Grad class: Philosophy of Mathematics

Philosophy of Mathematics

This class will take up some recent work in the philosophy of mathematics (but won’t presuppose any prior knowledge in this field). We’ll begin with some much-discussed epistemological problems:

Week 1: epistemological challenges

Benacerraf, Mathematical Truth, Journal of Philosophy 70 (1973), 403–20.
(*) Linnebo, Epistemological Challenges to Mathematical Platonism, Philosophical Studies 129 (2006), 545–574.
Potter, What is the Problem of Mathematical Knowledge, in Leng et al (eds) Mathematical Knowledge (OUP, 2007), 16–32

If you only have time for one article, read the one marked (*).

In weeks 2–8, we’ll focus primarily on two ostensibly very different approaches to the ontology and epistemology of mathematics: (i) the neo-logicist programme developed especially by Hale and Wright and (ii) fictionalist approaches to mathematics, variously advocated by Field, Azzouni, and others.

Week 2: abstraction and the bad company problem

Wright, The Philosophical Significance of Frege’s Theorem in The Reason’s Proper Study (OUP 2001)
(*) Studd, Abstraction Reconceived, BJPS (2016)
Weir, An Embarassment of Riches, NDJFL (2003)

Week 3: the Julius Caesar problem

(*) MacBride, The Julius Caesar Problem: More Problematic than Ever, in Modality and Identity (OUP 2006) [preprint]

Cook and Ebert, Abstraction and Identity

Linnebo, To be is to be an F

A useful additional resource for weeks 3–4 is MacBride’s survey article: Speaking with Shadows

[FWIW: some of my views are outlined in this talk.]

Week 4: rejectionist responses to neo-logicism

Field, Critical notice of Wright’s Frege’s Conception of Numbers as Objects

(*) Wright, Field and Fregean Platonism

Eklund, Neo-Fregean Ontology

Week 5: indispensability

Melia, Weasling away the Indispensability Argument, Mind 109 (2000)

Colyvan, Mathematics and Aesthetic Considerations in Science, Mind 111 (2002)

Melia, Response to Colyvan, Mind 111 (2002)

(*) Baker, Are there Genuine Mathematical Explanations of Physical Phenomena?,

Mind 114 (2005)

Week 6: Field’s programme

Field, Realism and Anti-Realism about Mathematics, Philosophical Topics, 13 (1982)
(*) Macbride, Listening to Fictions: a Study of Fieldian Nominalism, BJPS 50 (1999)

Week 7: eliminative structuralism

Benacerraf, What Numbers could not be
Hellman, Mathematics without Numbers: Towards a Modal-Structural interpretation, chapter 1
(*) Hellman, Three Varieties of Mathematical Structuralism

Week 8: easy road nominalism

Yablo, Does Ontology Rest on a Mistake, PAS Sup. 72 (1998)
(*) Colyvan, There is No Easy Road to Nominalism, Mind 119 (2010)
Azzouni, Taking the Easy Road out of Dodge, Mind 121 (2012)