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


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*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.
A: 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.)
A: 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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*http://www.cs.nyu.edu/pipermail/fom/
A: 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.
A: 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:


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*S. Awodey: Structure in mathematics and logic: a categorical perspective. Philosophia Mathematica (3), vol. 4 (1996), pp. 209--237. (See also the subsequent discussion.)

*F. W. Lawvere's "Down with "Foundations"! Up with algebra!" and "Why are we concerned? II" posting on the categories mailing list, which include further references.
Perhaps other, more knowledgable readers, can suggest additional references in this direction.
A: 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. 
A: 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.
A: For a break from the dry analyticity of most of the philosophy of mathematics, try Alain Badiou's "Number and numbers". 
A: Hermann Weyl's Levels of Infinity: Selected Writings on Mathematics and Philosophy, 2012, Dover Publications. It is lucid, rich in concepts, without symbolic tears. 
A: I think since the fundamentals of the philosophy of mathematics have been well-established above, I'd like to submit Albert Lautmann's Mathematics, Ideas and the Physical Real and the more recent (and 'continental') Synthetic Philosophy of Contemporary Mathematics by Fernando Zalamea.
A: 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...?
A: 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)
