## 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 âFive Booksâ style*. But hereâs a moreÂ expansive draft list of suggestions.*

Letâs begin with an entry-level book first published twenty years ago but not yet superseded or really improved on:

- Stewart Shapiro,Â
*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 âThe Â Big Threeâ, 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 âThe Contemporary Sceneâ, 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 Colyvanâs *An Introduction to the Philosophy of Mathematics* (CUP, 2012) is far too rushed to be useful. And I would say much the same of Ăystein Linneboâs *Philosophy of MathematicsÂ *(Princeton UP, 2017). David BostockâsÂ *Philosophy of Mathematics: An IntroductionÂ *(Wiley-Blackwell, 2009) is more accessible, but — apart from a chapter on predicativism — covers similar ground to the earlier parts of Shapiroâs 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 CertaintyÂ*(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 Hilbertâs alternative response to theÂ âcrisis in foundationsâ, and from the impact of GĂ¶delâs work on the logicist and Hibertian programmes. Giaquinto explores this with enviable clarity: thisÂ 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*Â 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 (Iâve not known a good student who didnât enjoy it).

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