## Philosophy of mathematics — a reading list

*A few people recently have quite independently asked me to recommend some introductory reading on the philosophy of mathematics. I have in fact previously posted here *a short list in the ‘Five Books’ style*. But here’s a more expansive draft list of suggestions.*

Let’s begin with an entry-level book first published twenty years ago but not yet superseded or really improved on:

- Stewart Shapiro,
*Thinking About Mathematics*(OUP, 2000). After introductory chapters setting out some key problems and sketching some history, there is a group of chapters on what Shapiro calls ‘The Big Three’, meaning the three programmatic ideas that shaped so much philosophical thinking about mathematics for the first half of the twentieth century — i.e. varieties of logicism, formalism, and intuitionism. Then there follows a group of chapters on ‘The Contemporary Scene’, on varieties of realism, fictionalism, and structuralism. This might be said to be a rather conservative menu — but then I think this is just what is needed for a very first introduction to the area, and Shapiro writes with very admirable clarity.

By comparison, Mark Colyvan’s *An Introduction to the Philosophy of Mathematics* (CUP, 2012) is far too rushed to be useful. And I would say much the same of Øystein Linnebo’s *Philosophy of Mathematics *(Princeton UP, 2017). David Bostock’s *Philosophy of Mathematics: An Introduction *(Wiley-Blackwell, 2009) is more accessible, but — apart from a chapter on predicativism — covers similar ground to the earlier parts of Shapiro’s book, but has little about more recent debates.

A second entry-level book, narrower in focus, that can also be warmly recommended is

- Marcus Giaquinto,
*The Search for Certainty*(OUP, 2002). Modern philosophy of mathematics is still in part shaped by debates starting well over a century ago, springing from the work of Frege and Russell, from Hilbert’s alternative response to the “crisis in foundations”, and from the impact of Gödel’s work on the logicist and Hibertian programmes. Giaquinto explores this with enviable clarity: this is really exemplary exposition and critical assessment. A terrific book.

Then, before moving on, I have to mention that most accessible of modern classics:

- Imre Lakatos,
*Proofs and Refutations*(originally published in 1963-64, and then in expanded book form by CUP, 1976). Textbooks tend to present developed chunks of mathematics in a take-it-or-leave-it spirit, the current polished surface hiding away the earlier rough versions, the conceptual developments, the false starts.*Proofs and Refutations*makes for a wonderful counterbalance. A classic exploration in dialogue form of the way that mathematical concepts are refined, and mathematical knowledge grows. We may wonder how far the morals that Lakatos draws can be generalized; but this remains a fascinating read (I’ve not known a good student who didn’t enjoy it).

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