Abstract logical structuralism
Article Sidebar
Main Article Content
Abstract
Structuralism has recently moved center stage in philosophy of mathematics. One of the issues discussed is the underlying logic of mathematical structuralism. In this paper, I want to look at the dual question, namely the underlying structures of logic. Indeed, from a mathematical structuralist standpoint, it makes perfect sense to try to identify the abstract structures underlying logic. We claim that one answer to this question is provided by categorical logic. In fact, we claim that the latter can be seen—and probably should be seen—as being a structuralist approach to logic and it is from this angle that categorical logic is best understood.
Article Details
References
Awodey, S., 1996. Structure in mathematics and logic: a categorical perspective. Philos. Math. (3) [Online], 4(3), pp.209–237. Available at: https://doi.org/10.1093/philmat/4.3.209.
Barr, M. and Wells, C., 1990. Category Theory for Computing Science [Online], Prentice Hall International Series in Computer Science. New York: Prentice Hall International, pp.xvi+432. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=1094561>.
Barr, M. and Wells, C., 2005. Toposes, triples and theories. Repr. Theory Appl. Categ. [Online], (12), pp.x+288. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=2178101>.
Baues, H.-J., 2002. Atoms of topology. Jahresber. Deutsch. Math.-Verein., 104(4), pp.147–164.
Bierman, G.M., 1995. What is a categorical model of intuitionistic linear logic? Typed Lambda Calculi and Applications (Edinburgh, 1995) [Online]. Vol. 902, Lecture Notes in Comput. Sci. Springer, Berlin, pp.78–93. Available at: https://doi.org/10.1007/BFb0014046.
Birkhoff, G., 1940. Lattice Theory [Online]. New York: American Mathematical Society, pp.v+155. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=0001959>.
Blute, R. and Scott, P., 2004. Category theory for linear logicians. Linear Logic in Computer Science [Online]. Vol. 316, London Math. Soc. Lecture Note Ser. Cambridge: Cambridge Univ. Press, pp.3–64. Available at: https://doi.org/10.1017/CBO9780511550850.002.
Boileau, A. and Joyal, A., 1981. La logique des topos. J. Symbolic Logic [Online], 46(1), pp.6–16. Available at: https://doi.org/10.2307/2273251.
Borceux, F., 1994. Handbook of Categorical Algebra. 2 [Online]. Vol. 51, Encyclopedia of Mathematics and its Applications. Cambridge: Cambridge University Press, pp.xviii+443. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=1313497>.
Caramello, O., 2014. Topologies for intermediate logics. MLQ. Mathematical Logic Quarterly [Online], 60(4-5), pp.335–347. Available at: https://doi.org/10.1002/malq.201200059.
Caramello, O., 2018. Theories, sites, toposes. Oxford: Oxford University Press.
Corry, L., 2004. Modern Algebra and the Rise of Mathematical Structures. Second. Basel: Birkhauser Verlag, pp.xvi+451.
Ehresmann, C., 1967. Sur les structures algebriques. C. R. Acad. Sci. Paris Ser. A-B [Online], 264, A840–A843. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=217140>.
Ehresmann, C., 1968. Esquisses et types des structures algebriques. Bul. Inst. Politehn. Ia，si (N.S.) [Online], 14(18)(fasc., fasc. 1-2), pp.1–14. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=238918>.
Eilenberg, S. and Mac Lane, S., 1945. A General Theory of Natural Equivalences. Trans. Amer. Math. Soc., 58, pp.231–294.
Freyd, P., 1966. Representations in Abelian Categories. Proceedings of the Conference on Categorical Algebra, La Jolla, Calif., 1965. New York: Springer-Verlag, pp.95–120.
Girard, J.-Y., 1987. Linear logic. Theoret. Comput. Sci. [Online], 50(1), p.101. Available at: https://doi.org/10.1016/0304-3975(87)90045-4.
Halmos, P.R., 1954. Polyadic Boolean algebras. Proc. Nat. Acad. Sci. U.S.A. [Online], 40, pp.296–301. Available at: https://doi.org/10.1073/pnas.40.5.296.
Halmos, P.R., 1956a. Algebraic logic. I. Monadic Boolean algebras. Compositio Math. [Online], 12, pp.217–249. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=78304>.
Halmos, P.R., 1956b. Algebraic logic. II. Homogeneous locally finite polyadic Boolean algebras of infinite degree. Fund. Math. [Online], 43, pp.255–325. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=86029>.
Halmos, P.R., 1956c. Algebraic logic. III. Predicates, terms, and operations in polyadic algebras. Trans. Amer. Math. Soc. [Online], 83, pp.430–470. Available at: https://doi.org/10.2307/1992883.
Halmos, P.R., 1956d. The basic concepts of algebraic logic. Amer. Math. Monthly [Online], 63, pp.363–387. Available at: https://doi.org/10.2307/2309396.
Halmos, P.R., 1962. Algebraic Logic [Online]. New York: Chelsea Publishing Co., p.271. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=0131961>.
Hellman, G., 2001. Three varieties of mathematical structuralism. Philos. Math. (3) [Online], 9(2), pp.184–211. Available at: https://doi.org/10.1093/philmat/9.2.184.
Hellman, G., 2003. Does category theory provide a framework for mathematical structuralism? Philos. Math. (3) [Online], 11(2), pp.129–157. Available at: https://doi.org/10.1093/philmat/11.2.129.
Hellman, G. and Shapiro, S., 2019. Mathematical Structuralism [Online], Elements in the Philosophy of Mathematics. Cambridge: Cambridge University Press, p.92. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=3967722>.
Henkin, L., Monk, J.D. and Tarski, A., 1971. Cylindric Algebras. Part I. With an Introductory chapter: General Theory of Algebras [Online]. Amsterdam; London: North-Holland Publishing Co., pp.vi+508. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=0314620>.
Henkin, L. and Tarski, A., 1961. Cylindric algebras. Proc. Sympos. Pure Math., Vol. II [Online]. Providence, R.I.: American Mathematical Society, pp.83–113. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=0124250>.
Johnstone, P.T., 2002. Sketches of an Elephant: A Topos Theory Compendium. Vol. 1 [Online]. Vol. 43, Oxford Logic Guides. New York: The Clarendon Press, Oxford University Press, pp.xxii+468+71. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=1953060>.
Jónsson, B. and Tarski, A., 1951. Boolean algebras with operators. I. Amer. J. Math. [Online], 73, pp.891–939. Available at: https://doi.org/10.2307/2372123.
Jónsson, B. and Tarski, A., 1952. Boolean algebras with operators. II. Amer. J. Math. [Online], 74, pp.127–162. Available at: https://doi.org/10.2307/2372074.
Kapulkin, K. and Lumsdaine, P.L., 2018. The homotopy theory of type theories. Adv. Math. [Online], 337, pp.1–38. Available at: https://doi.org/10.1016/j.aim.2018.08.003.
Krömer, R., 2007. Tool and Object [Online]. Vol. 32, Science Networks. Historical Studies. Basel: Birkhäuser Verlag, pp.xxxvi+367. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=2272843>.
Lafont, Y., 1988. The linear abstract machine. Vol. 59, 1-2, pp.157–180. Available at: https://doi.org/10.1016/0304-3975(88)90100-4.
Lair, C., 1981. Catégories modelables et catégories esquissables. Diagrammes [Online], 6, pp.L1–L20. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=684535>.
Lair, C., 2001. Éléments de théorie des esquisses. I. Graphes à composition. Diagrammes [Online], 45/46, pp.3–33. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=2102295>.
Lair, C., 2002. Éléments de théorie des esquisses. II. Systèmes tensoriels et systèmes enrichis de graphes à composition. Diagrammes [Online], 47/48, p.34. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=2104013>.
Lair, C., 2003. Éléments de théorie des esquisses. III. Esquisses. Diagrammes [Online], 49/50, p.58. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=2104014>.
Lambek, J., 1968a. A Fixpoint Theorem for Complete Categories. Math. Z., 103, pp.151–161.
Lambek, J., 1968b. Deductive Systems and Categories. I. Syntactic Calculus and Residuated Categories. Math. Systems Theory, 2, pp.287–318.
Lambek, J., 1969. Deductive Systems and Categories II. Standard Constructions and Closed Categories. In: P.J. Hilton, ed. Category Theory, Homology Theory, and their Applications, I. Vol. 86, Lecture Notes in Mathematics. Berlin: Springer-Verlag, pp.76–122.
Lambek, J., 1972. Deductive Systems and Categories III. Cartesian Closed Categories, Intuitionistic Propositional Calculus, and Combinatory Logic. In: F.W. Lawvere, ed. Toposes, Algebraic Geometry and Logic. Vol. 274, Lecture Notes in Mathematics. Berlin: Springer-Verlag, pp.57–82.
Lambek, J. and Scott, P.J., 1988. Introduction to Higher Order Categorical Logic [Online]. Vol. 7, Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, pp.x+293. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=939612>.
Landry, E. and Marquis, J.-P., 2005. Categories in Context: Historical, Foundational and Philosophical. Philos. Math. (3), 13(1), pp.1–43.
Lawvere, F.W., 1963. Functorial semantics of algebraic theories. Proc. Nat. Acad. Sci. U.S.A. [Online], 50, pp.869–872. Available at: https://doi.org/10.1073/pnas.50.5.869.
Lawvere, F.W., 1966. Functorial Semantics of Elementary Theories. Journal of Symbolic Logic, 31, pp.294–295.
Lawvere, F.W., 1967. Theories as Categories and the Completeness Theorem. Journal of Symbolic Logic, 32, p.562.
Lawvere, F.W., 1969a. Adjointness in Foundations. Dialectica, 23, pp.281–296.
Lawvere, F.W., 1969b. Diagonal arguments and cartesian closed categories. Category Theory, Homology Theory and their Applications, II (Battelle Institute Conference, Seattle, Wash., 1968, Vol. Two) [Online]. Berlin: Springer, pp.134–145. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=0242748>.
Lawvere, F.W., 1970. Equality in hyperdoctrines and comprehension schema as an adjoint functor. Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVII, New York, 1968) [Online]. Providence, R.I.: Amer. Math. Soc., pp.1–14. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=0257175>.
Lawvere, F.W., 1971. Quantifiers and sheaves. Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1 [Online], pp.329–334. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=0430021>.
Linton, F.E.J., 1966. Some aspects of equational categories. Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965). New York: Springer, pp.84–94.
Lurie, J., 2019. Ultracategories. [unpublished].
Mac Lane, S., 1965. Categorical Algebra. Bull. Amer. Math. Soc., 71, pp.40–106.
Mac Lane, S., 1998. Categories for the Working Mathematician. 2nd ed. Vol. 5, Graduate Texts in Mathematics. New York: Springer-Verlag.
Mac Lane, S. and Moerdijk, I., 1994. Sheaves in Geometry and Logic [Online], Universitext. New York: Springer-Verlag, pp.xii+629. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=1300636>.
Maietti, M.E., 2010. Joyal’s arithmetic universe as list-arithmetic pretopos. Theory Appl. Categ. [Online], 24, No. 3, 39–83. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=2610177>.
Makkai, M., 1988. Strong conceptual completeness for first-order logic. Ann. Pure Appl. Logic [Online], 40(2), pp.167–215. Available at: https://doi.org/10.1016/0168-0072(88)90019-X.
Makkai, M., 1990. A theorem on Barr-exact categories, with an infinitary generalization. Ann. Pure Appl. Logic [Online], 47(3), pp.225–268. Available at: https://doi.org/10.1016/0168-0072(90)90036-2.
Makkai, M., 1997a. Generalized sketches as a framework for completeness theorems. I. J. Pure Appl. Algebra [Online], 115(1), pp.49–79. Available at: https://doi.org/10.1016/S0022-4049(96)00007-2.
Makkai, M., 1997b. Generalized sketches as a framework for completeness theorems. II. J. Pure Appl. Algebra [Online], 115(2), pp.179–212. Available at: https://doi.org/10.1016/S0022-4049(96)00008-4.
Makkai, M., 1997c. Generalized sketches as a framework for completeness theorems. III. J. Pure Appl. Algebra [Online], 115(3), pp.241–274. Available at: https://doi.org/10.1016/S0022-4049(96)00009-6.
Makkai, M. and Paré, R., 1989. Accessible Categories: The Foundations of Categorical Model Theory [Online]. Vol. 104, Contemporary Mathematics. Providence, R.I.: American Mathematical Society, pp.viii+176. Available at: https://doi.org/10.1090/conm/104.
Makkai, M. and Reyes, G.E., 1977. First Order Categorical Logic [Online], Lecture Notes in Mathematics, Vol. 611. Berlin; New York: Springer-Verlag, pp.viii+301. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=0505486>.
Marquis, J.-P., 2009. From a Geometrical Point of View [Online]. Vol. 14, Logic, Epistemology, and the Unity of Science. Springer, Dordrecht, pp.x+309. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=2730089>.
Marquis, J.-P., 2015. Mathematical abstraction, conceptual variation and identity. In: P.-E.B.G.H.W.H.P. Schroeder-Heister, ed. Logic, Methodology and Philosophy of Science, Proceedings of the Fourteen International Congress. London: College Publications, pp.299–322.
Marquis, J.-P., 2016. Stairway to Heaven: The Abstract Method and Levels of Abstraction in Mathematics. The Mathematical Intelligencer, 38(3), pp.41–51.
Marquis, J.-P. and Reyes, G.E., 2012. The history of categorical logic: 1963–1977. Sets and Extensions in the Twentieth Century [Online]. Vol. 6, Handb. Hist. Log. Amsterdam: Elsevier/North-Holland, pp.689–800. Available at: https://doi.org/10.1016/B978-0-444-51621-3.50010-4.
McKinsey, J.C.C. and Tarski, A., 1944. The algebra of topology. Ann. of Math. (2) [Online], 45, pp.141–191. Available at: https://doi.org/10.2307/1969080.
McKinsey, J.C.C. and Tarski, A., 1946. On closed elements in closure algebras. Ann. of Math. (2) [Online], 47, pp.122–162. Available at: https://doi.org/10.2307/1969038.
McKinsey, J.C.C. and Tarski, A., 1948. Some theorems about the sentential calculi of Lewis and Heyting. J. Symbolic Logic [Online], 13, pp.1–15. Available at: https://doi.org/10.2307/2268135.
Melliès, P.A., n.d. A Functorial Excursion Between Algebraic Geometry and Linear Logic.
Melliès, P.-A., 2009. Categorical semantics of linear logic. In: P.-L. Curien, H. Herbelin, J.-L. Krivine and P.-A. Méllies, eds. Interactive models of computation and program behaviour, Panoramas et Synthèses 27. Société Mathématique de France, pp.1–196.
Mostowski, A., 1949. Sur l’interpretation géométrique et topologique des notions logiques. Library of the Tenth International Congress of Philosophy, Amsterdam, August 11–18, 1948, Vol. I, Proceedings of the Congress [Online]. publisher unknown, pp.767–769. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=0028277>.
de Paiva, V., 2014. Categorical semantics of linear logic for all. Advances in Natural Deduction [Online]. Vol. 39, Trends Log. Stud. Log. Libr. Springer, Dordrecht, pp.181–192. Available at: https://doi.org/10.1007/978-94-007-7548-0_9.
Pavlovi´c, D., 1992. On the structure of paradoxes. Arch. Math. Logic [Online], 31(6), pp.397–406. Available at: https://doi.org/10.1007/BF01277482.
Pitts, A.M., 1989. Conceptual completeness for first-order intuitionistic logic: an application of categorical logic. Ann. Pure Appl. Logic [Online], 41(1), pp.33–81. Available at: https://doi.org/10.1016/0168-0072(89)90007-9.
Rasiowa, H., 1951. Algebraic treatment of the functional calculi of Heyting and Lewis. Fund. Math. [Online], 38, pp.99–126. Available at: https://doi.org/10.4064/fm-38-1-99-126.
Rasiowa, H., 1955. Algebraic models of axiomatic theories. Fund. Math. [Online], 41, pp.291–310. Available at: https://doi.org/10.4064/fm-41-2-291-310.
Rasiowa, H. and Sikorski, R., 1950. A proof of the completeness theorem of Gödel. Fund. Math. [Online], 37, pp.193–200. Available at: https://doi.org/10.4064/fm-37-1-193-200.
Rasiowa, H. and Sikorski, R., 1953. Algebraic treatment of the notion of satisfiability. Fund. Math. [Online], 40, pp.62–95. Available at: https://doi.org/10.4064/fm-40-1-62-95.
Rasiowa, H. and Sikorski, R., 1955. An application of lattices to logic. Fund. Math. [Online], 42, pp.83–100. Available at: https://doi.org/10.4064/fm-42-1-83-100.
Rasiowa, H. and Sikorski, R., 1963. The Mathematics of Metamathematics [Online], Monografie Matematyczne, Tom 41. Pa´nstwowe Wydawnictwo Naukowe, Warsaw, p.522. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=0163850>.
Reck, E.H. and Schiemer, G., eds., 2020. The Prehistory of Mathematical Structuralism. Oxford: Oxford University Press.
Riehl, E., 2017. Category Theory in Context. Mineola: Dover Publications.
Rodin, A., 2014. Axiomatic Method and Category Theory [Online]. Vol. 364, Synthese Library. Studies in Epistemology, Logic, Methodology, and Philosophy of Science. Cham: Springer, pp.xii+285. Available at: https://doi.org/10.1007/978-3-319-00404-4.
Seely, R.A.G., 1989. Linear logic, ∗-autonomous categories and cofree coalgebras. Categories in Computer Science and Logic (Boulder, CO, 1987) [Online]. Vol. 92, Contemp. Math. Amer. Math. Soc., Providence, RI, pp.371–382. Available at: https://doi.org/10.1090/conm/092/1003210.
The Univalent Foundations Program, 2013. Homotopy Type Theory—Univalent Foundations of Mathematics [Online]. Princeton, N.J.: The Univalent Foundations Program; Institute for Advanced Study (IAS), pp.xiv+589. Available at: <https://mathscinet.ams.org/mathscinet-getitem?mr=3204653>.
Yanofsky, N.S., 2003. A universal approach to self-referential paradoxes, incompleteness and fixed points. Bull. Symbolic Logic [Online], 9(3), pp.362–386. Available at: https://doi.org/10.2178/bsl/1058448677.