프레게 제한― 수의 정의와 적용 가능성
Frege's Constraint: Definitions of Numbers and their Applicability
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초록
나는 이 글에서 프레게 제한에 대한 라이트의 해석이 옳지 않다는 것을 보인다. 우선 나는 그가 산수 개념의 적용에 대한 반성을 통해 그 개념에 대한 선천적 지식을 얻을 수 있다고 여긴 사례를 검토한다. 이 검토를 기초로 나는 산수 개념의 적용을 지배하는 원리가 있고, 프레게 제한은 바로 그런 원리가 성립하도록 해당 개념을 정의하라는 요구임을 보인다. 둘째로 나는 라이트의 해석은 수학의 적용에 대한 아주 좁은 견해에 의존한다는 것을 보인다. 나는 산수의 적용에 대한 프레게의 견해를 살펴본 후에 실수가 적용될 수 있는 양을 산수량, 순수 공간량 및 물리량 세 부류로 나눈다. 나는 실수 적용 원리는 순수 공간량만 아니라 물리량에 대해서도 성립한다는 것을 보인다. 이에 근거해서 나는 실수 이론을 확립하는 데에도 여전히 프레게 제한은 유효하다고 결론짓는다.
Abstract
In this paper I consider Wright's interpretation of Frege's constraint on definitions of arithmetical concepts. Wright thinks that this constraint requires us to incorporate the applications of arithmetical concepts into their definitions. According to him, we don't have to observe Frege's Constraint in establishing a field of mathematics if we should think of it structually. And he asserts that we should think of it structually unless there is an a priori route through which we can obtain some knowledge of basic concepts of it by our reflection on their empirical applications. On the basis of this interpretation, he asserts that, unlike number theory and geometry, we don't have to observe Frege's Constraint in establishing real analysis, since we cannot obtain any knowledge of the identity of a particular real number when we reflect on any procedure of measurement or calculation of empirical quantities. I show that Wright's assessment of Frege's Constraint on real analysis is not correct. At first, I investigate an example which he regard as our reflection on empirical applications of arithmetical concepts by which we can obtain some a priori knowledge of them. On the basis of it, I try to show that there are some principles which govern the application of arithmetical concetps, and that what Frege's Constraint requires us is that we should define these concepts in such a way that those principles hold. Secondly, I show that Wright's interpretation of Frege's constraint depends on too narrow a conception of applications of arithmetic. After I investigate Frege's conception of applications of arithmetics, I devide quantities to which real numbers can be applied into three kinds, i.e., arithmetical quantities, pure spatial quantities, and physical quantities. I show that application principles of real numbers hold not just with pure spatial quantities but also with physical quantities. I conclude that Frege's Constraint is tenable in establishing real analysis.
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