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

**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's theory of Forms and Aristotle's critique of it.

Mathematics as synthetic a priori. Radical empiricism.

**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.

**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.

**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)]

**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!

Home | Syllabus | Overview |
Schedule | Assignments | Links | (c) Dirk Schlimm 11/12/08 |