Philosophy of Mathematics, teaching
Sebastian Sequoiah-Grayson's Homepage
Topics
- Discuss Cantor's notion of set, its axiomatization in ZFC, and the role of set theory as a foundation for mathematics.
- Asses the extent to which Frege succeeded in establishing his claim that the truths of arithmetic were analytic.
- Can Mathematics be naturalized? Should we want it to be? In your answer, make explicit reference to Nominalism.
- Intuitionism in Mathematics and Logic: Explicate and Assess.
- What is Structuralism and is it defensible?
- How should we distinguish predicative functions from impredicative functions?
- Is higher-order logic truly logic, and is second-order logic distinct from set-theory?
Discuss Cantor's notion of set, its axiomatization in ZFC, and the role of set theory as a foundation for mathematics.
Readings:
- Boolos, G., (1971): The Iterative Conception of Set, in Boolos, G., Logic, Logic, and Logic, pp. 13-29, Harvard, 1998.
- Cantor, G. (1899): Letter to Dedekind, in van Heijenoort, J. (ed.): From Frege to Godel: A Source Book in Mathematical Logic, pp. 113-17, Harvard, 1967. You should also investigate the Peano, Von Neumann, and Zermelo papers in this volume.
- Copi, Irving, M. (1979): Symbolic Logic, 5th edition, Macmillan.
- Enderton, H. B. (1977): Elements of Set Theory, Academic Press.
- Halmos, P. R. (1960): Naive Set Theory, D. Van Nostrand Company.
- Levy, A. (1979): Basic Set Theory, Springer Verlag.
Discuss Cantor's notion of set, its axiomatization in ZFC, and the role of set theory as a foundation for mathematics.
Readings:
- Boolos, G. (1987): The Consistency of Frege's Foundations of Arithmetic, in Boolos, Logic, Logic, and Logic, pp. 183-201, Harvard, 1988.
- Boolos, G. (1997): Is Hume's Principle Analytic?, in Boolos, Logic, Logic, and Logic, pp. 301-15, Harvard, 1988.
- Boolos, G., and Heck, R. (1997): Die Grundlagen der Arithmetik, ยงยง82-83. in Boolos, Logic, Logic, and Logic, pp. 315-38, Harvard, 1988.
- Frege, G. (1884): Die Grundlegen der Arithmetik, English translation by J. L. Austin, 2nd revised edition, Blackwell, 1959.
- Frege, G. (1893, 1903): Grundgesteze der Arithmetik, English translation by M. Furth, UoC Press.
- Hale, R. L. V. (1999): Critical Review: Frege's Philosophy of Mathematics, review of Demopoulos (ed.) Frege's Philosophy of Mathematics, Philosophical Quaterly, pp. 92-104.
- Wright, C. (1999): Is Hume's Principle Analytic?, in Hale, B. and Wright, C. (eds.) pp. 307-34, The Reason's Proper Study: Essays Towards a Neo-Fregean Philosophy of Mathematics, Clarendon, 2001.
Can Mathematics be naturalized? Should we want it to be? In your answer, make explicit reference to Nominalism.
Readings:
- Maddy, P. (1997): Naturalism in Mathematics, Oxford University Press.
- Field, H. (1980): Science Without Numbers, Princeton University Press.
- Burgess, J. (1990): Epistemology and Nominalism, in A. D. Irvine (ed.): Physicalism in Mathematics, Dorcrecht: Kluwer Academic Publishers, pp. 1-15.
- Burgess, J. (1998): Occam's Razor and Scientific Method, in M. Schirn (ed.): Philosophy of Mathematics Today, Oxford University Press, pp. 195-214.
- Burgess, J. and Rossen, G. (1997): A Subject With No Object, Oxford University Press.
- Goodman, N. and Quine, W. V. O. (1947): Steps Towards a Constructive Nominalism, Journal of Symbolic Logic, 12, 105-122.
Intuitionism in Mathematics and Logic: Explicate and Assess.
Readings:
- Beal, J. C., and Restall, G. (2006): Logical Pluralism, Oxford University Press.
- Brouwer, L. E. J. (1912): Intuitionism and Formalism, in A. Heyting (ed.): Collected Works, pp. 123-38, Amsterdam, North-Holland.
- Dummet, M. (2000): Elements of Intuitionism, 2nd edition, Oxford University Press.
- Dummet, M. (1973): The Philosophical Basis of Intuitionistic Logic, in Truth and Other Enigmas, pp. 215-47, (1978).
- Heyting, A. (1974): Intuitionistic Views on the Nature of Mathematics, Synthese, 27, 79-91.
- Placek, T. (1999/2006): Mathematical Intuitionism and Intersubjectivity, (Kluwer/Springer).
- van Stigt, W. P. (1990): Brouwer's Intuitionism, Amsterdam: North-Holland.
What is Structuralism and is it defensible?
Readings:
- Hale, B. (1996): Structuralism's Unpaid Debts, Philosophica Mathematica, (3) 4, pp. 124-47.
- Hellman, G. (2001): Three Varieties of Mathematical Structuralism, Philosophica Mathematica, (3) 9, pp. 184-211.
- Parsons, C. (1990): The Structuralist View of Mathematical Objects, Synthese, 84, pp. 303-46.
- Shapiro, S. (1997): Philosophy of Mathematics: Structure and Ontology, Oxford University Press.
How should we distinguish predicative functions from impredicative functions?
Readings:
- Feferman, S. (1998): In the Light of Logic, Oxford University Press.
- Feferman, S. (1964): Systems of Predicative Analysis, Journal of Symbolic Logic, 29, pp. 1-30.
- Nelson, E. (1986): Predicative Arithmetic, Princeton University Press.
- Russell, B. (1906): On Some Difficulties in the Theory of Transfinite Numbers and Order, in D. Lackey (ed.): Essays in Analysis, pp. 135-64, George Braziller.
Is higher-order logic truly logic, and is second-order logic distinct from set-theory?
Readings:
- Isaacson, D. (1985): Arithmetical Truth and Hidden Higher-Order Concepts, in Logic Colloquium 85, edited by The Paris Logic Group, Amsterdam: North-Holland, pp. 147-69, (1987).
- Boolos, G. (1975): On Second-Order Logic, Journal of Philosophy, 72, pp. 509-27, (1975), reprinted in Boolos, G. Logic, Logic, and Logic, pp. 37-54, Harvard, (1998).
- Quine, W. V. O. (1975): Philosophy of Logic, 2nd edition: (1986), Harvard.
- Shapiro, S. (1999): Do Not Claim Too Much: Second-Order Logic and First-Order Logic, Philosophica Mathematica, (3) 7, 42-64.
- Shapiro, S. (1996): The Limits of Logic: Higher-Order Logic and the Lowenheim-Skolem Theorem, Aldershot UK, Dartmouth Publishing Company. [There are many relevant papers in this collection.]
- Vaananen, J. (2001): Second-Order Logic and the Foundations of Mathematics, Bulletin of Symbolic Logic, 7, pp. 504-520.
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