• Title/Summary/Keyword: G$\ddot{o}$del

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A Historical Background of Mathematical Logic and $G{\ddot{o}}del$ (수리논리학의 역사적 배경과 괴델)

  • Park, Chang-Kyun
    • Journal for History of Mathematics
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    • v.21 no.1
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    • pp.17-28
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    • 2008
  • This Paper introduces a historical background of mathematical logic. Logic and mathematics were not developed dependently until the mid of the nineteenth century, when two streams of logic and mathematics came to form a river so that brought forth synergy effects. Since the mid-nineteenth century mathematization of logic were proceeded while attempts to reduce mathematics to logic were made. Against this background $G{\ddot{o}}del's$ proof shows the limitation of formalism by proving that there are true arithmetical propositions that are not provable.

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The Mathematical Foundations of Cognitive Science (인지과학의 수학적 기틀)

  • Hyun, Woo-Sik
    • 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.

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G$\ddot{o}$del's Critique of Turings Mechanism (튜링의 기계주의에 대한 괴델의 비평)

  • Hyun Woosik
    • Journal for History of Mathematics
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    • v.17 no.4
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    • pp.27-36
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    • 2004
  • This paper addresses G$\ddot{o}$del's critique of Turing's mechanism that a configuration of the Turing machine corresponds to each state of human mind. The first part gives a quick overview of Turing's analysis of cognition as computation and its variants. In the following part, we describe the concept of Turing machines, and the third part explains the computational limitations of Turing machines as a cognitive system. The fourth part demonstrates that Godel did not agree with Turing's argument, sometimes referred to as mechanism. Finally, we discuss an oracle Turing machine and its implications.

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A reconstruction of the G$\ddot{o}$del's proof of the consistency of GCH and AC with the axioms of Zermelo-Fraenkel set theory

  • Choi, Chang-Soon
    • Journal for History of Mathematics
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    • v.24 no.3
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    • pp.59-76
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    • 2011
  • Starting from a collection V as a model which satisfies the axioms of NBG, we call the elements of V as sets and the subcollections of V as classes. We reconstruct the G$\ddot{o}$del's proof of the consistency of GCH and AC with the axioms of Zermelo-Fraenkel set theory by using Mostowski-Shepherdson mapping theorem, reflection principles in Tarski-Vaught theorem and Montague-Levy theorem and the fact that NBG is a conservative extension of ZF.

20세기 수학의 패러다임 - 20세기 전.후반의 수리철학을 중심으로

  • 박창균
    • Journal for History of Mathematics
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    • v.9 no.2
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    • pp.22-29
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    • 1996
  • 20세기 수리철학은 전기와 후기로 나누어 볼 수 있다. 수학의 기초에 대한 논의가 활발하던 시기인 20세기 초로부터 G$\ddot{o}$del의 불완전성 정리가 발표된 30여년간을 전기로 하고, 후기를 실재론과 반실재론읜 대립과 새로운 수리철학이 대두된 최근 30여년간의 기간이라 한다면, 본고는 이미 잘 알려진 전기의 수학기초론 보다는 후기의 달라진 수리철학을 수학의 '새로운 패러다임' 으로 규정하고 그 내용과 성격을 살펴본다.

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G$\ddot{o}$del의 부완전성정리와 수학적 진리

  • 김용국;김빙남
    • Journal for History of Mathematics
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    • v.1 no.1
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    • pp.71-75
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    • 1984
  • Whether the complete Hilbert program could be carried out was rendered very doubtful by results due to Godel. These results may be roughly characterized as a demonstration that, in any system broad enough to contain all the formulas of a formalized elementary number theory, there exist formulas that neither can be proved nor disproved within the system. In this paper, Godel's incompleteness theorem is explained roughly moreover formul system and machines being refered, related to his theory.

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Mathematical truth and Provability (수학적 참과 증명가능성)

  • Jeong, Gye-Seop
    • Korean Journal of Logic
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    • v.8 no.2
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    • pp.3-32
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    • 2005
  • Hilbert's rational ambition to establish consistency in Number theory and mathematics in general was frustrated by the fact that the statement itself claiming consistency is undecidable within its formal system by $G\ddot{o}del's$ second theorem. Hilbert's optimism that a mathematician should not say "Ignorabimus" ("We don't know") in any mathematical problem also collapses, due to the presence of a undecidable statement that is neither provable nor refutable. The failure of his program receives more shock, because his system excludes any ambiguity and is based on only mechanical operations concerning signs and strings of signs. Above all, $G\ddot{o}del's$ theorem demonstrates the limits of formalization. Now, the notion of provability in the dimension of syntax comes to have priority over that of semantic truth in mathematics. In spite of his failure, the notion of algorithm(mechanical processe) made a direct contribution to the emergence of programming languages. Consequently, we believe that his program is failure, but a great one.

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Application of array comparative genomic hybridization in Korean children under 6 years old with global developmental delay

  • Lee, Kyung Yeon;Shin, Eunsim
    • Clinical and Experimental Pediatrics
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    • v.60 no.9
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    • pp.282-289
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    • 2017
  • Purpose: Recent advancements in molecular techniques have greatly contributed to the discovery of genetic causes of unexplained developmental delay. Here, we describe the results of array comparative genomic hybridization (CGH) and the clinical features of 27 patients with global developmental delay. Methods: We included 27 children who fulfilled the following criteria: Korean children under 6 years with global developmental delay; children who had at least one or more physical or neurological problem other than global developmental delay; and patients in whom both array CGH and G-banded karyotyping tests were performed. Results: Fifteen male and 12 female patients with a mean age of $29.3{\pm}17.6months$ were included. The most common physical and neurological abnormalities were facial dysmorphism (n=16), epilepsy (n=7), and hypotonia (n=7). Pathogenic copy number variation results were observed in 4 patients (14.8%): 18.73 Mb dup(2)(p24.2p25.3) and 1.62 Mb del(20p13) (patient 1); 22.31 Mb dup(2) (p22.3p25.1) and 4.01 Mb dup(2)(p21p22.1) (patient 2); 12.08 Mb del(4)(q22.1q24) (patient 3); and 1.19 Mb del(1)(q21.1) (patient 4). One patient (3.7%) displayed a variant of uncertain significance. Four patients (14.8%) displayed discordance between G-banded karyotyping and array CGH results. Among patients with normal array CGH results, 4 (16%) revealed brain anomalies such as schizencephaly and hydranencephaly. One patient was diagnosed with Rett syndrome and one with $M{\ddot{o}}bius$ syndrome. Conclusion: As chromosomal microarray can elucidate the cause of previously unexplained developmental delay, it should be considered as a first-tier cytogenetic diagnostic test for children with unexplained developmental delay.

Turing's Cognitive Science: A Metamathematical Essay for His Centennial (튜링의 인지과학: 튜링 탄생 백주년을 기념하는 메타수학 에세이)

  • Hyun, Woo-Sik
    • Korean Journal of Cognitive Science
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    • v.23 no.3
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    • pp.367-388
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    • 2012
  • The centennial of Alan Mathison Turing(23 June 1912 - 7 June 1954) is an appropriate occasion on which to assess his profound influence on the development of cognitive science. His contributions to and attitudes toward that field are discussed from the metamathematical perspective. This essay addresses (i)Turing's mathematical analysis of cognition, (ii)universal Turing machines, (iii)the limitations of universal Turing machines, (iv)oracle Turing machine beyond universal Turing machine, and (v)Turing test for cognitive science. Turing was a ground-breaker, eager to move on to new fields. He actually opened wider the scientific windows to the mind. The results show that first, by means of mathematical logic Turing discovered a new bridge between the mind and the physical world. Second, Turing gave a new formal analysis of operations of the mind. Third, Turing investigated oracle Turing machines and connectionist network machines as new models of minds beyond the limitations of his own universal machines. This paper explores why the cognitive scientist would be ever expecting a new Turing Test on the shoulder of Alan Turing.

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