You may very well know the Five Books website, where a wide-ranging cast of contributors are asked “to make book recommendations in their area of work and explain their choices in an interview”. The recommendations are often quirky, sometimes even slightly bizarre, but rarely without interest. It’s an illuminating and fun project.
There are a good number of topics in philosophy covered at Five Books, including some quite narrow ones. Though I am guessing that, reasonably enough, they are not going to get round to dealing with a topic of comparatively limited interest like the philosophy of mathematics. That thought sets me wondering: what five books in the area would I recommend in the same spirit as a Five Books posting. So here’s my selection — my selection today, at any rate: another day I might feel differently! (My commentaries, unlike the often wide-ranging interviews that accompany Five Books recommendations, are brisk. And I suppose that these might be said to be pretty conservative choices. But I make no apologies for that!)
Modern philosophy of mathematics is still shaped by debates starting over a century ago, springing from the work of Frege and Russell, and also from Hilbert’s alternative response to the “crisis in foundations”, and from the impact of Gödel’s work on the logicist and Hibertian programmes. All this is covered brilliantly and at relatively modest length in Marcus Giaquinto’s The Search for Certainty: A Philosophical Account of Foundations of Mathematics (OUP, 2002) This is not just engaging and reliable but is written with very enviable clarity. (By all means, then go back to reading Frege’s Grundlagen, or dip into e.g. Bertrand Russell’s Introduction to Mathematical Philosophy. But you won’t find a better initial guide to those foundational debates than Giaquinto.)
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. So Imre Lakatos’s Proofs and Refutations (originally 1963/4: CUP 1976) 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 generalised; but this remains a fascinating read.
The next book is a (too rare) example of a philosopher writing a mathematics book, engaging head-on with the conceptual issues the mathematics throws up. Michael Potter’s Set Theory and Its Philosophy (OUP 2004) is exactly what you need to read before trying to think about which brand of set theory (if any) to buy and why, or about the sense in which set theory is foundational, etc. The interplay between the mathematical and the philosophical here is very illuminating.
Many philosophers know a bit about arithmetic and set theory: but there is a lot more to mathematics than that. If you want to engage philosophically with mathematics more widely, you need to have some sense of what is going on in some other areas of mathematics and to understand something of how these areas hang together (remember Sellars’s words about philosophy concerning itself with how things hang together …). I can’t think of a better place to start than with Saunders Mac Lane’s Mathematics: Form and Function (Springer 1986). You will need some, a little, mathematics to cope with this: but then you can’t hope to do the philosophy of X without knowing something about X! And this book is a remarkable achievement, written by a great mathematician with a genuine concern for some of the philosophical issues in the vicinity.
Thanks to the Stanford Encyclopedia and various series of Companions and Handbooks, it has never been easier to get up to speed with (fairly) recent work in various areas of philosophy. That’s certainly true in this area, thanks to Stewart Shapiro, ed., The Oxford Handbook of Philosophy of Mathematics and (Its) Logic (OUP 2005). Oddly the ‘its’ is missing from the book’s official title, but the essays here only talk about aspects of logic of concern to the philosophy of maths, and the ‘its’ is rightly there in the title of Shapiro’s own editor’s introduction. So here are 26 essays on aspects of the philosophy of mathematics and on relevant logical matters, written by a star cast, and — unlike many collections of this kind — at a pretty consistent level of accessibility and quality, and in some cases offering essays on opposing sides of major debates. Perhaps the overall coverage is slightly conservative in the choice of topics: but there is still a huge amount of interest here. If you don’t find a good proportion of these essays engaging and worthwhile, then mainstream philosophy of mathematics perhaps just isn’t for you.
So, with many regrets about what I’ve had to leave out, there are my suggestions for five books — I’d be very intrigued to hear yours!
Michael Potter’s books “Sets: An Introduction”, and “Set Theory and Its Philosophy” are (one or the other) pulled out of my shelves roughly weekly. (The last: SAI for an excursion into lattice theory.) There can’t be many professional philosophers with his level of mathematical competence; this reminds me of Michael Dummett. Full of intellectual protein. (And occasionally mistakes!)
Of which professional mathematicians could it be said that there are few with such a high level of philosophical competence? Mathematicians often talk about philosophy as if it was a plague to be eradicated from mathematics. Yet they themselves often express “philosophical” opinions. These can be toe-curlingly awful, and aren’t often better than just unintelligible.
This amateur loved Giaquinto and Potter but has been stymied by the Handbook. Every month or so, I pull the latter off the shelf, read a page or two, and then give up till the next month. It (first printing) does look nice on the shelf, though!
I wonder if the trouble with the Handbook, for some readers, might be that many of the essays are written assuming quite a lot of other philosophical background? So although relatively introductory to the philosophy of mathematics, they are not necessarily that accessible?
Penelope Maddy, Naturalism in Mathematics
John Stillwell, Reverse Mathematics: Proofs from the Inside Out
John P. Burgess and Gideon Rosen, A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics.
Timothy Gowers, June Barrow-Green, and Imre Leader (eds), The Princeton Companion to Mathematics
David Tall, How Humans Learn to Think Mathematically
In brief: Yes, it was a toss-up on my list between Mac Lane and the amazing Princeton Companion (not that they are straightforward alternatives). When Burgess and Rosen came out I found it (at the time) disappointing — maybe I should take another look. I did like Burgess’s Fixing Frege and that was a near miss. Stillwell’s book strikes me, though, as an opportunity missed.
I’m curious now about what opportunity you feel Stillwell missed. Perhaps it will encourage someone to write the book his could have been. (So far as I know, Stillwell’s is the only introductory book on reverse mathematics, and I wanted something about it in my list. I have to confess I haven’t finished reading it, though, so I might agree with you in the end.)
My introduction to the discipline was Stephen Körner’s The Philosophy of Mathematics. Lakatos’ Proofs and Refutations is my alll time favourite book in any discipline. I’m rather fond of Davis/Hersh The Mathematical Experience.