Starting with a review of the kinds of questions a foundation for mathematics should address, this paper provides a critique of set theoretical foundations, a proposal that multiple interconnected categorical foundations would be an improvement, and a way of recovering set theory within a categorical approach.
@article{PHSC_2005__9_2_5_0, author = {Neff Stout, Lawrence}, title = {Upsetting the foundations for mathematics}, journal = {Philosophia Scientiae}, volume = {9}, year = {2005}, pages = {5-21}, language = {en}, url = {http://dml.mathdoc.fr/item/PHSC_2005__9_2_5_0} }
Neff Stout, Lawrence. Upsetting the foundations for mathematics. Philosophia Scientiae, Tome 9 (2005) pp. 5-21. http://gdmltest.u-ga.fr/item/PHSC_2005__9_2_5_0/
[1] Mac Lane set theory. Slides from ASL presentation, personal communication, 2002.
, , and 2002.-[2] Toposes and Local Set Theories, Oxford: Oxford U. Press, 1988. | MR 972257 | Zbl 0649.18004
1988.-[3] Set Theory and the Continuum Hypothesis, New York and Amsterdam: Benjamin, 1966. | MR 232676 | Zbl 0182.01301
1966.-[4] Aspects of topoi, Bulletin of the Australian Mathematical Society, 1972. | Zbl 0252.18001
1972.-[5] Re: Fom: {n: n notin f(n)}, e-mail to FOM list, August 30 2002, Archived at: http://www.cs.nyu.edu/pipermail/fom/2002-August/005787.html
2002.-[6] Topoi: the Categorial Analysis of Logic, Amsterdam, New York, and Oxford: North Holland, 1979. | MR 551362 | Zbl 0434.03050 | Zbl 0528.03039
1979.-[7] Monoidal closed categories, weak topoi and generalized logics, Fuzzy Sets and Systems, 1991. | MR 1123574 | Zbl 0734.03035
1991.-[8] Categorical Logic and Type Theory, Studies in Logic and the Foundations of Mathematics, 141, Amsterdam: Elsevier, 1999. | MR 1674451 | Zbl 0911.03001
1999.-[9] Topos Theory, London, New York, and San Francisco: Academic Press, 1977. | MR 470019
1977.-[10] Algebraic Set Theory, London Mathematical Society Lecture Notes, Number 220, Cambridge: Cambridge University Press, 1995. | Zbl 0847.03025
and 1995.-[11] Synthetic Differential Geometry, London Mathematical Society Lecture Notes Series, Number 51 . Cambridge U. Press, 1981. | MR 649622 | Zbl 0466.51008
1981.-[12] Elementary Toposes, in Lecture Notes, Number 30, Aarhus: Aarhus Universitet Matematisk Institut, 1971. | MR 342578 | Zbl 0251.18015
and 1971.-[13] Where Mathematics Comes From: How the embodied mind brings mathematics into being, New York: Basic Books, 2000. | MR 1794854 | Zbl 0987.00003
and 2000.-[14] Lambek, Joachim and Scott, P.J. 1986.- Higher Order Categorical Logic, Cambridge studies in advanced Mathematics, Number 7, Cambridge: Cambridge U. Press, 1986. | MR 856915 | Zbl 0596.03002
[15] Introduction, Toposes, Algebraic Geometry and Logic, Lecture Notes in Mathematics, Number 274, Berlin, Heidelberg and New York: Springer Verlag, 1972. | MR 376798 | Zbl 0233.00005
1972.-[16] Mathematics: Form and Function, New York, Berlin, Heidelberg, Tokyo: Springer Verlag, 1990. | Zbl 0675.00001
1990.-[17] Towards a categorical foundation of mathematics, in Johann A. Makowsky and Elena V. Ravve, (eds.), Logic Colloquium '95, Lecture Notes in Logic, 11 153-190. Association for Symbolic Logic, Springer Verlag, 1998. | MR 1678360 | Zbl 0896.03051
1998.-[18] Category theory and the foundations of mathematics: philosophical excavations, Synthese, 103(3):421-447, 1995. | MR 1350266 | Zbl 1058.00505
1995.-[19] What is required of a foundation for mathematics, Philosophia Mathematica (3), 2, 16-35, 1994. | MR 1257682 | Zbl 0796.03004
1994.-[20] Quasitopoi, logic and heyting-valued models, Journal of Pure and Applied Algebra, 42, 141-164, 1986. | MR 857564 | Zbl 0597.18001
1986.-[21] Sur les quasitopos, Cahiers de Topologie et Géométrie Différentielle, 18, 181-218, 1977. | Numdam | MR 480693 | Zbl 0401.18002
1977.-[22] Non-standard Analysis, revised edition, Princeton Landmarks in Mathematics, Princeton: Princeton U. Press, 1996. | MR 1373196 | Zbl 0843.26012
1996.-[23] What is foundations of mathematics? On web page at http://www.math.psu.edu/simpson/hierarchy.html, 1996, still active Sept. 9, 2002.
1996.-[24] The logic of unbalanced subobjects in a category with two closed structures, in U. Höhle S.E. Rodabaugh, E.P. Klement, (eds.), Applications of Category Theory to Fuzzy Subsets, Dordrecht, Boston, London: Kluwer, 1991. | MR 1154569 | Zbl 0760.03021
1992.-[25] Notes on Topoi and Quasitopoi, Singapore: World Scientific, 1991. | MR 1094373 | Zbl 0727.18001
1991.-