What is the nature of mathematical knowledge, as compared to knowledge of the natural world? What, if any, is the connection between the two? What role does mathematics play in empirical sciences such as physics? What role does philosophy play in clarifying the foundations of mathematics? Do abstract objects, such as numbers, exist? Is mathematics somehow true of our world, or is it merely an ingenious language devised by humans to address all sorts of problems?

In this class, we will address these questions and study how leading philosophers and mathematicians have attempted to answer them, giving special attention to the influential schools of logicism, formalism, and intuitionism. No prior college mathematics or philosophy is presupposed, although both will be helpful. Since it offers a focal point for many issues raised in the class, I will give a self-contained introduction to set theory.

Course materials such as lecture notes, handouts, etc will be made available as they will be used in class.

• Topic 0: Organization of course
• Topic 1: Introduction
• Topic 2: From Plato to Mill
• Topic 3: Set Theory
• Topic 4: Logicism
• Topic 5: Formalism
• Topic 6: Transfinite mathematics, the Löwenheim-Skolem Theorem, and Gödel's Theorem (no lecture notes or slides, but see Jonathan Bain's website)
• Topic 7: Intuitionism
• Topic 8: Structuralism

Paper prompt:

• Paper prompt
• Grading rubric

Information concerning plagiarism and guides on how to write a smashing philosophy paper can be found in the sidebar of the top page of the teaching section. The leaflet concerning plagiarism is absolutely mandatory reading.

The following materials are mandatory for this course:

• Book: Stewart Shapiro, Thinking about Mathematics: The Philosophy of Mathematics, Oxford University Press, 2000. This book is available at the Price Center bookstore.
• A number of readings for this course are available from e-reserves: Link to this course's e-reserves page (the password is 'cw124')

The following articles are mandatory reading from the Stanford Encyclopedia of Philosophy (SEP), edited by Ed Zalta:

• Thomas Jech: Set Theory
• A D Irvine: Russell's Paradox

Note: These additional materials will not be tested in exams. They serve to give you some background or to offer some additional food for thought.

The Stanford Encyclopedia of Philosophy (SEP) is an excellent source for academically serious, yet relatively accessible survey articles on many, many topics in philosophy, including philosophy of mathematics. For this course, the following articles are relevant:

• Leon Horsten: Philosophy of Mathematics
• Oystein Linnebo: Platonism in the Philosophy of Mathematics
• Edward N Zalta: Frege's Logic, Theorem, and Foundations for Arithmetic
• Alan Weir: Formalism in the Philosophy of Mathematics
• Patricia Blanchette: The Frege-Hilbert Controversy
• Richard Zach: Hilbert's Program
• Joan Moschovakis: Intuitionistic Logic
• Rosalie Iemhoff: Intuitionism in the Philosophy of Mathematics

A relatively new, but outstanding, source of very accessible material to many issues covered in this class are the Philosophy Bites podcasts of top philosophers interviewed by David Edmonds and Nigel Warburton. They are absolutely free. So next time you ride to school, make sure to upload some of them beforehand to your iPod! Relevant for this class are for example:

• Edward Craig - What is Philosophy? (Edward Craig, editor of the Routledge Encylopedia of Philosophy and author of Philosophy: A Very Short Introduction gives an interesting angle on the nature of philosophy, how it relates to other kinds of thinking, and what makes good philosophy good.)
• Michael Dummett on Frege (Gottlob Frege was one of the founders of the movement known as analytic philosophy. In this episode of the Philosophy Bites podcast Frege expert Michael Dummett explains why he is so important for philosophy.)
• More to come...

Midterm: The class average was 19.27 out of 30.

• (3) Make sure to answer this question as it is relevant to this course.
• (5) Make sure to look this up if you didn't get it right.
• (8) Many only said that a set is non-denumerable if it cannot be brought into bijection with the natural numbers. This is clearly insufficient as it would make the singleton set whose only element was 1 non-denumerable. You either have to add in 'with a proper subset of' before 'the natural numbers' or else additionally stipulate that the set be infinite.
• (9) Many didn't get this one, it is both in the reading and the slides. Make sure to look it up!
• (11b) Too many of you didn't get this easy question right--there should be 2^3 = 8 entries in your list.
• (13) Some of you gave a procedure which wouldn't solve the problem in a finite number of steps; but the idea was to give a room assignment that would do the job in just one step, i.e., everyone gets to move at most once.
• (15) Only few got that one; the answer to at least the first half of the question is on the middle of page 5 of the handout on Set Theory.

Paper: The class average was 16.2 out of 30.

• Three students wrote on logicism, six on formalism, and none on intuitionism.
• Many of you received unnecessary deductions, e.g. for giving insufficient references and missing bibliographies.
• Another common way to lose points was to not go beyond the material in class and the readings--I expected at least some independent thought.
• Among those who wrote on logicism, the Caesar problem was often not precisely understood.
• Among those who wrote on formalism, many spent too much time introducing the different kinds of formalism rather than just characterize the general program and then say more about Hilbert and give a critical assessment of the approach.

Final exam: The class average was 25.25 out of 40.

• The answers to most questions can be looked up in your notes or the texts we read.
• (13) asked for Russell's construction of the natural numbers. So explain how the construction works! Only very few got that right.
• (21) For the essay, I really expected some critical evaluation of the position you are taking, including, in particular, how you respond to at least some straightforward objections to your view!