Instructor: Dr Benedikt Löwe
Vakcode: MoLPM6
Time: Wednesday, 1315
Place: P.016
ECTS credit points: 6
Course language: English
Intended Audience: M.Sc. students of Logic and Mathematics,
M.A. students of Philosophy
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).
 JaffeQuinn, 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, Settheoretic 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 nondeductive 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.113
 William P. Thurston,
On proof and progress in mathematics, Bulletin of the American Mathematical Society 30 (1994),
p.161177
 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.323
 September 17. Presentations & Discussion. Plato's Rationalism, and Aristotle.
Presentation. Nicola Di Giorgio.
 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.161180
 Janet Folina, Pictures, Proofs, and 'Mathematical Practice': Reply to James Robert Brown,
British Journal for Philosophy of Science 50 (1999), p.425429
 Ian Dove, Can Pictures Prove, Logique et Analyse 45 (2002), p.309340
 October 1. Presentations & Discussion.
Kant and Mill.
Presentation. Jonathan Shaheen.
"Testing
your approximate number sense" (New York Times)
 October 8. Presentations & Discussion. Logicism.
Presentation. Kian MintzWoo. Frank Nebel.
 October 15. Presentations & Discussion. Formalism.
Presentation. Christian Geist, Rob Uhlhorn.
 October 22. EXAM WEEK. No class.
 October 29. Presentations & Discussion. Intuitionism.
Presentation. Stefanie Kooistra, Maurice Pico de los Cobos.
 November 5. Class cancelled.
 November 12. Presentations & Discussion. Platonism:
Gödel and Quine.
Presentation. Alexandru Marcoci.
 November 19. Presentations & Discussion. Maddy:
Settheoretic
realism and settheoretic
naturalism.
Presentation. Lorenz Demey, Kian MintzWoo.
 Shapiro, p.220225

Penelope Maddy, Some Naturalistic Reflections on Set Theoretic Method, Topoi 20
(2001), p.1727

Penelope Maddy, Settheoretic naturalism, Journal of Symbolic Logic 61 (1996),
p. 490514
 Penelope Maddy, Three forms of naturalism,
in: Stewart Shapiro (ed.), The Oxford Handbook of Philosophy of Mathematics and
Logic, Oxford University Press 2005,
p.437459
 Lieven Decock, A Lakatosian approach to the QuineMaddy debate, Logique et Analyse 45 (2002),
p.249268
 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.287301
 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.135151
 December 15. 1517, 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.301340
 Robert S.D. Thomas, Mathematics and Fiction II: Analogy, Logique et Analyse 45
(2002),
p.185228
 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.4359
 December 17, 1719, P.016. Presentations & Discussion.
Proof.
Presentation. Yacin Hamami, Matthew WamplerDoty.
 Yehuda Rav, Why Do We Prove Theorems?, Philosophia Mathematica 7 (1999), p. 541
 Don Fallis, Intentional Gaps in Mathematical Proofs, Synthese 134 (2003), p.4569
 Don Fallis, What do Mathematicians want? Probabilistic Proofs and the epistemic goals of
mathematicians, Logique et Analyse 45 (2002), p.373388
 Aaron Lercher, What is the Goal of Proof?, Logique et Analyse 45 (2002), p.389395
 Don Fallis, Response to "What is the Goal of Proof?", Logique et Analyse 45 (2002), p.397398
Last update : 9 December 2008