早稲田大学高等研究所の秋吉と申します。
以下の要領でワークショップ「Computations, Proofs, and Intuitions: A Workshop on Philosophy of Mathematics」(WIAS Top Runners’ Lecture Collection of Science)開催のご案内をさせて頂きます。
ご都合よろしければぜひご参加ください。
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ワークショップご案内：“Computations, Proofs, and Intuitions: A Workshop on Philosophy of Mathematics”
We are pleased to invite you to a workshop titled “Computations, Proofs, and Intuitions: A Workshop on Philosophy of Mathematics” (WIAS Top Runners’ Lecture Collection).
ワークショップ「Computations, Proofs, and Intuitions: A Workshop on Philosophy of Mathematics」(WIAS Top Runners’ Lecture Collection of Science)開催のご案内をさせて頂きます。参加自由です。お気軽にご参加ください。
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Title: Computations, Proofs, and Intuitions: A Workshop on Philosophy of Mathematics
「計算，証明，直観」：数学の哲学ワークショップ
Date: September 18 (Fri.), 2015
Time: 10:00 – 18:00
Place: International Conference Center (Meeting Room 2, 3rd floor), Waseda University.
日時: 2015年9月18日(金) 10:00 – 18:00
会場: 早稲田大学早稲田キャンパス 国際会議場 3階第2会議室
（https://www.waseda.jp/top/access/waseda-campus#anc_8）
http://www.waseda.jp/top/assets/uploads/2015/08/waseda-campus-map.pdf
* Bldg. 18 on the map.
PROGRAM
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10:10-11:10 “Game Theory and "Symbolic" Logic”
Speaker: Mamoru Kaneko (Waseda University)
11:10-12:10 “Proof theory of the lambda-calculus”
Speaker: Masahiko Sato (Kyoto University)
12:10-14:00 Lunch Break
14:00-15:20 “The concept of computation - an axiomatic characterization”
Speaker: Wilfried Sieg (Carnegie Mellon University)
15:20-15:40 Break
15:40-16:40 “Kant on mathematical intuition: from an educational point of view” Speaker: Yasuo Deguchi (Kyoto University)
16:40-17:00 Break
17:00-18:00 “Aspects of the notion of computability”
Speaker: Makoto Kikuchi (Kobe University)
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世話人: 秋吉亮太（早稲田高等研究所助教）
主催: 早稲田高等研究所
問合せ先: 秋吉亮太
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ABSTRACTS
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[Speaker] Mamoru Kaneko (Waseda University)
[Title] Game Theory and "Symbolic" Logic
[Abstract]
In game theory, a player chooses/adjusts his behavior based on his understanding of the game situation and his decision/prediction criterion. In logic, a (ideal) mathematician calculates/proves a target theorem from his assumptions. An engine for such adjustments and calculations is logical inference. Such behavior is of a highly symbolic nature. However, it has been customary in the fields of mathematics as well as game theory that real targets are actual models but not symbolic expressions (axiomatizations). This attitude should be reversed when we take game theory as a serious study of human behavior/decision-making in social contexts including considerations of experiential sources for individual beliefs. This is very compatible with the basic idea of “symbolic” logic. In this presentation, we discuss various problems related to this interpretation.
[Speaker] Masahiko Sato (Kyoto University)
[Title] Proof theory of the lambda-calculus
[Abstract]
We present a new representation of
lambda-terms as a subalgebra
of a free algebra. As elements of the free
algebra, lambda-terms are
constructed without employing the abstraction
operation, and this
construction of lambda-terms enables us to study
lambda-terms as
natural finitary objects.
In this setting, we will develop the
lambda-calculus by defining
reductions as derivations and study the proof
theory of the
lambda-calculus. We will develop the
theory in the Minlog proof
assistant developed by Helmut Schwichtenberg.
[Speaker] Wilfried Sieg (Carnegie Mellon University)
[Title] The concept of computation - an axiomatic characterization
[Abstract]
Computations are pervasive in contemporary science and life; they are visible everywhere. However, the concept of computation emerged in an almost invisible corner of our intellectual life. I will describe the context for the emergence of computability as a crucial notion in mathematics and logic with a normative philosophical component.
My analysis of the emergence of this concept provides a novel perspective on the central methodological issue that surrounds computations, the “Church-Turing Thesis”. We focus on calculable functions on natural numbers and mechanical operations on syntactic configurations.
The latter analysis leads to boundedness and locality conditions that motivate axioms for computable dynamical systems. Models of these axioms are all reducible to Turing machines. Cellular automata and a variety of artificial neural nets can be shown to satisfy them.
Finally, I draw connections and point out directions for fascinating work. As to connections, I will emphasize that my novel perspective is rooted in the radical transformation of mathematics of the 19th century; especially, in the new form of structural axiomatics introduced by Dedekind and Hilbert.
[Speaker] Yasuo Deguchi (Kyoto University)
[Title] Kant on mathematical intuition: from an educational point of view
[Abstract]
Kantian notion of mathematical intuition has been criticized, notably by Frege, as too psychological or private to be the proper base of mathematics. This talk will challenge such an allegation by paying attentions to its historical backgrounds, especially that of German mathematical education. First, it focuses on Kantian notion of arithmetical intuition, and identifies one of its main resources; ’Segner’s arithmetic'. Since Vaihinger published his influential commentary of the first critique, Kantians of many variants have almost unanimously believed that it was one of his books; i.e., ‘Principle’. But this talk claims that it is his another book; i.e., ‘Lectures’. Segner’s ‘Lecture’ rather than ‘principle’ occupies a significant position in German history of mathematical education: it is a complement to the pedagogical tradition, so called, 'formal cultivation’ that was initiated by Ch. Wolff. In ‘Lectures', Segner employed such intuitive representations of numbers as points and asterisks, to make the rigid formality of mathematical thinking more approachable to the younger audiences. Since Segner’s intuitive examples of numbers and arithmetic operations were intended to be used in the context of classroom education, they should be publicly available for both teachers and students, and therefore visible and even manipulatable. Based on those observations, this talk claims that Kantian notion of mathematical intuition inherited this visibility, manipulability and public nature of Segner’s exemplars, and is not to be interpreted as being psychological or private.
[Speaker] Makoto Kikuchi (Kobe University)
[Title] Aspects of the notion of computability
[Abstract]
There are two important subclasses of the set of partial recursive functions. One is the set of primitive recursive functions and the other is the set of general or total recursive functions. The notion of computability had been scrutinized and expanded in 1930's and nowadays it is widely believed that we have succeeded in formulating the notion of computability accurately and adequately by reaching the concept of partial recursive functions. In this talk, we observe the two gaps of the three classes of computable functions and argue that the former expansion from primitive recursive functions to general recursive functions is somewhat mathematical while the latter enlargement from general recursive functions to partial recursive functions is rather philosophical. We discuss also consequences of observations of the discontinuity in the latter transition in the notion of computability.