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Let’s begin with an entry-level book first published twenty years ago but not yet superseded or really improved on: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

Next, having got from Shapiro a sense of some of the core problems, you should certainly sample some classic papers. Here are two extremely useful sourcebooks taking us up to the turn of the century:

- Paul Benacerraf & Hilary Putnam (eds),
*Philosophy of Mathematics: Selected Readings*(CUP: NB you want the 1983 2nd edition of this classic collection). - Dale Jacquette (ed),
*Philosophy of Mathematics: An Anthology*(Blackwell, 2002).

Then, for further discussions of debates old and new, the obvious next place to look is:

- Stewart Shapiro (ed.)
*The Oxford Handbook of Philosophy of Mathematics and Logic*(OUP, 2005). The editor’s introductory essay is in fact called ‘Philosophy of mathematics and its logic’, which should surely also have been the whole*Handbook*’s title — for of the twenty-six essays here, twenty are straight philosophy of mathematics, and the logic essays are mostly closely relevant to mathematics too. Large handbooks of this general type can often be very mixed bags, containing essays of decidedly varying quality; but this one really is a triumph. Some of the essays are very substantial, and as I recall it none is a makeweight. There are often pairs of essays taking divergent approaches (e.g. to contemporary logicism, to intuitionism, to structuralism). Of course, there are variations in the accessibility of the individual essays: but Shapiro seems to have done wonderful work in keeping his very well-selected authors under control! So for any serious student now — perhaps beginning graduate student — this must be*the*place to start explorations of issues in more recent philosophy of mathematics.

Following up interesting-seeming references in the *Handbook* essays will enable you to begin to explore the then-state-of-play in various areas in as much detail as you want: I therefore needn’t add more references here to important earlier work by a whole range of philosophers. And so, with a gesture towards the amazing resource that is the *Stanford Encyclopaedia of Philosophy — *which has some characteristically excellent long entries on various topics in the philosophy of mathematics, all with many further references — we could stop an introductory list at this point. Except I should perhaps mention another giant handbook. After all, if you are interested in the philosophy of maths, it helps to know some maths! For a guide with some wonderfully lucid essays, see the masterful

- Timothy Gowers (et al., eds)
*The Princeton Companion to Mathematics*(Princeton UP, 2008).

As I said, I have surely provided more than enough introductory reading! Still, let’s ask: what has been published since around the time of the *Handbook *which is both of note and is also reasonably accessible? There was a short collection edited by Otávio Bueno and Øystein Linnebo called *New Waves in the Philosophy of Mathematics* (Palgrave, 2009), which has moderate interest. Some of the papers collected in Paolo Mancosu (ed.) *The Philosophy of Mathematical Practice *(OUP, 2008) are worth reading. And of course, the journal *Philosophia Mathematica* continues to publish many good articles. But what of books?

Ian Hacking’s *Why is There Philosophy of Mathematics At All* (CUP, 2014) is an idiosyncratic though quite engaging ramble by an always-interesting philosopher. Penelope Maddy, who has a paper on ‘naturalism’ about mathematics in the *Handbook* has published two characteristically readable works developing her position, *Second Philosophy* (OUP, 2007) and *Defending the Axioms: On the Philosophical Foundations of Set Theory* (OUP, 2011). One position under-represented in the *Handbook* is outright fictionalism: Mary Leng’s *Mathematics and Reality* (OUP, 2010) mounts a defence. She argues that we have no reason to believe that mathematical objects exist, and then takes on the task of explaining how it can still be that mathematics can be crucially useful in the formulation of scientific theories. You might well not end up agreeing: but thinking through Leng’s lucidly presented arguments will force you to get clear about a range of central issues.

We are now perhaps going up a level in difficulty. Charles Parsons has been one of the most insightful philosophers of mathematics for half a century. His early collection of papers *Mathematics in Philosophy* (OUP, 1983) is still well worth reading. But his long-awaited major book *Mathematical Thought and Its Objects* (CUP, 2008) is quite tough going — it is not easy to work out the subtle position he is trying to develop as he negotiates his way between different kinds of structuralism. But there is a good deal to think about here.

Every philosopher of mathematics should at some point read Michael Dummett*’*s *Frege: Philosophy of Mathematics* (Duckworth, 1991). The later neo-Fregean project of Crispin Wright and Bob Hale’s *The Reason’s Proper Study* (OUP 2001) has rather run out of steam. But it did inspire important work on Frege — see in particular Richard Heck’s two books of papers *Frege’s Theorem* (OUP, 2011) and *Reading Frege’s Grundgesetze* (OUP, 2012). For more on post-Fregean themes see also the more approachable John Burgess, *Fixing Frege* (Princeton UP, 2005).

Philosophers continue to worry about the foundations of set theory — having, let’s hope, done their initial homework on Michael Potter’s excellent *Set Theory and Its Philosophy* (OUP, 2004), and perhaps also on José Ferreirós’ historical *Labyrinths of Thought* (Birkhäuser, 2007). We’ve mentioned Maddy’s work. For a new discussion see Luca Incurvati’s lucid *Conceptions of Set and the Foundations of Mathematics* (Cambridge, 2020). A smaller number of philosophers worry about the foundations of category theory: you’ll find some scattered papers with further references in *Philosophia Mathematica*.

My sense, though, is that after a period in which the philosophy of mathematics really flourished, there has perhaps been something of a lull more recently. However let me finish by mentioning a stand-out recent achievement, a little more wide-ranging than just philosophy of mathematics (though a considerably bumpier ride than perhaps the authors intended, so only just squeezing into what started out as an introductory list!) — namely Tim Button & Sean Walsh, *Philosophy and Model Theory* (OUP, 2018).

*And let me leave it there for the moment. What would you add?*

RichardI’m looking forward to Joel David Hamkins upcoming book “Lectures on the Philosophy of Mathematics.” Although based on the number of topics listed in the table of contents (available on his website), I suspect it won’t be as beginner-friendly as Shapiro’s book.

A book I like for the classic topics of logicism/finitism/intuitionism is “Philosophies of Mathematics” by Alexander George and Daniel Velleman.

Peter FI have been following Joel Hamkins’ current lecture series on the philosophy of mathematics and am looking forward to reading the book they’re based on (coming out early next year). I have always appreciated his willingness to take the philosophy side of things very seriously while still being up front about himself being a mathematician first and having a mathematician’s perspective, and so I am hoping that the book will also go on my list.

I’d mention four other books. First, I agree with Rowsety Moid’s comment on the other post; I am a big fan of Stillwell’s Reverse Mathematics. I think he does a great job of illustrating why reverse mathematics should matter to the general mathematician. I recommended the book to a PhD in pure mathematics candidate friend of mine who had no background in foundations beyond the usual demonstration of a set theoretic reductions and he found it fascinating and especially well motivated, particularly the idea of a base theory and why RCA_0 fits that role. Personally, I am also a big fan because Stillwell uses Smullyan’s Elementary Formal Systems as his model of computation which I think are severely underrepresented in the literature given their conceptual clarity and elegance.

Secondly, Sean Morris’ recent Quine, New Foundations, and the Philosophy of Set Theory. I was wondering if you were familiar with the book, I recently searched for any posts that might mention it but found none. I think that it is an incredible follow up, for people interested in the philosophy of set theory in particular, after reading Potter’s great book, or perhaps as a third after reading one of Maddy’s books. Ultimately the book is an argument for a pluralism about set theory, not in the now conventional multiverse sense, but from a Quinean perspective that we aren’t bound to the iterative conception of set because a conceptual analysis of the essence of set is fundamentally misguided, and therefore set theories that supposedly embody that notion are not our only option. I am not a Quinean, disagreement with him forms the backbone of many of my philosophical positions and interests, but I think Quine was a one in a trillion thinker who (almost) always had clear and precise arguments for his positions, and always had something interesting to say, and so I thoroughly enjoy grappling with Quinean positions, so I really did love this book.

Thirdly, Agustin Rayo’s recent On The Brink of Paradox, Highlights from the Intersection of Philosophy and Mathematics. This is less a book aimed to give an overview of the philosophy of mathematics, and more so a book aimed to show how foundational and philosophical issues can come up in various places in pure and applied mathematics through paradoxes. I watched videos of the lectures from one semester of Rayo’s teaching of the class and I really enjoyed the wide coverage of material. Rayo has said that the class is very popular and gets students from many different majors, so it’s a great opportunity to expose people from many different STEM or STEM adjacent backgrounds to the philosophy of mathematics and various paradoxes that can arise in their own fields (like probability theory or physics). So I think the book is great for people from non philosophy backgrounds to first engage with some of the more interesting and confounding areas in the subject. Personally, my favorite sections are those dealing with omega, and reverse omega, sequence paradoxes like Yablo’s paradox.

And that brings me to my final book, Roy Cook’s The Yablo Paradox. This is probably a ‘special topic’ type of reading, but it is a topic that I find endlessly fascinating and important. The book is a a treatise on the Yablo paradox, but also on the general nature of (non)self referential semantic paradoxes and the intensional aspects of ‘saying things’ with a formalism. In particular, and close to my own research, I love the use of graph theoretic tools to study the semantic paradoxes and giving a precise mathematical treatment of the complexities that come out of semantic interpretations of formalisms. The main questions the book addresses are what do we mean by saying a given semantic paradox is self referential, is the Yablo paradox an example of such, and can we come up with any others that are unobjectionably non self referential. Lots of fun with omega sequences and arithmeticization of formal languages.

Even if just short responses, of course I am curious of your opinions on each of these, but mostly I’d appreciate some elucidation of what issues you have with Stillwell’s book and if you have read Morris’ book on Quine.

Peter SmithReverse Mathematics. My first impression really was of an opportunity missed — but perhaps I just approached it with the wrong expectations. I’ll have to take another look.Rowsety MoidIt depends on what the opportunity is, but for some values of ‘opportunity’, I think Stillwell’s

Reverse Mathematicscan be an opportunity missed and nonetheless be a good and useful book for someone who wants an introductory treatment of RM (and has enough maths / logic experience that they won’t find it too difficult) (a course in analysis?).Anyway, I think there should be something about RM in the reading list, and I don’t know of a better alternative.

Chris GrantAs a mathematician who is not philosophically-minded but sort of wishes he were, I found Potter’s and Giaquinto’s books delightful but The Oxford Handbook of Philosophy of Mathematics and Logic utterly impenetrable. A few times a year, I pull the latter off the shelf, grit my teeth, and make it through a few pages, absorbing very little.

Peter SmithI guess I can understand that as a reaction from someone who doesn’t have a background in philosophy. The

Handbookis, I think, very much addressed to philosophical readers who will be familiar with the sorts of discussions in those other two collections of articles that I mentioned (and who are also coming primed with some familiarity of wider debates in epistemology, philosophical logic etc.).Owen GriffithsSome excellent recommendations here. I’m also looking forward to Hamkins’ book. And very much looking forward to A.C. Paseau’s advanced introduction ‘What is Mathematics About?’ which also grew out of Oxford Philosophy of Mathematics lectures. It’s forthcoming with OUP, I believe.

BenjaminHi, I am a math student who is interested in the philosophy of mathematics. I recently finished Shapiro’s Thinking About Mathematics and really enjoyed it. However, I have tried to read Benacerraf and Putnam’s anthology, but I lack the formal logic background that seems necessary to read some parts of it. Would you advise me to first study formal logic and then try to read it or to just read another book?

Peter SmithI think that it is difficult to progress very far into the philosophy of mathematics without some background in logic. But it is difficult to say in the abstract what you need! — that will depend a bit on the bits of the philosophy of mathematics which you want to follow. Perhaps a survey book like Robert Wolf’s A Tour Through Mathematical Logic might at least help for orientation. And then you can follow up whatever seems most relevant by looking at my Study Guide …

Doug MounceGiven that *philosophos* originally meant a “way” I would recommend Polya’s How to Solve It as an exposition on heuristics.

danielamJames Franklin’s “An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure” is worth mentioning.

WolfgangThanks for the survey of articles!

There is a unique one I have to strongly recommend: Zalamea’s “Synthetic Philosophy of Contemporary Mathematics”. It’s a bit of a terse read, and I have only skimmed parts.

Many contemporary philosophers of mathematics get hung up on the ‘basics’, i.e., metaphysics of mathematics, nature of axiom systems, etc.

Zalamea corrects for this by taking a closer look at the mathematics as it is actually practiced in the world by mathematicians of the past century to the present day- the nature of the problems looked at and being solved.

BenceI think Science and Sanity by Alfred Korzybski is also solid.