# Essential reads in the philosophy of mathematics and set theory

I am graduate student and have a decent understanding of logic and set theory.

Recently I have got interested in the philosophy of mathematics and set theory. I have read a number of papers by Penelope Maddy and Saharon Shelah, but I am wondering what other papers or books I should read. Any help will be appreciated.

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This looks to me like it should be community wiki. –  Charles Staats Apr 18 '12 at 13:27

Benacerraf and Putnam's Philosophy of Mathematics: Selected Readings is a pretty standard (as these things go) collection of seminal papers in the philosophy of mathematics generally, and in the philosophy of set theory in particular (Part IV). Looking farther afield, you could use Maddy as a guide to the literature and go through some of this syllabus, which largely builds around that volume.

You don't say exactly what papers of Maddy's you've read, so maybe this next isn't useful, but I remember getting a lot out of her Naturalism in Mathematics many moons ago, and maybe you'd prefer a single, focused work to a bevy of papers. Rather than a survey, this book takes a particular philosophical stance, and uses it to give a sustained argument against the idea of adopting $V=L$ as a foundational axiom. Along the way, Maddy situates her position among the traditional philosophy of math literature (e.g. Quine), while also dealing substantially with the set-theoretic issues/technicalities that necessarily intertwine with any attempts to do something serious.

Beyond the works already mentioned, if you seek current philosophical work that draws directly on the set-theoretic state-of-the-art, my humble suggestion is to look to folks like Peter Koellner (disclaimer: former advisor) and MO-superstar Joel David Hamkins.

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David Corfield's Towards a Philosophy of Real Mathematics is an excellent read, and also likely to stretch you mathematically. It takes up the theme mentioned in Andrej's answer: mathematics is a great deal more than set theory, so philosophy of mathematics should be a great deal more than philosophy of set theory. (But I understand that you're specifically interested in philosophy of set theory, and of course there's nothing wrong with that.)

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Thanks for the mention, but linking to a negative review might not be the best way to proceed. How about Brian Davies' review in Notices of the AMS ams.org/notices/201110/rtx111001454p.pdf. –  David Corfield Apr 18 '12 at 15:52
Sorry, David! I'd read that review before, but today I only skimmed it, and I didn't notice the negative parts. But I do think it contains some accurate (and uncritical) description of what's in your book - would you agree? That's why I chose it. Anyway, MO readers now have a choice of reviews to click on. –  Tom Leinster Apr 18 '12 at 21:24

The Stanford Encyclopedia of Philosophy is a good resource, especially for 'analytic' approaches. See the Philosophy of Mathematics entry, and links at the bottom of the page. You might also want to browse through the dedicated journal Philosophia Mathematica. If you're interested in approaches which look to broaden the range of questions asked by philosophers about mathematics, you could try Mancosu (ed.) The Philosophy of Mathematical Practice.

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I second the recommendation of Mancosu's anthology. There's a review of it by Timothy Bays in the March 2012 issue of the Notices. –  Timothy Chow Apr 18 '12 at 14:16

Philosophy of mathematics seems to focus primarily on set theory, which is probably a historical accident (just like it is a historical accident that set theory became the prevalent "foundation" in the 20th century). If you want to see things from other perspectives you could read things like:

Perhaps other, more knowledgable readers, can suggest additional references in this direction.

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Colin McLarty's "Exploring Categorical Structuralism" (cwru.edu/artsci/phil/PMExploring.pdf) is another follow-up in the Awodey/Hellman series. Also, one can find in the FOM archives many instances through the years of Harvey Friedman and Vaughan Pratt debating the merits of category-theoretic foundations. –  Ed Dean Apr 18 '12 at 8:32
I second Ed's recommendation. This is one of my favourite papers. The FOM archive discussions on categorical set theory contain some real poison, though: it's amazing the depths to which they sometimes sank. (I'm not pointing at Friedman or Pratt here.) –  Tom Leinster Apr 18 '12 at 9:57
For me Stephen Simpson was the star of those discussions. But I certainly wouldn't recommend reading old FOM archives. Most of it is very boring, and the interesting parts are detrimental to mental health. –  Andrej Bauer Apr 18 '12 at 17:00

Last fall I taught a course in the Philosophy of Set Theory at NYU and you can find the reading list available on my web page. This course was more narrrowly focused on the question of realism and pluralism than some of the other syllabi mentioned in the other excellent answers here.

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For the philosophy of Mathematics side, rather than the set theory side, I'd suggest:

Philosophy of Mathematics: Selected Readings; by Benacerraf & Putnam

From Frege to Godel: A Source Book in Mathematical Logic; edited by Jean van Heijenoort

Logic, Logic, and Logic; by George Boolos

Fixing Frege; by John Burgess

Foundations without Foundationalism; by Stewart Shapiro

New Waves in the Philosophy of Mathematics; edited by Linnebo and Bueno

For their historical interest:

Foundations of Arithmetic; by Gottlob Frege

An Introduction to Mathematical Philosophy; by Bertrand Russell

One book on the set theory side that I can recommend:

Set Theory and its Philosophy; by Michael Potter

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Paul Benacerraf's "What Numbers could not be" and "Mathematical Truth" are interesting. Shapiro's book Thinking about Mathematics is a good introduction to philosophy of math and goes through some history (Plato, Kant, Mill, Frege) before getting into stuff from the last century.

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"What Numbers could not be" is perhaps my favorite essay in the philosophy of mathematics. (To put this in context, I've probably read between 50 and 100 such essays.) In particular it is witty and amusing, which in my experience is quite rare for this genre of writing. –  Pete L. Clark Jun 27 '13 at 0:34

Maybe not quite the topic you're after, but I really enjoyed Lakatos, Proofs and Refutations.

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I'm a bit surprised not to see any Tarski references yet. Tarski's work on model theory was highly philosophically motivated and gave us the semantic definition of truth (one of the fundamental philosophical programmes in metaphysics and/or epistemology). This model theory is what put proof theory on a firm basis, gave the semantics of computation and programming languages their rigorous modern form, provided the language in which we could speak accurately about independence and other foundational relationships. It brings meaning to the possible worlds interpretations of modality, and actually gives meaning to the idea of mathematical interpretations in full generality.

If you are going to get into philosophers like Putnam and Kripke, Tarski is the prerequisite. Physical or experience-based foundationalists like the computational constructivists and the ultrafinitists are making a fundamental application of Tarski's view that model theory was actually the foundations of science in general and staking out positions that the meaning of mathematical statements must be found in experience. Tarski (as many of the Lvov-Warsaw logicians) was heavily influenced here by phenomenology.

Also, I would look at contributions from Feferman, Hintikka, Hellman, Zeilberger, and other modern foundationalists who also appear to be absent from the responses so far. I don't see how one can understand the modern philosophy of mathematics without understanding the very thoughtful approach of the various heretical controversies like Predicativism, Intuitionism and other schools of Constructivism, Ultrafinitism, etc.

Anyway, some suggestions:

• Logic, Semantics, Metamathematics by Tarski
• The Semantic Conception of Truth by Tarski

• Predicativism as a Philosophical Position by Hellman

• Does Mathematics Need New Axioms by Feferman
• On the Foundations of Mathematics by Brouwer
• Real Analysis is a Degenerate Case of Discrete Analysis by Zeilberger
• Logic and Belief by Hintikka

Also, I think anyone who takes the philosophic foundations of mathematics seriously should invest some good time with the Lvov-Warsaw logicians, who carried out some of the deepest early 20th century analysis here even as Gödel, Zermelo, et al. were transforming the foundations. Janiszewski, Leśniewski, Łukasiewicz, Mostowski, and others explored much of the ontological crises of mathematics in ways that focused a lot of early set theory and have influenced incredibly the modern investigations.

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For a break from the dry analyticity of most of the philosophy of mathematics, try Alain Badiou's "Number and numbers".

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The Foundations of Mathematics (FOM) mailing list has fair amount of interesting discussion with references worth following, plus a lot of lameness like anything else on the internet. Browsing its archives (so you can skip around easily) is usually fun and can help you find more stuff to look into. Location:

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Hermann Weyl's Levels of Infinity: Selected Writings on Mathematics and Philosophy, 2012, Dover Publications. It is lucid, rich in concepts, without symbolic tears.

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Borges 'The Library of Babel' is a beautiful meditation on all sorts of philosophical positions around the 'idea' of infinity, epistemology, the sociology of science, set theory paradoxes. Its literature & not philosophy though, or is it the other way around...?

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Here are 3 ones:

Critique of Pure Reason (Immanuel Kant) (It seems there are sections that are more mathematically relevant than others.)

The Provenace of Pure Reason (William Tait)

Philosophies of Mathematics (Alexander George and Daniel Velleman)

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I'm not sure why your answer was voted down, but I do think that Kant is a horrible place to look for mathematical philosophy. In attempting to demonstrate the synthetic a priori, he uses basic mathematical statements that are now easily seen to not be a priori. His examples in Euclidean geometry are challenged by the fact that our geometry is apparently non-Euclidean. The logic of statements about the world appears to be a nondistributive orthomodular logic. Number relies on distinguishability, a much more troubling issue than Kant believed... (cont.) –  ex0du5 Jun 28 '13 at 15:53
I'm not saying it isn't instructive to learn from Kant the direction of much of that philosophy, but I wouldn't look to his work as being seriously defensible today. And I don't think it's purpose was to benefit mathematics either, simply to use it in a different programme. But if one wants to focus on the mathematical content of his work, I would recommend his Prolegomena before CPR. It is more focused on the examples of his philosophy, including much arithmetic, logic, and geometry. (And the occasional claim that the inverse-square law of forces is required by the area of spheres). –  ex0du5 Jun 28 '13 at 15:58
While today it is possible (even fashionable) to disagree with much of what Kant wrote about mathematics, his influence on the subject was tremendous. Brouwer, Hilbert definitely knew their Kant and were strongly influenced by his ideas. –  Christian Remling Nov 9 '14 at 19:36
Studying the philosophy of mathematics and completely ignoring Kant is like reading Dante while having no idea who Vergil was. –  Christian Remling Nov 9 '14 at 19:39