프레게와 비유클리드 기하학
Frege and non-euclidean geometries
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나는 이 글에서 프레게 유고집의 단편 「유클리드 기하학에 관하여」(1899-1906)에 나오는 유클리드 기하학과 비유클리드 기하학의 양립 불가능 논제에 대해 다음 세 논제를 확립하려 한다. (1) 그 논제는 유클리드 기하학과 비유클리드 기하학의 양립 가능성을 보이는 힐버트의 『기하학의 기초』(1899)의 논의와 상충하지 않는다. (2) 그 논제는 『산수의 기초』(1884)에 제시된 견해로서 유클리드 기하학과 비유클리드 기하학이 서로 다른 영역의 대상들을 다룬다는 견해와도 상충하지 않는다. 그리고 (3) 그 논제는 벨트라미, 클라인 및 포앙카레의 비유클리드 기하학의 일관성에 대한 논증과도 충돌하지 않는다. 도리어 그 논제는 프레게의 뜻과 지시체 이론만 아니라 모형론의 관점에서도 메타이론적 논의가 의존하는 기본적인 의미론적 사실로 간주되어야 한다.
Abstract
In this paper, I try to settle the following three claims on Frege’s thesis of his “On the Euclidean geometry”(1899-1906) in his Nachgelassene Schriften which says that Euclidean geometry is not compatible with non-Euclidean geometries: (1) It is not contradictory to Hilbert’s conceptions of reinterpretations of geometrical terms in which Euclidean geometry can be regarded as being compatible with non-Euclidean geometries. (2) It is not contradictory also to Frege’s thought that whereas Euclidean geometry deals with objects intuitable in the Euclidean space, non-Euclidean geometries deal with abstract objects belonging to the realm of conceptual thoughts. And (3) it is not contradictory also to Beltrami, Klein and Poincare’s conceptions of Euclidean models of non-Euclidean geometries. It should be regarded rather as a basic semantical fact upon which metageometrical considerations depend not only from the viewpoint of Frege’s theory of sense and reference but also from the viewpoint of model theory.
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