Although this class has no required texts, many of the readings are available in the classic collection by Benacerraf and Putnam (Amazon, Google).
Jan. 12, Introduction
Jan. 19, Frege's logicism
Reading:
Gottlob Frege, "The Concept of Number", in Benacerraf and Putnam, pp. 130-159
Supplementary readings:
Gottlob Frege, Begriffsschrift (Concept Writing)
Gottlob Frege, "Über Sinn und Bedeutung" ("On Sense and Reference")
Parts of John Burgess, Fixing Frege
Jan. 26, Russell's logicism and Hilbert's formalism
Reading:
Carnap, "The Logicist Foundations of Mathematics", in Benacerraf and Putnam, pp. 41-52
von Neumann, "The Formalist Foundations of Mathematics", in Benacerraf and Putnam, pp. 61-65
Hilbert, "On the Infinite", in Benacerraf and Putnam, pp. 183-200
Supplementary readings:
Parts of John Burgess, Fixing Frege
Kurt Gödel, "Russell's Mathematical Logic", in Benacerraf and Putnam
Feb. 2, The indispensability argument
Reading:
Mark Colyvan, "Indispensability Arguments in the Philosophy of Mathematics", in Stanford Encyclopedia of Philosophy
David Liggins, "Quine, Putnam, and the 'Quine-Putnam' Indispensability Argument". 2008, Erkenntnis 68:1, 113-127.
Feb. 9, The Benacerraf problem
Reading:
Paul Benacerraf, "Mathematical Truth", in Benacerraf and Putnam, pp. 403-420
Oystein Linnebo, "Epistemological Challenges to Mathematical Platonism".
Supplementary readings:
Hartry Field, "Tarski's Theory of Truth"
Feb. 16, Field's nominalism
Reading:
Hartry Field, Ch. 1 of Realism, Mathematics, and Modality
Supplementary readings:
Parts of Hartry Field, Science Without Numbers
Feb. 23, Hale and Wright's neo-logicism
Bob Hale, "Is Platonism Epistemologically Bankrupt?" in Hale and Wright, The Reason's Proper Study
Mar. 2, Gödel's theorems
Reading: my notes on Gödel's theorems
Mar. 9 Gödel's program for new axioms
Reading:
Kurt Gödel, "What is Cantor's Continuum Problem?", in Benacerraf and Putnam
George Boolos, "The Iterative Conception of Set", in Benacerraf and Putnam
Mar. 16, Spring break - no class
Mar. 23 "Does Mathematics Need New Axioms?"
Reading:
Penelope Maddy, "Believing the Axioms. I"
Penelope Maddy, "Believing the Axioms. II" pp. 758-764
Supplementary readings:
Tony Martin, "Evidence in Mathematics"
Solomon Feferman, John Steel, Penelope Maddy, Harvey Friedman, "Does Mathematics Need New Axioms?"
Mar. 30, Lakatos
Reading:
Imre Lakatos, Proofs and Refutations, Chapter 1
Supplementary reading:
Imre Lakatos, Proofs and Refutations, Chapter 2
Apr. 6, Responses to Lakatos
Reading:
Mark Steiner, "The Philosophy of Mathematics of Imre Lakatos"
Jody Azzouni, "The Derivation-Indicator View of Mathematical Practice"
Apr. 13, Probabilistic proofs
Reading:
Don Fallis, "What do Mathematicians Want? Probabilistic Proofs and the Epistemic Goals of Mathematicians"
Don Fallis, "The Epistemic Status of Probabilisic Proofs"
Kenny Easwaran, "Probabilistic Proofs and Transferability"
Apr. 20, Diagrammatic reasoning
Reading:
John Mumma, "Proofs, Pictures, and Euclid"