthe earth (i.e., Geocentrism). Also, if the sun is assumed to be at center, the earth goes aroundthe sum (i.e., Heliocentrism). Hence,
The diﬀerence between Heliocentrism and Geocentrism
may be only a diﬀerence of how to take the coordinate system.

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
VaticanAnalyzing 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,(♯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) have done?
In order to answer to this question, we ﬁrst have to clarify the meaning of ”Heliocentrism vs. Geocentrism”.

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7.3 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” is somewhat 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 around the sum (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

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

Causality (and experiment)( Bacon, Galileo, Descartes, Newton)

Thus, this 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 the inconvenience is as follow:






Copernicus ⇒ Kepler ⇒ Galileo
However, these are not suﬃcient to decide ”Geocentrism vs. Heliocentrism”. It is a matter of 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 the worldview 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

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

Thus, Galileo thought it was a matter of measurement data, but the church thought it was a matter ofworldview. Therefore,• at the time (1633) of the trial of Galileo, the Church was convinced that it would not lose its argumentwith 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.

After all, Galileo was the active leader of the overthrow of the worldview of Aristotle, but he could notpropose 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 themost famous and important book in science. Three laws of Kepler were derived from three laws of dynamics
and the law of universal gravitation. Principia was written based on elementary geometry and not thediﬀerential and integral calculus. Why did Newton (= advocate of diﬀerential and integral calculus) notwrite Principia based on 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)−−−−−−−−−−→
paradigm shift

Causal relation
That is, we see:

Kinetic diﬀerential equation method: (Newton)152

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

Newtonian worldview

paradigm shift(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 paper is the worldviewism(cf. Sec. 1.3.1). As mentioned frequently up
to this point,(C1 ) The argument in ordinary language (or, in the motion function method (cf. Sec.2.3.3)) is fuzzy, andthus, ”Geocentrism vs. Heliocentrism” cannot be decided. Thus, we need a new worldview.
In Principia, Newton proposed Newtonian mechanics (i.e., Newtonian worldview) and showed that
(C2 ) When the motion of the sun and the planets is studied, the calculation becomeseasy under the assumption that the planets go around the sun.
Therefore, even the deﬁnitions ”center” and ”go around” depend on the worldview. After all, we conclude
that(D) ”Geocentrism or Heliocentrism” is not an issue that can be settled, no matter how accurate theobservations are. That is, ”Geocentrism vs. Heliocentrism” is not the problem of measurements, but
the problem of the worldview.
♠Note 7.8. Next is said to be the three major discoveries of modern science(♯1 ) Aristotelian worldview (purpose) −→ Newtonian worldview (causal relation)(♯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 to consider
that (♯2 ) is a great episode of the birth of (♯1 ).

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7.4 Principia; Newtonian worldview♠Note 7.9. Here, we have (cf. Classiﬁcation 1.11 [the classiﬁcation of philosophers]).
(♭1 ): the realistic worldview (physics)
Hērakleitos, Aristotle, Aristarchus, Archimedes, Eratosthenes, Ptolemaeus,Galileo, Newton, Einstein, · · ·(Although mathematics is not a worldview,Pythagoras, Eudoxus, Euclid)








 (♭21 ): the ﬁctional worldview (Western philosophy)
Plato, Scholasticism, Descartes, Locke, Leibniz, Berkeley, Hume, Kant, Husserl
(♭)






(♭22 ): the logical worldview (=the logical spirit=the spirit of ”Think logically!”)





Boole, Frege, Peirce, Saussure, Russell, Wittgenstein, Hempel, Quine, Popper








(♭ ): the mechanical worldview (statistics, quantum language) 23
Parmenides, Zeno, J. Bernoulli, statistics (e.g., Fisher), quantum language

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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 theground faster?
The only way to settle this issue would be to conduct an experiment. In fact, Galileo conducted thisexperiment on the Leaning Tower of Pisa, and discovered that ”two iron balls fall at the same time”. Thisis the famous ”Leaning Tower of Pisa legend”.
So far, so good.

But, in ”Dialogues concerning two new sciences”, Galileo said:• 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(G):”Dialogues concerning two new sciences” by Galileo Galilei ref. [24]Here, assume that(G1 ) 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 lightiron ball B connected by a string, fall?

Common sense tells me that(G2 ) The iron ball [A+B] will fall at a speed halfway between the heavy iron ball A and the lightiron ball B
However, this contradicts (G1 ). Since the iron ball [A+B] is heavier than the heavier iron ballA, according to the assumption (G1 ), the iron ball [A+B] should fall faster than the heavier ironball A. The assumption (G1 ) is wrong because it is strange that the falling speed increases only byconnecting with a string. Assuming that the lighter iron ball falls faster than the heavier one at (G 1 )is a contradiction as well. Therefore,(G3 ) without experiment ( and without Newtonian mechanics) , we can conclude a heavy iron balland 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 7.10. For convenience, consider two kinds of propositions such as(H1 ) analytic propositions: propositions grounded in meanings, independent of matters of fact.
(H2 ) synthetic propositions: propositions grounded in fact.
If so, we have the following question:(I) Is the proposition ”A heavy ball and a light ball fall at the same speed” an analytic proposition
or a synthetic proposition?
This will be discussed later ( Chap 11).

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157

Chapter 8Modern philosophy (Dualism): Descartes
The following is called the ﬂower of modern philosophy:Induction (due to Bacon)








BritishEmpiricism








Deduction
[Locke, Berkeley, Hume]
Descartes−→−→ Kant philosophy[The father of modern philosophy] Deduction (due to Descartes) [summing-up]





 Continental Rationalism 






(♯)

[Leibniz](Note that in spite of ”British Empiricism”, Locke, Berkeley, Hume were not concerned with experiment. Thus, I assume that they did not belong to the realistic worldview.)
Here, the(♯) is located as follows:
the realistic worldview (physics)

Aristotle· · ·Plato. . .















A.D.1000∼1500⇒ Scholasticism ⇒British empiricism
−−−−−−−−−−−−−−→ Newton → theory of relativity

Bacon;
Induction












Continental rationalism −−−−−−−−−−−−−−−−−−−−→
Descartes: Deduction

(♯)

−→ QL

the linguistic worldview (no experiment)

It is unthinkable now, but I think it was a time when philosophers were respected and believed that”philosophy is the king of science”. Even so,• Why were top elites in those days absorbed in the useless philosophy?
As mentioned before, I think that this is due to Platonic method of telling philosophy. Is ita desire to replicate the success of Euclidean geometry (i.e., logic about ﬁgure) in philosophy?
The ”science of monism” was completed by Newton. Thus, I think they wanted to pursue a”science of dualism” .
In this chapter, we discuss Descartes, British empiricism, Continental rationalism.

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8.1 The theory of Ideas has expired

8.1 The theory of Ideas has expired
Let us review Platonic method of telling philosophy:
world is soLive sotherefore−−−−−−→ Ethics · morals

ﬁctional worldview
introduction·preface·preliminary ﬁction·prologue

main subject

Then, ethics, morals is main, and the ﬁctional worldview is only preface. Although it is desirable that theworldview in preface is related to a dualistic idealism (cf. Sec.3.4.1 ), but it didn’t matter even if it was
merely a fable. In fact, the theory of Ideas is just a fable. In an extreme case, using the psychological tricks
such as(A) ”Intellectual’s remark can be trusted”,”As for the beautiful woman, a heart is fair”,”We can trust the assertion of Kant who was too serious and stiﬀ.”, etc.
In the preface (=view of the world), it is enough to win the trust of the reader. A scientist is not expected
to be a person of character, but a moralist must be a person of character. Therefore, Kant had to be as
earnest and honest as a clock.
The main current of western philosophy adhered to Platonic method of telling philosophy. In Plato
philosophy, the worldview (=the theory of Ideas) is completely an allegory, and the main subject is due to
Socrates’ ethics. In cases of Augustinus and Scholasticism, the main subject is of course Christianity. Hence,
the worldview in the preface is not so important. For example, ”only the present exists” [ resp. ”barren
discussion: Existence of God, Plato or Aristotle?”] is the intellectual act of killing time in Augustinus
philosophy [resp. Scholasticism ]. However, after the age of geographical discovery and the Renaissance, themomentum for the renewal of fairy tales had increased. And Descartes thought that
The expiry date of the theory of Ideas (≈ a kind of fable) was expired.
Descartes, using self-referential cogito proposition ”I think, therefore I am”, proposed the worldview (i.e.,
Descartes philosophy (= mind-matter dualism)). Roughly speaking,the theory of Ideas −−−−−−−−→ mind-matter dualism(Plato)cogito turn

(8.1)

(Descartes)

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Chap. 8 Modern philosophy (Dualism): Descartes

The main problem in this chapter
Problem 8.1.

(A): Is the following model change progress?
Idea theory −−−−−−−−−→ mind-matter dualism
cogito turn(Descartes)(Plato)(8.2)[Note]: This problem is meaningless since the standard of ”progress” has not been set. Therefoe,in this paper, I replace this problem (A) with the next (B).

(B): Which is closer to quantum language, Plato or Descartes?

In this chapter, we devote ourselves to this problem.
The philosophy of worldview is only an anecdote, however, it should be eﬀective for general people (andthus, quantitative arguments are not desirable). After all, following Socrates’ self-referential statement ”Iknow that I know nothing”, Descartes also wanted to use self-referential trick ”I think, therefore I am”.
♠Note 8.1. There is a good reason for the birth of Newtonian mechanics, for example, the Age of
Discovery, positional notation (= the discovery of zero), etc. Newtonian mechanics was powerful,for example, and had the power to refute the theory of celestial motion. On the other hand, there
may not be a ﬁrm reason for the birth of Descartes philosophy as the continuation of Platonic methodof telling philosophy. My opinion is as follows.
(♯1 ) Under the Christian strong inﬂuence, it could not be free to discuss the ethics. And thus, western
philosophy devoted itself to the preface (i.e., worldview) than the main subject (i.e., ethics).
Platonic method of telling philosophy might be a clever strategy for western philosophy to coexist
with Christianity. The following question is signiﬁcant:(♯2 ) Is the cogito turn (8.1) progress, or a change in trend?
As mentioned in Preface, to answer this is one of the main purposes of this paper. In Chapter 13, we
will conclude that it is ”progress”, in spite that the change from Plato philosophy (i.e., Idea theory)
to Descartes philosophy (i.e., mind-matter dualism) is not inevitable.

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8.2 ”I think, therefore I am” is meaningless from the scientiﬁc point of view

8.2 ”I think, therefore I am” is meaningless from the scientiﬁc point ofview

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. One of Descartes’s most enduring legacies was his development of Cartesian or analytic
geometry, which uses algebra to describe geometry. There is even a plausible legend that Descartes,
who was sleeping in his bed, came up with the system in order to track the position of a ﬂy movingaround the ceiling.

y 6(2, 1)

1O

2

x

Figure 8.1: [Cartesian coordinates ]8.2.1 Discourse on the Method (1637)

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 the most famous philosophical proposition (= cogito proposition):”I think, therefore I am”
That is, Descartes think:I think that ’I think, therefore I am’

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Chap. 8 Modern philosophy (Dualism): Descartes
In spite that this is a most doubtful proposition (and thus, it is not in quantum language, as seen in Note
1.7 or, Proposition 8.3), Descartes believed that it was the unquestionable truth.
And further, bearing in mind Euclid’s success in geometry, Descartes was convinced that
• Every statements derived from the cogito proposition are absolutely trusted
This is just ”The method of rightly conducting one’s reason and of seeking truth in the sciences”. Therefore,
after all, Bacon and Descartes are in opposite positions. That is,(A1 ) Bacon emphasized the importance of experiments, and so asserted the inductive method. He wasoriented to physics (and further, British empiricism)(A2 ) Descartes believed ”good sense”, and so asserted the deductive method. He was oriented to metaphysics (and further, Continental rationalism)
In this paper, the following ﬁgure is actually drawn so far repeatedly.
the realistic worldview (physics)
Aristotle· · ·Plato. . .

British empiricism
−−−−−−−−−−−−−−→ Newton → theory of relativity
Bacon;
Induction
A.D.1000∼1500 ⇒ Scholasticism ⇒












Continental rationalism −−−−−−−−−−−−−−−−−→ Descartes,Locke,Leibniz,...,Kant → QL
Descartes: Deduction








However, the part:the linguistic worldview (non-physics)

Continental rationalism(♯1 ) ”−−−−−−−−−−−−−−−−→ Descartes,Locke,Leibniz,...,Kant ”
Descartesis usually rewritten by the following (♯2 ), which is called ”the ﬂower of modern philosophy”
 Induction (due to Bacon) 






British Empiricism








Deduction
[Locke, Berkeley, Hume]−→(♯2 )Descartes−→ Kant philosophy Deduction (due to Descartes) [The father of modern philosophy][summing-up] Continental Rationalism 







[Leibniz](Note that in spite of ”British Empiricism”, Locke, Berkeley, Hume were not concerned with experiment.
Thus, I assume that they did not belong to the realistic worldview.)
Descartes asserted:
Proclaim 8.2. The ﬁrst principle (= cogito proposition) in philosophy
Now,(A) Descartes doubted everything. And he arrived in the cogito proposition which has no doubted room.
That is, he arrived in(B) I think, therefore I am.
And, he proclaimed that the cogito proposition (B) is the ﬁrst principle in philosophy.

8.2.2* Two (scientiﬁc or non-scientiﬁc) interpretations of ”I think, therefore I am”
What is described in this section has been mentioned many times in the references [44]∼[73].
Recall the linguistic Copenhagen interpretation:

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8.2 ”I think, therefore I am” is meaningless from the scientiﬁc point of view

we will show two (scientiﬁc or non-scientiﬁc) interpretations of the cogito proposition: ”I think, therefore I
am”.
Proposition 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 is easy. 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.
[(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 have the situation:(C2 ) Tom thinks ”Tom thinks, therefore Tom is”
which means that”observer = Tom”,”matter = Tom”,

That is, this violates the linguistic Copenhagen interpretation. Therefore, the (C 2 ) is not in quantum
language, that is, it is just a play on words.
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. That is,
This play on words ushered in the modern era.
It is obvious that the above (ii) is the recommended answer.

♠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”
Thus, it can be said that Descartes ignored the teachings of Wittgenstein (i.e., ”What we cannot speak
about we must pass over in silence”).

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Chap. 8 Modern philosophy (Dualism): Descartes

8.3 Descartes’ strategy
8.3.1 From cogito to dualistic idealism
The most important key-word in Descartes’ philosophy is ”I”. 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

Def. 1.7

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” (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-matter dualism) is”body”(=”sensory organ”) ,”matter”(C)”I”(=”mind”, ”brain”) ,For completeness, it should be noted that this is not a consequence of the cogito proposition. That is thecogito proposition is the reason added afterwards. The cogito proposition is nothing more than a catch copy.
Principle 8.4. The principle of Descartes’ philosophy is(D) Think of everything in terms of mind-matter dualism and idealism.
That is, Descartes proposed the ﬁctional worldview the called
which starts from the three key-words

mind-matter dualism andidealism←−[A](mind)”I”(=mind, brain, observer) ,(medium)[B]−→”body”,[C](matter)”matter”(=sensory organ, measuring instrument)

For the deﬁnitions of ”idealism” and ”idealism”, see Deﬁnition 1.8, i.e., idealism = the spirit such thatthinking takes precedence over experimentation (since experiments on the mind are almost impossible).
Also, dualism=the worldview concerning measurement.
That is, we see”I think, therefore I am” (= mind-matter dualistic idealism)(♯)

Live sothereforeﬁctional worldview (literary truth, pseudo- truth) −−−−−−→ Ethics · morals
main subject

introduction·preface·ﬁction

Although Descartes did not emphasize measurement (≈ cognition, epistemology), the importance of cognition was emphasized by his successors (Locke, Hume, etc.).

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8.3 Descartes’ strategy

Also, ethics, moral, etc. are not emphasized in (♯), but Descartes asserted ”The existence of God isderived”. As seen in Sec. 3.4, I think that the term ”therefore” in (♯) cannot be relied upon.
And, Descartes proposed the following two problems;
Problem 8.5. [Descartes problem; mind-body problem, qualia problem](D1 ) mind-body problem:
How are ”body” and ”mind” connected?

(D2 ) subjectivity problem (= qualia problem):Is the world I perceive the same as the world you perceive?
For the answer of (D1 )and (D2 ) in the quantum mechanical worldview, see Sec. 12.4 and Sec. 9.5.1
respectively.
♠Note 8.4. (i): The above two problems have been touted as the two most diﬃcult problems inmodern philosophy. Without raising this issue, the prosperity of modern philosophy might nothave been realized. In fact, Kant, Husserl, and others have challenged these problems, and theirachievements were supported, earning them respect as great philosophers though their approaches
are rather pseudo-psychological. However, from the scientiﬁc point of view, the honest impression
would be that philosophers are just making a fuss about an unsolvable problem. No scientist thinksthat their achievements have ”guaranteed the objectivity of natural science.”(ii): For the answer of (D1 )and (D2 ) in the quantum mechanical worldview, see Sec. 12.7 and Sec.
9.5.1 respectively. Many philosophers may not be immediately convinced because our approach is
not traditional. However, I am sure that our approach is truly philosophical. That is, I think thatphilosophers should not regard Problem 8.5 as the problems in psychology and brain science.

Descartes might think as follows.
(F) The theory of Ideas has expired expiration date. Thus, in order to refresh philosophy, a new modelchange (or, a new wrapping paper) is needed such as Descartes’ problem (D). Even if this is a non-sense
problem, this model-change won’t fade for about 100 years.
In fact, the cogito turn:
Cogito Turn

Theory of Ideas −−−−−−−−→ Mind-matter dualism
Plato

Descartesis the biggest model-change in the history of philosophy.

♠Note 8.5. Descartes’ book: ”The method of rightly conducting one’s reason and of seeking truthin the sciences” begins with the non-scientiﬁc proposition ”I think, therefore I am”, which is notexperimentally veriﬁable (cf. Proposition 8.3. ). Some may ﬁnd this ironic. However, I’d like to thinkthat Descartes had it all ﬁgured out. That is because• Euclid’s ”Elements” begins with seven axioms, which cannot be proved.
Thus, I guess that,• As Euclid aimed at a science of ﬁgures, Descartes aimed at a science of dualism.
If so, I can expect that Western philosophy may have a destination.

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Chap. 8 Modern philosophy (Dualism): Descartes♠Note 8.6. As mentioned in ref. [71], I rewrite as follows. It is not true to consider that every
phenomenon can be described in terms of quantum language. Although readers may think that thefollowing can be described in measurement theory, but we believe that it is impossible. For example,
the followings
cannot be written by quantum language:⃝:tense—past,present, future —⃝2 : Heidegger’s saying ”In-der-Welt-sein”
 1


⃝
:
the
measurementofameasurement,⃝
3
4 : Bergson’s subjective time⃝:observer’s
space-time,

5


⃝
6 : Only the present exists (due to Augustinus(354-430))
If we want to understand the above words, we have to propose the other scientiﬁc languages (except
quantum language). We have to recall Wittgenstein’s sayings

The limits of my language mean the limits of my world
I consider that ⃝–1 ⃝6 are related to self-reference in the wide sense.

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8.4 The correspondence of key-words in Descartes philosophy and quantum language

8.4 The correspondence of key-words in Descartes philosophy andquantum language
The key-words of Descartes philosophy (= mind-matter dualism) is”I”(=”brain”, ”mind”), ”body”(=”sensory organ”), ”matter”
However, we cannot expect the substantial result even if we consider Descartes’ problem. However, the above
three key-words are essentially important in the relation with the quantum language: Now let us explain
this.

Or,
For example, consider:• Examine whether the hot or cold water in the bath and put your hands in the bathtub.
In this case, ”hand” is regarded as ”measuring instrument”. In the same sense, ”eye” is also regarded as”measuring instrument”. Conversely, Glasses, microscope, telescope, etc. is a kind of body (= sensory
organ). If so, we want to conclude thatbody (particularly, sensory organ) ≑ measuring instrument
In the above Descartes ﬁgure, slightly incomprehensible one may be”I”(=”brain”, ”mind”) ≑ measured value
However, it suﬃces to consider ”there is no measured value without brain”. For example when a needle ofa voltmeter just moved, it is only a physical phenomenon. Nevertheless a movement of this needle is read,and it’s sensed by a brain. Then, it for the ﬁrst time becomes ”measured value”. The reason that Descartes
philosophy is useless is as follows.
(A) In spite that three key-words ”mind”, ”body”, ”matter” are gathered, Descartes philosophy has nocomputable structure. This is only the ﬁctional worldview, and not the mechanical worldview.
Table8.1

Key-words in each worldview (cf. Assertion 1.14 )mind-matter dualism[A](= mind)[B](between A and B)

Platoactual world

Idea

DescartesI, mind, brainbodyquantum language

measured value

measuring instrument(= observable)[C](= matter)/[Idea world]/[matter]state[system]That is, using the following change:

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Chap. 8 Modern philosophy (Dualism): Descartes
 


 





I =⇒ measured value , body =⇒ measuring instrument , matter =⇒ system
we get the computable worldview (i.e., Axioms 1 and 2 in Sec. 1.1.1), i.e., quantum language.
If so, the following problem is essential.
Problem 8.6. Descartes’ model-change:
the theory of Ideas −−−−−−−−−→ Descartes philosophy
model-change

That is, it is sure that Descartes’ model-change is supported by manycan be regarded as a progress?
people. Almost people certainly believe that science makes progress, that is, science development is not
fashionable change. However, there may be a lot of opinions about philosophy. We completely agree that
it is fun to think of dualistic idealism (= dualistic metaphysical world). However, we have the question:• Did western philosophy make essential progress?
which is essentially the same as• Does dualistic idealism (= dualistic metaphysical world) deserve to study as science?
Brief explanation of Problem 8.6:
Although this will be answered throughout this paper, I add a simple explanation as follows. Our answer
to Problem 8.6 is ”essential progress”. As the reason mentioned in Assertion 1.14 of Sec.1.4, the following
key-words converge to ”observable” such as
Idea →(Plato)body(Descartes)→ secondary qualities → · · · → observable(=measuring instrument)(Locke)(quantum language)

It should be noted that there is no settlement (i.e., ”essential progress” or ”fashionable change”?) without
ultimate goal (= quantum language). Therefore, this will be summarized in the ﬁnal chapter (=Postscript).

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169

Chapter 9Modern philosophy(Locke, Leibnitz,
Berkeley, Hume)The fact may be represented by the ﬁgure.
the realistic worldview (physics)
Aristotle· · ·Plato. . .

British empiricism
−−−−−−−−−−−−−−→ Newton → theory of relativity Bacon; Induction
A.D.1000∼1500 ⇒ Scholasticism ⇒












Continental rationalism −−−−−−−−−−−−−−−−−→ Descartes,Locke,Leibniz,...,Kant → QL
Descartes: Deduction








the linguistic worldview (no experiment)

However, the part:
Continental rationalism(♯1 ) ”−−−−−−−−−−−−−−−−→ Descartes,Locke,Leibniz,...,Kant ”
Descartesis usually rewritten by the following (♯2 ), which is called ”the ﬂower of modern philosophy”
 Induction (due to Bacon) 






British Empiricism








Deduction
[Locke, Berkeley, Hume](♯2 )Descartes−→−→ Kant philosophy[The father of modern philosophy]
Deduction(due
toDescartes)[summing-up]








Continental
Rationalism[Leibniz](Note that, in spite of ”British Empiricism”, Locke, Berkeley, Hume were not concerned with experiment. Thus, I assume that they did not belong to realistic worldview.)
It’s unbelievable now, but it was a time when philosophy was believed to be the ”king of the academy,”
and a time when philosophy was respected. Even so,• Why were top elites in those days absorbed in the useless philosophy?
As I have said many times, I think that this is due to Euclid’s success and Platonic method of telling
philosophy. .

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

9.1 Locke (1632 - 1704): The father of British Empiricism
9.1.1 ”An Essay Concerning Human Understanding” by Locke (1689)

There may be a reason to consider that(A) The role of Descartes was the elimination of such ”spiritual power” and ”supernatural being”, and toprepare the social environment of the appearance of Newton. That is, Descartes was only the opening
performer. In this sense, ”I think, therefore I am” (the existence of ”I”) was only the side show of
the opening performer.
And so,(B) The role of Descartes, as the opening performer, had been ﬁnished by the appearance of Newton.
Even if there was such history, it wasn’t strange. However, strangely, there were people who took ”theexistence of I” or ”Descartes Figure 1.2” seriously. For example, John Locke (1632 - 1704) thought asfollows.
(C) :”An Essay Concerning Human Understanding” by Locke (1689)
Locke is the successor of Descartes philosophy. He philosophically (i.e., without experiments) discussed the Descartes ﬁgure (i.e., the relation among ”I”(=”brain”, ”mind”), ”body”(=”sensory organ”), ”matter” ). He is called ”The father of British Empiricism (≈ epistemology)”. Since Isaac
Newton published his ”Principia Mathematica” (1686), John Locke humbly wrote in his ”Essay
Concerning Human Understanding” (1689) as follows.
(D1 ) ...
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, ...
However, his true intention might be next.
(D2 ) In the ﬁeld of ”matter” of Descartes ﬁgure, activity of Newton is remarkable. However,concerning the relation among ”I”(=”brain”, ”mind”), ”body”(=”sensory organ”), ”matter”,
he wanted to reach the summit.
♠Note 9.1. I think that ”the incomparable Mr. Newton” is the most important key-word in
the whole of modern philosophy.
If so,(E) It was too early more than 300 years to study ”epistemology” in science in earnest.
Thus, it is impossible to expect the result. However,(F) If we think that the work of philosophers is ”model-change”, then the achievements of Locke isenormous.
In fact, modern philosophy had a very easy environment to work in.
In other words, philosophers only
had to think about how to deal with Newtonian mechanics.
♠Note 9.2. If the above (D) is true, Locke may have been trying to create another kind of physics thatis diﬀerent from Newtonian mechanics. If so, his theory should belong to dualistic realism. However,170

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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)
Locke and others (Descartes, Leibniz, Hume, Kant, etc.) were not concerned with experimental
veriﬁcation (cf. for our deﬁnition of ”realism”, see Deﬁnition 1.7). Thus, in this paper, we think that
Descartes-Kant philosophy belongs to dualistic idealism. This was confused in my previous paper[65].

9.1.2 ”Tabula rasa”, primary quality and secondary quality
9.1.2.1 ”tabula rasa”
Tabula rasa is a Latin phrase often translated as ”blank paper” in English, that is,(G) The ”brain circuit” is a blank paper state at the start, but we look and hear in various ways, then”concept (= complex brain circuit)” is made.
Present-day brain science may say:
”It’s equal to say nothing by such general opinion.”
however, at any rate, the (G) is the starting point of British Empiricism.
♠Note 9.3. (i): Maybe Locke wasn’t claiming ”tabula rasa” in the scientiﬁc sense. I think he wasfollowing Bacon’s empiricism and insisting on a ”tabula rasa”.
(ii): Considering ”language” and not ”cognition”, we say tat(♯) ”ordinary language” is like tabula rasa
When a baby was born, a baby doesn’t know ordinary language at all (i.e., a baby is with tabula rasastate). The baby is acquiring ordinary language by trial and error.

9.1.2.2 Primary quality and secondary quality
According to Locke,(H1 ) primary quality (i.e., inherent nature (=primary quality)) · · · weight, temperature, length, etc.
(H2 ) secondary quality (i.e., sensations of inherent nature)· · · sweet, red, hot, salty, etc.
That is,(I) :Locke’s worldview
The world is composed of two (i.e., ”matter” and ”mind (= observer)”. ”Matter” has inherent nature(= primary quality), ”observer” has body (=”sensory organ”). Through the sensory organ, secondary
quality (sweet, red, hot, salty, etc.) is felt by our brain.

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

In terms of quantum language, we say:primary quality=⇒state,secondary quality=⇒observable (= measuring instrument)
as seen in the table below.
Table9.1

The key-words of worldviews (cf. Assertion 1.14 )mind-matter dualism[A](= mind)[B](between A and B)

Platoactual world

Idea

DescartesI, mind, brain

body

Locke

mind

quantum language

measured value

secondary quality
observable[C](= matter)/[idea world]/[matter]primary quality[matter]state[system]

Here,(J) Locke represents the most important concept in dualistic idealism as the term ”secondary quality”.
The terms such as Idea, body, etc. may be not comprehensive. 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 dualism.
If so, we may aﬃrmatively answer Problem 8.6, i.e.,
Can the direction:”Descartes −−−−−−−−−→ Locke” be regarded as progress?
model-change

That is, we may assert that
(Idea)(secondary quality)(body)
progress

progressmodel-change

model-change

Plato −−−−−−−−−→ Descartes −−−−−−−−−→

Locke(measuring instrument)
progress−−−−−−−−−→ quantum language
model-change

if ”to make progress” is deﬁned by ”to come near quantum language” (cf. Assertion 1.14). It should be
noted that this result cannot be conﬁrmed without an understanding of quantum language. Therefore, thiswill be summarized in the ﬁnal chapter (=Postscript)♠Note 9.4. (i): The polar star can be regarded as a measuring instrument such as a kind of compass.
(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 the right hand as
the measuring instrument, I feel the existence of my left hand. On the contrary, if I regard theleft hand as the measuring instrument, I feel the existence of my right hand.
It may be interesting, however, I do no know whether such thing is worth arguing.

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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)9.2 Dramatic presentation of ”British Empiricism vs. Continental
Rationalism”
Modern philosophy became popular through the following rival relation:
British Empiricism vs. Continental Rationalism

That is,(A)

Descartes−→[founder of epistemology][Locke, Berkeley, Hume]














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


























[tabula rasa, experience ]British Empiricism[Leibniz][”Critique of Pure Reason”]−→

Kant philosophy[summing-up, compromise]

There may be many opinions on this conﬂict structure. For example, some may consider that(A’) the above (A) is the staging of a conﬂict structure to make Kant

into a hero.

Let us explain the above (A):Modern philosophyBritish Empiricism ”tabula rasa”(= blank paper)”An Essay concerning Human Understanding” (by Locke, 1690) says that(B) He eliminated the possibility of innate knowledge before experience. Human being is born as the. (Locke,
Berkeley, Hume, · · · )blank state (”tabula rasa”)Continental Rationalism nativism”New Essays on Human Understanding” (by Leibniz, 1703) says that(C) nativism (= Anti-”tabula rasa”). the human mind as it is at birth, with ideas or thoughts in it.
(Leibniz,· · · )173

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9.2 Dramatic presentation of ”British Empiricism vs. Continental Rationalism”
That is, from”An Essay concerning Human Understanding” vs. ”New Essays on Human Understanding”
the rival relation:
British Empiricism[”tabula rasa”] vs. Continental Rationalism[nativism]
began. After nearly 100 years of twists and turns,
Appearance of Kant (Critique of Pure Reason: 1781)
And(D) Kant has integrated ”tabula rasa vs. nativism”
It is well known such above story that regards Kant as a hero.

9.2.1 Would Leibniz be serious for this argument (i.e., nativism) ?

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 and important contributions to theﬁelds of metaphysics, epistemology, logic, philosophy of religion, as well as mathematics, physics, geology,
jurisprudence, and history. Everyone may have the following question:(E) Why did such a genius participate in a nonsense argument (i.e., ”nativism vs. tabula rasa”) ?
It is clear that Locke’s theory is too extreme, and thus, it is a matter of course that Leibniz did not completely
agree with ”tabula rasa”. However,(F1 ) However, the story that Leibniz disputed Locke in ”New Essays on Human Understanding” is too
exaggerated.
The argument about ”nativism vs. tabula rasa” is non-sense in the following sense:(F2 ) Even if future brain science will make a decision favorable to one of them (i.e., ”nativism vs. tabularasa” ), it is independent of Leibniz’s (or, Locke’s) evaluation. That is because ”Continental Rationalism vs. British Empiricism” is regarded as a pre-science problem. What is the most important is
to form the ground on which the dualistic idealism (or the problem concerning brain) can be argued

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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)scientiﬁcally. Without the ground, it is useless even if they said something
For example, from the scientiﬁc point of view, atomism due to Democritus (BC.460-BC.370) is nonsense. It is poem. .
Ordinary people might have wanted to ﬁnd the big name of ”Genius Leibniz” in the debates of ”British
Empiricism vs. Continental Rationalism”. Maybe they just enjoyed a play of the name as ”the dawn ofmodern times” as an entertainment.
In this and next chapters, the readers will ﬁnd the miracle fact :• as geniuses ( Descartes, Locke, Leibniz, Berkeley, Hume, Kant, etc) played with words, theirarguments gradually became closer and closer to quantum languag
That is,♠Note 9.5. Considering ”language” and not ”cognition”, then, we say that(♯1 ) mathematics is nativism
That is because mathematics is based on set theory. That is, any mathematical theorem can bederived from Zermelo=Fraenkel Axioms.
Also, recall Note 9.3, in which we say tat(♯2 ) ”ordinary language” is like tabula rasa
That is because ”ordinary language” is not based on some axioms.

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9.3
* Leibniz-Clarke Correspondence: What is space-time?

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

This section is published in the following:• ref. [73]: S. Ishikawa; 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)• ref. [74]; S. Ishikawa; Leibniz-Clarke correspondence, Brain in a vat, Five-minute 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)

The problems (“What is space?” and “What is time?”) are the most important in modern science as well176

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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)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

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

Linguistic Copenhagen interpretation

9.3.1.1 Space in quantum language(How to describe “space” in quantum language)(Cf. the W ∗ -algebraic formulation in (A2 ) in Sec 1.5) is used in this section.)
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 classicalsystemand 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 ) ⊆ L∞ (Rq × Rp ) ⊆ B(L2 (Rq × Rp )]Also, concerning quantum system, we have the quantum basic structure:(C2 )[C(L2 (Rq ) ⊆ B(L2 (Rq ) ⊆ B(L2 (Rq )]Summing up, we have the basic structure∞2
 (C1 ): classical [C0 (Rq × Rp ) ⊆ L (Rq × Rp ) ⊆ B(L (Rq × Rp )](C) [A ⊆ A ⊆ B(H)](C2 ): quantum [C(L2 (Rq ) ⊆ B(L2 (Rq ) ⊆ B(L2 (Rq )]Since we always start from a basic structure in quantum language, we consider that
How to describe “space” in quantum language⇔ How to describe [(A):space] by [(C):basic structure](9.1)

This is done in the following steps.
Assertion 9.1. [The linguistic Copenhagen interpretation concerning ”space”]
How to describe “space” in quantum language(D1 ) Begin with the basic structure:[A ⊆ A ⊆ B(H)]177

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9.3
* Leibniz-Clarke Correspondence: What is space-time?
(D2 ) Next, consider a certain commutative C ∗ -algebra A0 (= C0 (Ω)) such that
A0 ⊆ A(D3 ) Lastly, the spectrum Ω (≈ Sp (A∗ )) is used to represent “space”.
Therefore, in quantum language, we see• space is a kind of state of a ”thing”.
For example,(E1 ) in the classical case (C1 ):[C0 (Rq × Rp ) ⊆ L∞ (Rq × Rp ) ⊆ B(L2 (Rq × Rp ))]we have the commutative C0 (Rq ) such that
C0 (Rq ) ⊆ L∞ (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 )) ⊆ 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)9.3.1.2 Time in quantum language(How to describe “time” in quantum language)
In what follows, let us explain “time” in measurement theory (= quantum language). This is easily donein the following steps.
Assertion 9.2. [The linguistic Copenhagen interpretation concerning ”time”]
How to describe “time” in quantum language(F1 ) Let T be a tree in Axiom 2 in Sec. 1.1. (Don’t mind the ﬁniteness or inﬁnity of T ) For each t ∈ T ,consider the basic structure:[At ⊆ At ⊆ B(Ht )](F2 ) Next, consider a certain linear subtree T ′ (⊆ T ), which can be used to represent “time”.
Therefore, in quantum language, we see• time is an order of occurring in succession which changes one after another.

9.3.2 Leibniz-Clarke Correspondence

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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)
Please read the following while looking at the above ﬁgure.
The above argument urges us to recall Leibniz-Clarke Correspondence (1715–1716: cf. [1]), which isimportant to know both Leibniz’s and Clarke’s (=Newton’s) ideas concerning space 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-time exists.
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 of occurring in succession
which changes one after another.
Therefore, I regard this correspondence as
Newton (≈ Clarke)←→vs.

(realistic view)

Leibniz(linguistic view)which should be compared to
Einstein(realistic view)

←→
vs.

Bohr
(linguistic view)(also, recall Note 4.3).
Again, we emphasize that Leibniz’s relationalism in Leibniz-Clarke correspondence is clariﬁed in quantum language, and it should be regarded as one of the most important parts of the linguistic Copenhagen
interpretation of quantum mechanics.
Many scientists may think that
Newton’s assertion is understandable, in fact, his idea was inherited by Einstein. On the other,
Leibniz’s assertion is incomprehensible and literary. Thus, his idea is not related to science.
However, recall the classiﬁcation of the world-description (Classiﬁcation 1.9):
(space-time in physics)(successors: Einstein, etc.)· · · realistic space-time 1 : Newton, Clarke(realistic worldview)“What is space-time?”

O











O:
2

(space-time in measurement theory)

Leibniz(linguistic worldview)···

linguistic space-timeO(i.e., spectrum, tree)

O“How should space-time be represented?”in which Newton and Leibniz respectively devotes himself to 1 and 2 . Although Leibniz’s assertion is notclear, we believe that
• Leibniz found the importance of “linguistic space and time” in science,
Also, it should be noted that(♯1 ) Newton proposed the scientiﬁc language called Newtonian mechanics,
on the other hand,
Leibniz could not propose a scientiﬁc language
After all, we conclude that(♯2 ) the cause of philosophers’ failure is not to propose a language.
Talking cynically, we say that(♯3 ) Philosophers continued investigating “linguistic interpretation” (=“how to use Axioms 1 and 2”)

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9.3
* Leibniz-Clarke Correspondence: What is space-time?
without language (i.e., Axiom 1(measurement:§2.7) and Axiom 2(causality:§10.3)).
Table 1.1 : realistic worldview vs. idealistic worldview
dispute ⧹ [R] vs. [L]⃝:a motion
Greece⃝:Ancientb⃝:c Problem of universals⃝:
d space-time⃝:
e quantum theory⃝:philosophyof sciencefRealistic worldview

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

Hērakleitos
Aristotle”Nominalismus”(Ockham)
NewtonEinstein
CarnapParmenides
Plato”Realismus”(Anselmus)
Leibniz
Bohr
Quine

⃝
a is my ﬁction, ⃝c is a confusion. ⃝d is the Leibniz=Clarke correspondence (cf. Sec. 9.3.2), ⃝e is BohrEinstein debates. Quantum language is proposed as one of answers to Bohr-Einstein debates(cf. ref.
[71]). Quantum language is proposed as one of answers to Bohr-Einstein debates(cf. ref. [71]). ⃝:f
Quine understood 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. 13.3).
♠Note 9.6. (i): Spinoza (1632-1677) is a popular philosopher, but he has nothing to do with quantum
language. Therefore, I did not mention Spinoza.
(ii): Leibniz is an undisputed genius. I think Leibniz is the only one who can discuss space-timeon equal terms with Newton. Unfortunately, however, I cannot understand his ”monadology”. As I
made clear in this section, Leibniz’s space-time is the space-time of quantum language. If Leibniz had
discussed his space-time in his monadology, I think I could have understood his monadology. And
I hope that with some modiﬁcation of the monadology, it would have become clear that ”monad =state” equals quantum language.I hope that researchers in monadology will study the relationshipbetween quantum language and monadology.

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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)9.4 Subjective idealism: Berkeley, ”To be is to be perceived”

9.4.1 Priest: Berkeley
Berkeley (1685 -1753) is famous as follows.
(A1 ) Berkeley is a priest, and he interpreted Locke’s primary quality as the state of things that come froma supernatural power such as a god. Thus his philosophy is called subjective idealism.

Table9.2

The key-words of worldviews (cf. Assertion 1.14 )mind-matter dualism[A](= mind)[B](between A and B)

Platoactual world

Idea

DescartesI, mind, brain

body

Locke

mind

Berkeleyquantum language

secondary quality
secondary qualitymindmeasured value

observable[C](= matter)/[idea world]/[matter]primary quality[matter]
/
[God]
state
[system]Thus I don’t think that
Locke −−−−−−→ Berkeley
progressthough there may be people who want to believe that.
However, he is an important ﬁgure in the following sense.
(x)” is not complete(A2 ) Berkeley indicated that Newton’s deﬁnition of diﬀerentiation ”lim h→0 f (x+h)−fh(A3 ) He said ”To be is to be perceived”, which represented the essential spirit of dualism. Also, ”If a tree
falls in a forest and no one is around to hear it, does it make a sound?” is said to be due to Berkeley.
Summing up, Berkeley was always the standpoint of anti-Newton (= anti-realism).
• If we think that modern philosophy (from Descartes to Kant) has signiﬁcance as a buﬀer zone ofChristianity with Newtonian mechanics, we can conclude that Berkeley is honest.
♠Note 9.7. 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 (17891857), Weierstrass (1815-1897), etc.). When I think from now on, there was a possibility that Leibniz
discovered it, but he was busy in the other things.

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9.4 Subjective idealism: Berkeley, ”To be is to be perceived”

9.4.2 (A3 ): To be is to be perceived
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” isemphasized as follows (cf. Sec. 1.1).
(=quantum language)
measurement theory[Axiom 2][Axiom 1]=

measurement + causal relation
linguistic Copenhagen interpretation

+

(9.2)

[the manual to use Axioms 1 and 2]

But, it is Genius Newton (and Einstein) that neglect this absolute truth (H 1 ). In fact, Newtonian mechanics
is formulated as follows.
[Newtonian kinetic equation]

Newtonian mechanics = No measurement +causal relation(9.3)

Here, note that Newton removed ”measurement” from (9.2) in spite of the maxim that there is no science
without measurement. The insightfulness of Newton is surprising. A genius isn’t confused by ”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, considered that the exclusion was a weak pointof 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 in next section.

9.4.3 ”Einstein-Tagore Meeting” and ”Bohr-Einstein debates”
Concerning ”realistic worldview vs. idealistic worldview”, Einstein-Tagore (poet, thinker in India) meetingin 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, yetthat table remains where it is*1 .”In the above, Tagore’s assertion is similar to Berkeley’s ”To be is to be perceived”, which belongs to thesituation of dualistic idealism(=idealistic worldview).
On the other hand, Einstein’s saying:*1 Einstein often said this kind of statement at various places, for example, ”Does the moon disappear when I’m

not looking at it?”182

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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)(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 (9.3) is signiﬁcant. In this paper, we are not concerned with Bohr-Einsteindebates in quantum mechanics (in order to solve this problem, I proposed quantum language), (cf. ref. [71])).
However, Bohr-Einstein debates is similar to the above. Thus, summing up, we see:realistic worldview vs. idealistic worldview (cf. Table 1.1 in Assertion 1.12)
Realistic worldview

Idealistic worldview[monism, realism, no measurement ][dualism, idealism, measurement ]NewtonNewton (and Clarke)
Einstein
Einstein

Berkeley
Leibniz
TagoreBohr

Now, concerning Bohr-Einstein debates, The impression that Einstein lost now has been left, but the author
does not think so (cf. ref. [71]).
♠Note 9.8. Omitting ”Newton vs. Berkeley” and ”Einstein vs. Tagore” in the above table, I repeatedly
mention the following table (cf. Assertion 1.12):

Table 1.1 : realistic worldview vs. idealistic worldview
dispute ⧹ [R] vs. [L]
Realistic worldview
Idealistic worldview(monism, realism, no measurement)(dualism, idealism, measurement)

Hērakleitos
Parmenides⃝:
a motion
Aristotle
Plato
Greece⃝:Ancientb⃝:
Problemofuniversals”Nominalismus”(Ockham)”Realismus”(Anselmus)c⃝:
NewtonLeibnizd space-time⃝:
EinsteinBohre quantum theoryof science
CarnapQuine⃝:philosophyf⃝a is my ﬁction, ⃝c is a confusion. ⃝d is the Leibniz=Clarke correspondence (cf. Sec. 9.3.2),⃝
e is Bohr-Einstein debates. Quantum language is proposed as one of answers to Bohr-Einsteindebates(cf. ref. [71]). Quantum language is proposed as one of answers to Bohr-Einstein debates(cf.
ref. [71]). ⃝:f Quine understood 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. Chap. 12).
♠Note 9.9. In Japan, I learned the dualistic proposition: ”If a tree falls in a forest and no one is aroundto hear it, does it make a sound?” in a Zen dialogue (i.e., a question-and-answer exchange between
Zen priests and their followers). Zen is one school of Buddhism. In modern Japan, most people may
think that Zen monologue is a kind of wordplay.

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9.4 Subjective idealism: Berkeley, ”To be is to be perceived”

9.4.4 Bohr-Einstein debates: Do the laws of physics require measurement?

For Bohr-Einstein debates, I discussed in ref. [71]. 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 as follow: Newtonianmechanics and quantum mechanics are formulated as follows:(♯1 ) Newtoinan mechanics =Nothing+Causality(Newtonian equation)

On the other hand, quantum mechanics is formulated as follows.
Measurement(♯2 ) quantum mechanics =+(Born’s quantum measurement)

Causality(Heisenberg (and Schrödinger) equation)

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 Einsteinsays the following (♯4 ), everyone has no choice but to shut up.
(♯4 ) Then, did the laws of physics not work before the birth of humankind?
Thus, I think that Bohr-Einstein debates is not settled yet.

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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)9.5 Qualia problem and Brain in a vat argument
This section was written with reference to the following.
• [73]: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)

9.5.1* The problem of qualia
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 found in any of the tests, we can conclude
that each ”pink” felt by Jack and Betty is the same.
♠Note 9.10. In science, ”absolute” or ”100In other words, no matter how carefully you investigate,
you can never be 100[(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 that(♭2 ) According to the linguistic Copenhagen interpretation (cf. (E1 ) in Sec. 1.1.2), you (= Betty) are
not qualiﬁed to be an observer. Even if you are qualiﬁed to be an observer, you should perform thesame 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].

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9.5 Qualia problem and Brain in a vat argument

Then, Betty says:(♭3 ) OK. I (=Betty) can understand. In short, the measurements I can only make against myself are notscientiﬁc measurements. Or, a measurement that only I can make is not a measurement. Is it OK?
Then, the scientist says to Betty:(♭4 ) That’s right.
♠Note 9.11. In the book ”The astonishing hypothesis” (by F. Click (the most noted for being a codiscoverer of the structure of the DNA molecule in 1953 with James Watson)), Dr. Click 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 assembly of nerve cells andtheir 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, even consciousness
can be measured in the future. However, no matter how much brain science develops, I believe that
there is no solution to the qualia problem other than the one described above.

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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)9.5.2* Brain in a vat argument
Let us introduce a famous problem ”Brain in a vat” due to H. Putnam (cf. ref. [100]) as follows:

Figure 8.1: [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 yourthe data to you brain to fool you into believing that you are still walking around a forest.
Then, you may say;(C2 ) ”Am I a brain in a vat?” Or, ”Can I check whether I am a brain vat or not?”And you think:(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, since (C3 ) is true, you may say:(C4 ) ”I cannot know if I have a limb or not.”Then, we have the following problem:
Problem 9.3. [The problem concerning ”Brain in a vat”]• Is the (C4 ) true?
[Answer]:
The (C3 ) clearly is true since (C1 ) is assumed. However, (C4 ) is not true. That is because(♯1 ) if you are in Case[1], you ﬁnd that you have a limb.
(♯2 ) if you are in Case[2], you also ﬁnd that you have a limb (under the assumption (C 1 )).
Thus, (C4 ) is wrong.
[Alternative explanation]:
For completeness, let’s rephrase the same thing as follows. You ask someone ”Do I have a limb?” If they
reply, ”Of course you have,” you can be sure that you have a limb. In short,• you only have to believe in the measurement results.
Or,• To be is to be perceived.
This is just the linguistic Copenhagen interpretation (cf. Sec. 9.4.3: Einstein-Tagore meeting).
♠Note 9.12. Recall the worldviewism (in Sec 1.3.1), which says(♯) Without the principle (i.e., the worldview), we can’t say anything
Under the worldview called the quantum mechanical worldview (i.e., quantum language), we haveseen the followings.
• What is probability (or, measurement, causality) ? cf. Sec. 1.1.1)• The solution of Zeno paradox (Flying arrow), (cf. Sec. 2.4.2)• The solution of Zeno paradox (Achilles and a tortoise), (cf. ref. [45], or Sec. 2.4.3)

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9.5 Qualia problem and Brain in a vat argument( ”to solve Zeno paradox” = ”to understand DST-formula” (cf. Sec. 2.4.3 )• the measurement theoretical understanding of Plato’s allegory of the sum , (cf. Sec. 3.3.2)• Plato’s Idea theory≈Zadeh’s fuzzy theory≈Sausuure’s linguistic theory (cf. Sec. 3.5.3)• Syllogism holds in classical systems, but not in quantum systems (cf. Sec. 4.3.2)• Only the present exists (cf. Sec. 6.1.2)• What is the problem of universals? (cf. Sec. 6.5.1)• What is Geocentrism vs. Heliocentrism? After all, the worldviewism (cf. Sec. 7.4.2)• Two (scientiﬁc or non-scientiﬁc) interpretations of I think, therefore I am.(cf. Sec. 8.2.2)• The problem of qualia (cf. Sec. 9.5.1)• Brain in a vat argument (cf. Sec. 9.5.2)

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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)9.6 Hume; skeptic who didn’t measure, ”A Treatise of Human Nature”

9.6.1 The review of Descartes
Let us review Descartes and Locke.
(A1 ) Descartes found the indisputable truth, i.e., cogito proposition ”I think, therefore I am”. Therefore,
he thought that everything derived from cogito proposition can be trusted. That is, he started from”the existence of I”.
The purity of Descartes is mind-blowing, but in the ﬁrst place ”I think, therefore I am” and ”the existenceof I” is suspicious (cf. Note 1.7, or Proposition 8.3 in Sec.8.2). Hence, the following is also suspicious:(A2 ) ”The existence of I” is certain. Therefore, the matters that I perceive exist. And further, Descartes
introduced ”body (= sensor organ)” which mediates between ”I” and ”matter”. After all, he reachedand discussed ”mind-matter dualism” (= Descartes problem 8.5), that is, ”the problem of mind-matter
dualism” and ”mind-body problem”.
Although Descartes problem 8.5 is, from the scientiﬁc point of view, a barren discussion, Descartes philosophy
was supported a lot of people. Since the philosophy of worldview is a kind of fashion or ”model-change”, tobe supported by many people is the most important.
(A3 ) If Descartes and Locke asserted that• there is a possibility that mind-matter dualism (with keywords ”matter”, ”I (= mind, brain)”,”body(=secondary quality)”, ”matter”) succeeds.
then, I think that they are, from the quantum linguistic point of view, true.
If so, we may aﬃrmatively answer Problem 8.6, i.e.,
Can the direction:”Descartes −−−−−−−−−→ Locke” be regarded as progress?
model-change

The following ﬁgure is near to the linguistic Copenhagen interpretation,

That is, we can assert that
progress

progressmodel-change

model-change

Plato −−−−−−−−−→ Descartes −−−−−−−−−→ Locke

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9.6 Hume; skeptic who didn’t measure, ”A Treatise of Human Nature”


9.6.2 Hume’s straying Less than brain science ; Hume’s wordplay
Descartes philosophy is a philosophy which has the risk of entering the science. In fact, Hume approached thezone of science. In ”A Treatise of Human Nature” (1739), Hume pointed out the leap in logic of ”Therefore”
in the above (A2 ). As Hume says, it is sure that ”the existence of matter” cannot be derived from ”theexistence of I”*2 . Also, it is not guaranteed that ”matter I perceive” is equal to ”true matter”. Thus, theexistence of ”matter” is doubtful. However, it is sure that I feel so. Hume states that(B) ” a bundle of perceptions ” (= brain circuit) exists
That is,(C) ”matter” and ”causal relation” are a kind of bundle of perceptions

I think that the above ”(A1 )→(B)→(C)” is self-referential in the wide sense. That is,• ”(A1 )→(B)→(C)” is a kind of psychological illusion as having been able to understand all events by
the word ”bundle of perceptions”.
To enjoy such convinced form may also be a pleasure of philosophy. Thus,(D) It is said Hume’s philosophy is the goal of British Empiricism
Hume took the faultﬁnding of Descartes, and Hume has entered into wrong direction ”brain science”. The
research of ”the bundle of perceptions” belongs to brain science.
(E) If Hume was a scientist, he was too early for 300 years
The cause of victory of Galileo was a ”telescope”. Hume studied ”brain science” without measuring instrument in spite that Hume thought that he himself is a philosopher and not scientist. Hence, I thinkthat
regress

Descartes −−−−−→ Locke −−−−→ Humeprogress

However, Hume was revived by Kant (cf. next Chapter 10: Kant: Copernican revolution). Then, I want to
say that
Descartes −−−−−→ Locke −−−−−→ Hume −−−−−→ Kant
progress

progress

progress

9.6.3 Hume; The causality problem
It is a matter of course that the representation 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 theit has ”causality”. Hume argues that scientiﬁc wisdom is a product of ”habitual thinking” , notobjective truth.

*2 This kind of logic is a typical self-reference (cf. Note 1.7, or Proposition 8.3 in Sec.8.2). Thus, Hume’s logic (orgenerally, philosophical logic, wordplay) in ordinary language cannot be trusted. That is, it is only a wordplay.

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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)

Most readers may regard this as ”a kind of rhetoric”, however, several readers may be convinced in ”Nowthat you say that, it may be so.” Surely, since you are looking through the 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, and when 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, in this book, I adopted the friendly story to
the utmost to Kant. Hume served as a bridge between Locke and Kant. Thus we think that

This will be discussed again in the next chapter (cf. Section 10.3 What is causality?).
♠Note 9.13. In the book ”The astonishing hypothesis” (by F. Click (the most noted for being a
co-discoverer of the structure of the DNA molecule in 1953 with James Watson)),Dr. Click 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 assembly of nerve cells andtheir 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.
Also, since the title of his book is ”The astonishing hypothesis”, Click must have felt that(♯3 ) Descartes philosophy is based on dualistic realism.

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9.6 Hume; skeptic who didn’t measure, ”A Treatise of Human Nature”
However, the deﬁnition of ”realism” is diﬀerent from our deﬁnition (Deﬁnition 1.8), where we consider
that realism is a worldview based on experimental veriﬁcation. It is clear that Descartes, Locke,
Hume were not concerned with about experimental validation. That is, they are not scientists but
philosophers. Therefore, it may be ”astonishing” by common sense, but it’s not ”astonishing” byour deﬁnition. In Chapter 11, we will show that the monistic realism and the dualistic idealism (i.e.,quantum language) are compatible.

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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)9.7 Hume’s problem of induction in the quantum mechanical worldview♠Note 9.14. 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 the followings:(♯1 ) Newtonian mechanics has been right to this day. So, will the Newtonian mechanics be righttomorrow?
(♯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 Humesuspected the justiﬁcation of induction. It cannot be said that these are meaningless. In fact, thefollowing (♯3 ) ( which is regarded as the particular case of (♯1 )) was just Einstein’s interest.
(♯3 ) Newtonian mechanics has been right to this day. However, Is Newtonian mechanics correct whenparticles move at very high speeds?
However, Einstein was praised for advocating the theory of relativity. He would not have been
evaluated if he had only doubted as in (♯3 ). It is important to doubt, but if there is no result, it is aword-play. I think many philosophers have enjoyed wordplay on Hume’s problem of induction.
Thus, in this section, we devote ourselves to
Hume’s problem of induction in the framework of quantum language.

9.7.1* The solution of Hume’s problem of induction(Cf. the W ∗ -algebraic formulation in (A2 ) in Sec 1.5) is used in this section.)
This section was written with reference to the following.
• [75]:Ishikawa, S: Philosophy of science for scientists; The probabilistic interpretation of science Journalof quantum information science, Vol. 9, No.3 , 140-154,
DOI: 10.4236/jqis.2019.93007(https://www.scirp.org/Journal/paperinformation.aspx?paperid=95447)
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 )Here, our present problem is(B) Can such induction (= inductive inference) as above be justiﬁed?
Recall that our spirit in this paper is ”No scientiﬁc arguments without worldview”, or, ”true justiﬁcation”= ”justiﬁcation under a certain worldview”.
♠Note 9.15. In Note 7.1, we say that Bernoulli’s achievement (i.e., the discovery of the law of large
numbers) equals Galileo’s achievement. That is, Scientiﬁc pioneer in the realistic worldview · · · Galileo

Scientiﬁc pioneer in the idealistic worldview · · · J. Bernoulli

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9.7 Hume’s problem of induction in the quantum mechanical worldview

The reason that we consider so is that Galileo=Newton mechanics and Bernoulli’s law of large numbers
have the power to predict the future. Further we think that there are essentially only two of these
theories that have the power to predict the future. Therefore, we are convinced that Hume’s problem
of induction and Bernoulli’s law of large numbers are closely related as mentioned below.
In this section we show that the justiﬁcation is easily solved in our quantum mechanical worldview. If we
expect a scientiﬁc answer to Hume’s problem, we must start with the scientiﬁc deﬁnition of ”the uniformity
principle of nature”, i.e., the following Deﬁnition 9.4 [ The uniformity principle of nature]. Some may feelthat the uniformity principle of nature (i.e., the condition in Deﬁnition 9.4) is quite diﬀerent from what
Hume thought. However, we think that it is impossible to propose the diﬀerent quantitative deﬁnition of theuniformity principle of nature that leads to a result like Theorem 9.6 [Inductive reasoning] (i.e., If similar
measurements are performed, the similar measured values are obtained ).
Deﬁnition 9.4. [The uniformity principle of nature] For simplicity, consider a classical basic structure[C(Ω) ⊆ L∞ (Ω, ν) ⊆ B(L2 (Ω, ν))] such that Ω is compact and ν(Ω) = 1. (Cf. the W ∗ -algebraic formulationin (A2 ) in Sec 1.5) is used in this section) A family of measurements {ML∞ (Ω,ν) (Oi := (X, F, Fi ), S[ωi ] ) | i =−n, −n + 1, ..., −1, 0, 1, 2, ..., N } is said to satisfy the uniformity principle of nature (concerning µ), if there
exists a probability space (X, F, µ) such that
[Fi (Ξ)](ωi ) = µ(Ξ)∀Ξ ∈ F, ∀i = −n, −n + 1, ..., −1, 0, 1, 2, ..., N

Remark 9.5. [No scientiﬁc arguments without worldview] The uniformity principle of nature is not aprinciple (= axiom) of a worldview, and thus it should be deﬁned as an assumption under a certain world
description. The argument without world description leads contraction as follows.
(C1 ) It worked well, assuming the uniformity principle of nature up to now.
(C2 ) So it will continue to work.
(C3 ) Thus, the uniformity principle of nature can be justiﬁed.
This is not true, since it is a cyclic argument. That is, the above is a wordplay.

Under the above deﬁnition, we assert the following theorem (essentially the same as the law of large numbers), which should be regarded as the fundamental theorem in philosophy of science. Also, recall thatthe law of large numbers is illustrated as follows.

Theorem 9.6. [Inductive reasoning, the quantum linguistic solution of Hume’s problem of induction]. Let

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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)[C(Ω) ⊆ L∞ (Ω, ν) ⊆ B(L2 (Ω, ν))] be a basic structure such that Ω is compact and ν(Ω) = 1. Assume thata family of measurements {ML∞ (Ω,ν) (Oi := (X, F, Fi ), S[ωi ] ) | i = −n, −n + 1, ..., −1, 0, 1, 2, ..., N } satisﬁes

×N

the uniformity principle of nature ( concerning µ). Let (x−n , x−n+1 , ..., x−1 , x0 , x1 , ..., xN ) ∈i=−n X be aNN∞measured value by the parallel measurement
i=−n ML (Ω,ν) (Oi := (X, F, Fi ), S[ωi ] ). Then, we see that♯{k | xk ∈ Ξ, k = −n, −n + 1, ..., −1, 0}≈ µ(Ξ)(= [Fi (Ξ)](ωi ))nΞ ∈ F, i = −n, −n + 1, ..., −1, 0, 1, 2, ..., N(9.4)

where n is suﬃciently large. Here ♯[Θ] is the number of elements in a set Θ. Roughly speaking, from thequantum linguistic point of view, I think thatinductive reasoning ≈ the law of large numbers
Proof. Let Ξi ∈ F (i = −n, −n + 1, ..., −1, 0, 1, ..., N ). Axiom 1 [measurement] (in Section 1.1) saysthat the probability that a measured value (x−n , x−n+1 , ..., x−1 , x0 , x1 , ..., xN ) obtained by the parallelNN
N∞
measurement N
i=−n Ξi is given byi=−n [Fi (Ξi )](ωi )i=−n ML (Ω,ν) (Oi := (X, F, Fi ), S[ωi ] ) belongs to×N
i=−n µ(Ξi ).

××

Thus, the sequence {xi }Ni=−n can be regarded as independent random variables with the=identical distribution µ. Hence, using the law of large numbers, we can immediately get the formula (9.4).
Also, this theorem is a direct consequence of the law of large numbers for parallel measurements (cf. refs.
[42], or § 4.2 in ref.[71]).
♠Note 9.16. The reader may wonder why philosophers have always failed to formulate inductive
reasoning? Because they did not ﬁrst declare a worldview and then formulate induction under thatworldview. We cannot say anything without a worldview ( such as quantum language), that is,there is no formulaion without a worldview
Remark 9.7. (i): Recall that the law of large numbers (which is almost equivalent to Theorem 9.6) says
that”frequency probability” = ”the probability in Axiom 1 [measurement] (in Section 1.1)”(cf. ref. [42]), though the probability in Axiom 1 [measurement] (in Section 1.1) has the several aspects.
Also, note that the law of large numbers in statistics (cf. ref. [83]) has already been accepted as thefundamental theorem in science. Therefore, even if Theorem 9.6 ([Inductive reasoning]+(9.4)) is called thefundamental theorem in philosophy of science, we don’t think it’s exaggerated. We believe that our proposal(i.e., Theorem 9.6) is completely true in our worldview. Thus, we think that the solution of Hume’s problem
of induction was practically already found as the law of large numbers. In the framework of our worldview,
we are convinced that the above is the deﬁnitive solution to Hume’s problem. However, there may be anotheridea if some start from another worldview. Hence, as described at the end of this paper, we hope that many
philosophers propose various mathematical foundations of scientiﬁc philosophy, in which Hume’s problem of
induction are discussed from the various viewpoints.
(ii): In Deﬁnition 9.4 [The uniformity principle of nature] and Theorem 9.6 [Inductive reasoning], we consider
the family of measurements {ML∞ (Ω,ν) (Oi := (X, F, Fi ), S[ωi ] ) | i = −n, −n + 1, ..., −1, 0, 1, 2, ..., N }. Thismay be too general. Usually, it suﬃces to consider that {ML∞ (Ω,ν) (Oi : = (X, F, F ), S[ωi ] ) | i = −n, −n +1, ..., −1, 0, 1, 2, ..., N }, i.e., F = Fi (−n ≤ ∀i ≤ N ).
N0∞(iii): It may be understandable to consider two measurements:
i=−n ML (Ω,ν) (O := (X, F, Fi ), S[ωi ] ) andNN∞
i=1 ML (Ω,ν) (O := (X, F, Fi ), S[ωi ] ). The reason that we do not consider two measurements is due to the
linguistic Copenhagen interpretation (G1 ), i.e., only one measurement is permitted.
Example 9.8. [Coin tossing]. Let us discuss the unfair coin tossing as the most understandable exampleof Theorem 9.6 [Inductive reasoning]. Consider a basic structure [C(Ω) ⊆ L∞ (Ω, ν) ⊆ B(L2 (Ω, ν))]. 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 L∞ (Ω, ν) such that
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 )

(9.5)

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That is, a family of measurements {ML∞ (Ω,ν) (O := (X, 2X , F ), S[ωi ] ) | i = −n, −n + 1, ..., −1, 0, 1, 2, ..., N }×Nsatisﬁes the uniformity principle of nature (concerning µ). Let (x−n , x−n+1 , ..., x−1 , x0 , x1 , ..., N ) ∈i=−n XNN
X∞
be a measured value obtained by the parallel measurement
i=−n ML (Ω,ν) (O := (X, 2 , F ), S[ωi ] ), i.e.,
inﬁnite coin throws. Here, Theorem 9.6 [Inductive reasoning] say that it is natural to assume that, forsuﬃciently large n,(x−n , x−n+1 , ..., x−1 , x0 ) = (T| H H T H H H{zT T ..... T H H})

(9.6)

n+1

(where the number of Hs ≈ 2n/3, T s ≈ n/3)
Then we can believe that we see that xi = H with probability 2/3 [ resp. xi = T with probability 1/3] foreach i = 1, 2, ..., N . It should be noted that even without knowing (9.5), we can conclude that if we know
(9.6).
Remark 9.9. It should be noted that the above example shows that Theorem 9.6 [Inductive reasoning] (orequivalently, the law of large numbers), like Newton’s kinetic equation, has the power to predict the future.
This is the reason that Hume’s problem of induction keeps attracting much researcher’s interest for a longtime. If the justiﬁcation of Hume’s problem of induction is solved, it should be the most fundamental theory
in science. Thus, we are convinced that our assertion (i.e., the law of large numbers ≈ the justiﬁcationof Hume’s problem of induction ) is true. As said in Note 9.16, many philosophers have not considered
induction under a certain worldview. This is why they failed.
In Sec. 11.4, we summarize as follows.
Logical worldview

Quantum mechanical worldview[ no measurement ][ measurement ]deduction (cf. Sec 11.4)←→

measurement

abduction (cf. Sec 11.4)←→

inferenceinduction (cf. Sec 9.7)←→

the law of large numbers

Here, the logical worldview (=the logical spirit=the spirit of ”Think logically!”) may not be a true worldview.

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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)9.8* grue paradox cannot be represented in quantum language(Cf. the W ∗ -algebraic formulation in (A2 ) in Sec 1.5) is used in this section.)♠Note 9.17. I’ve read several books that explain Goodman’s arguments concerning ”grue Paradox”(cf. ref. [25]), but I’ve never been impressed with them. Rather, I think that the 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 that grue Paradox evokes the inevitability ofthe quantum mechanical worldview. That is, grue paradox can not be understood without quantum
language.
This section was written with reference to the following.
• [75]:Ishikawa, S: Philosophy of science for scientists; The probabilistic interpretation of science Journalof quantum information science, Vol. 9, No.3 , 140-154,
DOI: 10.4236/jqis.2019.93007(https://www.scirp.org/Journal/paperinformation.aspx?paperid=95447)♠Note 9.18. The summary is as follows. If our understanding of inductive reasoning under the quantum
mechanical worldview (mentioned in the above) is true, that is,(♯)
Hume’s problem of induction ≈ the law of large numbers,
we can easily solve the grue paradox (cf. ref. [25]). The grue paradox arises, since the grue paradox
is due to the fact that the conditions for the law of logarithms to hold are not met. That’s all I’m
going to say below.

Let us mention it as follows.
Consider a basic structure [C(Ω) ⊆ L∞ (Ω, ν) ⊆ B(L2 (Ω, ν))]. Let Ωg , Ωb be the open subsets of the state
space Ω such that Ωg ∩ Ωb = ∅. And put Ωo = Ω \ (Ωg ∪ Ωb ). Let O = (X ≡ {g, b, o}, 2X , F ) be the observable
in L∞ (Ω, ν) such that
[F ({g})](ω) = 1 (ω ∈ Ωg ),= 0 (ω ∈ Ω \ Ωg )[F ({b})](ω) = 1 (ω ∈ Ωb ),= 0 (ω ∈ Ω \ Ωb )[F ({o})](ω) = 1 − [F ({g})](ω) − [F ({b})](ω) (ω ∈ Ω)

(9.7)

where ”g”, ”b”, ”o” respectively means ”green”, ”blue”, ”others”.
Let {e−n , e−n+1 , ..., e−1 , e0 , e1 , e2 , ..., eN } be the set of (green) emeralds. And assume that ωi (∈ Ω◦g ) is thestate of emerald ei (i = −n, −n + 1, ..., −1, 0, 1, 2, ..., N ).
A family of measurements {ML∞ (Ω,ν) (Oi := (X, 2X , F ), S[ωi ] ) | i = −n, −n + 1, ..., −1, 0, 1, 2, ..., N } clearlysatisﬁes the uniformity principle of nature, that is, there exists an probability space (X, 2X , µ) such that
[F (Ξ)](ωi ) = µ(Ξ)∀Ξ ∈ 2X , ∀i = −n, −n + 1, ..., −1, 0, 1, 2, ..., N

where µ({g}) = 1, µ({b, o}) = 0.
N
Let (x−n , x−n+1 , ..., x−1 , x0 , x1 , ..., xN ) ∈i=−n X be a measured value obtained by the parallel measureNNX∞ (Ω,ν) (O := (X, 2 , F ), S[ω ] ). We see, of course, that xi = g (i = −n, −n + 1, .., −1, 0).
ment
M
L
i
i=−n
And thus, we can believe, by Theorem 9.6 [Inductive reasoning], that x1 = x2 = ... = xN = g. For the sake ofcompleteness, note that we can predict x1 = x2 = ... = xN = g only by the data x−n = x−n+1 = ... = x0 = g.
This is usual arguments concerning Theorem 9.6 [Inductive reasoning].

×

On the other hand, Goodman’s grue paradox is as follows (cf. ref. [25]).
(D1 ) Deﬁne that Y has a grue property iﬀ Y is green at time i such that i ≤ 0 and Y is blue at time isuch that 0 < i. Suppose that we have examined the emeralds at −n, −n + 1, ... − 1, 0, and found themto all be green (and hence also grue ). Then, ”so-called inductive reasoning” says that emeralds at
1, 2, ..., N have the grue property (and hence blue) as well as green. Thus, a contradiction is gotten.

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9.8
* grue paradox cannot be represented in quantum language

However, we think that this (D1 ) cannot be described in quantum language. If we try to describe the (D 1 ),we may consider as follows.
(D2 ) Let {e−n , e−n+1 , ..., e−1 , e0 , e1 , e2 , ..., eN } be the set of emeralds. Let ωi (∈ Ω◦g ) be the state of emeraldei (i = −n, −n + 1, ..., −1, 0), and let ωi (∈ Ω◦b ) be the state of emerald ei (i = 1, 2, ..., N ). However,it should be noted that a family of measurements {ML∞ (Ω,ν) (Oi := (X, 2X , F ), S[ωi ] ) | i = −n, −n +1, ..., −1, 0, 1, 2, ..., N } does not satisfy the uniformity principle of nature. That is because[F ({g})](ωi ) = 1 (i = −n, −n + 1, ..., 0),[F ({g})](ωi ) = 0 (i = 1, 2, ..., N )Hence Theorem 9.6 [Inductive reasoning] cannot be applied.
Or,(D3 ) Let {e−n , e−n+1 , ..., e−1 , e0 , e1 , e2 , ..., eN } be the set of emeralds. And let ωi (∈ Ω◦g ) is the state of
emerald ei such that ω = ωi (i = −n, −n + 1, ..., −1, 0, 1, 2, ..., N ). Let Oi = (X, 2X , Fi ) be theobservable (i = −n, −n+1, ..., −1, 0, 1, 2, ..., N ) such that Oi is the same as O(= (X ≡ {g, b, o}, 2X , F ))in (9.7) (if i = −n, −n + 1, ..., −1, 0), and Oi = (X, 2X , Fi ) (if 0, 1, 2, ..., N ) is deﬁned by Fi ({g}) =F ({b}), Fi ({b}) = F ({g}), Fi ({o}) = F ({o}). However, in this case, it should be noted that a familyof measurements {ML∞ (Ω,ν) (Oi := (X, 2X , Fi ), S[ωi ] ) | i = −n, −n + 1, ..., −1, 0, 1, 2, ..., N } does notsatisfy the uniformity principle of nature. That is because[Fi ({g})](ωi ) = [F ({g})](ωi ) = 1(i = −n, −n + 1, ..., 0),[Fi ({g})](ωi ) = [F ({b})](ωi ) = 0(i = 1, 2, ..., N )Hence Theorem 9.6 [Inductive reasoning] cannot be applied.
Therefore Goodman’s grue paradox (D1 ) cannot be described in quantum language.
Remark 9.10. We believe that there is no scientiﬁc argument without scientiﬁc worldview. Thus, we canimmediately conclude that Goodman’s discussion (D 1 ) is doubtful since his argument is not based on any
scientiﬁc worldview. In this sense, the above arguments (D 2 ) and (D3 ) may not be needed. That is, theconfusion of grue paradox is due to lack of the understanding of Hume’s problem of induction in the linguistic
quantum mechanical worldview, and not lack of the term ”grue” is non-projectible (cf. ref. [25]). If we wantto solve grue paradox only, we may add a condition like ”projectible property”. However, what philosophy
should do is to propose a certain worldview and solve various paradoxes under the worldview at once. Thatis, we think that to solve Goodman’s grue paradox is to answer the following:(E) Propose a worldview! And further formulate Hume’s induction as the fundamental theorem in theworldview! In this formulation, conﬁrm that Goodman’s paradox is eliminated naturally.
What I did is this.
Remark 9.11. Readers think that the grue paradox is too unnatural. I agree. That is, the grue paradox islike playing football and then before you know it, you’re playing rugby. If the rules change in the middle ofa game, there is no sport. I think that this is due to that N. Goodman (1906-1998) belonged to the schoolof analytic philosophy. Analytic philosophy does not emphasize much of the worldviewism (it cf. Sec. 1.3.1)
such that
conclusion

world is so(♯3 )thereforeworldview −−−−−−→ discussions, calculation (= practical logic), properties
premisesubject

Thus, the grue paradox is a paradox resulting from the failure to follow world descriptivism. I think thatthe worldviewism should be emphasized even in analytic philosophy. Thus, I think that the grue paradox is
unproductive for us, if the weaknesses of analytic philosophy are not pointed out. Also, see Wittgenstein’s
paradox in Sec. 12.2.3.

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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)♠Note 9.19. Here, we have (cf. Classiﬁcation 1.11 [the classiﬁcation of philosophers]).
(♭1 ): the realistic worldview (physics)
Hērakleitos, Aristotle, Aristarchus, Archimedes, Eratosthenes, Ptolemaeus,Galileo, Newton, Einstein, · · ·(Although mathematics is not a worldview,Pythagoras, Eudoxus, Euclid)








 (♭21 ): the ﬁctional worldview (Western philosophy)
Plato, Scholasticism, Descartes, Locke, Leibniz, Berkeley, Hume, Kant, Husserl
(♭)






(♭22 ): the logical worldview (=the logical spirit=the spirit of ”Think logically!”)





Boole, Frege, Peirce, Saussure, Russell, Wittgenstein, Hempel, Quine, Popper








(♭ ): the mechanical worldview (statistics, quantum language) 23
Parmenides, Zeno, J. Bernoulli, statistics (e.g., Fisher), quantum language

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Chapter 10

Kant: Copernican revolution
The following is usually called the ﬂower of modern philosophy (i.e., epistemology, or dualistic idealism):[tabula rasa, experience]








BritishEmpiricism







 [Locke, Berkeley, Hume] (♯) Descartes −→−−−−−−−−−−−−−−−−→ Kant philosophyCopernican revolution[apriori,nativism][summing-up]








Continental
Rationalism[Leibniz]Thus, it implies that Kant is the greatest philosopher in modern philosophy.
Kant completely followed Platonic method of telling philosophy (i.e., the ﬁctional worldview: cf. Sec.
1.3 ) as follows.
(♯1 ): world is so•

ﬁctional worldview (literary truth, pseudo-truth)
preface, introduction, (ﬁctional)premise, expedient(♯2 ):you should do sotherefore−−−−−−→

ethics, morals
main subject

That is, Kant executed
the following:(♯1 ) world is so is secondary,”Critique of Pure Reason (1781)”:(♯2 ) you should do so is main theme”Critique of Practical Reason (1788)”, ”Critique of Judgment(1790)”
Hence, it is generally believed that Kant’s philosophy has been a great success. That is because many
people believe that
• Kant’s (♯1 ) is powerful enough to guarantee the legitimacy of the great Newtonian mechanics.
Hence, Kant can be trusted. Therefore, the (♯2 ) is also reliable.
In this sense, Kant is the perfecter of Platonic method of telling philosophy. However, I think that• the above (♯1 ) ”Critique of Pure Reason” is about Kant’s dream .
This is because he was only imagining things that could not be conﬁrmed without experimentation.
But, a miracle happens. The Copernican revolution in (♯1 ) is very similar to quantum language (cf.
Chapter 11).
Kant must have dreamed of a good relationship between philosophy and natural sciences. Speciﬁcally,
he dreamed that philosophy would provide the basis for Newtonian mechanics. I think he wanted towrite about that dream in the Critique of Pure Reason. So I think this book is a kind of enlightenment
book that promotes a good relationship between philosophy and natural science. And I think thisbook had a tremendous impact on the acceptance of natural science by the general public. Thus, I
think that the goal of the modern philosophy (♯) was achieved by Kant. That is, I believe he said thefollowing.
• Philosophy is greater than Newtonian mechanics.
So let’s not be afraid of science, but getalong with it.

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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, is one of the most inﬂuential
philosophers in the history of Western philosophy. His main work is ”Critique of Pure Reason (1781)”,”Critique of Practical Reason (1788)”, ”Critique of Judgment(1790)”, whose theme is respectively ”truth”(i.e., ”pseudo- truth” in the sense of this paper), ”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 caseof philosophy, the three (”truth, goodness, and beauty”) are one. As I said before, concerning ”good” and”beauty”, most philosophers have similar arguments. Therefore, they appeal a clear diﬀerence in the partof ”true”. It is the wisdom of Western philosophy to keep fresh by model change.
That is, he followed Platonic method of telling philosophy as follows.
”Critique of Practical Reason (1788)””Critique of Judgment(1790)””Critique of Pure Reason (1781)”(♯)

therefore−−−−−−→

ﬁctional worldview (literary truth)
preface, introduction, (ﬁctional)premise, expedientethics, morals,aesthetics
main subject

Of course, the interest of this text is concentrated to the worldview (i.e., ”Critique of Pure Reason” ).
Have said many times in this text, philosophy of the worldview is a ”preface” of (♯). Thus, it must be builtwith as ”meticulous logic” as a detective novel.

In fact,• ”Critique of pure reason” is very good as a preface,because it succeeded in impressing the reader that philosophy is not inferior to science. Considering the stateof the world at the time, if Kant was a smart philosopher, the conclusion was bound to be ”the coexistenceof philosophy and science.” What kind of discussion is used to draw this conclusion is the showcase of Kant’s
skill.
♠Note 10.1. As mentioned frequently up to this point, the ﬁctional 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, we must use the terms such as ”logic”,”reason”, etc. in the ﬁctional worldview, for example, ”Critique of Pure Reason”, ”Tractatus Logico
philosophicus (=TLP)”, etc. I think that Platonic method of telling philosophy (i.e., The ﬁctional

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Chap. 10 Kant: Copernican revolution
worldview) is a survival strategy for Western philosophy to co-exist with Christianity.
♠Note 10.2. It is often held that Kant lived a very strict and disciplined life, leading to an oft-repeatedstory that neighbors would set their clocks by his daily walks. In fact, he may have been a cheerfuland sociable man. However, the image of Kant as strict and honest promotes an understanding ofKant’s philosophy. For the philosopher, his image itself is a part of his work. Therefore, if you haveseen a painting of Kant chatting with friends 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 amodest and non-joking philosopher.

10.1.2 The purpose of ”Critique of Pure Reason” (1781)

It is usually said that(A) with the advent of Newton, the natural sciences have grown exponentially. People believe in thatsystematic 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” , notobjective 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 were shattered by
Hume.

Thus, I think that(B) His purpose is to answer the following• Why does Newtonian mechanics hold?
though his challenge failed.
♠Note 10.3. (i): Next is Einstein’s words:

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10.1 Critique of Pure Reason• 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 one of the great
realizations of Immanuel Kant that the postulation of a real external world would be senselesswithout this comprehensibility. (on page 292 of ref. [18]: Ideas and Opinions, Crown Publishers,
Inc . . New York 1954)

Note that the above (B) is essentially the same as the following:• Why is the world comprehensible?
With Einstein’s endorsement, it seems certain that Kant is the discoverer of ”the most incomprehensible problem in scientiﬁc history,” even if his theory is inadequate from a scientiﬁc point of view. I
don’t think any researcher would challenge this today, because everyone would think that this kind ofproblem is impossible to solve. This is a problem to enjoy not understanding, not a problem to solve.
(ii): I don’t think Hume’s point was so serious for scientists at the time. But I can understand thatmany philosophers of the time wanted to think so. I also believe that the intellectuals of the time werealso interested in knowing the various perspectives on the relationship between Newtonian mechanics
and philosophy. Therefore, Kant’s real goal was to provide philosophy fans with the answers thatthey wanted, not the truth. Thus, I think that
• Kant’s true purpose was to write a literary work on the theme of ”Why does Newtonian mechanics
hold?”(iii): As seen in Chap. 12 later, I assert that the purpose of Wittgenstein’s TPL (i.e., ref. [117]:”Tractatus Logico-Philosophicus”) is to explain• Why does logic work in our usual world?

10.1.3 Thing-in-itself, Copernican revolution; from copy theory to constitution theory
In order to solve the problem (B), Kant thought that(C) We can understand the ”world” only through the human perception. Also, cats can understand the”world” only through the cat perception. Thus,• There is ”cat’s world” for cats. and further, there is ”butterﬂy’s world” for butterﬂies.

If there are aliens whose cognitive ability is ﬁner than ours, their world is diﬀerent ours. Although
the diﬀerence of the worlds is made by that of the cognitive 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.

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Chap. 10 Kant: Copernican revolution

That is, Kant proposed so called Copernican revolution such that”copy theory” −−−−−−−−−−−−−−−−−→ ”constitution theory”
Copernican revolution (transcendental) idealism
realism
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:(♯1 ) Why can we make and understand Newtonian mechanics?
(♯2 ) Is there a question we can’t answer?

(F): [A rough sketch of Kant’s Answer for (♯1 )]: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 through sensibility based on
concepts such as quantity and cause-eﬀect relationships.

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10.1 Critique of Pure Reason

These abilities are a priori, that is, they are innate and must be common to all human beings. And throughexperience and training, these abilities will increase. Hence, our habitual thinking is also common to all
human beings at its core, and in this sense, it is universal. Therefore, we can understand Newton mechanics( though it depends on our sensibility and understanding).
[About (♯2 )]: 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 provides proofs for these by deriving antinomies, but I think this is more word play than proof.
♠Note 10.4. Of course, the above solution makes little sense from a scientiﬁc point of view. To be
precise, it does not even reach the level of being able to judge right from wrong. If Kant claims theabove to be a scientiﬁc hypothesis, he must provide a way to experimentally test it. However, froma philosophical point of view, Kant’s explanation was suﬃcient. That is, his diﬃcult and seemingly
elaborate arguments were supported by the general public of philosophy lovers. Probably, they
thought that
• Even Einstein said that he didn’t understand this diﬃcult question, so Kant’s explanation in”Critique of Pure Reason” was suﬃcient. And thus, Kant is reliable.
And the following is also reliable.
”Critique of Practical Reason (1788)””Critique of Judgment(1790)””Critique of Pure Reason (1781)”•

ﬁctional worldview (literary truth)
preface, introduction, (ﬁctional)premise, expedient

therefore−−−−−−→

ethics, morals,aesthetics
main subject

It should be noted that the above proof is not scientiﬁc. However, it is sure that Many philosophical fansagreed with Kant’s explanation. Further, they regarded the above proof as the following (I)Now, it is usually said that
 [tabula rasa, experience ] 










BritishEmpiricism







[”Critique of Pure Reason”]
 [Locke, Berkeley, Hume] (I)Descartes−→
Kant philosophy
−→




[founder of epistemology][summing-up,
compromise][nativism, a priori ]








 Continental Rationalism 






[Leibniz]206

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Chap. 10 Kant: Copernican revolution
In this sense, Kant is sometimes said to have integrated British Empiricism and Continental Rationalism.

Also, we see the following table:

Table10.1

The key-words of worldviews (cf. Assertion 1.14 )mind-matter dualism[A](= mind)[B](between A and B)

Platoactual world

Idea

DescartesI, mind, brain

body

Locke

mind

Kant

phenomenon

quantum language

measured value

secondary quality
cognitionobservable[C](= matter)/[idea world]/[matter]primary quality[matter]/[thing-in-itself]state[system]Seeing the above table, some may think:(J) Is it true?
Locke −−−−−−→ Kant
progress
This will be answered in Problem 10.4 later.

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10.2* Summary: What is causality?

10.2 * Summary: What is 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 be said that ”the purpose was to the point.”
For human beings to discover that the essence of movement and change is ”causal relationship”, we had to
wait for the appearance of Galileo, 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 ”birthof modern science”.
the birth of worldview

Movement”purpose”

the birth of modern science−−−−−−−−−−−−−−−−−−−−→(Hērakleitos, Parmenides, Zeno) Aristotle :(About 1500 years)

Causality( Galileo, Bacon, Descartes, Newton)♠Note 10.5. I cannot emphasize too much the importance of the discovery of the term: ”causality”.
That is,(♯) 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.

10.2.1 Four answers to ”what is causality?”
As mentioned above, about ”what is an essence of movement and change?”, it was once settled with theword ”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 the outstanding problems
in the present. Below, I arrange the history of the answer to this problem.
(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 asfollows. :208

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Chap. 10 Kant: Copernican revolution”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 tothis. 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 theworld. And when we think that ”something” in the world is ”causality”, we should just believethat it has ”causality”.

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 theCopernican revolution”), that is,”cognition constitutes the world.”
which is considered that the cognition circuit of causality is installed in the brain, and when it is stimulatedby ”something” and reacts, ”there is causal relationship.”209

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10.2* Summary: What is causality?

♠Note 10.6. 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 honest Kant made such an exaggerated description?” It is a matterof course that Kant had great conﬁdence such that it was the greatest discovery in the history ofphilosophy. I agree to his opinion. For additional explanation about this, see Problem 10.4 later.
Also, see Section 11.5.2.
(C) [Mathematical causality(Dynamical system theory)]:

Since dynamical system theory has developed as the mathematical technique in engineering, 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(t) = v1 (ω1 (t), ω2 (t), . . . , ωn (t), t)

dt
 dω

2
(t) = v2 (ω1 (t), ω2 (t), . . . , ωn (t), t)
dt(10.1)
·
·
·
·
··


 dωn
(t) = vn (ω1 (t), ω2 (t), . . . , ωn (t), t)dt
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 confusionof mathematics and worldview, it is quite useful. In this sense, I think that (C) should be evaluated more.
(D) [Linguistic causal relationship (MeasurementTheory)]:210

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Chap. 10 Kant: Copernican revolution

The causal relationship of measurement theory is decided by the Axiom 2 (causality; Sec. 1.1) ofChap. 1. If I say in detail,:
Although measurement theory consists of the two Axioms 1 and 2, it is the Axiom 2 that isconcerned with causal relationship. When describing something in quantum language (i.e., alanguage called measurement theory) and using Axiom 2 (causality; Sec. 1.1) , we think thatthing 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
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 ?”.

♠Note 10.7. Consider the following problems:(♯1 ) What is time (space, causality, probability, etc.) ?
There are two ways to answer.
 (a): To show the deﬁnition of XX(♯2 ) The answer of ”What is XX ?”(b): To show how to use the term ”XX”
In this note, the answer to the question (♯1 ) is presented from the linguistic point of view (b).

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10.3 Summary; Descartes=Kant philosophy

10.3 Summary; Descartes=Kant philosophy
10.3.1 Hume might be more scientiﬁc than Kant
I think, from the scientiﬁc point of view, that Hume’s theory and Kant’ theory are not so diﬀerent, where

andThat is, Hume was scientiﬁc, so I think he was hesitant to say anything imaginary. Kant, on the other hand,
was eloquent about imaginary things.
Thus I think that• ”From Hume to Kant” is not progress from the scientiﬁc point of view. Rather, Hume may have morescientiﬁc sense.

10.3.2 Why was Kant so successful?
Why was Kant so successful? I’ll give you four reasons as follows.

10.3.2.1 [I]: Platonic method of telling philosophy
Kant followed this method exactly such as”Critique of Practical Reason (1788)””Critique of Judgment(1790)””Critique of Pure Reason (1781)”(A)

therefore−−−−−−→

ﬁctional worldview (”truth”)
preface, introduction, (ﬁctional)premise, expedientethics, morals,aesthetics
main subject

Recall that philosophy is a literary art, not a science In ”truth, goodness, and beauty,” he was well awarethat ”true” is not ”scientiﬁcally true”,”majority true”.
Recall that
♠Note 10.8. The rule of philosophy (≈ idealism) is as follows:(♯1 ) Only discussion, no experimentation.
(♯2 ) the winner will be determined by popularity vote of the general public.

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Chap. 10 Kant: Copernican revolution

10.3.2.2 [II]: Kant had a good understanding of modern philosophy
Kant understood the meaning of the following (not in the East) genealogy of modern Western philosophy.
 Induction (due to Bacon)British Empiricism

Deduction
[Locke, Berkeley, Hume]−→(B)
Descartes Deduction (due to Descartes)[The father of modern philosophy] Continental Rationalism[Leibniz]

The following is my opinion. Kant might think that• the controversy: British Empiricism vs. Continental Rationalism is only apparent controversy.
Kant understood the meaning of the following (not in the East) genealogy of modern Western philosophy.
This controversy is only a superﬁcial dispute. What they cared about most was the success of science (byCopernicus, Galileo, and Newton). In other words, modern philosophy also functioned as a buﬀer between”church versus science”. Moreover, emotionally, philosophy was closer to the Church (e.g., Berkeley).

The general public was hoping that someone would declare that they had proven the superiority of philosophy
over science, even if it was a lie. So Kant decided that the main theme of his book ”Critique of Pure
Reason” would be to lay the foundation for Newtonian mechanics. For this, Kant prepared the following
two magic words that would never be defeated by science no matter how advanced it was.

”Thing-in-itself””Coperenican revolution”

Christianity would also have expected early modern philosophy to have played a role in competing with
Newtonian mechanics. This strategy was a great success, because the public was looking forward to theemergence of a philosopher who could stand up to Newton. The general public received the following.
(C) Kant’s book is too diﬃcult to understand, but apparently ”philosophy is greater than Newtonianmechanics.” Of course, I think that this is preferable to ”the general public thinks science is allpowerful.”

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10.3 Summary; Descartes=Kant philosophy♠Note 10.9. In view of the above, we can understand what the modest and humble Kant meant
when he spoke of a ”Copernican revolution”. The honest Kant could not hide his real intention to
re-invert the relationship between philosophy (+church) and science. To increase the eﬀectiveness of”Copernican revolution”, Kant may have created an image of humility and modesty. I don’t know
Kant’s intention, but in the end, ”Kant’s image” contributed to the great success of Kant’s philosophy.
Also, the freshness is gone it had been about 150 years since Descartes’ ”Discourse on the Method (1630),”and it was time for someone to make a stopping point. Continued to the next section.

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Chap. 10 Kant: Copernican revolution

10.4 Is Kant a progress from Descartes ?
10.4.1 The inevitability of Kant’s appearance
We think thatwhen it comes to the 1770s, the expiration date of epistemology was running out.
Therefore, many people might want to say(D) Newtonian mechanics moved the world. Does the world move by epistemology? It was too early for
300 years? After all, is the ”epistemology” important or not ?
(E) We are tired of the epistemology. Someone please ﬁnish the epistemology nicely.
This is the atmosphere of 1770’s, in which Kant appeared. Thus, Kant was accepted easily.
♠Note 10.10. The immature state of a discipline at the time of its birth is sometimes regarded as philosophy. For example, in ancient Greece, mathematics, astronomy, and atomic theory were philosophies.
It is also reasonable to consider the epistemology of Locke and Hume as the birth of brain science.
Kant, however, rejected this trend as follows.
(F1 ) 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-sensefrom the philosophical point of view. We think that
(F2 ) 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’d like to believe that• 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 ofphilosophy.
That is,the Copernican revolution = the discovery of ”true idealism”

This will be also emphasized again in the next chapter.
Thus, Kant follows Platonic method of telling philosophy such that
”Critique of Practical Reason (1788)””Critique of Judgment(1790)””Critique of Pure Reason (1781)”(G)

therefore−−−−−−→

ﬁctional worldview (literary truth)
preface, introduction, (ﬁctional)premise, expedientethics, morals, aesthetics
main subject

This was a great success. Today, no scientist is interested in ”Critique of Pure Reason (1781)”. However,215

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10.4 Is Kant a progress from Descartes ?
”Critique of Practical Reason (1788)” is a must for any ethicist. Thus, I think that the (G) is the mosttypical example of Platonic method of telling philosophy.
And I think ”Critique of Pure Reason” had a tremendous impact on the acceptance of natural science by
the general public. Thus, I think that the goal of the modern philosophy was achieved by Kant This implies
the end of Grand Narratives (i.e., epistemology), and the start of ”non-physical law (i.e., the philosophy ofproverb)” such as(H1 ) Bentham(1789): ”the greatest happiness of the greatest number”(H2 ) Hegel (1770 - 1831) : ”thesis-antithesis-synthesis”(H3 ) Darwin(1809 - 1882) :”the survival of the ﬁttest”(H4 ) Nietzsche(1844 - 1900) : ”God is dead”
It is natural, since the job of the philosopher is to make non-physical laws (= golden sayings, proverbs,
copies) that are supported by many philosophy enthusiasts.

10.4.2 Is Kant a progress from Descartes ?
I think that(I) Nevertheless, many readers must think that the Copernican revolution is far too

overrated.
In the next chapter, I will show that this is a fair estimation. Here, I will say somewhat more about thisissue as follows.
Problem 10.4. It is sure that both Descartes worldview and Kant worldview are useless. Thus, thefollowing problem may not be easy.
(J1 ) Is the following reform progress?
mind-matter dualistic idealism

Descartes’ worldview (literary truth)

mind-matter dualistic idealism−−−−−→ Kant worldview (literary truth)
progress(Descartes, Locke, Leibniz, Hume)?

(Kant)

Namely, what parts of Descartes’ worldview have advanced?
Brief Answer: The diﬀerence between the two are as follows.
(♯1 ) ”Idealism” in Descartes’ worldview means ”a theory that is possible in the distant future but not
currently experimental ”
for example, Hume’s theory is regarded as the immature early state of brain science.
Therefore, it is possible to think that Descartes-Hume’s theory belongs to pre-science (i.e., a kindof realism ) rather than philosophy (like Democritus atom theory).
(♯2 ) ”Idealism” in Kant’s worldview means ”transcendental idealism”, that is, ”idealism made by theCopernican revolution”.
Therefore, Kant’s theory belongs to philosophy ( ≈ true idealism), not to science ( ≈ realism).
That is,(J2 )(♯1 ):pre-science, a kind of realism
mind-matter dualistic idealism

Descartes’ worldview (literary truth)(♯2 ):philosophy, (transcendental) idealism
mind-matter dualistic idealism
Copernican revolution−−−−−−−−−−−−−−−→
progress

Kant worldview (literary truth)(Descartes, Locke, Leibniz, Hume)(Kant)

Thus, I believe that”transcendental idealism” is true idealism.
The precise answer will be presented in Answer 11.17 of Chapter 11 (i.e., quantum language is also transcendental idealism). Thus, we can conclude that the (J) is yes.

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Chap. 10 Kant: Copernican revolution(K): my answer: Now we proved −−→ and −−→ below.
⃝⃝
19(i.e., ”becoming(K1 ) If ”to make progress” is deﬁned by ”to come near quantum language” moreand more like quantum language” we can say that the time series ⃝can be1 - ⃝9 - ⃝11 ⃝13
regarded as progress, that is,

The precise answer ⃝9 will be presented in Answer 11.17 of Chapter 11 (i.e., quantum language isalso transcendental idealism).
That is,(K2 ) the time line:⃝
is not only a refreshing change, but also
1 - ⃝9 - ⃝11 ⃝13

a progress.

♠Note 10.11. Physics and science make up a theory while making modiﬁcations by the result of
the experiment. Thus, physics and science can expect sound development. On the other hand, thephilosophy of worldview is metaphysics, which cannot be determined by experiments. Thus, thequestion ”Did the philosophy of worldview make a progress?” is not easy to answer. That is because,
if we consider that• the western philosophy was able to keep freshness for a long time by renewing a preface part (i.e.,
the worldview), much like a car model change.
then, we must conclude that the philosophy of worldview does not make a progress. However, in thispaper, we assert that(♯1 ) the philosophy of worldview has been making a progress. And moreover, it ﬁnally converges toquantum language.
More precisely, we assert that(♯2 ) If ”to make progress” is deﬁned by ”to come near quantum language” (cf. Assertion 1.5), thenthe philosophy of worldview has been making a progress.
Remark 10.5. Recall Remark 3.13 such as
Introduction part (Pseudoscientiﬁc theories)
z
}|
{

Plato



Idea theory








DescartesI think, therefore I am 


Kant

thing-in-itself
Analytic philosophy

mathematical logic



















commonTherefore−−−−−−−→ spirit, ethics, moral

It was the method of the great philosophies (i.e., Plato’s way of telling philosophy) to ﬁnd something217

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10.4 Is Kant a progress from Descartes ?
impossible to doubt, and then declare that they had ”deduced” various things from it. Kant’s ”thing-initself” is also something impossible to doubt. No matter what someone says about Kant’s ”thing-in-itself”,
there is no experimental data to contradict it. If this is the case, then the rest is up to the power of literature.
It is no wonder even if philosophers after Kant thought this way. In this sense, Kant is also the founderof literature about ”thing itself”. The most successful were probably A. Schopenhauer (1788 - 1860) and
Husserl, though I have not understood their writings. I am sure they are great philosophers, but I do notthink they are up to par with Kant, since the discoverer of ”thing-in-itself” is Kant.
Remark 10.6. Someone may want to consider the following progress:
Plato −−−−−→ Descartes −−−−−→ Kant −−−−−→ Husserl
progress

progress(10.2)

progress

This is because it is widely accepted that Husserl overcame the heights of Kant. In terms of literary
enjoyment, I think the above is correct. However, in this text, the symbol ”X −−−−−→ Y is deﬁned by• ”Y” is closer to quantum language than ”X”
Thus, we consider that ”Husserl” is located as follows.

progress

Rather, I think Husserl was enjoying the word ”rigor” from a literary point of view without mathematics.
The idea of starting with the obvious does not sit well with science. Husserl’s phenomenology is one of thegreat achievements of Western philosophy, but it has little to do with science.
However, as mentioned in Note 11.3, we must follow the following rules of philosophy (= idealism).
(♯1 ) Only discussion, no experimentation.
(♯2 ) the winner will be determined by popularity vote of the general public.

♠Note 10.12. Here, we have (cf. Classiﬁcation 1.11 [the classiﬁcation of philosophers]).

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Chap. 10 Kant: Copernican revolution(♭1 ): the realistic worldview (physics)
Hērakleitos, Aristotle, Aristarchus, Archimedes, Eratosthenes, Ptolemaeus,Galileo, Newton, Einstein, · · ·(Although mathematics is not a worldview,Pythagoras, Eudoxus, Euclid)








 (♭21 ): the ﬁctional worldview (Western philosophy)
Plato, Scholasticism, Descartes, Locke, Leibniz, Berkeley, Hume, Kant, Husserl
(♭)






(♭22 ): the logical worldview (=the logical spirit=the spirit of ”Think logically!”)





Boole, Frege, Peirce, Saussure, Russell, Wittgenstein, Hempel, Quine, Popper








 (♭23 ): the mechanical worldview (statistics, quantum language)
Parmenides, Zeno, J. Bernoulli, statistics (e.g., Fisher), quantum language219

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221

Chapter 11

Linguistic philosophy ( Before TLP)
Throughout this text, we consider that linguistic philosophy = analytic philosophy. In this ﬁnal
chapter, we devote ourselves to thedualism−−−−−−−−−→
measurement

O+O in the following Figure 0 (i.e., Figure 0
13

14

in preface).
Figure 0 (in preface )the realistic worldview (physics): monism(monism)−−→ Newton(realism)

Socrates
Aristotle

O:Greek0philosophy
Parmenides, Zeno
Plato

Schola-O−−−−−→ 1
sticismOOrelativity−→ theory−−−−−−−−−−→ 32Oquantum−→ mechanics −−−−−−−−→ 4









(unsolved)

O TOE:
5

−−→

O

O
O

O

OO

theory of
everything(quantum phys.)language−−−−−−−−−−−−−−−−−
→7 




dualism8−−−−−−−−→ −−−−−−−−−→ 6 statistics

quantization
applied math.
system theoryclassical mechanical worldview



(=MT)
(mind-matter dualism) (Copernican revolution))





QL:
quantum
14
DescartesKant
9mechanical−−→language−−→
Locke,...
−−→−−−−−−−→ 10turn
quantum mechanicalﬁctional worldview

ﬁctional worldview
worldview


11



turn
linguistic





?
(logic in philo.))(logic, set theory)Wittgensteindualism
Boole, Frege, 12 Saussurre−−−−−−−−−→ 13
measurement
Russell,Cantor −−→ Zadeh

O

O

O

math.

logical worldview

the idealistic worldview (no experiment, metaphysics): dualism
Figure 0: The history of the world-descriptions
Philosophy (≈ dualistic idealism) has progressed toward QL( i.e., 0 → 1 → 9 → 11 → 13 → 14 )O O O O O O

I think that(♯1 ) Frege may have just tried to apply mathematical logic to our world, without a ﬁrm reason, justby strong belief.
(♯2 ) Wittgenstein sought to ﬁnd out why mathematical logic works in our world, but his considerations
were non-theoretical and too literary.
But nevertheless, I would like to conclude that they are the two founders of analytic philosophy. The
reason is that quantum language shows that their direction was correct.

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Frege and Russell asserted that• from ”logic as detective novel” to ”logic as Euclid’s Element”

dualism

However, philosophy is not mathematics. Thus, we need the next step ” −−−−−−−−−→
dualism

In order to explain ” −−−−−−−−−→
measurement

O+O” .
13

14

O+O” , I ﬁrst need to make sure that the reader understands
measurement

13

14

my understanding of Wittgenstein’s picture theory (in TLP (i.e., ref. [117]: ”Tractatus LogicoPhilosophicus”)). For this, in Sec. 11.7, I will propose ”Fuzzy logic in QL”, which is characterized as
the simultaneous justiﬁcation of Wittgenstein picture theory and Zadeh’s fuzzy sets theory. That is,(♯3 ) ”Fuzzy logic in QL (in Sec. 11.7)” =”Wittgenstein picture theory”+”Zadeh’s fuzzy sets theory”
And I see that

where ”progress” means ”getting closer to QL”. To assert the above, we must ﬁnd a dualistic idealismin Tractatus Logico-Philosophicus (abbreviated as TLP), i.e.,(♯4 ) ”logic” should be regarded as a ”binary measurement” (See Sec.11.7; Fuzzy logic ...)For this, we must rebuild the core of analytic philosophy. In any case, I think my ﬁction above requires
strict reader checking.

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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
1 (if P is true)
ϕ(P ) =
0 (if P is wrong (i.e., not true))(A1 ) Assume that P, P1 , P2 are propositions. Then, ¬P , P1 ∧ P2 , P1 ∨ P2 , P1 → P2 are propositions.
And it holds that
1 (if ϕ(¬P ) = 0)ϕ(¬P ) =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)
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.

Remark 11.2. The most important point to note above is that ”proposition” is not deﬁned. In mathematics, we can deﬁne a ”mathematical proposition” in set theory. However,(♯1 ) In non-mathematics, the lack of deﬁnition of ”propositions” is a fatal weakness of analytic
philosophy .
In this sense, analytic philosophy is not ”logical”. Also recall Carnap=Quine debate, which stems from thelack. Wittgenstein tried to overcome this and wrote [117]: ”Tractatus Logico-Philosophicus (abbreviatedas TLP)”, but he could not do it.
TLP was successful as an enlightening book on analytic philosophy,
but incomplete as a theory.

♠Note 11.1. In the above, it should be noted that(♯1 ) the most diﬃcult and important problem is ”What is ’P is true’ ?” (or, equivalently, ”What is
a proposition?”). Therefore, in general, the existence of a proposition is not guaranteed.
In mathematics (= set theory), we know the answer for ”What is a mathematical proposition?”.
Thus, mathematics is a typical example of symbolic logic at work. Therefore, our present problem
is(♯2 ) Is there any other world besides mathematics where symbolic logic works? ( This question is
our main interest in this chapter)
Although it is important that symbolic logic also works in logic gate, our interest here is not

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11.1 Elementary knowledge of symbolic logic in mathematics
engineering.

(iii): Thus, I think that, in the 100-year history of analytic philosophy, the most important question:”What is a proposition?” has always been ambiguous.
Exercise 11.3. The truth table below is well known. Wittgenstein was one of the contributors to the
popularization of truth tables.
(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)[|an − a| < ϵ(∀n > N )]This was due to the great mathematicians Cauchy and Weierstrass etc.

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Chap. 11 Linguistic philosophy ( Before TLP)

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 is used in ﬁelds where
detailed data is available as follows.

In motto terms, this ﬁgure says the following.
(B)
which is common sense that everyone knows.
Thus, we see that
(C1 ) ”ordinary logic” =”rigorous theory” is not true.
though the following is of course true:(C2 ) ”mathematical logic” =”rigorous theory””Logic for the humanities, statistics for the sciences”♠Note 11.2. I belong to a math department and know many brilliant mathematicians, but most ofthem know nothing more about logic than I mentioned above. I was surprised that philosophers knew
diﬃcult theorems of mathematical logic (e.g., Gödel’s incompleteness theorem, Löwenheim–Skolem
theorem, Banach-Tarski theorem, etc.). It is curious that the overwhelmingly successful worldview
is the (classical and quantum) mechanical worldview, yet many philosophers are indiﬀerent to it. Of
course, this is due to the misunderstanding (C1 ).
Frege and Russell might have deliberately exploited the confusion between (C 1 ) and (C2 ) to promote analytic
philosophy. As I have said many times in this text, Plato, Descartes, and Kant began their philosophy by”aﬃrming lies”. Wittgenstein tried to resolve the confusion between (C 1 ) and (C2 ), but he failed in the end.
In Sec. 12.2, we prove that(D) ”ordinary logic” =”rough argument in QL”
That is, in statistics, errors are exposed and discussed, on the other hand, in ordinary logic, errors are hidden
and are not discussed openly.

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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. [78]: Ishikawa, S., (2021) Fuzzy Logic in the Quantum Mechanical Worldview ; Related toZadeh, Wittgenstein, Moore, Saussure, Quine, Lewis Carroll, etc. JAMP, Vol. 9, No.3 , 140-154,(https://www.scirp.org/Journal/paperinformation.aspx?paperid=95447)
Now let us introduce probabilistic symbolic logic, which is a slight generalization of mathematical symbol
logic mentioned in the previous section. Probabilistic symbolic logic plays an essential role in fuzzy logic in
QL ( in Sec. 11.7). Since I am convinced that this fuzzy logic is what Wittgenstein had wanted to insist,I encourage readers to read 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 high school students.
However, it should be noted that this probabilistic symbolic logic, like mathematical symbolic logic, is not
suﬃcient for philosophy in the sense that it does not have a deﬁnition of ”proposition”.

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 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.
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.
(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):In the above, it should be noted that the deﬁnition of ”proposition” is not written. For
example, is Descartes’ ”I think, therefore I am” a scientiﬁc proposition? This is one of the most important
problems in philosophy of science. The deﬁnition of scientiﬁc proposition is proposed in Sec. .
(ii): Zadeh often emphasized that fuzziness and probability are diﬀerent concepts. However, I believe thatfuzziness without a probability interpretation cannot be a scientiﬁc concept. In this sense, the above (E)is essential. Thus, the diﬀerence between fuzziness and probability are mood. As mentioned before, in thispaper ”probabilistic logic” in quantum language is called ”fuzzy logic.

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11.2.2 Quasi-product probability
In addition, assume another proposition P2 such that Prob[P2 ; {T }] = p2 , Prob[P2 ; {F }] = 1 − p2 . Thus, wehave 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 }),(µ1 ×qp µ2 )({T, F } × {T }) = µ2 ({T }),(µ1 ×qp µ2 )({T, F } × {F }) = µ2 ({F }),and

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 )({(F, T )})(≡ µ2 ({F }) − α)(µ1 × µ2 )({(F, F )})(≡ 1 − µ1 ({F }) − µ2 ({F }) + α)(µ1 ×qp µ2 )({(T, T )})(≡ α)(µ1 × µ2 )({(T, F )})(≡ µ1 ({F }) − α)
qp

qp

where
max{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. 0.3] in the sameqp
way for the two questions P1 and P2 . Then, the µ1µ2 is considered as the ordinary product probabilityµ1µ2 such that

×

×

qp

(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).
\µ2 ({T })µ2 ({F })Table 11.2: product probability measure µ1µ1 ({T })µ1 ({T }) × µ2 ({T })(≡ α)µ1 ({T }) × µ2 ({F })(≡ µ1 ({F }) − α)×qp µ2 (= µ1 × µ2 )µ1 ({F })µ1 ({F }) × µ2 ({T })(≡ µ2 ({F }) − α)µ1 ({F }) × µ2 ({F })(≡ 1 − µ1 ({F }) − µ2 ({F }) + α)(ii):[Standard situation when P1 = P2 ] It is natural to think that the respondent selected in (C) will
qp
give the same answer to the same question (i.e., P1 and P1 ). In this case we see µ1µ1 such that
qp
qp
(µ1
µ1 )({(T, T )}) = p1 and (µ1µ1 )({(F, F )}) = 1 − p1 . Thus,( if (x1 , x2 ) = (T, T )qp
 p1
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).
\

×qp

Table 11.3: quasi-product probability measure µ1µ1µ1 ({T })µ1 ({F })µ1 ({T })µ1 ({F })(µ1(µ1

×qpqpµ1 )({(T, T )}) = p1
× µ1 )({(T, F )}) = 0
227qpµ1 )({(F, T )}) = 0×
qp(µ1 × µ1 )({(F, F )}) = 1 − p1(µ1

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×qpµ1 will be used. HowIn this paper, unless otherwise stated, this quasi-product probability measure µ1
ever, if so, it is not necessary to ask the same question twice; we only need to ask it once. For further disqp
qp
cussion, see Example 11.8 later. Also, thisi=1,2 µi (= µ) is easily extended to the case that
i=1,2,...,n µi(where µi = µ, i = 1, 2, ..., n). such thatn
( if (xi )nqp p1
i=1 = (T )i=1nn1 − p1 ( if (xi )i=1 = (F )n(µ)({(xi )i=1 }) =i=1
i=1,2,...,n0( others )

×

×

×

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 ”,
P1 ∨ P2 = ”P1 or P2 ”,
P1 → P2 = ”¬P1 or P2 ”,¬P = not P
If we write the above (i.e., P1 ∧ P2 , ¬P1 ∧ P2 , P1 ∧ ¬P2 , ¬P1 ∧ ¬P2 ) in a table, we get 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 )})p2̄ = µ2 ({F })≡ p1 − α)≡ 1 − p1 − p2 + α)
where max{p1 + p2 − 1, 0} ≤ α ≤ min{p1 , p2 }. Note that Tables 11.1∼11.3 are related to only quasi-productprobability µ1 ×qp µ2 , but Table 11.4 is related to µ1 ×qp µ2 and logical symbol (i.e., ∧, ¬).
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 =¬P1 P1 → P2 P1 ∧ P2 P1 ∨ P2
i=1,2 µi
qp
T
T
p12 =
F
T
T
T
i=1,2 µi ({(T, T )})
qp
µ
({(T,
F
)})
T
F
p12̄ =
F
F
F
T
i
i=1,2
qp
µ
({(F,
T
)})
F
T
p1̄2 =
T
T
F
T
i=1,2 iqpFF
p1̄2̄ =
T
T
F
F
i=1,2 µi ({(F, F )})

×

×
×
×
×

Thus, it will be enough to explain hoe to use the ”probability column” as follows. For example, consider theabove 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 )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
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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 is the formal deﬁnition of logic symbols( i.e., ¬, ∧, ∨, →).

Example 11.8. [Probabilistic truth table ((i): P1 ̸= P3 , (ii): P1 = P3 )](i) (Simplest case: P1 ̸= P3 ): The following table (i.e., Table 11.6) is the preparation of the next (ii) (i.e.,
Tables 7, 8, 9). Consider the truth table of [P1 → P2 ] ∧ P3 as follows.
Table 11.6: Probabilistic Truth Table ( Elementary propositions P1 , P2 , P3 )qp
P1 P2 P3
probability: p =[P1 → P2 ] ∧ P3
i=1,2,3 µi
qpTTTp123 =
T
i=1,2,3 µi ({(T, T, T )})qp
TTF
p123̄ =
F
i=1,2,3 µi ({(T, T, F )})qp
TFT
p12̄3 =
F
i=1,2,3 µi ({(T, F, T )})qp
TFF
p1̄2̄3̄ =F
i=1,2,3 µi ({(T, F, F )})qp
F
T
T
p1̄23 =
T
i=1,2,3 µi ({(F, T, T )})
qp
µ
({(F,
T,
F
)})
F
T
F
p1̄23̄ =Fi
i=1,2,3qp
FFT
p1̄2̄3 =T
i=1,2,3 µi ({(F, F, T )})qp
FFF
p1̄2̄3̄ =F
i=1,2,3 µi ({(F, F, F )})

×

×
×
×
×
×
×
×
×

This says that the proposition [P1 → P2 ] ∧ P3 is the map from{T, F }3 → {T, F } such that
T if (x1 , x2 , x3 ) = (T, T, T ), (F, T, T ), (F, F, T )[[P1 → P2 ] ∧ P3 ](x1 , x2 , x3 ) =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 (if x1 ̸= x3 ). Thus, puttingqp×qp
i=1,2 µi ({(x1 , x2 )}) = ×i=1,2,3 µi ({(x1 , x2 )} × {T.F }), we see the following: ( Note that Table 11.6= Table
11.7 except the probability column).
Table 11.7: Probabilistic Truth Table ( Overlapping elementary propositions P1 , P2 , P3 (= P1 ))qp
P1 P2 P3 (= P1 )probability: p =[P1 → P2 ] ∧ P3 (= P1 )i=1,2 µiqp
TTT
p123 (= p12 ) =T
i=1,2 µi ({(T, T )})T
TF
p123̄ (= 0)
F
qp
T
F
T
p12̄3 (= p12̄ ) =F
i=1,2 µi ({(T, F )})T
F
F
p12̄3̄ (= 0)F
F
TT
p1̄23 (= 0)
T
qp
F
T
F
p1̄23̄ (= p1̄2 ) =F
i=1,2 µi ({(F, T )})F
F
T
p1̄2̄3 (= 0)
T
qp
F
F
F
p1̄2̄3̄ (= p1̄2̄ ) =
F
µ
({(F,
F
)})
i=1,2 i

×

×

×

×

×

Since the case of the probability 0 (i.e., p123̄ = p1̄23 = p12̄3̄ = p1̄2̄3 = 0 ) can be omitted, we have the following
table:
Table 11.8: Probabilistic Truth Table (Overlapping elementary propositions P1 , P2 , P3 (= P1 ))qp
P1 P2 P3 (= P1 )probability: p =[P1 → P2 ] ∧ P3 (= P1 )i=1,2 µiqp
TTT
p123 (= p12 ) =
T
µ
({(T,
T
)})
i=1,2 iqp
TFT
p12̄3 (= p12̄ ) =F
i=1,2 µi ({(T, F )})qp
F
TF
p1̄23̄ (= p1̄2 ) =F
i=1,2 µi ({(F, T )})qp
FFF
p1̄2̄3̄ (= p1̄2̄ ) =
F
µ
({(F,
F
)})
i=1,2 i

×
×
×
×

×

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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 =[P1 → P2 ] ∧ P1
i=1,2 µi
qp
T
T
p12 =
T
i=1,2 µi ({(T, T )})qp
TF
p12̄ =
F
i=1,2 µi ({(T, F )})qp
F
T
p1̄2 =
F
i=1,2 µi ({(F, T )})qp
FF
p1̄2̄ =
F
i=1,2 µi ({(F, F )})

×

×
×
×
×

Hence, we conclude that Tables 7, 8 and 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).
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 for the ”probability column”.
Table 11.10: Probabilistic Truth Table ( Elementary propositions P1 , P2 )qp
P1 P2 probability: p =[P1 → P2 ] ∧ P1
i=1,2 µi
qp
T
T
p12 =
T
i=1,2 µi ({(T, T )})
qp
µ
({(T,
F
)})
T
F
p12̄ =
F
i
i=1,2
qp
F
T
p1̄2 =
F
i=1,2 µi ({(F, T )})qp
FF
p1̄2̄ =
F
i=1,2 µi ({(F, F )})

×
×
×
×

×

[P1 → P2 ] ∧ P1 → P2
T
T
T
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 non-overlapping 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 ̸=
Pj (∀i, j such that i ̸= j, cf. (H) above), and consider the probability space ({T, F }, 2{T,F } , µi ) such thatµi ({T }) = Prob[Pi ; {T }], µi ({F }) = Prob[Pi ; {F }]. Consider a quasi-product probability space ({T, F }n ,qp
P({T, F }n ), i=1,2,...,n µi ) such that

×

qp

(

× µi )({T, F }k−1 × {x} × {T, F }n−k ) = Prob[Pk ; {x}]i=1,2,...,n(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 logical sample space.
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 that×qp

Prob[P ; {T }] = (× µi )(P −1 ({T })))
i=1,2,...,nAlso, it is clear that the above Example 11.9 implies the following corollary.
Corollary 11.11. [Tautology in probabilistic logic] Let P be a proposition constructed from 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 syllogism.
(♯) Since Socrates is human being, and human being is mortal, it follows that Socrates is mortal.

However 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. A discussion of mathematics alone,
without a worldview, is powerless.

Therefore, the above Corollary 11.11 does not guarantee that (♯) holds. This will be discussed in Sec. 12.2.

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11.3 George Boole (1815-1864), Gottlob Frege (1848-1925) and Bertrand Russell (1872-1970)

11.3 George Boole (1815-1864), Gottlob Frege (1848-1925) andBertrand Russell (1872-1970)
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 by him. Also, Gottlob Frege (1848-1925)
was a German philosopher, logician, and mathematician. He is called the father of analytic philosophy ,concentrating on the philosophy of language, logic, and mathematics. Particularly, it’s said that the predicate
logic was proposed by him. Bertrand Russel (1872-1970) was mathematician, philosopher, logician, social
critic in England. With A. N. Whitehead he wrote Principia Mathematica, an attempt to create a logicalbasis for mathematics.

Recall Remark 10.5 such asz

introduction part
}|
Plato

Idea theory
Descartes(A)

I think, therefore I am
Kant

thing-in-itself
Kant’s philosophy provided the foundation for Newtonian mechanics

{












common Therefore−−−−−−−−−→ spirit, ethics, moraltrust me













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 things from it.
Following the example of Kant, B. Frege and B. Russell must have thought the following (or they musthave thought that the general public would support them if they said the following ):(B1 ) Mathematical logic provides the foundation for mathematics. Mathematics is just one 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
Therefore−−−−−−−−−→ ”Think everything logically!”(B2 )mathematical logic
trust logic
Logic is so powerful that it is the foundation of mathematics

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Chap. 11 Linguistic philosophy ( Before TLP)I think that Frege and Russell did not inherit Kant’s philosophy, but they learned how to spread philosophy
from Kant’s methods.

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 ofeveryday life are not suﬃciently abstract. Only mathematics and mathematical 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 of course he knew that
”Einstein was not familiar with mathematical logic”. So why did he make the above statement? My guessis as follows. He knew the method of philosophy (i.e., Plato’s way of telling philosophy), that is,
Philosophy is ”asserted ﬁction” that is, to say ”Therefore” in the (B), even if it’s a lie.
Moreover, I think Russell was a man of integrity, and he could not keep lying, so he made his student
Wittgenstein say,
(D1 )”The ’Therefore’ in (B) is a lie.”(D2 )”Russell’s (C) is a lie, since he cannot deﬁne ”non-mathematical proposition”

For example, is ”the ﬁst proposition ”I think, therefore, I am” a non-mathematical proposition?
Otherwise, Russell’s generosity to Wittgenstein cannot be understood.
♠Note 11.3. The rule of philosophy (≈ idealism) is as follows:(♯1 ) Only discussion, no experimentation.
(♯2 ) the winner will be determined by popularity vote of the general public.

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11.3 George Boole (1815-1864), Gottlob Frege (1848-1925) and Bertrand Russell (1872-1970)
Therefore, philosophers are not necessarily in pursuit of truth. Sometimes it is necessary to bluﬀ alittle to gain a lot of support. Russell did not mean to say that logic is greater than mathematics.
However, ordinary enthusiasts of philosophy may have felt from Russell’s words and actions that”logic is greater than mathematics.”

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Chap. 11 Linguistic philosophy ( Before TLP)

11.4 Peirce (1839-1914): Abduction

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. [96](Vol. II p.375), as the typical example of ”abduction”, Peirce mentioned as follows.
• Fossils are found; say, remains like those of ﬁshes, but far in the interior of the country. To explain
the phenomenon, we suppose the sea once washed over this land.
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 thefollowing fugure in Sec. 1.5.3:

This section was written with reference to the following.
• [75]:Ishikawa, S: Philosophy of science for scientists; The probabilistic interpretation of science Journalof quantum information science, Vol. 9, No.3 , 140-154,
DOI: 10.4236/jqis.2019.93007(https://www.scirp.org/Journal/paperinformation.aspx?paperid=95447)
The purpose of this section is to explain the following table:
Logical worldview

Quantum mechanical worldview[ no measurement (Qualitative, logical approach))][ measurement ( Quantitative, probabilistic approach)]deduction (cf. Sec 11.4)←→

measurement

abduction (cf. Sec 11.3)←→

inference (maximum likelihood method)

induction (cf. Sec 9.7)←→

the law of large numbers

where  the logical approach(♯)
the probabilistic approach(rough, handy)(exact, calculation required)235

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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 clearlyseen in the following timeline(i) Born’s discovery “the probabilistic interpretation of quantum mechanics” in ref. [6] (1926)(ii) Tractatus Logico-Philosophicus by L. Wittgenstein in ref. [117] (1921)(iii) Fisher’s great book “Statistical Methods for Research Workers” in ref. [22] (1925)
These three are among the major works of the top ten most inﬂuential writings of the 20th century.
Therefore, I believe that Peirce’s work was ahead of its time.

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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.
[Deduction]b ′1 ) are often omitted.
A typical example of deduction is as follows: In the following, (A′1 ) and (A(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 toblack 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 (♯1 ).
However, this has not solved everything. We have the following problem:(♯2 ) Is there a more natural solution than the above solution ( due to the logical worldview (=thelogical spirit=the spirit of ”Think logically!”))?
This new solution will be introduced in the following section.
[Abduction]
C.S, Peirce (cf. ref. [96]) 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 ′1 ) All the beans in that bag B2 are white and black ﬁfty-ﬁfty (or generally, the ratio of white beans to(Ablack beans is p/(1 − p)): [bag B2 −→”w”(≈ white) or ”b”(≈ black)]b 2 ) This bean (from B1 or B2 (i.e., it is not known whether it is B1 or B2 ) ) is white: [”w”(≈ white)](Ab 3 ) Therefore, this bean is from this bag B1 : [bag B1 ](A

♠Note 11.6.
 It is clear that the following is not a tautology:(♯1 )[B1 −→ w] ∧ [B2 −→ [w ∨ b]] ∧ w −→ [B1 ]237

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11.4 Peirce (1839-1914): Abductionb 3 ) can not be understood since this (♯3 ) is not a tautology.
Thus, the above conclusion (Ab 3 ) has a point as follows. For simplicity, assume that 0 ≤ p ≪ 1 (However, the above conclusion (Ai.e., most of the beans in the B2 are black).
b 2 ), assume that this white bean is from the bag B2 .
(♯2 ) After (AIf so, this is a very rare event that has happened since most of the beans in the B 2 are black. The
assumption (♯2 ) is unreasonable. That is, it is reasonable to consider that this white bean is from theb 3 ).
bag B1 , namely, (ANow from the theoretical point of view, we have two ways to think the above problem.
invent another symbolic logic in which (♯1 ) is derived
1 ⃝:To⃝:To
propose a diﬀerent worldview than the logical worldview (=the logical spirit
(♯3 )
2


=the
spiritof ”Think logically!”).
b 3 ) in the new worldview.
And to derive (AI think that ⃝isimpossible.
⃝
1
2 will be introduced in the following section.
[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, 0 < p < 1).
(A
Assume that we do not know the value p (0 < p < 1).
e 2 ) When we took 20 beans out of this bag B3 , every bean was white.
(Ae 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.

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. Consider theFirst, we will show that the abduction [(Astate space Θ = {θ1 , θ2 } with the discrete topology, and the classical basic structure [C(Θ) ⊆ L∞ (Θ, ν) ⊆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 and 1000(1 − p) black beans
belong to the bag B2 (where 0 < p < 1). Thus we have the observable O = ({w, b}, 2{w,b} , F ) in L∞ (Θ, ν)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 ML∞ (Θ,ν) (O := ({w, b}, 2{w,b} , F ), S[θi ] ), i = 1, 2. For example, Axiom 1[measurement] (in Section 1.1) says that
(B1 ) [measurement]: The probability that the measured value w is obtained by ML∞ (Θ,ν) (O :=({w, b}, 2{w,b} , F ), S[θ1 ] ) is equal to 1
This is the same as the deduction (i.e., (A1 )–(A3 )).
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 ML∞ (Θ,ν) (O :=(B({w, b}, 2{w,b} , F ), S[∗] ), which do you infer, [∗] = θ1 or [∗] = θ2 ?
Fisher’s maximum likelihood method Theorem 1.19 [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 (A238

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b 2 ) is the quantum linguistic representation of abduction (i.e., ( Ab 1 )–(Ab 3 )). For the
Therefore, the above (Bbsake of completeness, note that (B1 ) and (B2 ) are in reverse problem (cf. Remark 1.20). That is, we havethe following correspondence:
Correspondence 11.13.
(B1 ): measurement−−−−−−−−−−→ [(A1 ),(A′1 ),(A2 ) −→(A3 )]deduction
Axiom 1 [measurement] (in Section 1.1) simpliﬁed form
yreverse
yreverseb 2 ): inference(B(Fisher’s maximum likelihood method)(11.1)b 1 ),(Ab ′1 ),(Ab 2 ) −→(Ab 3 )]−−−−−−−−−−→ [(Asimpliﬁed form
abductionThus, the scientiﬁc meaning of abduction can be completely clariﬁed in the translation from logic toquantum language. Lastly we should mention thate 3 ) the above (Ae 1 )-(Ae 3 ) (i.e., inductive reasoning) are already discussed in quantum language (cf. Section(B
9.7: Hume’s problem of induction).
After all, I think that• logic is useful for making broad arguments.
However, a logical approach is of course also important since quantitative data is not available in most cases
on a daily basis. In fact, I think that Detective Sherlock Holmes was a great master of Peirce’s abduction.
As a slogan,the humanities are logic, and the sciences are statistics.
which is similar tologic for rough arguments, statistics for precise arguments.
This will be proved in Sec. 12.2.
♠Note 11.7. Philosophers may like logic (i.e., the logical worldview (=the logical spirit=the spirit of”Think logically!”)), thus, they may want to think that deduction, abduction and induction are kindsof logic. However, I think that Peirce’s argument in Sec. 11.4 is not natural. On the other hand, we
think that our quantum mechanical formulation is reasonable as follows.
(C1 )deduction······
measurement(possible but unnatural)

(C2 )
(C3 )

abduction(theoretically impossible)

induction(theoretically impossible)(possible and natural)· · · · · · inference (Fisher’s maximum likelihood method)(possible and natural)· · · · · · the law of large numbers(possible and natural)That is,
Logical worldview

Quantum mechanical worldview[ no measurement ][ measurement, probability ]deduction (cf. Sec 11.3)←→

measurement

abduction (cf. Sec 11.3)←→

inferenceinduction (cf. Sec 9.7)←→

the law of large numberswhere the logical approach(♯)(rough, handy)

the probabilistic approach (exact, calculation required)
Here, recall Sec. 9.8, i.e., the grue paradox arises in the logical worldview
method), and not in quantum mechanical worldview.

(or precisely, logical

Finally, note the following:239

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11.4 Peirce (1839-1914): Abduction• In the case that quantitative data is available,, ”quantum mechanical worldview” is superior to”logical worldview”. However, ”logical worldview” has the advantage that it can be used without
using mathematical formulas. Therefore, from a daily point of view, it may be said that ”logicalworldview” is superior.

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11.5 Bertrand Russell: ﬁve-minute hypothesis, McTaggart’s paradox,
Moore’s paradox
This section consists of excerpts from the following paper:• Ref. [78]: Ishikawa, S., (2021) Fuzzy Logic in the Quantum Mechanical Worldview ; Related toZadeh, Wittgenstein, Moore, Saussure, Quine, Lewis Carroll, etc. JAMP, Vol. 9, No.3 , 140-154,(https://www.scirp.org/Journal/paperinformation.aspx?paperid=95447)
Bertrand Russell (1872-1970) was a great intellectual and one of the founders of analytic philosophy alongwith Gottlob Frege (and student Ludwig Wittgenstein). Russell is a multi-talented philosopher who has
produced many achievements. However, his greatest achievement may have been the production of the”genius Wittgenstein”.

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 ( orequivalently, 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.
Also, it may be interesting to read ref. [109] for the delicate relationship between Russell and McTaggart.

11.5.1 Russell’s paradox in set theory
As mentioned frequently in this text, I think that mathematical logic is one of ﬁelds of mathematics 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’s paradox (or,Russell-Zermelo paradox).
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 ”type theory” (i.e., akind of axiomatic set theory), which was published as the Principia Mathematica on the foundations ofmathematics 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 of eight axioms by Zermelo and Fraenkel plus an axiom called the Axiom of Choice.

*1 I it is assumed that ”the set of all sets” is a set, the it leads contradiction. This is called Cantor’s paradox.

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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’ ?”. Therefore, It isbelieved 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. [117]: ”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 ofaxiomatic set theory (e.g., Zermelo-Fraenkel set theory), in which ”mathematical proposition” can becompletely deﬁned.
This (B) should be compared with the following (C).
(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 development of 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. [117]: ”Tractatus Logico-Philosophicus”) in which
only ”What we cannot speak about we must pass over in silence” is written. This implies that Wittgenstein’scentral 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
(i.e., the linguistic Copenhagen interpretation ≈ the quantum mechanical world view ≈ quantum language).
This will be discussed in later ( Sec. 11.7).
♠Note 11.8. Note that
Russell’s paradox(1902) −−−−−−−−−−−−−→Zermelo-Fraenkel set theory
about 20 years later

Descartes’ ”I think therefore I am”(1637) −−−−−−−−−−−−−−→linguistic Copenhagen interpretaabout 350 years latertion
This is due to the fact:• There was a strong belief in mathematics that paradox should be avoided at all costs. On theother hand, philosophers have enjoyed and exploited paradoxes.

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11.5.2 * Five-minute hypothesis

Bertrand Russell was a very eloquent philosopher who communicated many thought-provoking things to thegeneral public. For a layman like me, he is a very helpful philosopher. If the following quote had not comefrom 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” a wholly unreal past. There is no logicallynecessary 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. [106] ”The Analysis of Mind”, p.223, Bertrand Russell
This section was written with reference to the following.
• [73]: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)
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. [106]), is asfollows.
(A1 ) the universe was created ﬁve minutes ago. Or equivalently, the universe was created ten 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 (E 2 )in Sec. 1.1.2 says that this (A1 ) is not a statement in quantum language. Thus, the (A 1 ) is not scientiﬁc,that is, there is no experiment to verify the statement (A1 ).

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Therefore, we can conclude that
(A2 ) ”the universe was created ﬁve minutes ago” is not a proposition in QL.

Some may want to relate this hypothesis to skepticism (cf. ref. [106]), However we do not think that thisdirection is productive.

Remark 11.15. (i): Also, the above (A1 ) should be compared to the following (A2 )(A3 ) The universe was created in A.D. 2010. (Or equivalently, now is A.D. 2020, and the universe wascreated ten years ago.)
This (A3 ) can be denied by experiment, that is, it is diﬀerent from the fact. Thus, this is a proposition inquantum language.
(ii): Consider the following situation:

In the above, we say that(♯1 ) Russell’s statement is scientiﬁc, but it is wrong.
(♯2 ) Wittgenstein is making a logical error. Thai is, he misunderstands Russell’s scientiﬁc statements as
unscientiﬁc.

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11.5.3 * McTaggart’s paradox
This section was written with reference to the following.
• [73]: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)
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. [90], 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 (E 2 ) in
Sec. 1.1.2.)(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).

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 (B 2 ) (since the properties of A-series are not clear),
I agree to him if his assertion is (B4 ) (cf. ref. [47]). That is, I agree that McTaggart noticed ﬁrst thatobserver’s time is not scientiﬁc. Recall the linguistic Copenhagen interpretation (E 2 ) in Sec. 1.1.2:• 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 should be estimated more highly.
Therefore, we can say that(C) McTaggart was also related to the linguistic Copenhagen interpretation (≈ the problem

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such that ”What is a ’propositin’ ?”)246

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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?” was the 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 a pseudo-proposition
such as ”It is raining, but I do not believe it is raining”. It would be great if we could make a theoryof 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 notmuch possibility that a big theory would be born. Of course, the future is still undecided, and it’s too early
to say for sure.

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 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”.

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Although the sentence: ”It is raining, but I do not believe it is raining” can be interpreted in many ways,
but let’s discuss the following two ([I] and [II]) scientiﬁc interpretations[I]: Thus,(E1 ) ”I” in ”It is raining, but I do not believe it is raining” is a violation of rule (d). Thus, if ”I” is ”Mr.
X”, this should be rewritten by”It is raining, but Mr. X does not believe it is raining”(E2 ) Since ”Audience should not go on stage”, the measurer can be anyone except Mr. X.
That is, this implies that(F) the above ”It is raining, but I do not believe it is raining” is the measured value of the measurementfor(the state of the weather, the state of Mr. X’s mind).
[II]: ”It is raining, but I do not believe it is raining” can be interpreted by ”I know it is raining, but I do notbelieve it is raining”. Similarly, this should be rewritten by ”Mr. X knows it is raining, but Mr. X does notbelieve it is raining”. If this can be interpreted by ”Mr. X knows it is raining, but Mr. X does not hope it
is raining”, we says as follows.
(G) ”Mr. X knows it is raining, but Mr. X does not hope it is raining” is the measured value of themeasurement for(the state of Mr. X’s knowledge of the weather, the state of Mr. X’s wish of the weather).
The above two (F) and (G) are scientiﬁc interpretation of Moore’s paradox. However, The important thingis to propose a non-scientiﬁc understanding within everyday language, rather than a scientiﬁc understanding
through a linguistic Copenhagen interpretation. However, I don’t think we have obtained satisfactory resultson this yet.
If I were to write the author’s prediction, I would say that the results are probably not conclusive.
However, I believe that with the use of AI, the school of everyday language will make steady progress.
Therefore, we can say that(H) ”It is raining, but I do not believe it is raining” is not a proposition in QL.

11.5.4.3 Wittgenstein was interested in this paradox
However, the purpose here is not to examine this paradox in detail. What I have focused on is the following(I) 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 and insisted 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?” And Wittgenstein must have thought: (R 1 ) is ”what we cannot speak
about”. Furthermore, he must have thought Descartes’ cogito proposition ”I think, therefore I am” ( whichis called the ﬁrst principle of philosophy) was not also a proposition (cf. the linguistic Copenhagen interpretation (E1 ) in Sec. 1.1.2).
At that time, Russell, Moore, and McTaggart were professors in the Department of Philosophy at Cambridge University. Moore’s paradox, Russell’s ”ﬁve-minute hypothesis,” and McTaggart’s ”non-existence oftime” (cf. ref. [73]) were all based on the question such that ”What is a scientiﬁc proposition?” (or equivalently ”What is the Copenhagen interpretation?”). This was the environment in which analytic philosophy( or symbolically, ”Wittgenstein”) was born.

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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 arethings at the beginning, and that we give each thing a name, just like we put a label on it. However, thatis not the case. Rather, Saussureian linguistics says that we understand the order of things by the act ofgiving them names. I think that his theory is almost the linguistic version of the Copernican revolution ofKant. That is,copy theory (realism)

Saussure’s discovery
things −→ words −−−−−−−−−−−−−−−−−−−−−−−−−→

constructionism (transcendental idealism)

Copernican revolution in language

words −→ things(11.2)

Saussure’s linguistics

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 a country 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 akind of worldview (cf. (D) in Preface).
In the above, I’d like to assume that (A1 )∼(A3 ) are about the same.

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11.6 Saussure: Copernican revolution in language

11.6.2 Several Copernican revolutions♠Note 11.9. The original Copernican revolution is of course as follows (cf. Chap. 7):

Thus, the original Copernican revolution means
Developing science

realismthe original Copernican revolution

More advanced science−−−−−−−−−−−−−−−−−−−−−−−−−→
scientiﬁc progress

Ptolemaic system

realism
Copernican system////

However, the philosophical Copernican revolution means
copy theory

realismconstructionism
the philosophical Copernican revolution−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ transcendental idealism
scientiﬁc progress

And thus, there are several ”Copernican revolution”. In this text, the term is used to represent the turnfrom ”realism” to idealism”. That is, the Copernican revolution is common to Kant, Saussure, quantum
language as follows.
Assertion 11.16. Recall our main ﬁgure:250

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Chap. 11 Linguistic philosophy ( Before TLP)OOOThe above 9 , 11 , 13 are realized in the following diagram:
copy theory

the world −→ cognition
Descartes=LockeO

9 :Copernican revolution−−−−−−−−−−−−−−−−−−−−−→
in cognition

constructionism

cognition −→ the world
Kant

Oy
11

copy theory

constructionism

Copernican revolution−−−−−−−−−−−−−−−−−−−→
in language

things −→ words
languagewords −→ things
Saussure, Wittgenstein

Oy
13

copy theory

phenomena −→ QM
quantum mechanics (=QM)

OOb Copernican revolution−−−−−−−−−−−−−−−−−−−−−→
in quantum mechanicsxa 
quantumization

copy theory

phenomena −→ CMclassical mechanics (=CM)

linguistic turnmechanical turnconstructionism

QL −→ phenomena
quantum language (=QL)

Oy
c

O

d Copernican revolution−−−−−−−−−−−−−−−−−−−−−→
in classical mechanics

classicization

constructionism

statistics −→ phenomena
statistics

OOO(i) It is very diﬃcult to discover QL in the ” 9 , 11 , 13 ” path.
(ii) I think the discovery of statistics is the most diﬃcult. I don’t know how Fisher and others discovered
statistics. ” d ” alone does not tell us the source of the probability. From now on, even ordinary
people may have discovered statistics through the ” a → b → c ” path.
(iii) In the above, we see several ”Copernican revolution”. Thus we 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 ”true idealism”

O

O O O

where we think that ”true idealism”=”scientiﬁc idealism”=”transcendental idealism”.
♠Note 11.10. Recall the quotes in ”Tractatus Logico-Philosophicus” ( by Wittgenstein ) such that(♯) The limits of my language mean the limits of my world.
which implies the constructionism 11 .

O

Answer 11.17. [ The answer of Problem 10.4: Is ”From Descartes to Kant” progress?]
Now we can answer Problem 10.4, i.e.,•

Descartes philosophy−−−−−→ Kant philosophy
progress(Descartes, Locke, Leibniz, Hume)?

(Kant)

by the ﬁrst line of the diagram above, that is,copy theory

O

constructionism

9 :Copernican revolution
the world −→ cognition −−−−−−−−−−−−−−−−−−−−−→ cognition −→ the worldin cognition

Descartes=Locke

Kant

where ”to make progress” is deﬁned by ”to come near quantum language” (cf. Assertion 1.5).

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11.6 Saussure: Copernican revolution in language

11.6.3 * The quantitative expressions of ”signiﬁed”
Now let us explain the terms ”signiﬁer” and ”signiﬁed”, which were introduced by Saussure.
Deﬁnition 11.18. [”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 scientiﬁc standpoint.
In fact, Saussure’s idea does not play an important role in science.

Thus, I will propose Deﬁnition 11.19, in which Saussure’s idea is realized in quantum language as follows.
Deﬁnition 11.19. [Membership function (= Fuzzy set, cf. [39, 40, 41, 42, 121])] Let Ω be a state space. A
continuous function m : Ω → [0, 1] (i.e., the closed interval in the real R) 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 membership functions mD : Ω → [0, 1] of dogs as follows. Suppose that there are 100
zoologists and the following question is made them.
(D) Is this animal with the sate ω1 (∈ Ω) a dog or not?
The answer
is as follows.

70 zoologists say that this bird is a dog.
(E)
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 membershipfunction mD : Ω → [0, 1] of dogs is deﬁned by the interpolation method (which may be rather subjective).
Put ΩD = {ω ∈ Ω | mD (ω) = 1}, which is called the dog state class.

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Chap. 11 Linguistic philosophy ( Before TLP)

mD

1

Ω

ΩD
Figure 11.1: [membership function mD : Ω → [0, 1], and ΩD ]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 11.20. [”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 11.11. (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, As mentioned in Remark 3.12, Saussure’s theory is
locatedin the following.
•linguistic version−−−−−−−−−−−→

Idea
Plato’s idea theory

signiﬁer, signﬁedSaussure’s linguistic theory

body, secondary quality

membership function

Descartes, J. Locke

Zadeh’s fuzzy theoryquantiﬁcationycognitiveyversion

quantiﬁcationj
yidentiﬁcation

observable (= measuring instrument)
quantum language

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 ofWestern philosophy as the history of dualistic idealism.

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255

Chapter 12

Linguistic philosophy ( After TLP)
I agree with the following Dummett’s words: (cf. ref. [15]: Origins of analytical philosophy):(♯1 ) If we identify the linguistic turn as the starting-point of analytical philosophy proper, there canbe no doubt that, to however great an extent Frege, Moore and Russell prepared the ground, thecrucial step was taken by Wittgenstein in the Tractatus Logico-philosophicus (= TLP) of 1922 (127 page)That is because I, roughly speaking, think that
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 cannot speak about” (=”proposition”)?
(♯4 ) Why does logic work in our daily lives?
Although he failed in this task, his aspirations were enthusiastically praised by the general public,
and he laid the foundation for today’s analytic philosophy.
I agree with opinion (♯1 ) (=(♯2 )) because the philosopher’s job is not to solve problems, but to present
them.

In Sec. 12.1, I answer the above questions (♯3 ) and (♯4 ) in QL. And thus, the reader can be sure that
Wittgenstein’s direction was the right one.

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12.1* Fuzzy logic in QL; My scientiﬁc understanding of ”Tractatus Logico-Philosophicus (=TLP)”

12.1 * Fuzzy logic in QL; My scientiﬁc understanding of ”Tractatus
Logico-Philosophicus (=TLP)”This section consists of excerpts from the following paper:• Ref. [78]: Ishikawa, S., (2021) Fuzzy Logic in the Quantum Mechanical Worldview ; Related toZadeh, Wittgenstein, Moore, Saussure, Quine, Lewis Carroll, etc. JAMP, Vol. 9, No.3 , 140-154,(https://www.scirp.org/Journal/paperinformation.aspx?paperid=95447)

12.1.1 Wittgenstein and Zadeh

In 1965, L.A. Zadeh proposed an engineering concept called ”fuzzy sets”( cf. ref. [121]), which was
enthusiastically supported by some engineers. However, R.E. Kalman did not recognize ”fuzzy sets” as ascientiﬁc 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 thepast have discovered certain interesting things, formulated their ﬁndings in a non-fuzzy way, and therefore
we have progressed in science” (cf. ref. [122]).

Even now, more than 50 years later, I don’t think Kalman’s claim can be denied. In fact, the conceptof ”fuzzy sets” has not yet acquired the status of more than a convenient engineering 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 (whichis essentially the same as statistics ) is considered as the mathematical representation of the (classical)
worldview. That is,(classical mechanical worldview)(A)

Dynamical system theory = state equation + measurement equation(= statistics )( causality)(probability)(See ⃝
6 in Figure 0 (in preface) )On the other hand, quantum language is characterized as follows:(B)

Quantum language

O





(quantum mechanical worldview)= Axiom 2 +(causality)(= measurement theory)(See 14 in Figure 0 (in preface) )Axiom 1
(measurement)+ linguistic ( Copenhagen ) interpretation(how to use Axioms 1 and 2)

which is a mathematical representation of quantum mechanical worldview (cf. refs. [44], [71], [66]). And we
assert that ”scientiﬁc concepts”=”concepts within the quantum mechanical worldview”. If so, and if ”fuzzysets” is a scientiﬁc concept, ”fuzzy sets” must be completely understood in quantum language.

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Chap. 12 Linguistic philosophy ( After TLP)In ref. [117] (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], [85]). It is a fact, however, that TLP was accepted by much of the general public. I think the generalpublic felt that Boole and Frege’s ”symbolic logic” was just mathematics, and that this was not suﬃcientas philosophy. In other words, they expected TLP to answer the question, such that ”Why does logic workin our world?” or ”What is the deﬁnition of non-mathematical proposition?.” Wittgenstein’s motivation of
TLP would have been to answer these questions, but he was too poet and dreamer. Also, TLP was publishedin 1921, and Wittgenstein was unaware of the discovery of quantum mechanics (around 1925).
In 1965, L.A. Zadeh proposed an engineering concept called fuzzy sets (cf. ref. [121]), which was notrecognize as a scientiﬁc concept by some excellent scientists. And furthermore, Zadeh couldn’t convince
them of his idea. However, ref. [121] was one of the most cited papers in the 20th century. This implies thatmany engineers felt that mathematical set is not enough, expected more from his fuzzy set .
I think the situation of Wittgenstein and Zadeh is very similar in the sense that they were ardently
supported by a large number of the general public even though they were not supported by the best experts.
I would rather trust the senses of the many general public than the senses of a few experts. And what
Wittgenstein and Zadeh lacked, I believe, was a worldview.
Therefore, their claims are vague and incomprehensible, but if I understand them under the quantum
mechanical worldview, I think they are claiming almost the same thing.
In this section, we will explain it. That is, we simultaneously justify Wittgenstein’s TLP and Zadeh’s
fuzzy sets theory.

12.1.2 Easy example; classical system
Now let us explain ”Fuzzy logic in QL” (i.e., my understanding of TLP). Let us start from easy example asfollows.
Although our theory is valid for quantum systems as well as classical systems, in this Sec. 4.1 we explain
our idea in classical systems (i.e., the 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
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.
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”)

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
Also, note that Prob[¬P1 ; {T }] = Prob[P1 ; {F }] = 1 − p1 .

257

(where 0 ≤ p1 ≤ 1)

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12.1* Fuzzy logic in QL; My scientiﬁc understanding of ”Tractatus Logico-Philosophicus (=TLP)”The above will be formulated in terms of QL as follows. Let Ω a state space, which is a compact space 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
by a state ω, which is an element of the state space Ω. Thus, we have the map ωb : T → Ω. That is, thequantitative propertyofatomatotisrepresented
by
ω
b
(t).
For
example,itsuﬃcesto consider Ω such thatG)

ω = ω (1) (= weight), ω (2) (= diameter), ω (3) (= color value),ω (4) (= calorie), ω (5) (= sugar content), ..., ω (N ) (= ...) ∈ 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 givenby δω (G1 )(= C(Ω)∗ ⟨δω , G1 ⟩C(Ω) = G1 (ω))
This is the quantum linguistic representation of the above (E). That is, we identify a proposition 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 as ambiguous as ”the setof all dinosaurs”. However, for the sake of convenience, here we use the term ”the set of all tomatoes”. This
problem is the same as that of the Hempel’ raven paradox (i.e., ”the set of all ravens” leads to contradiction).
For further discussion about this, see refs. [66], [75].
(J): If we want to consider another proposition P2 (=MC(Ω) (O1 , S[δω′ ] ) ) such as
P2 = ”that tomato is red”,¬P2 = ”that tomato is not red”,

we must deﬁne MC(Ω) (O1 , S[δω ] ) ∧ MC(Ω) (O1 , S[δω′ ] ) (ω ̸= ω ′ ). This will be explained in Sec.4.3.
(K): If we want to consider both tomato’s world Ω1 and apple’s world Ω2 , it suﬃces to start from the tensor
space C(Ω1 ) ⊗ C(Ω2 ) (=C(Ω1 × Ω2 )). This will be also explained in Sec.4.3.

12.1.3 Fuzzy logic in QL
Let’s start with the following deﬁnition.

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Chap. 12 Linguistic philosophy ( After TLP)
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 afuzzy 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(L) a (TF)-measurement MA (O, S[ρ] ) is true with probability ρ(G({T }))Or,(M) Prob[MA (O, S[ρ] ); {T }] = ρ(G({T })) (= A∗ ⟨ρ, G({T })⟩A )Also, G({T })(∈ A) is called the membership function of O (cf. Deﬁnition 11.19).
♠Note 12.1. Our interest is ” Kant philosophy −−−−−−→ Analytic philosophy ”, and thus,
progressthe main concept ”proposition” in analytic philosophy must be formulated in the frameworkof dualistic idealism. This is realized in the above deﬁnition.

That is, Wittgenstein took over the baton of ”dualistic idealism
philosophy)” from Kant (cf. ref. [78], or Sec. 12.1).

(i.e., the mainstream of

In general, we must consider many propositions {Pi = MAi (Oi , S[ρi ] ) : i = 1, 2, ..., n}. In this section, wedevote ourselves to the following simple case:(N1 ) A is ﬁxed, i.e., A1 = A2 = ... = An
(N2 ) O1 , O2 , ..., On commute,(N3 ) a state ρ is only one, i.e., ρ1 = ρ2 = ... = ρn
For the general case, we discuss in next section (i.e., Sec.12.1.4). That is, in this section, we devote 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 is an easy consequence of the simple case as seen in thenext 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. 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))
qp(Ξ ∈ {T, F }, i = 0, 1, 2), and (×qp
G2 ))(Ξ1 × Ξ2 ) (Ξ1 , Ξ2 ∈ {T, F }).
i=1,2 µi )(Ξ1 × Ξ2 ) = (ρ(G1 ×(i): Put i = 1, 2. Deﬁne ¬MA (Oi , S[ρ] ) such that
¬MA (Oi , S[ρ] ) = MA (π ¬ Oi , S[ρ] )where the map π ¬ {T, F } → {T, F } is deﬁned by π ¬ (T ) = F, π ¬ (F ) = T . Clearly it holds that Prob[¬MA (Oi ,259

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12.1* Fuzzy logic in QL; My scientiﬁc understanding of ”Tractatus Logico-Philosophicus (=TLP)”

S[ρ] ); {T }] = ρ(Gi ({{F })). = µi ({F }).
(ii): 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 ) = 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 )}).
(iii): 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 })) =µ2 )({(T, T ), (T, F ), (F, T )}).

(µ1 ×qp(iv): 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 ×qpµ2 )({(T, T ), (F, F ), (F, T )}).
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 ̸= Oj (∀i, j such that i ̸=j) (i.e., non-overlapping condition in Theorem 11.10) and Oi (i = 1, 2, ..., n) commute. Fix the quasiqpnn
product observable ×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). And consider the quasi-productqp
qp{T,F }
measurement, Gi ), S[ρ] ) = MA ( i=1,...,n Oi = ({T, F }n , P({T, F }n ),i=1,...,n MA (Oi = ({T, F }, 2
qp
i=1,...,n Gi ), S[ρ] ). Put

×

×

×

•µi ({xi }) = ρ(Gi ({xi }))



qp

•

(xi ∈ {T, F }, i = 1, 2, .., n)qp
× µi ({(x1 , x2 , ..., xn )}) = ρ (×qp
i=1,...,n Gi )({(x1 , x2 , ..., xn )} = (×i=1,...,n µi )({(x1 , x2 , ..., xn )})i=1,...,n(∀(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, 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 that×qp

Prob[P ; {T }] = (× µi )(P −1 ({T })))
i=1,2,...,nRemark 12.6. Since the linguistic Copenhagen interpretation says that ”only one measurement is permitqp
qp{T,F }
ted”, we only take the measurement:, Gi ), S[ρ] ) = MA ( i=1,...,n Oi =i=1,...,n MA (Oi = ({T, F }, 2
qp({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, these measurements
qp{T,F }are included in the quasi-product measurement, Gi ), S[ρ] ).
i=1,...,n MA (Oi = ({T, F }, 2

×

×

×

×

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Example 12.7. [The QL version of Table 11.10]. Replacing P1 and P2 with MA (O1 , S[ρ] ) and MA (O2 , S[ρ] ),we get the following Table 11.11, i.e., the QL version of Table 11.10.
Table 11.11: Probabilistic Truth Table ( Elementary propositions MA (O1 , S[ρ] )(= P1 ), MA (O2 , S[ρ] )(= P2 ))probability:
MA (O1 , S[ρ] ) MA (O2 , S[ρ] )qp[P1 → P2 ] ∧ P1
[P1 → P2 ] ∧ P1 → P2(= P1 )(= P2 )p=i=1,2 µi
qp
µ
({(T,
T
)})
T
T
p12 =
T
T
i=1,2 iqpTFp12̄ =
FT
i=1,2 µi ({(T, F )})qp
F
T
p1̄2 =
F
T
µ
({(F,
T
)})
i
i=1,2
qp
F
F
p1̄2̄ =
µ
({(F,
F
)})
F
T
i=1,2 i

×
×
×
×

×

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,(O) syllogism (i.e., [[P1 → P2 ] ∧ [P2 → P3 ]] → [P1 → P3 ] always holds.

Remark 12.9. We have two results such that(i) in ref. [76], 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.
Thus, readers may think that (i) and (ii) are contradictory. However these are not contradictory sinceCorollary 12.8 requires that O1 , O2 , O3 commute. On the other hand, in ref. [76], the commutativity of O1
and O3 is not required. The most important one, of course, is Corollary 12.8.
Remark 12.10. [ Cogito proposition is not a proposition in QL; cf. refs. [44], [67]]: Examine the cogito
proposition ”I think, therefore I am”, in which it is natural to consider that”observer” = ”I” = ”measurement object (=system)”This is against the linguistic Copenhagen interpretation (E 1 ) ”No observer can measure itself” in Sec. 1.1.2.
Therefore, the cogito proposition is not a proposition in QL. The fact that the ﬁrst proposition of philosophy
is not a proposition is interesting.

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12.1* Fuzzy logic in QL; My scientiﬁc understanding of ”Tractatus Logico-Philosophicus (=TLP)”

Thus we see the following non-sense table:
Table 11.11′ : Probabilistic Truth Table ( non-propositions ”I think”(= P1′ ), ”I am”(= P2′ ))probability:”I think””I am”
qp′′[P1′ → P2′ ] ∧ P1′[P1′ → P2′ ] ∧ P1′ → P2′
p=(= P1 )(= P2 )
i=1,2 µi?
?
p12 =?
?
?
?
?
p12̄ =?
?
?
?
?
p1̄2 =?
?
?
?
?
p1̄2̄ =?
?
?

×Remark 12.11. The linguistic Copenhagen interpretation (E0 ) in Sec.1.1.2 says that(♯1 ) Don’t talk about what you can’t measure!
Thus, as shown in ref. [63], we see:Also, the (♯1 ) is equivalent to Berkely’s saying(♯2 ) To be is to be perceived:

This implies the following ﬁgure:262

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12.1.4 General caseIn previous section, we devote ourselves to {MA (Oi = ({T, F }, 2{T,F } , Gi ), S[ρ] ) : i = 1, 2, ..., n} under theconditions (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
j=1

I) ⊗ Gi (Ξi ) ⊗ (

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} satisﬁes that
Here, note that {Pbi = MNnj=1 Aj (O
j=1 j
N
n
b=
(P1 ) A
Ai is ﬁxed,
j=1

b 1, Ob 2 , ..., Ob n commute,(P2 ) On(P3 ) 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. (i): From the theoretical point of view, some may want to extend the above result toinﬁnite tensor product. For this, we must prepare the W ∗ -algebraic formulation of QL. I think challenging
this problem is a good exercise for the reader.
(ii): In this paper, we devote ourselves to binary logic (i.e., {T, F }-valued logic). If we want to consider
many valued logic (i.e., X-valued logic), we can start from X-value observable. In this case, it is clear that”fuzzy many valued proposition”=”Axiom 1”. Therefore, we can see the following equivalence:(♯1 ) Why does fuzzy logic work in our usual world?
(♯2 ) Why does Axiom 1 (measurement in QL) work in our usual world?
Example 12.13. [Hempel’s raven problem (cf. [75]); 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 sweet observable and the red observable. It is natural toconsider that ”Any sweet tomato is red” is deﬁned by(Q) SW ⊆ RD

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where SW = {ω ∈ Ω|GSW (ω) = 1} and RD = {ω ∈ Ω|GRD (ω) = 1}. In order to examine (Q) (i.e., toanswer the problem: ”Is the (Q) a proposition?”), it suﬃces to check the following:
Prob[MC(Ω) (ORD , S[δω ] ); {T }] = 1(∀ω ∈ SW )For simplicity’s sake, we assume SW = {ω1 , ω2 , ..., ωn }. Recall the linguistic Copenhagen
interpretation (L 1 )Nnsuch that only one measurement is permitted. Thus, consider the tensor spaceC(Ω)= C(Ωn ), andi=1
the tensor product measurement
MC(Ωn ) (nO

ORD = ({T, F }n , P{T, F }n ,

n
O

i=1

where
(

n
O

GRD )({(x1 , x2 , ..., xn )}) =i=1

GRD ), S[δ(ω ,ω ,...,ωn ) ] )
1

2

i=1
n
O

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 can conclude that (Q) istrue. Also, as an analogy of (ii) in Deﬁnition 12.4, we may consider 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, (Q) is equivalent toProb[MC(Ωn ) (π ∧ (⊗ni=1 ORD ), S[δ(ω ,ω ,...,ωn ) ] ); {T }] = 1
1

2

Thus, (Q) is a fuzzy proposition. If SW is an inﬁnite set, we must prepare the inﬁnite tensor algebra (i.e.,the W ∗ -algebraic formulation of QL (cf. refs. [71], [76])). We omit it in this paper, since this is simply amatter of mathematics.

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Chap. 12 Linguistic philosophy ( After TLP)

12.2 Wittgenstein; the biggest star of analytic philosophy.
12.2.1 What did he wanted to insist on in TLP?

Ludwig Wittgenstein (1889 - 1951), who was the student of B. Russel, was the most famous philosopher (inanalytic philosophy) in the 20th century.
Wittgenstein wrote two books, i.e.,
Tractatus Logico-Philosophicus (=TLP)(J)

the former Wittgenstein
1921

Philosophisch Investigations (=PI)
teacher−−−−−−→

the later Wittgenstein

gardener

1953

The former is concerned with world description, while the latter is literary. Thus, I think that the above issimilar 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, expedientethics, morals, aesthetics
main subject

Thus we devote ourselves to TLP.
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 evaluationof TLP has been settled yet. I think that TLP is theoretically insuﬃcient. In fact, in the last 100 years, nophilosopher has been able to read TLP in a theoretical way.
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 already thought thethoughts which are expressed in it — or similar thoughts.
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 is certain that
(D1 ) and (D2 ) are the most important in practical logic.
If so, I would be in a very favorable situation. Because I already know the answer to this problem.

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(E) About 20 years ago, I wrote papers on ”Why does logic work in our world?” (cf. refs. [39, 40,
41, 42])At the time, I was not aware that ”Why logic does work in our world?” was a question of philosophy.
After reading (G), I immediately rewrote the following papers.
(F) Ref. [76]: 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)(G) Ref. [78]: Ishikawa, S., (2021) Fuzzy Logic in the Quantum Mechanical Worldview ; Related toZadeh, Wittgenstein, Moore, Saussure, Quine, Lewis Carroll, etc. JAMP, Vol. 9, No.3 , 140-154,(https://www.scirp.org/Journal/paperinformation.aspx?paperid=95447) This is essentially thesame as Sec. 117 in this text.
Thus, I can assert the following:
Answers to the above (D)(D1 ) What is ”proposition”? (i.e., what is the deﬁnition of ”proposition”?)(D2 ) Why does logic work in our usual world?
Answer 12.14. The above problems (D) are already solved in the previous section.
That is;(♯1 ) Problem (D1 ) is already answered in Deﬁnition 12.3.
That is,Also,(♯2 ) Problem (D2 ) is clear since QL is more powerful than statistics.
Furthermore, as seen in Remark 12.12, Fuzzy logic in QL has the same power as Axiom 1.
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 personalitythat feels status in reading diﬃcult books that are impossible to understand. I would like to believethat Wittgenstein did not write TLP for those people. I tend to like the following 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, since Wittgenstein conﬁdently concluded”What we cannot speak about we must pass over in silence” in TLP, I thought he assumed that he had deﬁned”what we can speak about (= proposition)” in TLP. If this is true, it is the worst, since without deﬁnition,
this can be said of anyone. However, I was relieved when I read the following ( 38 in [118])• ”The basic of Russell’s logic, as also of mine in the TLP, is that what a proposition is illustrated bya few commonplace examples, and then pre-supposed as understood in full generality”
When I read this, I knew I could trust Wittgenstein. So, the following 6.54 in TLP means that Wittgensteinknew that TLP was like a ladder that could be thrown down.
• 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 from logical positivism
because he had an intuition that TPL was powerless to science.
The fact that his TLP is incomplete is inevitable. No matter how much of a genius Wittgenstein was, itwould 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. [117] (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 have reached QL. I
think that von Neumann’s [113] and Davies’ [14] are indispensable for QL. If von Neumann had taken aninterest in TLP, though, the history of analytic philosophy might have been diﬀerent.

§

§

§

♠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. Descartes and Kant are not logical either.
(ii): As mentioned in the previous section ( Sec. 12.1), I believe that QL is the central theory ofanalytic philosophy, which is also obvious if you look at List 13.1 in Chap.13. Of course, Kripke’s
possible worlds semantics is one of greatest theories in analytic philosophy. However, I have the same
opinion ( for Kripke’s possible worlds semantics ) as Dummett as follows(♯2 ) 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 for possible-worlds semantics was just a
mistake. It is occurred because Kripke succeeded in 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)

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12.2 Wittgenstein; the biggest star of analytic philosophy.

12.2.2 The power of Wittgenstein’s word: Linguistic turn
However, in this paper, I want to assert that Wittgenstein is one of the greatest philosophers (Plato,
Descartes, Kant, etc.). That is because he said the following sayings (H 1 ) - (H3 ):(H1 ) ”The limits of my language mean the limits of my world.”(H2 ) ”What we cannot speak about we must pass over in silence”(H3 ) ”Language-game”
The above is just the spirit of quantum language. Instead of my poor explanation of the spirit of quantum
language, I prefer to saying(I) ”The spirit of quantum language is represented by the above (H 1 ) - (H3 )”Seeing the above (H1 )∼ (H3 ), 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 oflanguage.
♠Note 12.5. For each great discovery, an anecdote (or, a catch copy, 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 10.6)(♯6 ) Wttgenstein· · · · · · primary school teacher, Gardener, Architect, Guardian: Russell (cf.
Sec.12.1.1)(♯7 ) Einstein· · · · · · Elevator(♯8 ) quantum mechanics· · · · · · Heisenberg’s uncertainty principle (cf.ref. [36], or, Note 4.1 of ref. [71])Here, the (♯8 ) is my opinion Has Heisenberg’s uncertainty relation ever been used eﬀectively in physics?
(Sec. 4.3 in ref. [71]).

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12.2 Wittgenstein; the biggest star of analytic philosophy.

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)(J)

the former Wittgenstein
1921

Philosophisch Investigations (=PI)
teacher−−−−−−→

the later Wittgenstein

gardener

1953

TLP is concerned with world description, while PI consists of a mishmash of miscellaneous things that could
not be written in the TLP. 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 with psychology andbrain science. Psychology (or, cognitive science) is an important discipline, but it is a discipline in which
experimentation and observation are essential, and it is not really Wittgenstein’s cup of tea. I have an
opinion that the ”philosophy of mathematics” should be left to mathematicians. That is, I believe thatphilosophers should concentrate on ”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 point of 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; Sec.
I have attempted this paradox several times, but none of them have been satisfactory. However, some things
should be mentioned below.
For skepticism, the most important question is: ”What should we be skeptical about?” I think. The objectof 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 + linguistic (Copenhagen) interpretation(= measurement theory)(Axiom 1)(Axiom 2)

(how to use Axioms 1 and 2)

since Axioms 1 and 2 are kinds of spells (i.e., incantation, magic words, metaphysical statements) as mentioned in Sec. 1.1.1. Also, recall the following sprit of ”Copenhagen interpretation” (cf. Note 1.3):

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Chap. 12 Linguistic philosophy ( After TLP)(i) Stop being bothered!
(ii) Shut up and calculate!
In this spirit, we have solved many unsolved problems in this paper and ref. [71]. However, I think it istime for us to be skeptical about QL.
Thus I believe that the following problem is important:(♯) How are the rules of QL learned?
Note that QL is a kind of idealism ( or, a kind of metaphysics ). I myself learned quantum language as ananalogy after learning quantum mechanics. Therefore, I initially thought that it would be diﬃcult to learn
a quantum language only after learning quantum mechanics. However, in my graduate laboratory, students
with no 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 canbe 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. AI (= artiﬁcial intelligence) that understands mathematics better than humans will be areality 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.
About Wittgenstein’s paradox, S. Kripke said in ref. [86] as follows.
(K) Wittgenstein has invented a new form of scepticism. Personally I am inclined to regard it as the mostradical and original sceptical problem that philosophy has seen to date, one that only a highly unusual
cast of mind could have produced.
I agree to Kripke’s opinion, but I think that Wittgenstein’s paradox is not only a problem for philosophy.
It is a problem for psychology, neuroscience, cognitive science, and AI. However, I was not sure what
Kripke’s quus meant, so I fear I am misreading ref. [86].

12.2.3.3 (iv): Private language
Private language is a language that records internal experiences such as sensations, emotions, will, andthoughts entirely for one’s own use.
Words belonging to this language refer only to direct internal phenomena, and their meanings are determined independently of externally observable expressions and actions, so they cannot be understood byothers.
Thus private language does not satisfy the linguistic Copenhagen interpretation. For example, we alreadydiscussed ”I think”, ”I am”, Qualia problem, etc.

However, theoretical research on private languages seems to be diﬃcult, and the PI only lists examples ofprivate languages.

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12.3 Quine’s analytic-synthetic distinction and Popper’s falsiﬁability in QL;

12.3

Quine’s analytic-synthetic distinction and Popper’s falsiﬁability in
QL;

This section consists of excerpts from the following paper:• Ref. [78]: Ishikawa, S., (2021) Fuzzy Logic in the Quantum Mechanical Worldview ; Related toZadeh, Wittgenstein, Moore, Saussure, Quine, Lewis Carroll, etc. JAMP, Vol. 9, No.3 , 140-154,(https://www.scirp.org/Journal/paperinformation.aspx?paperid=95447)

12.3.1 Quine’s analytic-synthetic distinction in QL;
12.3.1.1 What is a (non-mathematical) proposition?
Rudolf Carnap was a philosopher who was active in Europe before 1935 and in the United States thereafter.
He was a major member of the Vienna Circle and an advocate of logical positivism. He is considered ”oneof the giants among twentieth-century philosophers.
Carnap reconsidered the dichotomy of analytic proposition/comprehensive proposition since Kant in
the framework of linguistic and factual factors. Carnap reconsidered the dichotomy of analytic proposition/comprehensive proposition since Kant in the framework of linguistic and factual factors. According tohim, an analytic statement is a statement of the syntax of the language we use to formulate it. According tohim, an analytic statement is true only according to the syntactic or semantic rules of the language we use toestablish it, that is, according to linguistic factors alone. In other words, it is a statement that is judged to
be true only by linguistic factors. In contrast In contrast, a comprehensive statement is a statement whosetruth is not determined by linguistic factors alone, but depends on factual factors. In other words, it is astatement whose truth depends on factual factors.
For example,• analytic proposition: ”Horses are animals”, ”Unmarried men are single”, ”1 + 3 = 5”
• synthetic proposition: ”The horse is fast”, ”Jack is single”,♠Note 12.7. Actually, I am not sure about the above distinction. I think the fact that there is so272

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Chap. 12 Linguistic philosophy ( After TLP)
much discussion about Carnap=Quine debate suggests that this is an important issue. That is, thisquestion ultimately comes down to ”What is a proposition? After Wittgenstein, I think the mostimportant question in analytic philosophy is ”What is a proposition?
In ref. [101], Willard Van Orman Quine published the essay ”Two Dogmas of Empiricism” in which heargued that the analytic-synthetic distinction is untenable. The argument at bottom is that there are no”analytic” truths, but all truths involve an empirical aspect. The question of ”analytic vs. synthetic” is oneof the most important philosophical questions since Kant.
Quine’s philosophy is based on logical positivism. He was originally a follower of the philosophy of R.
Carnap, a central member of the Vienna School, who developed the ideas of logical positivism most rigorously.
However, he later became the sharpest critic of Carnap’s philosophy (and logical positivism). In this way,
he formed his own philosophy.
It is well known that it was the concept of ”analyticity” that Quine attacked most vigorously. One ofthe doctrines that Carnap and others defended that Quine regarded as dogma was the idea that a clear line
could be drawn between analytic and synthetic truths. Analytic truths can be characterized as truths about
the meaning of words alone. Quine, however, does not recognize the concept of ”meaning” of words in theﬁrst place. Therefore, of course, he does not recognize ”truth by the meaning of words alone” either.

12.3.1.2 My answer
Many philosophers have entered the fray on this issue, but all of them argued without a ”deﬁnition of
proposition”. Since QL has the deﬁnition of proposition, we can easily solve it. That is, since ”proposition
in QL”=”(TF)-measurement (= experimental veriﬁcation)”, we can naturally conclude that all propositions
are synthetic in QL.

Of course, diﬀerent worldviews (i.e., theoretical systems) can have diﬀerent ideas, and our proposal is not the
only correct one. For example, in theoretical system called mathematics, it is clear that every proposition isanalytic.
I hope that many readers will make various proposals. I think that philosophers of analytic philosophy
should argue the problem after presenting a worldview. I think that logic without a worldview is just abranch of mathematical logic.

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12.3 Quine’s analytic-synthetic distinction and Popper’s falsiﬁability in QL;

12.3.2 Popper’s falsiﬁability in QL
From the quantum linguistic point of view, as asserted in Assertion 1.3, 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.
little about Karl Popper (1902-1994).

Thus let me mention a

12.3.2.1 Popper’s falsiﬁability
This may be common to many philosophers of science, if not Popper, but their work is ”enlightening” andit is diﬃcult for us in science to understand its true meaning. Popper’s books on probability theory andquantum mechanics are also a little far from the interests of science, so I do not fully understand them.
Popper is famous for ”falsiﬁability” as follows. Falsiﬁability is the following view of science (cf. ref. [97]).
(A) 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. In other words, truth must always be
subjected to experiments that negate it. And if the denying experiment is conﬁrmed, the truth must
be denied.
Many scientists would take this claim for granted. For example,(B) 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.
However, there seem to be various arguments against falsiﬁability (A) in the philosophy of science as well.
For example, some may say as follows:(C1 ) The objectivity of a scientiﬁc theory is guaranteed by the majority vote of a group of highly qualiﬁed
scientists.
I think ordinary scientists would agree that this may be true.
However, I don’t think most scientists areinterested in the ”deﬁnition of scientiﬁc truth.
(C2 ) The objectivity of a scientiﬁc theory is guaranteed by the majority vote of a group of highly qualiﬁed
scientists.
Philosophers of science may ask, ”Why aren’t you interested in being involved in this essence of science?”
However, scientists would say, ”Science cannot be understood without experiments and calculations.” So,although it is a ”parallel line state that does not intersect forever”, the purpose of this text is to break this.

12.3.2.2 Falsiﬁability in philosophy
As mentioned in Remark 3.13, many excellent philosophers follow Euclid as follows.
introduction part (unquestionable truth)z}|{DescartesI think, therefore I am 








Kant
thing-in-itself
Analytic philosophy

mathematical logic













commonTherefore−−−−−−−→ ethics, moral

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That is, they wanted to start from ”unquestionable truth” since they were afraid of being pointed out theerror of their ways. They started with an unfalsiﬁable claims (i.e., ”I think, therefore I am”, ”thing-initself”, ”mathematical logic”), which are out of QL. However, I think that they should have kept in mind
the following word*1 .
• ”The biggest risk is not taking any risk”
On the other hand, QL starts from Axioms 1 and 2, which is a kind of ”incantation” (i.e., ”spell”,”metaphysical statements”), i.e.,(=quantum language)[Axiom 2][Axiom 1]
measurement theory = measurement + causal relation+

linguistic Copenhagen interpretation[the manual to use Axioms 1 and 2](*)

Here, Axioms 1 and 2 are metaphysical statements, whose truth are uncertain.
I have my doubts about the following classiﬁcation.
(D1 ) [Science]: Newtonian mechanics, quantum mechanics, the theory of relativity,...
(D2 ) [Pseudoscience]: individual psychology, psychoanalysis, ...

12.3.2.3 Falsiﬁability in QL
In conclusion, I suggest the following:(E) View falsiﬁability (A) as part of the Copenhagen interpretation ! (cf. (E0 ) in Sec. 1.1.2.2)
This is because we can see(A) ≈ ”there is no science without measurements”≈ To be is to be perceived

Let me explain (E) below.
(F) 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 (=pseudoscience)” and ”what we can speak about (=science). Otherwise, the argument is meaningless. It
is with this in mind that Popper formulated falsiﬁability.

*1 This is known as the word of Mark Zuckerberg, the founder of facebook

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12.3 Quine’s analytic-synthetic distinction and Popper’s falsiﬁability in QL;
On the other hand, QL has the form:[Axiom 2](=quantum language)[Axiom 1]
measurement theory = measurement + causal relation+

linguistic Copenhagen interpretation[the manual to use Axioms 1 and 2](*)

This came about in the following way. Above, Axiom 1 and Axiom 2 alone lead to a messed up conclusion.
For example,(G1 ) There’s a cat that you can’t tell if it’s alive or dead, and the moment you see it, it’s conﬁrmed thatthe cat is alive or dead.
(G2 ) An incomprehensible non-proposition such as ”I think, therefore I am” becomes a major proposition
as ”the ﬁrst proposition of philosophy”.
It is the ”linguistic Copenhagen interpretation” that contains such pseudoscientiﬁc propositions and keeps
them out of the open. In short,(H) ”linguistic Copenhagen interpretation” draws a line between scientiﬁc and pseudoscientiﬁc propositions.
Therefore, we can think that(F) ≈ (H)
If so, all scientists will understand the meaning of falsiﬁcationism.
♠Note 12.8. Recall the linguistic Copenhagen interpretation ( particularly, (E 7 ) in Sec. 1.1.2). Thatis,(I) (= (E7 ) in Sec. 1.1.2); most maxims of the philosophers (particularly, the dualistic idealism) canbe regarded as expressions in linguistic Copenhagen interpretation.
That is because philosophers have been investigating dualistic idealism ( without Axioms 1 and 2 ).
♠Note 12.9. (i): ”Why can’t scientists and philosophers understand each other?” is a quite importantquestion, but I think as follows. As I have said in this text in general, philosophers have been discussing
the Copenhagen interpretation without knowing Axioms 1 and 2 in (*). This is why scientists could
not understand the philosophy of science.
(ii):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 philosophical problems, only trivial puzzles.
All the while, Wittgenstein, seated by the ﬁreplace, had, says Popper, ”been nervously playing withthe poker”. After an exchange of views on ethics, Wittgenstein asked Popper to give an example ofan ethical rule. Popper replied, ”Not to threaten visiting lecturers with pokers.” Wittgenstein thenthrew away the poker and stormed oﬀ.

This episode implies an adversarial relationship between Wittgenstein and Popper. However, I think
Popper just did the next.
• using falsiﬁcationism, he drew a line between scientiﬁc propositions (= what we can speak about)and pseudoscientiﬁc propositions (= what we cannot speak about).

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12.4

Lewis Carroll’s logical paradox in fuzzy logic

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 a mathematician, photographer, inventor,
and Anglican deacon.

This section consists of excerpts from the following paper:• Ref. [78]: Ishikawa, S., (2021) Fuzzy Logic in the Quantum Mechanical Worldview ; Related toZadeh, Wittgenstein, Moore, Saussure, Quine, Lewis Carroll, etc. JAMP, Vol. 9, No.3 , 140-154,(https://www.scirp.org/Journal/paperinformation.aspx?paperid=95447)
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?”
Premise a : (P1 → P2 )Premise b : P1
Conclusion: z P2
the Tortoise says: ”I can understand a and b . However, why do a and b imply z ?”Achilles says: ”I see, then, the following OK?”
Premise a : (P1 → P2 )Premise b : P1
Premise c : a and b are ”true”, therefore z is ”true”
Conclusion: z P2
the Tortoise says: ”I can understand Premises a , b and c . However, why do a , b and c concludez ?”Achilles says: ”I see, then, the following OK?”
Premise a : (P1 → P2 )Premise b : P1
Premise c : a and b are ”true”, therefore z is ”true”
Premise d : a , b and c are ”true”, therefore z is ”true”
Conclusion: z P2

O
O

O

O
O
OO
O

O

O

O

O

O
OO

O
O
OO O
OOO O
O

the Tortoise says:

O

O

O

O

O

OO

O

O

OO O

O

OOO

O

”I can understand Premises a , b , c and d . However, why do a , b , c and dO

conclude z ?”... (Inﬁnite regress)
This is Lewis Carroll’s Paradox
Several philosophers have tried to resolve Carroll’s paradox. For example, Bertrand Russell discussed theparadox brieﬂy in ref. [107]. His opinion is as follows.
(A1 ) The above a , b and z are propositions, on the other hand, c and d are inference rules. And,
therefore, both the Tortoise and Achilles are confusing this.

OO

O

O

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12.4 Lewis Carroll’s logical paradox in fuzzy logic
I agree to his opinion. I think that their confusion is due to the lack of a deﬁnition of proposition. 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 11.12 (= Table 11.11). Therefore, logic is a simple calculation ofelementary arithmetic.
Table 11.12: Probabilistic Truth Table ( Elementary propositions MA (O1 , S[ρ] )(= P1 ), MA (O2 , S[ρ] )(= P2 ))probability:
MA (O1 , S[ρ] ) MA (O2 , S[ρ] )qp(= P1 )(= P2 )
p=
[P1 → P2 ] ∧ P1
[P1 → P2 ] ∧ P1 → P2
i=1,2 µi
qp
T
T
p12 =
T
T
i=1,2 µi ({(T, T )})qp
TF
p12̄ =
µ
({(T,
F
)})
F
T
i=1,2 iqpFTp1̄2 =
FT
i=1,2 µi ({(F, T )})qp
FF
p1̄2̄ =
FT
i=1,2 µi ({(F, F )})

×
×
×
×

×

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”.

O

O

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. Note that the above c and d are not written inthis table.
Rather, in the case of mathematics, the deﬁnition of ”proposition” is so clear that I think it is rare formathematicians 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 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 of interest to people
for over 100 years is that it is closely related to the question ”what is a scientiﬁc proposition?”. Carroll’s
logical paradox is not the childish problem that most mathematicians think it is.

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12.5 Flagpole problem in the quantum mechanical worldview
This section was written with reference to the following.
• [75]:Ishikawa, S: Philosophy of science for scientists; The probabilistic interpretation of science Journalof 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 fuzzy logic in QL (in Sec. 12.2) is practical logic within the quantum mechanicalworld view. As I have said many times in this text, philosophers like ”logic” (or, the word ”logic”) too much.
They always want to ”be logical”. I think that the following fact is incomprehensible.
• 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=thespirit of ”Think logically!”), but in the quantum mechanical worldview.
The ﬂagpole problem below is caused by philosophers’ excessive love of logic.

12.5.1 * 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 a young generation of philosophers of science. He is also known
for the raven paradox (also known as ”Hempel’s paradox”). Recalling that(A) our spirit is ”( quantum ) mechanics” rather than ”logic”.
the reader should read this section. Most scientists know Newtonian mechanics, but not mathematical logic.

12.5.1.1 Flagpole problem in Hempel’s model (= the deductive-nomological model (DN model))

•

Sun
−− 1

Flag
ω00
α◦

|

ω10

2

Figure 11.2: [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 ω00 meters tall (0 ≤ ω00 ≤ 1. The ﬂagpole casts a
shadow ω10 meters long. Suppose that we want to explain the length of the ﬂagpole’s shadow. On Hempel’s
model, the following explanation is suﬃcient.
(B1 ) 1. The sun is at an elevation angle α◦ in the sky.
2. Light propagates linearly.
3. The ﬂagpole is ω00 meters high.
Then,
4. The length of the shadow is ω10 = ω00 / tan α◦ = 2ω00

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12.5 Flagpole problem in the quantum mechanical worldview
This is a good explanation of ”Why is that shadow 2ω00 meters long?”
Similarly, we may consider as follows.
(B2 ) 1. The sun is at an elevation angle α◦ in the sky.
2. Light propagates linearly.
3. The length of the shadow is ω10
Then,
4. The ﬂagpole is ω00 (= (tan α◦ )ω10 = ω10 /2) meters tall.
However, this is not suﬃcient as the explanation of ”Why is the ﬂagpole ω00 (= ω10 /2) meters tall?”Thus we have the ﬂagpole problem as follows:(B3 ) [ Flagpole problem]
Why do we feel that the solution (B2 ) is unnatural?
My opinion is as follows.
(C) the above explanations (B1 ) and (B2 ) rely on DN model (i.e., deductive-nomological model), which isdue to a kind of the logical worldview (=the logical spirit=the spirit of ”Think logically!”) (i.e., thespirit 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.

12.5.1.2 Flagpole problem by the quantum linguistic explanation
In what follows, we discuss the ﬂagpole problem in terms of quantum language.
Consider two times t = 0, ϵ (0 < ϵ ≪ 1). For simplicity, put ϵ = 1. Consider a basic structure [C(Ωt ) ⊆L∞ (Ωt , νt ) ⊆ B(L2 (Ωt , νt ))] (cf. the W ∗ -algebraic formulation in (A2 ) in Sec 1.5) is used in this section),where Ω0 = [0, 1] is the state space (in which the length of the ﬂagpole is assumed to be represented) attime 0 (where the closed interval in the real line R), Ω1 = [0, 2] is the state space (in which the length of the
shadow is assumed to be represented) at time 1 and the νt is the Lebesgue measure.
Since the sun is at an elevation angle α◦ in the sky, it suﬃces to consider to the causal map ϕ0,1 : Ω0 → Ω1such that ϕ0,1 (ω0 ) = 2ω0 (∀ω0 ∈ Ω0 ). Thus, we can deﬁne the causal operator Φ0,1 : L∞ (Ω1 ) → L∞ (Ω0 )such that(Φ0,1 f1 )(ω0 ) = f1 (ϕ(ω0 )) (∀f1 ∈ L∞ (Ω1 ), ω0 ∈ Ω0 )Let Oe = (X, F, Fe ) be the exact observable in L∞ (Ω1 , ν1 ) (cf. [?, 75]). That is, it satisﬁes that X =Ω1 , F = BΩ1 (i.e., the Borel ﬁeld in Ω1 ), [Fe (Ξ)](ω1 ) = 1 ( if ω1 ∈ Ξ), = 0 (otherwise).
Thus, we have the measurement ML∞ (Ω0 ,ν0 ) (Φ0,1 Oe = (X, F, Φ0,1 Fe ), S[ω0 ] ). Then we have the following
0statement(D1 ) [Measurement]; the probability that the measured value x(∈ X) obtained by the measurement
ML∞ (Ω0 ,ν0 ) (Φ0,1 Oe = (X, F, Φ0,1 Fe ), S[ω0 ] ) is equal to 2ω00 is given by 1.
0which is the quantum linguistic representation of (B 1 ). That is, we consider that the (B1 ) is the simpliﬁed
form (or, the rough representation) of (D1 ). Also,(D2 ) [Inference]; Assume that the measured value ω10 (∈ X) is obtained by the measurement ML∞ (Ω0 ,ν0 )(Φ0,1 Oe = (X, F, Φ0,1 Fe ), S[∗] ). Then, we can infer that [∗] = ω10 /2
which is the quantum linguistic representation of (B 2 ). That is, we consider that the (B2 ) is the simpliﬁed
form (or, the rough representation) of (D2 ). Thus, we conclude that ”scientiﬁc explanation” is to describeby quantum language. Also, we have to add that the ﬂagpole problem is not trivial but signiﬁcant, since
this is never solved without Axiom 1 [measurement] ( in Chap. 1) and Axiom 2 [ causality] (in Chap. 1 )(i.e., the answers to the problems ”What is measurement ?” and ”What is causality ?”).

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Remark 12.16. Recall the following the mainstream of western philosophy:⃝:keywords[ cognition, causality]
1

Epistemology

Linguistic turn⃝:keyword[logic]2mechanical turn−−−−−−−−−−→ Logical worldview −−−−−−−−−−−→
progress

progress⃝:keywords[measurement, causality]
3

Quantum Language(Descartes, ..., Kant)

where we see that”⃝:
1 keywords [ cognition, causality]” ≈ ”⃝:
3 keyword [measurement, causality]”Thus, ”⃝:logic”
is too speciﬁc. Thus, I am skeptical of the logical worldview (=the logical spirit=the spirit
2of ”Think logically!”). As mentioned often in this text, I believe that(♯1 ) quantum language is the language of science (cf. Assertion 1.3).
(♯2 ) mathematical logic is the language of mathematics.

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12.6 * Hempel’s raven problem

12.6 * Hempel’s raven problem
12.6.1 Is ”the set of all tyrannosaurus” meaningful? : the set theoretical worldview
Let us explain ”the set theoretical worldview”, (which is a kind of ”the logical worldview (=the logicalspirit=the spirit of ”Think logically!”)”). Logic and set theory are similar, 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., the beginning of modern
mathematics):
naive set theory
axiomatic set theory
CantorZFC-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” in this section (cf. Note 1.13). Therefore, we assume
that ”set” is deﬁned by ”a collection of things”. This is the same as the use of ”set” in everyday language.
General people (including philosophers) may think(♯1 ) 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 into a mistake.
In this text, this is called the set theoretical worldview. However, I am skeptical of the set theoreticalworldview (♯1 ). That is, I think that(♯2 ) 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,(♯3 ) Is ”the set of all tyrannosaurus” meaningful? Or, is ”the set of all raven” meaningful? If it is notmeaningless, how do we represent ”All ravens are black”?
This (♯3 ) is Hempel’s raven problem.

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12.6.2 Hempel’s raven problem in the set theoretical worldview
First, let us review the traditional arguments concerning Hempel’s raven problem (cf. refs. [31, 32]). Thus,
we start from the followings:(A0 ) 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 logically denoted by(A1 ) ”Every raven is black” : (∀x)[x ∈ R −→ x ∈ B] i.e., R ⊆ B ⊆ U,
Also, this is logically equivalent to the following contraposition:(A2 ) ”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):(A3 ) Why is the actual veriﬁcation of (A2 ) much more diﬃcult than the actual veriﬁcation of (A 1 )?
(A4 ) Why can the truth of ”(A1 ): any raven is black” be known by (A2 ), i.e., without seeing a raven also
at once?
(A5 ) Is it possible to experimentally verify ”Every raven is black”?
These may be so called Hempel’s raven paradox. However, there is a reason to consider that ”the set of allravens” is as ambiguous as ”the set of all tyrannosaurs”. If so, that is, if the above (A 0 ) is ambiguous, allother (A1 )-(A5 ) are also ambiguous. That is, (A3 )-(A5 ) are not scientiﬁc problems.
Now we think that the most essential problem concerning Hempel’s raven problem is as follows:(B) What is the scientiﬁc meaning of ”Every raven is black”?
In order to study this problem, we must prepare the quantum linguistic formulation of ornithology, underwhich the meaning of ”Every raven is black” will be clariﬁed in this section. We believe that the above
problems cannot be solved without measurement theory since the above problems includes the terms ”actual
veriﬁcation” and ”experimentally verify” which are closely related to measurement.
Remark 12.17. Just to be sure, in this paper 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” = ”∀”).

12.6.3 Hempel’s raven problem in the quantum mechanical worldview

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12.6 * Hempel’s raven problem
In this section we slightly improve our result in• Ref. [75]: Ishikawa, S., (2019) Philosophy of science for scientists; The probabilistic interpretation ofscience, Journal of quantum information science, Vol. 9, No.3 , 140-154,(https://www.scirp.org/Journal/paperinformation.aspx?paperid=95447)
We think that Hempel’s raven problem raises the problem of ”What is the scientiﬁc meaning of ‘Every
raven is black’?”. In order to answer this problem, we must prepare the quantum linguistic formulation ofornithology
The arguments below are essentially the same as Saussure’s ”signiﬁer” and ”signiﬁed” in Sec. 11.5.3.
Deﬁnition 12.18. [Membership function (= Fuzzy set, cf. [39, 40, 41, 42, 121]), Observable] Let Ω be astate space. For simplicity, we always assume that Ω is compact. A continuous function m : Ω → [0, 1] (i.e.,the closed interval in R). is called a membership function. Also, we consider the following correspondence:m : Ω → [0, 1]
membership function←−−−−−−−−−−−→ O = ({y, n}, 2{y,n} , F )correspondence(12.1)

observablesuch that
F ({y})](ω) = m(ω)

F ({n})](ω) = 1 − m(ω),(∀ω ∈ Ω)

where ”y” and ”n” respectively means ”yes” and ”no”.
Deﬁnition 12.19. [Membership functions of black birds and ravens] Let Ω be a state space. A continuousfunction m : Ω → [0, 1] (i.e., the closed interval in R) is called a membership function. Assume that thestate (i.e., quantitative property) of any bird can be expressed by a point in the state space Ω. Deﬁne themembership functions mB : Ω → [0, 1] of black birds and the membership function mR : Ω → [0, 1] of ravens
as follows. Suppose that there are 100 specialists of ornithology and the following question is made them.
(C) Is this bird with the sate ω1 (∈ Ω) a black bird or not?
The answer
is as follows.

70 specialists say that this bird is a black bird.
(D)
30 specialists say that this bird is not a black bird.

Then the value of mB (ω1 ) is deﬁned by 0.7. For many birds with the state ωi (i = 2, 3, ...N ), repeating
the experiment in the same way, the value of mB (ωi ) (i = 2, 3, ...N ) is determined. And the membershipfunction mB : Ω → [0, 1] of black birds is deﬁned by the interpolation method (which may be rather
subjective). Similarly we get the membership function mR : Ω → [0, 1] of ravens.
Deﬁnition 12.20. (i): [Raven state class, Black bird state class]: PutΩB := {ω ∈ Ω | mB (ω) = 1},ΩR := {ω ∈ Ω | mR (ω) = 1}

which is respectively called a black bird state class and a raven state class (see Fig. 11.3 below).
(ii): [Raven, Black bird]: If the state of a certain bird belongs to Ω R [ resp. ΩB ], this bird of a certain
is called a raven [ resp. a black bird ]. It is not asked whether this bird exists really. This bird may be
extirpated like tyrannosaurs. Moreover, this bird may be a biology newly made by genome edit.
(iii): [”Every raven is a black bird”]: We say ”Every raven is black”, if it holds that Ω R ⊆ ΩB . (see Fig.
11.6 later).

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mB

1
mR

Ω

Ω
ΩR

ΩB

Figure 11.3: [Raven state class ΩR , black bird state class ΩB ]Deﬁnition 12.21. [Raven observable, Black bird observable]: Using the above membership functions, wedeﬁne two observables (i.e., Black bird observable, Raven observable) OB = ({y, n}, 2{y,n} , FB ), OR =({y, n}, 2{y,n} , FR ) in C(Ω), such that
FB ({y})](ω) = mB (ω)

FB ({n})](ω) = 1 − mB (ω),

FR ({y})](ω) = mR (ω)

FR ({n})](ω) = 1 − mR (ω)(∀ω ∈ Ω)

where ”y” and ”n” respectively means ”yes” and ”no”. Thus, a membership function can be identiﬁed witha binary observable.
Since we assume that any bird is characterized by a certain point in the state space Ω, it is natural toconsider that systematic ornithology is formulated as follows.
Formulation [I] [The quantum linguistic formulation of systematic ornithology [I]]:(E1 ) Ravens are characterize by the membership function mR : Ω → [0, 1] (or, equivalently, the observable OB = ({y, n}, 2{y,n} , FB )). If the state ω of a bird belongs to ΩR , then the bird is called araven.
(E2 ) Black birds are characterize by the membership function mR : Ω → [0, 1]. (or, equivalently, theobservable OR = ({y, n}, 2{y,n} , FR )). If the state ω of a bird belongs to ΩB , then the bird iscalled a black bird.
Interpretation 12.22. [The probabilistic interpretation of membership functions] We add the following
probabilistic interpretation to this formulation [I]: For example, again consider Deﬁnition 12.19, and moreover, the statement (D). i.e.,
70 specialists say that this bird with a state ω0 is a black bird.
′(D )30 specialists say that this bird with a state ω0 is not a black bird.

If we choose one person from the 100 specialists at random, the probability that he/she says that this bird isblack is given 0.7. Such a measurement is represented by the symbol MC(Ω) (OB = ({y, n}, 2{y,n} , FR ), S[ω0 ] ).
Therefore, we can use Axiom 1 [measurement] (in Section 1.1) as follows.

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12.6 * Hempel’s raven problem(F1 ) for a bird with a state ω0 (∈ ΩR ), the probability that the measured value y [ resp. n] is obtained by
the measurement MC(Ω) (OR = ({y, n}, 2{y,n} , FR ), S[ω0 ] ) is equal to [FR ({y})](ω) [ resp. [FR ({n})](ω)].
(F2 ) for a bird with a state ω1 (∈ ΩB ), the probability that the measured value y [ resp. n] is obtained by themeasurement MC(Ω) (OB = ({y, n}, 2{y,n} , FB ), S[ω1 ] ) is equal to [FB ({y})](ω1 ) [ resp. [FB ({n})](ω1 )].

12.6.4 A priori proposition: ”Any small black bird is black”
Next consider the following ﬁgure:

mB

1
mR

mSB
Ω

ΩB
ΩR

Ω
ΩSB

Figure 11.4: [Raven state class ΩR , black bird state class ΩB , small black bird state class ΩSB ]That is, we add the small black bird observable:
Deﬁnition 12.23. (i): [Membership function of small black birds]: Deﬁne the membership function mSB :Ω → [0, 1] of small black birds such as Deﬁnition 12.19.
(ii):[Small black bird state class] The small black bird state class Ω SB is deﬁned by {ω ∈ Ω | mSB (ω) = 1}.
(iii): [Small black bird]: If the state of a certain bird belongs to Ω SB , this bird is called a small black bird.
(iv): [”Every small black bird is black” ]: We say ”Every small black bird is black” if it holds that
ΩSB ⊆ ΩB
Note that this necessarily holds without actual veriﬁcation since it is assumed that a small black bird isdeﬁned by a black bird such that it is small. Thus, ”Every small black bird is black” is a priori proposition.
(v): [Small black bird observable]: And deﬁne Small black observable OSB = ({y, n}, 2{y,n} , FSB ) such that
[FSB ({y})](ω) = mSB (ω), [FSB ({n})](ω) = 1 − mSB (ω), (∀ω ∈ Ω).
Thus, we have the new formulation, which is a development of Formulation [I] (i.e., The quantum linguistic
formulation of systematic ornithology [I]] ):Formulation [II] [The quantum linguistic formulation of systematic ornithology [II]]:(G1 ) Ravens are characterize by the membership function mR : Ω → [0, 1]. A bird with a state ωR suchthat ωR ∈ ΩR is called a raven.
(G2 ) Black birds are characterize by the membership function mR : Ω → [0, 1]. A bird with a state ωBsuch that ωB ∈ ΩB is called a black bird.
(G3 ) Small black birds are characterize by the membership function mSB : Ω → [0, 1]. A bird with astate ωSB such that ωSB ∈ ΩSB is called a small black bird.
(G4 ) It holds that ΩSB ⊆ ΩB , i.e., Every small black bird is black. This is a priori statement, which is
directly derived from Deﬁnition 12.23 (iv).
Exercise 12.24. It is easy to see that the above (G4 ):ΩSB ⊆ ΩB says that286

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(H) Let ui (i = 1, 2, ..., N ) be a small black bird with the state ωi (∈ ΩSB ), which is denoted by ωe (ui ). For
each small black bird ui , the probability that the measured value y is obtained by the measurement
MC(Ω) (OB = ({y, n}, 2{y,n} , FB ), S[eω(ui )] ) is equal to 1.
According to the linguistic Copenhagen interpretation (E 4 ) in Sec. 1.1: ”Only one measurement is permitted”, the above (H1 ) is formally written as follows.
(I) the probability that the measured value (y, y, y, ..., y ) is obtained by the parallel measurement
{z
}
|
N
NN
{y,n}
, FB ), S[eω(ui )] ) is equal to 1.
i=1 MC(Ω) (OB = ({y, n}, 212.6.5 A posteriori proposition: ”Every raven is black”
12.6.5.1

Popper’s falsiﬁcationism in measurement theory♠Note 12.10. As mentioned in Sec. 12.3.2, 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 consideredscience, it must be falsiﬁable. If it is false, it can be shown through observation and experiment tobe false. However, I think that the meaning of ”observation and 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 is black). Since this is apriori 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. [97]) such that(K) ”ΩR ⊆ ΩB ” should be accepted, if many experiments which deny ”ΩR ⊆ ΩB ” are conducted 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} (See Figure 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 11.5: [ω0 ∈ ΩR \ {ω | mB (ω) = 0}, ρ0 ({ω ∈ Ω | 0 < mB (ω) < 1}TΩR }) ≈ 0, i.e., negligible. ][Test II]: 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.
N{y,n}, FB ),
That is, we take the parallel mixed measurement (cf. [?, 75]) 100
i=1 MC(Ω) (OB :=({y, n}, 2m∗S[∗] (ρ0 )), where a mixed state ρ0 (∈ S (C(Ω) )) satisﬁes ρ0 (ΩR ) = 1 and ρ0 (ΩR \ ΩB ) = 0.03. Here,287

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12.6 * Hempel’s raven problem
Twe, 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:N
100
{y,n}
(M2 ) By the parallel mixed measurement (cf. [?, 75])
, FB ), S[∗] (ρ0 )), ai=1 MC(Ω) (OB :=({y, n}, 2measured 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 mixed measurement
{z
}
|
100
N100
{y,n}
, FB ), S[∗] (ρ0 )) is given by (97/100)100 (< 0.048). That is, thei=1 MC(Ω) (OB := ({y, n}, 2probability 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 ravenis black)”. When we can’t do such test many times and still deny ”Ω R ⊆ ΩB (i.e., any raven is black)”,
according to Popper’s falsiﬁcationism, we will believe this.

If we believe in ”ΩR ⊆ ΩB (i.e., any raven is black)”, we can propose the following new formulation:
Formulation [III] [The quantum linguistic formulation of systematic ornithology [III]]:(N1 ) Ravens are characterize by the membership function mR : Ω → [0, 1]. The deﬁnition of ravens isgiven by a raven state class ΩR as shown in Deﬁnition 12.20 (ii).
(N2 ) Black birds are characterize by the membership function mR : Ω → [0, 1]. The deﬁnition of blackbirds is given by a raven state class ΩB as shown in Deﬁnition 12.20 (ii).
(N3 ) Small black birds are characterize by the membership function mSB : Ω → [0, 1]. The deﬁnitionof small black birds is given by a raven state class ΩSB as shown in Deﬁnition 12.23 (iii).
(N4 ) It holds that ΩSB ⊆ ΩB , i.e., Every small black bird is black. This is a priori statement, which is
directly derived from Deﬁnition 12.23 (iv).
(N5 ) It holds that ΩR ⊆ ΩB , i.e., Every raven bird is black, This is a posteriori statement, which isguaranteed in the sense of Popper’s falsiﬁcationism (or, statistical hypothesis testing)Thus we see the progress of ornithology (i.e., Formulation [I][III] ).

progress
progress=⇒ Formulation [II] =⇒ Formulation

mB

1

mR

mSB
Ω

ΩB

Ω

ΩR
ΩSB

Figure 11.6: [All ravens are black: (ΩR ⊆ ΩB ) ]288

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12.7 Three approaches to the mind-body problem
The mind-body problem is the most famous problem in Descartes philosophy. There are two standing
positions on whether we see this as a problem of science or a problem of philosophy.
(A) [Science]:If we are in the position of existential monism (i.e., the scientiﬁc position), i.e., if we expecta scientiﬁc solution, then the mind-body problem is a problem of brain science, AI, and cognitive
science.
(B) [Philosophy]: if we consider the mind-body problem 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 (B). This section was written with reference to the following.
(C) [67] 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 is natural to study
the mind-body problem in quantum language. This will be discussed in Section 9.4.4 (The third approach).

12.7.1 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 inorder to propose Descartes philosophy (i.e., mind-matter dualism). That is, his argument is as follows.
(D) If the existence of ”I” is deduced from the cogito proposition, 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-matter dualism) is(E)”I”(=”mind”), ”body”(=”sensory organ”), ”matter”
Here, we have the following problem:(F): The mind-body problem in dualistic idealism
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.

12.7.2 The ﬁrst approach; Cognitive scientiﬁc approach
As mentioned in Note 9.11, Dr. Click (the most noted for being a co-discoverer of the structure 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 sense of personal identityand free will, are in fact no more than the behavior of a vast assembly of nerve cells and their associatedmolecules.
From the scientiﬁc point of view, I agree to his opinion (G 1 ). (i.e., the denial of the substance dualism ).
Therefore, I believe that the following will be realized.

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12.7 Three approaches to the mind-body problem(G2 )Descartes(Kant,Husserl)−−−−−→−−−−−→ Brain science −−−−−→

Philosophy of mind

progress

progress

progress(based on cognitive science)

Solution(the mind-body problem)

This may imply that
(G3 ) the mind-body problem will be solved in science
However, it should be noted that the (H1 ) (i.e., the denial of the substance dualism ) and the dualistic
idealism (i.e., quantum language) do not contradict each other. That is because quantum language says:(H) Describe any monistic phenomenon (such as (G1 ) ) by dualistic language (=quantum language) !

12.7.3 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.
That is, the sentence ”How are ‘mind’ and ‘body’ connected?” is meaningless in Descartes=Kant epistemology. Thus, there may be a reason to consider that the mind-body problem is just ”what we cannot speak
about”. Therefore, according to Wittgenstein’s famous saying ”What we cannot speak about we must passover in silence” (in [117]), some may conclude that we must speak nothing about the problem (F). That is,
the mind-body problem is an illusory problem. However, I think, by (J) and (K) mentioned in the following
section, that this second approach is not only non-productive but also wrong. As mentioned before, I thinkthat the Wittgenstein’s next assertion is non-productive:(I2 ) philosophical problems arise from insuﬃcient attention to the variety of natural language use.

12.7.4 * The third approach; Quantum linguistic solution to the mind-body problem
It should be noted that(J) the demarcation problem (i.e., how to distinguish between ”what we cannot speak about” and ”what
we can speak about”) depends on language.
For example, the proposition ”the earth goes around the sun” cannot be written in mathematics but in the
Newtonian mechanical language. Note that both ”the limits of my language mean the limits of my world”
and ”the limits of your language mean the limits of your world” are true. Therefore,(K) in order to solve the mind-body problem in dualistic idealism, we should create the language in which
the mind-body problem can be regarded as ”what we can speak about”
Without this challenge (K), we cannot obtain the solution to the mind-body problem (F). In this sense, thesecond approach in Section 9.4.3 may be shallow.
Concerning the causality problem (i.e., What is causality?), we already answered it in Problem 1,1 (andNote 1.2), that is,”The solution to the causality problem”⇔”Axiom 2”(12.2)

Similarly, I can give the solution to the mind-body problem (i.e., How are ”mind” and ”body” connected?)
as follows.
Answer 12.25. [The solution to the mind-body problem]; The correspondence of the key-words(Assertion 1.14) says that(measured value, measuring instrument)(mind, body)the correspondence of the key-words

Descartes −−−−−−−−−−−−−−−−−−−−−−−−−→
linguistic turn

Quantum language

This says:
the epistemological mind-body problem

Descartes
How are ”mind” and ”body” connected?
the epistemological mind-body problem−−−−−−−−−→
linguistic turnquantum language(12.3)

How are ”measured value” and ”measuring instrument” connected?

Also, recall Figure 1.2 as follows:290

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If so, we can easily solve the mind-body problem (in the right-hand side of formula (12.3) that is,
The solution to the mind-body problem
The solution to the mind-body problem is just Axiom 1 (in Sec.1.1.1), that is, the relation between”measured value” and ”measuring instrument” is given as follows.
• With any system S, a basic structure [A ⊆ N ⊆ B(H)] can be associated in which measurement
theory of that system can be formulated. When the observer takes a measurement of anobservable (or, by a measuring instrument) O=(X, F, F ) for a system S[ρ] i.e., a system S witha state ρ), the probability that a measured value x (∈ X) obtained by the measurement belongsto Ξ (∈ F) is given by ρ(F (Ξ))(≡ A∗ (ρ, F (Ξ))N ).
Therefore,(L) ”The solution to the mind-body problem”⇔”Axiom 2”
Hence, by this (L) and the formula (12.2), we have the following equivalences:to propose quantum language⇔to propose Axiom 1 (measurement) and Axiom 2 (causality)⇔to solve the mind-body problem and the causality problem
and further, we want to add:
⇔to build a ﬁrm theory in dualistic idealism (i.e., metaphysics)(12.4)

If so, then my next desire has been fulﬁlled.
• ”mind-body problem” should be the greatest problem in philosophy.

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293

Chapter 13

Postscript: Can QL be post-analytic
philosophy?
13.1

Philosophy (of worldviews) has progressed towards quantum
languageIn this text, I discussed the history of western philosophy, i.e.,
the time series ⃝
0 - ⃝
1 - ⃝
9 - ⃝11 ⃝13 ⃝14
in the following ﬁgure
Assertion 0.1 ( in Preface ):the realistic worldview (physics): monism(monism)−−→ Newton(realism)

Socrates
Aristotle

O:Greek0philosophy
Parmenides, Zeno
Plato

Schola-O−−−−−→ 1
sticismOOrelativity−→ theory−−−−−−−−−−→ 32Oquantum−→ mechanics −−−−−−−−→ 4









(unsolved)

O TOE:
5

−−→

O

O
O

O

OO

theory of
everything(quantum phys.)language−−−−−−−−−−−−−−−−−
→7 




dualism−−−−−−−−→ 8 −−−−−−−−−→ 6 statistics

quantization
applied math.
system theoryclassical mechanical worldview



(=MT)
(mind-matter dualism) (Copernican revolution))





QL:
quantum
14
DescartesKant
9mechanical−−→language−−→
Locke,...
−−→−−−−−−−→ 10turn
quantum mechanicalﬁctional worldview

ﬁctional worldview
worldview


11



turn
linguistic





?
(logic in philo.))(logic, set theory)Wittgensteindualism
Boole, Frege, 12 Saussurre−−−−−−−−−→ 13
measurement
Russell,Cantor −−→ Zadeh

O

O

O

math.

logical worldview

the idealistic worldview (no experiment, metaphysics): dualism
Figure 0: The history of the world-descriptions
Philosophy (≈ dualistic idealism) has progressed toward QL( i.e., 0 → 1 → 9 → 11 → 13 → 14 )O O O O O OKSTS/RR-21/001
August 23, 2021

13.2 If the above (A) and (B) are true, almost all unsolved problems concerning dualistic idealism
have to be solved in QL

And I conclude that(A) If ”to make progress” is deﬁned by ”to come near quantum language” (i.e., ”becoming more andmorelike quantum language” we can say that the time series ( of philosophies of dualistic idealism)⃝
can be regarded as progress, that is,
0 - ⃝
1 - ⃝
9 - ⃝11 ⃝13 ⃝14

Therefore,(A′ ) the time line:

progress.



⃝
1 - ⃝
9 - ⃝11 ⃝13

is not only a refreshing change, but also a

Thus, quantum language, roughly speaking, has the four aspects as follows.

The four aspects of quantum language
7 : the linguistic turn of quantum mechanics
(i.e., the true color of the Copenhagen interpretation) 8 : the dualistic (i.e., measurement theoretical) reconstruction of statistics
10 : the ﬁnal goal of dualistic idealism=
(i.e., the linguistic and mechanical turn of the Descartes=Kant epistemology) 13 : the measurement theoretical understanding of analytic philosophy

(i.e., fuzzy logic in QL)

O
O
O
O

Thus, I assert that QL is the uniﬁed theory of dualistic idealism

I believe, from the scientiﬁc point of view, that
(B1 ) quantum language is the ﬁnal destination of the genealogy of Western philosophy.
And(B2 ) a scientiﬁc perfection of dualism and idealism is realized by quantum language
In order to assert the (B) (=(B1 ), (B2 )), we proved the following proposition throughout this paper:

13.2 If the above (A) and (B) are true, almost all unsolved problems
concerning dualistic idealism have to be solved in QL
13.2.1 The list of my answers of unsolved problems
If (A) and (B) (=(B1 ) + (B2 )) are true, it is natural to consider:(C) many unsolved problems raised in the 2500 year history of dualistic idealism can besolved within the framework of quantum languages.
My results concerning quantum language are summarized in the following two texts

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(D1 ): This text: History of western philosophy from the quantum theoretical
point of view; [Ver. 3]

(D)



(D2 ): Ref. [71]: The linguistic Copenhagen interpretation of quantum mechanics:
Quantum language [Ver 5]
The (C) is shown throughout this paper as follows.
List 13.1.
(D1 )Almost famous unsolved problems concerning dualistic
idealism can be solved in QL• What is probability (or, measurement, causality) ? cf. Sec. 1.1.1)• The solution of Zeno paradox (Flying arrow, Achilles and a tortoise), (cf.ref. [45], or Sec. 2.4)( ”to solve Zeno paradox” = ”to understand DST-formula” (cf. Sec. 2.4.3 )• the measurement theoretical understanding of Plato’s allegory of the sum , (cf. Sec. 3.3.2)• Plato’s Idea theory≈Zadeh’s fuzzy theory≈Sausuure’s linguistic theory (cf. Sec. 3.5.3)• 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)• Only the present exists (cf. Sec. 6.1.2)• What is the problem of universals? (cf. Sec. 6.5.1)• What is Geocentrism vs. Heliocentrism? After all, the worldviewism (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. 9.3)
• The problem of qualia (cf. Sec. 9.5.1)• Brain in a vat argument (cf. Sec. 9.5.2)• The solution of Hume’s problem of induction (cf. Sec. 9.7.1)• grue paradox cannot be represented in quantum language (cf. Sec. 9.8)
• What is causality? (cf. Sec. 10.2)• 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)• My scientiﬁc understanding of ”Tractatus Logico-Philosophicus (=TLP)” and ”Zadeh’s fuzzy sets”(cf. ref. [78], 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 picture theory must belong to295

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13.2 If the above (A) and (B) are true, almost all unsolved problems concerning dualistic idealism
have to be solved in QL
dualistic idealism in order to assert (A) (= Assertion 1.5)
• Lewis Carroll’s paradox (cf. Sec. 12.4)• Flagpole problem, (cf. Sec. 12.5)• Hempel’s raven paradox (cf. Sec. 12.6)• the mind-body problem (i.e., How are mind and body connected?), (cf. Sec. 12.7)
Also, for the solutions of unsolved problems in quantum mechanics, statistical mechanics, statistics andprobability theory, see ref. [71]), that is,(♯) ref. [71]: 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)(D2 ) The list of my answers for scientiﬁc unsolved problems in (D2 )ref. [71]; 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)• Kolmogorov’s extension theorem in quantum language (Sec.4.1 in ref. [71]) (Sec.4.1 in ref. [71])• The law of large numbers in quantum language (Sec.4.2 in ref. [71])• the discovery of Heisenberg’s uncertainty relation (cf. ref. [36], or, Sec. 4.3 in ref. [71])Has Heisenberg’s uncertainty relation ever been used eﬀectively in physics? (Sec. 4.3 in ref. [71])• Bell’s inequality holds in both classical and quantum systems (Sec. 4.5.2 in ref. [71])• Measurement theoretical formulation of measurement, inference, control (Sec. 5.2 in ref. [71])• Monty-Hall problem in quantum language (non-bayesian approach) (Sec.5.5 in ref. [71])• Two envelope problem in quantum language (non-bayesian approach) (Sec.5.6 in ref. [71])• Conﬁdence interval and statistical hypothesis test (Chapter 6 in ref. [71])• Analysis of variance (Chapter 7 in ref. [71])• Syllogism holds in classical systems, but not in quantum systems (Sec.8.6 and Sec.8.7 in ref. [71])• Mixed measurement theory (Bayesian measurement theory) (Chap. 9 in ref. [71])• The measurement theoretical characterization of the wave-function collapse (= projection pustulate)(cf. ref. [63], or Sec.11.2 in ref. [71])(von Neumann-Lüders Projection Postulate can be justiﬁed in QL: ref. [63])• The measurement theoretical characterizations of de Broglie’s paradox, quantum Zeno eﬀect,
Schrödinger’s cat, Wigner’s friend, Wheeler’s delayed choice experiment, Hardy Paradox, quantum eraser (Sec.11.3∼Sec.11.8 in ref. [71])• The measurement theoretical characterizations of double-slit experiment, Wilson cloud chamber(Sec.12.2, Sec.12.3 in ref. [71])• The measurement theoretical characterizations of regression analysis (Sec.13.2 in ref. [71])• The measurement theoretical characterizations of Brownian motion, Zeno’s paradox (Sec.14.2 ,296

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Chap. 13 Postscript: Can QL be post-analytic philosophy?
Sec.14.4 in ref. [71], also, see [45])• The measurement theoretical characterizations of least-squares method (Chap.15 in ref. [71])• The measurement theoretical characterizations of Kalman ﬁlter (Chap.16 in ref. [71])• The measurement theoretical characterizations of equilibrium statistical mechanics (Chap.17 in ref.
[71])
• The measurement theoretical characterizations of psychological tests (Chap.18 in ref. [71])• The measurement theoretical characterizations of belief (Chap.19 in ref. [71])• The mathematical foundation of science (Hempel’s raven paradox) (Chap.20 in ref. [71])If you look at the tables above, you will notice the following.
(E) The philosophies up to analytic philosophy ( 0 - 1 - 9 - 11 - 13 ) merely posed important unsolved problems. It is QL that has solved these problems.
In this sense, we can again assert the (B) (=(B1 )+ (B2 )):(B1 ) quantum language is the ﬁnal destination of the genealogy of Western philosophy.
And(B2 ) a scientiﬁc perfection of dualism and idealism is realized by quantum language
To put it another way,

OOOOO

(F1 ) QL has, for the ﬁrst time, proven that these philosophies ( ”Plato” →”Descartes ”→ ”Kant” → ”Wittgenstein”) are not just word games.
In other words. we can conclude that

(F2 ) These philosophies were pursuing problems that had scientiﬁc answers.
That is,(F3 ) the strangeness of philosophy (i.e., dualistic idealism) is due to thestrangeness of the Copenhagen interpretation of quantum mechanics

The above ﬁgure explains the following:(F4 ) Why could QL solve almost all the unsolved problems of dualistic idealism(Descartes, Kant, analytic philosophy)?

13.3 Can QL be post-analytic philosophy?
I think most philosophy enthusiasts have thought about the following.
(G1 ) Does the future of analytic philosophy look rosy?
In regards to this, Dr. Hawking said in his best seller book [27]:297

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13.3 Can QL be post-analytic philosophy?

• However, in the nineteenth and twentieth centuries, science became too technical and mathematical
for the philosophers, or anyone else except a few specialists. Philosophers reduced the scope of theirinquiries 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 ofphilosophy from Aristotle to Kant!
I’m sure there are many thoughts on this opinion of Hawking. I think the following;(G2 ) analytic philosophy has expired!
It has been 100 years since the emergence of analytic philosophy. Compared to modern mathematics,
physics and statistics, which emerged around the same time as analytic philosophy, I think it is diﬃcult
to say that analytic philosophy has developed smoothly. If the (G 2 ) is true and history (Plato→Descartes→Kant→Wittgenstein) repeats , everyone may think that
(G3 ) the current situation of philosophy is a once-in-a-century (or, once-in-two-centuries) opportunity.
I thought so too. And I believe that QL is a strong candidate for post-analytic philosophy because, as I haveshown in List 13.1, it has solved most of the open problems of dualistic idealism. However, I am not evensure that the term ”post-analytic philosophy” will take hold.
In fact, I also think(G4 ) the future of theoretical physics and mathematics is also not optimistic.
In theoretical physics, there has been too great a gap between theory and experiment. One of the reasons
why Hawking did not win the Nobel Prize in Physics is that ”his theory could not be experimentally veriﬁed”.
Also, it took more than 50 years to verify the existence of the Higgs boson. This means that Einstein-classphysicists can only be recognized after death.
Mathematics is in a more serious situation. This is because mathematicians must solve the Riemannconjecture before a quantum computer can be launched in earnest. And there will be a situation wheremathematicians will not be able to understand the proof of the Riemann prediction solved by computers.
It’s hard to predict the future, but I think that(H) In the not too distant future, there will come an era in which humans cannot beat quantum computerswhen it comes to logic.
Therefore, what I am thinking now is the following.
(I) It is lucky to discover QL ( and solve many problems in List 13.1) before the quantum computer will
be in full swing.
The Copenhagen interpretation is so powerful that it explains the principle of quantum computers. If I am
allowed to bluﬀ, I may say that(J)

QL is greater than QC (= quantum computer).

Since I am an amateur in the history of Western philosophy, I can make many mistakes in this book. However,I would like to conﬁdently argue that QL is the scientiﬁc ﬁnal form of dualistic idealism. I hope that readers
read ref. [71] quickly, and examine the power of QL.

298

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299
August 2021
Shiro ISHIKAWA*1
ishikawa@math.keio.ac.jp*1 For the further information concerning quantum language,

see home page: http://www.math.keio.ac.jp/~ishikawa/indexe.html

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301

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[115] Weinberg, Steven ”To Explain the World: The Discovery of Modern Science” HarperCollins Publishers (2015),(2015)[116] J. A. Wheeler, The ’Past’ and the ’Delayed-Choice Double-Slit Experiment’, pp 9-48, in A.R. Marlow, editor,
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Index
abduction, 235, 237
abstract ego (von Neumann), 123
Aesop(BC.620 - BC.510), 67
allegory of the cave, 69
allegory of the sun, 72
ambiguity of ordinary language
Arguments for the existence of God(Anselmus), 130
Geocentrism and Heliocentrism, 153
Zeno’s paradoxes, 54
analytic philosophy(=linguistic philosophy), 221
analytic-synthetic distinction, 273
Anselmus(1033 - 1109), 27, 90, 135, 180, 183
antinomy, 54
Archimedes(BC287 - BC212), 109
Aristarchus(BC310 - BC230), 107
Aristotle(BC384 - BC322), 29, 88, 136
asserted ﬁction, 65
Augustinus(354-430), 123, 165
axiom of choice, 105
Axiom 1 [measurement], 3, 31
Axiom 2[causality], 3, 31
Bacon(1561 - 1626), 92, 208
Banach-Tarski theorem, 105
Bell’s inequality, 183
Bergson, Henri-Louis(1859-1941), 165
Berkeley, George(1685-1753) , 181
J. Bernoulli(1654 - 1705), 144, 193
Bohr(1885 - 1962), 27, 90, 135, 180, 183
Bohr-Einstein debates, 6, 182–184
Boole, George(1815-1864), 232
Born(1882-1970), 236, 267
British Empiricism, 170
buoyancy, 109
C ∗ -algebraic formulation, 30
Cantor’s paradox, 241
Carnap, Ludorf (1891-1970) , 273
Carroll, Lewis(1838-1898), 277
Carroll’s logical paradox, 277
causal relation, 2, 146, 152
F. Click (1916 – 2004), 186
cogito proposition, 11, 104, 160, 162, 189
cogito turn, 158
Continental Rationalism, 173
lCopenhagen interpretation(=linguistic Copenhagen
interpretation), 4
Copernican revolution, 125, 191, 209
Critique of Pure Reason (Kant), 202
Democritus (BC.460-BC.370), 175
Descartes(1596-1650), 5, 160
Descartes ﬁgure, 5, 166
Discourse on the Method (Descartes), 160
discovery of zero, 128dogmatic slumber, 191, 209
DST-formula, 57
dualism(=mind-matter dualism), 27

Dummett, David (1925-2011) , 255, 267
eidos(Aristotle), 28, 29, 89, 136
Einstein(1879 - 1955), 27, 90, 135, 180, 183, 266
Einstein-Tagore meeting, 182
Eratosthenes(BC275 - BC194), 114
ethics · morals, 62
Euclid(BC330 - BC275) , 103
Eudoxus(BC.400 BC.347), 44
falsiﬁability, 8, 274, 287
ﬁctional worldview (the main current of western
philosophy), 21
Fisher(1890-1962), 236, 267
Five-minute hypothesis, 243
ﬂagpole problem, 279
Frege, G.(1848-1925), 232
Fuzzy logic, 256
Galapagos philosophy, 85
Galileo(1564 - 1642), 92, 144, 193, 208
Gauss(1777 - 1855), 129
Geocentrism(Ptolemaeus), 116
grue paradox, 197
habitual thinking (=worldview), 53
Harari.Y.N(1976 -), 67
Hawking(1942 –2018), 23, 297
Heidegger(1889-1976), 165
Heisenberg’s uncertainty principle, 110, 268
Heliocentrism, 148
Heliocentrism(Aristarchus), 107
Hempel, C.G(1905-1997), 279
Hērakleitos(BC.540 -BC.480), 49, 91
Hume, David (1711-1776) , 189, 193
Hume’s problem of induction, 193
Husserl (1859-1938) , 218hyle(Aristotle), 28, 29, 89, 136
I think, therefore I am., 160
Idea(Plato), 28, 29, 136, 166, 172
I know that I know nothing, 63
induction, 145
inductive reasoning, 145
Kant(1724-1804), 191, 209
Kelvin(1824 –1907), 19
J, Kepler(1571 - 1630), 146, 149
Kripke’s quus, 271
law of large numbers, 144, 193
leaning Tower of Pisa, 150
Leibniz(1646-1716), 174, 177
Leibniz-Clarke Correspondence, 177
linguistic philosophy(=analytic philosophy), 221
linguistic interpretation(=linguistic Copenhagen
interpretation), 4
List of unsolved problem, 15

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306
J. Locke (1632 - 1704), 170
logical spirit (=the spirit of Think logically!) (analytic
philosophy) , 23
logical positivism, 272
logical spirit (=the spirit of Think logically
logical worldview, 20
McTaggart. J.M.E.(1866–1921), 245
McTaggart’s paradox, 245
measurement, 3Merleau-Ponty(1908-1961), 172
metaphysics, 19
mind-body problem, 164, 289
modal logic, 24
model-change, 84, 164
Moore, E.J. (1873 –1958) , 247
Moore’s paradox, 247
Motion function method, 52
naive set theory, 19nativism, 173
necessity of dualism, 77
necessity of idealism, 76
Newton(1643-1727), 92, 179, 208
non-Euclidean revolution, 105
Ockham(1285 - 1347), 27, 90, 135, 180, 183
Ockham’s razor, 138
Only the present exists, 123
paradigm shift, 142
parallel postulate, 103
Parmenides(born around BC. 515), 27, 91
Peirce, C.S.(1839-1914), 235
PI(= Philosophisch Investigations), 265
Plato(BC427 - BC347), 29, 136, 166
Plato’s beard, 138
Popper(1902-1994), 8, 274, 287
population, 28, 29, 136
practical logic, 24predicate logic, 223
primary quality, secondary quality, 28, 29, 136, 172
Principia, 152
Problem of universals, 27, 90, 135, 180, 183
projection postulate, 9, 124
Proof of the existence of God, 130
propositional logic, 223
Protagoras(BC490 - BC420), 62
Ptolemaeus(AD.83 - AD.168), 116
pure Pythagoreanism, 47
purpose, 91
Pythagoras(BC582 - BC496), 41
QL(=quantum language), iii
qualia problem, 185
qualia (=subjectivity) problem, 164
quantum computer, 298
quasi-product observable, 32
Quine, Willard Van Orman (1908-2000) , 273
raven problem, 282
realistic worldview(physics), 21
Russell. B.(1872 - 1970), 79, 90, 241
Russell’s paradox, 241
Saussure, F.(1857-1913), 249
Scholasticism, 126

Schrödinger’s cat, 296
mechanical worldview(statistics, quantum language), 24
secondary quality, primary quality, 172
self-reference
cogito proposition, 11
subjective time, 124
set theoretical worldview, 19
signiﬁed, 252
signiﬁer, 252
Socrates(BC469 - BC399), 62
spectrum, 178
state equation, 210
subjective time, 123
subjectivity (= qualia) problem, 164
subjectivity problem, 164
syllogism, 261
syllogism(Aristotle), 94
tabula rasa (blank paper), 171
temporal logic, 24
tense, 123
tensor quasi-product observable, 32
Thalēs(BC640 - BC546), 36
thing-in-itself, 207
Thomas Aquinas(1225 - 1274), 29, 136
time theory(Augustinus), 123
TLP(=Tractatus Logico-Philosophicus), iii, 255
transcendental idealism, 216, 251
trial of Galileo, 149
von Neumann(1903- 1957) , 123
von Neumann-Lüders Projection Postulate, 296
W ∗ -algebraic formulation, 30
wave function collapse, 9, 123, 124
Weinberg(1933 -2021), 37, 67, 88, 93, 116, 146, 148
What is space-time?, 53
Whitehead(1861 - 1947), 37, 84
Wittgenstein(1889 - 1951), 16, 24, 165
Wittgenstein’s paradox, 270
worldviewism, 17
Zadeh, A. L.(1921 -2017), 257
Zadeh’s fuzzy set, 81, 256
Zeno(BC490 - BC430), 54
Zeno’s paradoxes , 54

KSTS/RR-21/001
August 23, 2021

KSTS/RR-21/001
August 23, 2021

Department of Mathematics
Faculty of Science and Technology
Keio University
Research Report2018[18/001] Shiro Ishikawa,
Leibniz-Clarke correspondence, Brain in a vat, Five-minute hypothesis, McTaggart’s
paradox, etc. are clarified in quantum language,
KSTS/RR-18/001, September 6, 2018 (Revised October 29, 2018)[18/002] Shiro Ishikawa,
Linguistic Copenhagen interpretation of quantum mechanics: Quantum Language[Ver. 4 ],KSTS/RR-18/002, November 22, 20182019
[19/001] Sumiyuki Koizumi,
Contribution to the N. Wiener generalized harmonic analysis and its application to
the theory of generalized Hilbert transforms,
KSTS/RR-19/001, August 2, 2019
(Second edition, September 20, 2019)(Third edition, December 6, 2019)[19/002] Shiro Ishikawa,
Hempel’s raven paradox, Hume’s problem of induction, Goodman’s grue paradox,
Peirce’s abduction, Flagpole problem are clarified in quantum language,
KSTS/RR-19/002, September 12, 2019 (Revised November 1, 2019)[19/003] Shiro Ishikawa,
Linguistic Copenhagen interpretation of quantum mechanics: Quantum Language[Ver. 5 ],KSTS/RR-19/003, December 26, 20192020
[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