Research on Brouwer's Philosophy and the notion of continuum in Intuitionism

布劳威尔哲学与直觉主义连续统概念研究

基本信息

批准号：
15520026
负责人：
KANEKO Hiroshi
金额：
$ 1.28万
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：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

项目摘要

In this research project, we could establish the following results. (1) It was usual until now to investigate formal systems of intuitionistic mathematics and logic independently from Brouwer's philosophy because it was thought that his philosophy is intrinsically subjective and solipsistic. However, according to our research, it is showed that we need not interpret Brouwer's criticism of language in mathematics as a rejection of communication but we should take his claims as a requirement for undetachability of propositional content and its construction process. And according to this interpretation we can understand Brouwer's intuitionistic philosophy and his intuitionistic mathematics in the unified way. (2) Brouwer's requirement for the undetachability of content and process is based on his view on mathematics that mathematics is not a system of conceptual operations but a system of action. From this view on mathematics it turns out that Brouwer has two different notions of possibil … More ity---conceptual possibility and possibility of developing actions. Then it is showed that Brouwer uses the difference of the notions of possibility as a device in order to introduce the notion of choice sequence into his mathematics. Therefore, it turns out that the undetachability requirement provides with one of the basic devices to intuitionistic analysis. (3) But then why must Brouwer require the undetachability of content and process? Although we could not arrive at the final answer, our tentative answer is as follows. Brouwer's undetachability requirement is based on the requirement for preserving personal process through which we arrive at mathematical results. So it is dear that Brouwer takes the preservation of personal process as valuable. Our answer to the question what is the epistemic significance of preservation of personal process is that it is also a requirement of preserving mathematical perspective. Therefore, Brouwer's skepticism to linguistic communication is not necessarily interpreted as a slide to subjectivism. We can understand his dawns as a strong requirement for communication, that is, the requirement of the share of perspective Less

在这个研究项目中，我们可以得出以下结果：（1）迄今为止，独立于布劳威尔的哲学来研究直觉数学和逻辑的形式系统是很常见的，因为人们认为他的哲学本质上是主观的和唯我论的。我们的研究表明，我们不必将布劳威尔对数学语言的批评解释为对交流的拒绝，而应该把他的主张视为命题内容及其构造过程的不可分离性的要求，并且根据这种解释我们可以理解。布劳威尔的直觉主义哲学和他的直觉主义数学的统一。 (2)布劳威尔对内容和过程的不可分离性的要求是基于他的数学观，即数学不是概念运算的系统而是行动的系统。事实证明，布劳威尔有两种不同的可能性概念——概念可能性和发展行动的可能性。然后表明，布劳威尔将可能性概念的差异作为一种手段。为了将选择序列的概念引入到他的数学中，事实证明，不可分离性要求提供了直观分析的基本手段之一。 (3) 但是，为什么布劳威尔必须要求内容和过程的不可分离性呢？无法得出最终答案，我们的初步答案如下：布劳威尔的不可分离性要求是基于保存个人过程的要求，通过这个过程我们得出数学结果。我们对保存个人过程的认知意义是什么这个问题的回答是，它也是保存数学观点的要求。因此，布劳威尔对语言交流的怀疑不一定被解释为主观主义的滑坡。理解他的黎明是对沟通的强烈要求，也就是对那份观点的要求 Less