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

在本研究项目中，我们可以得出以下结果。(1)直到现在，研究形式系统的直觉主义数学和逻辑一直独立于布劳威尔的哲学，因为人们认为他的哲学本质上是主观的和唯我主义的。然而，根据我们的研究表明，我们不必将布鲁尔对数学语言的批评解读为对交际的拒绝，而应该将他的主张视为对命题内容及其构建过程不可拆分的要求。通过这种解读，我们可以统一地理解布鲁尔的直觉主义哲学和他的直觉主义数学。(2)布劳威尔对内容和过程不可分割性的要求基于他的数学观，即数学不是一个概念运算系统，而是一个行动系统。从这种数学观点来看，布劳威尔对可能性…有两种不同的概念更多的真实性-概念上的可能性和发展行动的可能性。然后，证明了布劳威尔利用可能性概念的差异作为一种手段，将选择序列的概念引入他的数学中。因此，不可拆卸性要求为直观分析提供了基本手段之一。(3)但为什么布劳威尔必须要求内容和过程的不可分离性？虽然我们不能得出最终的答案，但我们的初步答案如下。布劳威尔的不可拆卸性要求是基于保存个人过程的要求，通过个人过程我们可以得到数学结果。因此，布劳威尔认为保存个人过程是有价值的，这是可贵的。我们对保存个人过程的认识意义是什么这一问题的回答是，它也是保存数学观点的要求。因此，布劳威尔对语言交际的怀疑并不一定被解读为滑向主观主义。我们可以将他的黎明理解为对沟通的强烈要求，也就是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