Phil 146

Philosophy of Mathematics


Fall 2015

Professor Paolo Mancosu

Office: 230 Moses Hall

Phone: 510-296-4325


Class meets: TTh 11.00-12.30, 20 Wheeler

Office hours: T 9:00-10:30

Graduate instructor: Richard Lawrence

Course Description

This is an introduction to the classics of philosophy of mathematics with emphasis on the debates on the foundations of mathematics. Topics to be covered: infinitist theorems in seventeenth century mathematics; the foundations of the Leibnizian differential calculus and Berkeley’s ‘Analyst’; Kant on pure intuition in arithmetic and geometry; the arithmetization of analysis (Bolzano, Dedekind); Frege’s logicism; the emergence of Cantorian set theory; Zermelo’s axiomatization of set theory; Hilbert’s program; Russell’s logicism; Brouwer’s intuitionism; Gödel’s incompleteness theorems.

Prerequisites: Phil 12A (or equivalent) and another course in philosophy


Week 1: Infinitistic theorems in XVIIth century mathematics.

Week 2: The foundations of the Leibnizian differential calculus and Berkeley’s Analyst.

Week 3: Kant on pure intuition in arithmetic and geometry.

Week 4: The arithmetization of analysis: Bolzano’s proof of the intermediate value theorem

Week 5: Dedekind’s theory of irrational numbers.

Week 6: Dedekind’s theory of natural numbers.

Week 7: Frege’s Begriffsschrift.

Week 8: Frege’s The Foundations of Arithmetic.

Week 9: Frege’s The Foundations of Arithmetic.

Week 10: Frege’s The Foundations of Arithmetic.

Week 11: The emergence of Cantorian set theory and the mathematical theory of the infinite; Zermelo’s axiom of choice and his axiomatization of set theory; semi-intuitionism.

Week 12: Hilbert’s program I (axiomatization).

Week 13: Russell’s type theory

Week 14: Brouwer’s intuitionism

Week 15: Hilbert’s program II (proof theory); Gödel’s results


Frege, The Foundations of Arithmetic, Northwestern University Press.

Dedekind, Essays on the Theory of Numbers, Dover.

P. Mancosu, ed., From Brouwer to Hilbert: The debate on foundations of mathematics in the 1920s, Oxford University Press, 1998.

Packet of readings (I) for the first 4 weeks: available at Copy Central (on Bancroft) at the beginning of the course.

Packets of readings (II, III): will be available at Copy Central in late September.

Course requirements

Three papers, to count for 25%, 35%, and 40% of the course grade, respectively. The first paper (approximately 3 pages) will be due in fifth week. The second paper (approximately 5 pages) will count as a midterm and will be due in ninth week. The third paper (approximately 10 pages) will count as the final. Graduate students in philosophy may use this course, with previous agreement with me, for satisfying the formal philosophy course requirement (i.e. a course in the 140 series or equivalent). In this case, they will also be required to complete a few exercise sets.

Updated on 2015-08-07 12:45:00 -0700 by Paolo Mancosu