# 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**

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*

### 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**

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?"

### Contemporary issues in the epistemology of mathematics

**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"