PHIL 20020: Philosophy of Mathematics

Second Semester 2005 - 2006

 Unit Director:  Řystein Linnebo

http://seis.bris.ac.uk/~plxol/Courses/PHIL20020/Syllabus.htm

Pure mathematics appears to be very different from the empirical sciences: It appears not to rely on experience but to be completely a priori; its truths appear to be necessary rather than contingent; and it appears to be concerned with abstract objects rather than concrete (spatiotemporal, causally efficacious) ones. These three features of pure mathematics - its apparent apriority, necessity, and concern with abstract objects - give rise to some deep and extremely interesting philosophical questions. Are these features to be taken at face value? If so, how are they to be understood? In particular, how are these features to be reconciled with a scientific world view? Alternatively, if the special features of mathematics are not taken at face value, can we give an alternative explanation of mathematics which nevertheless does justice to mathematical practice and mathematical experience?

We will discuss a number of classical and contemporary approaches to these questions and related ones. Topics to be discussed include the following.

• Some traditional philosophical views of mathematics (Plato, Kant)

• Is mathematics reducible to 'pure logic?' (Frege, Russell)

• Is mathematics just a formal game with uninterpreted symbols? (Curry, Hilbert)

• Are mathematical truths just useful conventions? (Hempel)

• Is mathematics a science of mental constructions? (Brouwer, Heyting)

• Is mathematics empirical after all, just unusually general and abstract? (Quine)

• If there are abstract mathematical objects, how can we know about them? (Benacerraf, Gödel, Maddy)

• Can sense be made of mathematics without postulating mathematical objects? (Field)

• Are mathematical objects just points in mathematical structures? (Benacerraf, Resnik)

Literature

Students are encouraged to obtain copies of the following two books:

• Stewart Shapiro, Thinking about Mathematics (Oxford UP, 2000)

• Paul Benacerraf and Hilary Putnam, Philosophy of Mathematics: Selected Readings 2^nd ed. (Cambridge UP, 1983)

The former is an excellent introduction to the subject. The latter is a classic anthology containing most of the articles we will study.

PROGRAMME

1. Two traditional philosophical views of mathematics
- Plato, excerpts from Meno
- Kant's Critique of Pure Reason, B-Edition Introduction, sections I-V
- Shapiro, pp. 51-63, 73-91

Optional
- Shapiro, ch.s 1 and 2
- Kant's Critique of Pure Reason, 'The Discipline of Pure Reason in Its Dogmatic Use' (Part II, Ch. 1, Section 1; especially A712/B740-A724/B752)

2. Frege's logicism
- Frege's Foundations of Arithmetic, excerpts in B&P
- Shapiro, pp. 107-115, 133-138

3. Russell's logicism
- Russell's Introduction to Mathematical Philosophy, excerpts in B&P
- Russell, 'The Regressive Method in Philosophy,' repr. in Lackey ed. Essays in Analysis by Bertrand Russell (George, Allen & Unwin, 1973), pp 272-83
- Shapiro, pp. 115-124

4. Formalism
- Curry, 'Remarks on the Definition and Nature of Mathematics,' in B&P
- Hilbert, 'On the Infinite', in B&P
- Shapiro, ch. 6

5. Conventionalism
- Hempel, 'On the Nature of Mathematical Truth,' in B&P
- Quine, 'Truth by Convention' and 'Carnap on Logical Truth,' both in B&P
- Shapiro, pp. 124-133

6. Intuitionism
- Heyting, 'The Intuitionist Foundations of Mathematics' and 'Disputation,' both in B&P
- Shapiro, ch. 7

7. Empiricist Platonism
- Quine, 'Two Dogmas of Empiricism' (esp. final two sections), in his From a logical Point of View (Harvard UP, 1953); Pursuit of Truth (Harvard UP, 1990), Section 40; From Stimulus to Science (Harvard UP, 1995), ch. 5
- Shapiro, pp. 212-20

Optional
- Colyvan, 'Indispensability Arguments in the Philosophy of Mathematics,' Stanford Encyclopedia of Philosophy

8. Benacerraf's epistemological challenge to platonism
- Benacerraf, 'Mathematical Truth', in B&P
- Shapiro, pp. 24-33
- Gödel, 'What is Cantor's Continuum Problem' (esp. the Supplement), in B&P
- Shapiro, pp. 201-11

Optional
- Penelope Maddy, Realism in mathematics (Oxford UP, 1990), pp. 1-5, 28-35, 58-75, 150-9.
- Shapiro, pp. 220-4

9. Nominalism
- Hartry Field, 'Realism and Anti-Realism about Mathematics,' in his Realism, Mathematics, and Modality (Blackwell, 1989) or in W.D. Hart (ed.), The Philosophy of Mathematics (Oxford UP, 1996)
- Shapiro, pp. 226-237, 243-249

10. Benacerraf's metaphysical challenge
- Benacerraf, 'What Numbers Could Not Be,' in B&P
- Shapiro, ch. 10

Optional
- Resnik, 'Mathematics as a Science of Patterns: Ontology and Reference,' Nous 15 (1981), pp. 529-550