PHIL 459.001 – Philosophy of Mathematics

Instructor: Marc Lange. This course meets T 1:00 – 3:30 p.m. in CW 213.

Ever since there has been philosophy, mathematics has been recognized as raising deep philosophical questions. (Purportedly, Thales was both the first philosopher and the first mathematician.) These questions include

(i) Are some mathematical claims true and, if so, what makes them true?

(ii) How do we know mathematics – including (a) is it knowable a priori and, if so, how do we manage to gain epistemic access to mathematical facts that way, and (b) what is the role of reasoning falling short of proof in mathematics?

(iii) What is the relationship between logic and mathematics?
(iv) Are mathematical truths necessary and independent of the mind?

(v) What makes mathematical truths applicable to the empirical world and what is the relation of mathematics to science?

Many of these questions acquired particular urgency in the late nineteenth century as imaginary numbers and new geometries were discovered and as there was greater concern to place mathematics on a rigorous, true, and certain foundation.

This course will examine some of these questions and some notable attempts to answer them. It is a survey course and presupposes no previous acquaintance with the philosophy of mathematics. (It starts from scratch and so it may – at least in places – be too elementary for some interested students. On the other hand, that I will take the responsibility for filling in all of the requisite background may enable the course to work well for some students with minimal relevant background.)

This course will include a survey of famous “-isms” in the philosophy of math, including Platonism, Logicism, Intuitionism, Formalism, Conventionalism, Empiricism, Nominalism, and Structuralism. This course will also (time-permitting) engage with some more recent topics in the philosophy of mathematics, such as the nature of explanation in mathematics and non-deductive reasoning in mathematics.

This course is intended to be useful to both some undergraduates and some graduate students, but the written work expected will differ for the two groups of students. There may also be some time for graduate student presentations (or even undergraduate student presentations) near the end of the course.

Written work for graduate students consists of the usual thing: a term paper of 15-25 pages on some topic of the student’s choice that makes direct contact with material discussed in the course.

Written work for undergraduate students consists of (i) a final term paper (in lieu of a final exam) of no more than 12 pages (typed, double-spaced, 8 ½ x 11 inch paper, 12-point type) due at the time at which the Registrar has set the final exam for the course, on one of the topics I will set; (ii) a brief oral presentation of the term paper during the period that the Registrar has set aside for the final exam; and (iii) two “midterm” exams (take-home, open-note) each consisting of several questions (each requiring a 1-2 pp. answer) occurring about 1/3 and 2/3 of the way through the semester.