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.
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.
- Plato and Aristotle (Shapiro, Ch. 3)
Plato's theory of Forms and Aristotle's critique of it.
- Kant and Mill (Shapiro, Ch. 4)
Mathematics as synthetic a priori. Radical empiricism.
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.
- Logicism (Shapiro, Ch. 5)
Arithmetization of Analysis. Frege, Russell, Carnap.
- Hilbert's "Grundlagen der Geometrie" (1899)
(Eves, Ch. 4 and 6)
Formal axiomatics. Independence results. Models.
- Formalism (Shapiro, Ch. 6)
The Frege-Hilbert Debate. Hilbert's Programme. Gödel's Incompleteness Theorems.
- Intuitionism (Shapiro, Ch. 7)
Brouwer, Heyting, Dummett.
Twentieth century positions regarding mathematical ontology
- Algebraic structures (Eves, Ch. 5)
Group theory.
- Realism (Shapiro, Ch. 8)
Platonism (Gödel, Quine), set-theoretic realism.
- Nominalism (Shapiro, Ch. 9)
Fictionalism.
- Structuralism (Shapiro, Ch. 10)
Formal axiomatics.
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!
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The following books are on Course Reserve (3 hour loan) in the
Humanities & Social Sciences Library (McLennan-Redpath).
Sources:
- 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.
Anthologies:
- 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.
Monographs:
- 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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