Does category theory break set theory?

• A. Blass. The interaction between category theory and set theory. Contemporary Mathematics, 30:5–29, 1984. http://www.math.lsa.umich.edu/~ablass/interact.pdf
• S. Feferman. Categorical foundations and foundations of category theory. Logic, Foun- dations of Mathematics and Computability Theory, 1:149–169, 1977.math.stanford. edu/~feferman/papers/Cat_founds.pdf
• S. Feferman and G. Kreisel. Set-theoretical foundations of category theory. InReports of the Midwest Category Seminar III (Lecture Notes in Mathematics 161), pages 201–
  247. 1969. dx.doi.org/10.1007/BFb
• A. Grothendieck et al. “Univers” in Expos ́e I (Prefaisceaux) of SGA4. dx.doi.org/ 10.1007/BFb
• S. Mac Lane. Foundations for categories and sets. Category Theory, Homology Theory and their Applications. II. (Lecture Notes in Mathematics 92), 92:146–164,
  1969. dx.doi.org/10.1007/BFb
• S. Mac Lane. One universe as a foundation for category theory. InReports of the Midwest Category Seminar III (Lecture Notes in Mathematics 161), pages 192–200.
  1969. dx.doi.org/10.1007/BFb
• S. Mac Lane. Categorical algebra and set-theoretic foundations. InAxiomatic set theory, pages 231–240. 1971. http://www.princeton.edu/~hhalvors/teaching/phi536_ s2011/maclane1971.pdf
• F. Muller. Sets, classes and categories. British Journal for the Philosophy of Science. 52, pages 539–573. 2001. dx.doi.org/10.1093/bjps/52.3.
• V. Rao. On doing category theory within set theoretic foundations. In What is category theory, pages 275–290. 2006.
• M.A. Shulman. Set theory for category theory. Arxiv preprint arXiv:0810.1279, 2008.

2 Does category theory provide an alternative foundation of mathematics?

• J.L. Bell. Category theory and the foundations of mathematics. British Journal for the Philosophy of Science, 32(4):349–358, 1981. http://www.jstor.org/stable/
• G. Blanc and A. Preller. Lawvere’s basic theory of the category of categories. The Journal of Symbolic Logic40(1), pages 14–18. 1975. http://www.jstor.org/stable/
• G. Blanc and M.R. Donnadieu. Axiomatisation de la cat ́egorie des cat ́egories. Cahiers Topologie G ́eom. Diff ́erentielle17(2), pages 135–170. 1976.
• P. Dedecker. Category theory and the foundations of mathematics. InMathematical logic and formal systems : a collection of papers in honor of Professor Newton C.A. da Costa. pages 67–84.
• G. Kreisel. Appendix II. Category theory and the foundations of mathematics. In Reports of the Midwest Category Seminar. III(LNM 106), pages 233–247.
• F.W. Lawvere. The category of categories as a foundation for mathematics. In Proceedings of the Conference on Categorical Algebra. pages 1–20. Springer, 1966. See also Math Review # MR0207517 (34 #7332) by J.R. Isbell. http://www.princeton.edu/ ~hhalvors/lawvere1965.pdf
• Ø. Linnebo and R. Pettigrew. Only up to isomorphism? Category theory and the foun- dations of mathematics. 2010. seis.bris.ac.uk/~rp3959/papers/onlyuptoiso. pdf
• M. Makkai. Towards a categorical foundation of mathematics. InLogic Colloquium, volume 95, pages 153–190. 1998. projecteuclid.org/euclid.lnl/
• J. Mayberry. What is required of a foundation for mathematics?. Philosophia Math- ematica, (3), Volume 2, Special Issue, “Categories in the Foundations of Mathematics and Language”, pages 16–35. 1994.
• C. McLarty. Axiomatizing a category of categories. The Journal of Symbolic Logic 56(4), pages 1243–1260. 1991. http://www.jstor.org/stable/

Does category theory help structuralism?

• S. Awodey. Structure in mathematics and logic: A categorical perspective.Philosophia Mathematica, 4(3):209, 1996.
• S. Awodey. An answer to Hellman’s question: does category theory provide a frame- work for mathematical structuralism? Philosophia Mathematica54–64, 2004.philmat. oxfordjournals.org/cgi/reprint/12/1/54.pdf
• J. Bain. Category-theoretic structure and radical ontic structural realism. ls.poly. edu/~jbain/papers/CatTheoROSR.pdf
• J.P. Burgess. Putting structuralism in its place. http://www.princeton.edu/~jburgess/ Structuralism.pdf
• J.C. Cole. Mathematical structuralism today. Philosophy Compass, 5(8):689–699,
• L. Corry. Nicholas Bourbaki and the concept of mathematical structure. Synthese, 92(3): 315–348, 1992. dx.doi.org/10.1007/BF
• J. Couture and J. Lambek. Philosophical reflections on the foundations of mathemat- ics. Erkenntnis, 34(2):187–209, 1991.
• G. Hellman. Does category theory provide a framework for mathematical structural- ism? Philosophia Mathematica11(2), pages 129–157. 2003. dx.doi.org/10.1093/ philmat/11.2.
• G. Hellman. Three varieties of mathematical structuralism. Philosophia Mathematica, 9(2):184, 2001.
• G. Hellman. Structuralism. The Oxford handbook of philosophy of mathematics and logic, 1(9):536–563, 2005.
• G. Hellman. What is categorical structuralism? The Age of Alternative Logics, pages 151–161, 2006.
• E. Landry. Category theory as a framework for anin reinterpretation of mathematical structuralism. The Age of Alternative Logics, pages 163–179, 2006.
• E. Landry. Reconstructing Hilbert to construct category theoretic structuralism. 2009.
• H. Leitgeb and J. Ladyman. Criteria of identity and structuralist ontology.Philosophia Mathematica, 2007.
• S. Mac Lane. Structure in mathematics. Philosophia Mathematica, 4(2):174, 1996.
• M. Makkai. On structuralism in mathematics. In Language, Logic, and Concepts, pages 43–66. cognet.mit.edu/library/books/chapter?isbn=0262600463&part= chap
• C. McLarty. Exploring categorical structuralism. Philosophia Mathematica, 12(1):37, 2004.
• C. McLarty. What structuralism achieves. The Philosophy of Mathematical Practice, 1(9):354–370, 2008.
• E. Palmgren. Category theory and structuralism. 2009. www2.math.uu.se/ ~palmgren/CTS-fulltext.pdf
• M. Pedroso. On three arguments against categorical structuralism. Synthese, 170(1):21– 31, 2009. dx.doi.org/10.1007/s11229-008-9346-
• S. Shapiro. Mathematical structuralism. Philosophia Mathematica, 4(2):81, 1996.
• S. Shapiro. Categories, structures, and the Frege-Hilbert controversy: The status of meta-mathematics. Philosophia Mathematica, 13(1):61, 2005.

4 Miscellaneous

• S. Awodey. From sets, to types, to categories, to sets.
• J.L. Bell. From absolute to local mathematics. Synthese69(3), pages 409–426. 1986. dx.doi.org/10.1007/BF
• A. Blass. Why sets? Pillars of computer science, pages 179–198, 2008. dx.doi.org/ 10.1007/978-3-540-78127-1_
• E. Engeler and H. R ̈ohrl. On the problem of foundations of category theory. Dialectica 23(1), pages 58-66. 1969. dx.doi.org/10.1111/j.1746-8361.1969.tb01179.x
• S. Feferman. Enriched Stratified systems for the foundations of category theory. in What is Category Theory? (G. Sica, ed.) Polimetrica. pages 195–203. 2006. math.stanford.edu/~feferman/papers/ess.pdf
• G. Kreisel. Observations of popular discussions on foundations. In Axiomatic set theory. American Mathematical Society. pages 183–190. 1971.
• J. Lambek and P. Scott. Reflections on a categorical foundations of mathematics. http://www.site.uottawa.ca/~phil/papers/LS11.final.pdf
• E. Landry. Category theory: the language of mathematics. Philosophy of Science 66(3), pages 14–27. 1999.
• E. Landry and J.-P. Marquis. Categories in context: historical, foundational and philosophical. Philosophia Mathematica13(1), pages 1–43. 2005. dx.doi.org/10. 1093/philmat/nki
• S. Mac Lane. Mathematical models: A sketch for the philosophy of mathematics. American Mathematical Monthly, 88(7):462–472, 1981.
• J.P. Marquis. Category theory and the foundations of mathematics: philosophical ex- cavations. Synthese, 103(3):421–447, 1995.springerlink.com/index/LNL7084Q448756Q6. pdf
• J.P. Marquis. Categories, sets and the nature of mathematical entities. The Age of Alternative Logics, pages 181–192, 2006.
• C. McLarty. Learning from questions on categorical foundations. Philosophia Mathe- matica, 13(1):44, 2005.
• C. McLarty What does it take to prove Fermat’s last theorem?. Bulletin of Symbolic Logic, 16 pages 359–377. 2010. jstor.org/stable/
• C. McLarty. Weak set theory for Grothendieck’s number theory. arxiv.org/abs/
• I. Moerdijk. Sets, topoi and intuitionism. Philosophia Mathematica, 6(2):169, 1998.
• N.H. Williams. On Grothendieck universes. Compositio Mathematica, 21(1): pages 1–3. 1969. archive.numdam.org/article/CM_1969__21_1_1_0.pdf
• Numerous articles inFoundational Theories of Classical and Constructive Mathemat- ics, forthcoming March 2011 from Springer. ISBN 9789400704305