• Proceedings of the Korean Society for Cognitive Science Conference
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• 2000.05a
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• pp.322-328
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• 2000

This study discusses cognition as a computable mapping in cognitive system and relates G del's Incompleteness Theorems to the computability of cognition from a metamathematical perspective. Understanding cognition as a from of computation requires not only Turing machine models but also neural network models. In previous studies of computation by cognitive systems, it is remarkable to note how little serious attention has been given to the issue of computation by neural networks with respect to G del's Incompleteness Theorems. To address this problem, first, we introduce a definition of cognition and cognitive science. Second, we deal with G del's view of computability, incompleteness and speed-up theorems, and then we interpret G del's disjunction on the mind and the machine. Third, we discuss cognition as a Turing computable function and its relation to G del's incompleteness. Finally, we investigate cognition as a neural computable function and its relation to G del's incompleteness. The results show that a second-order representing system can be implemented by a finite recurrent neural network. Hence one cannot prove the consistency of such neural networks in terms of first-order theories. Neural computability, theoretically, is beyond the computational incompleteness of Turing machines. If cognition is a neural computable function, then G del's incompleteness result does not limit the compytational capability of cognition in humans or in artifacts.

• Journal for History of Mathematics
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• v.15 no.3
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• pp.75-82
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• 2002

G$\ddot{o}$del's metamathematical program from Incompleteness to Speed-up theorems shows the necessity of ever higher systems beyond the fixed formal system and devises the relative consistency.

• Annual Conference on Human and Language Technology
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• 1995.10a
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• pp.258-263
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• 1995

마음과 기계의 관계에 대한 $G{\ddot{o}}del's$의 선언결론(disjunctive conclusion)은 마음과 정보처리형식체계의 논리적 동치성을 함의하고 있다. 그리고 $G{\ddot{o}}del's$의 불완전성 정리(Incompleteness Theorems)에 따르면 마음과 정보처리형식체계의 논리적 동치성은 무모순이며, 동치성 반증의 이론은 그 모델을 가질 수 없다.

• Journal for History of Mathematics
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• v.13 no.1
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• pp.137-141
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• 2000

This paper discusses Godel's Disjunctive Conclusion in terms of cognitive science and his Incompleteness Theorems from a metamathematical perspective.

• 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.

• Korean Journal of Logic
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• v.21 no.1
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• pp.97-133
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• 2018

In his unpublished article, "Is Mathematics Syntax of Language?," $G{\ddot{o}}del$ criticizes what he calls the 'syntactical interpretation' of mathematics by Carnap. Park, Chun, Awodey and Carus, Ricketts, and Tennant have all reconstructed $G{\ddot{o}}del^{\prime}s$ arguments in various ways and explored Carnap's possible responses. This paper first recreates $G{\ddot{o}}del$ and Carnap's debate about the nature of mathematics. After criticizing most existing reconstructions, I claim to make the following contributions. First, the 'language relativity' several scholars have attributed to Carnap is exaggerated. Rather, the essence of $G{\ddot{o}}del^{\prime}s$ critique is the applicability of mathematics and the argument based on 'expectability'. Thus, Carnap's response to $G{\ddot{o}}del$ must be found in how he saw the application of mathematics, especially its application to science. I argue that the 'correspondence principle' of Carnap, which has been overlooked in the existing discussions, plays a key role in the application of mathematics. Finally, the real implications of $G{\ddot{o}}del^{\prime}s$ incompleteness theorems - the inexhaustibility of mathematics - turn out to be what both $G{\ddot{o}}del$ and Carnap agree about.