Frege's Foundations of Arithmetic, Teaching
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Gratitude extended to Dan Isaacson for compiling the following:
Essay Questions
- Expound Mill's account of mathematics and logic, and assess Frege's criticisms of Mill.
- Frege declares that never losing sight of the distinction between concept and object is one of three fundamental principles he kept to in the Foundations (Introduction, p. X ). Expound Frege's distinction between concept and object and assess its importance for his Foundations of Arithmetic.
- Expound Frege's definition of natural number, and consider whether, on the basis of such a definition, we can know that the numbers exist.
- Assess Frege's claim that, "arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one" (Foundations of Arithmetic, §87, p. 99). Include some account of Frege's development of logic in Begriffsschrift, and his definition in Part III of Begriffsschrift of "y follows in the φ-series after x", and his adoption of Basic Law V in Grundgesetze and its inconsistency.
Expound Mill's account of mathematics and logic, and assess Frege's criticisms of Mill.
Readings
- John Stuart Mill, A System of Logic, Ratiocinative and Inductive: Being a connected view of the principles of evidence and the methods of scientific investigation, London, 1843, Book II, Chapters III, V, VI, VII, Book III, Chapter XXIV, §§3-9.
- Gottlob Frege, Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der Zahl, Breslau, 1884; English translation by J.L. Austin, The Foundations of Arithmetic: a logico-mathematical enquiry into the concept of number, B.H. Blackwell, Oxford 1950, revised edn 1953, §§7-10, 16-17, 23-25, (pp. 9-17, 22-24, 29-33).
- Karl Britton, "The nature of arithmetic: a reconsideration of Mill's views", Proceedings of the Aristotelian Society 48 (1947-48), II, pp. 1-12.
- Glenn Kessler, "Frege, Mill, and the Foundations of Arithmetic", The Journal of Philosophy 77 (1980), pp. 65-79.
- Philip Kitcher, "Arithmetic for the Millian", Philosophical Studies 37 (1980), pp. 215-236.
- Philip Kitcher, "Mill, mathematics and the naturalist tradition", John Skorupski (ed), The Cambridge Companion to Mill, Cambridge University Press, 1998, pp. 57-111.
- Michael D. Resnik, "Mill's empiricism", Chapter 4 of Frege and the Philosophy of Mathematics, Cornell University Press, Ithaca, New York, 1980, pp. 137-160.
- Stewart Shapiro, "Mill", Section 4.3 of Thinking about mathematics: The philosophy of mathematics, Oxford University Press, 2000, pp. 91-102.
- John Skorupski, "Empiricism in logic and mathematics", Chapter 5 of John Stuart Mill, Routledge, London, 1989, pp. 126-166.
Frege declares that never losing sight of the distinction between concept and object is one of three fundamental principles he kept to in the Foundations (Introduction, p. X ). Expound Frege's distinction between concept and object and assess its importance for his Foundations of Arithmetic.
Readings:
- Gottlob Frege, The Foundations of Arithmetic: a logico-mathematical enquiry into the concept of number (1884), English translation by J.L. Austin B.H. Blackwell, Oxford, revised edn 1953, §§ 46-53, the footnote in §66, and §97.
- Gottlob Frege, "Function and concept" (1891), English translation by P.T. Geach, Peter Geach and Max Black (eds), Translations from the Philosophical Writings of Gottlob Frege, Basil Blackwell, Oxford 1960, pp. 21-41, or Brian McGuinness (ed), Gottlob Frege Collected Papers on Mathematics, Logic, and Philosophy, Basil Blackwell, 1984, pp. 137-156.
- Gottlob Frege, "On concept and object" (1892), English translation by P.T. Geach, Geach and Black (eds), op. cit., pp. 42-55; McGuinness (ed), op. cit., pp. 182-194.
- Tyler Burge, "Frege on extensions of concepts 1884-1903", Philosophical Review 93 (1984), pp. 3-34.
- Anthony Kenny, "Function, concept and object", Chapter 6 of Frege: An Introduction to the Founder of Modern Analytic Philosophy, Penguin Books, 1995, pp. 100-125.
- Harold Noonan, "Function and concept" and "On concept and object", Chapter 4 of Frege: A Critical Introduction, Polity Press, Cambridge, 2001, pp. 133-167.
- Joan Weiner, "Function and concept" and "On concept and object", Frege, Oxford University Press, 1999, pp. 72-90, 105-116.
Expound Frege's definition of natural number, and consider whether, on the basis of such a definition, we can know that the numbers exist.
Readings:
- Gottlob Frege, The Foundations of Arithmetic: a logico-mathematical enquiry into the concept of number 1884; English translation by J.L. Austin, , B.H. Blackwell, Oxford 1950, revised edn 1953, chapter 4, "The concept of number", pp. 67-98
- Herbert Enderton, Elements of Set Theory, Chapter 4: Natural Numbers, Academic Press, New York, 1977, pp. 66-73.
- Donald Gillies, Frege, Dedekind, and Peano on the Foundations of Arithmetic, Chapter 7: "Frege's Logicism", Van Gorcum, Assen, The Netherlands, 1982, pp. 45-49.
- Anthony Kenny, Frege, Chapter 5: "The Foundations of Arithmetic, II", Penguin Books, London, 1995, pp. 78-99.
- Charles Parsons, "Frege's theory of number", Max Black (ed.) Philosophy in America, Cornell University Press, 1965, pp. 180-203; reprinted, with a Postscript, in Charles Parsons, Mathematics in Philsoophy: Selected Essays, Cornell University Press, 1983, pp. 150-175, and in William Demopoulos (ed.), Frege's Philosophy of Mathematics, Harvard University Press, 1995, pp 182-210.
- Joan Weiner, Frege, Chapter 4: "Defining the numbers", Oxford University Press (Past Masters), pp. 49-71.
- Crispin Wright, Frege's Conception of Numbers as Objects, Chapter 4: "Number theory and logic", Aberdeen University Press, Scots Philosophical Monographs No. 2, 1983, pp. 130-169.
- Michael Dummett, Frege: Philosophy of Mathematics, Chapter 10, ‘Frege's strategy', Duckworth, 1991, pp. 111-124.
Assess Frege's claim that, "arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one" (Foundations of Arithmetic, §87, p. 99). Include some account of Frege's development of logic in Begriffsschrift, and his definition in Part III of Begriffsschrift of "y follows in the φ-series after x", and his adoption of Basic Law V in Grundgesetze and its inconsistency.
Readings:
- Gottlob Frege, Begriffsschrift , eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle, 1879; English translation by Stephen Bauer-Mengelberg, "Begriffschrift, a formula language, modeled upon that of arithmetic, for pure thought", in Jean van Heijenoort (ed.), From Frege to Gödel: a source book in mathematical logic 1879-1931, Harvard Unversity Press, 1967, pp. 1-82 (reprinted in Frege and Gödel: two fundamental texts in mathematical logic, Harvard University Press, 1970); English translation by T.W. Bynum, "Conceptual notation, a formula language of pure thought modelled upon the formula language of arithmetic", in T.W. Bynum (ed.), Gottlob Frege Conceptual Notation and related articles, Oxford University Press, 1972, pp. 101-203.
- Paul Benacerraf, "Frege, the last Logicist", P.A. French, T.E. Uehling, H.K. Wettstein (eds), The Foundations of Analytic Philosophy, Midwest Studies in Philosophy, Vol. VI, University of Minesotta Press, 1981, pp. 17-35; ; reprinted in William Demopoulos (ed.), Frege's Philosophy of Mathematics, Harvard University Press, 1995, pp. 41-67.
- George Boolos, "Reading the Begriffsschrift", Mind 94 (1985), pp. 331-44; reprinted in William Demopoulos (ed.), op.cit., pp. 163-181; and in George Boolos, Logic, Logic, and Logic, Harvard University Press, 1998, pp. 155-170.
- Anthony Kenny, "Concept Script", Frege, Chapters 2 and 3, Penguin Books, London, 1995, pp. 12-49.
- William and Martha Kneale, "Frege's general logic: The Begriffsschrift", The Development of Logic, Oxford University Press, 1962, Chapter 8, §1, pp. 478-493.
- Harold W. Noonan, Frege: a critical introduction, Chapter 2 "Logic", Polity Press, Cambridge, 2001, pp.36-79.
- Bertrand Russell, letter to Frege 16 June 1902, and Gottlob Frege, letter to Russell 22 June 1902, English translation in Jean van Heijenoort (ed.), From Frege to Gödel: a source book in mathematical logic 1879-1931, Harvard University Press, 1967, pp. 124-128.
- Joan Weiner, "The philosopher behind the last logicist", The Philosophical Quarterly 34 (1984), pp. 242-264; reprinted in Crispin Wright (ed.), Frege: Tradition & Influence, Basil Blackwell, Oxford, 1984, pp. 57-79.
- Joan Weiner, "Frege's new logic", Frege, Chapter 3, Oxford University Press, 1999, pp. 25-48.
- Edward N. Zalta., "Frege's Logic, Theorem, and Foundations for Arithmetic", The Stanford Encyclopedia of Philosophy (Summer 2005 Edition), Edward N. Zalta (ed.).
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