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.

在这个研究项目中，我们旨在阐明以下两点。所谓的希尔伯特计划与戈德尔第二次不完整定理之间的关系以及布鲁维尔对希尔伯特的批评如何对希尔伯特计划的历史发展产生影响。结果，首先，我们可以证明官方观点说，希尔伯特·波格拉姆（Hilbert Pro-Gram）由于戈德尔（Godel）的第二次不完整定理而失败，因此不会批判性地证明。该结果基于迈克尔·德特​​尔夫森（Michael Detlefsen）的先前研究。但是，我们应该从这个结果中得出的不仅仅是希尔伯特计划的复兴，而是戈德尔的第二次不收入定理会关注直觉问题，即语言（或形式上）表达式之间的基本问题以及我们的含义。秒，我们可以证明，布鲁维尔对希尔伯特形式主义的批评，尤其是布鲁维尔对数学中语言方法的批评，在希尔伯特有限主义的复杂过程中至关重要。应当指出的是，希尔伯特计划的许多策略都可以解释为对布鲁维尔批评的响应。

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。

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。

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）。