브라우워의 직관주의와 구성
Brouwer’s Intuitionism and Construction
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필자는 이 글에서 브라우워의 직관주의가 초기 구성주의에 대한 철학적설명일 수 있다는 점을 보이고, 그것을 계기로 브라우워식 구성 개념이고전수학에 대해 어떤 함축을 갖는지 살펴보고자 한다. 이를 위해 첫째, 수학은 정신 활동이라는 브라우워의 주장을 몇 가지 관련 개념들을 중심으로 재구성한 후, 둘째, 그것이 어떤 식으로 구성주의 형성에 기여하는지를 살펴볼 것이다. 셋째, 선택열(choice sequence)에 의한 실수 정의를 설명함으로써 브라우워식 구성의 특징을 알아본 후, 그러한 구성 개념에 따르는 것은 고전수학의 진리 개념을 받아들이지 않는 것에 해당한다는 논의와 선택열에 의한 실수 구성이 고전적 정의의 적절한 대안이라는 논의를 함께 살펴볼 것이다. 마지막으로, 브라우워의 실수 구성은 수학적으로허용될 수 없다는 비트겐슈타인의 반론을 검토해볼 것이다.
Abstract
This paper purposes to show that Brouwer’s intuitionism can be regarded as a philosophical account of the early constructivism, a trend exemplified by the works of Kronecker and Poincaré, and then to consider its implications on classical mathematics. For these, I first reconstruct Brouwer’s claim that mathematics is a mental activity. Next, I argue that the claim plays an important role in the development of constructivism. Thirdly, after showing that choice sequences define real numbers, I examine the argument that the choice sequence method leads to undermining of the classical notion of truth and may be regarded as an alternative way to the classical definition of real numbers. Finally, I argue that Wittgenstein’s objection to Brouwerian real number construction is not successful.
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