著者
岡本 賢吾
出版者
日本科学哲学会
雑誌
科学哲学 (ISSN:02893428)
巻号頁・発行日
vol.40, no.2, pp.23-39, 2007-12-25 (Released:2009-05-29)
参考文献数
15

Two remarkable results attained by Domain Theory, which serves as mathematical foundations for denotational semantics of programming languages, are explained and considered from philosophical viewpoints: 1) the analysis of recursion by the fix-point semantics and 2) the introduction of the notion of continuity and of compact elements. In particular, the author finds them conceptually illuminating in that firstly, they succeed in making explicit those unnoticed semantic elements lying behind the syntax of the languages which play essential roles in the construction and execution of recursive programs, and that secondly, they show the way to reconstruct various ordinary classical mathematical structures by virtue of complementing approximation processes to their infinite noncompact elements.
著者
岡本 賢吾
出版者
日本科学哲学会
雑誌
科学哲学 (ISSN:02893428)
巻号頁・発行日
vol.36, no.2, pp.103-118, 2003-12-30 (Released:2009-05-29)
参考文献数
3

Mathematical structures are identified with classes (in naive set theories, which were based on Comprehension Principle), or with sets (in axiomatic set theories, which adopted the principle in its restricted form). By analyzing abstraction operators and the set-theoretical diagonal arguments, the author indicates both classes and sets could be best regarded as certain self-applicable functions, treated as objects on their own. On the other hand, contemporary higher-order type theories, guided by the Curry-Howard Isomorphism, identify mathematical structures with propositions. According to this conception, formal derivation of a judgment counts both as the proof of a proposition and as the construction of a structure. The author examines its significance to the study of how language contributes to the construction of mathematical structures.
著者
岡本 賢吾
出版者
The Philosophy of Science Society, Japan
雑誌
科学哲学 (ISSN:02893428)
巻号頁・発行日
vol.34, no.1, pp.7-19, 2001

Frege's well-known thesis that arithmetic is reducible to logic leaves unexplained what is the gain of the reduction and what he means by logic in principle. First, the author contends that the real interest of the reduction consists in a form of conceptual reduction: it frees us from the ordinary naive conception of numbers as forming extremely peculiar genus and replaces it with a very general and basic conception of them. Second, it is pointed out that Frege's concept of logic involves two elements. One is based on the iteratability of the operation of abstraction and naturally leads him to accept a sort of denumerably higher order logical language. The other is based on the so-called comprehension principle. Each of the two elements could be said to be logical in some sense but they are inconsistent with each other. Still, we can learn much from his attempt to search for as extensive and global a conception of logic as possible.