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課程名稱 |
邏輯與數學基礎
Logic and Foundations of Mathematics |
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開課學期 |
102-2 |
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授課對象 |
文學院 哲學研究所 |
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授課教師 |
楊金穆 |
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課號 |
Phl7760 |
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課程識別碼 |
124EM7350 |
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班次 |
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學分 |
3 |
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全/半年 |
半年 |
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必/選修 |
選修 |
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上課時間 |
星期四6,7,8(13:20~16:20) |
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上課地點 |
哲研討室二 |
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備註 |
C領域。
總人數上限：15人 |
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課程簡介影片 |
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核心能力關聯 |
核心能力與課程規劃關聯圖 |
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課程大綱
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課程概述 |
自Frege以來，分析哲學的傳統邏輯的研究不但與語言哲學的進展密切相關，在尋找穩固的數學基礎方面，邏輯也扮演了舉足輕重的角色。哲學家必須處理種種相關的形上學問題，例如：「如何證成（或否認）數學對象的確存在？」「如何理解所謂的數學真理？」等等涉及數學的基礎及預設的重要課題。文獻上也的確出現了許多各種不同的哲學 立場，試圖回答這些問題。由此可見，邏輯不只研究正確的推理，也可被視為關於數學基礎的研究。
本課程將會沿著早期數理邏輯的發展，研讀早期邏輯發展史上的經典文章（大約涵蓋從Frege到Kripke這段期間的重要著作），希望能藉此掌握這段時期的研究概況。本課程會特別著重邏輯與數學哲學的關係以及種種相關議題。
大致而言，課程首先會從Frege1984年的名著《算術基礎》（Foundations of Arithmetic) 開始，討論數學能否成功地化約成邏輯，接著會研究Ruseell 的類型理論（type theory）、數學哲學中的邏輯主義、直覺主義、形式主義等三種主要的立場、著名的Godel不完備定理證明（Godel’s incompleteness theorem）、以及Tarski、Gentzen、Turing、Church、Kripke等人的代表性著作。
本課程屬於進階課程，學生最好應先具備一階邏輯相關知識（尤其是Tarski的形式語意學以及Henkin的完備性定理證明）。
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課程目標 |
本課程將會沿著早期數理邏輯的發展，研讀早期邏輯發展史上的經典文章（大約涵蓋從Frege到Kripke這段期間的重要著作），希望能藉此掌握這段時期的研究概況。本課程會特別著重邏輯與數學哲學的關係以及種種相關議題。
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課程要求 |
輪流上台報告（約30分鐘）。
每週繳交2頁的讀書摘要。
期末繳交14頁（約4000字）的期末報告。
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預期每週課後學習時數 |
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Office Hours |
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指定閱讀 |
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參考書目 |
Frege, Gottlob (1884), The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number, translated by J. L. Austin, 2nd revised edition, Oxford: Basil Blackwell, 1986.
Russell, B., ‘The doctrine of types’, Appendix B to The Principles of Mathematics, Cambridge: Cambridge University Press, 1903, pp. 534-540.
Uhrquhart, Alasdair, ‘The theory of types,’ in The Cambridge Companion to Bertrand Russell, Nicholas Griffin ed., Cambridge: Cambridge University Press, 2003, pp. 286-309.
Poincare, J. H. (1900), ‘Intuition and logic in mathematics’, in From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols, William B. Ewald ed., Oxford: Oxford University Press, 1996, pp.1012-20.
Brouwer (1912), ‘Intuitionism and formalism’, English translation by A. Dresden, Bull. Amer. Math. Soc., 1913, 20: 81–96, reprinted in Benacerraf and Putnam, 1983: 77–89; also reprinted in Heyting ed., 1975: 123–138; reissued by Bulletin (New Series) of the American Mathematical Society, 1999, 37/1: 55-64.
Heyting, A. (1931), ‘The intuitionist foundations of mathematics’, original German version ‘Die intuitionistische Grundlegung der Mathematik’, Erkenntnis 2, pp. 106-115; English version in Philosophy of Mathematics, P. Benacerraf and H. Putnam eds., 2nd edition (1st edition, 1964), Cambridge: Cambridge University Press, 1983, pp. 52-61.
Hilbert, David (1927), ‘The foundations of mathematics’, in From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Jean van Heijenoort ed., 3rd printing with corrections (1st edition, 1967), Cambridge MA.: Harvard University Press, 1976, pp. 464-80.
Brouwer (1927), ‘Intuitionistic reflections on formalism’, English translation in van Heijenoort, 1976: 490–492. [Brouwer lists four topics on which intuitionism and formalism might enter into a dialogue. Three of the topics involve the law of excluded middle.]
Von Neumann, J. (1931/[1983]), ‘The formalist foundations of mathematics’, in Benacerraf and Putnam, (1983): 61-65.
Godel, K., ‘Uber formal unentscheidbare Satze der Principia Mathematica und verwandter Systeme, I’, Monatshefte fur Mathematik und Physik, 1931, 38: 173-98. English translation as ‘On formally undecidable propositions of Principia Mathematica and related systems I’, in Kurt Godel Collected works, Vol. I, Solomon Feferman ed., Oxford: Oxford University Press, 1986, pp. 144-195.
Floyd, J., ‘Wittgenstein on philosophy of logic and mathematics’, Chapter 4 of Shapiro, 2005: 75-128. [Shapiro, S. ed., The Oxford Handbook of Philosophy of Mathematics and Logic, Oxford: Oxford University Press, 2005.]
Tarski, A. (1933/5), ‘The concept of truth in formalized languages’, in Tarski, 1983, pp. 152–278.
Henkin, Leon (1949), ‘The completeness of the first-order functional calculus’, Journal of Symbolic Logic, 1949, Vol. 14: 159-66; reprinted in The Philosophy of Mathematics, J. Hintikka ed., Oxford: Oxford university Press, 1969, pp. 42-50.
Gentzen, Gerhard (1934/5), ‘Untersuchungen uber das logische Schliesen’, Mathematische Zeitschrift, 39:176-210 and 405-431. Translated as ‘Investigations into Logical Deduction’, in The Collected Papers of Gerhard Gentzen, M. Szabo (ed.), Amsterdam: North-Holland, 1969, pp.68–131.
Beth, E. W. (1955), ‘Semantic entailment and formal derivability’, Mededelingen van de Koninklijke Nederlandse Akademie van Wetenschappen, Afdeling Letterkunde, N.R. (Nieuwe Reeks, Amsterdam), 1955, 18/13: 309-342; reprinted in The Philosophy of Mathematics, J. Hintikka ed., Oxford: Oxford university Press, 1969, pp.9-41.
Turing, A. M. (1936), ‘On computable numbers, with an application to the Entscheidungsproblem’, Proceedings of the London Mathematical Society, s2/42: 230–65.1936–37; reprinted in The Essential Turing, B. J. Copeland ed., Oxford: Clarendon Press, 2004, pp. 58-90. A correction ibid., s2-43 (1936): 544–546 (1937).
Church, A., ‘An unsolvable problem of elementary number theory’, American Journal of Mathematics, 1936, 58: 345–363.
Church, A., ‘A note on the Entscheidungsproblem’, Journal of Symbolic Logic, 1936, 1: 40–41.
Kripke, S., ‘A completeness theorem in modal logic’, Journal of Symbolic Logic, 1959, 24:1-14.
Kripke, S., ‘Semantical considerations on modal logic’, Acta Philosophica Fennica, 1963, 16: 83-94.
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評量方式
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