2004 Fiscal Year Final Research Report Summary
Philosophical Appraisal of the Representative Standpoints in the Foundations of Logic and Mathematics: Realism, Constructivism and Physicalism
| Project/Area Number |
13410002
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Philosophy
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| Research Institution | Tokyo Metropolitan University |
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Principal Investigator |
OKAMOTO Kengo Tokyo Metropolitan University, Philosophy, Associate Professor, 人文学部, 助教授 (00224072)
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| Co-Investigator(Kenkyū-buntansha) |
TANAJI Nobuharu Tokyo Metropolitan University, Philosophy, Professor, 人文学部, 教授 (20112469)
MATSUSAKA Youichi Tokyo Metropolitan University, Philosophy, Associate Professor, 人文学部, 助教授 (50244398)
NOMOTO Kazuyuki Souka University, literature, Professor, 文学部, 教授 (70007714)
IIDA Takashi Keio University, literature, Professor, 文学部, 教授 (10117327)
KANEKO Hiroshi Senshu University, literature, Professor, 文学部, 教授 (60191988)
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| Project Period (FY) |
2001 – 2004
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| Keywords | Philosophy of Logic / Philosophy of Mathematics / Realism / Constructivism / Physicalism / Type Theory / Category Theory / Set Theory |
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| Research Abstract |
Today, Realism(Platonism), Constructivis and Physicalistic Nominalism are three representative standpoints in the philosophical foundations of logic and mathematics. While they have their own merits and insights, they all suffer from certain philosophical difficulties peculiar to each of them. One would have to concede that those difficulties are urging us to reconsider the familiar fundamental questions in this field seriously: How should we identify the basic principles of logic and mathematics, and how should we give them adequate explanations and conceptual justifications. We seem to be simply at a loss as to how to answer these questions.
On the other hand, strongly influenced by the rapid growth of theoretical computer sciences, studies in logic and foundations of mathematics have gone through remarkable reformations and innovations for recent decades. Among the most notable developments are (i)systems of Typed Lambda Calculus, (ii)Category theory and (ii) sub-structural logics. T
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hese theories throw various new lights on the questions of the fundamental principles, thereby more or less invalidating the opposition between the three standpoints mentioned above, at the same time requiring renewal of interpretations of those historical figures such as Frege, Cantor, Hilbert and Brower.
Unfortunately, philosophers have been, and stall seem to be, very slow to understand and appreciate those innovative, far-reaching insights and proposals provided by the new developments in logic and foundations of mathematics. We attempt to build basic conceptual frameworks in which one could develop extensive philosophical analysis and critical examinations of those insights and proposals. Results of our present research could be summarize by the following two theses: First, we need to establish adequate principles for constructing those formalized languages (syntax) that are sensitive to dynamics of uses of language (egg making judgments, executing computations, etc). Second, in accordance with these syntactical principles, we need to find out relevant principles for constructing those mathematical structures that are flexible enough to admit of various kinds of indeterminacy and patiality. Less
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