Historical and Logical Investigation into Hilbert Program as a philosophy of mathematics

作为数学哲学的希尔伯特纲领的历史和逻辑研究

• 批准号: 13610014
• 负责人: KANEKO Hiroshi
• 金额: $ 1.41万
• 依托单位: Senshu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13610014/
• 关键词: Hilbert program; formalism; Hilbert; incompleteness theorem; Brouwer; Intuitionism; Philosophy of mathematics; 直感主義; 証明論; 心的構成; 構成主義

In this research project, we aimed at the explication of the following two points; how is the rela-tion between the so-called Hilbert program and Godel's second incompleteness theorem and how did Brouwer's criticism against Hilbert exert an influence on the historical development of Hilbert program. As a result, first of all we could show that the official view which said that Hilbert pro-gram faild because of Godel's second incompleteness theorem would not critically demonstrated. This result is based on Michael Detlefsen's prior research. But what we should derive from this result is not simply the revival of Hilbert program but the insight that Godel's second incom-pleteness theorem would be concerned with the problem of intensionality, that is, the fundamental problem of discripancy between linguistic (or formal) expressions and what we mean by them. Sec-ondly, we could show that Brouwer's criticism against Hilbertian formalism, especially Brouwer's criticism of the linguistic methods in mathematics, was fundamentally important in the process of sophistication of Hilbert's finitism. It should be noted that many strategies of Hilbert program are interpretable as responces to Brouwer's criticism.

本研究项目的目的是阐明所谓的希尔伯特纲领与哥德尔第二不完备性定理之间的关系以及布劳威尔对希尔伯特纲领的批评对希尔伯特纲领的历史发展产生了怎样的影响。因此，首先，我们可以证明，希尔伯特程序失败是由于哥德尔第二不完全性定理的官方观点不会被批判地证明。这一结果是基于Michael Detlefsen之前的研究。但是我们从这个结果中得到的不仅仅是希尔伯特纲领的复兴，而是要认识到哥德尔第二不完备性定理将涉及内涵问题，即语言（或形式）表达式与我们所指的内容之间的区分的基本问题。其次，我们可以看到，布劳威尔对希尔伯特形式主义的批判，特别是布劳威尔对数学中语言方法的批判，在希尔伯特有限论的完善过程中起着根本性的重要作用。应该指出的是，希尔伯特程序的许多策略可以解释为对布劳威尔批评的回应。

项目成果

Hiroshi KANEKO: "Brouwer's Conception on Language, Mind and Mathematics"Annals of the Japan Association for Philosophy of Science. Vol.11No.1. (2003)

金子浩：《布劳威尔关于语言、心灵和数学的构想》日本科学哲学协会年鉴。

金子洋之: "ブラウワーにおける言語と数学"日本科学哲学会『科学哲学』. 34-1. 21-35 (2001)

Hiroyuki Kaneko：“Brouwer 中的语言和数学”日本科学哲学会 34-1（2001）。

Hiroshi Kaneko: "'Hilbert's Program and Incompleteness Theorems-A survey of works of Michael Detlefsen'"Ikiita-Tetsugaku. No.7. 46-73 (2002)

Hiroshi Kaneko：“‘希尔伯特纲领和不完备性定理-迈克尔·德特​​勒夫森作品综述’”Ikiita-Tetsugaku。