Upsetting the foundations for mathematics

Lawrence Neff Stout

Philosophia Scientiae (2005)

  • Volume: 9, Issue: 2, page 5-21
  • ISSN: 1281-2463

Abstract

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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.

How to cite

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Neff Stout, Lawrence. "Upsetting the foundations for mathematics." Philosophia Scientiae 9.2 (2005): 5-21. <http://eudml.org/doc/103760>.

@article{NeffStout2005,
abstract = {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.},
author = {Neff Stout, Lawrence},
journal = {Philosophia Scientiae},
language = {eng},
number = {2},
pages = {5-21},
publisher = {Éditions Kimé},
title = {Upsetting the foundations for mathematics},
url = {http://eudml.org/doc/103760},
volume = {9},
year = {2005},
}

TY - JOUR
AU - Neff Stout, Lawrence
TI - Upsetting the foundations for mathematics
JO - Philosophia Scientiae
PY - 2005
PB - Éditions Kimé
VL - 9
IS - 2
SP - 5
EP - 21
AB - 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.
LA - eng
UR - http://eudml.org/doc/103760
ER -

References

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  1. [1] Awodey, Steve, Butz, Carsten, Simpson, A. and Streicher, Th. 2002.— Mac Lane set theory. Slides from ASL presentation, personal communication, 2002. 
  2. [2] Bell, J.L.1988.— Toposes and Local Set Theories, Oxford: Oxford U. Press, 1988. Zbl0649.18004MR972257
  3. [3] Cohen, Paul J.1966.— Set Theory and the Continuum Hypothesis, New York and Amsterdam: Benjamin, 1966. Zbl0182.01301MR232676
  4. [4] Freyd, Peter1972.— Aspects of topoi, Bulletin of the Australian Mathematical Society, 1972. Zbl0252.18001
  5. [5] Friedman, Harvey2002.— 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. [6] Goldblatt, R.1979.— Topoi: the Categorial Analysis of Logic, Amsterdam, New York, and Oxford: North Holland, 1979. Zbl0528.03039MR551362
  7. [7] Höhle, Ulrich1991.— Monoidal closed categories, weak topoi and generalized logics, Fuzzy Sets and Systems, 1991. Zbl0734.03035MR1123574
  8. [8] Jacobs, Bart1999.— Categorical Logic and Type Theory, Studies in Logic and the Foundations of Mathematics, 141, Amsterdam: Elsevier, 1999. Zbl0911.03001MR1674451
  9. [9] Johnstone, Peter T.1977.— Topos Theory, London, New York, and San Francisco: Academic Press, 1977. Zbl0368.18001MR470019
  10. [10] Joyal, Andre and Moerdijk, Ieke1995.— Algebraic Set Theory, London Mathematical Society Lecture Notes, Number 220, Cambridge: Cambridge University Press, 1995. Zbl0847.03025
  11. [11] Kock, Anders1981.— Synthetic Differential Geometry, London Mathematical Society Lecture Notes Series, Number 51 . Cambridge U. Press, 1981. Zbl0466.51008MR649622
  12. [12] Kock, Anders and Wraith, Gavin C.1971.— Elementary Toposes, in Lecture Notes, Number 30, Aarhus: Aarhus Universitet Matematisk Institut, 1971. Zbl0251.18015MR342578
  13. [13] Lakov, George and Núñez, Raphael2000.— Where Mathematics Comes From: How the embodied mind brings mathematics into being, New York: Basic Books, 2000. Zbl0987.00003MR1794854
  14. [14] Lambek, Joachim and Scott, P.J. 1986.— Higher Order Categorical Logic, Cambridge studies in advanced Mathematics, Number 7, Cambridge: Cambridge U. Press, 1986. Zbl0596.03002MR856915
  15. [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. Zbl0233.00005MR376798
  16. [16] Mac Lane, Saunders1990.— Mathematics: Form and Function, New York, Berlin, Heidelberg, Tokyo: Springer Verlag, 1990. Zbl0675.00001
  17. [17] Makkai, Michael1998.— 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. Zbl0896.03051MR1678360
  18. [18] Marquis, Jean-Pierre1995.— Category theory and the foundations of mathematics: philosophical excavations, Synthese, 103(3):421–447, 1995. Zbl1058.00505MR1350266
  19. [19] Mayberry, John1994.— What is required of a foundation for mathematics, Philosophia Mathematica (3), 2, 16–35, 1994. Zbl0796.03004MR1257682
  20. [20] Monro, G. P.1986.— Quasitopoi, logic and heyting-valued models, Journal of Pure and Applied Algebra, 42, 141–164, 1986. Zbl0597.18001MR857564
  21. [21] Penon, Jacques1977.— Sur les quasitopos, Cahiers de Topologie et Géométrie Différentielle, 18, 181–218, 1977. Zbl0401.18002MR480693
  22. [22] Robinson, Abraham1996.— Non-standard Analysis, revised edition, Princeton Landmarks in Mathematics, Princeton: Princeton U. Press, 1996. Zbl0843.26012MR1373196
  23. [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. [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. Zbl0760.03021MR1154569
  25. [25] Wyler, Oswald1991.— Notes on Topoi and Quasitopoi, Singapore: World Scientific, 1991. Zbl0727.18001MR1094373

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