Philosophy of Mathematics 2008/2009; 1st Semester Institute for Logic, Language & Computation Universiteit van Amsterdam

Objectives. This course is both a course for philosophers (with sufficient formal skills) to learn something about a particular and peculiar branch of philosophy of science dealing with abstract entities, and a course for logicians to see connections between the history and mathematical investigation of logic and their applications in philosophy.

Contents. Philosophy of Mathematics is perceived as different from Philosophy of Science. The main reason for this is that mathematics is seen as special. It is special as it deals with abstract objects using the method of deduction; as a consequence of this, it generates the only certain knowledge we have in science. Or does it? Is mathematics as special as we tend to think it is? We'll start off by some modern accounts of the special nature of mathematics, then go through the history of philosophy of mathematics to see the standard approaches (Platonism, Logicism, Formalism, Intuitionism, Naturalism). After this, we return to contemporary discussions of the nature of mathematics.

Format. Student presentations, plenary discussions, term paper (see the example for a paper structure).

Study material.

• Stewart Shapiro, Thinking about Mathematics, Oxford University Press 2000 (amazon.de).
• Jaffe-Quinn, Theoretical Mathematics: PDF file
• Thurston, Proof and Progress in Mathematics: PDF file
• Brown, Proofs and Pictures: PDF file
• Folina, Pictures: PDF file
• Maddy, Some naturalistic reflections: PDF file
• Maddy, Set-theoretic naturalism: PDF file
• Maddy, Three forms of naturalism: PDF file
• Aberdein, The uses of argument: PDF file
• Rav, Why do we prove theorems?: PDF file
• Fallis, Intentional Gaps: PDF file

Classes:

• September 3. Technicalities. "Is mathematics special?"
• September 10. "Theoretical Mathematics"; a proposal for non-deductive mathematics. Mathematics as a social practice.
  
  • Arthur Jaffe, Frank Quinn, "Theoretical mathematics": Toward a cultural synthesis of mathematics and theoretical physics, Bulletin of the American Mathematical Society 29 (1993), p.1-13
  • William P. Thurston, On proof and progress in mathematics, Bulletin of the American Mathematical Society 30 (1994), p.161-177
  • Jody Azzouni, How and Why Mathematics is Unique as a Social Practice, in: Bart Van Kerkhove, Jean Paul Van Bendegem (eds.), Perspectives on Mathematical Practices, Bringing Together Philosophy of Mathematics, Sociology of Mathematics, and Mathematics Education, Springer 2007 [Logic, Epistemology, and the Unity of Science 5], p.3-23
• September 17. Presentations & Discussion. Plato's Rationalism, and Aristotle.
  Presentation. Nicola Di Giorgio.
  • Shapiro, p.49-72
• September 24. Presentations & Discussion. Pictures and Proofs..
  Presentation. Pablo Cubides Kovacsics, Lisa Fulford.
  • James Robert Brown, Proofs and Pictures, British Journal for Philosophy of Science 48 (1997), p.161-180
  • Janet Folina, Pictures, Proofs, and 'Mathematical Practice': Reply to James Robert Brown, British Journal for Philosophy of Science 50 (1999), p.425-429
  • Ian Dove, Can Pictures Prove, Logique et Analyse 45 (2002), p.309-340
• October 1. Presentations & Discussion. Kant and Mill.
  Presentation. Jonathan Shaheen.
  • Shapiro, p.73-103
  "Testing your approximate number sense" (New York Times)
• October 8. Presentations & Discussion. Logicism.
  Presentation. Kian Mintz-Woo. Frank Nebel.
  • Shapiro, p.107-139
• October 15. Presentations & Discussion. Formalism.
  Presentation. Christian Geist, Rob Uhlhorn.
  • Shapiro, p.140-171.
• October 22. EXAM WEEK. No class.
• October 29. Presentations & Discussion. Intuitionism.
  Presentation. Stefanie Kooistra, Maurice Pico de los Cobos.
  • Shapiro, p.172-197.
• November 5. Class cancelled.
• November 12. Presentations & Discussion. Platonism: Gödel and Quine.
  Presentation. Alexandru Marcoci.
  • Shapiro, p.201-220.
• November 19. Presentations & Discussion. Maddy: Set-theoretic realism and set-theoretic naturalism.
  Presentation. Lorenz Demey, Kian Mintz-Woo.
  • Shapiro, p.220-225
  • Penelope Maddy, Some Naturalistic Reflections on Set Theoretic Method, Topoi 20 (2001), p.17-27
  • Penelope Maddy, Set-theoretic naturalism, Journal of Symbolic Logic 61 (1996), p. 490-514
  • Penelope Maddy, Three forms of naturalism, in: Stewart Shapiro (ed.), The Oxford Handbook of Philosophy of Mathematics and Logic, Oxford University Press 2005, p.437-459
  • Lieven Decock, A Lakatosian approach to the Quine-Maddy debate, Logique et Analyse 45 (2002), p.249-268
• November 26. Class cancelled.
• December 3. Class cancelled.
• December 10. Presentations & Discussion. Informal Logic and Argumentation Theory in Mathematics.
  Presentation. Elke Ballemans, Bjarni Hilmarsson.
  • Andrew Aberdein, The uses of argument in mathematics, Argumentation 19 (2005), p.287-301
  • Andrew Aberdein, The Informal Logic of Mathematical Proof, in: Bart Van Kerkhove, Jean Paul Van Bendegem (eds.), Perspectives on Mathematical Practices, Bringing Together Philosophy of Mathematics, Sociology of Mathematics, and Mathematics Education, Springer 2007 [Logic, Epistemology, and the Unity of Science 5], p.135-151
• December 15. 15-17, P.016. Presentations & Discussion. Mathematics and Narrative.
  Presentation. Sam van Gool, Charlotte Vlek.
  • Robert S.D. Thomas, Mathematics and Fiction I: Identification, Logique et Analyse 43 (2000), p.301-340
  • Robert S.D. Thomas, Mathematics and Fiction II: Analogy, Logique et Analyse 45 (2002), p.185-228
  • Robert S.D. Thomas, The Comparison of Mathematics with Narrative, in: Bart Van Kerkhove, Jean Paul Van Bendegem (eds.), Perspectives on Mathematical Practices, Bringing Together Philosophy of Mathematics, Sociology of Mathematics, and Mathematics Education, Springer 2007 [Logic, Epistemology, and the Unity of Science 5], p.43-59
• December 17, 17-19, P.016. Presentations & Discussion. Proof.
  Presentation. Yacin Hamami, Matthew Wampler-Doty.
  • Yehuda Rav, Why Do We Prove Theorems?, Philosophia Mathematica 7 (1999), p. 5-41
  • Don Fallis, Intentional Gaps in Mathematical Proofs, Synthese 134 (2003), p.45-69
  • Don Fallis, What do Mathematicians want? Probabilistic Proofs and the epistemic goals of mathematicians, Logique et Analyse 45 (2002), p.373-388
  • Aaron Lercher, What is the Goal of Proof?, Logique et Analyse 45 (2002), p.389-395
  • Don Fallis, Response to "What is the Goal of Proof?", Logique et Analyse 45 (2002), p.397-398