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프레게와 힐버트 – 메타이론적 증명
Frege and Hilbert on the Nature of Metatheoretical Proofs
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초록
나는 이 논문에서 공리이론에 관한 힐버트의 메타이론적 증명에 대한 프레게의 평가를 검토한다. 나는 이에 대해 널리 주장되는 해석을 거부하고 다음 두 주장이 참이라는 것을 보이려 한다. 첫째로 프레게는 힐버트가 하려던 일은 애초에 문장형식으로서 공리들 사이의 상호 무모순성 및 독립성을 증명하려는 것이었고, 힐버트는 이 일에 성공하였다고 생각하였다. 둘째로 프레게는 힐버트의 메타이론적 연구를 논리학의 가치있는 발전으로 높이 평가하였지만, 특수과학으로서 유클리드 기하학의 공리들 사이의 무모순성이나 독립성을 보이는 것으로 간주되어서는 안 된다고 생각하였다.
Abstract
In this paper, I reconsider Frege’s evaluations of Hilbertian metatheoretic proofs of axiomatic theories. I try to show the following two theses in opposition to what many commentators assert: (1) Frege thought not only that Hilbert regarded his metatheoretic proofs as being purported to show the mutual consistency and independence of axioms of his formal theories, and but also that he succeeded in it. (2) Although Frege admitted that Hilbert’s metatheoretic investigation made a valuable contribution to logic, he rejected the alleged thesis that Hilbert’s proof had shown the mutual independence of axioms of Euclidean geometry as a special science.
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