This will be a graduate-level lecture seminar on the Philosophy of Mathematics, run jointly by Professor Timothy Williamson and myself, held during Trinity term 2019 at Oxford University. We shall meet every Tuesday 2-4 pm during term in the Ryle Room at the Radcliffe Humanities building.

We shall discuss a selection of topics in the philosophy of mathematics, based on the readings set for each week, as set out below. Discussion will be led each week either by Professor Williamson or myself.

In the classes led by Williamson, we shall discuss issues concerning the ontology of mathematics and what is involved in its application. In the classes led by me, we shall focus on the philosophy of set theory, covering set theory as a foundation of mathematics; determinateness in set theory; the status of the continuum hypothesis; and set-theoretic pluralism.

**Week 1** (30 April)

Discussion led by Williamson. Reading: Robert Brandom, ‘The significance of complex numbers for Frege’s philosophy of mathematics’, *Proceedings of the Aristotelian Society* (1996): 293-315 https://www.jstor.org/stable/pdf/4545241.pdf

**Week 2** (7 May)

Discussion led by Hamkins. Reading: Penelope Maddy, *Defending the Axioms: On the Philosophical Foundations of Set Theory*, OUP (2011), 150 pp.

**Week 3** (14 May)

Discussion led by Hamkins. Reading: Donald Martin, ‘Multiple universes of sets and indeterminate truth values’, *Topoi* (2001): 5-16. https://link.springer.com/article/10.1023%2FA%3A1010600724850

**Week 4** (21 May)

Discussion led by Hamkins. Reading: Chris Freiling, ‘Axioms of symmetry: throwing darts at the real number line’, *Journal of Symbolic Logic* (1986) https://www.jstor.org/stable/pdf/2273955.pdf. And: Solomon Feferman, ‘The Continuum Hypothesis is neither a definite mathematical problem nor a definite logical problem’, https://math.stanford.edu/~feferman/papers/CH_is_Indefinite.pdf

**Week 5** (28 May)

Discussion led by Hamkins. Reading: his ‘The set-theoretic multiverse’, *Review of Symbolic Logic* (2012): 416-449

https://doi.org/10.1017/S1755020311000359. And: Penelope Maddy, ‘Set-theoretic foundations’, *Contemporary Mathematics* (2017). http://philsci-archive.pitt.edu/13027/1/MaddyFoundations.pdf

**Week 6** (4 June)

Discussion led by Williamson. Reading: Cian Dorr, ‘Of numbers and electrons’, *Proceedings of the Aristotelian Society* (2010): 133-181. https://onlinelibrary.wiley.com/doi/epdf/10.1111/j.1467-9264.2010.00282.x

**Week 7** (11 June)

Discussion led by Williamson. Reading: Otávio Bueno and Mark Colyvan, ‘An inferential conception of the application of mathematics’, *Noûs* (2011): 345-374. https://onlinelibrary.wiley.com/doi/epdf/10.1111/j.1468-0068.2010.00772.x

**Week 8** (18 June)

Discussion led by Williamson. Reading: his ‘Alternative logics and applied mathematics’, *Philosophical Issues* (2018): 399-424. https://onlinelibrary.wiley.com/doi/epdf/10.1111/phis.12131