[logic-ml] ワークショップ「Computations, Proofs, and Intuitions: A Workshop on Philosophy of Mathematics」

Tue Sep 1 15:22:49 JST 2015

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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10:00-10:10 Opening Remarks

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

georg.logic at gmail.com

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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.
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