PHILOSOPHY OF MATHEMATICS

 

Course Aims:

The course provides an overview of selected topics in contemporary philosophy of mathematics, focusing on metaphysical and epistemological questions such as: Are some mathematical claims true and, if so, what makes them true? Are mathematical truths necessary and independent of the mind? How do we have knowledge of mathematical truths? Is this knowledge a priori, and if so, how do we have epistemic access to mathematical facts? The course explores several attempts to answer these questions, including mathematical realism, intuitionism, naturalism, modalism, (neo-)Logicism, and structuralism. We will examine these positions critically, and look at the relationships between them.  

 

Course Texts:

Philosophy of Mathematics: Selected Readings, second edition, edited by Benacerraf and Putnam (BP)

Thinking About Mathematics, by Stewart Shapiro (SS)

Additional readings will be made available.
 

Assessment:

 

An essay of 2,000 - 3,000 words, to be handed in at the end of the semester.

 

Course Schedule:

 

Week 1: Introduction​​

- Background

- ​SEP http://plato.stanford.edu/entries/philosophy-mathematics/

- SS chapters 1 and 2

 

Week 2: Platonism/Realism

- Required

- ​BP pp. 470-485, Kurt Gödel "What is Cantor’s Continuum Problem?"

BP pp. 403-420, Paul Benacerraf "Mathematical Truth"

- Background

​- SEP http://plato.stanford.edu/entries/platonism-mathematics/

- SS chapter 8, pp. 201-211

 

Week 3: Intuitionism

- Required

- BP pp. 66-76, Arend Heyting "Disputation"

- BP pp. 97-129, Michael Dummett "The philosophical basis of intuitionistic logic"

- Background

- SS chapter 7

- SEP http://plato.stanford.edu/entries/intuitionism/

- SEP http://plato.stanford.edu/entries/logic-intuitionistic/

Week 4: Naturalism

- Required

- W. V. Quine "Two Dogmas of Empiricism" Philosophical Review, v. 60, pp. 20-43, 1951

- Penelope Maddy "Three Forms of Naturalism" in Stewart Shapiro, Oxford Handbook of Philosophy of Mathematics and Logic, Oxford UP, pp. 437–459, 2005

- Background

- SEP http://plato.stanford.edu/entries/naturalism-mathematics/

- SS chapter 8, pp. 212-224

Week 5: Logicism

- Required

- Gottlob Frege, Foundations of Arithmetic, Intro, §§1-4, 45-69, 87-91, 104-09

- Richard Heck "Frege’s Theorem: An Introduction"

http://rgheck.frege.org/pdf/published/IntroToFregesTheorem.pdf

- Background

- SEP http://plato.stanford.edu/entries/logicism/

- SS chapter 5, pp. 107-124

Week 6: Neo-Logicism

- Required

- Roy Cook "New Waves on an Old Beach" in Otávio Bueno and Øystein Linnebo New Waves in the Philosophy of Mathematics, Palgrave Macmillan, pp. 13-34

- George Boolos "Is Hume’s Principle Analytic?" in George Boolos Logic, Logic, and Logic, Harvard UP, pp. 301-314, 1998

- Background

- SEP http://plato.stanford.edu/entries/logicism/

- SS, chapter5 , pp. 133-138

Week 7: Thin objects

- Required

- Øystein Linnebo "Metaontological Minimalism", Philosophy Compass, v. 7, pp. 139-151, 2012

- Øystein Linnebo, [New book], chapters 1-2

- Background

- Bob Hale & Crispin Wright "The Metaontology of Abstraction" in David Chalmers Metametaphysics, Oxford UP, pp. 178-212, 2009

Week 8: Structuralism overview

- Required 

- Erich Reck & Michael Price, "Structures and Structuralism in Contemporary Philosophy of Mathematics", Synthese, v. 125, pp. 341-383, 2000

- BP pp. 272-294, Paul Benacerraf "What numbers could not be"

- Background

- Geoffrey Hellman "Structuralism" in Stewart Shapiro Oxford Handbook of Philosophy of Mathematics and Logic, OxfordUP, pp. 536-562, 2005

- SS chapter 10

Week 9: Non-eliminative structuralism

- Required

- Stewart Shapiro Philosophy of Mathematics: Structure and Ontology, Oxford UP, 1997, chapter 3

- Michael Resnik "Mathematics as a Science of Patterns: Ontology and Reference" Noûs, 1981

 

Week 10: Modalism 

- Required

- BP pp. 295-311, Hilary Putnam "Mathematics without Foundations"

- Geoffrey Hellman Mathematics without Numbers, Oxford UP, 1989, chapter 1

Week 11: Identity of indiscernibles

- Required

- Steward Shapiro "Structure and Identity" in Fraser MacBride Identity and Modality, Oxford UP, pp. 109-45, 2006

- Hannes Leitgeb & James Ladyman "Criteria of Identity and Structuralist Ontology", Philosophia Mathematica, v. 16, pp. 388-396, 2008

- Background

- SEP http://plato.stanford.edu/entries/identity-indiscernible/

Week 12: Structural abstraction

- Required

- Øystein Linnebo & Richard Pettigrew "Two Types of Abstraction for Structuralism", Philosophical Quarterly, v. 64, pp. 267-283, 2014

- Georg Schiemer and John Wigglesworth "The Structuralist Thesis Reconsidered"

Week 13: General discussion

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