Phil 411: Topics in Philosophy of Logic and Mathematics, Fall 2008


Philosophical reflections on mathematics and particular episodes from the history of mathematics will be presented and discussed side by side in this course. The examples from mathematical practice serve as illustrations for the subject matter the philosophical reflections are about, and, at the same time, they serve as proving ground for adequateness of the philosophical claims about mathematics.

  • Introduction (Shapiro, Ch. 1 and 2; Eves, Ch. 9)
    Questions and positions in philosophy of mathematics.

    Additional readings:
    - Paul Benacerraf, "Mathematical Truth", Journal of Phil. 70:661-680, 1973 [JSTOR, on reserve (Benacerraf and Putnam)]
    - Eugene Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", Communications in Pure and Applied Mathematics 13(1):1-14, 1960 [online]
Traditional positions in philosophy of mathematics / The development of geometry
  • Euclid's Elements (Eves, Ch. 1 and 2)
    Origins of Greek mathematics. Material axiomatics. Euclid's definitions and axioms. The Pythagorean Theorem.

    Additional readings:
    - Euclid, "Elements" [online, on reserve]
    - William Dunham, "Journey through Genius", 1990. Ch. 2 "Euclid's Proof of the Pythagorean Theorem", 27-60, [on reserve]

  • Plato and Aristotle (Shapiro, Ch. 3)
    Plato's theory of Forms and Aristotle's critique of it.

    Additional readings:
    - Henry Mendell, "Aristotle and Mathematics", Stanford Encyclopedia of Philosophy [online]

  • Kant and Mill (Shapiro, Ch. 4)
    Mathematics as synthetic a priori. Radical empiricism.

    Additional readings:
    - Immanuel Kant, "Prolegomena to any Future Metaphysics", 1783 [online]
    - Philip Kitcher, "Kant and the Foundations of Mathematics", Phil. Review 84:23-50, 1975 [JSTOR]
    - Michael Friedman, "Kant's Theory of Geometry", Phil. Review 94(4):455-506, 1985 [JSTOR]
Nineteenth and early twentieth century positions in philosophy of mathematics
  • Non-Euclidean geometry (Eves, Ch. 3 and 4.4-4.5)
    Other developments: Analytic and projective geometry.

    Additional readings:
    - A brief historical summary of the development of Non-Euclidean geometry [online]

  • Logicism (Shapiro, Ch. 5)
    Arithmetization of Analysis. Frege, Russell, Carnap.

    Additional readings:
    - Gottlob Frege, "On concept and object", translated by P.T.Geach and M.Black in: Mind, 60(238):168-180, 1951. [JSTOR]
    - Gottlob Frege, "Foundations of Arithmetic" [on reserve]
    - Rudolf Carnap, "Empiricism, Semantics, and Ontology", Revue Internationale de Philosophie, 4:20-40, 1950. [on reserve (Benacerraf and Putnam)]

  • Hilbert's "Grundlagen der Geometrie" (1899) (Eves, Ch. 4 and 6)
    Formal axiomatics. Independence results. Models.

    Additional readings:
    - Hilbert, "Les principes fondamentaux de la géométrie", 1900 [online (French)]
    - Hilbert, "Foundations of Geometry" [on reserve]

  • Formalism (Shapiro, Ch. 6)
    The Frege-Hilbert Debate. Hilbert's Programme. Gödel's Incompleteness Theorems.

    Additional readings:
    - Frege-Hilbert correspondence [handout]
    - Brown, "Philosophy of mathematics: An introduction to the world of proofs and pictures", 1999. Ch. 8 "What is a definition?" [on reserve]

  • Intuitionism (Shapiro, Ch. 7)
    Brouwer, Heyting, Dummett.

    Additional readings:
    - Michael Detlefsen, "Brouwerian Intuitionism", Mind 99(396):501-534, 1990 [JSTOR]
    - Dirk Schlimm, "Against against Intuitionism", Synthese 147(1):171-188, 2005. [online]
    - For a comparsion between classical and intuitionistic logic, see Jeremy Avigad's notes on "Classical and Constructive Logic" [online]
Twentieth century positions regarding mathematical ontology
  • Algebraic structures (Eves, Ch. 5)
    Group theory.

    Additional readings:
    - Rothman, "The short life of Évariste Galois", 1987 [handout]

  • Realism (Shapiro, Ch. 8)
    Platonism (Gödel, Quine), set-theoretic realism.

    Additional readings:
    - William Tait, "Truth and Proof: The Platonism in Mathematics", Synthese 69:341-370, 1986 [on reserve (Hart)]
    - Mark Balaguer: "Platonism in Metaphysics", Stanford Encyclopedia of Philosophy [online]
    - Mark Colyvan, "Indispensability Arguments in the Philosophy of Mathematics", Stanford Encyclopedia of Philosophy [online]
    - Řystein Linnebo, "Epistemological Challenges to Mathematical Platonism" [online]

  • Nominalism (Shapiro, Ch. 9)

    Additional readings:
    - Stephen Yablo, "Go Figure: A Path Through Fictionalism", Midwest Studies in Philosophy [online]

  • Structuralism (Shapiro, Ch. 10)
    Formal axiomatics.

    Additional readings:
    - Paul Benacerraf, "What numbers could not be", The Philosophical Review 74:47-73, 1954 [JSTOR, on reserve (Benacerraf and Putnam)]
    - Charles Parsons, "The structuralist view of mathematical objects", Synthese, 84:303-346, 1990 [on reserve (Hart)]
    - Erich Reck and Michael Price, "Structures and Structuralism in Contemporary Philosophy of Mathematics", Synthese, 125:341-383, 2000. [online]
New directions in philosophy of mathematics
  • Challenging foundationalism

    - Imre Lakatos, "A renaissance of empiricism in the recent philosophy of mathematics?" British Journal for the Philosophy of Science, 27(3):201-223, 1976 [JSTOR, on reserve (Tymoczko)]
    - Judith Grabiner, "Is Mathematical Truth Time-Dependent?", American Mathematical Montly 81(4):354-365, 1974 [JSTOR, on reserve (Tymoczko)]

  • Proofs and refutations

    - Imre Lakatos, "Proofs and Refutations (I)" British Journal for the Philosophy of Science, 14(53):1-25, 1963 [JSTOR]; "Part II", 14(54):120-139, 1963 [JSTOR]; "Part III", 14(55):221-245, 1963 [JSTOR]; "Part IV", 14(56):296-342, 1964 [JSTOR]

  • The development of mathematical knowledge

    - Philip Kitcher, "The Nature of Mathematical Knowledge", 1983. Ch. 7 "Mathematical change and scientific change" [on reserve]
    - Emily Grosholz, "A New View of Mathematical Knowledge (Review of Philip Kitcher, The Nature of Mathematical Knowledge)", British Journal for the Philosophy of Science, 36:71-78, 1985 [JSTOR]
    - William Thurston, "On proof and progress in mathematics", Bulletin of the American Mathematical Society 30(2):161-177, 1994 [online, on reserve (Tymoczko)]

Cognitive aspects of mathematics
  • The cognitive basis of mathematical knowledge

    - Marinella Capelletti and Valeria Giardino, "The cognitive basis of mathematical knowledge", in: Mary Leng, Alexander Paseau, and Michael Potter, "Mathematical Knowledge". Oxford: Oxford University Press, 2007. pp.74-83.
    - Peter Gordon, "Numerical Cognition Without Words: Evidence from Amazonia", Science 306(5695):496-499, 2004 [online]
    - Marc D. Hauser, Noam Chomsky, and W. Tecumseh Fitch, "The faculty of language: What is it, who has it, and how did it evolve?", Science 298:1569-1579, 2002 [online]

  • The philosophy of embodied mathematics

    - George Lakoff and Raphael Núñez, "Where Mathematics Comes From", 2002.
    - Review by Bonnie Gold, 2001. [MAA Online]
    - Reply from the authors, 2001 [MAA Online]
    - Glenn Parsons and James R. Brown, "Platonism, Metaphor, and Mathematics", Dialogue XLIII(I):46-66, 2004.


Additional readings marked with a are highly recommended!

The following books are on Course Reserve (3 hour loan) in the Humanities & Social Sciences Library (McLennan-Redpath).


  • Heath, Sir Thomas L. (ed.), "The thirteen books of Euclid's Elements, translated from the text of Heiberg, with introd. and commentary by Sir Thomas L. Heath", New York, Dover Publications, 1956.
  • Frege, Gottlob, "The foundations of arithmetic: A logico-mathematical enquiry into the concept of number" (English translation by J. L. Austin), Evanston, Ill: Northwestern University Press, 1968.
  • Hilbert, David, "The foundations of geometry" (authorized translation by E.J. Townsend), La Salle, Ill.: Open Court, 1950.
  • Van Heijenoort, Jean (ed.), "From Frege to Gödel: A source book in mathematical logic, 1879-1931", Cambridge: Harvard University Press, 1967.
  • Benacerraf, Paul and Putnam, Hilary (eds.), "Philosophy of mathematics: Selected readings", New York: Cambridge University Press, 1983.
  • Hart, W.D. (ed.), "The philosophy of mathematics", Oxford: Oxford University Press, 1996.
  • Hintikka, Jaakko (ed.), "The philosophy of mathematics", London: Oxford U.P., 1969.
  • Tymoczko, Thomas (ed.), "New directions in the philosophy of mathematics: An anthology", Princeton, N.J. : Princeton University Press, 1998.
  • Brown, James Robert, "Philosophy of mathematics: An introduction to the world of proofs and pictures", New York: Routledge, 1999.
  • Dunham, William, "Journey through genius: the great theorems of mathematics", New York: Wiley, 1990.
  • Eves, Howard Whitley, "An introduction to the foundations and fundamental concepts of mathematics", New York: Holt, Rinehart and Winston, 1965.
  • Grosholz, Emily, "Representation and productive ambiguity in mathematics and the sciences", New York: Oxford University Press, 2007.
  • Kitcher, Philip, "The nature of mathematical knowledge", New York: Oxford University Press, 1983.
  • Shapiro, Stewart, "Thinking about mathematics : The philosophy of mathematics", New York: Oxford University Press, 2000.

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