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Marquis, Jean-Pierre

Category theory and the foundations of mathematics: philosophical excavations. (English) Zbl 1058.00505

Synthese 103, No. 3, 421-447 (1995).

Cited in 5 Documents

MSC:

00A30 Philosophy of mathematics
03A05 Philosophical and critical aspects of logic and foundations
18B99 Special categories
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[1] Barr, M. and Wells, C.: 1985,Toposes, Triples and Theories, Ergenbnisse Math. Wiss. 278, Springer-Verlag, New York. · Zbl 0567.18001
[2] Barr, M. and Wells, C.: 1988, ?The Formal Description of Data Types Using Sketches?, inMathematical Foundations of Programming Language Semantics, LNCS 298, Springer-Verlag, New York, pp. 490-527.
[3] Barr, M. and Wells, C.: 1990,Category Theory for Computing Science, Prentice Hall, New York. · Zbl 0714.18001
[4] Beeson, M. J.: 1985,Foundations of Constructive Mathematics, Springer-Verlag, New York. · Zbl 0565.03028
[5] Bell, J. L.: 1981, ?Category Theory and the Foundations of Mathematics?,British Journal for the Philosophy of Science 32, 349-58. · Zbl 0474.18001 · doi:10.1093/bjps/32.4.349
[6] Bell, J. L.: 1986, ?From Absolute to Local Mathematics?,Synthese 69, 409-26. · doi:10.1007/BF00413980
[7] Bell, J. L.: 1988,Toposes and Local Set Theories, Oxford University Press, New York. · Zbl 0649.18004
[8] Bénabou, J.: 1985, ?Fibered Categories and the Foundations of Naive Category Theory?,Journal of Symbolic Logic 50(1), 10-37. · Zbl 0564.18001 · doi:10.2307/2273784
[9] Blass, A.: 1984, ?The Interaction Between Category Theory and Set Theory?,Contemporary Mathematics 30, 5-29. · Zbl 0551.18001
[10] Boileau, A. and Joyal, A.: 1981, ?La logique des Topos?,Journal of Symbolic Logic 46(1), 6-16. · Zbl 0544.03035 · doi:10.2307/2273251
[11] Boolos, G.: 1971, ?The Iterative Conception of Set?,Journal of Philosophy 68, 215-32. · doi:10.2307/2025204
[12] Bunge, M.: 1979,A World of Systems, D. Reidel, Dordrecht.
[13] Bunge, M.: 1985,Philosophy of Science and Technology, D. Reidel, Dordrecht.
[14] Chapman, J. and Rowbottom, F.: 1992,Relative Category Theory and Geometric Morphisms, Oxford University Press, Oxford. · Zbl 0751.18001
[15] Cole, J. C.: 1973, ?Categories of Sets and Models of Set Theory?,Proceedings of the Bertrand Russell Memorial Logic Conference, 351-99.
[16] Corry, L.: 1992, ?Nicolas Bourbaki and the Concept of Mathematical Structure?,Synthese 92, 315-48. · Zbl 0752.00004 · doi:10.1007/BF00414286
[17] Couture, J. and Lambek, J.: 1991, ?Philosophical Reflections on the Foundations of Mathematics?,Erkenntnis 34, 187-209. · doi:10.1007/BF00385720
[18] Dedekind, R.: 1901,Essays on the Theory of Numbers, W. W. Beman, trans., 2nd ed. 1909, Open Court, Chicago. · JFM 32.0185.01
[19] Faith, C.: 1973,Algebra Rings, Modules, and Categories, Vol. I, Springer-Verlag, New York. · Zbl 0266.16001
[20] Feferman, S.: 1969, ?Set-Theoretical Foundations of Category Theory?,LNM 106, Springer-Verlag, 201-47. · Zbl 0213.03001
[21] Feferman, S.: 1977, ?Categorical Foundations and Foundations of Category Theory?, in R. Butts (ed.),Logic, Foundations of Mathematics and Computability, D. Reidel, pp. 149-169. · Zbl 0401.18001
[22] Feferman, S.: 1985, ?Working Foundations?,Synthese 62, 229-54. · doi:10.1007/BF00486048
[23] Field, H.: 1989,Realism, Mathematics, and Modality, Basil Blackwell, Oxford.
[24] Fourman, M.: 1980, ?Sheaf Models for Set Theory,?Journal of Pure and Applied Algebra 19, 91-101. · Zbl 0446.03041 · doi:10.1016/0022-4049(80)90096-1
[25] Goguen, J. A.: 1991, ?A Categorical Manifesto?,Mathematical Structures in Computer Science 1, 49-68. · Zbl 0747.18001 · doi:10.1017/S0960129500000050
[26] Goguen, J. and Burstall, R.: 1984, ?Introducing Institutions?, in E. Clarke and D. Kozen (eds.),Logics of Programs, LNCS, 164, New York: Springer-Verlag, pp. 221-56. · Zbl 0543.68021
[27] Goguen, J. and Burstall, R.: 1986, ?A Study in the Foundations of Programming Methodology: Specifications, Institutions, Charters and Parchments?, in D. Pitt, S. Abramsky, A. Poigne and D. Rydeheard (eds.),Proceedings, Conference on Category Theory and Computer Programming, LNCS, 240, Springer-Verlag, New York, 313-33. · Zbl 0615.68002
[28] Goldblatt, R.: 1984,Topoi, 2nd ed., Studies in Logic and the Foundations of Mathematics, vol. 98, North-Holland. · Zbl 0528.03039
[29] Goldfarb, W.: 1988, ?Poincaré Against the Logicists?, in W. Aspray and P. Kitcher (eds.),History and Philosophy of Modern Mathematics, University of Minnesota Press, Minneapolis, pp. 61-81. · Zbl 1366.01024
[30] Griffiths, P. and Harris, J.: 1978,Principles of Algebraic Geometry, John Wiley and Sons, New York. · Zbl 0408.14001
[31] Hallett, M.: 1984,Cantorian Set Theory, Oxford University Press, Oxford.
[32] Hartshorne, R.: 1977,Algebraic Geometry, Springer-Verlag, New York. · Zbl 0367.14001
[33] Hatcher, W. S.: 1982,The Logical Foundations of Mathematics, Pergamon Press, New York. · Zbl 0504.03001
[34] Heinzmann, G.: (ed.), 1986,Poincaré, Russell, Zermelo et Peano, Alben Blanchard, Paris.
[35] Hilbert, D.: 1927, ?The Foundations of Mathematics?, in J. van Heijennort (ed.),From Frege to Gödel, 1967, Harvard University Press, Cambridge, MA. · JFM 53.0849.03
[36] Hylton, P.: 1980, ?Russell’s Substitutional Theory?,Synthese 45, 1-31. · Zbl 0517.01019 · doi:10.1007/BF00413984
[37] Hylton, P.: 1990,Russell, Idealism, and the Emergence of Analytic Philosophy, Oxford University Press, New York.
[38] Irvine, A.: 1989, ?Epistemic Logicism and Russell’s Regressive Method,Philosophical Studies 55, 303-27. · doi:10.1007/BF00355328
[39] Isbell, J.: 1967, ?Review of Lawvere 1966,Mathematical Reviews,34(7332), 1354-5.
[40] Johnstone, P. T.: 1977,Topos Theory, Academic Press, London.
[41] Johnstone, P. T.: 1987,Notes on Logic and Set Theory, Cambridge University Press, Cambridge. · Zbl 0642.03001
[42] Kitcher, P.: 1983,The Nature of Mathematical Knowledge, Oxford University Press, Oxford. · Zbl 0519.00022
[43] Kock, A. and Reyes, G. E.: 1977, ?Doctrines of Categorical Logic?, in J. Barwise (ed.),Handbook of Mathematical Logic, North-Holland, Amsterdam.
[44] Kuyk, W.: 1977,Complementarity in Mathematics, D. Reidel, Boston. · Zbl 0361.02007
[45] Lambek, J.: 1992, ?Are the Traditional Philosophies of Mathematics really Incompatible??, Preprint, McGill University. · Zbl 0803.03002
[46] Lambek, J. and Couture, J.: 1991, ?Philosophical Reflections on the Foundations of Mathematics?,Erkenntnis 34, 187-209. Lambek, J. and Scott, P. J.: 1986,Introduction to Higher Order Categorical Logic, Cambridge University Press. · doi:10.1007/BF00385720
[47] Lawvere, W.: 1966, ?The Category of Categories as a Foundation for Mathematics?,Proceedings of La Jolla Conference on Categorical Algebra, Springer, New York, pp. 1-20. · Zbl 0192.09702
[48] Lawvere, W.: 1969, ?Adjointness in Foundations?,Dialectica 23, 281-96. · Zbl 0341.18002 · doi:10.1111/j.1746-8361.1969.tb01194.x
[49] Lawvere, W.: 1975, Introduction ofToposes, Algebraic Geometry and Logic, LNM 274, Springer-Verlag, New York, pp. 1-12.
[50] Levy, I.: 1979,Basic Set Theory, Springer Verlag, New York.
[51] Mac Lane, S.: 1969, ?One Universe as a Foundation for Category Theory?,LNM 106, Springer-Verlag, New York, pp. 192-9. · Zbl 0211.32202
[52] Mac Lane, S.: 1971, Categories for the Working Mathematician, Springer-Verlag. · Zbl 0232.18001
[53] McLarty, C.: 1990, ?The Uses and Abuses of the History of Topos Theory?,British Journal for the Philosophy of Science 41(3), 351-76. · Zbl 0709.18002 · doi:10.1093/bjps/41.3.351
[54] Maddy, P.: 1990,Realism in Mathematics, Clarendon Press, Oxford.
[55] Makkai, M. and Paré, R.: 1989, Accessible Categories: The Foundations of Categorical Model Theory, Contemporary Mathematics 104, American Mathematical Society, Providence. · Zbl 0703.03042
[56] Makkai, M. and Reyes, G.: 1977,First-Order Categorical Logic, LNM 611, Springer-Verlag, New York. · Zbl 0357.18002
[57] Marquis, J.-P.: 1993, ?Russell’s Logicism and Categorical Logicisms?, inRussell and Analytic Philosophy, A. Irvine and G. Weddekind (eds.), University of Toronto Press, Toronto, pp. 293-324.
[58] Mathias, A. R. D.: 1992, ?The Ignorance of Bourbaki?,The Mathematical Intelligencer 14(3), 4-13. · Zbl 0764.01009 · doi:10.1007/BF03025863
[59] Mayberry, J.: 1977, ?On the Consistency Problem for Set Theory: An Essay on the Cantorian Foundations of Classical Mathematics (I)?,The British Journal for the Philosophy of Science 28, 1-34. · Zbl 0395.03032 · doi:10.1093/bjps/28.1.1
[60] Meseguer, J.: 1989, ?General Logics?,Logic Colloquium 87, in H.-D. Ebbinghauset al. (eds.), North-Holland, pp. 275-329.
[61] Mitchell, W.: 1972, ?Boolean Topoi and the Theory of Sets?,Journal of Pure and Applied Algebra 2, 261-74. · Zbl 0245.18001 · doi:10.1016/0022-4049(72)90006-0
[62] Moerdijk, I. and Reyes, G.: 1991,Models for Smooth Infinitesimal Analysis, Springer-Verlag, New York. · Zbl 0715.18001
[63] Osius, G.: 1974, ?Categorical Set Theory: A Characterization of the Category of Sets?,Journal of Pure and Applied Algebra 4, 79-119. · Zbl 0282.02027 · doi:10.1016/0022-4049(74)90032-2
[64] Parsons, C.: 1983,Mathematics in Philosophy, Cornell University Press. · Zbl 0514.05037
[65] Piaget, J.: 1977, ?General Conclusions?,Epistemology and Psychology of Functions, by J. Piaget, Jean-Blaise Grize, Alina Szeminska and Vinh Bang, transl. by, F. X. Castellanos and V. D. Anderson, D. Reidel, Boston, pp. 167-96.
[66] Poincaré, H.: 1909, ?La Logique de l’Infini?, In. G. Heinzmann (ed.),Poincaré, Russell, Zermelo et Peano, 1986, Blanchard, Paris.
[67] Resnik, M.: 1981, ?Mathematics as a Science of Patterns: Ontology and Reference?,Noûs 15, 529-50. · Zbl 1366.00041
[68] Resnik, M.: 1982, ?Mathematics as a Science of Patterns: Epistemology?,Noûs 16, 95-105. · Zbl 1366.00042
[69] Resnik, M.: 1988, ?Mathematics from the Structural Point of View?,Revue Internationale de Philosophie 42(167), 400-24.
[70] Russell, B.: 1897,An Essay in the Foundations of Geometry, 2nd ed. 1956, Dover, New York. · Zbl 0075.15301
[71] Russell, B.: 1903,The Principles of Mathematics, W. W. Norton and Co, New York. · JFM 34.0062.14
[72] Russell, B.: 1907, ?The Regressive Method of Discovering the Premises of Mathematics?, in D. Lackey (ed.),Essays in Analysis, 1973, George Allen and Unwin, London, pp. 272-83.
[73] Shafarevich, I. R.: 1977,Basic Algebraic Geometry, Springer-Verlag, New York. · Zbl 0362.14001
[74] Shapiro, S.: 1989, ?Logic, Ontology and Mathematical Practice?,Synthese 79, 13-50. · Zbl 0757.03003 · doi:10.1007/BF00873255
[75] Shoenfield, J.: 1977, ?Axioms of Set Theory?, in J. Barwise (ed.),Handbook of Mathematical Logic, North-Holland, pp. 322-44.
[76] Steiner, M.: 1989, ?The Applications of Mathematics to Natural Sciences?,The Journal of Philosophy LXXXVI(9), 449-80. · doi:10.2307/2026759
[77] Tieszen, R. L.: 1989,Mathematical Intuition, Kluwer Academic Publishers, Dordrecht. · Zbl 0686.03003
[78] Wang, H.: 1974,From Mathematics to Philosophy, Routledge and Kegan Paul, London. · Zbl 0554.03002
[79] Wang, H.: 1983, ?The Concept of Set?, in P. Benacerraf and H. Putnam (eds.),Philosophy of Mathematics, 2nd ed., Cambridge: Cambridge University Press, pp. 520-70.
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