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

在这个研究项目中，我们可以建立以下结果。(1)迄今为止，人们通常把直观数学和逻辑的形式系统独立于布劳威尔的哲学进行研究，因为人们认为他的哲学本质上是主观的和唯我论的。然而，我们的研究表明，我们不需要将布劳威尔对数学语言的批评解释为对交流的拒绝，而应该将他的主张理解为命题内容及其构建过程不可分离性的要求。根据这种解释我们可以统一地理解布劳威尔的直觉主义哲学和他的直觉主义数学。(2)布劳尔对内容和过程不可分离性的要求是基于他的数学观，即数学不是一个概念操作系统，而是一个行动系统。从这一数学观点出发，我们发现布劳尔有两种不同的可能性概念：概念可能性和发展行为的可能性。然后表明，为了将选择序列的概念引入到他的数学中，布劳威尔使用了可能性概念的差异作为一种手段。因此，不可分离性要求为直观分析提供了一种基本手段。(3)但是为什么浏览器必须要求内容和过程的不可分离性？虽然我们不能得出最后的答案，但我们暂时的答案如下。browwer的不可分离性要求是基于保留我们得到数学结果的个人过程的要求。因此，布劳尔认为个人过程的保存是有价值的，这一点是值得肯定的。对于保存个人过程的认识论意义是什么这个问题，我们的答案是，它也是保存数学视角的要求。因此，布劳尔对语言交际的怀疑并不一定被解释为滑向主观主义。我们可以把他的黎明理解为对沟通的强烈要求，即对分享视角的要求较少

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