구문론으로서의 수학: 괴델의 비판과 카르납의 과학적 철학
Mathematics as Syntax: Gödel’s Critique and Carnap’s Scientific Philosophy
0
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
괴델은 ‘수학은 언어의 구문론인가?’라는 미출간 논문에서 그가 ‘구문론적 해석'이라고 부르는 카르납의 관점을 비판한다. 박일호, 전영삼, 어워디와 캐러스, 리켓츠, 테넌트 등은 괴델의 논변을 여러 가지 방식으로 재구성하고, 카르납의 가능한 대응을 검토해 왔다. 이 논문은 수학의 성격에 대한 괴델과 카르납의 논쟁을 재현한 뒤 대부분의 기존 논의를 비판하고 다음과 같은 새로운 기여를 하려 한다. 먼저 여러 학자가 카르납의 견해로 지적한 ‘언어 상대성’이 과장되었다고 주장한다. 오히려 괴델 비판의 핵심은 수학의 적용 문제이며 ‘기대가능성’에 기초한 논변이다. 따라서 카르납이 수학의 적용, 특히 과학에의 적용을 어떻게 보았는지를 논의하여 괴델에 응답한다. 그 과정에서 기존 논의가 간과한 카르납의 ‘대응 원리’가 핵심적인 역할을 한다고 주장한다. 마지막으로 괴델 불완전성 정리의 진정한 함축인 수학의 소진불가능성은 카르납과 괴델 자신이 정확히 동의하는 부분이라고 주장한다.
Abstract
In his unpublished article, “Is Mathematics Syntax of Language?,” Gödel criticizes what he calls the ‘syntactical interpretation’ of mathematics by Carnap. Park, Chun, Awodey and Carus, Ricketts, and Tennant have all reconstructed Gödel’s arguments in various ways and explored Carnap’s possible responses. This paper first recreates Gödel and Carnap’s debate about the nature of mathematics. After criticizing most existing reconstructions, I claim to make the following contributions. First, the ‘language relativity’ several scholars have attributed to Carnap is exaggerated. Rather, the essence of Gödel’s critique is the applicability of mathematics and the argument based on ‘expectability’. Thus, Carnap’s response to Gödel must be found in how he saw the application of mathematics, especially its application to science. I argue that the ‘correspondence principle’ of Carnap, which has been overlooked in the existing discussions, plays a key role in the application of mathematics. Finally, the real implications of Gödel’s incompleteness theorems – the inexhaustibility of mathematics – turn out to be what both Gödel and Carnap agree about.
참고문헌 (35)
- 박일호, "괴델과 카르납: 수학은 언어의 구문론인가?", 과학철학, 10(1), pp.27~56 (2007), 한국과학철학회
- 전영삼, "괴델 이후의 힐베르트와 카르납: 체계 상대성 문제를 중심으로", 과학철학, 12(2), pp.111~150 (2009), 한국과학철학회
- Awodey, S., "Logical Empiricism: Historical and Contemporary Perspectives", pp.57~64 (2003), Pittsburgh University Press
- Awodey, S., "Carnap Brought Home: The View from Jena", pp.179~200 (2004), Open Court
- Awodey, S., "Kurt G ödel: Essays for his Centennial", pp.252~274 (2010), Cambridge University Press
- Berto, F., "There's Something About G ödel: The Complete Guide to the Incompleteness Theorem" (2009), Wiley-Blackwell
- Carnap, R., "Eigentliche und uneigentliche Begriffe", Symposion, 1, pp.355~374 (1927)
- Carnap, R., "Logical Syntax of Language" (1937), Kegan Paul
- Carnap, R., "Testability and Meaning-Continued", Philosophy of Science, 4(1), pp.1~40 (1937)
- Carnap, R., "Foundations of Logic and Mathematics", International Encyclopedia of Unified Science, 1(3) (1939), The University of Chicago Press
- Carnap, R., "Logical Foundations of Probability" (1950), The University of Chicago Press
- Carnap, R., "his Meaning and Necessity: A Study in Semantics and Modal Logic", pp.205~221 (1950), The University of Chicago Press
- Carnap, R., "Dear Carnap, Dear Van: The Quine-Carnap Correspondence and Related Work", pp.427~432 (1990), University of California Press
- Carnap, R., "Minnesota Studies in the Philosophy of Science 1", pp.38~76 (1956)
- Carnap, R., "The Philosophy of Rudolf Carnap" (1963), Open Court
- Carnap, R., "Philosophical Foundations of Physics: An Introduction to the Philosophy of Science" (1966), Basic Books
- Carus, A. W., "Carnap and Twentieth-Century Thought: Explication as Enlightenment" (2007), Cambridge University Press
- Franzén, T., "G ödel's Theorem: An Imcomplete Guide to Its Use and Abuse" (2005), A K Peters
- Friedman, M., "History and Philosophy of Modern Mathematics", pp.82~94 (1988), University of Minnesota Press
- Friedman, M., "Reconsidering Logical Positivism", pp.198~233 (1999), Cambridge University Press
- Friedman, M., "Carnap’s Logical Syntax of Language", pp.236~249 (2009), Palgrave-Macmillan
- Demopoulos, W., "Cambridge Companion to Carnap", pp.248~272 (2007), Cambridge University Press
- Gödel, K., "Gödel, Collected Works, vol. III: Unpublished Essays and Lectures", pp.304~323 (1951), Oxford University Press
- Gödel, K., "G ödel, Collected Works, vol. III: Unpublished Essays and Lectures", pp.334~362 (1953), Oxford University Press
- Gödel, K., "G ödel, Collected Works, vol. IV: Correspondence A-G", pp.442~450 (2003), Oxford University Press
- Goldfarb, W., "G ödel, Collected Works, vol. III: Unpublished Essays and Lectures", pp.324~334 (1995), Oxford University Press
- Goldfarb, W., "Origins of Logical Empiricism", pp.213~230 (1996), University of Minnesota Press
- Goldfarb, W., "Science and Subjectivity: The Vienna Circle and Twentieth Century Philosophy", pp.61~78 (1992), Akademie-Verlag
- Kuhn, T., "The Structure of Scientific Revolutions" (1970), Chicago University Press
- Ricketts, T., "Reading Putnam", pp.176~200 (1994), Blackwell
- Ricketts, T., "Cambridge Companion to Carnap", pp.200~225 (2007), Cambridge University Press
- Sarkar, S., "‘The Boundless Ocean of Unlimited Possibilities’: Logic in Carnap's Logical Syntax of Language", Synthese, 93, pp.191~237 (1992)
- Stein, H., "Was Carnap Wrong After All?", Synthese, 93, pp.275~295 (1992)
- Tennant, N., "Carnap, Gödel, and the Analyticity of Arithmetic", Philosophia Mathematica, 3, pp.100~112 (2008)
- Wagner, P., "Carnap’s Ideal of Explication and Naturalism" (2012), Palgrave-Macmillan