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

在这个研究项目中，我们旨在说明以下两点；所谓希尔伯特计划和哥德尔第二不完备定理之间的关系是怎样的？布劳维尔对希尔伯特的批评对希尔伯特计划的历史发展产生了怎样的影响？因此，首先我们可以证明，官方认为希尔伯特规划因为哥德尔第二不完备定理而失败的观点是不能被严格证明的。这个结果是基于Michael Detlefsen之前的研究。但是我们应该从这个结果中得到的不仅仅是希尔伯特计划的复兴，而是哥德尔的第二个不完备定理将与密集性问题有关的洞察力，也就是说，语言（或形式）表达与我们通过它们表达的意思之间的区别的基本问题。其次，我们可以证明，布劳尔对希尔伯特形式主义的批评，尤其是对数学语言方法的批评，在希尔伯特有限主义的成熟过程中具有根本性的重要意义。应该指出的是，希尔伯特计划的许多策略都可以解释为对布劳威尔批评的回应。

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.