• Korean Journal of Logic
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• v.15 no.1
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• pp.155-190
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• 2012

As far as Hilbert's Program is concerned, there seems to be important differences in the development of Wittgenstein's thoughts. Wittgenstein's main claims on this theme in his middle period writings, such as Wittgenstein and the Vienna Circle, Philosophical Remarks and Philosophical Grammar seem to be different from the later writings such as Wittgenstein's Lectures on the Foundations of Mathematics (Cambridge 1939) and Remarks on the Foundations of Mathematics. To show that differences, I will first briefly survey Hilbert's program and his philosophy of mathematics, that is to say, formalism. Next, I will illuminate in what respects Wittgenstein was influenced by and criticized Hilbert's formalism. Surprisingly enough, Wittgenstein claims in his middle period that there is neither metamathematics nor proof of consistency. But later, he withdraws his such radical claims. Furthermore, we cannot find out any evidences, I think, that he maintained his formerly claims. I will illuminate why Wittgenstein does not raise such claims any more.

• Journal for History of Mathematics
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• v.22 no.3
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• pp.31-44
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• 2009

Anyone wishing to understand cognitive science, a converging science, need to become familiar with three major mathematical landmarks: Turing machines, Neural networks, and $G\ddot{o}del's$ incompleteness theorems. The present paper aims to explore the mathematical foundations of cognitive science, focusing especially on these historical landmarks. We begin by considering cognitive science as a metamathematics. The following parts addresses two mathematical models for cognitive systems; Turing machines as the computer system and Neural networks as the brain system. The last part investigates $G\ddot{o}del's$ achievements in cognitive science and its implications for the future of cognitive science.