[日本語(Japanese)]        

What is the Copenhagen interpretation? --- Linguistic Interpretation of Quantum Mechanics; Quantum language
(by Shiro. ISHIKAWA; Keio university)


CONTENTS
§ 1 [Animistic worldview] → [Mechanical worldview] → [Linguistic worldview]

We have a more than 3000-year history concerning the research of worldviews. And this study has been always located at the central position of science and philosophy. The first paradigm shift (i.e., the leap from "the animistic worldview (life dwells on things) " of the Middle Ages) is the mechanical worldview, which was owed to Galileo, Bacon, Descartes, Newton, and so on. This paradigm shift is very powerful, and hence, it opened the door from the Middle Ages to the Modern Ages (see ① in Fig.1). Here, the mechanical word view is characterized as.

(A): Study every (non-physical) science, modeled on physics. It is simple, but, as the norm of the best description of the world, it has dominated the modern science until today.
If it be so, everyone may dream to propose the scond paradigm shift, however, it may not be easy. In fact, it is not difficult to enumerate shattered dreams such as catastrophe theory, fuzzy theory, complex system theory, chaos theory and so on. However, It is not stopped that we strive against instinct of liking to tell "grand narratives". That is, we cannot stop to investigate the paradigm shift.
Here we assert that the linguistic worldview in Fig. 1 due to the quantum language (=measurement theory), which is the final stage of the linguistic series in Fig. 1, is the scond paradigm shift.



§ 2 The beginning of things: Heisenberg's uncertainty principle

Our starting point is the discovery ( 20 years ago) of the mathematical formulation of Heisenberg's uncertainty principle (cf. ref. [1]). This discovery urged us to investigate the following problem:


(B):        What is quantum mechanics?     Or, what is the standard interpretation of quantum mechanics?


Quantum mechanics has various interpretations ( such as the Copenhagen interpretation, the many worlds interpretation, etc.). This fact may be negligible in applications. But it is serious for the formulation of Heisenberg's uncertainty principle. That is, the "errors" depend on the interpretation (e.g., the wavefunction collapse is accepted or not?) ). Thus, we now think that we should have argued about Heisenberg's uncertainty relation under a certain firm interpretation of quantum mechanics in ref. [1].



§ 3 In the beginning was the word --- the miraculous power of language ---

Quantum mechanics was born about 90 years ago. However, we may be possessed with the surprising prejudice. That is, we may convince "Quantum mechanics is physics." Writing the conclusion first, we say that the linguistic worldview (i.e., the in Fig. 1) is,

(C):   In the beginning there exists the language called "quantum language". And world is described and constructed by the language.

(cf. Refs.[2, 3]). Although the explanation of quantum language is omitted here, it suffices to consider that it is composed of two spells (concerning "measurement" and "causality") and the linguistic interpretation. What we can do in quantum language (=measurement theory) is only to trust in man's linguistic competence. Fig 1 says that quantum language has the following three aspects:

[⑦ Fig 1]: The ture colors of the Copenhagen interpretation. That is, the Copenhagen interpretation does not belong to physics!    (Even if "the true quantum mechanics" exists in the direction of ⑤, I believe that it is the relativistic quantum mechanics that does not need "interpretation")
[⑧ Fig 1]: The final goal of the dualistic idealism ( Descartes=Kant philosophy)
[⑨ Fig 1]: Theoretical statistics of the future


This language has a great power to describe ordinary phenomena as well as quantum phenomena. For example, economics is created by describing economical phenomena in quantum language.

Also,

  • economics is created by describing economical phenomena in quantum language.
  • quantum mechanics is created by describing quantum phenomena in quantum language.
  • psychology is created by describing psychological phenomena in quantum language.
  • ・・・・・・・・・・・・・

(see Fig 2). We assert that this is the true picture of quantum mechanics, and we are convinced that this is just the answer to the question (B). Therefore, the "⑦ →" in Fig. 1 is a historical circumstance. and thus, the true direction is the "← ⑦"(where we consider the usual quantum mechanics and not the quantum physics beyond the ⑤ in Fig. 1). That is, (C)=[Fig.2]=(D), namely,
(D):  
Thus, the linguistic worldview (C) (= in Fig. 1) asserts that

  • Science (more precisely, non-physical science) is to describe the world by quantum language.

Therefore,

  • the metaphysics called quantum language is located in the center of science.

Again note that the linguistic worldview (C) (= in Fig. 1)is due to the great human's power of linguistics.

If quantum mechanics is regarded as physics, we cannot understand the reason that there are several paradoxes and interpretations in quantum mechanics. For example, we think that Schrödinger's cat does not live in the linguistic worldview since it can not be described by quantum mechanics. (Recall Wittgenstein's saying: "The limits of my language mean the limits of my world"). Also, we consider that the dualism is not fit for physics. Such idea is not only due to us but also Einstein, who never regarded quantum mechanics as physics.

§ 4 Mechanical worldview (A) vs. linguistic worldview (C)

For example, some philosophers has been studying "Achilles and the tortoise (cf. the above (D))" during about 2500 years. If you think that they are too stupid to understand the geometric series, you are an ordinary person, i.e., the believer of the mechanical worldview (A). That is, you do not know that the philosophers has been making efforts to discover another worldview rather than the mechanical worldview (A) (cf. [11]). Also, under the mechanical worldview (A), we cannot answer the question "What is statistics? or "What is (non-physical) space-time?" That is, the quantum language is necessary to answer these questions (cf. [7-9, 12-14]). Moreover, we cannot understand the dualistic idealism (in the main stream of the philosophy) without the linguistic worldview (C) (cf. [4,14]). This is the reason that there are no understandable books concerning dualistic idealism. Since unsolved problems are easily solved under the (C), we believe that the linguistic worldview (C) is superior to the mechanical worldview(A). Also, it should be noted that the mechanical worldview (A) has no room in Fig. 1. Thus, it is improper to consider the mechanical worldview (A) as a kind of worldview.
Dr. Hawking said in his best seller book [A Brief History of Time: From the Big Bang to Black Holes, Bantam, Boston, 1990]:
(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 the analysis of language." What a comedown from the great tradition of philosophy from Aristotle to Kant!

We think that this is not only his opinion but also most scientists' opinion. However, we do not agree to his opinion, since we know the development such as "Kant ⇒ Wittgenstein ⇒ The linguistic worldview " in Fig. 1.

§ 5 The end of grand narratives --- 3000-year final answer ---

Now we believe that the proof of "the ⑩ in Fig. 1" has been almost completed. Therefore, the countdown of the two ends of grand narratives (⑤ and ⑩ in Fig.1) started. In the above sense (i.e., the linguistic series acquires the linguistic worldview (C)), we dare to declare

"the happy end of grand narratives"

However, science has already rushed into post modern time[=the ages of small narratives = the ages of sciences(i.e., engineering, social sciences, bio, etc.) in (D)]. Thus, we think that science is not over.



[References]
The refs. [2,3,4,24,27] may be easiest to understand. The [24 (or 23)] clarified the similaities and differences between so called Copenhagen interpretation and the linguistic interpretation. Also, [8] is the text for undergraduate students. Almost all my outcomes are written in the preprints [26, 28] (the lecture note of the master course in Dept of Math. Keio university).

My favorite papers are [25,28], which can be read without the knowledge of quantum language [26].


[1]: S. Ishikawa, "Uncertainty Relations in Simultaneous Measurements for Arbitrary Observables," Rep. Math. Phys., Vol. 29, No. 3, pp. 257-273 (1991) doi: 10.1016/0034-4877(91)90046-P, [PDF download]
[2]: S. Ishikawa, "A New Interpretation of Quantum Mechanics," Journal of Quantum Information Science, Vol. 1 No. 2, 2011, pp. 35-42. doi: 10.4236/jqis.2011.12005 ( download free)
[3]: S. Ishikawa, "The linguistic interpretation of quantum mechanics," arXiv:1204.3892v1[physics.hist-ph] , (2012) ( download free)
[4]: S. Ishikawa, "Quantum Mechanics and the Philosophy of Language: Reconsideration of Traditional Philosophies," Journal of quantum information science, Vol. 2, No. 1, 2012, pp.2-9. doi: 10.4236/jqis.2012.21002 ( download free)
[5]: S. Ishikawa, "Fuzzy Inferences by Algebraic Method," Fuzzy Sets and Systems, Vol. 87, No. 2, 1997, pp.181-200. doi: 10.1016/S0165-0114(96)00035-8 , [PDF download]
[6]: S. Ishikawa, "A Quantum Mechanical Approach to Fuzzy Theory," Fuzzy Sets and Systems, Vol. 90, No. 3, 1997, pp. 277-306. doi: 10.1016/S0165-0114(96)00114-5 , [PDF download]
[7]: S. Ishikawa, "Statistics in measurements," Fuzzy sets and systems, Vol. 116, No. 2, 141-154 (2000). doi:10.1016/S0165-0114(98)00280-2 , [PDF download]
[8]: S. Ishikawa, "Mathematical Foundations of Measurement Theory," Keio University Press Inc. 335pages, 2006. http://www.keio-up.co.jp/kup/mfomt/
[9]: S. Ishikawa, "A Measurement Theoretical Foundation of Statistics," Applied Mathematics, Vol. 3, No. 3, 2012, pp. 283-292. doi: 10.4236/am.2012.33044 ( download free)
[10]: S. Ishikawa, "Ergodic Hypothesis and Equilibrium Statistical Mechanics in the Quantum Mechanical World View," World Journal of Mechanics, Vol. 2, No. 2, 2012, pp. 125-130. doi:10.4236/wjm.2012.22014 ( download free)
[11]: S. Ishikawa, "Zeno's paradoxes in the Mechanical World View," arXiv:1205.1290v1 [physics.hist-ph] , (2012) ( download free)
[12]: SS. Ishikawa, "Monty Hall Problem and the Principle of Equal Probability in Measurement Theory," Applied Mathematics , Vol. 3, No. 7, 2012, pp. 788-794. doi:10.4236/am.2012.37117 ( download free)
[13]: S. Ishikawa, "What is statistics?; The Answer by Quantum Language," arXiv:1207.0407v1 [physics.data-an] , 2012 ( download free)
[14]: S. Ishikawa: "Measurement Theory in the Science of Philosophy," arXiv:1209.3483v1[physics.hist-ph] ,( 2012) (download free)
[15]: S. Ishikawa: "Heisenberg uncertainty principle and quantum Zeno effects in the linguistic interpretation of quantum mechanics," http://arxiv.org/abs/1308.5469[quant-ph] ,( 2013) (download free)
[16]: S. Ishikawa: "A quantum linguistic characterization of the reverse relation between confidence interval and hypothesis testing," http://arxiv.org/abs/1401.2709[math.ST] ,( 2014) (download free)
[17]: S. Ishikawa: "ANOVA (analysis of variance) in the quantum linguistic formulation of statistics," http://arxiv.org/abs/1402.0606[math.ST] ,( 2014) (download free)
[18]: S. Ishikawa: "Regression analysis in quantum language," http://arxiv.org/abs/1403.0060[math.ST] ,( 2014) (download free)
[19]: S. Ishikawa, K. Kikuchi: "Kalman filter in quantum language," http://arxiv.org/abs/1404.2664[math.ST] ,( 2014) (download free)
[20]: S. Ishikawa: "The double-slit quantum eraser experiments and Hardy's paradox in the quantum linguistic interpretation," http://arxiv.org/abs/1407.5143[quantum-ph] ,( 2014) (download free)
[21]: S. Ishikawa: "The Final Solutions of Monty Hall Problem and Three Prisoners Problem," arXiv:1408.0963v1 [stat.OT] ,( 2014) (download free)
[22]: S. Ishikawa: "The two envelopes paradox in non-Bayesian and Bayesian statistics," arXiv:1408.4916v4 [stat.OT] ,( 2014) (download free)
[23]: S. Ishikawa: "Linguistic interpretation of quantum mechanics: Quantum language," KSTS/RR-15/001 [Reseach Report; Keio Math] ,( 2015), 416 pages (download free) or, KSTS/RR-15/001 [S. Ishikawa]
[24]: S. Ishikawa, "Linguistic interpretation of quantum mechanics; Projection Postulate," Journal of Quantum Information Science, Vol. 5 No. 4, 2015, pp. 150-155. DOI: 10.4236/jqis.2015.54017 (download free)   
Also, see Reseach Report; Keio Math [KSTS/RR-15/009](S. Ishikawa ). ( download free) ,arXiv:1511.07777 [physics.gen-ph] ,( 2015), (download free)
[25]: S. Ishikawa: "Linguistic interpretation of quantum mechanics: Quantum language [ver. 2]," Research Report, Keio Math. [KSTS/RR-16/001] ,( 2016), 426 pages (download free) or, Research Report, Keio Math. [KSTS/RR-16/001] (S. Ishikawa)
The html version is seen in the right side bar. Also, this preprint is the draf of the following book.
Shiro Ishikawa, "Linguistic Interpretation of Quantum Mechanics - Towards World-Description in Quantum Language -" Shiho-Shuppan Publisher 2016, ( 405 pages)
[26]: S. Ishikawa: "History of Western Philosophy from the quantum theoretical point of view," Reseach Report; Keio Math [KSTS/RR-16/005] ,( 2016), 141 pages (download free) or, Reseach Report; Keio Math [KSTS/RR-16/005](S. Ishikawa )    or,   Philpapers (2016)
[27]: S. Ishikawa: "A final solution to the mind-body problem by quantum language" Journal of quantum information science, Vo.7 No.2 2017, 48-56 (download free), The preprint is as follows: Reseach Report; Keio Math [KSTS/RR-17/003](S. Ishikawa )(download free),
[28]: S. Ishikawa: "History of Western Philosophy from the quantum theoretical point of view, Versin 2" Reseach Report; Keio Math [KSTS/RR-17/004](S. Ishikawa ) ,( 2017), 139 pages (download free)










Shiro Ishikawa, "Linguistic Interpretation of Quantum Mechanics : Quantum language [ver.2]" KSTS/RR-16/001 (2016, Research Report in Keio Math)
CONTENTS
:[PDF]+ [English HTML]+ [Japanese HTML]

0.0 Home page
0.1: Preface
1.0: 1.0:Feynman's question1.0:ファインマン博士への解答
1.1:Quantum language(1.1:量子言語)
1.2(1):Axioms 1 and 2 (measurement and causality ) and interpretation: 【1.2(1):言語ルール1と2(測定と因果関係)とその解釈】
1.2(2):Linguistic interpretation【1.2(2):言語的(コペンハーゲン)解釈】
1.2(3):Summary【1.2(3):要約】
1.3:Example (Hot or Cold?)【1.3:例(熱い? 冷たい?)】
2.0: Axiom 1( measurement ); Abstract 2.0: 言語ルール1(測定);アブストラクト
2.1: Basic structure $[{\mathcal A} \subseteq$ $ \overline{\mathcal A} \subseteq B(H)]$( General Theory)【2.1: 基本構造$[{\mathcal A} \subseteq$ $ \overline{\mathcal A} \subseteq B(H)]$(一般論)】
2.2: Quantum basic structure $[{\mathcal C}(H) \subseteq$ $ B(H) \subseteq B(H)]$【2.2: 量子基本構造$[{\mathcal C}(H) \subseteq$ $ B(H) \subseteq B(H)]$】
2.3: Classical basic structure $[C_0(\Omega ) \subseteq$ $ L^\infty ( \Omega, \nu ) \subseteq B(H)]$【2.3: 古典基本構造$[C_0(\Omega ) \subseteq$ $ L^\infty ( \Omega, \nu ) \subseteq B(H)]$】
2.4: State and observable【2.4: 状態と観測量】
2.5: Examples of observables【2.5: 観測量の例】
2.6: System quantity【2.6: システム量】
2.7: Axiom 1 ; No science without measurements【2.7: 言語ルール1;測定無くして、科学無し】
2.8: Classicalexamples ( urn problem, etc.) 【2.8: 古典系の例(壺問題, etc.) 】
2.9: Stern=Gerlach experiment【2.9: シュテルン=ゲルラッハの実験】
2.10: de Broglie paradox【2.10:ド・ブロイのパラドックス】
3.0: Linguistic interpretation; Abstract 3.0:言語的解釈;アブストラクト
3.1: The linguistic interpretation【3.1:言語的(コペンハーゲン)解釈】
3.2: Tensor operator algebra【3.2:テンソルは作用素代数】
3.3.1: Only one observable【3.3.1:観測量は一つだけ】
3.3.2: state doesnot move【3.3.2:状態は動かない】
3.3.3: Only one state 【3.3.3:状態は一つだけ 】
4.0: Linguistic interpretation; quantum systems4.0:言語的解釈;量子系
4.1: Kolmogorov extension theorem【4.1:コルモゴロフの拡張定理】
4.2: The law of large numbers【4.2:大数の法則】
4.3.1: Why is Heisenberg's uncertainty principle famous?【4.3.1:ハイゼンベルグの不確定性原理は何故有名なのか?】
4.3.2: Mathematical formulation of Heisenberg's uncertainty principle【4.3.2:ハイゼンベルグの不確定性原理の数学的定式化】
4.3.3: except approximately simultaneous measurement【4.3.3:ハイゼンベルグの不確定性原理が破れる場合】
4.4: EPR-paradox【4.4: EPR-パラドックス】
4.5: Bell's inequalirty 【4.5: ベルの不等式がわからない】
5.0: Fisher statistics (abstract)5.0:フィッシャー統計学;アブストラクト
5.1: Urn problem【5.1:壺問題】
5.2: Fisher's maximum likrlihoof method【5.2:フィッシャーの最尤法】
5.3: Examples of Fisher's maximum likrlihoof method【5.3:フィッシャーの最尤法の例】
5.4: Moment method 【5.4:モーメント法】
5.5: Monty Hall problem: High school student puzzle 【5.5:モンティホール問題:高校生パズル】
5.6: Two envelope problem: High school student puzzle 【5.6:二つの封筒問題:高校生パズル】
6.0: Confidence interval and statistical hypothesis testing ( Abstract )6.0:信頼区間と仮説検定:アブストラクト
6.1: Review: classical quantum language 【6.1:復習: 古典量子言語】
6.2: The reverse relation between confidence interval and statistical hypothesis 【6.2:信頼区間と仮説検定の逆関係】
6.3(1): Population mean (Confidence interval and statistical hypothesis testing)【6.3(1):母平均(I:信頼区間と仮説検定)】
6.3(2): Population mean (Confidence interval and statistical hypothesis testing) 【6.3(2):母平均(II:信頼区間と仮説検定) 】
6.4(1): Population variance (Confidence interval and statistical hypothesis testing)【6.4(1):母分散(I:信頼区間と仮説検定)】
6.4(2): Population variance (Confidence interval and statistical hypothesis testing) 【6.4(2):母分散(II:信頼区間と仮説検定)】
6.5: Difference of population means (Confidence interval and statistical hypothesis 【6.5:母平均の差(信頼区間と仮説検定)】
6.6: Student $t$-distribution of population mean【6.6:母平均のスチューデントの$t$-分布】
7.0: ANOVA(Analysis of variance) ( Abstract )7.0:ANOVA(分散分析):アブストラクト
7.1: Zero way ANOVA (= Student $t$-distribution )【7.1: 零元分散分析(=スチューデントの$t$-分布)】
7.2: The one way ANOVA 【7.2: 一元分散分析】
7.3(1): The two way ANOVA【7.3(1):二元分散分析(I)】
7.3(2): The two way ANOVA 【7.3(2):二元分散分析(II)】
7.4: Supplement (Gauss integral ) 【7.4:補遺(ガウス積分) 】
8.0: Practical logic - Do you believe in syllogism?8.0:実践論理 - 三段論法を信じますか?
8.1: Marginal observable and quasi-product observable 【8.1:辺観測量と擬積観測量】
8.2: Properties of quasi-product observables【8.2: 擬積観測量の性質】
8.3: The definition of "implication "【8.3:"含意"の定義】
8.4: Cogito-- I think, therefore I am【8.4: コギト---われ思う、故にわれ在り】
8.5: Combined observable -- Only one measurement is permitted【8.5: 結合観測量 -- 測定は一回だけ】
8.6: Syllogism-- Does Socrates die? 【8.6: 三段論法-- ソクラテスは死ぬか? 】
8.7: Syllogism does not hold in quantum systems【8.7: 三段論法は量子系ではNG】
9.0: Mixed measurement theory ($\supset$Bayesian statistics)9.0:混合測定($\supset$ベイズ統計):アブストラクト
9.1: Mixed measurement theory ( Bayesian statistics ) 【 9.1:混合測定($\supset$ベイズ統計)】
9.2: Simple examples in mixed measurements【9.2:混合測定の簡単な例】
9.3: St. Petersburg two envelope problem【9.3:セントペテルスブルグの二つの封筒問題】
9.4: Bayesian statistics is to use Bayes theorem【9.4:ベイズ統計とはベイズの定理を使うこと】
9.5: Two envelope problem (Bayes' method)【9.5:二つの封筒問題(ベイズの方法)】
9.6:Monty Hall problem ( Bayesian approach )【9.6:モンティホール問題(ベイズの方法)】
9.7:Monty Hall problem ( The principle of equal weight ) 【9.7:モンティホール問題(等確率の原理) 】
9.8: Averaging information ( Entropy )【9.8:平均情報量(エントロピー )】
9.9: Fisher statistics: Monty Hall problem [three prisoners problem]【9.9:フィッシャー統計: モンティホール問題 [三囚人の問題]】
9.10:Bayesian statistics: Monty Hall problem [three prisoners problem] 【9.10:ベイズ統計: モンティホール問題 [三囚人の問題]】
9.11: Equal probability}: Monty Hall problem [three prisoners problem] 【9.11:等確率の原理: モンティホール問題 [三囚人の問題] 】
9.12: Bertrand's paradox( "randomness" depends on how you look at)【9.12:ベルトランのパラドックス( "ランダム"は見方次第)】
10.0: Causality (Abstract)10.0:因果関係:アブストラクト
10.1: The most important unsolved problem---what is causality?【10.1:因果関係とは何か?---現代科学の最重要問題】
10.2: Causality---Mathematical preparation 【10.2:因果関係---数学的準備】
10.2.2: Simple example---Finite causal operator is represented bymatrix【10.2.2:簡単な例---行列表示】
10.3:Axiom 2---Smoke is not located on the place which does not have fire【10.3:言語ルール2---火のないところに煙は立たない】
10.4: Kinetic equation (in classical mechanics and quantum mechanics)【10.4:(古典系と量子系の)運動方程式】
10.5: Exercise:Solve Schrödinger equation by variable separation method【10.5:演習:変数分離法によるシュレーディンガー方程式の解】
10.6:Random walk and quantum decoherence【10.6:酔歩と量子デコヒーレンス】
10.7: Leibniz=Clarke Correspondence: What is space-time? 【10.7:時空とは何か?---ライプニッツ=クラーク論争】
11.0: Measurement and causality (Abstract)11.0:単純測定と因果関係:アブストラクト
11.1: The Heisenberg picture and the Schrödinger picture【11.1:ハイゼンベルグ描像とシュレーディンガー描像】
11.2: Wave function collapse ( = Projection postulate )【11.2:射影公理 ( 波束の収縮)】
11.3:de Broglie's paradox(non-locality=faster-than-light)【11.3:ド・ブロイのパラドックス(非局所性=光より速い何かがある)】
11.4: Quantum Zeno effect【11.4:量子ゼノン効果】
11.5: Schrödinger's cat, Wigner's friend and Laplace's demon【11.5:シュレーディンガーの猫、ウィグナーの友人、ラプラスの悪魔】
11.6: Wheeler's Delayed choice experiment: "Particle or wave?" is a foolish question【11.6:ウィーラーの遅延選択実験: "粒子 or波?"は愚問】
12.0: Realized causal observable in general theory12.0:実現因果観測量
12.1: Finite realized causal observable【12.1: 有限実現因果観測量】
12.2 Double-slit experiment 【12.2 二重スリット実験 】
12.3: Wilson cloud chamber in double slit experiment【12.3: ウィルソンの霧箱】
12.4: Two kinds of absurdness ---idealism and dualism【12.4: 二元論と観念論のトンデモ性 ---】
13.0: Fisher statistic (II)13.0:フィッシャー統計 (II)
13.1: "Inference = Control" in quantum language【13.1: "推定 = 制御"と思え】
13.2: Regression analysis【13.2: 回帰分析=推定+実現因果観測量】
14.0: Regression analysis14.0:古典因果関係+言語解釈
14.1: Infinite realized causal observable in classical systems【14.1: 無限実現因果観測量】
14.2: Is Brownian motion a motion?【14.2: ブラウン運動とは何か?】
14.3: The Schrödinger picture of the sequential deterministic causal operator【14.3: 決定的因果作用素列のシュレーディンガー描像】
14.4 : Zeno's paradoxes---Flying arrow is not moving 【14.4 : ゼノンのパラドックス---アキレスと亀】
15.0: Least-squares method and Regression analysis15.0 :最小二乗法と回帰分析:アブストラクト
15.1 The least squares method【15.1 最小二乗法:簡単すぎて難しい】
15.2: Regression analysis in quantum language【15.2: 最小二乗法から回帰分析へ】
15.3: Regression analysis(distribution , confidence interval and statistical hypothesis testing)【15.3: 回帰分析(分布,信頼区間,仮説検定)】
15.4: Generalized linear model【15.4: 一般線形モデル】
16.0: Kalman filter 16.0:カルマンフィルター:アブストラクト Abstract (16.0: Kalman filter)
16.1: Bayes=Kalman method (in $L^\infty(\Omega, m)$)【16.1: ベイズ=カルマンの方法(in $L^\infty(\Omega, m)$)】
16.2: Problem establishment (concrete calculation)【16.2: カルマンフィルターの問題設定(具体的計算)】
16.3: Bayes=Kalman operator【16.3: ベイズ=カルマン作用素】
16.4: Calculation: prediction part 【16.4: 計算: 予測部分 】
16.5: Calculation: Smoothing part【16.5: 計算: 平滑化部分】
17.0: Equilibrium statistical mechanics

17.1: Equilibrium statistical mechanics (Causality)
17.2: Equilibrium statistical mechanics (Probability)
18.0: The reliability in psychological test 17.0:心理統計:アブストラクト
18.1: Reliability in psychological tests 【17.1: 心理テストの信頼性 】
18.1.3: Reliability coefficient【17.1.3: 心理統計の信頼性係数】
18.2: Correlation coefficient: How to calculate the reliability coefficient
19.0: How to describe "brief"18.0: "信念"の確率化:アブストラクト
19.1: Belief, probability and odds【18.1: 信念, 確率, オッズ】
19.2: The principle of equal odds weight【18.2: 等確率の原理(II)】
20.1: Two kinds of ( realistic and linguistic ) world- views19.1 あとがき: 二つの世界記述法(実在的と言語的)
20.2: The summary of quantum language 【19.2: 量子言語のまとめ 】
20.3: Quantum language is located at the center of science【19.3: 科学をするとは、量子言語で話すこと】