Research Report

KSTS/RR-20/001
November 15, 2020

History of Western Philosophy fromthe quantum theoretical point of view; [Ver. 3]

byShiro Ishikawa

Shiro Ishikawa
Department of Mathematics
Keio University

Department of Mathematics
Faculty of Science and Technology
Keio University
c
⃝2020
KSTS
3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522 Japan

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History of Western Philosophy from the quantum
theoretical point of view; [Ver. 3]
Shiro ISHIKAWA1 (Mail: ishikawa@math.keio.ac.jp)
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 thatquantum language is the only scientiﬁcally successful theory in dualistic idealism. Hence, thisturn encourages us to understand Western philosophies (Parmenides, Plato, Descartes, Locke,
Berkeley, Hume, Kant, Wittgenstein, etc.) within the framework of quantum language. This isdone in this paper. That is, I show that most unsolved philosophical problems in dualism aresolved within the framework of quantum language.

Preface:

Did philosophy make progress?

Quantum language (i.e., QL (=MT = measurement theory), the linguistic Copenhagen
interpretation of quantum mechanics, the quantum mechanical worldview) proposed by myself
is a language that is inspired by the Copenhagen interpretation of quantum mechanics, andmoreover, 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 [The location of quantum language in the history of world-description (cf. ref.[43,66]) ]
the realistic worldview (physics): monism(monism)2−−→ Newton ⃝relativity
3 → theory −−−−−−→⃝


(unsolved)
5 TOE:theory of⃝−→ everythingquantum(quantum phys.)4→ mechanics −−−−−−→⃝0⃝:Greek
Schola1philosophy −−−−→⃝
languagesticism
7 
−
−
−
−
−
→
⃝

(mind-matter
dualism)Parmenides, Zenolinguistic(Copernican revolution)

Plato


(analytic)Descartes


(=MT)

Locke,... ⃝(scientiﬁc)6mechanical 8−→ philosophy −−−−−−→⃝−→ Kant
10 QL:quantum⃝turn−→ language
ﬁctional worldview
logical worldview
quantum mechanical
worldviewdualism
9 −−−−−→⃝
statistics

system theorySocrates
Aristotle(realism)classical mechanical worldview

the idealistic worldview (no experiment): dualism
Figure 0.1: The history of the world-descriptions

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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The part of quantum theory
already studied in the following:

[

⃝
2 - ⃝
7 - ⃝
10

]

[
]
and statistics ⃝in the above were
9 - ⃝
10

(A) ref. [66]: 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 - ⃝
6 - ⃝
8 - ⃝
10 ] , which is almost
equal to the history of western philosophy. This part ⃝was not detailed
0 - ⃝
1 - ⃝
6 - ⃝
8 - ⃝
10
in my lecture, but it will be a good preparation to read (A) (since (A) may be too hard). Thatis, this paper is written such as it can be read without the mathematical preparation.
I think that the following is one of the most fundamental problems of western philosophy:
The progress problem (B)(B) Do the time series[]⃝mean the progress of philosophy?
0 - ⃝
1 - ⃝
6 - ⃝
8

In this paper I assert that
The answer (C) of the progress problem (B)(C1 ) If ”to make progress” is deﬁned by ”to come near quantum language” (i.e., ”becom[
ing more and morelikequantum
language”
we
can
say
that
the
time
series
⃝→
0
]
⃝→
⃝→
⃝→
⃝
can be regarded as progress, or more precisely,
1
6
8
10

mind-matter dualism
transcendental idealism
Copernican revolution
Plato −−−−−−−−−−−−−→ Descartes=Locke −−−−−−−−−−−−−→
Kant
Cogito turn (dualism)

progress

progresslinguistic philosophy
the quantum mechanical worldview
Saussuremechanical turn (scientiﬁc turn)linguistic turn−−−−−−−−→
Quantum language
Wittgenstein−−−−−−−−−−−−−−−−−−−−→
progress
progress (quantiﬁcation)
quantitative argument is possible(0.1)

The name of ”scientiﬁc turn” is due to the fact that quantitative argument is possible inquantum language and that it is only quantum language which scientists need in abovementioned. Also, it should be noted that quantum language is more useful than statistics.
Thus, we conclude that
(C2 ) a scientiﬁc perfection of dualism and idealism is realized by quantum language
which is surely one of the most important assertions in western philosophy. Here, in thispaper, it suﬃces to consider that ”idealism”≈”linguistic theory”≈”metaphysics” (cf. Definition 1.8)

2

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It is sure:• If anyone believes that he has[ found a theory that
10 he will certainly want
] goes beyond ⃝,to talk about the landscape ⃝seen from the theory.
0 - ⃝
1 - ⃝
6 - ⃝
8
Hence 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 arecaused by insuﬃcient understanding of idealism and dualism. Thus,• if the (C2 ) is true (i.e., if quantum language is overwhelmingly powerful compared toother immature philosophies), I can expect that most unsolved problems in dualism 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 ):(D) The list of my answers for philosophical unsolved problems• What is probability (or, measurement, causality) ? cf. Sec. 1.1.1)• Zeno paradox (Flying arrow), (cf. Sec. 2.4.2)• Zeno paradox (Achilles and a tortoise), (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. 4.3.3)• 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)• 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. 9.7.2)• What is causality? (cf. Sec. 10.3)• What is Peirce’s abduction? (cf. Sec. 11.3.1)• Five-minute hypothesis (cf. Sec. 11.4.1)• McTaggart’s paradox (cf. Sec. 11.4.2)• quantitative representation of ”Signiﬁer” and ”signiﬁed” (cf. Sec. 11.5.3)• A scientiﬁc understanding of Wittgenstein’s picture theory (cf. Sec. 11.6.2)• Wittgenstein’s paradox (cf. Sec. 11.6.3)• Flagpole problem, (cf. Sec. 11.7.1)• Hempel’s raven paradox (cf. Sec. 11.8)• the mind-body problem (i.e., How are mind and body connected?), (cf. Sec. 11.9.4)
3

KSTS/RR-20/001
November 15, 2020(♯) Also, for the solutions of unsolved problems in quantum mechanics, statistical mechanics,
statistics and probability theory, see ref. [66]). Particularly, I think that the following
two are important:• the discovery of Heisenberg’s uncertainty relation (Sec. 4.3 in ref. [66])• The clariﬁcation of the projection postulate (i.e., the wavefunction collapse) (Sec.
11.2 in ref. [66])Lastly I should add the following:• this paper can be read without the knowledge of quantum theory since the mathematicalarguments were left to ref. [66] mentioned in (A): KSTS/RR-19/003 (2019); 473 p(http://www.math.keio.ac.jp/academic/research pdf/report/2019/19003.pdf)• this paper is the revised version of ref. [61]: History of Western Philosophy from thequantum theoretical point of view [Ver. 2], Research Report (Department of mathematics,
Keio university, Yokohama), (KSTS-RR-17/004, 2017, 132 pages)(http://www.math.keio.ac.jp/academic/research pdf/report/2017/17004.pdf)
Roughly speaking, I say that(∗) this [Ver. 3] = [Ver. 2] + refs. [68, 70, 71]• My specialty is not philosophy but mathematical physics (i.e., quantum mechanics andmathematics). Thus this paper may have many mistakes and misconceptions. However, I
have a conﬁdence in only the insistence (C2 ):a scientific perfection of dualism and idealism
is realized by quantum language. Thus I hope that readers will improve this paper basedon the (C2 ).
I believe that philosophy is for amateur philosophy enthusiasts. Therefore, I have written thisbook so that readers can read it without prior knowledge. Also, readers are recommended togo reading this paper fast, and further, to advance reading [66]2 .
November 2020
Shiro Ishikawa

2

For the further information concerning quantum language, see my home page: http://www.math.keio.
ac.jp/∼ishikawa/indexe.html

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Contents
The sections containing solutions of famous unsolved problems are marked with the red *1

1 The outline 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 ?) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Linguistic Copenhagen interpretation (or in short, Linguistic interpretation) .
1.2 The history of worldview and our purpose . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Quantum language in the history of worldviews . . . . . . . . . . . . . . . . . .
1.2.2 Our purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 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 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Preview of the assertions in this paper . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Realistic worldview?, (ﬁctional, mechanical) worldview? . . . . . . . . . . . . .
1.4.2 Keywords: Monism and dualism . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Appendix: The mathematical foundations of quantum language . . . . . . . . . . . .
1.5.1 Mathematical preparations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.2 Measurement theory; Axiom 1 [Measurement] and Axiom 2 [Causality] . . . .
1.5.3 Mixed measurement theory; Axiom(m) 1 [Mixed measurement] and Axiom 2[Causality] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.4 Several results derived from the linguistic Copenhagen interpretation . . . . .

1
1
1
4
10
10
10
14
14
18
18
23
23
24
27
27
30
31
32

2 Ancient Greek philosophy (before Socrates)
37
2.1 Thalēs (BC.640 - BC.546) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.1.1 Thalēs: the ﬁrst philosopher: ” the arche (= the ﬁrst thing of all things) is water” 37
2.1.2 Thalēs’ ability at math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2 Pythagoras (BC.582 - BC.496) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2.1 The mathematical ability of Pythagorean religious organization . . . . . . . . 40
2.2.2 The arche (= the ﬁrst principle of all things) is number . . . . . . . . . . . . . 41
2.3 Hērakleitos and Parmenides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3.1 Hērakleitos(BC.540 - BC.480) . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3.2 Parmenides(BC.515 - unknown) . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3.3 Motion function method as a worldview . . . . . . . . . . . . . . . . . . . . . . 46
2.4 Zeno(BC490 - BC430): The Motion Paradox . . . . . . . . . . . . . . . . . . . . . . . 49
2.4.1 What is Zeno’s paradoxes?
Without a worldview, we cannot say anything . 49
2.4.2 * The solution⋆ about Zeno’s paradoxes (e.g., Flying arrow) in the motion function
method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

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CONTENTS
2.4.3

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

51

3 The Big Three in Greek Philosophy (Socrates, Plato)
3.1 Protagoras and Socrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Ethics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Magic proposition: I know I know nothing . . . . . . . . . . . . . . . . . . . .
3.2 Plato(BC.427 - BC.347) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 The theory of Ideas – Asserted ﬁction – . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Allegory of the cave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 The theory of anamnesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 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 . . . . .
3.4 Plato: The ﬁctional worldview (=Plato’s way of telling philosophy) . . . . . . . . . .
3.4.1 The necessity of idealism and dualism . . . . . . . . . . . . . . . . . . . . . . .
3.5 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Summary: Plato’s way of telling philosophy . . . . . . . . . . . . . . . . . . . . . . . .
3.6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55
55
55
57
59
59
62
64
65
65
66
68
68
71
71
72

4 The Big Three in Greek Philosophy (Aristotle)
4.1 Aristotle (BC.384 - BC.322) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Realistic worldview vs. idealistic worldview . . . . . . . . . . . . . . . . . . . .
4.1.2 Edios and Hyle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Why does the motion happen? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 From purpose to causality: Modern science started from the discovery of ”causality” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Practical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Aristotle’s syllogism in ordinary language . . . . . . . . . . . . . . . . . . . . .
4.3.2 Can we trust syllogism (A → B, B → C, therefore, A → C) ? . . . . . . . . .
4.3.3 * Syllogism holds in classical systems, but not in quantum systems . . . . . . .
4.3.4 Syllogism does not always hold in quantum systems . . . . . . . . . . . . . . .
4.3.5 Why does logic arise in classical QL? . . . . . . . . . . . . . . . . . . . . . . .

79
79
79
80
83

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

97
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99
99
100
102
102
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104
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105

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5.5
5.6

5.4.3 Principle of leverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Eratosthenes (BC.275 - BC.194) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 The biggest ancient observer . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Claudius Ptolemaeus (AD.83 - AD.168) . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.1 The ancient scientiﬁc collected studies . . . . . . . . . . . . . . . . . . . . . . .

106
108
108
109
109

6 The Middle Ages
113
6.1 Augustinus(AD. 354 - AD.430) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.1.1 Philosophy is a maidservant of theology . . . . . . . . . . . . . . . . . . . . . . 113
6.1.2 * ”Confessions” by Augustinus: Only the present exists . . . . . . . . . . . . . . 115
6.1.3 ”Subjective time” is a magic word which excites our delusion . . . . . . . . . . 117
6.2 Scholasticism –Graft Bamboo (=Aristotle) to a tree (=Plato) – . . . . . . . . . . . . 120
6.2.1 Aristotle’s philosophy spread to the Islamic world . . . . . . . . . . . . . . . . 120
6.2.2 Crusade expedition and Inﬂow of Islamic culture . . . . . . . . . . . . . . . . . 120
6.3 The discovery of zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.3.1 Positional notation (= the discovery of zero): Arabic numerals . . . . . . . . . 122
6.3.2 Arabic numerals and Roman numeral . . . . . . . . . . . . . . . . . . . . . . . 122
6.3.3 The explosion of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.4 The proof of the existence of God . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.4.1 Anselmus (1033 - 1109) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.4.2 Review: the worldviewism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.4.3 The inﬂow of Aristotle philosophy . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.5 Scholasticism; Problem of universals . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.5.1 * What is the problem of universals? . . . . . . . . . . . . . . . . . . . . . . . . 127
6.6 Thomas Aquinas (1225 - 1274); Scholasticism . . . . . . . . . . . . . . . . . . . . . . . 130
6.6.1 Grafting Bamboo (Aristotlelian science) to a Tree (Plato’s literature) . . . . . 130
6.7 Ockham’s razor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7 Early modern – Scientific revolution: From Geocentrism to Heliocentrism
135
7.1 Paradigm shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.2 Francis Bacon (1561 - 1626): The father of empiricism, Inductive reasoning . . . . . . 138
7.2.1 How to create science: The exclusion of idols (=prejudice, preconception) . . . 138
7.3 From Geocentrism to Heliocentrism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.3.1 What is ”Geocentrism vs. Heliocentrism”? . . . . . . . . . . . . . . . . . . . . 142
7.3.2 Somehow ”from Geocentrism to Heliocentrism” . . . . . . . . . . . . . . . . . 143
7.3.3 ”Geocentrism vs. Heliocentrism” is the problem of the worldview . . . . . . . 144
7.3.4 The Galileo legend; Leaning Tower of Pisa, Trial of Galileo . . . . . . . . . . . 145
7.4 Principia; Newtonian worldview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.4.1 Principia (1687) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.4.2 * What is ”Geocentrism vs. Heliocentrism”? After all, the worldviewism . . . . 147
8 Modern philosophy(Descartes)
149
8.1 The theory of Ideas has expired . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.2 ”I think, therefore I am” is meaningless from the scientiﬁc point of view . . . . . . . . 152
8.2.1 Discourse on the Method (1637) . . . . . . . . . . . . . . . . . . . . . . . . . . 152
8.2.2 * Two (scientiﬁc or non-scientiﬁc) interpretations of ”I think, therefore I am” . 154
8.3 Descartes’ strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.3.1 From cogito to dualistic idealism . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.4 The correspondence of key-words in Descartes philosophy and quantum language . . . 158

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CONTENTS
9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)161
9.1 Locke (1632 - 1704): The father of British Empiricism . . . . . . . . . . . . . . . . . . 162
9.1.1 ”An Essay Concerning Human Understanding” by Locke (1689) . . . . . . . . 162
9.1.2 ”Tabula rasa”, primary quality and secondary quality . . . . . . . . . . . . . . 163
9.2 Dramatic presentation of ”British Empiricism vs. Continental Rationalism” . . . . . 166
9.2.1 Would Leibniz be serious for this argument (i.e., nativism) ? . . . . . . . . . . 167
9.3* Leibniz-Clarke Correspondence: What is space-time? . . . . . . . . . . . . . . . . . . 169
9.3.1 “What is space?” and “What is time?” . . . . . . . . . . . . . . . . . . . . . . 169
9.3.2 Leibniz-Clarke Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
9.4 Subjective idealism: Berkeley, ”To be is to be perceived” . . . . . . . . . . . . . . . . 174
9.4.1 Priest: Berkeley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
9.4.2 (A3 ): To be is to be perceived . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
9.4.3 ”Einstein-Tagore Meeting” and ”Bohr-Einstein debates” . . . . . . . . . . . . 176
9.4.4 Bohr-Einstein debates: Do the laws of physics require measurement? . . . . . 177
9.5 Qualia problem and Brain in a vat argument . . . . . . . . . . . . . . . . . . . . . . . 179
9.5.1 * The problem of qualia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
9.5.2 * Brain in a vat argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
9.6 Hume; skeptic who didn’t measure, ”A Treatise of Human Nature” . . . . . . . . . . 183
9.6.1 The review of Descartes. . . . . . . . .] . . . . . . . . . . . . . . . . . . . . . . 183[
9.6.2 Hume’s straying Less than brain science ; Hume’s wordplay . . . . . . . . . . 184
9.6.3 Hume; The causality problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
9.7 Hume’s problem of induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
9.7.1 * The solution of Hume’s problem of induction . . . . . . . . . . . . . . . . . . 187
9.7.2 * Grue paradox cannot be represented in quantum language . . . . . . . . . . . . 191
10 Kant: Copernican revolution
195
10.1 ”Surely You’re Joking, Mr. Kant!”: Antinomy . . . . . . . . . . . . . . . . . . . . . . 196
10.1.1 Three Critiques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
10.1.2 Antinomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
10.2 Kant’s epistemology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
10.2.1 Kant’s compromise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
10.2.2 Thing-in-itself, Copernican revolution; from copy theory to constitution theory 200
10.2.3 ”Critique of Pure Reason (1781)” . . . . . . . . . . . . . . . . . . . . . . . . . 201
10.3* What is causality? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
10.3.1 Four answers to ”what is causality?” . . . . . . . . . . . . . . . . . . . . . . . 203
10.4 Summar; Descartes=Kant philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
10.4.1 Before Kant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
10.4.2 Is Kant a progress from Descartes ? . . . . . . . . . . . . . . . . . . . . . . . . 208
10.4.3 After Kant: Hegel and Husserl . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
11 Linguistic philosophy and quantum language
215
11.1 Linguistic philosophy, Analytic philosophy . . . . . . . . . . . . . . . . . . . . . . . . 216
11.1.1 Analytic philosophy; A philosophy without a worldview . . . . . . . . . . . . . 216
11.1.2 (A1 ), (A2 ) and (A3 ) are related? . . . . . . . . . . . . . . . . . . . . . . . . . . 218
11.2 George Boole (1815-1864) and Gottlob Frege (1848-1925) . . . . . . . . . . . . . . . . 223
11.3 Peirce (1839-1914) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
11.3.1 * What is Peirce’s abduction? . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
11.4 Bertrand Russell: ﬁve-minute hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . 232
11.4.1 * Five-minute hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

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CONTENTS
11.4.2 * McTaggart’s paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.5 Saussure: Copernican revolution in language . . . . . . . . . . . . . . . . . . . . . . .
11.5.1 Saussure’s linguistics: What comes ﬁrst, things or words? . . . . . . . . . . . .
11.5.2 Several Copernican revolutions . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.5.3 * The quantitative expressions of ”signiﬁer” and ”signiﬁed” . . . . . . . . . .
11.6 Linguistic turn: Wittgenstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.6.1 * My scientiﬁc understanding of ”Tractatus Logico-Philosophicus (=TLP)” . .
11.6.2 Which is more essential, quantum language or Wittgenstein’s picture theory?
11.6.3 * Wittgenstein’s paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.6.4 Main character Wittgenstein in Linguistic turn . . . . . . . . . . . . . . . . . .
11.6.5 Appendix: Linguistic turn ( From epistemology to quantum language) . . . .
11.7 Flagpole problem in the quantum mechanical worldview . . . . . . . . . . . . . . . . .
11.7.1 * The quantum linguistic solution of Flagpole problem . . . . . . . . . . . . . .
11.8* Hempel’s raven problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.8.1 Is ”the set of all tyrannosaurus” meaningful? : the set theoretical worldview .
11.8.2 Hempel’s raven problem in the set theoretical worldview . . . . . . . . . . . .
11.8.3 Hempel’s raven problem in the quantum mechanical worldview . . . . . . . . .
11.8.4 A priori proposition: ”Any small black bird is black” . . . . . . . . . . . . . .
11.8.5 A posteriori proposition: ”Every raven is black” . . . . . . . . . . . . . . . . .
11.9 Three approaches to the mind-body problem . . . . . . . . . . . . . . . . . . . . . . .
11.9.1 The mind-body problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.9.2 The ﬁrst approach; Cognitive scientiﬁc approach . . . . . . . . . . . . . . . . .
11.9.3 The second approach; Illusory problem? . . . . . . . . . . . . . . . . . . . . .
11.9.4 * The third approach; Quantum linguistic solution to the mind-body problem .

233
235
235
235
237
240
240
248
251
253
256
259
259
263
263
264
265
268
270
275
275
276
276
277

12 Postscript: Philosophy is One
281
12.1 Philosophy (of worldviews) has progressed towards quantum language . . . . . . . . . 281
12.2 QL has the power to solve almost all unsolved problems concerning dualistic idealism 283
12.2.1 The list of my answers of unsolved problems . . . . . . . . . . . . . . . . . . . 283
12.2.2 In the end . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286v

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Chapter 1
The outline of quantum language(=measurement theory)
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 ofthis chapter. However, if you are a graduate student, I recommend you to read this paper and”quantum language (ref. [66])” 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 (≈ analytic, linguistic philosophy),• the mechanical worldview (≈ statistics, quantum language).
as seen in Assertion 0.1 in Preface.

1.1

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

Axiom 1 (What is probability (or, measurement) ?) andAxiom 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 alanguage, 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
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1.1 The overview of quantum language(=measurement theory)is not almighty, for example, the theory of relativity is beyond the description of quantum
language.
The framework of measurement theory(=quantum language) is simple, that is, it is composed with two axioms (Axiom 1 concerning measurement and Axiom 2 concerning causal
relation) and the manual to use Axioms 1 and 2 (called the linguistic Copenhagen interpretation). That is, (cf. refs [42, 66] ),(=quantum language)[Axiom 2][Axiom 1]

measurement theory = measurement + causal relation+

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

Although it is not needed to read Axioms 1 and 2 for reading this paper, we, for completeness, 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.
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 measurementof an observable (or, by a measuring instrument) O=(X, F, F ) for a system S[ρ] i.e., a system S with a state ρ), the probability that a measured value x (∈ X) obtained by themeasurement belongs to Ξ (∈ F) is given by ρ(F (Ξ))(≡ A∗ (ρ, F (Ξ))N ).

And
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 .
5Here, 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.
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Chap. 1 The outline of quantum language (=measurement theory)
In this sense, quantum language is metaphysics. 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
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 createa language 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 thehistory of philosophy, philosophers failed to propose a suitable language. It should be noted that
Newton’s success is due to the proposal of ”the language called Newtonian mechanics”. Recall
that Newton mechanics does not answer the question ”What are mass, force, acceleration?”,but ”How should they be 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
point of view,(♯1 ) making a phirosophy called  (♯2 ) proposing a worldview called  (♯3 ) making a language called  where these are equivalent: i.e., (♯1 )≈(♯2 )≈(♯3 ).

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?”3

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1.1 The overview of quantum language(=measurement theory)
should be interpreted 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 and 2), what is important is to use Axioms 1 and 2 appropriately.
To this end, I recommend readers to read ref. [66] 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 alreadyknow how powerful it is to create a proper language as the great success of Newtonian mechanics.
In addition, 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?”

1.1.2

Linguistic Copenhagen interpretation (or in short, Linguistic
interpretation)

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), sothat, 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 needto make practice, and will master it only by trial and error.

♠Note 1.3. In Mermin’s book [85], he said
• If I were forced to sum up in one sentence what the Copenhagen interpretation says tome, it would be “Shut up and calculate”
This means that ”What is the Copenhagen interpretation? is unsolved problem. Thus, I assertthat the linguistic Copenhagen interpretation is the true Copenhagen interpretation (cf. ref.
[66] ).

However,(C3 ) if we want to master quantum language as quick as possible, we will need a good manual to use
the axioms.

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Chap. 1 The outline of quantum language (=measurement theory)
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, atﬁrst, understood in dualism (i.e., ”observer” and ”measuring object”) as illustrated in Figure 1.2.
a ⃝)”b in dualism
Figure 1.2. [Descartes Figure]: Image of ”measurement(=⃝+measuring instrument

observer(body, eye, etc.)[observable]• [measured value]
a project light⃝system(matter, measuring object)(I(=mind))[state]-b perceive the reaction⃝(i.e., the reﬂected light)

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

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

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.

5.

(matter, measuring object)

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1.1 The overview of quantum language(=measurement theory)♠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 justa physical phenomenon. Therefore, Axiom 1 (measurement) is not required. However, inthe 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 thebiggest scientiﬁc debate of the 20th century, but it’s not yet settled. My standing point is asfollows.
• 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 isalso a type of measuring instrument.
////

1.1.2.2

The linguistic Copenhagen interpretation [ (E1 )-(E7 ) ]The linguistic Copenhagen interpretation is ”the manual to use Axiom 1 and 2”. Thus, there arevarious 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 and the following (E1 )-(E7 ) in mind,
describe every phenomenon in terms of Axioms 1 and 2!
(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”. In addition, the measurement
should not be dependent on the observer. In other words, it makes no diﬀerence if I
measure it, if you measure it, or if Newton measures it.
♠Note 1.5. I’m not sure if the following analogy is appropriate, consider the followings:(♯1 ) I measure my body temperature with a thermometer.
(♯2 ) I feel my body feverish.

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Chap. 1 The outline of quantum language (=measurement theory)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 understandthat cogito proposition ”I think, therefore I am” in Chapter 8.

(E2 ) While ”matter” is in the space-time, the observer is not. Thus, the question: ”When,
where and by whom was the measured value obtained?” is out of the scope of measurement theory. Thus, words such as ”now,” ”here,” and ”I” should not be used in ascientiﬁc proposition. If you are going to use it, you need to be very careful. That is,there is no tense either in measurement theory or in science.
(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:[59] 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 notdescribed in quantum language.

(E5 ) There is no probability without measurement. Also, the measurement cannot be measured.
(E6 ) State never moves.
Schrödinger picture)Thus, we always use the Heisenberg’s picture (and not theand so on.
Since it is believed thatquantum language is the ﬁnal goal of dualistic idealism
8 in Figure 0.1 in Assertion 0.1 (in Preface), it is deduced that(cf. ⃝(E7 ) maxims of the philosophers (particularly, the dualistic idealism) can be regarded as
7

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1.1 The overview of quantum language(=measurement theory)
expressions in linguistic Copenhagen interpretation.
For example, 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.
Some people may wonder (E7 ). However, note(F1 ) Descartes=Kant philosophy and quantum language have the same purpose to establish theworldview,
and(F2 ) Descartes=Kant philosophy and quantum language have the same methodology of idealistic
dualism.
Then, it is natural to consider
maxims of philosophers ≈ the linguistic Copenhagen interpretation.
We assert the following.

Assertion 1.3. [descriptive power of quantum language](cf. ref. [66])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 isthe language of science”, we consider that(G2 ) Quantum language is the language of science,or,
to do sciences is to describe phenomena by quantum language
However, it should be noted that quantum language is not almighty, for example, 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 scientiﬁcally nonsense propositions from quantum language. As mentioned later, for example,

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Chap. 1 The outline of quantum language (=measurement theory)(♯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 isinconsistent with that the above (E1 ). Thus, cogito proposition is not a proposition in quantum
language. This will be discussed again in Proposition 8.2.
Next,(♯2 ) The hypothesis that the world was created five minutes ago (due to B. Russell in
Sec.11.4) 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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1.2 The history of worldview and our purpose

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 ofworld-description (cf. ref.[43, 66]) ]the realistic worldview (physics): monism(monism)

Socrates
Aristotle
0⃝:Greek2−−→ Newton ⃝(realism)

relativity
3→ theory −−−−−−→⃝
quantum
4→ mechanics −−−−−−→⃝

(unsolved)
theory of ⃝5−−→ everything(quantum phys.)Schola1philosophy −−−−−→⃝
sticism(mind-matter dualism)
Parmenides, Zeno
(Copernican revolution)Platolanguage
7 −−−−−−→ ⃝linguistic
(analytic)Descartes(scientiﬁc)(=MT)
Locke,...
6⃝mechanical 

8
−−→
Kant
−−→ philosophy −−−−−−−→⃝
quantum ⃝10
turn
ﬁctional worldview
logical worldview.
−−→ languagequantum mechanical
worldviewdualism
9 −−−−−−→⃝
statistics

system theoryclassical mechanical worldview

the idealistic worldview (no experiment): dualism
Figure 0.1: The history of the world-descriptions

Therefore, quantum language has the following three aspects:
The three aspects of quantum language
7 the linguistic turn of quantum mechanics
⃝:




(i.e., the true color of the Copenhagen interpretation)
8⃝: the ﬁnal goal of dualistic idealism=
(i.e., the linguistic and mechanical turn of the Descartes=Kant epistemology)
9 the dualistic reconstruction of statistics
⃝:

(1.2)

♠Note 1.8. I cannot answer the question: ”What is quauntum mechanics?”, since this is related
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.

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Chap. 1 The outline of quantum language (=measurement theory)
In this paper I assert thatHas western philosophy made progress?
Assertion 1.5. (A) If ”to make[progress” is deﬁned by ]”to come near quantum language”, we
can say that the time series ⃝→
⃝→
⃝→
⃝→
⃝
can be regarded as progress, or more
0
1
6
8
10
precisely,

Cogito turn (dualism)

Plato −−−−−−−−−−−−−→
progressmind-matter dualism
transcendental idealism
Copernican revolution
Descartes=Locke −−−−−−−−−−−−−→
Kant
progresslinguistic philosophy
the quantum mechanical worldview
Saussuremechanical turn (scientiﬁc turn)linguistic turn−−−−−−−−−−−−−−−−−−−−→
Wittgenstein
Quantum language−−−−−−−−→
progress
progressquantitative 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, we conclude 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. Definition 1.8)

It is sure:that he/she ﬁnds
(C) if someone believes
10 he/she certainly wants to talk about[] the theory beyond ⃝,the landscape ⃝
⃝
⃝
⃝
seen
from
the
theory.
0168This 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
and idealism is realized by quantum language] is true, we can expect that most unsolved problems
can be solved in the framework of quantum language.
In fact, in this paper we show the solutions of the following problems
List 1.6.
(D) The list of my answers for philosophical unsolved problems• What is probability (or, measurement, causality) ? cf. Sec. 1.1.1)• Zeno paradox (Flying arrow), (cf. Sec. 2.4.2)• Zeno paradox (Achilles and a tortoise), (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. 4.3.3)• Only the present exists (cf. Sec. 6.1.2)

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1.2 The history of worldview and our purpose• 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)• 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. 9.7.2)• What is causality? (cf. Sec. 10.3)• What is Peirce’s abduction? (cf. Sec. 11.3.1)• Five-minute hypothesis (cf. Sec. 11.4.1)• McTaggart’s paradox (cf. Sec. 11.4.2)• quantitative representation of ”Signiﬁer” and ”signiﬁed” (cf. Sec. 11.5.3)• A scientiﬁc understanding of Wittgenstein’s picture theory (cf. Sec. 11.6.2)• Wittgenstein’s paradox (cf. Sec. 11.6.3)• Flagpole problem, (cf. Sec. 11.7.1)• Hempel’s raven paradox (cf. Sec. 11.8)• the mind-body problem (i.e., How are mind and body connected?), (cf. Sec. 11.9.4)(♯) Also, for the solutions of unsolved problems in quantum mechanics, statistical mechanics,
statistics and probability theory, see ref. [66]). Particularly, I think that the following two areimportant:• the discovery of Heisenberg’s uncertainty relation (Sec. 4.3 in ref. [66])• The clariﬁcation of the projection postulate (i.e., the wavefunction collapse) (Sec. 11.2
in ref. [66])Since these problems are easily solved in quantum language, we can be convinced thatquantum language is the final goal of the dualistic idealism[⃝
0 - ⃝
1 - ⃝
6 - ⃝]
8♠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 missthe point. For example, Wittgenstein did not propose his language, but he left the maxim suchthat(♯2 ) ”The limits of my language mean the limits of my world”

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Chap. 1 The outline of quantum language (=measurement theory)
which is just the spirit of quantum language (as mentioned in Chapter 11).

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1.3 Realistic worldview and idealistic worldview

1.3

Realistic worldview and idealistic worldview

1.3.1

The worldviewism

The worldviewism isthe spirit which starts 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 greaterthan logic. 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
on a 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.
• ······♠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. 9.7.2)• Wittgenstein’s paradox (cf. Sec. 11.6.3)
////

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Chap. 1 The outline of quantum language (=measurement theory)
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] is a thing that can be expressed using these keywords)
Remark: For example, in Newtonian mechanics, ”velocity”, ”acceleration”, ”force”, etc. are exist.
Also, if we see Christianity as a kind of worldview, we say that God exists.
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 strictdeﬁnition in philosophy (not mathematics). And the more rigorous we try to say it, the more
imprecise it becomes.

As seen in Assertion 1.4, we have the following classiﬁcation of philosophies:

Classification 1.9. [(D): Classiﬁcation of worldviews]

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1.3 Realistic worldview and idealistic worldview(♭1 ) : the realistic worldview : Aristotle, Archimedes, Galileo, Newton, Einstein(monism)









(♭21 ): the ﬁctional worldview (≈ western philosophy)









Plato, Descartes, Locke, Kant







(♭22 ): the logical worldview (≈ analytic, scientiﬁc phil.)(♭2 ) : the idealistic worldview
Frege, Saussure, Russell, Wittgenstein...
(dualism)







(♭
):
the
mechanical worldview

23





statistics, quantum language
Throughout this paper, we explain that
cogitoCopernican(♭21 ): Plato −−−→ (♭21 ): Descartes −−−−−−−→ (♭21 ): Kant
turn
revolution






linguistic
mechanical→ (♭23 ): QL−−−−−→ (♭22 ): Wittgenstein −−−−−−−


turn
turn








(♭3 )Others (Thinking Tip, etc.) : Darwin’s theory of evolution, Hegel’s dialectic, etc.






 (♭ )mathematics
4////
I think mathematical logic in (♭22 ) should be classiﬁed in (♭4 ) , but for the sake of convenience, inthis text 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 the naive set theory (cf. [25])), it is naturally interpreted as ”a collection of things”.

♠Note 1.11. 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 badtheories. However, I think that at least quantum language (which is not mathematics) is a good
metaphysics. Hence, I think Kelvin is wrong.

♠Note 1.12. 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

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Chap. 1 The outline of quantum language (=measurement theory)(♯3 ) mathematical theory (including logic, set theory, category theory) has nothing to do withworldview.
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. 11, we will seethat this worldview 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 havethe idea (which may be called a logical worldview) that modal logic can be used to describe theworld. Thus, the relationship between logic and philosophy varies somewhat from researcher
to researcher. Also, it is sure that many philosophers consider that logic is one of ﬁeld ofphilosophy. 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.
In Chapter 11. we will see that the logical worldview 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

discussion

and(♭21 ) the fictional worldview (≈ western philosophy)
invisible man exists

don’t steal a glance
thereforeworldview (literary truth, pseudo- truth) −−−−−→ ethics, practical logic, discussions
main subject(ﬁctional)premiseand(♭22 ) the logical worldview (≈ Analytic philosophy)
Describe the world by symbolic logic, set theory
therefore

A → B, B → C, therefore, A → C

worldview (symbol logic, set theory) −−−−−→ calculation, discussions, properties
theorem

Axiomatic logic

and(♭23 ) the mechanical worldview (= the classical and quantum mechanical worldview)
Describe the world by (mechanical) language

worldview

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

language inspired by mechanics

discussions

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1.3 Realistic worldview and idealistic worldview

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

discussion

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,
Newtonian mechanics(F)

worldview
premise · law

Heliocentrism is true (cf. Chap. 7)
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:(G): The fictional worldview (western philosophy) (♭21 )The ﬁctional worldview ( in Plato’s way of telling philosophy) is as follows.
world is so(G0 )you should do so

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

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

Therefore,][(G1 ) world is so is secondary,[](G2 ) you should do so is main theme

For example,

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

don’t steal a glance
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.

1.3.3.2

The logical worldview (analytic philosophy) (♭22 )(I1 ): The logical worldview (axiomatic logic) (♭22 )As mentioned in Note 1.12, I think:(I1 ) the logical worldview (or, set theoretical worldview)
Describe the world by symbolic logic, set theory

understandable propositions (cf. Remark 1.10)
thereforeworldview (symbol logic, set theory) −−−−−→

scientiﬁc discussions
science, science philosophy

Axiomatic logic

Main philosophers (mentioned in this paper) are as follows.
Boole, Boole, Frege, Peirce, Russell, Wittgenstein, Hempel, Popper,...
though their positions are not unanimous.

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

AxiomsThis is the greatest achievement of mankind, which was completed by Cantor, Russell, 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.
However, I am somewhat skeptical of the logical (and set theoretical) worldview (I1 ) since ”mathematics” and ”our world” are completely diﬀerent.
But these (I1 ) and (I2 ) alone are too small for their research area. Therefore, the research area
was expanded as follows.
(I3 ) Think rationally, carefully and logically, or use words correctly! (i.e., analyze language !)(This is similar to Bacon’s assertion, cf. Sec. 7.2.) Thus, in this paper I assume that the term: ”logicalworldview” has three aspects (I1 ), (I2 ) and (I3 ).

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1.3 Realistic worldview and idealistic worldview♠Note 1.13. The term: ”logical worldview” has three aspects. (I1 ), (I2 ) and (I3 ). I think that(I2 ) is the greatest, but (I1 ) is doubtful. In Chap. 11, we will see that scientiﬁc philosophy (e.g.,
the ﬂagpole problem and Hemple’s raven paradox) highlighted the weakness of the logical (andset theoretical) worldview (I1 ). (In this sense, scientiﬁc philosophy is a good philosophy in (I1 ).)And thus, I proposed the quantum mechanical worldview (= quantum language). However, I
think that the (I3 ) may be the worst in the above if ”logically” means ”like mathematical logic(i.e., like symbolic logic)”. It should be noted that(♯1 ) Good mathematicians are not always able to consider the world logically.
(♯2 ) I. Newton could consider the world logically, in spite that he didn’t know symbolic logic.
I think that it is only in mathematics that mathematical logic is all-powerful. That is, I thinkthat Bacon=Descartes’ ”rationally” may be sounder than ”logically” in (I3 ) (i.e., analytic philosophy).

♠Note 1.14. As mentioned in Chapter 11, Dr. Hawking said in his best seller book [23]:• 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 their inquiries so much that Wittgenstein the most famous philosopher thiscentury, said ”The sole remaining task for philosophy is the analysis of language.” Whata 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 thinkthat one of 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)(1.4)
linguistic idealism♠Note 1.15. Readers should not confuse two logics (i.e., axiomatic logic and practical logic) suchthat(K1 ): axiomatic logic (i.e., mathematical logic) in (I1 ) 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.,

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Chap. 1 The outline of quantum language (=measurement theory)
world is so(A1 ) (=(A))

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

That is, ”logic” (precisely, ”practical logic”) is created by a worldview. As mentionedin (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 (e.g., modal logic,
temporal logic, etc.) were very successful as philosophy.
That is because these were powerless in solving unsolved philosophical problems, compared toquantum language (cf. List 1.6).

♠Note 1.16. The following is a common refrain among scientists:(♯1 ) In spite that philosophers like logic, they are illogical.
Now we can understand this paradox (♯1 ) as follows. That is,(♯2 ) philosophers are interested in axiomatic logic, but they don’t have much training in mastering practical logic under a certain worldview (e.g., Newtonian mechanics, statistics,etc.).
In fact, philosophers are well suited to syllogism, and scientists are good at calculating diﬀerential
equations.
Now we have the following question:(♭1 ) Why has there been such a gap between philosophers and scientists?
To answer this question may be our purpose of this text. That is, we think that(♭2 ) philosophers have no scientific worldview, but scientists have some scientificworldviews (i.e., Newtonian mechanics, statistics, etc.).
Then, we have the last question such as

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1.3 Realistic worldview and idealistic worldview(♭3 ) What kind of scientific worldview have philosophers (from Plato to Kant)
continued to desire?
We can answer this question as follows.
(♭4 ) What they have sought is a worldview on dualistic idealism., that is, quantum
language
The above will be explained throughout this text.

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 thewhole discipline. Therefore, philosophy is no longer regarded as the king of academia. the purpose of
my research is to restore philosophy through quantum language.
(M): The mechanical worldview (Statistics, quantum language) (♭23 )The mechanical worldview is as follows.
describe phenomena by the language

language is proposed like mechanics(M)

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

discussions

Therefore,][(M1 ) language is proposed , that is, ”establish the verbal system”[](M2 ) describe phenomena by the mechanical language

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

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

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 mechanical worldview] ](♭1 ) : the realistic worldview : Aristotle, Archimedes, Galileo, Newton, Einstein(monism)









(♭21 ): the ﬁctional worldview (≈ western philosophy)









Plato, Descartes, Locle, Kant







(♭
):
the
logical worldview (≈ analytic, scientiﬁc phil.)22 (♭ ) : the idealistic worldview
2Frege,
Saussure, Russell, Wittgenstein...
(dualism)







(♭ ): the mechanical worldview



 23


statistics, quantum language
Throughout
thispaper,
we
explainthat
cogitoCopernican(♭21 ): Plato −−−→ (♭21 ): Descartes −−−−−−−→ (♭21 ): Kant
turn
revolution






linguistic
mechanical→ (♭23 ): QL−−−−−→ (♭22 ): Wittgenstein −−−−−−−


turn
turn








(♭3 )Others (Thinking Tip, etc.) : Darwin’s theory of evolution, Hegel’s dialectic, etc.






 (♭ )mathematics
4////
I think mathematical logic in (♭22 ) should be classiﬁed in (♭4 ) , but for the sake of convenience, inthis text I may think of a ﬁeld of mathematics as a kind of worldview. That is because, comparedto people in science, 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
dispute  [R] vs. [L]

Realistic worldview

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

Hērakleitos⃝:
a motion
Aristotle⃝:Ancient
Greeceb⃝:c Problem of universals ”Nominalismus”(Ockham)
Newton⃝:d space-time⃝:
Einstein
e quantum theory

Parmenides
Plato”Realismus”(Anselmus)
LeibnizBohr⃝
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 de-23

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1.4 Preview of the assertions in this paper
bates(cf. ref. [66]).

1.4.2

Keywords: Monism and dualism

Recall Figure 1.2 (in Section 1.1.2: The linguistic Copenhagen interpretation) below:a ⃝)”bFigure 1.13. (Figure 1.2 in Section 1.1.2) [Descartes Figure]: Image of ”measurement(=⃝+in dualism:
dualism and monism
dualismmonismmeasuring instrument system, hyle, point massobserver(body, eye, etc.)[observable]• [measured value]a⃝projectlight(I(=mind))(matter, measuring object)[state]- [eidos]b⃝perceive
reaction

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:24

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

secondary quality

Berkeleymind

secondary quality

Kant
SaussureZadehphenomenon

recognition/

signiﬁed [signiﬁer]
fuzzy set (= membership function)

Wittgenstein

truth value[proposition]

statistics

sample ( space )quantum mechanics

measured value

quantum language

measured value[C](= matter)/[Idea world]/[universale in re]/[matter]primary quality[matter]/[God]/[thing-in-itself]/
logical space[object]
parameter[population]
state[particle]
state[system]/[ trial]observable[ measurement]
observable[ measurement][The key-words of the monism] The realistic worldview is monism, and its completed version isrealized 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 1.17. (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 the other 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 the above table.

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1.4 Preview of the assertions in this paper(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 isno ”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, inthis paper, the diﬀerence is not emphasized.
Corollary 1.15. Of the three (i.e., ”measured value”, ”observable”, ”state”), the ”observable” isthe most diﬃcult to understand. Thus, Let me summarize the history of ”observable” as follows.

•

linguistic version−−−−−−−−−−→

Idea

signiﬁer, signﬁed

Plato’s idea theorycongnitiveyversion

Saussure’s linguistic theory

body, secondary quality

membership function

Descartes, J. Locke

Zadeh’s fuzzy theory

quantiﬁcationquantiﬁcationyyidentiﬁcationjobservable (= measuring instrument)
quantum language

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

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

1.5

Appendix: The mathematical foundations of quantum language

Although we will explain ”quantum language (= measurement theory”) in this section, I may haveomitted too much. For the precise explanation, see [66]. Rather, it might be recommended to skip
this section.

1.5.1

Mathematical preparations

In this paper we are mainly interested in the spirit of quantum language, not the theoretical part of it.
For the theoretical aspect, see ref. [66]. However, I would like to write some theoretical explanations ofphilosophical problems in this paper. For this, I simply add the mathematical foundations of quantum
language in this section. Even if you skip this section, you will understand most of the text.
Following refs. [40, 42, 43, 66] (all our results until present are summarized in ref. [66]), 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 ∥F ∥B(H) = sup∥u∥H =1 ∥F u∥H ), and consider the triplet [A ⊆N ⊆ B(H)] (or, the pair [A, N]B(H) ), called a basic structure. Here, A(⊆ B(H)) is a C ∗ -algebra, and
N (A ⊆ N ⊆ B(H)) is a particular C ∗ -algebra (called a W ∗ -algebra) such that N is the weak closureof A in B(H).
The measurement theory (= ”quantum language” = ”the linguistic Copenhagen interpretation”) isclassiﬁed as follows.
 (A1 ): quantum system theory (when A = C(H))(A)
measurement theory =(= quantum language)(A2 ): classical system theory (when A = C0 (Ω))
The Hilbert space method for the mathematical foundations of quantum mechanics is essentially due
to von Neumann (cf. ref. [103]). He devoted himself to quantum (A1 ). On the other hand, in mostcases, we devote ourselves to classical (A2 ), and not (A1 ). However, the quantum (A1 ) is convenient
for us, in the sense that the idea in (A1 ) is often introduced into classical (A2 ).
When A = C(H), the C ∗ -algebra composed of all compact operators on a Hilbert space H, the (A1 )is called quantum measurement theory (or, quantum system theory), which can be regarded as the
linguistic aspect of quantum mechanics. Also, when A is commutative (that is, when A is characterizedby C0 (Ω), the C ∗ -algebra composed of all continuous complex-valued functions vanishing at inﬁnity on
a locally compact Hausdorﬀ space Ω (cf. refs. [91, 100, 108])), the (A2 ) is called classical measurement
theory.
Also, note that, when A = C(H), i.e., quantum cases,(B1 ) A∗ (= C(H)(∗ ) =)T r(H) (=trace class), N = B(H), N∗ = T r(H) (i.e., pre-dual space),
thus, T r(H) ρ, T B(H) = TrH (ρT ) (ρ ∈ T r(H), T ∈ B(H)).
Also, when A = C0 (Ω), i.e., classical cases,

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1.5 Appendix: The mathematical foundations of quantum language
2(B2 ) A∗ (= C0 (Ω)∗ ) = M(Ω) i.e., ”the space of all signed measures on Ω”, N = L∞ (Ω, ν)(⊆( B(L
) (Ω, ν))),1N∗ = L (Ω, ν), where ν is some measure on Ω (with the Borel ﬁeld BΩ , thus, L1 (Ω,ν) ρ, T L∞ (Ω,ν) =∫1∞Ω ρ(ω)T (ω)ν(dω) (ρ ∈ L (Ω, ν), T ∈ L (Ω, ν)) (cf. ref. [100]).

In most cases, we devote ourselves to a compact space Ω with a probability measure ν (i.e., ν(Ω) = 1)
and thus, C0 (Ω) is simply denoted by C(Ω).
Let A(⊆ N ⊆ 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(F )| | F ∈A such that ∥F ∥A (= ∥F ∥B(H) ) ≤ 1}. Deﬁne the mixed state ρ (∈ A∗ ) such that ∥ρ∥A∗ = 1 andρ(F ) ≥ 0 for all F ∈ A such that F ≥ 0. And deﬁne the mixed state space Sm (A∗ ) such that
Sm (A∗ )={ρ ∈ A∗ | ρ is a mixed state}.
Similarly , deﬁne the W ∗ -mixed state ρm (∈ N∗ ) such that ∥ρm ∥N∗ = 1 and ρm (F ) ≥ 0 for all F ∈ Nmsuch that F ≥ 0. And deﬁne the W ∗ -mixed state space S (N∗ ) such thatm

S (N∗ )={ρ ∈ N∗ | ρ is a W ∗ -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 ∈ Sm (A∗ ) and 0 < θ < 1” implies “ρ = ρ1 = ρ2 ”. Put
Sp (A∗ )={ρ ∈ Sm (A∗ ) | ρ is a pure state},
which is called a state space. It is well known (cf. ref. [100]) that Sp (C(H)∗ ) = {|u⟩⟨u| (i.e., theDiracnotation) | ∥u∥H = 1}, and Sp (C0 (Ω)∗ ) = {δω0 | δω0 is a point measure at ω0 ∈ Ω}, where∫∗pΩ f (ω)δω0 (dω) = f (ω0 ) (∀f ∈ C0 (Ω)). The latter implies that S (C0 (Ω) ) can be also identiﬁed withΩ (called a spectrum space or simply spectrum) such as
Sp (C0 (Ω)∗ ) ∋ δω ↔ ω ∈(state space)Ω(spectrum)

In this paper, Ω and ω(∈ Ω) is respectively called a state space and a state.
()
In Axiom 1 later ( or, more generally, in Axiom(m)( 1) ,) we need thevalueof A∗ ρ, G N , where()ρ ∈ Sm (A∗ ), G ∈ N. In quantum cases, we see that A∗ ρ, G N = T r(H) ρ, G B(H) . Thus, the value of
(
)
(
)
(
)
ρ, G N is clearly determined. However, in classical cases (i.e., A∗ ρ, G N = M(Ω) ρ, G L∞ (Ω,ν) ), weA∗
have to prepare the following deﬁnition.
(C1 ) [Essentially continuous in general cases] An element F (∈ N) is said to be essentially continuous
at ρ0 (∈ Sm (A∗ )), if there uniquely exists a complex number α such that
• if ρ (∈ N∗ , ∥ρ∥N∗ = 1, ρ ≥ 0) converges to ρ0 (∈ Sm (A∗ )) in the sense of weak∗ topologyof A∗ , that is,
ρ(G) −−→ ρ0 (G) (∀G ∈ A(⊆ N)),then ρ(F ) converges to α.
(C2 ) [Essentially continuous in quantum cases (i.e., C(H) ⊆ B(H) = B(H))] An element F (∈ B(H))
is said to be essentially continuous at |e⟩⟨e| (e ∈ H, ||e||H = 1), if there uniquely exists a complex
number α such that28

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Chap. 1 The outline of quantum language (=measurement theory)• if ρ (∈ T r(H), ∥ρ∥T r(H) = 1, ρ ≥ 0) converges to |e⟩⟨e| in the sense of weak∗ topology ofT r(H), that is,()()
ρ, G C(H) −−→ T r(H) |e⟩⟨e|, G C(H) (= ⟨e, Ge⟩H ) (∀G ∈ C(H)(⊆ B(H))),
T r(H)()
then T r(H) ρ, F B(H) converges to αω0 .

(C3 ) [Essentially continuous in classical cases] An element F (∈ L∞ (Ω, ν)) is said to be essentiallycontinuous at δω0 (∈ Sp (M(Ω))) (i.e., at ω0 (∈ Ω)), if there uniquely exists a complex numberαω0 such that
• if ρ (∈ L1 (Ω, ν), ∥ρ∥L1 (Ω,ν) = 1, ρ ≥ 0) converges to δω0 (∈ Sp (M(Ω))) in the sense of
weak∗ topology of M(Ω), that is,∫∫ρ(G)(=
G(ω)ρ(ω)ν(dω)) −−→
G(ω)δω0 (dω)(= G(ω0 )) (∀G ∈ C0 (Ω)(⊆ L∞ (Ω, ν))),
Ω

Ω

(1.6)
then ρ(F )(=∫Ω F (ω)ρ(ω)ν(dω)) converges to αω0 .

Deﬁne Fb (∈ L∞ (Ω)) such that
( if F is essentally continuos at ω) αωb
F (ω) =
F (ω) (otherwise)
Let D ∈ BΩ . Deﬁne χD ∈ L∞ (Ω, ν) such that
 1 ( if ω ∈ D)
χD (ω) =
0 (otherwise)b such that {ω ∈ Ω | χ
Then deﬁne D
bD (ω) = 1}
Assumption 1.16. (i): When we consider F (∈ L∞ (Ω, ν)), we always assume that F satisﬁes thatF = Fb.
b(ii): When we consider D(∈ BΩ ), we always assume that D satisﬁes that D = D.
The following deﬁnition is due to E.B. Davies (cf. ref. [13]).
Definition 1.17. [Observable] An observable O =(X, F, F ) in N is deﬁned as follows:(i) [σ-ﬁeld] X is a set, F(⊆ 2X ≡ P(X), the power set of X) is a σ-ﬁeld of X, that is, ”Ξ1 , Ξ2 , ... ∈F ⇒ ∪∞/ Ξ} ≡ Ξc , i.e., the complement of Ξ) ∈n=1 Ξn ∈ F”, ”Ξ ∈ F ⇒ X \ Ξ(≡ {x | x ∈ X, x ∈F”.
(ii) [Countable additivity] F is a mapping from F to N satisfying: (a): for every Ξ ∈ F, F (Ξ) is
a non-negative element in N such that 0 ≤ F (Ξ) ≤ I, (b): F (∅) = 0 and F (X) = I, where 0
and I is the 0-element and( the identity in N respectively. (c): for any countable decomposition){Ξ1 , Ξ2 , . . . , Ξn , ...} ∑of Ξ i.e., Ξ, Ξn ∈ F (n = 1, 2, 3, ...), ∪∞n=1 Ξn = Ξ, Ξi ∩ Ξj = ∅ (i ̸= j) , it∗holds that F (Ξ) = ∞n=1 F (Ξn ) in the sense of weak topology in N.
In classical case, an observable (X, F, F ) in L∞ (Ω, ν) is said to be continuous observable, if F (Ξ) is
essentially continuous on Ω (∀Ξ ∈ F).

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1.5 Appendix: The mathematical foundations of quantum language

1.5.2

Measurement theory; Axiom 1 [Measurement] and Axiom 2[Causality]

Quantum language ([35, 66]) is composed of two axioms (i.e., Axioms 1 and 2) as follows. With any
system S, a basic structure [A ⊆ N ⊆ B(H)] can be associated in which the measurement theory (A)of that system can be formulated. A state of the system S is represented by an element ρ(∈ Sp (A∗ ))and an observable is represented by an observable O =(X, F, F ) in N. Also, the( measurement of
the observable O for the)system S with the state ρ is denoted by MN (O, S[ρ] ) or more precisely,
MN (O :=(X, F, F ), S[ρ] ) . An observer can obtain a measured value x (∈ X) by the measurement
MN (O, S[ρ] ).
The Axiom 1 presented below is a kind of mathematical generalization of Born’s probabilistic
interpretation of quantum mechanics. And thus, it is a statement without reality.
Now we can present Axiom 1 in the W ∗ -algebraic formulation as follows.
Axiom 1 [ Measurement, the probabilistic interpretation of science ] (which is the same as Axiom
1 mentioned in Section 1.1)
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 MN (O :=(X, F, F ),S[ρ] ) i.e., a measurement of an observable (or, by a measuring instrument) O=(X, F, F ) for a system)
S[ρ] i.e., a system S with a state ρ) , the probability that a measured value x (∈ X) obtained by
the measurement belongs to Ξ (∈ F) is given by ρ(F (Ξ))(≡ A∗ (ρ, F (Ξ))N ) if F (Ξ) is essentiallycontinuous at ρ(∈ Sp (A∗ )).
This axiom gives answers to ”What is probability?” and ”What is measurement?”. (Recall Problem
1.1.)p
Remark 1.18. (i): In quantum( cases) (i.e., the cases that ρ ∈ S (T r(H))) ⊆ T r(H), F (Ξ) ∈ B(H)),
the probability ρ(F (Ξ))(= T r(H) ρ, T B(H) = TrH (ρT )) is always deﬁned (cf. (E1 )). That is, the F (Ξ)
is always essentially continuous at any ρ ∈ Sp (T r(H))). On the other hand, in the classical cases (i.e.,the cases that ω ∈ Ω, F (Ξ) ∈ L∞ (Ω, ν)), it is not guaranteed that F (Ξ) is essentially continuous at ω(∈ Ω)). Thus, put Fω0 = {Ξ ∈ F : F (Ξ) is essentially continuous at ω0 }. If Fω0 = F, the measurement
ML∞ (Ω,ν) (O :=(X, F, F ), S[ω0 ] ) makes the sample probability space (X, F, [F (·)](ω0 )), which is usualin statistics. Thus, roughly speaking, statistics starts from ”sample probability space”, on the other
hand, quantum language starts from ”measurement”.
(ii): In quantum cases, Axiom 1 is well known as Born’s probabilistic interpretation of quantum
mechanics. In classical case, it is natural to ask ”Why does probability appear?”. For the answer ofthis question, see Chapter 1 (Section 1.3) page 12 in ref. [66]: Example: measurement of ”Cold orHot”.
(iii): Axiom 1 is the quantitative realization of the spirit: ”there is no science without measurements”.
And, we think that Axiom 1 means the probabilistic interpretation of science since it is a kind ofmathematical generalization of Born’s probability interpretation of quantum mechanics.

Example 1.19. [(i): projective observable] An observable O = (X, F, F ) in N is called a projective
observable if it holds that F (Ξ) = F (Ξ)2 (∀Ξ ∈ F). The following exact observable is typical inprojective observables.
[(ii): Exact measurement, exact observable] Consider a basic structure [C(Ω) ⊆ L∞ (Ω, ν) ⊆ B(L2 (Ω, BΩ , ν))].
Deﬁne the exact observable Oe = (X(= Ω), F(= BΩ ), Fe ) in L∞ (Ω, ν) such that
[Fe (Ξ)](ω) = 1 (ω ∈ Ξ ∈ F),

30= 0 (ω ∈/ Ξ ∈ F).
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Chap. 1 The outline of quantum language (=measurement theory)
Let ω 0 ∈ Ω. Thus, we have the exact measurement ML∞ (Ω,ν) (Oe = (X, F, Fe ), S[ω0 ] ). Then we havethe following statement(E) Let D(⊆ X = Ω) be any open set such that ω 0 ∈ D. Then Axiom 1 says that the probability
that the measured value x(∈ X) obtained by the measurement ML∞ (Ω,ν) (Oe = (X, F, Fe ), S[ω0 ] )belongs to D is given by 1.
This implies that x = ω 0 , since D is arbitrary open set such that ω 0 ∈ D. Also, it should be
noted that Fe (Ξ) is not essentially continuous at ω 0 if ω 0 ∈ ∂(Ξ), i.e., the boundary of Ξ. Thus,(X, F, [Fe (·)](ω 0 )) is not always a probability space. However, note that there exists a probability
space (X, F, µ) such that [Fe (Ξ)](ω 0 ) = µ(Ξ) (∀Ξ(∈ F) such that ω 0 ∈/ ∂(Ξ) ), though the uniquenessis not guaranteed. The probability space (X, F, µ) is often called the sample space of the measurement
ML∞ (Ω,ν) (Oe = (X, F, Fe ), S[ω0 ] )Next, we explain Axiom 2. Let [A1 , N1 ⊆ B(H1 )] and [A2 , N2 ⊆ B(H2 )] be basic structures. A
continuous linear operator Φ1,2 : N2 (with weak∗ topology) → N1 (with weak∗ topology) is called aMarkov operator, if it satisﬁes that (i): Φ1,2 (F2 ) ≥ 0 for any non-negative element F2 in N2 , (ii):Φ1,2 (I2 ) = I1 , where Ik is the identity in Nk , (k = 1, 2). In addition to the above (i) and (ii), we
assume that Φ1,2 (A2 ) ⊆ A1 and sup{∥Φ1,2 (F2 )∥A1 | F2 ∈ A2 such that ∥F2 ∥A2 ≤ 1} = 1.
It is clear that the dual operator Φ∗1,2 : A∗1 → A∗2 satisﬁes that Φ∗1,2 (Sm (A∗1 )) ⊆ Sm (A∗2 ). If it
holds that Φ∗1,2 (Sp (A∗1 )) ⊆ Sp (A∗2 ), the Φ1,2 is said to be deterministic. If it is not deterministic, it
is said to be non-deterministic. Also note that, for any observable O2 :=(X, F, F2 ) in N2 , the (X, F,
Φ1,2 F2 ) is an observable in N1 .
Definition 1.20. [Sequential causal operator; Heisenberg picture of causality]
Let (T, ≤) be a tree
like semi-ordered set such that “t1 ≤ t3 and t2 ≤ t3 ” implies “t1 ≤ t2 or t2 ≤ t1 ”. The family {Φt1 ,t2 :Nt2 → Nt1 }(t1 ,t2 )∈T 2 is called a sequential causal operator, if it satisﬁes that
5(i) For each t (∈ T ), a basic structure [At ⊆ Nt ⊆ B(Ht )] is determined.
(ii) For each (t1 , t2 ) ∈ T52 , a causal operator Φt1 ,t2 : Nt2 → Nt1 is deﬁned such as Φt1 ,t2 Φt2 ,t3 = Φt1 ,t3(∀(t1 , t2 ), ∀(t2 , t3 ) ∈ T52 ). Here, Φt,t : Nt → Nt is the identity operator.
Now we can propose Axiom 2 (i.e., causality). (For details, see ref. [66].)
Axiom 2[Causality] (which is the same as Axiom 2 mentioned in Section 1.1)
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 .
5

When parameters t1 , t2 (t1 < t2 ) are regarded as time, we usually consider that a causal operator
Φt1 ,t2 : Nt2 → Nt1 represents ”causality”. Thus, this axiom gives an answer to ”What is causality?”.
That is, we consider that, if Axiom 2 is used in the quantum linguistic representation of a phenomenon,
causality exists in the phenomenon.

1.5.3

Mixed measurement theory; Axiom(m) 1 [Mixed measurement]
and Axiom 2 [Causality]

Measurement theory is classiﬁed as follows:31

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1.5 Appendix: The mathematical foundations of quantum language(F)measurement theory(F1 ): (pure) measurement (= Axiom 1 + Axiom 2) in Sections 1.1.1 or 1.5.2
this corresponds to ”Fisher’s statistics=
 (F2 ): mixed measurement (=Axiom(m) 1 + Axiom 2)this corresponds to ”Bayesian statistics

Let us explain the mixed measurement theory. Axiom 2 and the linguistic Copenhagen interpretation
are common in both (pure) measurement theory and mixed measurement theory. Thus, it suﬃces to
explain Axiom(m) 1 (mixed measurement) as follows.
Mixed measurement is composed of two axioms (i.e., Axiom(m) 1 and Axiom 2) as follows. With
any system S, a basic structure [A ⊆ N ⊆ B(H)] can be associated in which the mixed measurement
theory (F2 ) of that system can be formulated. A mixed state of the system S is represented by anelement ρm (∈ Sm (A∗ )) and an observable is represented by an observable O =(X, F, F ) in N. Also,
the mixed measurementof the observable O for the system S with the() mixed state ρm is denotedby MN (O, S[∗] (ρm )) or more precisely, MN (O :=(X, F, F ), S[∗] (ρm )) . An observer can obtain ameasured value x (∈ X) by the measurement MN (O, S[∗] (ρm )).
The Axiom(m) 1 presented below is a kind of mathematical generalization of Born’s probabilistic
interpretation of quantum mechanics. And thus, it is a statement without reality.
Now we can present Axiom(m) 1 in the W ∗ -algebraic formulation as follows.
Axiom(m) 1 [ Mixed measurement, the probabilistic interpretation of science ]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 an observer takes a mixed measurement
MN (O :=(X, F, F ), S[∗] (ρm )) i.e., a measurement of an observable (or, by a measuring instrument))
O=(X, F, F ) for a system S[∗] (ρm ) i.e., a system S with a mixed state ρm , the probability thata measured value x (∈ X) belongs to Ξ (∈ F) is given by ρ(F (Ξ))(≡ N∗ (ρm , F (Ξ))N ) if F (Ξ) is
essentially continuous at ρ(∈ Sm (A∗ )).

1.5.4

Several results derived from the linguistic Copenhagen interpretation

1.5.4.1

Parallel measurement

As the further explanation of parallel measurement in the linguistic Copenhagen interpretation (E4 )in Sec. 1.1.2, we have to add the following deﬁnition.
Definition 1.21. [Parallel measurement (cf. [66])] Though the parallel measurement can be deﬁned inboth classical and quantum systems, we, for simplicity, devote ourselves to classical systems as follows.
Let [C(Ω) ⊆ L∞ (Ω.ν) ⊆ [B(L2 (Ω, ν))] be a classical basic structure, where we assume, for simplicity,that Ω is compact space and ν is a measure such that ν(Ω) = 1 and ν(D) > 0 (∀open set D ⊆ Ω).
Consider a family of measurements {ML∞ (Ω,ν) (Oi := (Xi , Fi , Fi ), S[ωi ] ) | i = 1, 2, ..., N }. However,the linguistic Copenhagen interpretation (G1 ) says ”Only one measurement is permitted”.
Therefore,⊗N∞ (Ω,ν) (Oi :=instead of the family of measurements, we consider the parallel measurement
ML
i=1(Xi , Fi , Fi ), S[ωi ] ), which is deﬁned byN⊗

ML∞ (Ω,ν) (Oi := (Xi , Fi , Fi ), S[ωi ] )

i=1

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

=M

×

N

L∞ (

i=1 Ω,N( ⊗ i=1 Oi := (× Xi ,  i=1 Fi , ⊗ i=1 Fi ), S[(ω1 ,ω2 ,...,ωN )] )N⊗N

i=1 ν)

i=1

where ×i=1 Ω is the ﬁnite product compact space of Ωs,N

Also, ×N
i=1 Fi (⊆ P(

N

N

⊗N

i=1 ν is the ﬁnite product probability of νs.
N
i=1 Xi )) is the inﬁnite product σ-ﬁeld, i.e., the smallest σ-ﬁeld that includes{ N× Ξi| Ξi ∈ Fi
i=1

And further, deﬁne the observable⊗N∞
i=1 Fi in L (}×Ni=1 Ω, ⊗Ni=1 ν) which satisﬁes that

N

N

i=1

i=1

[( ⊗ i=1 Fi )((× Ξi )](ω1 , ω2 , ..., ωN ) = ×[Fi (Ξi )](ωi ) ∀Ξi ∈ Fi , ωi ∈ ΩNThen, Axiom 1 [measurement] says that(G) the probability that a measured value obtained by the parallel measurement(Oi := (Xi , Fi , Fi ), S[ωi ] ) belongs to ×continuous at ωi (∀i = 1, 2, ..., N ).

N
i=1 Ξi is given by×⊗N

i=1 ML∞ (Ω,ν)
N
i=1 [Fi (Ξi )](ωi ), if Fi (Ξi ) is essentially

Remark 1.22. The above ﬁnite parallel measurement can be generalized to the case that the index
set Λ is inﬁnite. That is,⊗

ML∞ (Ω,ν) (Oλ := (Xλ , Fλ , Fλ ), S[ωλ ] )

λ∈Λ

=M ∞

×

L (

λ∈Λ Ω,⊗λ∈Λ ν)( ⊗ λ∈Λ Oλ := ( × Xλ ,  λ∈Λ Fλ , ⊗ λ∈Λ Fλ ), S[(ωλ )λ∈Λ ] )λ∈ΛThe existence of the parallel measurement is guaranteed in both classical and quantum systems. Cf.
§ 4.2 in ref. [66]. It is not so diﬃcult to extend the above ﬁnite parallel measurements to inﬁnite
parallel measurements for mathematicians. However, in this paper, we are not concerned with theinﬁnite parallel measurement. That is because our concern is not mathematics but foundations ofphilosophy of science.

1.5.4.2

Practical logic (Implication, Syllogism)

Definition 1.23. [Image observable, marginal observable] Consider the basic structure [A ⊆ N ⊆B(H)]. And consider the observable O = (X, F, F ) in N. Let (Y, G) be a measurable space, and letΘ : X → Y be a measurable map. Then, we can deﬁne the image observable Θ(O) = (X, F, F ◦ Θ−1 )in N, where F ◦ Θ−1 is deﬁned by(F ◦ Θ−1 )(Γ) = F (Θ−1 (Γ))(∀Γ ∈ G).

Here we add the following deﬁnition.
Definition 1.24. [Implication: cf. refs. [35, 66]] Consider a basic structure:[A ⊆ N ⊆ B(H)]33

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1.5 Appendix: The mathematical foundations of quantum language
Let O1 = (X1 , F1 , F1 ) and O2 = (X2 , F2 , F2 ) be observables in N. Let O12 = (X1 × X2 , F1  F2 , F12 )be an observable such that F1 (Ξ1 ) = F12 (Ξ1 × X2 ) and F2 (Ξ2 ) = F12 (X1 × Ξ2 ) (∀Ξ1 ∈ F1 , Ξ2 ∈ F2 ).
Let ρ ∈ Sp (A∗ ), Γ1 ∈ F1 , Γ2 ∈ F2 . Then, if it holds thatρ(F 12 (Γ1 ×(X2 \ Γ2 ))) = 0
this is denoted by[O1 ; Γ1 ]=⇒MN (O12 ,S[ρ] )[O2 ; Γ2 ] or, equivalently, [O1 ; X1 \ Γ1 ]⇐=MN (O12 ,S[ρ] )[O2 ; X2 \ Γ2 ]That is because the probability that a measured value (x1 .x2 ) obtained by the measurement MN (O12 :=(X1 × X2 , F1  F2 , F12 ), S[ρ] ) belongs to Γ1 × (X2 \ Γ2 ) is equal to 0.
Remark 1.25. [Syllogism] Using Deﬁnition 1.24, we showed that
• Syllogism always holds in classical systems (cf. ref. [35])• Syllogism does not always hold in quantum systems (cf. ref. [47], § 8.7 in ref. [66])1.5.4.3

Practical logic (Inference; Fisher’s maximum likelihood method)

We begin with the following notation:
Notation 1.26. [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
(H1 ) in most cases that the measurement ML∞ (Ω,ν) (O:=(X, F, F ), S[ω0 ] ) is taken, it is usual to thinkthat the state ω0 (∈ Ω) is unknown.
That is because(H2 ) 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 1.27. [Inference; Fisher’s maximum likelihood method (cf. ref. [38] or §5.2 in ref. [66]]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 beω0 (∈ 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. [66]).
Remark 1.28. [ Inference and Control cf. §5.2 in ref. [66]] The inference problem is characterized
as the reverse problem of measurements. That is, we consider thatML∞ (Ω,ν) (O:=(X, 2X , F ), S[ω0 ] )(I1 ) (state ω0 , observable O) −−−−−−−−−−−−−−−−−−−−−−−−−→ measured value x0
measurement (Axiom 1)
On the other hand34

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Chap. 1 The outline of quantum language (=measurement theory)
ML∞ (Ω,ν) (O:=(X, 2X , F ), S[∗] )(I2 ) (measured value x0 , observable O) −−−−−−−−−−−−−−−−−−−−−−−−→ state ω0
inference (reverse Axiom 1)
Thus, (I1 ) and (I2 ) are in reverse problem.
Also, we note, from the mathematical point of view, that inference problem (I3 ) and control problem(I4 ) are essentially the same as follows.
(I3 ) [Inference problem; statistics]: when measured value x0 is obtained, infer the unknown state ω0and(I4 ) [Control problem; dynamical system theory]: Settle the state ω0 such that measured value x0
will be obtained!
Thus, we think, from the theoretical point of view, that statistics and dynamical system theory are
essentially the same. Thus, we consider that statistics (= dynamical system theory) is the mathematical representation of classical mechanical worldview. On the other hand, quantum language isregarded as the mathematical representation of quantum mechanical worldview.

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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 allthings) is ⃝⃝” is standard. Here ”⃝⃝” is, for example, as follows.
Thalēs· · · · · · water

Democritus· · · · · · atom

Pythagoras· · · · · · numberHērakleitos· · · · · · motion, ﬁreParmenides· · · · · · logic, motion

Zeno· · · · · · logic, motion

In this chapter, we ﬁrst point out that Parmenides’ claim is very similar to the Copenhagen
interpretation of quantum mechanics. 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 paradoxes is due to the lack of the worldviewism (called the
motion function method) and not the mathematical problem concerning geometric series.

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 reasonof God. Myths have been handed down from generation to generation from our ancestors byoral and written word.
There is a great deal of maritime trade, and many ethnic groups come together to interact,
but it is rare to ﬁnd a region that is not united by force of arms because it is not a vast plain.
This situation was realized on the eastern Mediterranean coast (now called Greece, Turkey,
Syria, and Israel (Egypt)) around 1000 BC. In this region, various civilizations and cultures
have merged to create a new culture. Particularly noteworthy was the realization of uniﬁcation
not by force of arms, but by culture. In other words, ”philosophy as a synthesis of several
myths” and ”the alphabet as a synthesis of several letters” were born. That is,37

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2.1 Thalēs (BC.640 - BC.546)
 integration of several myths(A)=⇒ philosophy

integration of the several characters =⇒ Phoenician alphabet =⇒ alphabet

The alphabet is a phonetic alphabet because it was created with the intention of being acommon letter between diﬀerent ethnic groups. Egypt had an advanced civilization (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 the question ”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 causedby the vibration of the waterThis may be childish, but is an explanation that does not brought out the ”God” . This is thereason why Aristotle said Thalēs was the ”the ﬁrst philosopher.”♠Note 2.1. S. Weinberg (1933 -), An American physicist, won a Nobel Prize of physics in 1979,
said in his book [105] ”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 asphysicists 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, auniversal myth) in order to avoid conﬂict between diﬀerent peoples. This paper, however, may
be more radical. As far as to describing the world (not ethics), Plato, Descartes, Kant, etc. allshould be regarded as poets or ﬁction writers.

♠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 ”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 theﬁrst philosopher”, which is the same as the spirit of this paper mentioned later.

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:38

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Chap. 2 Ancient Greek philosophy (before Socrates)
Theorem 2.1. [central angle = β, inscribed angle= α] =⇒ β = 2αβαProof. It suﬃces to draw the additional line through the center.

♠Note 2.3. When Thales visited Egypt, there is a story that the king of Egypt was impressed byThales’ measuring the height of the pyramid in the way of measuring called triangulation. But,
I think it is unreliable. Three great pyramids in the Egyptian Giza desert (deceased persons are
Khufu, Khafre, Menkaure) erecting time of is the 2500 B.C. Thus I guess that the triangulation
was known in those days (2000 years before Thalēs) in Egypt. If so, Thalēs’ theorem should be
doubt whether it is due to Thalēs. However, even as a true prover was unknown, the abilityof 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)).

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2.2 Pythagoras (BC.582 - BC.496)

2.2

Pythagoras (BC.582 - BC.496)

2.2.1

The mathematical ability of Pythagorean religious organization

Pythagoras was a leader in the mathematics study group, which may be regarded as a kindof religious organization called Pythagorean religion. As the mathematical achievements, thefollowings are known:• the discovery of irrational numbers, the Pythagorean theorem, the construction of aregular pentagon
and so on.
Theorem 2.2. Finding of irrational members, e.g.,√

2 is an irrational number.

Theorem 2.3. (Pythagorean theorem): In △ABC, the followings are equivalent:(♯1 ) ∠A = 90◦(♯2 ) AB2 + CA2 = BC2
Construction 2.4. 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 beconstructed as follows (the ﬁgure below (right)). Thus, we easily get 1+2 5 .
A
B

E

C

D

√
5

C

D

♠Note 2.4. It could have been something extra, but I wrote the explanation of the constructionof a regular pentagon for beginners. Even university students in the department of mathematics
sometimes don’t know this.

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Chap. 2 Ancient Greek philosophy (before Socrates)♠Note 2.5. The above two (the discovery of irrational numbers and the Pythagorean theorem)
are one of the most important discoveries in mathematics. If the following episodes are true, we
can trust his mathematical ability.
(♯1 ) Pythagoras was killing the disciple, who found the irrational number, in order to hide theexistence of irrational numbers.
(♯2 ) When Pythagoras discovered the Pythagorean theorem, he celebrated it, oﬀered the sacriﬁce of the bull.
The two theorems, about 2000 years from the originally discovered to the scientiﬁc revolution(Descartes’ Analytical Geometry, etc.; 17th century), not been used even once with an essential
meaning. Nevertheless, their importance had been recognized in Pythagorean organization.
This suggests a very high mathematical level of Pythagorean organization. Even if I had beena member of the Pythagorean school, I don’t think I would have been able to recognize theimportance of these two theorems as worthy of (♯1 ) and (♯2 ) above.

♠Note 2.6. (i): Pythagoras is said to have known that the earth was round. At the time,
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 andthe method of exhaustion (as the quadrature). It is said that many of his theorems are written
in Euclid’s Elements.
Even in ancient China which had the great culture, the prevailing belief was that the Earth was
ﬂat and square, while the heavens were round, until the introduction of European astronomy in
the 17th century(ii): Also, ancient Greek mathematics should be praised. That is,
Pythagorean theorem(♭2 )Pythagoras
BC582-496

method 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.

2.2.2

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

The main spirit of Pythagorean religious organization is ”the ﬁrst principle of all things isnumber”.
Now let us explain the following principle (called Pythagoreanism in this paper):41

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2.2 Pythagoras (BC.582 - BC.496)(A):Pythagoreanism

The ﬁrst principle of all things is number.
That is, ”Describe the world using mathematics!”
The phrase: ”The arche (= ﬁrst principle) of all things is ⃝⃝” is a fashion in those days. Notethat ”water”, ”ﬁre” etc. are visible, but ”number” is not.
♠Note 2.7. 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.

♠Note 2.8. As mentioned in Note 2.6, 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 was a sphere. It is natural that the pure Pythagoreanism such as(♯) The world is written in only the language of mathematics.
is not true. I believe that the world and mathematics are not directly related (though religious
people might not think so). If it can be written in only the language of mathematics, it is justmathematics. However, this pure Pythagoreanism has carried over to the present day and isinhibiting the healthy development of our worldview (e.g., see ”The theory of probability” in[66], or ”Analytic philosophy” in Chap. 11).

Hence we have the following problem (i.e., the problem of worldview), which is the main
problem in this paper:
Problem 2.5. 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, thatis, in this paper, we devote ourselves to ”measurement (Axiom 1) and causality (Axiom 2) (cf.
Sec.1.1).
♠Note 2.9. Possibly in this time, distinction of mathematics and science was not clear. Whatis mathematics? We had to wait for Cantor’s set theory and axiomatic set theory (by Zermelo
and Fraenkel) (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 scientiﬁcrevolution 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:42

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Chap. 2 Ancient Greek philosophy (before Socrates)• Plato → Descartes → Kant → Wittgenstein
However, it can be called only mysteriousness that Pythagoreanism has not blossomed in themain stream of western philosophy. As emphasized later, our assertion isdualism + mathematics −→ Quantum languagethat is, we assert that
• Pythagoreanism blossoms as quantum language in western philosophy
if the following is accepted:
Pythagoreanism• Plato → Descartes → Kant → Wittgenstein −−−−−−−−−−→ Quantum languge
quantiﬁcation

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2.3 Hērakleitos and Parmenides

2.3

Hērakleitos and Parmenides

2.3.1

Hērakleitos(BC.540 - BC.480)

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

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” as follows.
(B) ”motion” is the most fundamental key-word in science.
If so, we can relate the (B) to Parmenides.

2.3.2

Parmenides(BC.515 - unknown)

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 logicallywith reason. Even if it appears to be moving, it’s just that the human being has thesense 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). Also, in case of quantum mechanics, its object is too small, is notseen. Thus, we cannot rely on the sense, but only calculation. We can completely consent
to Parmenides’ assertion (C2 ) in case of quantum mechanics.

♠Note 2.10. Thus, I may have an opinion such that(♯) It is not too much to say that Parmenides and Kolmogorov are the founders of the Copenhagen interpretation of quantum mechanics.

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Chap. 2 Ancient Greek philosophy (before Socrates)( where Kolmogorov is the founder of the modern theory of probability (cf. ref. [76]). 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’ assertion is similar to Hērakleitos’, that is,(D) ”motion” is the most fundamental key-word in science.
The following (E1 ) and (E2 ) are my ﬁction about the diﬀerence between Hērakleitos andParmenides:(E1 ) Since Hērakleitos said ”The ﬁrst principle of all things is ﬁre”, he seems to premise therealistic 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 2.11. 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]

Realistic worldview

Idealistic worldview(monism, realism, no measurement)(dualism, idealism, measurement)⃝:
Hērakleitos
a motion⃝:AncientAristotle
Greeceb⃝:c Problem of universals ”Nominalismus”(Ockham)
Newton⃝:d space-time
Einstein⃝:
e quantum theory

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Parmenides
Plato”Realismus”(Anselmus)
Leibniz
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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 allthings) is number”, Hence,(F) Mathematics is indispensable to describe the world. However, we need words 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”,
which is assumed to be due to Parmenides in this paper though the true discoverer cannot be
speciﬁed, (cf. Note 2.14 ).
(H): (Scientiﬁc linguistic) motion function method (due to Parmenides)
Principle 2.6. [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

t
♠Note 2.12. In the above, we should note that(♯1 ) ”Moving feeling” is erased since the above graph is ﬁxed.
If it is so, as Parmenides says, we think that(♯2 ) if we devote ourselves to logic or mathematics without relying on the sense, then 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, 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. [41],or Chap. 14 in ref.[66].

The motion function method is easy, and it is usually studied in elementary school as follows.
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Chap. 2 Ancient Greek philosophy (before Socrates)
Problem 2.7. An A spot and a B spot are 1400 meters away. Amy left the A spot for a B spotby 80 m per minute. Betty left the B spot for A spot at the same time by 60 m per minute.
How many minutes later will Amy and Betty meet?
[Answer] Amy’s motion function fA : R(time axis) → R(one dimensional space axis) is deﬁned
by fA (t) = 60t, Betty’s motion function fB : R(time axis) → R(one dimensional space axis) is
deﬁned by fB (t) = 1400 − 80t, Thus, solving fA (t) = fB (t), we see
60t = 1400 − 80t then, t = 10
Hence, after 10 minutes later, the two meet.

♠Note 2.13. 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)♠Note 2.14. (a): Note that the motion function method is derived from quantum language (cf.
ref. [39, 40, 66]). 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 thecomplete understanding of the concept of ”function” is after Leibniz. However, we think thatthe 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 ”Whatis time?” and ”What is space?” (the Leibniz=Clarke correspondence (cf. Sec. 9.3.2). For thequantum linguistic answers to this questions, see ref. [66].

♠Note 2.15. 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

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2.3 Hērakleitos and Parmenides• 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 mechanicalworldview is discovered by plural persons. For example, the discoverer of the theory ofprobability (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. [66]), we want to regard quantum theory, discovered by Heisenberg, SchrödingerandBorn, as a kind of mechanical worldview (and not realistic worldview).

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Chap. 2 Ancient Greek philosophy (before Socrates)

2.4

Zeno(BC490 - BC430): The Motion Paradox

In this section, from the quantum linguistic point of view, we study Zeno’s paradoxes, theoldest paradox in science.
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?
say anything

Without a worldview, we cannot

Zeno is a disciple of Parmenides. Thus, Zeno’s Paradoxes may a collaboration between thetwo.
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 thinkthat the ﬂying arrow expresses the essence of the problem exactly and is the best masterpiecein Zeno’s paradoxes.
Now we present Zeno’s paradoxes (i.e., ﬂying arrow) as follows:
Please taste the literary pleasure.

Paradox 2.8. [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,
then motion is impossible.
The literature-like proof of [Flying arrow is not at rest]• We have to accept that an arrow passes there. However, ”to pass there” is not equivalent”to exist there”. What is ”to pass there”? ”To pass there” is both ”to exit there” and”not to exist there”. Therefore, ﬂying arrow is not at rest.

Now we can answer the question ”What is Zeno’s paradoxes?”. That is, we consider that(A) Zeno’s paradoxes say ”Don’t trust on ‘logic’ unconditionally”, that is, ”Start
from a worldview and not logic”.
since Paradox 2.8 shows that ”antinomy” happens in the logic in ordinary language. If so, we
have to obey the worldviewism in Sec. 1.3.1. that is,• First declare the worldview, and discuss the world in the worldview. Namely,49

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2.4 Zeno(BC490 - BC430): The Motion Paradox
world is soconclusion
thereforeworldview −−−−−→ discussions, calculation, practical logic, properties
premisesubject

Seeing the literature-like proof in Paradox 2.8, 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 2.9. Propose a certain worldview, in which Zeno’s paradoxes (e.g., Flying arrow)
can be discussed.
[Answer]; This is answered in Answer2.10 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 2.16. The above is not to be taken for granted. As mentioned in Chap. 11 later, thespirit of analytic philosophy is ”Start from logic”, though the great tradition of philosophy from
Plato to Kant was always based on the worldviewism (cf. Chaps. 3-10). Thus, I am somewhat
skeptical of analytic philosophy since I believe that the above (B) is the fundamental spirit ofphilosophy.

2.4.2 * The solution⋆ about Zeno’s paradoxes (e.g., Flying arrow) in
the motion function method
If we obey the motion function method (in the mechanical worldview), we can easily solve
Zeno’s paradoxes (e.g., Flying arrow) as follows.
Answer 2.10. [An answer to Problem 2.9(scientiﬁc answer)] Under the motion function method (cf. Section 2.3.3) in the mechanical worldview, we discuss ”Flying arrow” asfollows.
• Consider the motion function x(t), that is, for each time t, the position x(t) of the arrowis corresponded. It is obvious that• ”for each time t, the position x(t) of the arrow is corresponded” do not imply thatthe motion function x(t) is a constant function.
Therefore, the arrow is not necessarily at rest.

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Chap. 2 Ancient Greek philosophy (before Socrates)

6

x(t)

-

t

♠Note 2.17. Recall that we were confused in Paradox2.8[Zeno’s paradoxes]. However, we could
easily solve it in Answer 2.10. Thus we should be surprised at the power of the motion function
method. If a certain worldview is determined, Zeno’s paradoxes can be solved. Thus, the motionfunction method is not necessarily determined uniquely. For example, it is a good exercise tosolve Zeno’s paradoxes under Newtonian mechanics or the theory of relativity.

♠Note 2.18. 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 [theclassiﬁcation of philosophers]):
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





Boole, Frege, Peirce, Saussure, Russell, Wittgenstein, Hempel, Popper








(♭23 ): the mechanical worldview (statistics, quantum language) 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 the same position. Also, Pythagoras has no position in the above since we decide thatmathematics is not a kind of worldview.

2.4.3

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

The idea of associating Zeno’s paradox (”Achilles and a tortoise”) with an inﬁnite series misses
the essence of Zeno’s paradox, because Parmenides and Zeno’s interest should be a worldview.
That is, ”Flying arrow” is the most important paradox in Zeno’s paradoxes.
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2.4 Zeno(BC490 - BC430): The Motion Paradox
However, since geometric series’ method is most famous, I add ”Achilles and a tortoise” asan appendix (cf. ref. [41], or Chap. 14 in ref.[66]).
Readers should also taste the literary pleasure in the following.
Paradox 2.11. [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 bebehind 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 motionfunction method ]For example, assume that the velocity vq (= v) [resp. vs (= γv)] of the quickest [resp. slowest]
runner is equal to v(> 0) [resp. γv (0 < γ < 1)]. And further, assume that the position of thequickest [resp. slowest] runner at time t = 0 is equal to 0 [resp. a (> 0)]. Thus, we can assume
that the position q1 (t) of the quickest runner and the position q2 (t) of the slowest runner attime t (≥ 0) is respectively represented by the following motion function:{
q1 (t) = vt
(2.1)
q2 (t) = γvt + a
Thus, it suﬃces to calculate formula (2.1).
Although it can be solved by various method, I present two methods as follows(i.e., (i) or (ii)):[(i): Algebraic calculation of (2.1)]:
Solving q1 (s0 ) = q2 (s0 ), that is,vs0 = γvs0 + aaa
we get s0 = (1−γ)v, the fast runner catches up with the slow
. That is, at time s0 = (1−γ)v
runner.
[(ii): Iterative calculation of (2.1)]:
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Chap. 2 Ancient Greek philosophy (before Socrates)k)aThus, we see that tk = (1−γ(k = 0, 1, ...). Then, we have that(1−γ)v() ( (1 − γ k )a (1 − γ k+1 )a )q1 (tk ), q2 (tk ) =
,
1−γ
1−γ
( a
a )
→
,
1−γ 1−γ(2.2)

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

q1 , q6
2
a
1−γ

..
.

..
..
.

(1−γ 3 )a1−γ

q2 (t) = γvt + a?

(1−γ 2 )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)v(= s0 )tGraph: q1 (t) = vt, q2 (t) = γvt + a[(iii): Conclusion]: After all, by the above (i) or (ii), we can conclude thata(♯) the quickest runner can overtake the slowest at time s0 = (1−γ)v.

♠Note 2.19. (♭1 ): Note that the above (ii) [= the solution using a geometric series] is anothersolution of (i). Thus, there was no need to use geometric series. The point is that there is adiﬀerence in the position of whether one considers the Achilles and a turtle paradox to be amathematical problem or a philosophical problem. Many philosophers have gotten into deadends by confusing the two positions. Philosophers should have considered it as a problem ofworld description.
(♭2 ): From the philosophical point of view, ”ﬂying arrow” is deﬁnitely better than ”Achilles anda tortoise”. However, as far as quizzes go, ”Achilles and a tortoise” is well done. ”Achilles anda 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 2.20. As mentioned in Preface, the purpose of this paper is to understand the history ofwestern philosophy from the quantum linguistic point of view. Thus,

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2.4 Zeno(BC490 - BC430): The Motion Paradox(♯1 ) We aren’t interested about how Zeno considered his paradoxes.
The established theory may say that(♯2 ) 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, although in this section we solved Zeno’s
paradoxes under the motion function method, our present problem is how to characterize themotion function method in the framework of quantum language. For the answer to ”What isspace-time in quantum language?”, see Sec. 9.3.2: (Leibniz=Clarke correspondence).

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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
of western 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.

3.1

Protagoras and Socrates

3.1.1

EthicsThe 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.

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3.1 Protagoras and Socrates

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 theSophists, preached that ”man is the measure of all things”. He argued for relativism,
which holds that there is no such thing as objective truth. In other words, he argued that only
the subjective judgment of each person is essential. Since the common sense of capitalism is”the average of subjective value” = ”price”
one might say that the sophist’s claim is reasonable and modern common sense.
However, Socrates (BC.469 - BC.399) had objected to this idea. He investigated that(B) How should we live?
And, he clariﬁed the following words:(C) ”goodness”, ”happiness”, ”virtue”, ”justice”, ”courage”, ”love” · · · · · ·That is, Socrates asserted that the investigation of the above words is also the central themeof philosophy. In the following dispute:”relativism (rational sophists) ” vs. ”absolutism (a man of faith: Socrates) ”
Socrates 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 same problem (B). In this sense, we can say that
(D1 ) The mathematics of Pythagoras was unparalleled in the world. Compared to this ,Socrates was common sense and mediocre.
It is a matter of course that
(D2 ) If these words mentioned in the above (C) didn’t spread, the human race might have 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 philosophyof 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).

Hence I agree that Socrates is called the father of philosophy.

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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)♠Note 3.1. In general, Ethical philosophy does not have any truth or universality. Society ofants has ”ethics and morals of ants”, and apes must have ”ethics and morals of apes”. The
ethical philosophy of the Neanderthals must have been quite diﬀerent from ours. Therefore, ourancestors must have destroyed the Neanderthals. Communication and contact with civilizedaliens 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 thesame 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 toan established theory, read Plato’s novel which makes Socrates a main character, we may havea variety of 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 in engineering 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 ”Iknow that I know nothing” (self-referential proposition, anti-Copenhagen interpretive
proposition) as a way to end the discussion. To put it in an extreme way,(E) End the discussion by saying something like a paradox that messes with the other person’s
brain.

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3.1 Protagoras and Socrates
Recall Figure 1.2 (in Section 1.1.2: The linguistic Copenhagen interpretation) below:
Figure 3.1. (= Figure 1.2 in
a ⃝)”b in dualism”measurement(=⃝+Section

1.1.2)[Descartes

measuring instrument

observer(body, eye, etc.)[observable]• [measured value]a⃝project
light

Image

of

system(matter, measuring object)(I(=mind))

Figure]:[state]-b⃝perceive
reaction

Note that(F) the self-referential propositions (e.g., ”I know that I know nothing”, ”I think, therefore Iam”, 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” in the above self-referential propositions. Thus, the self-referential propositions aretaboo 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 ofargumentation, taboo propositions are often valid. Taboo propositions are so unclear in their
meaning that listeners 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 know that he is a sophist. This is natural. Because philosophy is a kind ofliterature.

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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
in order to support Socrates.
If the propositions such as ”Man’s life is heavier than the Earth”, ”Love is forever”, ”Love
always overcomes money”, etc. are the objective truth, the occupation of the insurance company
does not hold. However, Socrates wanted to believe so. To help Socrates, Plato proposed theoccult 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?”)from worldview (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 totell philosophy” 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”
is believed as the objective truth. That is, there exists Idea (= the true form) inheavens. 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.
That is, Plato wanted to say that”love always overcomes money” is the objective truth in the world of Idea,
therefore, ”Believe in love!”
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3.2 Plato(BC.427 - BC.347)
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 the mainstream of Western philosophy, despite the twists and turns that followed. The
theory of ideas, which was supposed to be a logistic support of ethical philosophy (Socrates),
became the mainstream of the philosophy of worldview.
(E) Our human DNA prefers logic (or reasoning) as if the philosophy of ethics were derived
from the philosophy of worldview. That is,”the world is so”, therefore ”we should live so”
namely,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
we live?” is answered from the objective point of view. However, the human brain does notoperate on logic alone. ”Logic” cannot function without some kind of sensuous common soil.
Logic alone is not enough, what is important is ”logic in a common sensory soil” (=”practical logic”).
In other word, ”without common sensory soil, logic cannot work”, therefore ”theﬁrst thing to do is to form a common sensory soil.
If so, Plato’s method 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.

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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)♠Note 3.3. Here, ”logic in the common soil of the senses” has a similar meaning to ”logic undera certain description of the world”. In a broader sense, this ”logic” is the logic usually used byordinary people, such as the logic of Newton mechanics, the logic of a court case, the logic ofpolitics, the logic of family life, the logic of insurance solicitation, and so on. One of the various
kinds of logic is the logic of mathematical logic (= symbolic logic), which is so universal thateven 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 of economics. It is natural to expect that logic under the theory of Ideas promote ourunderstanding of Socrates and Plato ethics.
As mentioned in Chap. 11, 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 (= inference)
Practical logic plays an important role in worldviews. On the other hand, symbolic logic plays
an important role in mathematics. I am skeptical of analytic philosophy, in which symbolic logic
and practical logic are confused. I fee 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 is natural to be hesitant to teach morality to others face-to-face. I think Aesop was preachingethics 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.
(ii): In the book ”Sapiens: A Brief History of Humankind” (cf. [26], 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 ago⃝

3 : the Scientiﬁc revolution, beginning a mere 500 years ago⃝
1 implies that
The above ⃝• we 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 thatmen seek. There may not be the truth.
What people want is an ”asserted ﬁction”, not the truth.
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3.2 Plato(BC.427 - BC.347)
And this is the philosophy. I think that’s what Plato would have thought.

♠Note 3.5. S. Weinberg (1933 -), a physicist at the University of Texas, Austin, won a Nobel
Prize in 1979 for work that became a cornerstone of particle physics, said in his book [105] ”To
explain the word; 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 or even (aside perhaps from Zeno) seriously to justify their speculations.
Plato’s philosophy is a representative of idealism (cf. for the deﬁnition of ”idealism”, 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. Bacon (i.e., the father of British Empiricism) of the scientiﬁc revolution (the 17th
century) that the importance of observation was, for the ﬁrst time, emphasized. See Section 7.2♠Note 3.6. In order to avoid eternal argument, Socrates invented ”Socratic method”, that is, themagic 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 init, 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 arrangement of a human DNA,
However, this idea may not be within philosophy.

3.2.2

Allegory of the cave

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

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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)
Allegory 3.3. [Allegory of the cave]• idea world
Sun

U

ﬁre

quasi-object

Nshadow•

actual world prisonerA group of prisoners are looking at the shadows on the back wall of a cave. The shadows
just are reality since they have never experienced anything other than shadows. The shadows
represent physical reality. One day, one of the prisoners gets free. Namely, he is the philosopher,
the lover of wisdom. Let’s assume he is Socrates. He turns around and the ﬁrst thing he sees
is objects of stone 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 wallof 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 think that the world outside thecave 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, just as hewas 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 undertakea similar journey. Hence, there is a possibility that the prisoners would therefore reach out andkill anyone 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?
(See Review 3.10 later)quantum language

measured value

observable/[Idea world]state[system]

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3.2 Plato(BC.427 - BC.347)

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 lessthan recalling (= anamnesis).
Remark 3.5. If we believe in the theory of anamnesis, 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 (anamnesis)

actual world

anamnesis(See Review 3.10 later)quantum language

measured value

observable/[Idea world]state[system]

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

3.3

The allegory of the sun: Measurement theoreticalaspect of Idea theory

The analogy of the sun is found in ”Republic”, written by Plato. Upon being urged by Glaucon(Plato’s elder brother) to deﬁne goodness, a cautious Socrates professes himself incapable ofdoing so. Instead he draws an analogy and oﬀers to talk about ”the child of goodness”. For
the answer to ”Why a child?”, see Note 3.8.

3.3.1

The allegory of the sun

Note that the North Star can be also regarded as a measuring instrument for orientation. With
this in 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
of Good” is all about. No matter how much you open your eyes, you cannot see anything”visible” in the ”visible” world, such as a ﬂower, a tree, or a dog, without the light of the sun.
Figure A [Sun]observer(I(=mind))•−→ ????

observersystem(matter, object)(I(=mind))•

Sun [≈ Idea of Light]]

R system(matter, object)•−→ Tree!! 

⃝


⃝






reﬂected light

In the beginning, when it’s dark as shown on the left ﬁgure, you can’t see anything, even if
your vision is normal.
However, by developing the skills to use a measuring instrument called the ”Sun [= Idea
of Light],” you can see that it is a ”tree”.
Thus, in this fable, the ternary relation (”the beholder,” ”the sun (the mediating thing),””the thing to be seen”) are clear and very easy to understand.
In the same way, things that exist in the invisible realm, such as virtue and courage,
cannot be known unless one has the skill to use the measuring instrument called the ”idea of
the good”, even if one has reason.
Figure B [Idea][Idea of Good]]
observer(I(=mind))•−→ ????

observersystem(I(=mind))(matter, object)•

R system(matter, object)•−→ Good!!

⃝


⃝






reﬂected goodIn other words, if you learn how to use a measuring instrument called the sun, you can
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3.3 The allegory of the sun: Measurement theoretical aspect of Idea theory
make the object ”visible”. In the same way, the mastery of the measuring instrument, whichis 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 ofcompass (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).

3.3.2 * The measurement theoretical understanding of the allegoryof the sun
Recall Figure 1.2 [ Descartes ﬁgure] in Sec. 1.1.2, namely,a ⃝)”b in dualism
Figure 1.2 ; [Descartes Figure]: Image of ”measurement(=⃝+observermeasuring instrument system(body, eye, etc.)[observable](I(=mind))• [measured value]a⃝(matter, measuring object)-

project light[state]b perceive the reaction⃝(i.e., the reﬂected light)

Here, it should be noted that this is essentially the same as the ﬁgure [Allegory of the sun].
Summary 3.8. We see:66

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

philosophy

science

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

science→======←philosophy

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

Descartes Figure
in Figure 1.2

in Allegory 3.6

is equal to the measurement theoretical understanding of Allegory 3.6 [Allegory of thesun].
Thus, ”Descartes ﬁgure” should have been called ”Plato ﬁgure”.

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

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

you should do sothereforeﬁctional worldview (literary truth, pseudo- truth) −−−−−→ ethics, morals(A)main subject

preface, introduction, (ﬁctional)premise, expedient

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
the theory of Ideas.
In fact, epistemology (including Kant) is, from a scientiﬁc point of view, more an allegory thana brain science.
♠Note 3.7. Kant is highly regarded on today. As seen later (Chapter 10: Kant), the reason isthat Kant understood Plato’s intention perfectly.
”Critique of Practical Reason (1788)””Critique of Judgment(1790)””Critique of Pure Reason (1781)”•

therefore−−−−−→

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

where it should be noted that Critique of Pure Reason is not scientiﬁc.

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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)
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 themodern 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
introduction·preface·ﬁctioncommon sensory soil

3.4.1.2

main subject

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 the settings 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
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 −−−−→ Quantum language
progress

progress

progress

progress(3.1)

where −−−−→ means ”getting closer to quantum language”. However, some ask the following
progressquestion:69

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3.4 Plato: The ﬁctional worldview (=Plato’s way of telling philosophy)• 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 not necessarily 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 can expect that the series (3.1) is inevitable.

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

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) areneeded. 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:
1 ⃝)”
2
in
Figure 1.2 (in Section 1.1.2); [Descartes Figure]: Image of ”measurement(=⃝+dualism[A](mind)←− [B] −→[C](matter)(medium)

measuring instrument

observer(body, eye, etc.)[observable]• [measured value]
1⃝projectlightsystem(matter, measuring object)(I(=mind))(A)
measured value ,(observer, brain, mind)[state]-2⃝perceive
reaction(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”.
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3.5 Key words of 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
next?
(B)
(A)
(C)
.
measured value , observable (= measuring instrument) ,state(observer, brain, mind)(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:\⃝:Plato:cave
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/

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
the sun), 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 havethought 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., akind of measuring instrument).
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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)♠Note 3.8. I am not conﬁdent in the correspondence ⃝3 in Review 3.10 because the theory ofideas 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 all cases 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
not satisﬁed 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. ref. [98]),• 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, thisallegory may hinder our understanding of Idea theory.
However, it is interesting that the scientiﬁc explanation of Idea theory is similar to measurement
theory. Thus, in this paper, I choose ⃝3 as mentioned in Review 3.10.

♠Note 3.10. 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 isa kind of literature.
Therefore, I think that the emphasis on the relationship to mathematics hinders our understanding of 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 passthrough this gate”
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,
the importance 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 and quantitative arguments. For example, the importance of logic is emphasized in73

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3.5 Key words of dualism
courts. There must be many unknowns in many cases. Still, the judge must judge the person.
Therefore, the impression of ”logically judged” is very important.
However, ”mathematical, logical, reasonable” must be emphasized in philosophy. That is because the diﬀerence 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 the importance of experimentation, it won’t be confused with religion.

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){y,n}(OB = ({y, n}, 2, B), S[eω(T0 )] ), where ”y” [resp. ”n” ] means ”yes” [ resp. ”no”].
1

B({y})•[B({y})](ω0 )
ω•0

Ω
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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)
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 notedthat this example is essentially the same as Deﬁnition 11.16 (”signiﬁer” and ”signiﬁed”). Thatis, 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. [35, 36, 37, 38])
Or more precisely,(D)linguistic version−−−−−−−−−→

Idea
Plato’s idea theory

signiﬁer, signﬁedSaussure’s linguistic theory

secondary quality

membership function

John Locke

Zadeh’s fuzzy theoryquantiﬁcationycongnitiveyversion


yidentiﬁcation

quantiﬁcationjobservable
quantum language

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3.6 Summary: Plato’s way of telling philosophy

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 the main theme (i.e., ethics, moral).
In other words, consider the following ﬁgure, i.e., Plato’s way of telling philosophy:
world is so

ﬁctional worldview (literary truth, pseudo- truth)(A2 )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 (asthe support of the main assertion [ethics. moral]) is a kind of literature.
(Notice)therefore(A3 ) Some consider that the term: ”−−−−−→” implies that the ﬁctional worldview shouldbe ”logical”. Here, it should be noted that the ”logical” is similar to the ”logical” ofdetective story.

If we think about it simply, the ”introduction (= ﬁctional world view)” part isunnecessary. 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 3.13. 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., theﬁctional worldview) is changeable.
Therefore, the progress of Western philosophy (which was not realized in the East) canbe realized as follows.
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Chap. 3 The Big Three in Greek Philosophy (Socrates, Plato)• 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
the theory that ethics is a rule that keeps society alive. Depending on what kind of society weenvision, 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 philosophersand 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 notuniversal. 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
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
the United States. In this sense, the world description is not logical.
On the other hand,• the realistic worldview is the world standard (moreover, the universe standard)
Aristotle−→Archimedes−→Newton−→ · · ·• The logical worldview can’t be called the world standard since it is quite inﬂuenced by
Descartes=Kant philosophy.
Aristotle −→ Frege −→ Wittgenstein−→ · · ·( I think that philosophers need not be more logical than scientists. Thus, I can’t understand why 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)”, which is more poetic than logical. )• the mechanical worldview is also the world standard:
Parmenides·Zeno−−−−−−−−−−−−−−−−−−−−−−−−−−−−→Fisher(statistics)(motion function method)

Tunnel of two thousand several hundred years

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3.6 Summary: Plato’s way of telling philosophy
And, it is a matter of course that the above two (the realistic worldview and the mechanicalworldview ) are useful. It is remarkable that the philosophy of worldview (=the ﬁctional worldview) isn’t useful at 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., asmentioned in the above remark, Christianity is too strong.

♠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
is clearly 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





Boole, Frege, Peirce, Saussure, Russell, Wittgenstein, Hempel, Popper








(♭23 ): the mechanical worldview (statistics, quantum language) Parmenides, Zeno, J. Bernoulli, statistics (e.g., Fisher), quantum language
Spirit of Pythagoras is inherited, and Parmenides and Zeno have argued establishment of theworldview as science sincerely. However, Plato used the ﬁctional worldview as a means of theprotection of Socrates(ethic philosophy) and has dwarfed the worldview in non-scientiﬁc way.
But, as Whitehead said ”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,the scientiﬁc 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.

→ 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 wascompletely diﬀerent from Plato’s philosophy. He is called the father of all sciences (≈ the fatherof the realistic worldview). It is no exaggeration to say that philosophy was started by these
two men. That is, we see: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.

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[ (≈ thefather of] the realistic worldview). He could not accept Plato’s theory of Ideas(= asserted
ﬁction ). Namely,• Philosopher Plato preferred asserted ﬁction (without experiment) to truth(with experiment)
On the other hand,• Scientist Aristotle preferred truth (with experiment) to asserted ﬁction (=without experiment)
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4.1 Aristotle (BC.384 - BC.322)♠Note 4.1. S. Weinberg (1933 -), a physicist at the University of Texas, Austin, won a Nobel
Prize in 1979 for work that became a cornerstone of particle physics, said in his book [105] ”To
explain the word; 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 besaying 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

4.1.2

progress

progress

progress

Edios and Hyle

Aristotle proposed the concepts such as ”eidos” and ”hyle” as follows.
(A): Edios(Aristotle’s Idea) and hyle
Aristotle said that
• Edios (= Aristotle’s Idea = true form) is not in the heaven, but in hyle (= matter= particle).
1 ⃝)”
2
in
Figure 1.2 (in Section 1.1.2); [Descartes Figure]: Image of ”measurement(=⃝+dualism:
dualism and monism (= Aristotle’s worldview)
dualismmonismmeasuring instrument system, hyle, point massobserver(body, eye, etc.)[observable]• [measured value]
1⃝projectlight(I(=mind))(matter, measuring object)[state]- [eidos]2⃝perceive
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Chap. 4 The Big Three in Greek Philosophy (Aristotle)

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.
• 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. [98]),• 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 ofidealism. As the following diagram shows, Plato and Aristotle are water and oil. In fact,
Scholasticism (= a compromise between Plato and Aristotle) was not successful because Platoand Aristotle do not mix.

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4.1 Aristotle (BC.384 - BC.322)

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
dispute in 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]

Realistic worldview

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

Hērakleitos⃝:
a motion
Aristotle⃝:Ancient
Greeceb⃝:c Problem of universals ”Nominalismus”(Ockham)
Newton⃝:d space-time⃝:
Einstein
e quantum theory

Parmenides
Plato”Realismus”(Anselmus)
LeibnizBohr⃝
c is more of a confusion than a dispute. ⃝d is the Leibniz=Clarke correa is my ﬁction, ⃝spondence (cf. Sec. 9.3.2), ⃝e is Bohr-Einstein debates. Quantum language is proposed as
one of answers to Bohr-Einstein debates(cf. ref. [66]).

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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 thediscovery 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
and Parmenides:• Hērakleitos(BC.540 -BC.480): ”Everything ﬂows.”



(A)
• Parmenides (born around BC. 515): ”There is no movement.”(Zeno’s teacher)
I think the reader will have the following question.
• Why are their names still there, even though it was 2,500 years ago?
As I mentioned before, ”motion and change” is the most important keyword in science (=”worldview”), that is, I consider:(B)[The beginning of World description]= [The discovery of movement and change ] = HērakleitosParmenides

This is why their names are still there.

However, Aristotle (BC384–BC322) pursued an even more fundamental problem:(C) What is the essence of movement and change?
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4.2 Why does the motion happen?
and concluded as follows.
(D):Purpose (Aristotle)
Aristotle asserted that all the movements had the ”purpose”.
• For example, a stone falls because it has the purpose to go downward, 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”
for a long period of 1500 years or more.
We were freed from the spell of ”Purpose”, only after Galileo, Bacon, Descartes, and Newton
et al. discovered the essence of movement and change lies in “Causality”.
Scientiﬁc revolution from ”Purpose” to ”Causality”
is the greatest paradigm shift in the history of science. It is not an exaggeration even if we callthe shift ”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)

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 a sense. However, in the idealistic worldview, the problem ”What is
causality?” is not solved yet. For the complete answer to the problem, we had wait for theappearance of quantum language (Axiom 2 (causal relation) in Sec.1.1.1, also, see ref. [66]).
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. 10.3: What is causality?).

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Chap. 4 The Big Three in Greek Philosophy (Aristotle)♠Note 4.3. S. Weinberg (1933 -), a physicist at the University of Texas, Austin, won a Nobel
Prize in 1979 for work that became a cornerstone of particle physics, said in his book [105] ”To
explain the word; 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 through Aristotle’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)
understand the 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 thecausal 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 (Newtonianmechanics), 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

4.3.1

Aristotle’s syllogism in ordinary language

If the ecology of various animals is observed, it will be clear that the base of language wasdue to intimidation, solidarity, reproduction. Language was one of the strongest arms for thesurvival and breeding. 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,(A) ”rhythm and song”, ”logical structure”, ”quantity concept”, ”grammar”, ”tense”, ”character”, etc.
However, it was done gradually by many people, tens of thousands of years ago, and it is notpossible to identify the names of the contributors. But the surprise that ordinary language
had a logical structure is passed down as ”Aristotle’s syllogism,” Namely,
1⃝:
Aristotle’s syllogism in ordinary language(B) Since Socrates is human being, and human being is mortal, it follows that Socrates
is mortal.
Although this is quite famous, I 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 and remained authoritative for almost 2,000 years.
In fact, Immanuel Kant said that there was nothing else to invent after the work of Aristotle.

4.3.2

Can we trust syllogism (A → B, B → C, therefore, A → C) ?

Again recall Sec. 2.4 (Zeno’s paradoxes), in which we studied the fact:(E) the logic in ordinary language cannot be trusted. That is, we must start from the worldview.
That is, our belief is the worldviewism (i.e., the worldview ﬁrst ) as follows:(F): worldviewism (cf. Sec. 1.3.1)
The worldviewism has the following form:
conclusion

world is sotherefore(F) worldview −−−−−→ discussions, calculation, practical logic, properties (motion, etc.)
premisesubjectthat is, the spirit such that ”Start from worldview”

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Chap. 4 The Big Three in Greek Philosophy (Aristotle)
As mentioned before, Zeno’s paradoxes suggest the necessity of the worldviewism. After the
proclamation of the worldview (e.g., Newtonian mechanics, the theory of relativity, quantum
language, etc.) , we have to discuss Zeno’s paradoxes (cf. Sec. 1.3.1).
Of course, the worldviewism is indispensable to the arguments ”syllogism” as well as ”motion”. That is,(G1 ) After a worldview (i.e., language system) is decided, logic is decided automatically andnaturally. On the contrary, when a worldview (i.e., language system) is not decided, logic
is not decided. That is,
the worldview takes precedence over syllogism (or logic).
In this sense, it is not exaggeration even if we say ”language = logic”. If so, we may nod that
there is the meaning of both of ”language” and ”logic” for Greek ”logos”.
After all, I think (cf. Note 1.15) that(G2 ) • Axiomatic logic is closely related to mathematics• Practical logic is closely related to worldviews
Thus, I have an opinion that philosophers should be more interested in practical logic than inaxiomatic logic.

♠Note 4.4. The above arguments may be obvious. For example, consider the worldview called
Newtonian mechanics. It is a matter of course that
• After the declaration of the worldview (=the law of Newtonian mechanics), the language
system called Newtonian mechanics is established. That is,there is no calculation without Newton’s law
Hence, under the language system, the reason (e.g., calculation, logic) is formed.
Therefore, it is obvious that the worldviews takes precedence over syllogism (or logic).

4.3.3 *

Syllogism holds in classical systems, but not in quantum
systems

We have the following theorem:
Theorem 4.3. Consider quantum language as the worldview. Then,(H1 ) in classical cases, syllogism always holds(H2 ) in quantum cases, syllogism does not necessarily hold.
[Proof ]:For classical cases (H1 ), see Sec. 4.3.5. Also, see refs. [35, 66])
For quantum cases (H2 ), see Sec. 4.3.4. Also, see refs [47, 66]).
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4.3 Practical logic♠Note 4.5. Here, we have the following question:(♯) Why are ”Zeno’s paradoxes” and ”Aristotle’s syllogism” famous? Why are those discussed
repeatedly in philosophy?
General philosophers might feel ”something which doesn’t ﬁt nicely (i.e., the neglect of theworldviewism)” in ”Zeno’s paradoxes” and ”Aristotle’s syllogism”. This feeling is transmitted
more than 2,000 years. This is our answer to the above (♯).

♠Note 4.6. 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 of the realistic worldview. As mentioned before, I am skeptical of thelogical worldview, but the following diagram is usual in philosophers:
Arithtotle −−→ Boole −−→ Frege −−→ Russell −−→ ...
Aristotle’s syllogism 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 Boole, Frege, Peirce, Saussure, Russell, Wittgenstein, Hempel, Popper








(♭23 ): the mechanical worldview (statistics, quantum language) Parmenides, Zeno, J. Bernoulli, statistics (e.g., Fisher), quantum language
In science, quantitative discussion and computation become important. In this sense, we consider that physics (or, mechanics) is located in the center of science. On the other hand, thelogical worldview is rather qualitative, and therefore, the logical worldview is rather inﬂuential
in the ﬁeld of humanities. For example, trials are an area where the importance of logic is mostemphasized. Also, I think that philosophers are people who like logic most.

4.3.4

Syllogism does not always hold in quantum systems

Proof 4.4. The proof of Theorem 4.3 (H2 );(H2 ) : [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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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 be√
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 that√1 − 12 (p1 −p2 )2 − 12 (p1 +p2 )28ϵ|c
u0 (p1 , q2 )| = |
e 8σ
|
(4.2)
2πϵσ
where uc0 is the Fourier transform of u0

p2 6

q2 6
q1 − q2 + a = 0

a

a

-

O

-

O

q1

p1
p1 + 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
~∂
i∂q1
~∂
P2 =
i∂q2

Q 1 = q1 ,P1 =Q 2 = q2 ,(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 thatthe measured value (q10 , p02 , p02 )(∈ R3 ). That is,(q10 , p02 )

p02

=⇒

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

the momentum of A2

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

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

But, the above argument (particularly, “syllogism”) is not true. That is because(♯4 ) (Q1 ⊗ P2 ) × (I ⊗ P2 ) and (I ⊗ P2 ) × (P1 ⊗ I) ( therefore, O1 and O2 ) do not commute, andthus, the simultaneous observable does not exist.
Thus, we can not test the (♯3 ) experimentally.

4.3.5

Why does logic arise in classical QL?

The close relationship between measurement and logic was ﬁrst discussed in ref. [35].
• Ishikawa,S. Fuzzy inferences by algebraic method, Fuzzy Sets and Systems 87, 181–200 (1997)
doi:10.1016/S0165-0114(96)00035-8
Or, see ref. [71] ( In preparation)
The argument in this section are regarded as a slight variation of the argument in ref. [35].

4.3.5.1

The proof of Theorem 4.3 (H1 )Recall Theorem 4.3 (H1 ), that is,(H1 ) : [Syllogism holds in classical systems], i.e., [Classical logic (i.e., ∧, ∨, ¬) holds in classicalsystems]:
Or more precisely,[In a class of classical binary projective measurements, measurement has properties
like propositional logic]:
In this section we will devote ourselves to the above proof as follows.
Proof 4.5. The proof of Theorem 4.3 (H1 ): Consider a classical basic structure:[C0 (Ω) ⊆ L∞ (Ω, ν) ⊆ B(L2 (Ω, ν)]Here, assume that Ω is a locally compact space with a Borel measure ν on Ω such that ν(D) > 0 (for any open set D (⊆ Ω)). Consider many tomatoes, that is, roughly speaking, consider T as the setof 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, the quantitative property of a tomato
t is represented by ω(t). For example, it suﬃces to consider Ω as RN (= N -dimensional real space),
where N is suﬃciently large natural number. That is,(Ω ∋ ω = ω (1) (= weight), ω (2) (= diameter), ω (3) (= diameter), ω (4) (= color value),)ω (5) (= calorie), ω (6) (= sugar content), ..., ω (N ) (= ...) ∈ RN

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Consider a binary projective observable (i.e., {1, 0}-valued projection observable, or {x1 , x0 }-valued
projection observable ) O ≡ (X, 2X , F) in L∞ (Ω, ν), where X = {1, 0} ( or, X = {x1 , x0 }) andF (Ξ) = [F (Ξ)]2 (∀Ξ ∈ F) (cf, Example 1.19).
Further, recalling Assumption 1.16, we, for the sake of simplicity, assume that {ω ∈ Ω | [F ({1})](ω) =1, a.e.}(≡ Γ) is an open set such that
Γ = [Γ]◦ ,ν(Ω \ (Γ ∪ [Γc ]◦ )) = 0

where Dc is the complement of D, i.e., Ω \ D, D =”the closure of D”, D◦ =”the interior of D)”.
Note that we can assume that Γc = [Γc ]◦ in the sense of ”almost everywhere concerning ν”. The{1, 0}-valued projective observable O ≡ (X, 2X , F) in L∞ (Ω, ν) is also denoted byOΓ ≡ (X, 2X , F Γ )

(4.4)

Ω
Γ

♠Note 4.7. (i): Someone might say that the term ”the set of all tomatoes” is ambiguous. However,for the sake of convenience, here we use the term ”the set of all tomatoes”. In Sec. 11.8 [Hempel’s raven problem], we will discuss it precisely.
(ii): If we want to both tomato’s world Ω1 and apple’s world, it suﬃces to start from the productb
space Ω1 × Ω2 . Thus, we consider the world also is represented by a large state space Ω.

4.3.5.2

Classical logic ( i.e., not, and, or, implication)

Put X = {1, 0}. Put Γ =RD, or Γ =SW in the formula (4.4). Consider a binary projective observables ORD ≡ (X(= {1, 0}), 2X , F RD ) and OSW ≡ (X, 2X , F SW ) in L∞ (Ω, ν). Consider a measurement
ML∞ (Ω,ν) (ORD , S[bω(t)] ). That is, we consider that the following three are equivalent (i.e., Axiom 1 (measurement) says that (I1 ) ⇔ (I2 ). Also, (I3 ) is the expression of (I1 ) in ordinary language):(I1 ) A measured value 1 is obtained by the measurement ML∞ (Ω,ν) (ORD , S[bω(t)] ).
( strictly speaking, the probability that a measured value 1 is obtained by the measurement
ML∞ (Ω,ν) (ORD , S[bω(t)] ) is equal to 1. )(I2 ) ωb (t) ∈ RD(≡ {ω ∈ Ω | [F RD ({1})](ω) = 1})(I3 ) A tomato t is ”red”.
Similarly, consider a measurement ML∞ (Ω,ν) (OSW , S[bω(t)] ). That is, we consider that the following three
are equivalent:(J1 ) A measured value 1 is obtained by the measurement ML∞ (Ω,ν) (OSW , S[bω(t)] ).

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4.3 Practical logic(J2 ) ωb (t) ∈ SW(≡ {ω ∈ Ω | [F SW ({1})](ω) = 1})(I3 ) A tomato t is ”sweet”.
Ω

RD
SW

[ Not]
It is clear that the following four are equivalent:(K0 ) A measured value 0 is obtained by the measurement ML∞ (Ω,ν) (OSW , S[bω(t)] ).
(K1 ) A measured value 1 is obtained by the measurement ML∞ (Ω,ν) (Θ¬ (OSW ), S[bω(t)] ) ( which is also
denoted by ¬ML∞ (Ω,ν) (OSW , S[bω(t)] ) ), where Θ¬ : {1, 0} → {1, 0} is deﬁned by Θ¬ (1) = 0,Θ¬ (0) = 1 (cf. Deﬁnition 1.23).
(K2 ) ωb (t) ∈ [RD]c (≡ {ω ∈ Ω | [F SW ({0})](ω) = 1})(K3 ) A tomato t is not ”red”.
[And]
We see that the following four are equivalent:(L0 ) A measured value (1, 1) is obtained by the measurement ML∞ (Ω,ν) (OSW × ORD , S[bω(t)] ).
(L1 ) A measured value 1 is obtained by the measurement ML∞ (Ω,ν) (Θ∧ (OSW × ORD ), S[bω(t)] ) ( whichis also denoted by ML∞ (Ω,ν) (OSW , S[bω(t)] )∧ML∞ (Ω,ν) (ORD , S[bω(t)] ) ), where Θ∧ : {1, 0}2 → {1, 0}
is deﬁned by Θ∧ (1, 1) = 1, Θ∧ (1, 0) = Θ∧ (0, 1) = Θ∧ (0, 0) = 0.
∩
(L2 ) ωb (t) ∈ RD(≡ {ω ∈ Ω | [F SW ({1})](ω) = 1}) SW(≡ {ω ∈ Ω | [F RD ({1})](ω) = 1})(L3 ) A tomato t is ”red” and ”sweet”[Or]
We see that the following four are equivalent:(M0 ) A measured value (x1 , x2 ) obtained by the measurement ML∞ (Ω,ν) (OSW × ORD , S[bω(t)] ) belongsto {(1, 1), (1, 0), (0, 1)}(M1 ) A measured value 1 is obtained by the measurement ML∞ (Ω,ν) (Θ∨ (OSW × ORD ), S[bω(t)] ) ( whichis also denoted by ML∞ (Ω,ν) (OSW , S[bω(t)] )∨ML∞ (Ω,ν) (ORD , S[bω(t)] ) ), where Θ∨ : {1, 0}2 → {1, 0}
is deﬁned by Θ∨ (1, 1) = Θ∨ (1, 0) = Θ∨ (0, 1) = 1, Θ∨ (0, 0) = 0.
∪
(M2 ) ωb (t) ∈ RD(≡ {ω ∈ Ω | [F SW ({1})](ω) = 1}) SW(≡ {ω ∈ Ω | [F RD ({1})](ω) = 1})(M3 ) A tomato t is ”red” or ”sweet”

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Chap. 4 The Big Three in Greek Philosophy (Aristotle)[Implication]
We see that the following four are equivalent:(N0 ) A measured value (x1 , x2 ) obtained by the measurement ML∞ (Ω,ν) (OSW × ORD , S[bω(t)] ) belongsto {(1, 1), (0, 1), (0.0)}(N1 ) A measured value 1 is obtained by the measurement ML∞ (Ω,ν) (Θ→ (OSW × ORD ), S[bω(t)] ) ( whichis also denoted by ML∞ (Ω,ν) (OSW , S[bω(t)] )→ML∞ (Ω,ν) (ORP , S[bω(t)] ) ), where Θ→ : {1, 0}2 → {1, 0}
is deﬁned by Θ→ (1, 1) = Θ→ (0, 1) = Θ→ (0, 0) = 1, Θ→ (1, 0) = 0.
(N2 ) ωb (t) ∈ SWc (≡ {ω ∈ Ω | [F SW ({1})](ω) = 0}) ∨ RD(≡ {ω ∈ Ω | [F RD ({1})](ω) = 1})(N3 ) A tomato t is not ”sweet”, or it is ”red”
RD
Remark 4.6. When ω1 ̸= ω2 , it should be noted that the symbol ”ML∞ (Ω,ν) (OSW , S[ω1 ] )∧ML∞⊗(Ω,ν) (O ,S[ω2 ] )” is not yet deﬁned. This should be deﬁned by the parallel ML∞ (Ω,ν)⊗L∞ (Ω,ν) (Θ∧ (OSW ORD ),S[(ω1 ,ω2 )] ). More generally, the symbol ”∧λ∈Λ ML∞ (Ωλ ,νλ ) (OΓλ , S[ωλ ] )” is deﬁned by⊗M⊗λ∈Λ L∞ (Ωλ ,νλ ) (OΓλ , S[(ωλ )λ∈Λ ] )λ∈Λ

It might have been better to use parallel measurements ( and not simultaneous measurements ) all
the time to avoid confusion.
Remark 4.7. Note that propositional logic (i.e., ¬, ∧, ∨, →) and predict logic (i.e., ¬, ∧, ∨, →, ∀,∃) are essentially the same since P1 ∧ P2 ∧ P3 ∧ ... = (∀n)[Pn ] and P1 ∨ P2 ∨ P3 ∨ ... = (∃n)[Pn ]. Thus,
this paper distinguishes propositional logic and predict logic.

4.3.5.3

Syllogism

Further, consider a measurement ML∞ (Ω,ν) (ORP , S[bω(t)] ). That is, we consider that the following three
are equivalent:(O1 ) A measured value 1 is obtained by the measurement ML∞ (Ω,ν) (ORP , S[bω(t)] ).
(O2 ) ωb (t) ∈ RP(≡ {ω ∈ Ω | [F RP ({1})](ω) = 1})(O3 ) A tomato t is ”ripe”.

Theorem 4.8. [Syllogism]:b (t)(∈ Ω) be the state of t. Assume the followings:
Let t be a tomato, and let ω(P0 ) A measured value (x1 , x2 ) obtained by the measurement ML∞ (Ω,ν) (OSW × ORP , S[bω(t)] )belongs to {(1, 1), (0, 1), (0.0)}
which is equivalent to(P1 ) ML∞ (Ω,ν) (OSW , S[bω(t)] )→ML∞ (Ω,ν) (ORP , S[bω(t)] )b (t) ∈ RP(≡ {ω ∈ Ω | [F RP ({1})](ω) =(P2 ) ωb (t) ∈/ SW(≡ {ω ∈ Ω | [F SW ({1})](ω) = 1}) ∨ ω
1})(P3 ) A tomato t is not ”sweet”, or it is ”ripe”.
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4.3 Practical logic
and(P′0 ) A measured value (x2 , x3 ) obtained by the measurement ML∞ (Ω,ν) (ORP × ORD , S[bω(t)] )belongs to {(1, 1), (0, 1), (0.0)}
which is equivalent to(p′1 ) ML∞ (Ω,ν) (ORP , S[bω(t)] )→ML∞ (Ω,ν) (ORD , S[bω(t)] )which is equivalent tob (t) ∈ RD(≡ {ω ∈ Ω | [F RD ({1})](ω) =(P′2 ) ωb (t) ∈/ RP(≡ {ω ∈ Ω | [F RP ({1})](ω) = 1}) ∨ ω
1})(P′3 ) A tomato t is not ”ripe”, or it is ”red”.
Then the following holds:(Q0 ) A measured value (x1 , x3 ) obtained by the measurement ML∞ (Ω,ν) (OSW × ORD , S[bω(t)] )belongs to {(1, 1), (0, 1), (0.0)}(Q1 ) ML∞ (Ω,ν) (OSW , S[bω(t)] )→ML∞ (Ω,ν) (ORD , S[bω(t)] )b (t) ∈ RD(≡ {ω ∈ Ω | [F RP ({1})](ω) =(Q2 ) ωb (t) ∈/ SW(≡ {ω ∈ Ω | [F RD ({1})](ω) = 1}) ∨ ω
1})(Q3 ) A tomato t is not ”sweet”, or it is ”red”.

[Proof ]: Recalling the linguistic Copenhagen interpretation ”Only one measurement is permitted” ( in Sec. 1.1.2), we have enough to see the simultaneous observable OSW × ORP × ORD ,which uniquely exists (cf. Proposition 4 (i) ). Thus, we have the measurement ML∞ (Ω,ν) (OSW ×
ORD × ORP , S[bω(t)] ). Let (x1 , x2 , x3 ) be the measured value. We easily see that (x1 , x3 ) belongsto {(1, 1), (0, 1), (0.0)}. Thus, (Q) holds.
4.3.5.4

Elementary binary projective observables

Consider the state space Ω, which is ﬁnite ( or, countable ) with a metric d (i.e., d(ω1 , ω2 ) =1 (ω1 ̸= ω2 ), = 0 (ω1 = ω2 ). Further, assume that the Borel measure ν is deﬁned by the point
measure, i.e., ν({ω}) = 1 (∀ω ∈ Ω).
Deﬁnition 4.9. Let λ be any element of Ω. Putting Γ = {λ} in the formula (4.4), deﬁne theelementary binary projective observable O{λ} = (X(= {1, 0}), 2X , F {λ} ) in L∞ (Ω, ν) such that{
1 (if ω = λ){λ}[F ({1})](ω) =,[F {λ} ({0})](ω) = 1 − [F{λ} ({1})](ω)
0 (if ω ̸= λ)(∀ω ∈ Ω)
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Chap. 4 The Big Three in Greek Philosophy (Aristotle)
The measurement ML∞ (Ω,ν) (O{λ} , S[ω] ) (λ, ω ∈ Ω) is called an elementary measurement.
////
It is clear that it holds that
(R1 ) A measured value 1 is obtained by the elementary measurement ML∞ (Ω,ν) (O{λ} , S[ω] )⇐⇒ λ = ω(R2 ) A measured value 1 is obtained by the elementary measurement ML∞ (Ω,ν) (O{λ} , S[ω] )⇐⇒ λ ̸= ω
Theorem 4.10. Let Γ be a subset of Ω. And let ω ∈ Ω. Then we see thatML∞ (Ω,ν) (OΓ , S[ω] ) = ∨λ∈Γ ML∞ (Ω,ν) (O{λ} , S[ω] )♠Note 4.8. Many readers may not consider the above theorem to be particularly important. I havethe same view. However, the theorem was prepared in preparation for Sec. 11.6.1. Of course, thespirit of expressing complex observables in simple observables is quite important. In quantum
language, this spirit is realized by von Neumann’s spectral decomposition theorem ( and Holevo’s
theorem), that is, ”Any observable can be composed of binary projective observables”, cf. [29,
103].

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Chapter 5
Around Alexandria; Hellenistic period
Wisdom of pyramid building for thousands of years was accumulated by Egypt (Alexandria).
Bright people studied in Egypt from each place of the Mediterranean Sea coast to learn it. For
example,Euclid· · · geometry

Aristarchus· · · Heliocentrism

Archimedes· · · buoyancy, lever

Eratosthenes· · · the measurement of the earth

Ptolemaeus· · · Geocentrism

5.1

Around Alexandria; Hellenistic period

Inﬂuenced by the tradition of pyramid construction engineering, the studies of the Alexandrian
school are solid and scientiﬁc. Put diﬀerently, it could be said that there was no philosophicalappeal 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

−−−→

v.s. Sophists···−−−→ Scholasticism
Middle Ages

Many Greek philosophers were not willing to put their theories to work. But, in Hellenistic
period, Practical research was respected.
In Section 5.3, I will explain Heliocentrism of Aristarchus a little in detail as the preparationof Chap. 6, in which we say that97

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5.1 Around Alexandria; Hellenistic period• ”Geocentrism vs. Heliocentrism” is a metaphysical problem that cannot be put on blackand-white in the experiment. That is,(♯) ”Geocentrism vs. Heliocentrism” is a philosophical problem, and not the problemof truth or falsehood.

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Chap. 5 Around Alexandria; Hellenistic period

5.2

Euclid(BC.330 - BC.275)

5.2.1

Euclid geometry - Parallel postulate

Three great pyramids in the Egyptian Giza desert (deceased person is Khufu, Khafre, Menkaure)erecting time of is the 2500 BC. Since then more than 2,000 years later, Euclid (BC.330 BC.275) was born. Euclid is referred to as the ”father of geometry” who was active in Alexandria (the mouth of the Nile). His book ”Elements” is one of the most inﬂuential works in thehistory of mathematics. It has been estimated to be second only to the Bible in the number ofeditions published since the ﬁrst printing in 1482 AD. When I think from now,(A) Euclid advocated geometric axiomatization and considered the parallel postulate, andwas 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 thesame side that sum 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 • exstsβα•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.

♠Note 5.1. In ”Elements”, geometry is not only written but also algebra. For example, it isshown that prime numbers are inﬁnite. The proof is as follows.

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5.2 Euclid(BC.330 - BC.275)(♯) Assume that the set of prime numbers is ﬁnite, that is, {2, 3, 5, 7, ..., n}. Put
N = (2 × 3 × 5 × 7 × ... × n) + 1
Then, N is a prime number or it can be divided by the larger prime number than n. In
each case, it contradicts the assumption that n is the largest prime number.

5.2.2

non-Euclidean revolution

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

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

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Chap. 5 Around Alexandria; Hellenistic period

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

Then, we want to doubt axiom of choice, but a description of the mathematics largely
decreases when I do not accept axiom of choice. Hence, we usually accept axiom of choice.
Remark 5.1. I have an opinion that the main streem of Western philosophy is

Plato



dualism
Find the axiomatic approach to dualism=⇒Descartes-Kant philosophy−−−−−−−−−−−−→ ?
not yet completed
Euclidaxiomatc approach
If it were not for Euclid’s ”Elements” and the Bible, Western philosophy would be just like
Eastern philosophy.

♠Note 5.2. There was also tradition of pyramid construction, and Egypt was an advanced countryof mathematics. Pythagoras and Archimedes also learned geometry in Egypt. Then Alexandria
was an academic city as there was Alexandria library having 700,000 collection of books. After
Euclid, we know that
• Eratosthenes (BC.275 - BC.194) : He was determined to 46250km the whole circumferenceof the 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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5.3 Aristarchus (BC.310 - BC.230)

5.3

Aristarchus (BC.310 - BC.230)

5.3.1

the diameter of the moon : the diameter of the sun

Aristarchus (BC.310 - BC.230) was an ancient Greek astronomer and mathematician who
presented Heliocentrism. He calculated as follows.
Proposition 5.2.
(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 saysthat 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 to consider thatthe diameter of the earth ≈ the diameter of the earth’s shadow
Hence, measuring by eye, we see (A1 ).
The answer to (A2 ): Look at the lower left ﬁgure (the ﬁrst quarter moon). Note thatcos 87◦ ≈ 1/19. And using the fact that The sun and the moon are seen as the same size, we
can calculate:the diameter of the moon
the distance between the moon and the earth
1== cos 87◦ ≈the diameter of the sun
the distance between the sun and the earth
19
earth

lunar eclipse

earth’sshadow

87◦

sunﬁrst quarter moon

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 goesaround the small earth. It is sure that the small earth goes around the big sun.
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That is,(B2 ) Aristarchus proposed Heliocentrism
His argument is almost complete since the diﬀerence between ”the volume” and ”the mass” istrivial.
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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5.4 Archimedes (BC.287 - BC.212)

Archimedes (BC.287 - BC.212)

5.4

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 theweight 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 − [buoyacy]/g
where (g: gravitational constant,[buoyacy] =(5.1)

the shape of this matter is assumed a cone.)[]
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

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

f1

? ? ?
-



-



-

⇑ h



- buoyacy



-



S

(the density of water is ρ(= 1).)

6 6 6

f2

♠Note 5.3. A famous anecdote of the golden crown is the delicate anecdote that there is notconnected with Archimedes’ principle. In like there is a relationship, I try to write this in what
follows.

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Chap. 5 Around Alexandria; Hellenistic period• 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 aroundthe streets naked without even wearing clothes.
He found:(♯) Preparing the gold bullion of the weight same as the crown, compare the weight the goldbullion and the weight is the crown in water. Then, we can, by the (5.1), compare thevolume of the gold 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 10.7)(♯6 ) Wttgenstein· · · · · · primary school teacher, Guardian: Russell (cf. Sec.11.6.1)(♯7 ) Einstein· · · · · · Elevator(♯8 ) quantum mechanics· · · · · · Heisenberg’s uncertainty principle (cf. Note 4.1 of ref. [66])Here, the (♯8 ) is my opinion.

5.4.2

The tomb of Archimedes

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

4πr3
,
3

The surface area of the ball B = 4πr2

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

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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 the world”. In spite that he referred Aristarchus’ Heliocentrism in his book: ”TheSand 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 revolvearound 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
understand them, and he did not say ambiguous and unintelligible things like philosophy (Plato,etc.). This clarity is a factor in Archimedes’ popularity. As I have mentioned before, his words
pierce our hearts,• Heureka!”(=”I have found it!”)• Give me a lever long enough and a fulcrum on which to place it, and I shall move 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, sothey instructed him not to do any harm. However, when Archimedes was thinking about writing
a ﬁgure on the sand, he was almost taken away by the 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 ancient scientists” until the end of his life.

♠Note 5.6. Note that
• Archimedes did not speak ambiguous things like Plato’s philosophy.

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Therefore the work of Archimedes is quantitative, clear and easy to understand. Since politicalpower could interpret the vague philosophy conveniently, philosophy could inﬂuence to maintainharmony 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 asphere. Aristotle deduced that the lunar eclipse was the shadow of the Earth and believed that
the Earth was a sphere. When you look at the ocean in the distance, it looks like an arc, so
there were probably people who believed that the earth was a 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).
Eratosthenes measured the whole circumference of the earth as follows.
NP
A
θ

S

θ
sun light

SP

NP:North pole, SP:South pole, A:Alexandria, S:Syene(=Aswan)• Syene is on the tropic of cancer, thus, the sun is seen in right above at noon on thesummer solstice.
• Aswan is located just south of Alexandria. The distance =AS=925km.
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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Chap. 5 Around Alexandria; Hellenistic period

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”, he adopted Aristotle’s Geocentrism (i.e., the sun goes around theearth). 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 observationand experiment as important most.

•Saturn•Sun•Saturn•VenusMoon•• •Mercury •

Earth

•
Moon

Jupiter•Earth•Venus• •Mercury •Jupiter

Sun

•Mars

•Mars

Geocentrism (=Ptolemaic system)

Heliocentrism (=Copernican system)

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5.6 Claudius Ptolemaeus (AD.83 - AD.168)♠Note 5.8. S. Weinberg (1933 -), a physicist at the University of Texas, Austin, won a Nobel
Prize in 1979 for work that became a cornerstone of particle physics, said of his book [105] ”To
explain the word; The discovery of modern science” as follows:• [in Chapter 8]: In one respect the work on this theory described in the Almagest is strikingly
modern in its methods. Mathematical models are proposed for planetary motions containing
various free numerical parameters, which are then found by constraining the predictions of
the models to agree with observation.
Decades ago, when I saw the planetarium when I was a kid, a commentator explained that
”Ptolemaeus asserted a foolish Geocentrism, and this was corrected by Copernicus.” However,when I went to the planetarium a few months ago, a commentator praised Ptolemaeus, saying,”His epicycle model is groundbreaking at the time.” It was great to hear a similar opinion to
Weinberg’s at a planetarium in the Far East island nation.

♠Note 5.9. Recall the following ﬁgure:poem

science

The arche is ⃝⃝−−−−→Socrates−−−−−→

Aristotle −−−−→ Hellenistic periodxstudent

student

Plato

−−−−→

···

−−−−→ Scholasticism
Middle Ages

v.s. Sophists

That is, Aristotle bridged the gap between this pseudoscience (≈ the arche is ⃝⃝) and thebase for the foundation that would become science. Archimedes is the discoverer of ”principleof 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 classiﬁcationof 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Boole, Frege, Peirce, Saussure, Russell, Wittgenstein, Hempel, 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 2.6):
geocentrism 1

the earth is round(C1 ) [astronomy]:

Pythagoras
BC582-496−→

EudoxusBC400-347

110

geocentrism 2−→

geocentrism 3

Aristotle−→ Ptolemaeus

BC384-322

AD83-168

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Pythagorean theorem(C2 ) [math]:

Pythagoras
BC582-496

method ofexhaustion−→

EudoxusBC400-347

111−→

Elements

quadrature

Euclid−→ Archimedes

BC330-275

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): Occam’s razor, ”Nominalismus” in Problem of universals

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 ofChristianity.
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6.1 Augustinus(AD. 354 - AD.430)(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 and Christianity have an aﬃnity. In other words, there are reasons tothink 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 to earth) .
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 was celebrated 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 maidservantof 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 toAugustine, we get the following

dualism \ three key-words[A](= mind)[B](Mediating of A and C)⃝:
3 Plato: sunRemark 3.7

actual world

Idea

Augustinus; Christianity

earthly city

churchquantum language

measured value

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

progress

progress

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Chap. 6 The Middle Ages
but we do not stick to this since Christianity is not related to a measurement theory.
It is unclear whether the Christian fathers (such as Augustine) took Plato’s philosophy seriously
or not. 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 )therefore

Live soﬁctional 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 ”descriptionof the world” in the introduction part can be any fairy tale or ﬁction. However, as I explainedin 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 into a solid theory.
It is the purpose of this paper to show this. Speciﬁcally, we show that
• Plato → Descartes → Rocke → Kant → Wittgenstein → Quantum languge
scientiﬁc theory

6.1.2 * ”Confessions” by Augustinus: Only the present exists
This section is written in the following reference:• [68]: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)
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6.1 Augustinus(AD. 354 - AD.430)
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 planand intention. What we ordinary people want to know most is, how should we live? Christian
Fathers, as God’s spokespersons, preached this to the people as the teachings of Christ. Therefore, Christian fathers, like God, had to be able to answer any questions immediately. Amongthem, the question that puzzled them were, ”how is the world made?”, ”if this world was madeby 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 ((D2 )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 understandingof the world for the sake of deepening our faith.

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,
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Chap. 6 The Middle Ages
(F) the future is in ”prediction”, the past is in ”memory”. There is what we can realize ”onlynow”.
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 methodof telling philosophy as follows.
Only present exists

Live joyfully now!
thereforeﬁctional worldview −−−−−→ Ethics·morals
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 (E2 ) in Sec. 1.1.2 says that this (K) is not a statement in quantum language. Thus, the(G) is not scientiﬁc, that is, there is no experiment to verify the (G).
Now,
• Augustinus’ tense (past, present, future) is a kind of sermon. But it may be interestingin 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 realm of science, but they continued to attract the interest of philosophers. For
example, Bergson, a philosopher of ”subjective time”, tried to challenge Einstein of ”theoryof relativity” to an argument. But he was rejected by Einstein, saying ”I don’t know thetime 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
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6.1 Augustinus(AD. 354 - AD.430)• 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 nonscientiﬁc 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). For the quantum linguistic understanding of ”wave function collapse”, see:[59] 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 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. Thisis a 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 can make is not a measurement. In science, ”I”, ”now” and ”here” are forbidden.
Thus, ’Now I am here’ is not a scientiﬁc proposition (i.e., a proposition in quantum language).
Enjoyment of wordplay is an important 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 notedthat these are not statements in quantum language, i.e., these violate the linguistic Copenhagen
interpretation.

♠Note 6.4. For completeness, let us rewrite as follows.
1 How shoul 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:
1 may be derived from ⃝.
2• the ”logic” which is dressed so that ⃝2 is a reason added later. We might be, by common sense, convinced thatIn this sense, the ⃝”the worldview was to describe the world plainly and with no ﬁction”. However, Plato andAugustinus consider that• the worldview is to create the world that it is convenient for faith or doctrine.

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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 the limits of my world”, and ﬁnally, quantum language),• the worldview is not to to describe the world plainly and with no ﬁction, but to create theworld that 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
it seems 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 andAxiom 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 thenorm of 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 ofAristotle spread to 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 andsupported Plato’s philosophy. The losers were driven to the Muslim world, and Aristotle’s
philosophy must have spread to the Muslim world by such a process.
Eastern Islam was centered in Baghdad which was famous on the Arabian Nights. 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 worldlearned 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. Suchpublic opinion had been drifting.
• The achievements of the Crusades do not rise by Plato’s way. Thus, let’s study Aristotle
at the tip of the Islamic culture!
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.
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That is, Scholasticism was born. As the typical persons of Scholasticism, we list up as
follows.
(B1 ) Anselmus (1033 - 1109) ”The father of Scholasticism”, ”Realismus”(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 inﬂuence increased. 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 Aristotelianphilosophies.
Of course, it is impossible to succeed this ”fusion”. That is because Plato philosophy andAristotle philosophy are ”oil (idealistic dualism) and water (realistic monism)”, and these are
diﬀerent categories (cf. Assertion 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 Christian culture 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,−1000,−15,−81.5,+3.1415+0.3333 · · · ,0,+39.045,+0.009876,

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 numeral

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 = MMMCMXCIX122

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6.3.3

The explosion of mathematics

European mathematics originally had the high potential of Euclidean geometry. With theintroduction of the positional notation (the discovery of zeros), computation became easier.
The word ”ALGEBRA (algebra)” is originally an Arabic word. The formula for the solutionof the quadratic equation : ax2 + bx + c = 0:
x=

−b ±

√

b2 − 4ac
2a

was also understood by the Arabic mathematician al-Khwarizmi (about 790 - about 850)
Mathematician Gauss(1777 - 1855) said(C) ”If genius Archimedes invented the positional notation, I am certain that the mathematics
must have progressed drastically.”
The positional notation triggered the ”math explosion” such that(D) Solution of algebraic equations, complex numbers, the function concept, betting of problem (probability), analytic geometry (Descartes coordinates), calculus, diﬀerential equations, linear algebra, number theory, etc.

♠Note 6.7. 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 sets⃝0
Of course, it is needless to say that the biggest discovery is ”⃝:thediscovery of natural numbers”.

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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 ofGod”. 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, themeaning of ”existence” is not clear. Therefore, there is no need to read the proofs seriously.

Proof 6.2. Anselmus: the proof of God’s Existence1⃝:
God is a being than which none greater can be imagined . That is, the greatest possible
being that can be imagined.
2⃝:
It is obvious that God exists as an idea in the mind.
3⃝:
A being that exists as an idea in the mind and in reality is greater than a being thatexists only as an idea in the mind.
4⃝:
Thus, if God exists only as an idea in the mind, then we can imagine something that isgreater than God. But we cannot imagine something that is greater than God.
5⃝:
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 argumentwith ”a set of all sets”, which is related to Russell’s paradox. Thus, the above may be a kind ofself-referential word play.
(ii): If the meaning of ”existence” is deﬁned in the sense of Deﬁnition 1.7, the existence of”God” is obvious. Because ”God” is the most important key-word in the Bible.

6.4.2

Review: the worldviewism

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

premisesubject

That is,(B) The worldviewism is the spirit ”Start from the worldview” or ”Start from the ﬁrm
premise!”.
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)• Zeno paradox (Flying arrow), (cf. Sec. 2.4.2)• Zeno paradox (Achilles and a tortoise), (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)

6.4.3

The inﬂow of Aristotle philosophy

Although we cannot understand Proof 6.2, we think that Anselmus did not say much morethan the following:(D1 ) ”Aristotle philosophy ﬂowed in via Islam, and I underwent the inﬂuence”.
That is,Aristotle −−−−→ syllogism −−−−→ logic −−−−→

Plato

proof


via Islamyinﬂow−−−−→ Augustinus −−−−→ God −−−−→ The existence of God

In other words, a priest who had studied Aristotle’s syllogism felt like he had ”proved theexistence of God” by using the syllogism.
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6.4 The proof of the existence of God
We can see that ”Proof of the Existence of God” is a product of Schola’s philosophy (areckless 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, andAristotle’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 ”proofof 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 themind control of ”stop thinking”.
After all, we see,(G) the history of Schola’s philosophy is that the ”eyes to see reality” of Aristotle’s stylegradually matured from the blind state of faith alone, and Descartes received the baton
and opened the curtain of the modern 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 thesigniﬁcance and motivation of this challenge.

♠Note 6.9. All scientists are interested to ”God”. ”What is God ?( = How about neuronal circuit
concerning belief?)” and ”What is subjective time? (= How about biological clock?) are one of
the most interesting problems in brain science.

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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 a dualistic
and idealistic description of the world, and Aristotle’s philosophy, which is a monistic andrealistic description of the world, are like water and oil (or, grafting a bamboo to a tree), andthus , naturally it was confusing. As shown in the previous section, the ﬁrst confusion was the”proof of God’s existence” of Anselmus. The second confusion is the ”Problem of universals”
in this section.
The problem of universals is the 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 thefollowing problem:
Do ”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 be answered from the quantum mechanical worldview ?
Let 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
Also, it is assumed that a membership function m : Ω → [0, 1] induces the observable Om =({y, n}, 2{y,n} , Fm ) in C(Ω), where Fm ({y})](ω) = m(ω)(∀ω ∈ Ω).
Recall Example 3.11 (and Remark 3.12), that is,(A2 )linguistic version−−−−−−−−−→

Idea

signiﬁer, signﬁed

Plato’s idea theorycongnitiveyversion

Saussure’s linguistic theory

secondary quality

membership function

John Locke

Zadeh’s fuzzy theoryquantiﬁcationyyidentiﬁcation

quantiﬁcationjobservable
quantum language

That is,(A3 ) ”Idea of honesty”(= ”Degree of honesty”)−−−−−−−→ ”Menbership 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

128[C](= matter)/[Idea world]
eidos[hyule]state[system]

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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. (viaIslam)Aristotle school (”Nominalismus”). The power in the church gradually shifted to
the Aristotle group. In this sense, 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 a little irrational since religion is not realistic.
Table 1.1 : realistic worldview vs. idealistic worldview
dispute  [R] vs. [L]

Realistic worldview

Idealistic worldview(monism, realism, no measurement)(dualism, idealism, measurement)⃝:
Hērakleitos
a motion⃝:AncientAristotle
Greeceb⃝:c Problem of universals ”Nominalismus”(Ockham)⃝:
Newtond space-time⃝:
Einstein
e quantum theory

Parmenides
Plato”Realismun”(Anselmus)
LeibnizBohr⃝
c is more of a confusion than a dispute. ⃝d is the Leibniz=Clarke correa is my ﬁction, ⃝isBohr-Einsteindebates.
Quantum
language is proposed as
spondence (cf. Sec. 9.3.2), ⃝e
one of answers to Bohr-Einstein debates(cf. ref. [66]).

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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 summingup of Scholasticism. He was the most important at the intermediate time of Scholasticism (or,Problem of universals), that is,(1033−1109)

Anselmus(”Realismus”)(1285−1347)(1225−1274)−→−→

Thomas Aquinas(compromise or fusion)

Ockham(6.1)(”Nominalismus”)

His proposal is the compromise of ”Realismus” (due to Plato) and Ockham (due to Aristotle), thus, his theory 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 thetheory 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

130/[Idea world]
eidos[hyle][universale in re]//[matter]state[system]

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♠Note 6.10. As mentioned in Sec.3.5, I am not conﬁdent in the above table, because the theoryof ideas and Scholasticism were not created with the intention of measurement. However, I willproceed with this policy of discussion in this paper.

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

Augustinus −→(”Realismus”)

Plato (+ Aristotle)

Anselmus(”Realismus”)

Plato + Aristotle−→

Thomas Aquinas(compromise or fusion)

Aristotle−−−−→
overstep

Ockham(”Nominalismus”)(6.2)
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
[the classiﬁcation of philosophers]).

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6.6 Thomas Aquinas (1225 - 1274); Scholasticism
(♭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Boole, Frege, Peirce, Saussure, Russell, Wittgenstein, Hempel, Popper








(♭23 ): the mechanical worldview (statistics, quantum language) Parmenides, Zeno, J. Bernoulli, statistics (e.g., Fisher), quantum language

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Chap. 6 The Middle Ages

6.7

Ockham’s razor

William of Ockham (1285 - 1347), a Scholastic philosopher or theologian born in Ockham in
England, is known as an advocate of Ockham’s razor(=the law of parsimony) in philosophy
and science.
(A):Ockham’s razor(=the law of parsimony)
Ockham’s razor is as follows:(A) Shave unnecessary assumptions with a razor!
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 Occam’s razor, Ptolemy claimed the theory of celestial motion from the observational data of the time.
And(D2 ) relying on Occam’s razor, Galileo claimed for a geodynamic theory from the observational
data of the time.
Occam’s Razor is an explanation added later and thus, powerless. But then again...
(E) Is there a case (other than mathematics) in which ”Occam’s Razor” is working eﬀectively?
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6.7 Ockham’s razor
This makes me skeptical of Occam’s razor.
Even in the Middle Ages, it is diﬃcult to imagine that people would have been impressedby 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 and does not know much about it. Occam, however, is the leadingedge of the Aristotelian sect.
Therefore, I guess that(F) What Ockham wanted to shave with a razor is the theory of Ideas(cf. (C) in Sec. 6.6). It makes sense, then. However, it would be too short-sighted to thinkthat it must be better to eliminate such hypotheses as the existence of ”something ideal in theheavens.
Plato must have thought that the idea theory was a fairy tale. It would be immature andfoolish to cut oﬀ the fairy tale.

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

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.

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7.1 Paradigm shift

Table 7.1. Scientiﬁc revolution; Chronological table
Before Galileo: The era of observation and experiment• 1450: Gutenberg’s printing press

inventor• 1453:The Eastern Roman Empire extinction (Constantinople surrender)• 1492: Columbus, discovery of the American Continental

navigator• 1498: Vasco da Gama, discovery of the sea route to India

navigator• 1500s: Leonardo da Vinci, ”Mona Lisa’s smile”

artist• 1510: Copernicus, Heliocentrism

scientist• 1510: Raﬀaello, ”The School of Athens”, Admiration to ancient Greece

artist• 1517: Luther, Protestant Reformation

religionist• 1519 - 20: Magellan, the ﬁrst circumnavigation of the Earth• 1540s: Michelangelo, ”The Last Judgment”

artist• 1600: G. Bruno, who supported heliocentrism along with Galileo, was burned at thephilosopher
stake by the Vatican• 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

illuminator• 1633 Galileo’s trial ”And yet it moves”

scientist

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

philosopher• 1670: Pascal, ”Pensèes”

enlightener• 1685 - 1750: Bachartist• 1687: Newton, ”Principia”

scientist• 1688: Glorious Revolution• 1690: John Locke, the father of British Empiricism, ”An Essay Concerning
philosopher
Human Understanding”, tabula-rasa, the secondary quality• around 1700: Jakob Bernoulli, the law of large numbers

mathematician• 1703: Leibniz, ”New Essays on Human Understanding”

philosopher• 1715 - 16: Leibniz-Clarke correspondence (cf. Sec. 9.3.2)• 1739: Hume, ”A Treatise of Human Nature”

philosopher• 1781: Kant, ”Critique of Pure Reason”

philosopher

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Chap. 7 Early modern – Scientiﬁc revolution: From Geocentrism to Heliocentrism♠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,
the proportion 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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7.2 Francis Bacon (1561 - 1626): The father of empiricism, Inductive reasoning

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 ”scientiﬁcrevolution” as follows.

the birth of worldview•

Movement

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

Or,

Aristotle· · ·Plato. . .

(physics)
the realistic worldview
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”.
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Chap. 7 Early modern – Scientiﬁc revolution: From Geocentrism to Heliocentrism(A):Induction principle (by bad idols), how to create scienceHis proposal is as follows.
1(A) ⃝:Exclusionof bad idols
23−→⃝:data,
collection by observation and experiment−→⃝:scientiﬁctheory, principle

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

Recall that Greek philosophy starts from ”principle”, e.g., ”the arche (= the ﬁrst principle of
all things) is ⃝⃝”, or, Euclid advocated geometric axiomatization, and a lot of theorems were
derived from Euclid axioms, that is,(B1 ) Mathematics:

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

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

which is the scientiﬁc method proposed by Bacon (who was called the father of empiricism).
7.2.1.1

Isaac Newton (the exclusion of bad idols) (1642-1727)

Newton said:”I frame no hypotheses”
And he practiced Bacon’s induction principle, and proposed Newtonian mechanics as follows.
12(C): ⃝exclude
bad idols (i.e., Aristotle’s purpose, Geocentrism) −→ ⃝Data
collection (due3to Tycho Brahe, Kepler, Galileo) −→ ⃝Science
theory (Newtonian mechanics)♠Note 7.2. This may be say in a philosophy side. We must add the next section (good idols).

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7.2 Francis Bacon (1561 - 1626): The father of empiricism, Inductive reasoning
7.2.1.2

Isaac Newton (good idols) (1642-1727)

Bacon’s induction principle is not simple. there is another way (by good idols) such as(D): Induction principle (by good idols)
Induction principle (by good idols) is as follows.
123(D) ⃝believe
good idols −→ ⃝Data
collection −→ ⃝Science
theoryNewton said:”I frame no hypotheses”
And he practiced Bacon’s induction principle (good idols), and proposed Newtonian mechanics
as follows.
12(E): ⃝believe
good idols (i.e., Causal relation) −→ ⃝Data
collection (due to Tycho Brahe,3
Kepler, Galileo) −→ ⃝Science
theory (Newtonian mechanics)♠Note 7.3. Although ironically,(♯) Bacon, who proposed the exclusion of idols, was also one of discoverers of ”good idols”
called ”causal relation”.
If so, what Bacon wanted to say really may be[] []bad idols = dogmatism in Scholasticism (or, Aristotle’s purpose)♠Note 7.4. S. Weinberg (1933 -), a physicist at the University of Texas, Austin, won a Nobel
Prize in 1979 for work that became a cornerstone of particle physics, said in his book [105] ”To
explain the word; 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 theseventeenth and eighteenth centuries would invoke Bacon as a counterweight to Plato andAristotle, .... It is not clear to me that anyone’s scientiﬁc work was actually changed forthe better by Bacon’s writing. Galileo did not need Bacon to tell him to do experiments,
and neither I think did Boyle or Newton.
From the pure scientiﬁc point of view, I almost agree to the above opinion. However, it should
be noted that the scientiﬁc revolution was not only achieved by scientists, but by the collective
power of philosophers, enlighteners, astronomers, mathematicians, adventurers, artists, educators, politicians, religionists, and the general public. The diﬃcult and important thing is to get
public support for the scientiﬁc revolution. I think Bacon did this work well. He enlightened
the method to create science (i.e., the importance of causality and experiment) to the generalpublic, not the scientists. As I will show in the next chapter, I consider as follows:140

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

Aristotle· · ·Plato. . .

(physics)
the realistic worldview
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) 







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 withexperiment. 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 byepicycle 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 hadreached the scientiﬁc level. For example, there is an opinion that
• At the time, Europe is in the cold period, the masses were hungry for ”the sun”. The
public was hungry for the sun central principle. Therefore, there is a foundation thatallows the germination of Heliocentrism.
In spite of the above, I want to assert that Copernicus is a great scientists. The reason I thinkso is written in the following Note 7.5.

•Saturn•Sun•Saturn•VenusMoon•• •Mercury •

Earth

•
Moon

Jupiter•Earth•Venus• •Mercury •Jupiter

Sun

•Mars

•Mars

Geocentrism (=Ptolemaic system)

Heliocentrism (=Copernican system)

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Chap. 7 Early modern – Scientiﬁc revolution: From Geocentrism to Heliocentrism♠Note 7.5. S. Weinberg (1933 -), An American physicist, won a Nobel Prize of physics in 1979,
said in his book [105] ”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 tolove Copernicus. Thus I would like to regard him as the great scientist.

In this paper, I discuss the next.
(A): What is ”Heliocentrism vs. Geocentrism”?
Now,(A) Note that motion is relative. Thus, if the earth is assumed to be at center, the sungoes around the earth (i.e., Geocentrism). Also, if the sun is assumed to be at center,the earth goes around the 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 Vatican
Analyzing the enormous data obtained by Tycho Brahe’s steady astronomical observation,
Kepler found the following laws:• Kepler’s laws of planetary motion:
1609:(♯1 ) The ﬁrst law of elliptical orbits,(♯2 ) The second law of areal velocity,
1619:
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7.3 From Geocentrism to Heliocentrism(♯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”.

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 adiﬀerence of how to take the coordinate system.
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Chap. 7 Early modern – Scientiﬁc revolution: From Geocentrism to Heliocentrism
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)−−−−−−−−−−−−−−−−−−−−−→ Causality (and experiment)(Hērakleitos, Parmenides, Zeno, Aristotle) Aristotle’s spirit :(About 1500 years)( 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 technologicalinnovation (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 ofcourse 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.
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7.3 From Geocentrism to Heliocentrism
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
Thus, Galileo thought it was a matter of measurement data, but the church thought it wasa matter of worldview. Therefore,• at the time (1633) of the trial of Galileo, the Church was convinced that it would not lose
its argument with Galileo and, at worst, could bring it into an endless dispute.
An endless dispute implies the win of the church. The church is not so stupid.
No way, the church did not think that Newton would appear
No one would have predicted the appearance of Newton.
After all, Galileo was the active leader of the overthrow of the worldview of Aristotle, but
he could not propose the new worldview. In that sense, the Galileo legend is just the beginningof Newton’s appearance

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

7.4

Principia; Newtonian worldview

7.4.1

Principia (1687)”PhilosophiæNaturalis Principia Mathematica” (in short, ”Principia”) , written by Newton(1687), is the most famous and important book in science. Three laws of Kepler were derived
from three laws of dynamics and the law of universal gravitation. Principia was written basedon elementary geometry and not the diﬀerential and integral calculus. Why did Newton (=advocate of diﬀerential and integral calculus) not write Principia based on diﬀerential andintegral calculus? Although there may be several opinions for this question, The work (basedon 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, theworldviewism

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[

Kinetic diﬀerential equation method: (Newton)]

That is, we see:(B): Paradigm shift [Newtonian mechanical worldview ]Now,(B) Aristotle’s worldview(purpose)−−−−−−−−→
paradigm shift

Newtonian worldview(causal relation)

Here, Newtonian worldview means Newtonian mechanics, that is,
Newton’s kinetic equation (i.e., the chain of causal relations)+ the law of universal gravitation
Recall that the main theme of this 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))
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7.4 Principia; Newtonian worldview
is fuzzy, and thus, ”Geocentrism vs. Heliocentrism” cannot be decided. Thus, we need a
new worldview.
In Principia, Newton proposed Newtonian mechanics (i.e., Newtonian worldview) and showedthat(C2 ) When the motion of the sun and the planets is studied, the calculation becomes easy 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
the observations are. That is, ”Geocentrism vs. Heliocentrism” is not the problem ofmeasurements, 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 ).

♠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Boole, Frege, Peirce, Saussure, Russell, Wittgenstein, Hempel, Popper








(♭23 ): the mechanical worldview (statistics, quantum language)Parmenides, Zeno, J. Bernoulli, statistics (e.g., Fisher), quantum language

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Chapter 8Modern philosophy(Descartes)
The following is called the ﬂower of modern philosophy:Induction (due to Bacon) 







British Empiricism








Deduction
[Locke, Berkeley, Hume]−→ Kant philosophy
Descartes
−→




[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 withexperiment. Thus, I assume that they did not belong to the realistic worldview.)
Here, the(♯) is located as follows:

Aristotle· · ·Plato. . .













the realistic worldview(physics)

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 believedthat ”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?
Namely, I think they wanted to pursue a ”science of dualism” (cf. Remark 3.9, Remark 5.1).
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 the worldview 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 notexpected 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 themain 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, the momentum forthe 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

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Chap. 8 Modern philosophy(Descartes)
The philosophy of worldview is only an anecdote, however, it should be eﬀective for general
people (and thus, quantitative arguments are not desirable). After all, following Socrates’ selfreferential statement ”I know 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 method of 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 tocoexist 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
11, 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 thescientiﬁc point of view

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 positionof a ﬂy moving around the ceiling.

y 6(2, 1)

1

-

O

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 RenéDescartes, 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”, hedecided ”Start from the unquestionable truth”. And he found the most famous philosophicalproposition (= cogito proposition):”I think, therefore I am”
That is, Descartes think:I think that ’I think, therefore I am’
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 8.2), 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
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Chap. 8 Modern philosophy(Descartes)
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 was oriented 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.

Aristotle· · ·Plato. . .

(physics)
the realistic worldview
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 (non-physics)
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]








Rationalism
Continental
[Leibniz](Note that in spite of ”British Empiricism”, Locke, Berkeley, Hume were not concerned withexperiment. Thus, I assume that they did not belong to the realistic worldview.)
Descartes asserted:
Proclaim 8.1. 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.

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

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 [40]∼[68].
Now we will show two (scientiﬁc or non-scientiﬁc) interpretations of the cogito proposition:”I think, therefore I am”.
Proposition 8.2. [(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. However, the interpretation (C1 ) is atrivial 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 (C2 ) is not inquantum 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.

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Chap. 8 Modern philosophy(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 Iam”
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 isperceived 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(C)”I”(=”mind”, ”brain”) ,”body”(=”sensory organ”) ,”matter”

For completeness, it should be noted that this is not a consequence of the cogito proposition.
That is the cogito proposition is the reason added afterwards. The cogito proposition is nothing
more than a catch copy.
Principle 8.3. 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 mind-matter dualism
and idealism which starts from the three key-words[A](mind)”I”(=mind, brain, observer) ,←−(medium)[B] −→”body”,[C](matter)”matter”(=sensory organ, measuring instrument)

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8.3 Descartes’ strategy
For the deﬁnitions of ”idealism” and ”idealism”, see Deﬁnition 1.8, i.e., idealism = the spirit
such that thinking takes precedence over experimentation (since experiments on the mind arealmost 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.).

Also, ethics, moral, etc. are not emphasized in (♯), but Descartes asserted ”The existenceof God is derived”.
And, Descartes proposed the following two problems;
Problem 8.4. [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 ) in the quantum mechanical worldview, see Sec. 11.9
For the answer of (D2 ) in the quantum mechanical worldview, see Sec. 9.5.1

Descartes might think as follows.
(F) The theory of Ideas has expired expiration date. Thus, in order to refresh philosophy, a
new model-change (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
Descartes

Plato

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

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Chap. 8 Modern philosophy(Descartes)♠Note 8.3. 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
not experimentally veriﬁable (cf. Proposition 8.2. ). Some may ﬁnd this ironic. However, I’dlike to think that 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.

♠Note 8.4. As mentioned in ref. [66], 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 thatthe following can be described in measurement theory, but we believe that it is impossible. For
example, the followings cannot be written by quantum language:⃝⃝
1 : tense—past, present, future —2 : Heidegger’s saying ”In-der-Welt-sein”



⃝
⃝
3 : the measurement of a measurement,
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 and quantum 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.

measuring instrument

observer(body, eye, etc.)[observable]• [measured value]
a project light⃝system(matter, measuring object)(I(=mind))[state]-b perceive the reaction⃝(i.e., the reﬂected light)

1 ⃝)”2 in dualism
Figure 8.1: [Descartes Figure]: Image of ”measurement(=⃝+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 kindof 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 whena needle of a voltmeter just moved, it is only a physical phenomenon. Nevertheless a movementof 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.
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Chap. 8 Modern philosophy(Descartes)(A) In spite that three key-words ”mind”, ”body”, ”matter” are gathered, Descartes philosophy has no computable structure. This is only the ﬁctional worldview, and not themechanical worldview.
Table 8.1: 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:
[]
[
] [
]
[
] [
]
[
]
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 8.5. Descartes’ model-change:
the theory of Ideas −−−−−−−→ Descartes philosophy
model-change

That is, it is sure that Descartes’ model-change is supcan be regarded as a progress?
ported by many people. Almost people certainly believe that science makes progress, thatis, 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 8.5:
Although this will be answered throughout this paper, I add a simple explanation as follows.
Our answer to Problem 8.5 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)

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

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

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

the realistic worldview(physics)
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]−→ Kant philosophy
Descartes
(♯2 )
−→




[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 withexperiment. 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 theacademy,” 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 oftelling 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 to prepare the social environment of the appearance of Newton. That is,
Descartes was only the opening performer. In this sense, ”I think, therefore I am” (theexistence 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 ofNewton.
Even if there was such history, it wasn’t strange.
However, strangely, there were people who took ”the existence of I” or ”Descartes Figure1.2” seriously. For example, John Locke (1632 - 1704) thought as follows.
(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 might thinkin his ”Essay Concerning Human Understanding” (1689) as follows.
(D) 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.
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 is enormous.
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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)♠Note 9.1. If the above (D) is true, Locke may have been trying to create another kind of physics
that is diﬀerent from Newtonian mechanics. If so, his theory should belong to dualistic realism.
However, Locke and others (Descartes, Leibniz, Hume, Kant, etc.) were not concerned withexperimental veriﬁcation (cf. for our deﬁnition of ”realism”, see Deﬁnition 1.7). Thus, in thispaper, we think that Descartes-Kant philosophy belongs to dualistic idealism. This was confusedin my previous paper [61].

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 9.2. (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): As mentioned later (cf. Note 10.3, Summary 11.31), considering ”language” and not”cognition”, then, in several languages (ordinary language, mathematics, Newtonian mechanics,
programing language, etc.), 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 tabularasa state). 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.
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9.1 Locke (1632 - 1704): The father of British Empiricism
That is,(I) :Locke’s worldview
The world is composed of two (i.e., ”matter” and ”mind (= observer)”. ”Matter” hasinherent nature (= primary quality), ”observer” has body (=”sensory organ”). Throughthe sensory organ, secondary quality (sweet, red, hot, salty, etc.) is felt by our brain.

In terms of quantum language, we say:primary quality=⇒state,secondary quality=⇒observable (= measuring instrument)
as seen in the table below.
Table 9.1: 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

secondary quality

quantum language

measured value

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
quantity”. The terms such as Idea, body, etc. may be not comprehensive. However,• ”secondary quantity” is a word we can understand. Hence, Locke’s achievement
should be honored.
Again, note that ”secondary quantity” is a word that forms the foundation of dualism.
If so, we may aﬃrmatively answer Problem 8.5, i.e.,
Can the direction:”Descartes −−−−−−−→ Locke” be regarded as progress?
model-change

That is, we may assert that164

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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)(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, this will be summarized in the ﬁnal chapter (=Postscript)♠Note 9.3. (i): The polar star can be regarded as a measuring instrument such as a kind ofcompass.
(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 righthand as the measuring instrument, I feel the existence of my left hand. On the contrary, if
I regard the left 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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9.2 Dramatic presentation of ”British Empiricism vs. Continental Rationalism”

Dramatic presentation of ”British Empiricism vs.
Continental Rationalism”

9.2

Modern philosophy became popular through the following rival relation:
British Empiricism vs. Continental Rationalism
That is,Deduction(A)

Descartes[The father of modern philosophy]

−→












Induction (due to Bacon)

British Empiricism[Locke, Berkeley, Hume]

















Deduction (due to Descartes) 





 Continental Rationalism 






−→ Kant philosophy[summing-up][Leibniz](Note that in spite of ”British Empiricism”, Locke, Berkeley, Hume were not concerned withexperiment. Thus, I assume that they did not belong to physics, but idealism.)
Let us explain the above:

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 isborn as the blank state (”tabula rasa”) . (Locke, Berkeley, Hume, · · · )[]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,· · · )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)
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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)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 andeighteenth centuries and is known as the last ”universal genius”. He made deep and important
contributions to the ﬁelds 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. tabularasa”) ?
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. tabula rasa” ), 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 scientiﬁcally. Without the ground, it is uselesseven if they said something
For example, from the scientiﬁc point of view, atomism due to Democritus (BC.460BC.370) is non-sense. It is poem. .
For the argument about ”Continental Rationalism vs. British Empiricism” from the linguistic point of view, see Summary 11.31 in Chap. 11.
Ordinary people might have wanted to ﬁnd the big name of ”Genius Leibniz” in the debatesof ”British Empiricism vs. Continental Rationalism”. Maybe they just enjoyed a play of thename as ”the dawn of modern times” as an entertainment.
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9.2 Dramatic presentation of ”British Empiricism vs. Continental Rationalism”♠Note 9.4. As mentioned later (cf. Note 10.3, Summary 11.31), 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 9.2, 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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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)9.3 *Leibniz-Clarke Correspondence: What is spacetime?

This section is published in the following:• ref. [68]: 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. [69]; 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 well as the traditional philosophies. In this section, we give the quantum linguistic answer to these problems. As seen later, our answer is similar to Leibniz’s relationalism
concerning space-time. In this sense, we consider that Leibniz is one of the discoverers of thelinguistic Copenhagen interpretation

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

9.3.1.1

Space in quantum language(How to describe “space” in quantum language)

In what follows, let us explain “space” in measurement theory (= quantum language).
For example, consider the simplest case, that is,(A)“space”=Rq (one dimensional space)

Since classical system and quantum system must be considered, we see (B1 ): a classical particle in the one dimensional space Rq(B)(B2 ): a quantum particle in the one dimensional space Rq
In the classical case, we start from the following state:(q, p) = (“position”, “momentum”) ∈ Rq × Rp
Thus, we have the classical basic structure:169

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9.3
* Leibniz-Clarke Correspondence: What is space-time?
(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 (C1 ): classical [C0 (Rq × Rp ) ⊆ L∞ (Rq × Rp ) ⊆ B(L2 (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)](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)170

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

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

Newton’s absolutism says that the space-time should be regarded as a receptacleof a “thing.” Therefore, even if “thing” does not exits, the space-time exists.
On the other hand,
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9.3
* Leibniz-Clarke Correspondence: What is space-time?
(H) [The metaphysical space-time]

Leibniz’s relationalism says that
(H1 ) Space is a kind of state of “thing”.
(H2 ) Time is an order of occurring in succession which changes one after another.
Therefore, I regard this correspondence as←→v.s.

Newton (≈ Clarke)(realistic view)

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

←→
v.s.

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 thelinguistic Copenhagen interpretation of quantum mechanics.
Many scientists may think that
Newton’s assertion is understandable, in fact, his idea was inherited by Einstein. On theother, 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):
1 : Newton, Clarke⃝(realistic world view)







2 :
 ⃝

(space-time in physics)· · · realistic space-time(successors: Einstein, etc.)“What is space-time?”(space-time in measurement theory)

Leibniz(linguistic world view)···

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

1 and ⃝.
2in which Newton and Leibniz respectively devotes himself to ⃝Although Leibniz’s

assertion is not clear, we believe that
• Leibniz found the importance of “linguistic space and time” in science,
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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)(♯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”) 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]

Realistic worldview

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

Hērakleitos⃝:
a motion⃝:AncientAristotle
Greeceb⃝:c Problem of universals ”Nominalismus”(Ockham)⃝:
Newtond space-time⃝:
Einstein
e quantum theory

Parmenides
Plato”Realismus”(Anselmus)
LeibnizBohr⃝
c is more of a confusion than a dispute. ⃝d is the Leibniz=Clarke correa is my ﬁction, ⃝spondence (cf. Sec. 9.3.2), ⃝e is Bohr-Einstein debates. Quantum language is proposed as
one of answers to Bohr-Einstein debates(cf. ref. [66]).

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

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 thatcome from a supernatural power such as a god. Thus his philosophy is called subjective
idealism.

Table 9.2: 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

secondary quality

Berkeleymind

secondary quality

quantum language

measured 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.
(A2 ) Berkeley indicated that the mathematical deﬁnition of lim 00 is not complete(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 of Christianity with Newtonian mechanics, we can conclude that Berkeley is honest.
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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)♠Note 9.5. 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 (byCauchy (1789-1857), 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.

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” is emphasized 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 (H1 ). 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 theexclusion was a weak point of Newtonian mechanics. And he said
To be is to be perceived
This is the golden rule of anti-Newtonianism (i.e., anti-physicalism, idealism). The opposing
structure: [Newton vs. Berkeley] continues to [Einstein vs. Bohr] as mentioned in next section.
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9.4 Subjective idealism: Berkeley, ”To be is to be perceived”

9.4.3”Einstein-Tagore Meeting” and ”Bohr-Einstein debates”

Concerning ”realistic worldview vs. idealistic worldview”, Einstein-Tagore (poet, thinker inIndia) meeting in 1930 is famous, in which they asserted as• Tagore: ”Truth is always limited by human perception.”• Einstein: ”Truth is independent of our consciousness, For instance, if nobody is in thishouse, yet that table remains where it is1 .”In the above, Tagore’s assertion is similar to Berkeley’s ”To be is to be perceived”, which
belongs to the situation of dualistic idealism(=idealistic worldview).
On the other hand, Einstein’s saying:(C1 ) if nobody is in this house, yet that table remains where it is (= Does the moon disappear
when I’m not looking at it?)is the same as(C2 ) Truth is independent of us (= realistic worldview)Thus, Einstein and Newton are similar, in the sense that
Truth is independent of human being (i.e., physics holds without measurement)
Therefore, it should be noted that (9.3) is signiﬁcant.
In this paper, we are not concerned with Bohr-Einstein debates in quantum mechanics(in order to solve this problem, I proposed quantum language), (cf. ref. [66])). 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. [66]).
1
Einstein often said this kind of statement at various places, for example, ”Does the moon disappear when
I’m not looking at it?”176

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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)♠Note 9.6. 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
a motion⃝:AncientAristotle
Greecebuniversals⃝:”Nominalismus”(Ockham)
Problemofc⃝:Newtond space-time⃝:
Einstein
e quantum theory

Parmenides
Plato”Realismus”(Anselmus)
LeibnizBohr⃝
c is more of a confusion than a dispute. ⃝d is the Leibniz=Clarke correspona is my ﬁction, ⃝dence (cf. Sec. 9.3.2), ⃝e is Bohr-Einstein debates. Quantum language is proposed as one ofanswers to Bohr-Einstein debates(cf. ref. [66]).

♠Note 9.7. In Japan, I learned the dualistic proposition: ”If a tree falls in a forest and no one
is around to 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.

9.4.4

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

For Bohr-Einstein debates, I discussed in ref. [66]. 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:
Newtonian mechanics and quantum mechanics are formulated as follows:(♯1 ) Newtoinan mechanics =Nothing+Causality(Newtonian equation)

On the other hand, quantum mechanics is formulated as follows.
(♯2 ) quantum mechanics =Measurement(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
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9.4 Subjective idealism: Berkeley, ”To be is to be perceived”(♯3 ) The moon is there whether one looks at it or not.
in Einstein and Tagore’s conversation.
So far, many experimental results support Bohr. However, if Einstein says the following(♯4 ), everyone has no choice but to shut up.
(♯4 ) Then, did the laws of physics not work before the birth of 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.
• [68]: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.
[(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 andalways 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 the same tests as in [(A):Scientiﬁc answer] for Jack and yourself as test
subjects, that is, various tests (colorblindness test, EEG measurement, electroretinogram,etc.). However, if so, it is the same as the situation [(A):Scientiﬁc answer].
Then, Betty says:
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9.5 Qualia problem and Brain in a vat argument(♭3 ) OK. I (=Betty) can understand. In short, the measurements I can only make against
myself are not scientiﬁc measurements. Or, a measurement that only I can make is not ameasurement. Is it OK?
Then, the scientist says to Betty:(♭4 ) That’s right.

♠Note 9.8. 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. Clicksaid 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 ofnerve cells and their associated molecules.
That is, he believed the monistic realism such as(♯2 ) the movement of the human spirit is also a kind of physical phenomenon.
I agree to his opinion. And I believe that with the development of brain science, 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.

9.5.2 * Brain in a vat argument
Let us introduce a famous problem ”Brain in a vat” due to H. Putnam (cf. ref. [92]) as follows:
Case[1]

Case[2]

•

• • •
• •
Brain

Figure 8.1: [Brain in a vat ]There is a possibility of the following.
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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)(C1 ) a mad scientist has removed your brain, and placed it into a vat of liquid to keep it alive
and active. The scientist has also connected your brain to a powerful computer, which
sends neurological signals to the brain in the way the brain normally receives them. Thus,
the computer is able to send your the data to you brain to fool you into believing thatyou 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 (C1 )).
Thus, (H4 ) is wrong.
[Alternative explanation]:
For completeness, let’s rephrase the same thing as follows. You ask someone ”Do I have alimb?” 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).

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9.5 Qualia problem and Brain in a vat argument♠Note 9.9. 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
have seen the followings.
• What is probability (or, measurement, causality) ? cf. Sec. 1.1.1)• Zeno paradox (Flying arrow), (cf. Sec. 2.4.2)• Zeno paradox (Achilles and a tortoise), (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 philosophy.
(A1 ) Descartes found the indisputable truth, i.e., cogito proposition ”I think, therefore I am”.
Therefore, 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 existence of I” is suspicious (cf. Note 1.7, or Proposition 8.2 in Sec.8.2). Hence, thefollowing 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 reached and discussed ”mind-matter dualism” (= Descartes problem 8.4),
that is, ”the problem of mind-matter dualism” and ”mind-body problem”.
Although Descartes problem 8.4 is, from the scientiﬁc point of view, a barren discussion,
Descartes philosophy was supported a lot of people. Since the philosophy of worldview isa kind of fashion or ”model-change”, to be 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 8.5, i.e.,
Can the direction:”Descartes −−−−−−−→ Locke” be regarded as progress?
model-change

That is, we may 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, Humeapproached the zone 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 bederived from ”the existence of I”2 . Also, it is not guaranteed that ”matter I perceive” is equal
to ”true matter”. Thus, the existence of ”matter” is doubtful. However, it is sure that I feelso. 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 )→(K)→(L)” is self-referential in the wide sense. That is,• ”(A1 )→(K)→(L)” is a kind of psychological illusion as having been able to understandall 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 notscientist. Hence, I think that
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

2
This kind of logic is a typical self-reference (cf. Note 1.7, or Proposition 8.2 in Sec.8.2). Thus, Hume’s logic(or generally, philosophical logic) 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)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 asfollows.
(♯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”, weshould just believe that the it has ”causality”.
Most readers may regard this as ”a kind of rhetoric”, however, several readers may be convincedin ”Now that 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. 10.3 [What is causality?]), that is,”recognition constitutes the world.”
which is considered that the recognition 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.
This will be discussed again in the next chapter (cf. Section 10.3 What is causality?).

♠Note 9.10. In the book ”The astonishing hypothesis” (by F. Click (the most noted for beinga co-discoverer of the structure of the DNA molecule in 1953 with James Watson)), Dr. Clicksaid 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 ofnerve cells and their associated molecules.
That is, he believed the monistic realism such as(♯2 ) the movement of the human spirit is also a kind of physical phenomenon.
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” by our 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♠Note 9.11. Philosophy has the habit of enjoying problems that are impossible to solve or beyond
the frame of science. 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 beright tomorrow?
(♯2 ) Quantum language have been very useful to this day. So, will quantum language be useful
tomorrow?
The justiﬁcation problem of these may be called Hume’s problem of induction, though David
Hume suspected the justiﬁcation of induction. That is, Hume’s problem of induction is out ofworldviewism. That is because, even if a great worldview is proposed, everybody can say :(♯3 ) Will the worldview be right tomorrow?
Therefore, thinking about these issues is a time-sucking pleasure, and while it may not be
advisable, what we should do is ﬁnding a problem that is similar to the above problems and canbe solved in the framework of quantum language.

9.7.1 * The solution of Hume’s problem of induction
This section was written with reference to the following.
• [70]:Ishikawa, S: Philosophy of science for scientists; The probabilistic interpretation ofscience Journal of quantum information science, Vol. 9, No.3 , 140-154,
DOI: 10.4236/jqis.2019.93007(https://www.scirp.org/Journal/paperinformation.aspx?paperid=95447)
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
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, ”truejustiﬁcation” = ”justiﬁcation under a certain worldview”.
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9.7 Hume’s problem of induction♠Note 9.12. In Note 7.1, we say that Bernoulli’s achievement (i.e., the discovery of the law oflarge numbers) equals Galileo’s achievement. That is, Scientiﬁc pioneer in the realistic worldview · · · Galileo

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

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 convincedthat 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 mechanicalworldview. 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 feel that the uniformity principle of nature (i.e., thecondition 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 the uniformity 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. A familyof 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 a principle (= 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.
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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)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 oflarge numbers), which should be regarded as the fundamental theorem in philosophy of science.
Theorem 9.6. [Inductive reasoning, the quantum linguistic solution of Hume’s problem of
induction]. Let [C(Ω) ⊆ L∞ (Ω, ν) ⊆ B(L2 (Ω, ν))] be a basic structure such that Ω is compactand ν(Ω) = 1. Assume that a family of measurements {ML∞ (Ω,ν) (Oi := (X, F, Fi ), S[ωi ] ) | i =−n, −n + 1, ..., −1, 0, 1, 2, ..., N } satisﬁes the uniformity principle of nature ( concerning µ). Let(x−n , x−n+1 , ..., x−1 , x0 , x1 , ..., xN ) ∈ ×i=−n X be a measured value by the parallel measurement⊗Ni=−n ML∞ (Ω,ν) (Oi := (X, F, Fi ), S[ωi ] ). Then, we see thatN♯{k | xk ∈ Ξ, k = −n, −n + 1, ..., −1, 0}≈ µ(Ξ)(= [Fi (Ξ)](ωi ))n)(Ξ ∈ 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.
Proof. Let Ξi ∈ F (i = −n, −n + 1, ..., −1, 0, 1, ..., N ). Axiom 1 [measurement] (in Section 1.1)
says that the probability that a measured value (x−n , x−n+1 , ..., x−1 , x0 , x1 , ..., xN ) obtained⊗NNby the parallel measurement
i=−n ML∞ (Ω,ν) (Oi := (X, F, Fi ), S[ωi ] ) belongs to ×i=−n Ξi isgiven by ×i=−n [Fi (Ξi )](ωi ) = ×i=−n µ(Ξi ). Thus, the sequence {xi }Ni=−n can be regarded asNN

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 consequenceof the law of large numbers for parallel measurements (cf. refs. [38], or § 4.2 in ref.[66]).
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. [38]), 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. [76]) has already been
accepted as the fundamental theorem in science. Therefore, even if Theorem 9.6 ([Inductive
reasoning]+(9.4)) is called the fundamental theorem in philosophy of science, we don’t thinkit’s exaggerated. We believe that our proposal (i.e., Theorem 9.6) is completely true in ourworldview. Thus, we think that the solution of Hume’s problem of induction was practicallyalready found as the law of large numbers. In the framework of our worldview, we are convinced189

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9.7 Hume’s problem of inductionthat 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 }. This may 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 ).
⊗(iii): It may be understandable to consider two measurements: 0i=−n ML∞ (Ω,ν) (O := (X, F, Fi ),⊗S[ωi ] ) and N
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
example of 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)

That is, a family of measurements {ML∞ (Ω,ν) (O := (X, 2X , F ), S[ωi ] ) | i = −n, −n + 1, ..., −1, 0,
1, 2, ..., N } satisﬁes the uniformity principle of nature (concerning µ). Let (x−n , x−n+1 , ..., x−1 , x0 ,⊗Nx1 , ..., N ) ∈ ×i=−n X be a measured value obtained by the parallel measurement N
i=−n ML∞ (Ω,ν)(O := (X, 2X , F ), S[ωi ] ), i.e., inﬁnite coin throws. Here, Theorem 9.6 [Inductive reasoning] saythat it is natural to assume that, for suﬃciently large n,(x−n , x−n+1 , ..., x−1 , x0 ) = (T| H H T H H H{zT T ..... T H H})

(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 withprobability 1/3] for each i = 1, 2, ..., N . It should be noted that even without knowing (9.5),
we can conclude that if we know (9.6).

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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)
Remark 9.9. It should be noted that the above example shows that Theorem 9.6 [Inductive
reasoning] (or equivalently, 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 long time. If the justiﬁcation of Hume’s problem
of induction is solved, it should be the most fundamental theory in science. Thus, we areconvinced that our assertion (i.e., the law of large numbers ≈ the justiﬁcation of Hume’s
problem of induction ) is true.

9.7.2 * Grue paradox cannot be represented in quantum language
This section was written with reference to the following.
• [70]:Ishikawa, S: Philosophy of science for scientists; The probabilistic interpretation ofscience Journal of quantum information science, Vol. 9, No.3 , 140-154,
DOI: 10.4236/jqis.2019.93007(https://www.scirp.org/Journal/paperinformation.aspx?paperid=95447)
If our understanding of inductive reasoning (mentioned in the above) is true, we can solve thegrue paradox (cf. ref. [21]). 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 subsetsof 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 the state 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 }clearly satisﬁes 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.
Let (x−n , x−n+1 , ..., x−1 , x0 , x1 , ..., xN ) ∈ ×i=−n X be a measured value obtained by the parallel⊗NX
measurement
i=−n ML∞ (Ω,ν) (O := (X, 2 , F ), S[ωi ] ). We see, of course, that xi = g (i =N

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9.7 Hume’s problem of induction−n, −n + 1, .., −1, 0). And thus, we can believe, by Theorem 9.6 [Inductive reasoning], thatx1 = x2 = ... = xN = g. For the sake of completeness, 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. [21]).
(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 i such that 0 < i. Suppose that we have examined the emeralds at−n, −n + 1, ... − 1, 0, and found them to 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.
However, we think that this (D1 ) cannot be described in quantum language. If we try todescribe the (D1 ), 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 stateof emerald ei (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 thestate of emerald ei such that ω = ωi (i = −n, −n + 1, ..., −1, 0, 1, 2, ..., N ). Let Oi =(X, 2X , Fi ) be the observable (i = −n, −n+1, ..., −1, 0, 1, 2, ..., N ) such that Oi is the sameas 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 family of measurements {ML∞ (Ω,ν) (Oi :=(X, 2X , Fi ), S[ωi ] ) | i = −n, −n + 1, ..., −1, 0, 1, 2, ..., N } does not satisfy the uniformity
principle of nature. That is because[Fi ({g})](ωi ) = [F ({g})](ωi ) = 1
192(i = −n, −n + 1, ..., 0),
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Chap. 9 Modern philosophy(Locke, Leibnitz, Berkeley, Hume)[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 can immediately conclude that Goodman’s discussion (D1 ) is doubtful since his argument is not based on any scientiﬁc worldview. In this sense, the above arguments (D2 ) and (D3 )may not be needed. That is, the confusion of grue paradox is due to lack of the understandingof Hume’s problem of induction in the linguistic quantum mechanical worldview, and not lackof the term ”grue” is non-projectible (cf. ref. [21]). Thus, 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 the worldview! In this formulation, conﬁrm that Goodman’s paradox is eliminatednaturally.
What I did is this.

Remark 9.11. Readers think that the Grue paradox is too unnatural. I agree. That is, theGrue paradox is like playing football and then before you know it, you’re playing rugby. If therules change in the middle of a game, there is no sport. I think that this is due to that N.
Goodman (1906-1998) belonged to the school of 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 that the worldviewism should be emphasized even in analytic philosophy. Thus, I thinkthat the Grue paradox is unproductive for us, if the weaknesses of analytic philosophy are notpointed out. Also, see Wittgenstein’s paradox in Sec. 11.6.3.

♠Note 9.13. Here, we have (cf. Classiﬁcation 1.11 [the classiﬁcation of philosophers]).

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9.7 Hume’s problem of induction(♭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Boole, Frege, Peirce, Saussure, Russell, Wittgenstein, Hempel, Popper








(♭23 ): the mechanical worldview (statistics, quantum language) 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] 







British Empiricism







 [Locke, Berkeley, Hume] −−−−−−−−−−−−−−−−→ Kant philosophy
Descartes −→





 Copernican revolution[a priori, nativism][summing-up]








Continental
Rationalism[Leibniz]

Kant completely followed Platonic method of telling philosophy (i.e., the ﬁctional worldview)
as follows.
world is so(♯)

you should do sothereforeﬁctional worldview (literary truth, pseudo- truth) −−−−−→ ethics, morals
preface, introduction, (ﬁctional)premise, expedient

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 attempt (♯1 ) at a metaphysical basis for the natural sciences is nothing less thana metaphysical basis for science in general, including Newtonian mechanics. Thus (♯2 ) isreliable.
Thus, Kant is the perfecter of Platonic method of telling philosophy. However, from the scientiﬁcpoint of view, the (♯1 ) is nothing but poetry.
But, a miracle happens. The (♯1 ) is very similar to quantum language (cf. Chapter 11).

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10.1 ”Surely You’re Joking, Mr. Kant!”: Antinomy

10.1”Surely You’re Joking, Mr. Kant!”: Antinomy

10.1.1Three Critiques

Immanuel Kant (1724 - 1804) 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., ”pseudotruth” in the sense of this paper), ”virtue”, ”beauty”. That is, he followed Platonic method oftelling philosophy as follows.
”Critique of Practical Reason (1788)””Critique of Judgment(1790)””Critique of Pure Reason (1781)”(A)

thereforeﬁctional worldview (literary truth) −−−−−→ ethics, morals,aesthetics
main subject

preface, introduction, (ﬁctional)premise, expedient

Of course, the interest of this paper is concentrated to the worldview (i.e., ”Critique of Pure
Reason” ). Have said many times in this book, philosophy of the worldview is only a ”preface”,
thus, it might be immature. However,(B) ”Critique of Pure Reason” might be perfect as a preface.

♠Note 10.1. As mentioned frequently up to this point, the ﬁctional worldview is really ”asserted
ﬁction”, however, we must pretend not to accept the ﬁctional worldview as ”non-logical”, or
we must accept it as ”logical in a wide sense”. That is because it must be prohibited thatthe diﬀerence between philosophy and religion becomes fuzzy. Therefore, we must use theterms 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 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-repeated story that neighbors would set their clocks by his daily walks. In fact, he may
have been a gregarious man. However, the image of Kant as strict and honest promotes anunderstanding of Kant’s philosophy. For the philosopher, his image itself is a part of his work.

10.1.2

Antinomy

Kant asserted that(C) There is a proposition P such that ”P is true” and ”P is not true”
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And he called such a proposition antinomy.
Antinomy 10.1. [Four Antinomies]
Kant assert that he ﬁnds the following four antinomies:(D1 ) The world has a beginning in time, and is also limited as regards space.
(D2 ) Every composite substance in the world is made up of simple parts, and nothing anywhere exists save the simple or what is composed of the simple.
(D3 ) Causality in accordance with laws of nature is not the only causality from which theappearances of the world can one and all be derived. To explain these appearances it isnecessary to assume that there is also another causality, that of Spontaneity.
(D4 ) There belongs to the world, either as its part or as its cause, a being that is absolutely
necessary.
[Notice] Propositions (D1 )-(D4 ) are only word play since the ”logic” in ordinary language
cannot be trusted, that is, it is not discussed under a certain worldview. Thus, the following
proof is not worth reading.
Proof of (D1 )Although each proposition (i.e., (D1 )-(D4 )), for example, according to Kant, let us show thatthe proposition (D1 ) concerning time is antinomy as follows.
1. If the world has no beginning, then for any time t an inﬁnite series of successive
states of things has been synthesized by t.
2. An inﬁnite series cannot be completed through successive synthesis.
3. The world has a beginning (is limited in time).
Therefore, Kant concludes that
• The proposition (D1 ) concerning time is antinomy

Some might read the above and say,
”Surely You’re Joking, Mr. Kant!”
With this antinomy, Kant may have wanted to show the ”limits of reason”.
However, I would like to think that Kant’s intention is worldviewism. That is, what Kant
had to say in these antinomies is this:• Without the principle (i.e., the worldview), we can’t say anything• A worldview with antinomies should not be adopted
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10.1 ”Surely You’re Joking, Mr. Kant!”: Antinomy
If so, I agree with Kant. As emphasized throughout this paper, we must again emphasize theimportance of worldviewism ”Begin with the worldview!” (cf. (B) in Sec.1.3.1).
Remark 10.2. Discussions without a worldview are pointless. For example, we already know
the followings.
(♯1 ) Zeno’s paradoxes can be clariﬁed in the motion function method (cf. Sec.2.4)(♯2 ) Syllogism does not hold in quantum mechanics (cf. Sec.4.3),(♯3 ) The statement ”Only ’now’ exists” violates the linguistic Copenhagen interpretation. (cf.
Sec.6.1),(♯4 ) The reason that Anselmus’ ”Arguments for the existence of God” is not clear is due tothe fact that it is not being discussed under a certain worldview. (cf. Sec.6.4),(♯5 ) The diﬀerence between Geocentrism and Heliocentrism was revealed only under Newtonian mechanics (cf. Chap.7),(♯6 ) The qualia problem is solved in quantum language (cf. Sec.9.5.1)(♯7 ) ”Brain in a vat” is solved in quantum language (cf. Sec.9.5.2)(♯8 ) The problem: ”What is space-time?” can be answered from the linguistic point of view.
(cf. Sec 9.2 [Leibniz-Clarke correspondence)(♯9 ) Hume’s problem of induction and Grue Paradox can be solved in quantum language. (cf.
Sec.9.7)(♯10 ) Kant’s Antinomy arises from an argument without a worldview (in this section).

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Chap. 10 Kant: Copernican revolution

10.2

Kant’s epistemology

10.2.1

Kant’s compromise

Now, it is usually said that(A) Kant’s ”Critique of Pure Reason” is a kind of compromise between Continental Rationalism and British Empiricism
That is,−→Descartes[The father of modern philosophy]












[Locke, Berkeley, Hume]












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











[tabula rasa, experience]

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

Kant philosophy[summing-up, compromise]

The meaning of ”compromise” is as follows.
A priori concept is, for example,(B) sensibility (= space-time perception) and understanding (=thought)
which is within Continental Rationalism and not ”tabula rasa”. But,(C) Cognition, judgment is going to be formed gradually through the experience
which is similar to British Empiricism.
♠Note 10.3. (= Summary 11.31) Consider ”language” and not ”cognition”. That is, consider
the following linguistic turn (cf. Chap.11):(cognition)(language)

mind-matter dualism(=epistemology) −−−−−−−−→ linguistic philosophy
linguistic turn(Descartes·Kant)(Wittgenstein)

Then, in several languages (ordinary language, mathematics, Newtonian mechanics, programing
language, etc.), we say tat(♯1 ) ”ordinary language” is like tabula rasa (i.e.,British Empiricism)(cf. Note 9.2)(♯2 ) Mathematics is nativism like Continental Rationalism(♯3 ) quantum language is like Kant’s compromiseThis will be again discussed in Summary 11.31 in Chap.11.

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10.2 Kant’s epistemology

10.2.2Thing-in-itself, Copernican revolution; from copy theory toconstitution theory

Kant thought that(D) We can understand the ”world” only through the human perception. Also, cats canunderstand the ”world” only through the cat perception. Thus,
There is ”cat’s world” for cats. and further, there is ”butterﬂy’s world” for butterﬂies.
If there are aliens whose cognition ability is ﬁner than ours, their world is diﬀerent ours.
Although the diﬀerence of the worlds is made by that of the cognition ability, it is surethere exists something, which is called ”thing-in-itself” by Kant.
That is, Kant thought as follows.
• we do not perceive the world as copy, but we perceive the world as it is constituted bycognition ability.
That is, Kant proposed so called Copernican revolution such thatfrom ”copy theory” to ”constitution theory”
namely,(E1 ) It’s not ”the world ﬁrst, recognition later”, but ”recognition ﬁrst, the world later”.
or(E2 ) Recognition is not about painting a photorealistic picture, but it is similar to painting an
abstract picture.
Also, we see the following table:
Seeing the above table, some may think:(E3 ) Is it true?
Locke −−−−−−→ Kant
progress
This will be answered in Problem 10.6 later.
♠Note 10.4. Many scientists may think of Kant’s discovery (= the Copernican revolution) as a
triﬂe, and may wonder, ”Why did modest Kant give it such an exaggerated name?”. Thus, it
is natural to consider:200

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Chap. 10 Kant: Copernican revolution

Table 10.1: 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

secondary quality

Kantphenomenon

recognition

quantum language

measured value

observable[C](= matter)/[idea world]/[matter]primary quality[matter]/[thing-in-itself]state[system]• Kant believed the Copernican revolution as the greatest discovery in the history of philosophy.
I agree with him. Namely, I consider thattransendenatal idealism

idealism as immature realism(F1 ) fake idealism (in pre-science) −−−−−−−−−−−−−→ true idealism (in philosophy)(Descartes, Locke, Leibniz, Hume)

Copernican revolution(Kant)

That is, I believe that
(F2 ) ”the Copernican revolution” = ”the discovery of philosophy”
For the further arguments, see Problem 10.6 later.

♠Note 10.5. As said in the linguistic Copenhagen interpretation (E3 ) in Sec. 1.1.2, ”measuring
instrument” is superior to ”matter (= thing-in itself)”. Recall Berkeley’s saying:• To be is to be perceived.
which is similar to ”Recognition is previous, the world is later”. However, it was Kant who fully
understood (F).

10.2.3”Critique of Pure Reason (1781)”

This section is the preparation of next chapter 11.
Explanation 10.3. (The preparation of Explanation 11.30 in Chap.11)”Critique of Pure Reason (1781)” : Extracted from Microsoft Encarta (DVD version,
Japanese (2009)——————————————————–The Critique of Pure Reason made it clear that the [cognitive] ability of human beings is not
merely to passively [copy] things in the world, but rather to actively work on the world and201

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10.2 Kant’s epistemology
create the objects of their recognition on their own (British Empiricism) . The creation does
not mean the creation from nothing. The world is already there in some way, and informationfrom this world, which can be given to us through our senses, is necessary for [recognition] to beestablished. However, this information, as it is, is chaotic and confusing. The [cognitive] abilityof human beings must give order to the information of these confused senses through a certain
form (i.e., a priori comprehensive judgment and recognition : Continental rationalism) that isinherent to them, and only then can they bring together the objects of uniﬁed recognition (i.e.,posteriori recognition). According to Kant, the form of [cognition] is a priori comprehensive
judgment and recognition. That is,(i) [Form of sensitivity(intuition)(Space-time (=R × R3 ))](ii) [Form of understanding(thinking)] (e.g., the notion of quantity, such as single or many,and the notion of relations, such as causality)
If so, then the propositions ”all things are in time and space” and ”everything follows acausal relation” apply unconditionally to all objects of experience, even though they cannot
be proved empirically. This is because the object can be constituted only in the form of space,time, and causal relations. This is similar to the case, for example, where the statement ”The
world is green” is considered to be a correct statement for all human beings if they all wear
green sunglasses.
(MSN; Encarta encyclopedia. 2009 DVD Japanese version(translated by the author)).

That is, Kant proposed:(G1 ) the Reconciliation between Rationalism and Empiricism:[tabula rasa, experience]








British Empiricism






[”Critique of Pure Reason”]

[Locke, Berkeley, Hume]
Kant philosophy
−→




[a
priori,nativism][summing-up, compromise]








Rationalism
Continental
[Leibniz](G2 ) the Copernican revolution:
Copy theory( the world is previous, recognition is later)−−−−−−−−−−−−→
Copernican revolution

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10.3 * What is causality?
As mention in Sec 4.2, Aristotle considered the cause of the movement to be the ”purpose” ofthe movement. Although this was what should be praised, it was not able to be said that ”thepurpose was to the point.” For human beings to discover that the essence of movement andchange 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 wecall it ”birth of 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 10.6. 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.

10.3.1

Four answers to ”what is causality?”

As mentioned above, about ”what is an essence of movement and change?”, it was once
settled with the word ”causality.” However, not all were solved now. We do not yet understand”causality” fully. In fact,
Problem 10.4. 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.
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10.3* What is causality?
(A) [Realistic causality]: Newton advocated the realistic describing method of Newtonianmechanics as a ﬁnal settlement of accounts of ideas, such as Galileo, Bacon, and Descartes,and he thought as follows. :”Causality” actually exists in the world. Newtonian equation described faithfully
this ”causality”. That is, Newtonian equation is the equation of a causal chain.
This realistic causality may be a very natural idea, and you may think that you cannot thinkin addition to this. In fact, probably, we may say that the current of the realistic causal
relationship which continues like
”Newtonian mechanics−→ Electricity and magnetism−→ Theory of relativity−→ · · · ”
is the mainstream of science.
However, there are also other ideas, i.e., three ”non-realistic causalities” as follows.
(B) [Cognitive causality]: David Hume, Immanuel Kant, etc. thought as follows. :We can not say that ”Causality” actually exists in the world, or that it does notexist in the world. And when we think that ”something” in the world is ”causality”,
we should just believe that 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 isKant’s famous ”Copernican revolution” (i.e., ”Kant was awakened from his dogmatic slumber
by Hume’s idea and came up with the Copernican revolution”), that is,”recognition constitutes the world.”
which is considered that the recognition circuit of causality is installed in the brain, and when
it is stimulated by ”something” and reacts, ”there is causal relationship.”♠Note 10.7. 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.

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Chap. 10 Kant: Copernican revolution
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 most greatest discovery in thehistory of philosophy. I agree to his opinion. For additional explanation about this, see Problem
10.6 later. Also, see Section 11.5.2.

(C) [Mathematical causality(Dynamical system theory)]:

Since dynamical systemtheory 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
·
·
·
·
··


 dωn
(t) = vn (ω1 (t), ω2 (t), . . . , ωn (t), t)
dt(10.1)

and, we think that(♯) the phenomenon described by the state equation has ”causality.”This is the spirit of dynamical system theory (= statistics). Although this is proposed under
the confusion of mathematics and worldview, it is quite useful. In this sense, I think that (C)should be evaluated more.
(D) [Linguistic causal relationship (MeasurementTheory)]:

The causal relationshipof measurement theory is decided by the Axiom 2 (causality; Sec. 1.1) of this chapter.
If I say in detail,:
Although measurement theory consists of the two Axioms 1 and 2, it is the Axiom 2that is concerned with causal relationship. When describing a certain phenomenonin quantum language (i.e., a language called measurement theory) and using Axiom
2 (causality; Sec. 1.1) , we think that the phenomenon has causality.

Summary 10.5. The above is summarized as follows.
(A) World is ﬁrst
(B) Recognition is ﬁrst
(C) Mathematics(buried into ordinary language) is ﬁrst
(D) Language (= quantum language) is ﬁrst

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10.3* What is causality?
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 10.8. 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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10.4

Summar; Descartes=Kant philosophy

10.4.1

Before Kant

Every thing started from Descartes ﬁgure(=Figure 1.2=Figure 8.1), i.e., ”mind”, ”body”, ”matter”. For example, in ”An Essay Concerning Human Understanding (1689)”, Locke might think
as follows.
(A) 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.
Leibniz (in ”The human being intelligence new discussion”1703) which advocated an objection in Locke can also be conscious of Newton. After all, Unproductive confrontation structure”Locke vs. Descartes·Leibniz” began. That is,[tabula rasa, experience]

British Empiricism[a priori, nativism]vs.

Continental Rationalism[Descartes, Leibniz][Locke, Berkeley, Hume]And further, through Berkeley, Hume, modern philosophy had been ﬂowering as follows.

[”Discourse on the Method”]−→

Descartes[The father of modern philosophy]












[Locke, Berkeley, Hume]












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











[tabula rasa, experience]

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

Kant philosophy[summing-up]

According to Platonic method of telling philosophy (cf. (G) in Sec. 1.3), we may expect
the following:(B): The ﬁctional worldview (western philosophy)
The ﬁctional worldview (=How to tell philosophy due to Plato) is as follows.
you should do soworld is so(B)therefore−−−−−→ ethics, morals

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

main subject

Therefore,][(B1 ) world is so is secondary,[](B2 ) you should do so is main theme

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10.4 Summar; Descartes=Kant philosophy
In spite of the above (B1 ), we think that(C) In modern philosophy, philosophers might be too eager to the preface (i.e., ﬁctional worldview).
This might be due to the fact that rivalry to Newtonian mechanics was too strong. Or, Christianity might hope that modern philosophy played a role of rivalry to Newtonian mechanics.

10.4.2Is Kant a progress from Descartes ?

10.4.2.1

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 ?
Thus, many people thought:(E1 ) 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.
The philosophy enthusiasts of the time might think:(E2 ) Kant’s book is diﬃcult to read, more than 600 pages long. Kant’s consideration of ”Understanding”, ”reason” and ”sensibility” is not even in the realm of scientiﬁc hypothesis.
It is more of a literary art than a science. Therefore, it is unlikely that scientists will
still be paying attention to it in 300 years’ time. However, we would be grateful if thisbrought an end to epistemology. Dr. Kant’s brain is as accurate as a clock, let us believehim. Since it’s a compromise of British empiricism and Continental Rationalism, none
will complain about it.
Thus, Kant theory was the conclusion declaration named ”uniﬁcation”.
(F1 ) Greatness of Kant is to have prevented that epistemology faces the direction of the brain
science by the showy name called ”Copernican revolution”.
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Chap. 10 Kant: Copernican revolution
That is, Kant understand that, even if the epistemology is clariﬁed by the brain science, thisis non-sense from 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 liketo 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 theterm: ”Copernican revolution”.
Thus, Kant follows Platonic method of telling philosophy such that
”Critique of Practical Reason (1788)””Critique of Judgment(1790)””Critique of Pure Reason (1781)”
thereforeﬁctional worldview (literary truth) −−−−−→ ethics, morals, aesthetics(G)preface, introduction, (ﬁctional)premise, expedient

main subject

This was a great success. Today, no scientist is interested in ”Critique of Pure Reason (1781)”.
However, ”Critique of Practical Reason (1788)” is a must for any ethicist. Thus, I think thatthe (G) is the most typical example of Platonic method of telling philosophy.
This implies the end of Grand Narratives (i.e., epistemology), and the start of ”non-physicallaw (i.e., the philosophy of proverb)” 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.
10.4.2.2

Is Kant a progress from Descartes ?

I think that(I) Nevertheless, many readers must think that the Copernican revolution is

far too overrated.
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10.4 Summar; Descartes=Kant philosophy
In the next chapter, I will show that this is a fair estimation. Here, I will say somewhat more
about this issue as follows.
Problem 10.6. It is sure that both Descartes worldview and Kant worldview are useless.
Thus, the following problem may not be easy.
(J) is the following reform progress?
mind-matter dualistic idealism

mind-matter dualistic idealism

Descartes’ worldview (literary truth) −−−−→ 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 ”theory without experimental veriﬁcation”(♯2 ) ”Idealism” in Kant’s worldview means ”transcendental idealism”, that is, ”idealism
made by the Copernican revolution”.
And, we can show that ”transcendental idealism” is similar to the idealism of quantum language. Thus, we can conclude that the (J) is yes. The precise answer will be presented in
Answer DDD11.12Answer of Chapter 11.
♠Note 10.9. 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,the philosophy of worldview is metaphysics, which cannot be determined by experiments. Thus,
the question ”Did the philosophy of worldview make a progress?” is not easy to answer. Thatis because, if we consider that• the western philosophy was able to keep freshness for a long time by renewing a prefacepart (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 this paper, we assert that(♯1 ) the philosophy of worldview has been making a progress. And moreover, it ﬁnally converges
to quantum language.
More precisely, we assert that(♯2 ) If ”to make progress” is deﬁned by ”to come near quantum language” (cf. Assertion 1.5),
then the philosophy of worldview has been making a progress.

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Chap. 10 Kant: Copernican revolution

After Kant: Hegel and Husserl

10.4.3

Recall that our purpose of this paper is to show that
Plato −−−−→ Descartes −−−−→ Kant −−−−→ Wittgenstein −−−−→ Quantum language
progress

progress

progress

progress(10.2)
where −−−−→ means ”getting closer to quantum language”.
progress

If so, readers may think, ”Why aren’t the names of great philosophers like Hegel and Husserl
mentioned?”
10.4.3.1

The reason that Hegel is omitted in (10.2)

Hegel (1770-1831) was a German philosopher and an important ﬁgure of German idealism. He
studied Kant’s work:”Critique of Practical Reason (1788)””Critique of Judgment(1790)””Critique of Pure Reason (1781)”(K)

thereforeﬁctional worldview (literary truth) −−−−−→ ethics, morals, aesthetics
preface, introduction, (ﬁctional)premise, expedient

main subject
therefore

And he might think that Kant’s explanation of the above ” the truth −−−−−→ the good ” isnot suﬃcient, that is, the relation between the truth (Critique of Pure Reason) and the good
(Critique of Practical Reason) is not clear. And thus, he wrote ”The Phenomenology of Spirit”(1807), which is his most widely discussed philosophical work. Compared to Kant’s book
”Critique of Practical Reason”, this work may be popular with general philosophy enthusiasts.
However, as I have noted many times in this paper, I think that there is little relationshipbetween truth and ethics. Also, note that our interest is concentrated to the worldview. Hence,
Hegel is farther from quantum language than Kant. Thus, we omit ”Hegel” in (10.2).
10.4.3.2

The reason that Husserl is omitted in (10.2)

Edmund Husserl (1859–1938) was the principal founder of phenomenology and thus one of themost inﬂuential German philosophers of the 20th century. He wrote ”Ideen” (1913), in which
he tried to reinforce Kant’s work (Critique of Pure Reason) from a psychological perspective
and to establish ”the foundation for various disciplines”. In fact Ideen is one of the mostinﬂuential books in 20th century. However, as I have noted many times in this paper, I thinkthat the truth in philosophy is rather literary. And thus, his phenomenology is rather literaryand not scientiﬁc. It was supposed to be a big theory (such as ”the foundation for various
disciplines”), but most scientists today are not interested in it. Also, note that our true interest
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10.4 Summar; Descartes=Kant philosophy
is concentrated to the (scientiﬁc) worldview. Hence, Husserl is farther from quantum language
than Kant. Thus, we omit ”Husserl” in (10.2).

10.4.3.3

It’s not that I have a low opinion of Hegel and Husserl.

It’s not that I have a low opinion of Hegel and Husserl. Rather, I don’t have the philosophicalbackground to be able to evaluate their accomplishments. Thus, I think that

progress′ *

Hegel

•
Plato −−−−−−−→ Descartes −−−−−−−→ Kant • −−−−−−−→ Wittgenstein −−−−−−−→ QL
progress(QL)
progress(QL)
progress(QL)• progress(QL)
progress′′ j

Husserl

where [progress(QL)] is deﬁned by ”to come near quantum language” (cf. Assertion 1.5). Also,
the above [progress′ ] and [progress′′ ] is respectively diﬀerent from [progress(QL)]. Of course, thedeﬁnition of ”progress” is not always determined uniquely. For example, Husserl may say that

progress′ *

Hegel

•
−
−
→
Plato −−−−−→
Descartes−
Kant• −−−−−−−→ Wittgenstein −−−−−−−→ QL−−
progress(QL)
progress′′
progress′′• progress(QL)
progress′′ j

Husserl

where [progress′′ ] is deﬁned by ”to come near Husserl”. However, it should be noted that
• Of the worldviews mentioned above, quantum language is the only one that is scientiﬁcallyuseful.

♠Note 10.10. 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Boole, Frege, Peirce, Saussure, Russell, Wittgenstein, Hempel, Popper








(♭23 ): the mechanical worldview (statistics, quantum language) Parmenides, Zeno, J. Bernoulli, statistics (e.g., Fisher), quantum language

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Chapter 11
Linguistic philosophy and quantum
language
The purpose of this Chapter is to show the following(♯1 )Answer 11.12

Descartes −−−−−−−−→ Kant −−−−−−−→
progress ⃝1

progress ⃝1linguistic phil.

−−−−−−−→ Quantum language
progress ⃝2

Saussure (Remark 11.13)
Wittgenstein (Sec. 11.6.3)

where −−−−−−−→ means ”getting closer to quantum language”.
progress ⃝1

Assume that ”linguistic philosophy” is classiﬁed as follows. ((♭122 ),(♭222 ) are mathematics and(♭522 ) is linguistics)
 12
 (♭22 ) foundation of math. (Cantor, Gödel), (♭22 ) math. (Bourbaki),




 3

 (♭22 ) analytic philosophy, symbolic logic (Frege, Russell, Wittgenstein,...)(♭22 )

(♭422 ) analystic, scientiﬁc philosophy (Hempel, Popper, ...)






 5
(♭22 ) linguistics, ordinary language (Saussure, Wittgenstein (except logic))
If so, the above (♯1 ) can be rewritten more precisely as follow:(♯2 ) Descartes=Kant −−−−−−−→
progress ⃝1

Analytic phil. (♭322 , ♭422 , ♭522 )(♭522 ):Saussure (Remark 11.13)
3(♭22 , ♭522 ): Wittgenstein (Sec. 11.6.3)(♭422 ):Hempel (Remark 11.42)

Also, for the ”−−−−−−−→”, see List 11.26.
progress ⃝2215−−−−−−−→ Quantum language
progress ⃝2

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11.1 Linguistic philosophy, Analytic philosophy

11.1

Linguistic philosophy, Analytic philosophy

11.1.1Analytic philosophy; A philosophy without a worldview

The expiration date of epistemology had expired in Kant’s time. Philosophers had to ﬁnd
their next subject. Some philosophers (Frege, Russel, etc.) have tried to create philosophy(i.e., analytic philosophy) from scratch, abandoning the great tradition of the dualistic idealism(from Plato to Kant). I respect their pioneering spirit, but whether or not they succeed isanother matter. That is, we have the following question:• Is analytic philosophy a true philosophy?
I think no one has answered this question yet.
Roughly speaking, we have the following diagram:
Diagram 11.1. [ A brief modern history around analytic philosophy]
Incompleteness theoremGödel (1931)
Turing (1936)

Tech.
−−−−−→(ϵ − δ)-deﬁnition of limit
Naive set theory
Karl Weierstrass
Cantor−−−−−→−−−−−→(1815-1897)(1891)

AI:(♭022 )great successTuring machinex
Axiomatic set theory
ZFC-Axiom(1921)xmath.
−−−−−→

found. math. (♭122 )modern math.(♭222 )great successlogical worldview
propositional logic
G. Boole(1847)

predicate logic−−−−−→ Gottlob Frege −−−−−→(1879)

Type theory
B. Russell(1903)
Saussure(1857-1913)

Analytic phil. (♭322 )phil.
−−−−−−−−−−→ Scientiﬁc phil.(♭422 )Wittgensteina little successlinguistics−−−−−−−−→

linguistics(♭522 )promising

Recall Classiﬁcation 1.9 [ the 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 (≈ analytic, scientiﬁc phil.)22(♭2 ) : the idealistic worldview
Frege, Saussure, Russell, Wittgenstein...
(dualism)







(♭
):
the
mechanical worldview

23






statistics,
quantum language(♭3 )Others (Thinking Tip, etc.) : Darwin’s theory of evolution, Hegel’s dialectic, etc.
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Chap. 11 Linguistic philosophy and quantum language
Further, the above ”the logical worldview (♭22 )” is classiﬁed as follows.
Classiﬁcation 11.2. [The location of the logical worldview] Assume that the above is classiﬁed as follows.
(A0 ): Artiﬁcial intelligence, ”Logic circuit may be more powerful than any worldview”(♭022 ) Computer; (Gödel, Turing, von Neumann, etc.) (A1 ): {mathematics: ”Describe math. by logic ( and set theory)!”(♭122 ) foundation of math. (Boole, Frege, Cantor, Russell, Gödel,...),

(♭222 ) math. (Bourbaki),(A2 ):the logical worldview (or precisely, the naive logical worldview) :(♭22 )

”Describe
everything by logically!”


{ 3


(♭22 ) analytic philosophy, symbolic logic (Frege, Russell, Wittgenstein,...)

(♭422 ) scientiﬁc philosophy (Hempel, Popper, ...)(A3 ): linguistics: ”Describe everything reasonably!”(♭522 ) linguistics, ordinary language (Saussure, Wittgenstein)
In this chaper, we devote (A2 ) and (A3 ) since (A0 ) and (A1 ) are mathematics and technology.

♠Note 11.1. In ancient Greece, mathematics was a ﬁeld of philosophy. Philosophers still tend
to confuse philosophical and mathematical problems. For example, as mentioned in Section
2.4, Zeno’s paradox is the problem of a worldview. However, there are still many philosophers
who think that Zenon’s paradox (Achilles and a tortoise) has been solved by considering it asa geometric series problem. I think that analytic philosophy is also confused between mathematics and philosophy. As mentioned in Diagram 11.1, it is Wittgenstein that made analyticalphilosophy independent of mathematics. That is,(♯1 )Philosophy

Symbolic logic, Set theory −−−−−−−−→ Analytic philosophy
Frege, Russell

Wittgenstein

It was enthusiastically supported by many amateur philosophy enthusiasts. However, I think
that, from the theoretical point of view, the following problem is not yet settled.
• Was Wittgenstein’s attempt successful?
As mentioned in this chapter, my opinion is ”There is no direct relationship between logic andthe world in analytic philosophy” and thus, ”The (♯1 ) is unpromising as a worldview”. There
are a lot of negative opinions for analytic philosophy (cf. [81]). My position has always been to
consider quantum language:(♯2 )lingiustic turn

Quantum mechanics −−−−−−−−→ Quantum language

And, in this section, I will say that the (♯2 ) is signiﬁcantly more productive than the (♯1 ). And
thus, I conclude that•

Analytic philosophy −−−−−→ Quantum language
progress

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11.1 Linguistic philosophy, Analytic philosophy♠Note 11.2. I am not proﬁcient in the use of the terms ”analytic philosophy”, ”scientiﬁc philosophy (=philosophy of science)”, or ”linguistic philosophy (=philosophy of language)”. We might
be forgiven for thinking that these three are the same. In this paper we consider the following.
But there is no strict distinction among the three.
• scientiﬁc philosophy · · · · · · the main theme is science• ”analytic philosophy”≈”linguistic philosophy”. The linguistic philosophy is broader, if I
may say so.
In short, I don’t know much about these things as philosophy. Most people agree that (A0 ), (A1 )and (A3 ) do not belong to philosophy. Even if someone argues that (♭22 ) has nothing to do withphilosophy, I may not dispute it. That is because analytic philosophy today names a style ofdoing philosophy (i.e., like (A3 ): ”Describe everything reasonably!” or ”Aim for argumentative
clarity and precision!”), not a philosophical program or a set of substantive views.

11.1.2(A1 ), (A2 ) and (A3 ) are related?

I predict the next progress:(quantum mechanical worldview)(A0 ) logical worldview −−−−−→
progressquantum language−−−−−→ AI
progress

Therefore, (A0 ) will be the most important after about 2050 AD. However, I omit it, since I
don’t know much about technology.
Let us (A1 ), (A2 ) and (A3 ) explain more precisely.
11.1.2.1(A1 ): mathematics: ”Describe math. by logic ( and set theory)!” inClassiﬁcation 11.2

Above, the part involving Cantor, Frege, Russell, Zermelo, Fraenkel and so on is the biggest
event in the history of mathematics, because they built mathematics with axiomatic set theory.
That is,Describe mathematics by logic, set theory!

(A1 ): (♭122 ) and (♭222 ):axiomatic logic, set theory
Cantor, Frege, Russell, Zermelo=Fraenkel and so on

therefore−−−−−→ Math. (Found. math. (♭122 ) and modern math. (♭222 ))This was one of the most successful ﬁelds of the 20th century. However, it should be noted thatthe above is mathematics and not philosophy ( i.e., worldview).
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Chap. 11 Linguistic philosophy and quantum language
11.1.2.2(A2 ):the logical worldview: ”Describe phenomena with logic!” in Classiﬁcation 11.2

The above success of (A1 ) led many philosophers to expect that science could be described bysymbolic logic and set theory as well. This is called the logical worldview (or, the set theoreticalworldview) in this text. That is,(A2 ) : (♭322 ) and (♭422 ): analytic philosophy”Describe phenomena with logic ( and set theory)!”symbolic logic, set theory

therefore−−−−−→ the world will be clear

the logical (set theorical) worldview

if the world is restricted to science, it is called ”scientiﬁc philosophy (♭422 )”.
Many philosophers might consider that
• (A1 ) is the greatest, and therefore, (A2 ) must be promising too.
This optimistic outlook, I believe, prompted the spreading of analytic philosophy. However,I think that the above (A2 ) did not achieve great success. There are many reasons for thisfailure, but I mention just one:(B1 ) the method of regarding a ﬁeld of mathematics (e.g., logic, set theory) as a worldview is
too straightforward.
(Also, see Remark 11.3 below.)
In fact, philosophers in scientiﬁc philosophy (♭422 ) found the attempt (A2 ) diﬃcult to succeed.
This will be discussed in Section 11.7 (Flagpole problem) and Section 11.8 (Hempel’s raven
problem). Thus, I conclude that(B2 ) analytic philosophy is a philosophy without a worldview (or precisely, without a goodworldview).
This means that the boundary line between analytic philosophy (A2 ) and linguistics (A3 ) is
not clear. Therefore, philosophers in analytic philosophy (♭322 ) were also interested in linguistics(♭522 ).
Remark 11.3. (i): There may be a variety of opinions (A2 ). What they (e.g., Frege, Cantor,
Russel, Zermelo, Frankel, etc.) revealed is in (A1 ) as follows.
(C) mathematics (or, symbolic logic) is not directly related to the world.
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11.1 Linguistic philosophy, Analytic philosophy
Thus, we need ”worldview”, which is a bridge between ”logic (mathematics)” and ”the world”.
That is,mathematics(logic) ←− worldview −→ world(♯1 )Therefore, any worldview should contain non-mathematical words as the key-words ( as wellas mathematical terms). For example, everyone knows thatmathematics ←−(♯2 )diﬀerential equation

Newtonian mechanics(space-time, force, mass, velocity, ...)−→ world

Also, ”the logical worldview” may be represented as loows.
logic, axiomatic set theory) ←−(♯3 )logic, naive set theory(set, a collection of things )−→ world(Also, again see Note 2.8: pure Pythagoreanism.)
In the above, don’t get nervous about the diﬀerence between elementary set theory (=naive set
theory= Cantor’s set theory, cf. [25]) and mathematical set theory (=axiomatic set theory).
Here, in elementary set theory, ”set” is deﬁned by ”a collection of things”. This is the sameas the use of ”set” in everyday language. Thus, using naive set theory, the above (♯3 ) can be
rewiritten bynothing

logic, naive set theory ←−(♯4 )−→ world

Since even most mathematicians are not familiar with mathematical set theory but elementary
set theory, it is more natural to think of (♯4 ) than (♯3 ). Therefore, there is a reason to consider
that the logical worldview (A2 ) is not a worldview, at least is is not a worldview in the true
sense.
(ii): Analytic philosophy can only be successful if any of modal logic, temporal logic, intuitionist
logic, fuzzy logic, quantum logic, etc. are successful in terms of application. But I think thispossibility is hopeless. There are various ﬁelds in mathematics, but I don’t think there is anecessity to start from logic. For example, there is no reason to adhere to symbolic logic when
expressing ”time” mathematically.
////
11.1.2.3(A3 ): linguistics: ”Consider everything logically!” in Classiﬁcation 11.2

It is a matter of course that linguistics is very important. And no one can disagree with thefollowing spirit.
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Chap. 11 Linguistic philosophy and quantum language(A3 ) Let’s use the word correctly!

Speak carefully!

Consider everything logically!

Try

to be reasonable!
This is not a philosophical assertion; it may be a moral slogan. This spirit might be enlightenedby Wittgenstein’s saying:(D) philosophical problems arise from insuﬃcient attention to the variety of natural languageuse.
The (A3 ) is having many meanings. However, the inﬂuence of Wittgenstein on the active
exchange between linguistics and philosophy was immeasurable. With the development of
automatic translation technology and other technologies in (A0 ), the future of linguistics is
promising.
However, it should be noted that neither linguistics nor mathematics is philosophy (i.e.,
worldview).
Remark 11.4. We have two types of logic (i.e., symbolic logic and practical logic) (cf. Note1.15), i.e.,{
symbolic logic (= axiomatic logic =mathematical logic) ( cf. Postulate 11.5)(E)practical logic (= inference)
However, in the logical worldview, we tend to confuse the two, or regard practical logic assymbolic logic. The most important ”logic” in philosophy is practical logic. Recall the worldviewism in Sec. 1.3.1 as follows.
b The worldviewism in Sec. 1.3.1(A):
The worldviewism has the following form:
world is sob1 ) worldview(Apremiseconclusion
therefore−−−−−→ practical logic (≈ inference, how to think, calculation), discussions, etc.
subject

That is,b2 ) Since the world is like this, think like this!
(A
That is, the worldviewism is the spirit such that ”Start from worldview”. I think thatb3 ) It’s not an exaggeration to say ”philosophy = worldviewism”.
(A
Thus we have three kinds of ”logic” as follows:
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11.1 Linguistic philosophy, Analytic philosophy(♯1 ) ”logic” in (A1 ) = ”logic” in (A2 ) = symbolic logic(♯2 ) ”logical” in (A3 )≈”reasonable”≈”careful”b(♯3 ) practical logic (A)In analytic philosophy, the importance of ”logic” (i.e., ”logic” in (♯1 )) is often emphasized. For
example, they may sometimes say that(♯4 ) Doing philosophy without knowing logic is like doing French literature without knowing
French.
However, it should be noted that this ”quote” is said under the conﬂation of symbolic logic andpractical logic.
In order to master analytical philosophy (i.e., the logical worldview), I think it’s enoughto know the logic and set theory taught in the ﬁrst year of university in the math department. It doesn’t matter if the philosopher cannot solve problem concerning the deﬁnition of”limn→∞ an = a”, which may be taught in the second year of university ( cf. Exercise 11.8 (iii))). Predicate logic, etc., is not available to most physicists either. Only mathematicians shouldbe able to use it. As mentioned in Remark 11.3 (ii), I think all attempts to use logic at thegraduate level have failed.
It is not the study of logic that is important, but the following.
• anaytic philosophy has been studied and supported under the confusion between symbolic
logic and practical logic.
In this sense, analytic philosophy is not a good theory. But still, I can say the following.
()
Answer 11.12
Kant −−−−−−−−→ linguistic, analytic philosophy −−−−−→
quantum language
progress(the logical worldview))

progress(the quantum mechanical worldview)

This will be discussed in this chapter.
////

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Chap. 11 Linguistic philosophy and quantum language

11.2

George Boole (1815-1864) and Gottlob Frege (18481925)

George Boole 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
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.
Postulate 11.5. [ Symbolic logic (i.e., Propositional logic and predicate logic)]
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))[Propositional logic] : G. Boole(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 means ”and”, ”or”, ”not”, ”implies”. Note that P1 → P2 isdeﬁned by ¬P1 ∨ P2 ”. Also, De Morgan’s laws say that P1 ∨ P2 = ¬(¬P1 ∧ ¬P2 ) holds.
Also, assume that Pθ (θ ∈ Θ ≡ {1, 2, ..., N }), then it clearly 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∈Θ[Predicate logic]: G. Frege(A2 ) The above ﬁnite set Θ(≡ {1, 2, ..., N }) can be extended to any inﬁnite set Θ.

♠Note 11.3. Some may think that the predicate logic (i.e., inﬁnite operations) is a slight generalization of the propositional logic (i.e., ﬁnite operations). However, the predicate logic is
powerfully expressive. In fact, Zermelo-Fraenkel Axiom asserts that all mathematical propositions can be described by only ”predicate logic” and ”∈”. That is, Postulate 11.5 completely
works for all mathematical propositions.

Remark 11.6. As mentioned in Note 11.3, Postulate 11.5 completely works for all mathematical propositions. However, Postulate 11.5 does not necessarily work for non-mathematical
propositions. Most propositions we use in our daily lives do not have a deﬁnite truth function.
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11.2 George Boole (1815-1864) and Gottlob Frege (1848-1925)
For example,• if P1 =”American are cheerful”, P2 =”Tyrannosaurs are violent”, P3 =”every raven isblack”, it is diﬃcult to determine the truth function ϕ(Pi ) (i = 1, 2, 3).
As seen in Sec 11.8 later, even the deﬁnition of ”P3 is true” is not easy. Thus, I doubt if thisP is even a proposition. Strictly speaking, I believe that(♯) Postulate 11.5 can only be applied to mathematical propositions. That is, predicate
logic is the language of mathematics.
In short, the deﬁnition of ”proposition” is not clear in non-mathematical ﬁelds. Hence, there
is some diﬃculty in using Postulate 11.5 in philosophy. Therefore, I am skeptical of the strict
use of Postulate 11.5 in philosophy.

Remark 11.7. The works of Boole, Frege and Russell belong to mathematics. It was Wittgenstein who arranged their work as philosophy such that Amateur philosophy enthusiasts supported

Analytic phil.
propositional logic

predicate logic

G. Boole−→ Gottlob Frege −→(1847)|(1879)

L. Wittgenstein
B. Russell −−−−−−−→ (Tractatus Logico-Philosophicus
philosophy(1903)(1921)

Type theory{z}

Confusion of mathematics and philosophy

Russell may have written the plot of the drama above, but Wittgenstein was a great actor
who played his role perfectly. And thus, after Wittgenstein, analytic philosophy has become amajor trend in Western philosophy. Wittgenstein tried to do something that was theoretically
impossible to do, so the assessment of him is mixed. In this text, I value him highly. Philosophy
is not a matter of logic, but of asserting in strong terms. In this sense, Wittgenstein understood
philosophy better than Frege and Russell did.
I think that Frege and Russell have been admired under the confusion of mathematics andphilosophy. Of course, there are other opinions. It can be also said that Frege and Russell, whoused the confusion of mathematics and philosophy to spread analytic philosophy, were morestrategic. On the other hand, Wittgenstein was too simple, in a good way. This will be again
discussed in Sec. 11.6.
////

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Chap. 11 Linguistic philosophy and quantum language
Exercise 11.8. (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

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

Thus, syllogism: (p → q) ∧ (q → r) → (p → r) is always true.
(ii):((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
(iii):[The elementary 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. As mentioned in Note
9.5, Even though Newton did not know the elementary use of such predicate logic, he was able
to make physics. I don’t know any physicist saying:• Doing physics without knowing logic is like doing French literature without knowing French.
I think the only correct one is• Doing mathematics without knowing logic is like doing French literature without knowing
French.

♠Note 11.4. I am not an expert in history of mathematics or history of philosophy, so I am
not familiar with G. Frege. However, in this text, I follow the opinions of the philosophers inanalytic philosophy, and regard him as a genius since Aristotle.

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11.3 Peirce (1839-1914)

11.3

Peirce (1839-1914)

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

11.3.1 * What is Peirce’s abduction?
Charles Sanders Peirce (1839-1914) was an American philosopher, logician, mathematician, andscientist 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).
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 (= inference)
However, in the logical worldview, we tend to confuse the two, or regard practical logic assymbolic logic.
For example, some philosophers have tried to construct worldviews using some ”symbolic
logic” (e.g., modal logic, temporal logic). These may be successful to some extent, but they
are unlikely to become mainstream analytical philosophy.
I think that Peirce’s inference can be also regarded as one of the confusions of the logicalworldview.
Let us explain it as follows.
Let P =”It rains”, Q=”the lawn in the garden gets wet.” Assume that P → Q is true. Since[[P −→ Q] ∧ P ] −→ Q] is true (see Exercise 11.8), it may be natural to consider that(♯1 ) It rained last night, so the lawn is wet this morning.
This may be correct, but what we like to think is next.
(♯2 ) the lawn is wet this morning, so it must have rained last night.
that is, [[P −→ Q] ∧ Q] −→ P ], which does not always hold in the sense of Boole’s logic
as seen in Exercise 11.8. Thus, Peirce wanted to propose the new symbolic logic in which[[P −→ Q]∧Q] −→ P ] is true. (He might be inspired by the success of non-Euclidean geometry.)
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His idea was not successful as mathematical logic, it (= the abduction) was popularized as a”thinking tip”. I think• ”If he had formulated his ideas as practical logic, he would have succeeded”.
This will be discussed in Sec. 11.3.1.1 (the logical worldview) and Sec. 11.3.1.2 (the quantum
mechanical worldview). And I will show that Peirce’s abduction has been supported under the
confusion between symbolic logic and practical logic.
11.3.1.1

Deduction, abduction and induction in ”logic” (i.e., in the logical worldview)

According to Peirce, three kinds of inferences (i.e., deduction, abduction, induction) are important. Let us explain deduction, abduction and induction as follows.
The reader will be convinced that Pearce believed that ”the language of science is logic.”[Deduction]
A typical example of deduction is as follows:)(b ′ ) are often omitted.
In the following, (A′1 ) and (A1(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 whitebeans to black beans is p/(1 − p) where 0 < p < 1):[bag B2 −→”w”(≈ white) or”b”(≈ black)](A2 ) This bean is from this bag B1 : [bag B1 ]
(A3 ) Therefore, this bean is white: [”w”(≈ white)]
It is, of course, obvious and ordinary.

♠Note 11.5. It is clear that the following is a tautology:
[
]
(♯1 )
[B1 −→ w] ∧ [B2 −→ [w ∨ b]] ∧ B1 −→ [w]Thus, the above conclusion (A3 ) can be understood as a consequence of this tautology (♯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)?
This new solution will be introduced in the following section.

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11.3 Peirce (1839-1914)[Abduction]
On the other hand, C.S, Peirce (cf. ref. [88]) proposed abduction. 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(Abeans to black beans is p/(1 − p)):[bag B2 −→”w”(≈ white) or ”b”(≈ black)]b 2 ) This bean (from 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:
[
]
(♯3 )
[B1 −→ w] ∧ [B2 −→ [w ∨ b]] ∧ w −→ [B1 ]b 3 ) can not be understood since this (♯3 ) is not a tautology.
Thus, the above conclusion (Ab 3 ) has a point. For simplicity, assume that 0 ≤ p ≪ 1 ( i.e.,
However, the above conclusion (Amost of the beans in the B2 are black).
b 2 ), assume that this white bean is from the bag B2 .
(♯4 ) After (AIf so, this is a very rare event that has happened since most of the beans in the B2 are black.
The assumption (♯4 ) is unreasonable. That is, it is reasonable to consider that this white beanb 3 ).
is from the bag B1 , namely, (ANow from the theoretical point of view, we have two ways to think the above problem.
⃝:Toinvent another symbolic logic in which (♯3 ) is derived
1



(♯5 )
⃝:To
propose a diﬀerent worldview than the logical worldview.
2b 3 ) in the new worldview.
And to derive (AI think that ⃝1 is impossible. ⃝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,(A
0 < p < 1). 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.
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Chap. 11 Linguistic philosophy and quantum language
11.3.1.2

Deduction, abduction and induction in quantum language (i.e., the quantum mechanical world view)

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. ConFirst, we will show that the abduction [(Asider the state 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 thefollowing 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 1.27 [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 (A229

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11.3 Peirce (1839-1914)b 2 ) is the quantum linguistic representation of abduction (i.e., (Ab 1 )–
Therefore, the above (Bb 3 )). For the sake of completeness, note that (B1 ) and (Bb 2 ) are in reverse problem (cf.
(A
Remark 1.28). That is, we have the following correspondence:−−−−−−−−→ [(A1 ),(A′1 ),(A2 ) −→(A3 )](B1 ): measurement
simpliﬁed form
deduction
Axiom 1 [measurement] (in Section 1.1)yreverse
yreverseb 2 ): inference(B(Fisher’s maximum likelihood method)(11.1)b 1 ),(Ab ′ ),(Ab 2 ) −→(Ab 3 )]−−−−−−−−→ [(A1simpliﬁed form
abductionThus, the scientiﬁc meaning of abduction can be completely clariﬁed in the translation fromlogic to quantum language.
Lastly we should mention thate 3 ) the above (Ae 1 )-(Ae 3 ) (i.e., inductive reasoning) are already discussed in quantum language(B(cf. Section 9.7: Hume’s problem of induction).

Remark 11.9. Recall the worldviewism in Sec. 1.3.1 such as(♯): The worldviewism
The worldviewism has the following form:
think like this! ( conclusion )world is sotherefore(♯1 ) worldview −−−−−→ practical logic (≈ inference, how to think, calculation), etc.
premisesubject

That is,(♯2 ) Since the world is like this, think like this!
Note that the formula (11.1 is the typical example of the worldviewism (♯1 ). That is,[The practical logic is produced by a worldview]

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Chap. 11 Linguistic philosophy and quantum language
the quantum mechanical worldview −−−−−−−−−→
therefore(quantum language)practical logic−−−−−−−−−→ [(A1 ),(A′1 ),(A2 ) −→(A3 )](B1 ): measurement
therefore
deduction
Axiom 1 [measurement] (in Section 1.1)yreverse
yreverseb 2 ): inference(B(Fisher’s maximum likelihood method)b 1 ),(Ab ′ ),(Ab 2 ) −→(Ab 3 )]−−−−−−−−−→ [(A1therefore
abduction

The above says that• a lot of information is lost from the phenomenon in order to express it in logical form if
it is possible.
In this sense, I think that• logic is useful for making broad arguments.
It’s common knowledge in science that, for us humans, we can’t have a rigorous discussion
without doing the calculation, not the logic. Of course, it should be note that the method of
calculation is based on a worldview.
I’m not completely denying Peirce’s abduction. In fact, I think that Detective Sherlock Holmes
was a great master of Peirce’s abduction.
////♠Note 11.7. Philosophers may like logic (i.e., the logical worldview), thus, they may want tothink that deduction, abduction and induction are kinds of logic. However, I think that theargument in Sec. 11.3.1.1 is not natural. On the other hand, we think that our quantum
mechanical formulation is reasonable as follows.
(C1 )
(C2 )
(C3 )

deduction(possible but unnatural)abduction(theoretically impossible)

induction(theoretically impossible)······

measurement(possible and natural)· · · · · · inference (Fisher’s maximum likelihood method)(possible and natural)· · · · · · the law of large numbers(possible and natural)Thus, I think that Peirce’s argument, from the theoretical point of view, highlighted the weaknesses of the logical worldview.

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11.4 Bertrand Russell: ﬁve-minute hypothesis

11.4

Bertrand Russell: ﬁve-minute hypothesis

Bertrand Russell (1872-1970) is a great British logician.
The naive set theory (i.e., Cantor’s set theory) involves contradictions. For example, Russell
showed Russell’s paradox such that
• if it is assumed that {U | U ∈/ U } is a set, then it leads contradiction.
To avoid such a paradox, Alfred North Whitehead and Bertrand Russell proposed ”type theory”(i.e., a kind of axiomatic set theory), which was published as the Principia Mathematica on
the foundations of mathematics in 1910–1913. Also, the axiomatic system of set theory wasdeveloped 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 andFraenkel plus an axiom called the Axiom of Choice.
Also, Russell was a great intellectual and one of the founders of analytic philosophy alongwith his predecessor Gottlob Frege (and student Ludwig Wittgenstein). From the analytic
philosophical point of view, his greatest achievement may have been the production of the”genius Wittgenstein”.
I know almost nothing about Russell’s philosophy by skipping over his book [96]. Thus, I
can understand the mathematician Russell, but not the philosopher Russell. Therefore, I can’t
speak for Russell’s philosophy
I will mention ”Five-minute hypothesis” in Sec. 11.4.1 and ”McTaggart’s paradox” in Sec.
11.4.2. Also, see ref. [99] for the delicate relationship between Russell and McTaggart.

11.4.1 * Five-minute hypothesis
This section was written with reference to the following.
• [68]: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)
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As mentioned frequently in this text, I think that mathematical logic is one of ﬁelds ofmathematics and thus, Russell’s study of logic has little to do with traditional Western philosophy (i.e., Descartes-Kant philosophy). Thus, in this text I am not interested in Russell’s
logical work.
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. [97]), is as follows.
(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:
Note that this hypothesis (A1 ) is related to ”tense”. Thus, the linguistic Copenhagen interpretation (E2 ) in Sec. 1.1.2 says that this (A1 ) is not a statement in quantum language. Thus,
the (A1 ) is not scientiﬁc, that is, there is no experiment to verify the statement (A1 ).
Some may want to relate this hypothesis to skepticism (cf. ref. [97]), However we do notthink that this direction is productive.

Remark 11.10. Also, the above (A1 ) should be compared to the following (A2 )(A2 ) The universe was created in A.D. 2010. (Or equivalently, now is A.D. 2020, and theuniverse was created ten years ago.)
This (A2 ) can be denied by experiment, that is, it is diﬀerent from the fact. Thus, this is a
proposition in quantum language.

11.4.2 * McTaggart’s paradox
This section was written with reference to the following.
• [68]: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)
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11.4 Bertrand Russell: ﬁve-minute hypothesis[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. [82], McTaggart asserted ”the Unreality of Time” as follows.
The sketch of McTaggart’s proof(B1 ) Assume that there are two kinds of times. i.e., ”observer’s time (A-series)” and ”objective time (B-series)”. (Note that this assumption is against the linguistic Copenhagen
interpretation (E2 ) in Sec. 1.1.2.)(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).
About this proof, there are various opinions also among philosophers. Although I cannot
understand the above part (B2 ) (since the properties of A-series are not clear), I agree to him if
his assertion is (B4 ) (cf. ref. [43]). That is, I agree that McTaggart noticed ﬁrst that observer’s
time is not scientiﬁc.
Recall the linguistic Copenhagen interpretation (E2 ) in Sec. 1.1.2:• (i.e.,(E2 ) 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 estimatedmore highly.

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Chap. 11 Linguistic philosophy and quantum language

11.5

Saussure: Copernican revolution in language

Ferdinand de Saussure (1857-1913) was a Swiss linguist. He is widely considered one of thefounders of 20th-century linguistics.

11.5.1

Saussure’s linguistics: What comes ﬁrst, things or words?

Let’s think a little more about the implications of Saussureian linguistics. We tend to thinkthat there are things at the beginning, and that we give each thing a name, just like we put
a label on it. However, that is not the case. Rather, Saussureian linguistics says that weunderstand the order of things by the act of giving them names. I think that his theory isalmost the linguistic version of the Copernican revolution of Kant. That is,copy theory

constructionism

Saussure’s discovery
things −→ words −−−−−−−−−−−−−−−−−−−−−−→ words −→ things
Copernican revolution in language(11.2)

Saussure’s linguistics

For example,(A1 ) I, who live in Japan, an island nation, know the names of many ﬁsh, but rarely distinguishbetween ”cow, bull, ox, calf,” etc.
(A2 ) Also, in Japan, the rainbow has seven colors (i.e., red, orange, yellow, green, blue, indigo,violet), and I was surprised when I ﬁrst heard that there is a country where the rainbow
does not have seven colors.
(A3 ) Also, I think that there is a possibility that quantum language changes our scientiﬁcworldview.

In the above, I’d like to assume that (A1 )∼(A3 ) are about the same.

11.5.2

Several Copernican revolutions

Saussure’s linguistics bridges Kant and quantum language as follows.

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11.5 Saussure: Copernican revolution in language
Assertion 11.11. Summing up, we can write as follows.
copy theory

constructionism

Copernican revolution in cognition
the world −→ cognition −−−−−−−−−−−−−−−−−−−−−−−−−−−→ cognition −→ the world⃝1
Kant
Descartes=Locke




turn
⃝
2 ylinguisticy(Wittgenstein)copy theory

things −→ words

constructionism

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

words −→ things
Saussure’s linguistics
⃝
3 ymechanical turnycopy theory

phenomena −→ QM

Copernican revolution in mechanics−−−−−−−−−−−−−−−−−−−−−−−−−−−−→

quantum mechanics (=QM)

constructionism

QL −→ phenomena
quantum language (=QL)(11.3)
Therefore, I think that• ”The Copernican revolution (due to Kant)” is the greatest discovery in thehistory of philosophy.
That is,the Copernican revolution = the discovery of ”true idealism”

where we think that ”true idealism”=”transcendental idealism”.

♠Note 11.8. As mentioned in Note 10.7, Kant named his discovery the Copernican revolution.
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 most greatest discovery in thehistory of philosophy. I believe that Kant said ”the Copernican revolution” because Kant himself
thought it was the greatest discovery in the history of philosophy. This is by no means Kant’s
big talk. I think that his conﬁdence is realized in Assertion 11.11.

Answer 11.12. [ The answer of Problem 10.6: Is ”From Descartes to Kant” progress?]
Now we can answer Problem 10.6, i.e.,•

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

(Kant)

The above formula (11.3) in Assertion 11.11 says that
Descartes −−−−−−−−−→ Kant −−−−−−−−−→ Saussure −−−−−−−−−→ QL⃝⃝⃝
1 in (11.3)
2 in (11.3)
3 in (11.3)
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Chap. 11 Linguistic philosophy and quantum language
Therefore, we can be convinced that
Descartes philosophy

Copernican revolution−−−−−−−−−−−−→(Descartes, Locke, Leibniz, Hume)

Linguistic turnprogress−−−−−−−−→
progress

Kant(Kant philosophy)

linguistics

mechanical turn−−−−−−−−−→ QL
progress(Saussure linguistics)(11.4)

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

Remark 11.13. I will restate what I wrote at the beginning of this chapter as follows.
 1 (♭22 ) foundation of math. (Boole, Frege, Russel, Cantor, Zermelo, Fraenkel, Gödel),...,






(♭222 ) math. (Bourbaki),





(♭322 ) analytic philosophy, symbolic logic (Frege, Russell, Wittgenstein,...)(♭22 )
(♭422 ) scientiﬁc philosophy (Hempel, Popper, ...)






 5
(♭22 ) linguistics, ordinary language (Saussure, Wittgenstein (except logic))
If so, the above formula (11.4) can be rewritten more precisely as follow:(♯2 ) Descartes=Kant −−−−→ linguistic phil. (♭322 ), (♭422 ),(♭522 ) →− Quantum language
progress⃝:a (♭522 ):Saussure (Remark 11.12)⃝:b (♭522 ): Wittgenstein (Sec. 11.6.3)⃝:
c (♭422 ):Hempel (Remark 11.42)⃝
b and ⃝
c will be discussed later.

11.5.3 * The quantitative expressions of ”signiﬁer” and ”signiﬁed”
Now let us explain the terms ”signiﬁer” and ”signiﬁed”, which were introduced by Saussure.
Deﬁnition 11.14. [”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 ”signiﬁe”.
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11.5 Saussure: Copernican revolution in language
Although Saussure’s proposal (i.e., the above deﬁnition) is very signiﬁcant, this violates theworldviewism (cf. Sec. 1.3.1). As emphasized frequently throughout the paper, I consider(C) Without a worldview, we can’t say anything solid
In fact, Saussure’s idea does not play an important role in science.
Thus, I propose Deﬁnition 11.14, in which Saussure’s idea is realized in quantum language asfollows.

Deﬁnition 11.15. [Membership function (= Fuzzy set, cf. [35, 36, 37, 38, 109])] Let Ω be astate space. A continuous function m : Ω → [0, 1] (i.e., the closed interval in the real R) is calleda membership function. Assume that the state (i.e., quantitative property) of any animal canbe 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 madethem.
(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 membership function 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.
Also, for the probabilistic interpretation of a membership function, see Interpretation 11.39.

With the above preparation, we get the following deﬁnitions.
Deﬁnition 11.16. [”signiﬁer” and ”signiﬁed” in quantum language](F1 ) ”signiﬁer”—–”dog”(F2 ) ”signiﬁed”—–mD : Ω → [0, 1] (or roughly, ΩD )238

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Chap. 11 Linguistic philosophy and quantum language
The above will play an essential role in Hempel’s raven prpblem in Sec. 11.8.

mD

1

Ω

ΩD
Figure 11.1: [m : Ω → [0, 1], and ΩD ]♠Note 11.9. (i); The above deﬁnition is essential to the solution of Hempel raven problem. See
Section 11.8(ii): From the quantum linguistic point of view, As mentioned in Remark 3.12, Saussure’s theory
is located in the following.
•linguistic version−−−−−−−−−−→
Idea
signiﬁer, signﬁed
Plato’s idea theorycongnitiveyversion

Saussure’s linguistic theory

body, secondary quality

membership function

Descartes, J. Locke

Zadeh’s fuzzy theoryquantiﬁcationyyidentiﬁcation

quantiﬁcationjobservable (= 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 thehistory of Western philosophy as the history of dualistic idealism.

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11.6 Linguistic turn: Wittgenstein

Linguistic turn: Wittgenstein

11.6

Ludwig Wittgenstein (1889 - 1951), who was the student of B. Russel, was the most famous
philosopher (in linguistic philosophy) in the 20th century.
His position is as follows.
Simbolic logic, Set theory

z

}|
{
a
⃝ : Gottlob Frege −→ B. Russell
predicate logic

Type theory












L. Wittgensteina
Logico-Philosophicus (=TLP) (1921)⃝:Tractatusb⃝:PhilosophicalInvestigations (1953)−−−−−−→

philosophySaussure: natural language 




b :
⃝

the logical worldview(Linguistics)

That is,(A) Frege and Russell asserted that symbolic logic is the foundation of mathematics. Also,
they wanted to assert that sybbolic logic is the foundation of science.
(B) Saussure discovered the Copernican revolution in linguistics(C) Wittgenstein asserted that ”logic” is the foundation of science (cf. Sec. 11.5.2).
However, I think that Wittgenstein’s attempt was a social success, but it may not have beena success in theory. That is because it is ambiguous whether his ”logic” is symbolic logic orpractical logic. This will be discussed in what follows.

11.6.1 * My scientiﬁc understanding of ”Tractatus Logico-Philosophicus(=TLP)”In the preface of his book ”Tractatus Logico-Philosophicus”, L. Wittgenstein said that(D) This book will perhaps only be understood by those who have themselves already thought
the thoughts which are expressed in it — or similar thoughts.
This is a very signiﬁcant sentence for me. That is because the (D) implies that, I can understandhis theory if what I have said in Sec 4.3.5 is ”similar thoughts”.

♠Note 11.10. ”Tractatus Logico-Philosophicus (= TLP)” is a curious book, one of the leading
philosophy books of the 20th century. It’s a book that encourages people to ”be logical,” yet it’s
not written logically. Furthermore, probably not a single reader has read this book logically. No
reader has understood this book, but it is a philosophical book that everyone is convinced thatit must have some great things to say. Recently, some people have even declared this theory to

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Chap. 11 Linguistic philosophy and quantum language
be nonsensical (cf. refs. [12, 78]). However, this happens frequently in the history of philosophy(cf. Note 3.10). It is not surprising, for example, that both ”Discourse on the Method” and”Critique of Pure Reason” are the same kind of thing. But something surprising happens in
TLP. In fact, TLP can be read really logically, as I will write below, as I will write below. In
this sense TLP is groundbreaking.

As mentioned in Remark 11.25 later, Wittgenstein’s picture theory is only a theory ofVenn diagram in a logical space (= set), which is the most elemental part in naive set theory.
However, since my purpose is to compare his theory and quantum language, I will explain his
theory by comparing it to quantum language.
Now let us review Sec.4.3.5: ”[In a class of classical binary projection measurements,
measurement has properties like propositional logic]”, using the word ”proposition”(without using the word ”measurement”).
Table 11.17. [Logic in quantum language (Sec. 4.3.5) vs. logic in Wittgenstein’s picture
theory (this section)]
Table: [ Proposition is a kind of measurement]
Logic in quantum language (Sec. 4.3.5)
Logic in Wittgenstein’s theory (Sec. 11.6.1)
Axiom 1 in Section 1.1
Deﬁnition 11.19
the linguistic Copenhagen interpretation( Naive set theory? )system, particle, object, the world, tomato
object, thing, the world, tomato
state space ( state )logical space ( case, fact,, atomic fact )measured value {1, 0}
truth value {T, F}
binary projection measurement
proposition
ML∞ (Ω,ν) (OΓ ≡ (X(= {1, 0}), 2X , F Γ ), S[bω(t)] )PΩ (Γ, S[bω(t)] )Theorem 4.8 (Syllogism in measurements)
Theorem 11.21 (Syllogism in propositions)
elemetary measurement
elementary proposition
Theorem 4.10
Theorem 11.24
Elementary measurements are not fundamental
Elementary propositions are fundamental
I encourage you to read the following, referring to the above table. If you understandSec. 4.3.5, you should be able to understand this section in a few minutes. The terms andconcepts are not ambiguous in Wittgentein’s theory. Using the above table, we can translate
the language of Wittgenstein’s theory into quantum language.
////
Seeing Table 11.17, assume that (Ω, ν), Γ, ωb (t), SW, RP, RD, ..are the same as in Sec. 4.3.5.
That is, Ω is a locally compact space with a Borel measure ν on Ω such that ν(D) > 0 ( forany open set D(̸= ∅) (⊆ Ω)). Here, note that we can assume that Ω is compact without loss of241

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11.6 Linguistic turn: Wittgenstein
generality (cf. Stone-Čech compactiﬁcation; ref. [15]). And, Γ(⊆ Ω) is an open set such that
Γ = [Γ]◦ ,ν(Ω \ (Γ ∪ [Γc ]◦ )) = 0

where Dc is the complement of D, i.e., Ω \ D, D =”the closure of D”, D◦ =”the interior ofD)”. where Dc is the complement of D, i.e., Ω \ D, D =”the closure of D”, D◦ =”the interiorof D)”. Note that we can assume that Γc = [Γc ]◦ = [Γ]c in the sense of ”almost everywhereconcerning ν”.
ΩΓConsider 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 logicalspace Ω (⇔state space Ω in Sec. 4.3.5). Thus, we have the injective map ωb : T → Ω. Thatis, the quantitative property of a tomato t is represented by ω(t). For example, it suﬃces toconsider Ω as RN (= N -dimensional real space), where N is suﬃciently large natural number( or, N = ∞). That is,(Ω ∋ ω = ω (1) (= weight), ω (2) (= diameter), ω (3) (= diameter), ω (4) (= color value),)ω (5) (= calorie), ω (6) (= sugar content), ..., ω (N ) (= ...) ∈ RN

♠Note 11.11. (=Note 4.7); Someone might say that the term ”the set of all tomatoes” is ambiguous. However, for the sake of convenience, here we use the term ”the set of all tomatoes”.
In Sec. 11.8 [ Hempel’s raven problem], we will discuss it precisely.

Remark 11.18. In Sec. 4.3.5, the binary projection measurement ML∞ (Ω,ν) (OΓ , S[bω(t)] ) follows
Axiom 1 ( and the linguistic Copenhagen interpretation ). On the other hand, the propositionPΩ (Γ, S[ω] ) is not related to Axiom 1. Thus, we need the deﬁnition below.
////
Deﬁnition 11.19. A pair (Γ, ω) (Γ ⊆ Ω, ω ∈ Ω) is called a proposition (i.e., ”a system witha state ω has Gamma-property”), which is denoted by PΩ (Γ, S[ω] ) (cf. a binary projection
measurement ML∞ (Ω,ν) (OΓ , S[ω]) ) in Sec. 4.3.5).
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Chap. 11 Linguistic philosophy and quantum language
Deﬁne as follows.

{(E1 ) a truth value of the proposition PΩ (Γ, S[ω] ) =T (true) ( if ω ∈ Γ)
F (false) ( if ω ∈/ Γ)That is, truth values {T, F}(= X) correspond to measured values {1, 0}(= X) (or, X = {x1 , x0 }).
If the true value of PΩ (Γ, S[bω(t)] ) is equal to ”T”, we may say that(E2 ) a tomato t has Γ-property.
////
11.6.1.1

Classical logic ( i.e., not, and, or, implication)

Consider a proposition PΩ (RD, S[bω(t)] ). Deﬁnition 11.18 says that the following three are equivalent (i.e., (I1 ) ⇔ (I2 ). Also, (I3 ) is the expression of (I1 ) in ordinary language):(I1 ) A truth value of the proposition PΩ (RD, S[bω(t)] ) is equal to ”T”.
(I2 ) ωb (t) ∈ RD
(I3 ) A tomato t is ”red”.
Similarly, consider a proposition PΩ (SW, S[bω(t)] ). Thus, we see that the following three are
equivalent:(J1 ) A truth value of the proposition PΩ (SW, S[bω(t)] ) is equal to ”T”.
(J2 ) ωb (t) ∈ SW
(J3 ) A tomato t is ”sweet”.
Ω

RD
SW

[ Not]
It is clear that the following four are equivalent:
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11.6 Linguistic turn: Wittgenstein(K1 ) A truth value of the proposition PΩ (SWc , S[bω(t)] ) is equal to ”T”( Here, PΩ (SWc , S[bω(t)] ) is also denoted by ¬PΩ (SW, S[bω(t)] ) ),(K2 ) ωb (t) ∈ [RD]c(K3 ) A tomato t is not ”red”.
[And]
We see that the following three are equivalent:(L1 ) A truth value of the proposition PΩ (SW∩RD), S[bω(t)] ) is equal to ”T” ( which is also

denoted by PΩ (SW, S[bω(t)] )∧PΩ (RD, S[bω(t)] ) )(L2 ) ωb (t) ∈ RD∩

SW

(L3 ) A tomato t is ”red” and ”sweet”[Or]
We see that the following three are equivalent:(M1 ) A truth value of the proposition PΩ (Θ∨ (SW∪RD), S[bω(t)] ) ( which is also denoted byPΩ (SW, S[bω(t)] )∨PΩ (RD, S[bω(t)] ) ) is equal to ”T”(M2 ) ωb (t) ∈ RD∪

SW

(M3 ) A tomato t is ”red” or ”sweet”[Implication]
We see that the following three are equivalent:(N1 ) A truth value of the proposition PΩ (SWc∪

RD, S[bω(t)] ) ( which is also denoted byPΩ (SW, S[bω(t)] )→PΩ (RP, S[bω(t)] ) ) is equal to ”T”,
(N2 ) ωb (t) ∈ SWc∪

RD(N3 ) a tomato t is not ”sweet”, or it is ”red”
Remark 11.20. It is clear that logical operation ( Boolean algebra; [¬, ∧, ∨, →]) holds in theabove.
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Chap. 11 Linguistic philosophy and quantum language
11.6.1.2

Syllogism

Further, consider a proposition PΩ (ORP , S[bω(t)] ). That is, we consider that the following three
are equivalent:(O1 ) A truth value of the proposition PΩ (ORP , S[bω(t)] ) is equal to ”T”.
(O2 ) ωb (t) ∈ RP(≡ {ω ∈ Ω | [F RP ({T})](ω) = 1})(O3 ) A tomato t is ”ripe”.
Theorem 11.21. [Syllogism]:b (t)(∈ Ω) be the state of t. Assume the followings:
Let t be a tomato, and let ω(P1 ) PΩ (SW, S[bω(t)] )→PΩ (RP, S[bω(t)] )which is equivalent to(P2 ) [b
ω (t) ∈/ SW] ∨ [b
ω (t) ∈ RP](P3 ) A tomato t is not ”sweet”, or it is ”ripe”.
And further, assume(P′1 ) PΩ (RP, S[bω(t)] )→PΩ (RD, S[bω(t)] )which is equivalent to(P′2 ) [bω (t) ∈/ RP] ∨ [b
ω (t) ∈ RD](P′3 ) A tomato t is not ”ripe”, or it is ”red”.
Then the following holds:(Q1 ) PΩ (SW, S[bω(t)] )→PΩ (RP, S[bω(t)] )(Q2 ) [b
ω (t) ∈/ SW] ∨ [b
ω (t) ∈ RD](Q3 ) A tomato t is not ”sweet”, or it is ”red”.
[Proof using Deﬁnition 11.19]
A simple calcultion shows that
[SWc ∪ RP] ∩ [PRc ∪ RD] = [SWc ∩ PRc ] ∪ [SWc ∩ RD] ∪ [RP ∩ PRc ] ∪ [RP ∩ RD]=[SWc ∩ PRc ] ∪ [SWc ∩ RD] ∪ [RP ∩ RD] ⊆ SWc ∪ RD
Hence, recalling (P2 ) and (P′2 ), we immediately see (Q2 ). and thus, (Q1 ), (Q3 ).

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11.6 Linguistic turn: Wittgenstein♠Note 11.12. Note that the proof of this theorem ( due to Deﬁnition 11.19 is simple comparedto Theorem 4.8 ( using Axiom 1 and the linguistic Copenhagen interpretation in Sec. 1.1.2).
That is, in the proof of Theorem 11.21, we do not need to check the existence and uniquenessof the simultaneous observable OSW × ORD × ORP ,The following exercise promote the reader’s understanding of ”Proposition”.
Exercise 11.22. Let Γ ⊆ Ω and Λ ⊆ Ω. Then, we have the following question.
• Is the statement ”Λ ⊆ Γ” a proposition?
[Answer]: Inspired by Remark 4.6 ( due to the linguistic Copenhagen interpretation: ”only one
measurement is permitted”), we get the following discussion:Λ⊆Γ⇐⇒(∀λ ∈ Λ)[λ ∈ Γ]⇐⇒(ωλ )λ∈Λ ∈ × Γλλ∈Λ(where ωλ := λ (∀λ ∈ Λ)), Γλ := Γ (∀λ ∈ Λ))⇐⇒PΩΛ [ × Γλ , S[(ωλ )λ∈Λ ] ](where ΩΛ is the Λ-dimensional produc space)λ∈ΛThus, Deﬁniytion 11.19 says that the statement ”Λ ⊆ Γ” is a proposition. Also, note that thisproblem is closely related to Hempel’s raven problem (cf. Sec. 11.8).
11.6.1.3

Elementary proposition

Consider the state space Ω, which is ﬁnite ( or, countable ) with a metric d (i.e., d(ω1 , ω2 ) =1 (ω1 ̸= ω2 ), = 0 (ω1 = ω2 ). Further, assume that the Borel measure ν is deﬁned by the point
measure, i.e., ν({ω}) = 1 (∀ω ∈ Ω).
Deﬁnition 11.23. [elementary proposition] Let λ be any element of Ω. Putting Γ = {λ},deﬁne the proposition PΩ ({λ}, S[ω] ), which is called an elementary proposition (⇔ elementarymeasurement in Sec. 4.3.5).
////
It is clear that it holds that
(R1 ) A truth value of the elementary proposition PΩ ({λ}, S[ω] ) is equal to ”T”⇐⇒ λ = ω
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Chap. 11 Linguistic philosophy and quantum language(R2 ) A truth value of the elementary proposition PΩ ({λ}, S[ω] ) is equal to ”F”⇐⇒ λ ̸= ω
Theorem 11.24. Let Γ be a subset of Ω. And let ω ∈ Ω. Then we see thatPΩ (Γ, S[ω] ) = ∨λ∈Γ PΩ ({λ}, S[ω] )That is, every proposition can be represented by the sum of elementary propositions. This is
not trivial since Exercise 11.22 is not trivial.
[Proof].
We see that
The true value of PΩ (Γ, S[ω] ) is equal to ”T”⇐⇒ω ∈ Γ⇐⇒∃λ(∈ Γ)[ω ∈ {λ}]⇐⇒The true value of ∨λ∈Γ PΩ ({λ}, S[ω] ) is equal to ”T”♠Note 11.13. The above is the main assertion in Wittgenstein’s theory. However many readers
may not consider the above theorem to be particularly important. I have the same view. In fact,
this theorem holds only in special cases such as Ω has a discrete topology. Of course, the spirit ofexpressing complex observables in simple observables is quite important. In quantum language,
this spirit is realized by von Neumann’s spectral decomposition theorem ( and Holevo’s theorem),that is, ”Any observable can be composed of binary projective observables”, cf. [29, 103].

Remark 11.25. In this section, I was devoted to understanding Wittgenstein’s theory incomparison to quantum language. However, it is more easy to understand Wittgenstein’s
theory in comparison to naive set theory as follows.
(S1 ) a set ( logical space ) Ω is not ”a collection of things”” but ”a collection of facts ( states)”()(S2 ) A proposition P is deﬁned by (ω, Γ) ∈ Ω × 2Ω .
{(S3 ) ”A truth value of a proposition P (= (ω, Γ))” =T (ω ∈ Γ)F (ω ∈/ Γ)This is all that matters. It is easy to understand Theorems 11.21 and 11.24 in the set theoreticalformulation. In summary, I can say the following.
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11.6 Linguistic turn: Wittgenstein(T) ”Wittgenstein’s picture theory” = ”The theory of Venn diagram in logical space Ω”
Thus, if he had used the element symbol ”∈” ( or ”Venn diagram”) in ”Tractatus LogicoPhilosophicus”, this book would have been easily understood by everyone. He may have thoughtthat using set theory to conclude that ”logic is important” would be a circular argument.
However, if his theory cannot be expressed clearly using mathematics, then I think his theoryis ”what we cannot speak about”.

♠Note 11.14. (i): Wittgenstein’s saying ”The limits of my language mean the limits of myworld is very famous. However, if so, quantum mechanical world is out of his world. That is
because the argument in this section is restricted to classical measurements and not quantum
measurements. However, his world is not so small. In fact, the world of equilibrium statisticalmechanics ﬁts best into his world (Ref. [46]).
(ii): TLP ends with the following sentence:
7”What we cannot speak about we must pass over in silence”

Just before this, it says
6.54 ”He must, so to speak, throw away the ladder, after he has climbed up on it”
I thought it was an impressive ending that could be interpreted in a variety of ways.

11.6.2Which is more essential, quantum language or Wittgenstein’s
picture theory?

Now let’s compare the discussion in Sec. 4.3.5 with the discussion in Sec. 11.6.1, that is,

Logic in quantum language (Sec. 4.3.5)
Axiom 1 in Section 1.1
the linguistic Copenhagen interpretation
system, particle, object, the world, tomato
state space ( state )measured value {1, 0}
binary projection measurement
ML∞ (Ω,ν) (OΓ ≡ (X(= {1, 0}), 2X , F Γ ), S[bω(t)] )Theorem 4.8 (Syllogism in measurements)
elementary measurement
Theorem 4.10
Elementary measurements are not fundamental

Logic in Wittgenstein’s theory (Sec. 11.6.1)
Deﬁnition 11.19( Naive set theory? )object, thing, the world, tomato
logical space ( case, fact,, atomic fact )truth value {T, F}
propositionPΩ (Γ, S[bω(t)] )Theorem 11.21 (Syllogism in propositions)
elementary proposition
Theorem 11.24
Elementary propositions are fundamental

The most signiﬁcant diﬀerence of the two is that248

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Chap. 11 Linguistic philosophy and quantum language(A) the argument in Sec. 4.3.5 ( Quantum language) is based on Axiom 1 and the linguistic
Copenhagen interpretation ( in Sec 1.1), which is connected to the world. Quantum
language asserts that• in a class of classical binary projection measurements, measurement hasproperties like propositional logic,thus,• measurement is the foundation of science, and logic is the foundation of mathematics
On the other hand,(B) the argument in Sec. 11.6.1 ( Wittgenstein’s picture theory) is based on Deﬁnition 11.19,
which is essentially the same as naive set theory ( as discussed in Remark 11.25). Thus, histheory cannot be regarded as disconnected from the world as mathematics is. Wittgenstein’s picture theory asserts that• logic is the foundation of science as well as mathematics.
which is an idea inherited from Frege and Russell. That is, Wittgenstein’s picture theory
made Russell and Frege’s dream come true.
(Whether their dream was on target is another matter. To be clear, Wittgenstein had
replaced the importance of measurement (=”there is no science without measurement”)
for the importance of logic (=”there is no science without logic”) .)That is, I think that Wittgenstein’s theory is, in a broad sense, a kind of mathematical theory,and thus, it cannot be regarded as a kind of worldview. If we have to use the word ”worldview”,(
We may sometimes call it the logical worldview ( or, set theoretical worldview). But this is)habitual, not formal. Without the worldview ( i.e., the world description method), logic (ormathematics) alone cannot produce anything but mathematical theory. I think that(C) Quantum language has the power to solve philosophical unsolved problems, but Wittgenstein theory does not.
This has been shown throughout this text as follows.
List 11.26. (= List 1.6)
(D1 ) The list of quantum linguistic answers for philosophical unsolved problems
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11.6 Linguistic turn: Wittgenstein• What is probability (or, measurement, causality) ? cf. Sec. 1.1.1)• Zeno paradox (Flying arrow), (cf. Sec. 2.4.2)• Zeno paradox (Achilles and a tortoise), (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. 4.3.3)• 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)• 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. 9.7.2)• What is causality? (cf. Sec. 10.3)• What is Peirce’s abduction? (cf. Sec. 11.3.1)• Five-minute hypothesis (cf. Sec. 11.4.1)• McTaggart’s paradox (cf. Sec. 11.4.2)• quantitative representation of ”Signiﬁer” and ”signiﬁed” (cf. Sec. 11.5.3)• A scientiﬁc understanding of Wittgenstein’s picture theory (cf. Sec. 11.6.2)
Current Point• Wittgenstein’s paradox 11.6.3• Flagpole problem, (cf. Sec. 11.7.1)• Hempel’s raven paradox (cf. Sec. 11.8)• the mind-body problem (i.e., How are mind and body connected?), (cf. Sec. 11.9.4)(D2 ) Also, for the solutions of unsolved problems in quantum mechanics, statistical mechanics,
statistics and probability theory, see ref. [66]). Particularly, I think that the following
three are important in physics:• the discovery of Heisenberg’s uncertainty relation (Ref. [32], or Sec. 4.3 in ref.
[66])
• The clariﬁcation of the projection postulate (i.e., the wavefunction collapse) (ref.
[59] , or Sec. 11.2 in ref. [66])• The measurement theoretical characterizations of equilibrium statistical mechanics(Ref. [46], or Chap.17 in ref. [66])

It is known that Wittgenstein wrote ”Tractatus Logico-Philosophicus” in order to solve all
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Chap. 11 Linguistic philosophy and quantum language
philosophical problems in one fell swoop. And he concluded that
(E1 ) philosophical problems arise from insuﬃcient attention to the variety of natural language
use.
that is,”Philosophical problems were not appropriate”.
However. I think that his saying is not fair. I think that(E2 ) these unsolved problems in (D) arise from a lack of our understanding of dualistic idealism.
(Recall my assertion that the formula (1.2), i,e., quantum language is the ﬁnal goal ofdualistic idealism.)
Thus, Wittgenstein’s picture theory cannot solve unsolved problems in (D), since his theory isbased on naive set theory ( which is not related to dualistic idealism).
Some may assert that
(E3 ) ”Similar thoughts” in Wittgenstein’s saing (D) in Sec. 11.6.1 might be an even greater
thought than quantum language.
This may be a possible. However, it should be noted that quantum language solved many
fundamental philosophical unsolved problems ( in List 11.26), but Wittgenstein’s picture theory
has no power to slove such propbelms. Thus, I think he was going for such theory like quantum
language.

11.6.3 * Wittgenstein’s paradox
11.6.3.1

Without a worldview, we cannot say anything

Around the time I conceived of the quantum language (e.g., [35, 36, 37, 38, 40] in quantum
mechanical worldview), almost all unsolved philosophical problems (philosophical paradoxes indualistic idealism) can be solved within the framework of a quantum language, and I thought.
In fact, they have been solved and are noted in List 11.26 (=1.6) [The list of our answers forphilosophical problems]. As mentioned in Note 1.10, without a worldview, we have several
paradoxes as follows.
(F1 ) Zeno’s paradoxes (in Sec. 2.4)
(F2 ) Hume’s problem of induction (and Grue paradox) (in Sec. 9.7)
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11.6 Linguistic turn: Wittgenstein(F3 ) Wittgenstein’s paradox
However, I believe I have solved (F1 ) by regarding it as the philosophical problem concerning
worldviews and not the mathematical problem concerning geometric series. Also, I solved (F2 ),considering it within the framework of quantum language. However, I still think of (F3 ) as amere play on words and a completely unproductive problem. Next let us explain it.
11.6.3.2

Wittgebstein’s paradox: The logical worldview is not a worldview

It might be reasonable to assume that analytical philosophy is a style of doing philosophy andhas nothing to do with the logical worldview. If so, we can understand Wittgenstein’s paradox
as follows.
Problem 11.27. Wittgenstein’s paradox (in Philosophical Investigations (1953))
Consider the following quiz:[Quiz]:
1, 2, 3, 4 5,
What number is X ?

6,

7,

8,

9,

10,

11,

12,

X, · · ·[Answer]: It may be natural to consider that X = 13. However, if I think of them as
numbers on a clock face. there is a reason to consider that X = 1. Thus, this quiz is
inappropriate as a question.

Most readers will ﬁnd this quiz trivial and boring. Thus, the true problem concerning Wittgenstein’s paradox is as follows.
(G) Why did Wittgenstein mention a trivial problem like the quiz above? Why didn’t Plato,
Descartes and Kant consider these problems?
[Answer]: This is already answered in Remark 9.11. That is, Wittgenstein’s paradox highlights
the weaknesses of analytic philosophy (i.e., analytic philosophy didn’t have a true worldview).
That is, the above quiz cannot be solved without the condition: ”the world is like this”.
Therefore, I think that Wittgenstein’s paradox is important in the sense that it symbolizes thefact that analytic philosophy does not have a worldview. ( I’m not sure if this will be generally
approved as the answer to Wittgenstein’s paradox, but I’ll list it in List 1.6)

♠Note 11.15. Readers may think that the above is too childish. However, it should be notedthat the above is essentially the same as ”Geocentrism vs. Heliocentrism” (cf. Chap. 7). Thatis, we can say nothing without a worldview.

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Chap. 11 Linguistic philosophy and quantum language

11.6.4Main character Wittgenstein in Linguistic turn

11.6.4.1

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 (H1 ) - (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 thespirit of quantum language, I prefer to saying(I) ”The spirit of quantum language is represented by the above (H1 ) - (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 Newton to be an
enlightener of the philosophy of language.
11.6.4.2

Wittgenstein advanced philosophy

I think that
Linguistic turnmechanical turn(J1 ) Epistemology −−−−−−−−→ linguistic phil. analytic phil. −−−−−−−−−→ Quantum Language(Descartes, ..., Kant)

regress (QL)

Frege
Russell[][
where −−−−−−−→ resp. −−−−−−−→ means ”getting closer to quantum language” resp.
progress (QL)
regress (QL)]
”getting away from quantum language” . That is because Frege and Russell’s assertion was
too mathematical. And so, their assertion is not related to dualistic idealism (≈ quantum
language).
However, I think that
(J2 )Descartes(Discourse on the Method)

Copernican revolution−−−−−−−−−−−−→
progress (QL)⃝
1

Linguistic turn−−−−−−−−−→
progress (QL)⃝2

Kant( Critique of Pure Reason)mechanical turn−−−−−−−−−→ Quantum Language

Wittgenstein
Tractatus Logico Philosophicus (=TLP)

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progress ⃝3

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11.6 Linguistic turn: Wittgenstein
That is because(K1 ) ”progress (QL)⃝”
1 is proved in Problem 10.6 in Chap 10.
(K2 ) ”progress (QL)⃝”;2 Recall the argument in Sec. 4.3.5 (”Why does classical logic hold inclassical systems?”) and Sec. 11.6.2 (My scientiﬁc understanding of ”Tractatus LogicoPhilosophicus (=TLP)”), which say that Wittgenstein’ picture theory can be completely
understood in QL. Thus, we can conclude that TLP is closer to QL than to ”Critique ofPure Reason, that is, progress (QL)⃝.
2(K3 ) ”progress ⃝”3 is clear if you examine List 11.26 in Chap 11.
Therefore, in spite of (J1 ), I assert that
Frege, Russell
Linguistic turnmechanical turn(L) Epistemology −−−−−−−−→ linguistic phil, analytic phil. −−−−−−−−−→ Quantum Language(Descartes, ..., Kant)

progress (QL)

progress

Saussure, Wittgenstein

Therefore, from the quantum mechanical point of view, of the three (i.e., Frege, Russell,
Wittgenstein), I can rate Wittgenstein the highest as a philosopher.
Even if that were the case, if quantum language were not good, everything would be meaningless. Of course, I believe that quantum language is going to get a lot of support.
Remark 11.28. (= Remark 11.13)): I will restate what I wrote at the beginning of this chapter
as follows.
 1(♭22 )








 (♭222 )
(♭322 )(♭22 )
(♭422 )






 5
(♭22 )

foundation of math. (Boole, Frege, Russel, Cantor, Zermelo, Fraenkel, Gödel),...,
math. (Bourbaki),analytic philosophy, symbolic logic (Frege, Russell, Wittgenstein,...)
scientiﬁc philosophy (Hempel, Popper, ...)
linguistics, ordinary language (Saussure, Wittgenstein (except logic))

If so, the above formula (11.4) can be rewritten more precisely as follow:(♯2 ) Descartes=Kant −−−−→ linguistic phil. (♭322 ), (♭422 ),(♭522 ) −→ Quantum language
progress⃝:a (♭522 ):Saussure (Remark 11.12)⃝:b (♭322 , ♭522 ): Wittgenstein (Sec. 11.6.3)⃝:
c (♭422 ):Hempel (Remark 11.42)⃝
c will be discussed later.

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Chap. 11 Linguistic philosophy and quantum language♠Note 11.16. 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 10.7)(♯6 ) Wttgenstein· · · · · · primary school teacher, Guardian: Russell (cf. Sec.11.6.1)(♯7 ) Einstein· · · · · · Elevator(♯8 ) quantum mechanics· · · · · · Heisenberg’s uncertainty principle (cf. Note 4.1 of ref. [66])Here, the (♯8 ) is my opinion.

Remark 11.29. [The key-words of the dualism] Now we can complete the table in Assertion 1.14. 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:\[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

secondary quality

Berkeleymind

secondary quality

Kant
SaussureZadehphenomenon

recognition/

signiﬁed [signiﬁer]
fuzzy set (= membership function)

Wittgenstein

truth value[proposition]

statistics

sample ( space )quantum mechanics

measured value

quantum language

measured value/[ trial]observable[ measurement]
observable[ measurement]

255[C](= matter)/[Idea world]/[universale in re]/[matter]primary quality[matter]/[God]/[thing-in-itself]/
logical space[object]
parameter[population]
state[particle]
state[system]

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11.6 Linguistic turn: Wittgenstein

11.6.5

Appendix: Linguistic turn ( From epistemology to quantum
language)

Kant’s ”Criticism of pure reason” may be given a severe evaluation in modern times, but it isrevived by linguistic turn (by Wittgenstein and others), i.e.,
epistemology −−−−−−−−→ linguistic philosophy(♯1 )linguistic turn

Kant

WittgensteinorCognition −−−−−−−−→(♯2 )linguistic turn

Kant

Languagequantum language

In this section, I asset that• the linguistic turn can be understood in (♯2 ) than (♯1 )That is, it suﬃces to change ”cognition” (in Critique of Pure Reason) ) to ”language”, that is,batch conversion”cognition” −−−−−−−−−→ ”language”
by which ”the spirit of Critique of Pure Reason” changes to ”the spirit of quantum language”
as follows.
Explanation 11.30. (Continued from Explanation 10.3 in Chap.10)
Explanation of the spirit of quantum language : i.e., from Critique of Pure Reason toquantum language (cf. [50])That is, read such as[the outline of ”Critique of Pure Reason”][Explanation 10.3][Explanation 11.30],[the spirit of quantum language]i.e.,——————————————————–[Critique of Pure Reason ][perception ability]

What was revealed in [Quantum language] is that human [linguistic ability] is not just to passively replicate things of the world. Rather it acts on the world actively and creates the subject[recognition]of its [language]. Even though we make it, the world is not necessarily completed from nothing[recognition]

like God. The world is already there in some form. In order for [description] to be established,
information from this world that can be obtained through a sense is necessary as a material.
[perception ability]

However, this information is only disorganized confusing as it is. Human [linguistic ability]
gives an orderly order to information of this confused sensation through a certain form that[recognition]is inherent in human beings. It is necessary to compile the subject of uniﬁed [language] for[Kant][a priori synthetic judgment]

the ﬁrst time. According to [quantum language], its format (=follows.
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Chap. 11 Linguistic philosophy and quantum language[Form of sensitivity(intuition)( space (=R × R3 ))](i)[Axioms 1 (measurement) ][Form of understanding(thinking)](ii) [Axioms 2 (Causality)]
If so, the proposition that ”everything is in time and space” or ”everything follows causality”
cannot be proved empirically. Nevertheless, it will apply unconditionally to all subjects ofexperience. The reason is that the object is composed for the ﬁrst time in a format such[space, time, causality, etc.][if we wear green sunglasses]

as [measurement, causality]. For example, [we know the only term ”green”], the remark that”the world is green” is similar to being regarded as a correct remark for all human beings.

Summary 11.31. Consider ”language” and not ”cognition”. That is, consider the following
linguistic turn:(cognition)(language)

mind-matter dualism(=epistemology) −−−−−−−−→ linguistic philosophy
linguistic turn(Descartes·Kant)(Wittgenstein)

Then, in several languages (ordinary language, mathematics, Newtonian mechanics, programing language, etc.), we say that (cf. Note 10.3)(♯1 ) ”ordinary language” is like tabula rasa (i.e.,British Empiricism)(♯2 ) Mathematics is like Continental Rationalism(♯3 ) quantum language is like Kant’s compromise
In quantum language, Axioms (measurement and causality) are ﬁrst declared, and thus, quantum language is not like ”tabula rasa”. However, the linguistic Copenhagen interpretation isgoing to be formed gradually through the experience, thus, quantum language is like Kant’s
compromise. As seen in the formula (1.1) in Chap. 1, we see that(=quantum language)[Axiom 1][Axiom 2]

measurement theory = measurement + causal relation+

linguistic Copenhagen interpretation[the manual to use Axioms 1 and 2]♠Note 11.17. If ”Why is our cognition possible?” is asked, then, we may have only answer suchthat ”Wonder of the human recognition ability”, which is represented by ”transcendent” (dueto Kant). And moreover, If ”Why is our language possible?” is asked, then, we may have only
answer such that ”Wonder of the human language ability”, which is represented by ”language

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11.6 Linguistic turn: Wittgensteingame” (due to Wittgenstein). That is,
Wonder of the human language ability

Wonder of the human recognition ability

congnition
Kant (transcendental)−−−−−−−−→
linguistic turn258

language
Wittgenstein (language game))

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Chap. 11 Linguistic philosophy and quantum language

11.7

Flagpole problem in the quantum mechanical worldview

This section was written with reference to the following.
• [70]:Ishikawa, S: Philosophy of science for scientists; The probabilistic interpretation ofscience Journal of quantum information science, Vol. 9, No.3 , 140-154,
DOI: 10.4236/jqis.2019.93007(https://www.scirp.org/Journal/paperinformation.aspx?paperid=95447)
Remark 11.32. I mentioned Classiﬁcation 11.2 as the classiﬁcation of the logical worldview
as follows:
Classification [The classiﬁcation of the logical worldview] = Classiﬁcation 11.2 ](A0 ): Artiﬁcial intelligence, ”Logic circuit may be more powerful than any worldview”(♭022 ) Computer; (Gödel, Turing, von Neumann, etc.)
 (A ): mathematics: ”Describe math. by logic ( and set theory)!”

1 {




(♭122 ) foundation of math. (Boole, Frege, Cantor, Russell, Gödel,...),

(♭222 ) math. (Bourbaki),(A2 ):the logical worldview (or precisely, the naive logical worldview) :(♭22 )

”Describe
science by logic ( and set theory)!”


{ 3



(♭
)
analytic
philosophy, symbolic logic (Frege, Russell, Wittgenstein,...)224 ) scientiﬁc philosophy (Hempel, Popper, ...)

(♭

22



(A
):
linguistics:”Describe everything reasonably!”
3



5
(♭22 ) linguistics, ordinary language (Saussure, Wittgenstein)(♭23 ): the ( classical, quantum) mechanical worldview

In this and the next sections, I will mention that C.G. Hempel pointed out that the logicalworldview (A2 ) is not ﬁt for science. And I show that quantum language (i.e., the quantum
mechanical worldview(♭23 )) is superior to the naive logical worldview (A2 ).

11.7.1 * The quantum linguistic solution of Flagpole problem
Carl Gustav Hempel (1905-1997) was a German philosopher. He was a major ﬁgure in logicalempiricism, 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 generationof philosophers of science. He is also known for the raven paradox (also known as ”Hempel’s
paradox”)
As I have said many times in this text, philosophers like ”logic” (or, the word ”logic”) too
much. They always want to ”be logical”.
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11.7 Flagpole problem in the quantum mechanical worldview
I think that he following fact is incomprehensible.
• Philosophers are more fond of ”logic” than mathematicians
That is, I assert that• the scientiﬁc explanation should not be described in the logical worldview, but in thequantum mechanical worldview.
This will be discussed in Sec. 11.7.1.1 and Sec. 11.7.1.2.
11.7.1.1

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 thelength 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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Chap. 11 Linguistic philosophy and quantum language
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-nomologicalmodel), which is due to a kind of the logical worldview (i.e., the spirit that scienceshould 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.
11.7.1.2

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 ))], where Ω0 = [0, 1] is the state space (in which
the length of the ﬂagpole is assumed to be represented) at time 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 tobe 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 → Ω1 such 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 )261

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11.7 Flagpole problem in the quantum mechanical worldview
Let Oe = (X, F, Fe ) be the exact observable in L∞ (Ω1 , ν1 ) (cf. Example 1.19). That is, itsatisﬁes 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[ω00 ] ). Then we havethe following statement(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[ω00 ] ) is equal to 2ω00 is given by 1.
which is the quantum linguistic representation of (B1 ). 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 (B2 ). 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 describe by quantum language. Also,
we have to add that the ﬂagpole problem is not trivial but signiﬁcant, since this is neversolved without Axiom 1 [measurement] (in Section 1.1)(measurement) and Axiom 2 (in Section
1.1)(causality) (i.e., the answers to the problems ”What is measurement ?” and ”What is
causality ?”).

Remark 11.33. Recall the following the mainstream of western philosophy:⃝:keywords[ cognition, causality]
1

Epistemology(Descartes, ..., Kant)

Linguistic turn⃝:keyword[logic]2mechanical turn−−−−−−−−−→ Logical worldview −−−−−−−−−−→
progress

progress⃝:keywords[measurement, causality]
3

Quantum Language

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. As mentioned2
often in this text, I believe that(♯1 ) quantum language is the language of science (cf. Assertion 1.3).
(♯2 ) logic is the language of mathematics.
Also, recall Remark 11.3.
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11.8 * Hempel’s raven problem
11.8.1

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”).
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 beginningof 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 1.12). Therefore, we assume that ”set” is deﬁned by ”a collection of things”. This is thesame 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 amistake.
In this text, this is called the set theoretical worldview.
However, I am skeptical of the set theoretical worldview (♯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 not meaningless, how do we represent ”All ravens are black”?
This (♯3 ) is Hempel’s raven problem.

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11.8* Hempel’s raven problem

11.8.2Hempel’s raven problem in the set theoretical worldview

First, let us review the traditional arguments concerning Hempel’s raven problem (cf. refs.
[27, 28]). 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” islogically 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(A1 )?
(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 all ravens” is as ambiguous as ”the set of all tyrannosaurs”. If so, that is, if theabove (A0 ) is ambiguous, all other (A1 )-(A5 ) are also ambiguous. That is, (A3 )-(A5 ) are notscientiﬁc problems.
Now we think that the most essential problem concerning Hempel’s raven problem is asfollows:(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, under which 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.
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Remark 11.34. 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” = ”∀”).

11.8.3Hempel’s raven problem in the quantum mechanical worldview

In this section we slightly improve our result in• Ref. [70]: Ishikawa, S., (2019) Philosophy of science for scientists; The probabilistic
interpretation of science, 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 meaningof ‘Every raven is black’?”. In order to answer this problem, we must prepare the quantum
linguistic formulation of ornithology
The arguments below are essentially the same as Saussure’s ”signiﬁer” and ”signiﬁed” inSec. 11.5.3.
Deﬁnition 11.35. [Membership function (= Fuzzy set, cf. [35, 36, 37, 38, 109]), Observable]
Let Ω be a state space. For simplicity, we always assume that Ω is compact. A continuousfunction 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]
menbership function←−−−−−−−−−−→ O = ({y, n}, 2{y,n} , F )correspondence(11.5)

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 11.36. [Membership functions of black birds and ravens] Let Ω be a state space.
A continuous function m : Ω → [0, 1] (i.e., the closed interval in R) is called a membership265

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11.8* Hempel’s raven problem
function. Assume that the state (i.e., quantitative property) of any bird can be expressed bya point in the state space Ω. Deﬁne the membership 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 membership function 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 11.37. (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 ΩR [ resp. ΩB ], this bird of acertain 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 ΩR ⊆ 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 11.38. [Raven observable, Black bird observable]: Using the above membershipfunctions, we deﬁne 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 with a binary observable.
Since we assume that any bird is characterized by a certain point in the state space Ω, it isnatural to consider 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 , thenthe bird is called a raven.
(E2 ) Black birds are characterize by the membership function mR : Ω → [0, 1]. (or, equivalently, the observable OR = ({y, n}, 2{y,n} , FR )). If the state ω of a bird belongs to ΩB ,then the bird is called a black bird.
Interpretation 11.39. [The probabilistic interpretation of membership functions] We add
the following probabilistic interpretation to this formulation [I]: For example, again consider
Deﬁnition 11.35, and moreover, the statement (D). i.e.,
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11.8* Hempel’s raven problem
{
′

(D )

70 specialists say that this bird with a state ω0 is a black bird.
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 is black 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.
(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 the measurement MC(Ω) (OB = ({y, n}, 2{y,n} , FB ), S[ω1 ] ) is equal to[FB ({y})](ω1 ) [ resp. [FB ({n})](ω1 ) ].

11.8.4

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 11.40. (i): [Membership function of small black birds]: Deﬁne the membershipfunction mSB : Ω → [0, 1] of small black birds such as Deﬁnition 11.35.
(ii):[Small black bird state class] The small black bird state class ΩSB is deﬁned by {ω ∈
Ω | mSB (ω) = 1}.
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Chap. 11 Linguistic philosophy and quantum language(iii): [Small black bird]: If the state of a certain bird belongs to ΩSB , 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 holdsthatΩSB ⊆ ΩB
Note that this necessarily holds without actual veriﬁcation since it is assumed that a small
black bird is deﬁned by a black bird such that it is small. Thus, ”Every small black bird isblack” 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 astate ωR such that ωR ∈ ΩR is called a raven.
(G2 ) Black birds are characterize by the membership function mR : Ω → [0, 1]. A bird witha state ωB such 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 a state 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 11.40 (iv).
Exercise 11.41. It is easy to see that the above (G4 ):ΩSB ⊆ ΩB says that(H) Let ui (i = 1, 2, ..., N ) be a small black bird with the state ωi (∈ ΩSB ), which is denoted byωe (ui ). For each small black bird ui , the probability that the measured value y is obtainedby the measurement MC(Ω) (OB = ({y, n}, 2{y,n} , FB ), S[eω(ui )] ) is equal to 1.
According to the linguistic Copenhagen interpretation (E4 ) 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 measure| {z }
N
⊗N
{y,n}
, FB ), S[eω(ui )] ) is equal to 1.
ment i=1 MC(Ω) (OB = ({y, n}, 2269

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11.8* Hempel’s raven problem

11.8.5

A posteriori proposition: ”Every raven is black”

11.8.5.1

Popper’s falsiﬁcationism in measurement theory♠Note 11.18. Karl Popper (1902-1994) was one of the 20th century’s most inﬂuential philosophersof science. Popper claims that, in order for something to be considered science, it must be
falsiﬁable. If it is false, it can be shown through observation and experiment to be 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 a priori statement, we can accept this statement without veriﬁcation by experiment.
In this section we will discuss the statement ”ΩR ⊆ ΩB ” (i.e., Every raven is black), which
is not a priori proposition but a posteriori proposition.
Hence, our problem is as follows:(J) How can we be sure of ΩR ⊆ ΩB (i.e., ”Every raven is black” )?
i.e., What should we do to be sure of ”ΩR ⊆ ΩB ”?.
In order to do it, we obey Popper’s falsiﬁcationism (cf, ref. [89]) 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 Figure11.5 below)
This test is quite natural, and thus, we should try this ﬁrst.

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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}∩ΩR }) ≈ 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⊗{y,n}ravens. That is, we take the parallel mixed measurement 100, FB ),
i=1 MC(Ω) (OB :=({y, n}, 2S[∗] (ρ0 )), where a mixed state ρ0 (∈ Sm (C(Ω)∗ )) satisﬁes ρ0 (ΩR ) = 1 and ρ0 (ΩR \ ΩB ) =∩0.03. Here, we, for simplicity, assume that ρ0 ({ω ∈ Ω | 0 < mB (ω) < 1} ΩR }) ≈ 0, i.e.,
negligible. (See Figure 11.5 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:⊗{y,n}(M2 ) By the parallel mixed measurement 100, FB ), S[∗] (ρ0 )), a meai=1 MC(Ω) (OB :=({y, n}, 2sured value (y, y, y, ..., y ) is obtained.
| {z }
100

Then, we calculate, by Axiom(m) 1 (mixed measurement) in Chapter 1,(M3 ) the probability that a measured value (y, y, y, ..., y ) is obtained by the parallel mixed| {z }
100

measurement
⊗100
{y,n}
, FB ), S[∗] (ρ0 )) is given by (97/100)100 (< 0.048). That is,
i=1 MC(Ω) (OB := ({y, n}, 2
the probability that (M2 ) is realized (i.e., we meet one hundred black ravens continuously)
is less than 0.048 (> (97/100)100 )).
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11.8* Hempel’s raven problem
Thus, if we believe (M0 ), a very rare thing (i.e., (M3 )) happened since probability 0.048 is quiterare. Therefore, we should doubt the hypothesis (M0 ). That is, we couldn’t deny ”ΩR ⊆ ΩB
(i.e., any raven is black)”. When we can’t do such test many times and still deny ”ΩR ⊆ ΩB
(i.e., any raven is black)”, according to Popper’s falsiﬁcationism, we will believe this.

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 deﬁnitionof ravens is given by a raven state class ΩR as shown in Deﬁnition 11.37 (ii).
(N2 ) Black birds are characterize by the membership function mR : Ω → [0, 1]. The deﬁnition of black birds is given by a raven state class ΩB as shown in Deﬁnition 11.37(ii).
(N3 ) Small black birds are characterize by the membership function mSB : Ω → [0, 1]. The
deﬁnition of small black birds is given by a raven state class ΩSB as shown in Deﬁnition
11.40 (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 11.40 (iv).
(N5 ) It holds that ΩR ⊆ ΩB , i.e., Every raven bird is black, This is a posteriori statement,
which is guaranteed in the sense of Popper’s falsiﬁcationism (or, statistical hypothesis
testing)Thus we see the progress of ornithology (i.e., Formulation [I]progress=⇒ Formulation [III] ).

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mB

1
mR

mSB
Ω

ΩR

ΩB

Ω
ΩSB

Figure 11.6: [All ravens are black: (ΩR ⊆ ΩB ) ]Remark 11.42. Let’s compare analytic philosophy (♭322 ) and scientiﬁc philosophy (♭422 ) inClassiﬁcation 11.2.
(A1 ): {mathematics Describe math. by logic ( and set theory)!
(♭122 ) foundation of math. (Cantor, Russell, Gödel),(♭222 ) math. (Bourbaki),(A2 ):theworldview: Describe science by logic ( and set theory)!
{ logical(♭322 ) analytic philosophy, symbolic logic (Frege, Russell, Peirce, Wittgenstein,...)(♭422 ) scientiﬁc philosophy (Hempel, Popper, ...)(A3 ): linguistics: Describe everything carefully!
(♭522 ) linguistics, ordinary language (Saussure, Wittgenstein)
Both analytic philosophy (♭322 ) and scientiﬁc philosophy (♭422 ) should have argued for the
logical worldview. However, scientiﬁc philosophy (♭422 ) has the following had the following
advantages:(♯1 ) The science was simple and therefore easy to think about.
(♯2 ) Scientists were more likely to be interested.
But, as a result, this advantages have become weaknesses od scientiﬁc philosophy. Thas isbecause the ﬂagpole problem ( in Sec. 11.7) and Hempel’s raven problem ( in Sec. 11.8) and soon highlighted the weaknesses of the logical worldview. It has become the scientiﬁc philosophy
that scientists are not interested in. It would have been a bit better if art (aesthetics, music,etc.) had been the subject of the research. I do, however, have a lot of respect for the philosophyof science. This is because it exposed the weaknesses of the logical worldview and showed me
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11.8* Hempel’s raven problem
List 11.26(♯2 ) Descartes=Kant −−−−−−→ Analytic phil. (♭322 , ♭422 , ♭522 ) −−−−−−→ Quantum language
progress ⃝1(♭522 ):Saussure (Remark 11.13)(♭322 , ♭522 ): Wittgenstein (Sec. 11.6.3)(♭422 ):Hempel (Remark 11.42)

progress ⃝2

See the top page in this chapter.

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11.9

Three approaches to the mind-body problem

The mind-body problem is the most famous problem in Descartes philosophy. There are twostanding 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 expect a 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) [62] Ishikawa,S., A Final solution to mind-body problem by quantum language, Journalof 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).

11.9.1

The mind-body problem

Now let us introduce the mind-body problem, which is said to be the greatest unsolved problem
in dualistic idealism.
In spite that the cogito proposition ”I think, therefore I am” is non-sense (cf. Sec. 8.2),
Descartes used it in order to propose Descartes philosophy (i.e., mind-matter dualism). Thatis, his argument is as follows.
(D) If the existence of ”I” is deduced from the cogito proposition, the existence of ”matter”(which is perceived by ”I”) is accepted. And further, the medium of ”I” and ”matter” isautomatically accepted as ”body (= sensory organ)”.
Therefore, the key-words of Descartes philosophy (= mind-matter dualism) is(E)”I”(=”mind”), ”body”(=”sensory organ”), ”matter”
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11.9 Three approaches to the mind-body problem
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.

11.9.2

The ﬁrst approach; Cognitive scientiﬁc approach

As mentioned in Note 9.8, 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”[11]) as follows.
(G1 ) You, your joys and your sorrows, your memories and your ambitions, your sense of
personal identity and free will, are in fact no more than the behavior of a vast assemblyof nerve cells and their associated molecules.
From the scientiﬁc point of view, I agree to his opinion (G1 ). (i.e., the denial of the substancedualism ). Therefore, I believe that the following will be realized.
(G2 ) Descartes −−−−−→ Philosophy of mind −−−−−→ Brain science −−−−−→(Kant,Husserl)

progress(based on cognitive science)progress

progress

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 ) andthe 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) !

11.9.3The 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.
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Chap. 11 Linguistic philosophy and quantum language
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 pass over in silence” (in [107]), some may conclude that wemust 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 think that the
Wittgenstein’s next assertion is non-productive:(I2 ) philosophical problems arise from insuﬃcient attention to the variety of natural languageuse.

11.9.4 * The third approach; Quantum linguistic solution to themind-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” aretrue. Therefore,(K) in order to solve the mind-body problem in dualistic idealism, we should create thelanguage 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, the second 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 (and Note 1.2), that is,”The solution to the causality problem”⇔”Axiom 2”(11.6)

Similarly, I can give the solution to the mind-body problem (i.e., How are ”mind” and”body” connected?) as follows.

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11.9 Three approaches to the mind-body problem
Answer 11.43. [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 −−−−−−−−−−−−−−−−−−−−→

Quantum language

linguistic turnThis says:
the epistemological mind-body problem

Descartes
How are ”mind” and ”body” connected?
the epistemological mind-body problem−−−−−−−−→
linguistic turnquantum language(11.7)

How are ”measured value” and ”measuring instrument” connected?

Also, recall Figure 1.2 as follows:
measuring instrument

observer(body, eye, etc.)[observable]• [measured value]
a project light⃝system(matter, measuring object)(I(=mind))[state]-b perceive the reaction⃝(i.e., the reﬂected light)

If so, we can easily solve the mind-body problem (in the right-hand side of formula (11.7) thatis,
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 whichmeasurement theory of that system can be formulated. When the observer takes ameasurement of an observable (or, by a measuring instrument) O=(X, F, F ) for asystem S[ρ] i.e., a system S with a state ρ), the probability that a measured value
x (∈ X) obtained by the measurement belongs to 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 (11.6), we have the following equivalences:to propose quantum language⇔to propose Axiom 1 (measurement) and Axiom 2 (causality)278

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Chap. 11 Linguistic philosophy and quantum language⇔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)(11.8)

If so, then my next desire has been fulﬁlled.
• ”mind-body problem” should be the greatest problem in philosophy.

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Chapter 12
Postscript: Philosophy is One
12.1

Philosophy (of worldviews) has progressed towards
quantum language

Recently I proposed ”quantum language” (or, ”QL”, ”measurement theory”, ”the linguistic
Copenhagen interpretation of quantum mechanics”, ”the quantum mechanical worldview”),
which was not only characterized as the metaphysical and linguistic turn of quantum mechanics
but also the linguistic turn of Descartes=Kant epistemology (≈ dualistic idealism).
I assert that this theory is located as follows in the history of world-descriptions.
Assertion 12.1. (= Assertion 0.1 in Preface= Assertion 1.5) [The location of quantum language in the history of world-description (cf. ref.[43, 66]) ]the realistic worldview (physics): monism(monism)2−−→ Newton ⃝relativity
3 → theory −−−−−−→⃝


(unsolved)

theory of5⃝−→ everythingquantum(quantum phys.)4→ mechanics −−−−−−→⃝0⃝:Greek
Schola1philosophy −−−−→⃝
languagesticism7−−− −−→ ⃝ (mind-matter dualism)
Parmenides, Zenolinguistic(Copernican revolution)

Plato


(analytic)Descartes


(=MT)

Locke,... ⃝(scientiﬁc)6mechanical ⃝quantum8−→ philosophy −−−−−−→⃝−→ Kant
10
turnlanguage−→ﬁctional worldview
logical worldview

mechanicalquantum
worldviewdualism
9 −−−−−→⃝
statistics

system theorySocrates
Aristotle(realism)classical mechanical worldview

the idealistic worldview (no experiment): dualism
Figure 0.1: The history of the world-descriptions
Thus, quantum language, roughly speaking, has the three aspects as follows.
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12.1 Philosophy (of worldviews) has progressed towards quantum language
The three aspects of quantum language
7 the standard interpretation of quantum mechanics
⃝:




(i.e., the true colors of the Copenhagen interpretation)thus, in this paper we consider that
”the linguistic Copenhagen interpretation”= ”the linguistic interpretation”
=”the Copenhagen interpretation”

8 the ﬁnal goal of the dualistic idealism (Descartes→Kant→Wittgenstein)⃝:epistemology = measurement theory”Generalized logic” in the sense that Wittgenstein’s picture theory ⊂ QL








9 theoretical statistics of the future⃝:
(i.e., the dualistic foumulation of statistics)
Thus, I assert that QL is the uniﬁed theory of (scientiﬁc) metaphysics♠Note 12.1. These pioneers are as follows.
7 Born’s discovery “the probabilistic interpretation of quantum mechanics” in [6] (1926)(a;⃝)
8 Tractatus Logico-Philosophicus by L. Wittgenstein in [107] (1921)(b;⃝)
9 Fisher’s great book “Statistical Methods for Research Workers” in [20] (1925)(c;⃝)
These three are among the major works of the top ten most inﬂuential writings of the 20th
century. Thus, it is surprising that the three (Born, Wittgenstein, Fisher) investigated the samething in the diﬀerent ﬁelds in the same age. That is, I have summarized these as one theory (=quantum language ) as follows.







7 → ref. [32]: Uncertainty relation ...
(a;⃝)
8 → ref. [35]: Fuzzy inferences ..
(b;⃝)
9 → ref. [36, 38]: Statistics in measurements...
(c;⃝)







−→

(= Measurement theory)

Quantum language
refs. [32],...., [40] ,..... [61, 66]

The above is the starting point of my work about quantum language.

////
I believe, from the scientiﬁc point of view, that
(A1 ) quantum language is the ﬁnal destination of the genealogy of Western philosophy.
And(A2 ) a scientiﬁc perfection of dualism and idealism is realized by quantum language282

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Chap. 12 Postscript: Philosophy is One
In order to assert the (A) (=(A1 ), (A2 )), we proved the following proposition throughout
this paper:
Proposition 12.2.
(B)

mind-matter dualism
transcendental idealism
Copernican revolution
Plato −−−−−−−−−−−−−→ Descartes=Locke −−−−−−−−−−−−−→
Kant
Cogito turn (dualism)
progress(QL)

progress(QL)

the quantum mechanical worldview
linguistic philosophy
quantum mechanical turn−−−−−−−−−−−−−−−−→
Wittgenstein
Quantum language−−−−−−−−→
progress(QL)
progress(see (D) later)quantitative argument is possible(12.1)
linguistic turnif ”−−−−−−−−−−→” is deﬁned by ”getting closer to quantum language”.
progress(QL)[Remark]:• As mentioned in Sec. 10.4.3, the deﬁnition of ”progress” varies.
• For the −−−−−−−−−−−−→, see Sec. 11.6.4.2 or, the (D) below.
progress(see (D) later)

12.2

QL has the power to solve almost all unsolved problems concerning dualistic idealism

12.2.1The list of my answers of unsolved problems

If (A1 ), (A2 ) and (B) are true, it is natural to consider:(C) many unsolved problems raised in the 2500 year history of dualistic idealism
can be solved within the framework of quantum languages.
My results concerning quantum language are summarized in the following two texts(D1 ): This text: History of western philosophy from the quantum theoretical
point of view; [Ver. 3]

(D)


(D2 ): Ref. [66]: The linguistic Copenhagen interpretation of quantum mechanics:
Quantum language [Ver 5]
The (C) is shown throughout this paper as follows.

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12.2 QL has the power to solve almost all unsolved problems concerning dualistic idealism(D1 ) The list of my answers for philosophical unsolved problems in (D1 )• What is probability (or, measurement, causality) ? cf. Sec. 1.1.1)• Zeno paradox (Flying arrow), (cf. Sec. 2.4.2)• Zeno paradox (Achilles and a tortoise), (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. 4.3.3)• 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)• 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. 9.7.2)• What is causality? (cf. Sec. 10.3)• What is Peirce’s abduction? (cf. Sec. 11.3.1)• Five-minute hypothesis (cf. Sec. 11.4.1)• McTaggart’s paradox (cf. Sec. 11.4.2)• quantitative representation of ”Signiﬁer” and ”signiﬁed” (cf. Sec. 11.5.3)• A scientiﬁc understanding of Wittgenstein’s picture theory (cf. Sec. 11.6.2)• Wittgenstein’s paradox (cf. Sec. 11.6.3)• Flagpole problem, (cf. Sec. 11.7.1)• Hempel’s raven paradox (cf. Sec. 11.8)• the mind-body problem (i.e., How are mind and body connected?), (cf. Sec. 11.9.4)(♯) Also, for the solutions of unsolved problems in quantum mechanics, statistical mechanics,
statistics and probability theory, see ref. [66]). Particularly, I think that the following
three are important in physics:• the discovery of Heisenberg’s uncertainty relation (Ref. [32], or Sec. 4.3 in ref.
[66])
• The clariﬁcation of the projection postulate (i.e., the wavefunction collapse) (ref.
[59] , or Sec. 11.2 in ref. [66])• The measurement theoretical characterizations of equilibrium statistical mechanics(Ref. [46], or Chap.17 in ref. [66])284

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Chap. 12 Postscript: Philosophy is One(D2 ) The list of my answers for scientiﬁc unsolved problems in (D2 )ref. [66]; 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. [66]) (Sec.4.1 inref. [66])• The law of large numbers in quantum language (Sec.4.2 in ref. [66])• the true discovery of Heisenberg’s uncertainty relation (Sec. 4.3 in ref. [66])• Bell’s inequality holds in both classical and quantum systems (Sec. 4.5.2 in ref. [66])• Measurement theoretical formulation of measurement, inference, control (Sec. 5.2 in ref.
[66])
• Monty-Hall problem in quantum language (non-bayesian approach) (Sec.5.5 in ref. [66])• Two envelope problem in quantum language (non-bayesian approach) (Sec.5.6 in ref.
[66])
• Conﬁdence interval and statistical hypothesis test (Chapter 6 in ref. [66])• Analysis of variance (Chapter 7 in ref. [66])• Syllogism holds in classical systems, but not in quantum systems (Sec.8.6 and Sec.8.7
in ref. [66])• Mixed measurement theory (Bayesian measurement theory) (Chap. 9 in ref. [66])• The measurement theoretical characterization of the wave-function collapse (= projection pustulate) (Sec.11.2 in ref. [66])• The measurement theoretical characterizations of de Broglie’s paradox, quantum Zenoeﬀect, Schrödinger cat, Wigner’s friend, Wheeler’s delayed choice experiment, Hardy
Paradox, quantum eraser (Sec.11.3∼Sec.11.8 in ref. [66])• The measurement theoretical characterizations of double-slit experiment, Wilson cloud
chamber (Sec.12.2, Sec.12.3 in ref. [66])• The measurement theoretical characterizations of regression analysis (Sec.13.2 in ref.
[66])
• The measurement theoretical characterizations of Brownian motion, Zeno’s paradox(Sec.14.2 , Sec.14.4 in ref. [66])• The measurement theoretical characterizations of least-squares method (Chap.15 in ref.
[66])
• The measurement theoretical characterizations of Kalman ﬁlter (Chap.16 in ref. [66])• The measurement theoretical characterizations of equilibrium statistical mechanics(Chap.17 in ref. [66])• The measurement theoretical characterizations of psychological tests (Chap.18 in ref.
[66])
• The measurement theoretical characterizations of belief (Chap.19 in ref. [66])• The mathematical foundation of science (Hempel’s raven paradox) (Chap.20 in ref. [66])285

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12.2 QL has the power to solve almost all unsolved problems concerning dualistic idealism

12.2.2In the end

The late Hawking (1942–2018) said in his best seller book [23]:(E) Philosophers reduced the scope of their inquiries so much that Wittgenstein the most
famous philosopher this century, said ”The sole remaining task for philosophy is theanalysis of language.” What a comedown from the great tradition of philosophy from
Aristotle to Kant!
Thus, Dr. Hawking was concerned about the divergence between philosophy and science. I
would not have extended quantum language into the realm of philosophy had I not read thissentence by Hawking. My dream was to have Hawking see this text, but it is now an unfulﬁlled
dream. I’m sorry to hear about Hawking’s passing.

I am not an expert in philosophy. However, I have a conﬁdence in only the insistence (A1 )and (A2 ).
(A1 ) quantum language is the ﬁnal destination of the genealogy of Western philosophy (dualistic idealism).
(A2 ) a scientiﬁc perfection of dualism and idealism is realized by quantum language
Thus I hope that readers will improve this text based on the (A1 ) and (A2 ). Also, readers will
be able to easily solve numerous unsolved problems outside of the lists (D1 ) and (D2 ) in Sec.
12.2.1.
Also, if you ﬁnd my mistake in the lists (D1 ) and (D2 ) in Sec. 12.2.1, please let me know
it. I will correct the mistake and make it public (probably, in the next version of this text).
Particularly, if there is another more excellent interpretation of Wittgenstein’s picture theory,
I would be interested to know about this.
The history of western philosophy in this text is my own ﬁction. Naturally, there may besome diﬀerences from the accepted theory. However, even if there are such diﬀerences, I opti286

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mistically believe that these are acceptable to many like-minded philosophy enthusiasts underNietzsche’s words: there are no facts, only interpretations. I wanted to make the interrelationship of unsolved problems in (D1 ) clear in my own ﬁction. In other words, I want to believe”Philosophy is One.”
It is my hope that many philosophy enthusiasts and philosophers will write improved versions of this text.
November 2020
Shiro ISHIKAWA1
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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View, World Journal of Mechanics, Vol. 2, No. 2, 2012, pp. 125-130. doi: 10.4236/wim.2012.22014.
(http://www.scirp.org/journal/PaperInformation.aspx?PaperID=18861#.VKevmiusWap )[47] S. Ishikawa, The linguistic Copenhagen interpretation of quantum mechanics,
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Index
linguistic philosophy, 216

F. Click (1916 – 2004), 180
cogito proposition, 9, 99, 152, 154, 183
cogito turn, 150
Continental Rationalism, 166
Copernican revolution, 119, 185, 204
Critique of Pure Reason (Kant), 196
Critique of Pure Reason (Microsoft Encart), 201

abduction, 227
abstract ego (von Neumann), 118
Aesop(BC.620 - BC.510), 61
allegory of the cave, 62
allegory of the sun, 65
ambiguity of ordinary language
Arguments for the existence of God(Anselmus), Democritus (BC.460-BC.370), 167
Descartes(1596-1650), 4, 5, 152
124
Descartes ﬁgure, 4, 5, 158
Antinomy (Kant), 197
Discourse on the Method (Descartes), 152
Geocentrism and Heliocentrism, 148discovery of zero, 122
Zeno’s paradoxes, 49
dogmatic slumber, 185, 204
Anselmus(1033 - 1109), 23, 82, 129, 173, 177
dualism(=mind-matter dualism), 24, 255
antinomy, 49
antinomy(Kant), 196
eidos(Aristotle), 25, 26, 81, 131a priori synthetic judgment, 202
Einstein(1879 - 1955), 23, 82, 129, 173, 177
Archimedes(BC287 - BC212), 104
Einstein-Tagore meeting, 176
Aristarchus(BC310 - BC230), 102
elementary measurement, 94, 95
Aristotle(BC384 - BC322), 26, 79, 131
elementary proposition, 246
asserted ﬁction, 59
Eratosthenes(BC275 - BC194), 108
Augustinus(354-430), 117, 157
ethics · morals, 55
axiom of choice, 100
Euclid(BC330 - BC275) , 99
Axiom 1 [measurement], 2
Eudoxus(BC.400 BC.347), 41
Axiom 2[causality], 2
Bacon(1561 - 1626), 84, 204
Banach-Tarski theorem, 100
basic structure, 30
Bell’s inequality, 176
Bergson, Henri-Louis(1859-1941), 157
Berkeley, George(1685-1753) , 174
J. Bernoulli(1654 - 1705), 137, 188
binary projective observable, 91
Bohr(1885 - 1962), 23, 82, 129, 173, 177
Bohr-Einstein debates, 6, 176, 177
Boole, George(1815-1864), 223
Born(1882-1970), 282
British Empiricism, 162buoyancy, 104causal relation, 2, 140, 147

ﬁctional worldview (the main current of western
philosophy), 18
Fisher(1890-1962), 282
Five-minute hypothesis, 232
ﬂagpole problem, 259
Frege, G.(1848-1925), 223
Galapagos philosophy, 78
Galileo(1564 - 1642), 84, 137, 188, 204
Gauss(1777 - 1855), 123
Geocentrism(Ptolemaeus), 109
grue paradox, 191
Harari.Y.N(1976 -), 61
Hawking(1942 –2018), 20, 286
Heidegger(1889-1976), 157
Heisenberg’s uncertainty principle, 105, 255

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Heliocentrism, 142
Heliocentrism(Aristarchus), 102
Hempel, C.G(1905-1997), 259
Hērakleitos(BC.540 -BC.480), 44, 83
Hume’s problem of induction, 187
Husserl (1859-1938) , 211hyle(Aristotle), 25, 26, 81, 131

non-Euclidean revolution, 100

McTaggart. J.M.E.(1866–1921), 234
McTaggart’s paradox, 234
measurement, 2Merleau-Ponty(1908-1961), 165
metaphysics, 16
mind-body problem, 156
mind-body problem , 275
modal logic, 21
model-change, 77, 156
Motion function method, 46

raven problem, 263
realistic worldview(physics), 18
Russell. B.(1872 - 1970), 73, 81, 232

observable, 29, 30
Ockham(1285 - 1347), 23, 82, 129, 173, 177
Only the present exists, 117

paradigm shift, 135parallel postulate, 99
Parmenides(born around BC. 515), 24, 83
I think, therefore I am., 152
Peirce, C.S.(1839-1914), 226
Idea(Plato), 25, 26, 131, 159, 164
Philosophical Investigations, 240
I know that I know nothing, 57
picture theory, 241
image observable, 33
Plato(BC427 - BC347), 26, 131, 159
induction, 139
Popper(1902-1994), 270
inductive reasoning, 138
population, 25, 26, 131
Kant(1724-1804), 185, 204
practical logic, 20, 21
Kelvin(1824 –1907), 16
primary quality, secondary quality, 25, 26, 131,
164
J, Kepler(1571 - 1630), 139, 144
Principia, 147
law of large numbers, 136, 187
Problem of universals, 23, 82, 129, 173, 177
leaning Tower of Pisa, 145
projection postulate, 7, 118
Leibniz(1646-1716), 167, 169projective observable, 30
Leibniz-Clarke Correspondence, 169
Proof of the existence of God, 124
linguistic interpretation(=linguistic Copenhagen in- Protagoras(BC490 - BC420), 56
terpretation), 4Ptolemaeus(AD.83 - AD.168), 109
linguistic philosophy, 240
pure Pythagoreanism, 42, 219, 220
linguistic turn, 256
purpose, 83
List of unsolved problem, 11, 249
Pythagoras(BC582 - BC496), 40
J. Locke (1632 - 1704), 162
logical worldview (analytic philosophy) , 19
qualia problem, 179
logical worldview, 17
qualia (=subjectivity) problem, 156

Saussure, F.(1857-1913), 235
Scholasticism, 120
mechanical worldview(statistics, quantum language),22
secondary quality, primary quality, 164
self-reference
cogito proposition, 9
subjective time, 118
naive set theory, 16naive set theory (=elementary set theory = Can- sequential causal observable, 31
tor’s set theory, 220
set theoretical worldview, 17
nativism, 166
signiﬁed, 237
necessity of dualism, 69
signiﬁer, 237
necessity of idealism, 69
Socrates(BC469 - BC399), 56
Newton(1643-1727), 84, 172, 204
spectrum, 170

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November 15, 2020state, pure state, mixed state, 28
state equation, 205
subjective time, 117
subjectivity (= qualia) problem, 156
subjectivity problem, 156
syllogism(Aristotle), 86
tabula rasa (blank paper), 163
temporal logic, 21
tense, 117
Thalēs(BC640 - BC546), 37
thing-in-itself, 200
Thomas Aquinas(1225 - 1274), 26, 131
time theory(Augustinus), 116
Tractatus Logico-Philosophicus (=TLP), 240transcendental idealism, 210, 236
trial of Galileo, 144
von Neumann(1903- 1957) , 118
wave function collapse, 7, 118
Weinberg(1933 -), 38, 62, 80, 85, 110, 140, 143
What is space-time?, 47
Whitehead(1861 - 1947), 38, 76
Wittgenstein(1889 - 1951), 12, 20, 157
Wittgenstein’s paradox, 252
world was created ﬁve minutes ago (Russell’s skepticism), 9
worldviewism, 14
Zadeh’s fuzzy set, 74
Zeno(BC490 - BC430), 49
Zeno’s paradoxes , 49

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KSTS/RR-20/001
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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