Phil 570

Epistemology of Mathematics

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

Traditional issues in mathematical epistemology

Jan. 19, Frege's logicism


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


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


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


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


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

The justification of set theory

Mar. 2, Gödel's theorems

Reading: my notes on Gödel's theorems

Mar. 9 Gödel's program for new axioms


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?"


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?"

Contemporary issues in the epistemology of mathematics

Mar. 30, Lakatos


Imre Lakatos, Proofs and Refutations, Chapter 1

Supplementary reading:

Imre Lakatos, Proofs and Refutations, Chapter 2

Apr. 6, Responses to Lakatos


Mark Steiner, "The Philosophy of Mathematics of Imre Lakatos"

Jody Azzouni, "The Derivation-Indicator View of Mathematical Practice"

Apr. 13, Probabilistic proofs


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


John Mumma, "Proofs, Pictures, and Euclid"