• Title, Summary, Keyword: 힐베르트

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유한주의와 철학적 해석

  • Park, Jeong-Il
    • Korean Journal of Logic
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    • v.4
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    • pp.37-62
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    • 2000
  • 괴테의 불완전성 정리와 그것의 역사적 배경을 이루었던 힐베르트의 프로그램이 어떤 관계에 놓이느냐 하는 쟁점은 전자가 후자를 논박하느냐 하는 문제로 요약될 수 있다. 전자는 수리논리학에서의 한 정리이고 후자는 어떤 철학적 해석이 요구되는 요소를 지니고 있다. 따라서 전자는 '그 자체만으로는' 후자를 논박할 수 없으며, 어떤 철학적 해석이 부여될 때에만 그런 일은 가능할 수 있다. 후자가 지니고 있는 철학적 요소 중에서 가장 중요한 것은 힐베르트의 "유한주의"의 개념이다. 이 개념에 대해서 어떤 철학적 해석이 부여되느냐에 따라, 전자는 후자를 논박할 수도 있고 그렇지 않을 수도 있다. 특히, 힐베르트의 "유한주의"에 대한 철학적 해석은 괴델의 "특수한 유한주의적" 해석과 겐첸의 "구성주의적 해석"으로 대변할 수 있으며, 각각 논박설과 반논박설의 근거를 제공하고 있다. 결국, 문제는 그러한 철학적 해석들이 힐베르트의 사유에 비추어 얼마나 정당한가 하는 점이다. 이 글에서 나는 겐첸의 해석이 괴델의 해석에 대해 대등하게 경합적일 뿐만 아니라, 어떤 점에서는 힐베르트의 사유에 비추어 더 정당하다는 것을 보이고자 한다.

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Wittgenstein on Hilbert's Program (비트겐슈타인과 힐베르트 프로그램)

  • Park, Jeong-Il
    • 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.

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On Hilbert's 'Grundlagen der Geometrie' (힐베르트의 저서 '기하학의 기초'에 관하여)

  • Yang, Seong-Deog;Jo, Kyeong-Hee
    • Journal for History of Mathematics
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    • v.24 no.4
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    • pp.61-86
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    • 2011
  • In this article we introduce old and new references for 'Grundlagen der Geometrie' written by Hilbert and summarize its contents. We then compare the 1902 English translation of the first (German) edition and the 1971 English translation of the 10th (German) edition focusing on the changes of the contents, terminologies, expressions, etc. We then finally discuss about the implications of these changes in translating mathematics classics into modern Korean and in creating mathematics books in modern Korean.

Axioms underlying area of triangle and volume of triangular pyramid and Hilbert't third problem (삼각형의 넓이와 삼각뿔의 부피에 내재된 공리와 힐베르트의 세 번째 문제)

  • Do, Jonghoon
    • Journal of the Korean School Mathematics Society
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    • v.18 no.4
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    • pp.371-385
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    • 2015
  • In this paper we investigate the axioms defining area and volume so that revisit area formula for triangle, volume formula for triangular pyramid, and related contents in school mathematics from the view point of axiomatic method and Hilbert's third problem.

힐베르트의 '수학 문제'에 관하여

  • 한경혜
    • Journal for History of Mathematics
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    • v.16 no.4
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    • pp.33-44
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    • 2003
  • This article shows Hilbert's Paris address within the context of his career and the mathematics of his day. And the text of Hilbert's speech will be considered in some detail, particularly those parts of it that reflects his vision of mathematics most clearly. At the end it is attempted to show the limitations of the standard view of Hilbert as an advocate of a formalist approach to mathematics.

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교과서에서 나타난 작도방법의 정당화

  • Kang, Mee-Kwang;Hwang, Seurgi-Gi
    • East Asian mathematical journal
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    • v.26 no.2
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    • pp.151-164
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    • 2010
  • This study is to provide improved teaching methods on classical geometric construction education by a straightedge and compass in school mathematics. In this paper, justifications of construction methods of Korean textbooks, for perpendicular bisector of an segment and angle bisector are discussed, which can be directly applicable to teaching geometric construction meaningfully. Based on these considerations, several implications for desirable teaching methods concerning geometric construction were suggested.

Pythagorean Theorem I: In non-Hilbert Geometry (피타고라스의 정리 I: 비-힐베르트 기하에서)

  • Jo, Kyeonghee;Yang, Seong-Deog
    • Journal for History of Mathematics
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    • v.31 no.6
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    • pp.315-337
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    • 2018
  • Pythagorean thoerem exists in several equivalent forms in the Euclidean plane, that is, the Hilbert plane which in addition satisfies the parallel axiom. In this article, we investigate the truthness and mutual relationships of those propositions in various non-Hilbert planes which satisfy the parallel axiom and all the Hilbert axioms except the SAS axiom.

힐베르트의 세 번째 문제

  • 한인기
    • Journal for History of Mathematics
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    • v.12 no.2
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    • pp.25-39
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    • 1999
  • In Euclidean plane geometry, areas of polygons can be computed through a finite process of cutting and pasting. The Hilbert's third problem is that a theory of volume can not be based on the idea of cutting and pasting. This problem was solved by Dehn a few months after it was posed. The purpose of this article is not only to study Hilbert's third Problem and its proof but also to provide basis for the secondary school mathematics.

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Philosophical Implication of Turing's Work -Concentrated on Halting Theorem- (튜링의 업적이 지닌 철학적 함의 -'멈춤정리'를 중심으로-)

  • Park, Chang-Kyun
    • Journal for History of Mathematics
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    • v.25 no.3
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    • pp.15-27
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    • 2012
  • This paper aims to examine Alan Turing's life at the centenary of his birth and to discuss a philosophical implication of his work by concentrating on halting theorem particularly. Turing negatively solved Hilbert's decision problem by proving impossibility of solving halting problem. In this paper I claim that the impossibility implies limits of reason, and accordingly that the marginality in cognition and/or in action should be recognized.