Upsetting the foundations for mathematics
Philosophia Scientiae, Volume 9 (2005) no. 2, pp. 5-21.

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.

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     author = {Neff Stout, Lawrence},
     title = {Upsetting the foundations for mathematics},
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     url = {http://archive.numdam.org/item/PHSC_2005__9_2_5_0/}
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Neff Stout, Lawrence. Upsetting the foundations for mathematics. Philosophia Scientiae, Volume 9 (2005) no. 2, pp. 5-21. http://archive.numdam.org/item/PHSC_2005__9_2_5_0/

[1] Awodey, Steve, Butz, Carsten, Simpson, A. and Streicher, Th. 2002.- Mac Lane set theory. Slides from ASL presentation, personal communication, 2002.

[2] Bell, J.L. 1988.- Toposes and Local Set Theories, Oxford: Oxford U. Press, 1988. | MR | Zbl

[3] Cohen, Paul J. 1966.- Set Theory and the Continuum Hypothesis, New York and Amsterdam: Benjamin, 1966. | MR | Zbl

[4] Freyd, Peter 1972.- Aspects of topoi, Bulletin of the Australian Mathematical Society, 1972. | Zbl

[5] Friedman, Harvey 2002.- 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

[6] Goldblatt, R. 1979.- Topoi: the Categorial Analysis of Logic, Amsterdam, New York, and Oxford: North Holland, 1979. | MR | Zbl

[7] Höhle, Ulrich 1991.- Monoidal closed categories, weak topoi and generalized logics, Fuzzy Sets and Systems, 1991. | MR | Zbl

[8] Jacobs, Bart 1999.- Categorical Logic and Type Theory, Studies in Logic and the Foundations of Mathematics, 141, Amsterdam: Elsevier, 1999. | MR | Zbl

[9] Johnstone, Peter T. 1977.- Topos Theory, London, New York, and San Francisco: Academic Press, 1977. | MR

[10] Joyal, Andre and Moerdijk, Ieke 1995.- Algebraic Set Theory, London Mathematical Society Lecture Notes, Number 220, Cambridge: Cambridge University Press, 1995. | Zbl

[11] Kock, Anders 1981.- Synthetic Differential Geometry, London Mathematical Society Lecture Notes Series, Number 51 . Cambridge U. Press, 1981. | MR | Zbl

[12] Kock, Anders and Wraith, Gavin C. 1971.- Elementary Toposes, in Lecture Notes, Number 30, Aarhus: Aarhus Universitet Matematisk Institut, 1971. | MR | Zbl

[13] Lakov, George and Núñez, Raphael 2000.- Where Mathematics Comes From: How the embodied mind brings mathematics into being, New York: Basic Books, 2000. | MR | Zbl

[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 | Zbl

[15] Lawvere, F.W. 1972.- Introduction, Toposes, Algebraic Geometry and Logic, Lecture Notes in Mathematics, Number 274, Berlin, Heidelberg and New York: Springer Verlag, 1972. | MR | Zbl

[16] Mac Lane, Saunders 1990.- Mathematics: Form and Function, New York, Berlin, Heidelberg, Tokyo: Springer Verlag, 1990. | Zbl

[17] Makkai, Michael 1998.- 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 | Zbl

[18] Marquis, Jean-Pierre 1995.- Category theory and the foundations of mathematics: philosophical excavations, Synthese, 103(3):421-447, 1995. | MR | Zbl

[19] Mayberry, John 1994.- What is required of a foundation for mathematics, Philosophia Mathematica (3), 2, 16-35, 1994. | MR | Zbl

[20] Monro, G. P. 1986.- Quasitopoi, logic and heyting-valued models, Journal of Pure and Applied Algebra, 42, 141-164, 1986. | MR | Zbl

[21] Penon, Jacques 1977.- Sur les quasitopos, Cahiers de Topologie et Géométrie Différentielle, 18, 181-218, 1977. | EuDML | Numdam | MR | Zbl

[22] Robinson, Abraham 1996.- Non-standard Analysis, revised edition, Princeton Landmarks in Mathematics, Princeton: Princeton U. Press, 1996. | MR | Zbl

[23] Simpson, Stephen G. 1996.- What is foundations of mathematics? On web page at http://www.math.psu.edu/simpson/hierarchy.html, 1996, still active Sept. 9, 2002.

[24] Stout, Lawrence N. 1992.- 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 | Zbl

[25] Wyler, Oswald 1991.- Notes on Topoi and Quasitopoi, Singapore: World Scientific, 1991. | MR | Zbl