2004 Fiscal Year Final Research Report Summary
Research on Brouwer's Philosophy and the notion of continuum in Intuitionism
| Project/Area Number |
15520026
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Philosophy/Ethics
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| Research Institution | Senshu University |
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Principal Investigator |
KANEKO Hiroshi Senshu University, Faculty of Letter, Professor, 文学部, 教授 (60191988)
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| Project Period (FY) |
2003 – 2004
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| Keywords | Brouwer / intuitionism / philosophy of mathematics / intuitionistic mathematics / constructivism / intuitionistic logic |
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| Research Abstract |
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
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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
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