Element contents
Mathematical Structuralism
Published online by Cambridge University Press: 06 December 2018
Summary
The present work is a systematic study of five frameworks or perspectives articulating mathematical structuralism, whose core idea is that mathematics is concerned primarily with interrelations in abstraction from the nature of objects. The first two, set-theoretic and category-theoretic, arose within mathematics itself. After exposing a number of problems, the Element considers three further perspectives formulated by logicians and philosophers of mathematics: sui generis, treating structures as abstract universals, modal, eliminating structures as objects in favor of freely entertained logical possibilities, and finally, modal-set-theoretic, a sort of synthesis of the set-theoretic and modal perspectives.
Keywords
Information
- Type
- Element
- Information
- Online ISBN: 9781108582933Publisher: Cambridge University PressPrint publication: 29 November 2018
References
Assadian, B. [2017]. “The Semantic Plights of the Ante-Rem Structuralist.” Philosophical Studies, https://doi.org/10.1007/s11098-017–1001-7.CrossRefGoogle Scholar
Awodey, S. [1996]. “Structure in Mathematics and Logic: A Categorical Perspective.” Philosophia Mathematica, 4(3): pp. 209–237.10.1093/philmat/4.3.209CrossRefGoogle Scholar
Awodey, S.[2004]. “An Answer to Hellman’s Question: ‘Does Category Theory Provide a Framework for Mathematical Structuralism?’.” Philosophia Mathematica, 12(1): pp. 54–64.CrossRefGoogle Scholar
Bell, J. L. [1986]. “From Absolute to Local Mathematics.” Synthese, 69(3): pp. 409–426.10.1007/BF00413980CrossRefGoogle Scholar
Beltrami, E. [1868a]. “Saggio di interpretazione della geometria non euclidea.” Giornale di matematiche, 6, 284–312. [French trans. in Annales scientifiques de 1’ecole Normale Superieure, (I)6 (1869), pp. 251–288.1.]Google Scholar
Beltrami, E. [1868b]. “Teoria fondamentale digli spazii di curvatura costante.” Annuli di mathematica pura ed applicata, (2)2, 232–255. [French trans. in Annales scientifiques de 1’ecole Normale Superieure, (1)6 (1869), pp. 347–375.1.]10.1007/BF02419615CrossRefGoogle Scholar
Benacerraf, P. [1965]. “What Numbers Could Not Be,” reprinted in Benacerraf, P. and Putnam, H. (eds.), Philosophy of Mathematics: Selected Readings (Second Edition), Cambridge University Press, 1983, pp. 272–294.Google Scholar
Benacerraf, P. [1965]. “What Numbers Could Not Be,” reprinted in Benacerraf, P. and Putnam, H. (eds.), Philosophy of Mathematics: Selected Readings (Second Edition), Cambridge University Press, 1983, pp. 272–294.Google Scholar
Bernays, P. [1967]. “Hilbert, David,” in Edwards, P. (ed.), The Encyclopedia of Philosophy, Volume 3, Macmillan Publishing Company and The Free Press, New York, pp. 496–504.Google Scholar
Bolzano, B. [1950]. Paradoxes of the Infinite. Steele, D. A. (trans.), Routledge & Kegan Paul, London.Google Scholar
Boolos, G. [1971]. “The Iterative Conception of Set.” In Boolos, G., Logic, Logic, and Logic, Harvard University Press, 1998, pp. 13–29.Google Scholar
Burgess, J. P. [1999]. “Review of Shapiro [1997].” Notre Dame Journal of Formal Logic, 40(2): pp. 283–291.Google Scholar
Burgess, J. P. and Rosen, G. [1997]. A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics. Oxford University Press.Google Scholar
Burgess, J. P., Hazen, A., and Lewis, D. [1991]. “Appendix on Pairing.” In Lewis, D., Parts of Classes, Blackwell, Oxford, pp. 121–149.Google Scholar
Cantor, G. [1932]. Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, Zermelo, E. (ed.), Springer, Berlin.Google Scholar
Coffa, A. [1986]. “From Geometry to Tolerance: Sources of Conventionalism in Nineteenth-Century Geometry.” In Colodny, R. G. (ed.), From Quarks to Quasars: Philosophical Problems of Modern Physics, Pittsburgh University Press, Pittsburgh, pp. 3–70.10.2307/jj.5973221.5CrossRefGoogle Scholar
Coffa, A. [1991]. The Semantic Tradition from Kant to Carnap. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Dedekind, R. [1872]. “Stetigkeit und irrationale Zahlen,” translated as “Continuity and Irrational Numbers.” In Beman, W. W. (ed.), Essays on the Theory of Numbers, Dover Press, New York, 1963, pp. 1–27.Google Scholar
Dedekind, R. [1888]. “Was sind und was sollen die Zahlen?,” translated as “The Nature and Meaning of Numbers.” In Beman, W. W. (ed.), Essays on the Theory of Numbers, Dover Press, New York, 1963, pp. 31–115.Google Scholar
Dedekind, R. [1932]. Gesammelte mathematische Werke 3, Fricke, R., Noether, E., and Ore, O. (eds.), Vieweg, Brunswick.Google Scholar
Demopoulos, W. [1994]. “Frege, Hilbert, and the Conceptual Structure of Model Theory.” History and Philosophy of Logic,” 15(2): pp. 211–225.10.1080/01445349408837233CrossRefGoogle Scholar
Dummett, M. [1991]. Frege: Philosophy of Mathematics. Harvard University Press, Cambridge, MA.Google Scholar
Feferman, S. [1977]. “Categorical Foundations and Foundations of Category Theory.” In Butts, R. E. and Hintikka, J. (eds.), Logic, Foundations of Mathematics, and Computability Theory, D. Reidel, Dordrecht, pp. 149–169.10.1007/978-94-010-1138-9_9CrossRefGoogle Scholar
Feferman, S. and Hellman, G. [1995]. “Predicative Foundations of Arithmetic.” Journal of Philosophical Logic, 24(1): pp. 1–17.10.1007/BF01052728CrossRefGoogle Scholar
Frege, G. [1879]. “Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens,” translated in van Heijenoort [1967], pp. 1–82.Google Scholar
Frege, G. [1884]. The Foundations of Arithmetic. J. L. Austin (trans.), 2nd Edition. Harper, New York, 1960.Google Scholar
Frege, G. [1903b]. “Über die Grundlagen der Geometrie.” Jahresbericht der Mathematiker-Vereinigung, 12, pp. 319–324, 368–375.Google Scholar
Frege, G. [1906]. “Über die Grundlagen der Geometrie.” Jahresbericht der Mathematiker-Vereinigung, 15, pp. 293–309, 377–403, 423–430.Google Scholar
Frege, G. [1967]. Kleine Schriften. Darmstadt, Wissenschaftlicher Buchgesellschaft (with I. Angelelli).Google Scholar
Frege, G. [1971]. On the Foundations of Geometry and Formal Theories of Arithmetic. E.-H. W. Kluge (trans.), Yale University Press, New Haven, Connecticut.Google Scholar
Frege, G. [1976]. Wissenschaftlicher Briefwechsel. Gabriel, G., Hermes, H., Kambartel, F., and Thiel, C. (eds.), Felix Meiner, Hamburg.10.28937/978-3-7873-2549-8CrossRefGoogle Scholar
Frege, G. [1980]. Philosophical and Mathematical Correspondence. Basil Blackwell, Oxford.Google Scholar
Freudenthal, H. [1962]. “The Main Trends in the Foundations of Geometry in the 19th Century.” In Nagel, E., Suppes, P., and Tarski, A. (eds.), Logic, Methodology and Philosophy of Science: Proceedings of the 1960 Congress, Stanford University Press, Stanford, pp. 613–621.Google Scholar
Goldblatt, R. [2006]. Topoi: The Categorial Analysis of Logic (Revised Edition). Dover Publications.Google Scholar
Goldfarb, W. D. [1979]. “Logic in the Twenties: The Nature of the Quantifier.” Journal of Symbolic Logic, 44(3): pp. 351–368.CrossRefGoogle Scholar
Goodman, N. [1977]. The Structure of Appearance. 3rd Edition. D. Reidel.10.1007/978-94-010-1184-6CrossRefGoogle Scholar
Grassmann, H. [1972]. Gessammelte mathematische und physicalische Werke 1. Engels, F. (ed.), Johnson Reprint Corporation, New York.Google Scholar
Hale, B. [1996]. “Structuralism’s Unpaid Epistemological Debts.” Philosophia Mathematica, (3)4: pp. 124–147.10.1093/philmat/4.2.124CrossRefGoogle Scholar
Hallett, M. [1990]. “Physicalism, Reductionism and Hilbert.” In Irvine, A. D. (ed.), Physicalism in Mathematics, Kluwer Academic Publishers, Dordrecht, Netherlands, pp. 183–257.10.1007/978-94-009-1902-0_9CrossRefGoogle Scholar
Hallett, M. [1994]. “Hilbert’s Axiomatic Method and the Laws of Thought.” In George, A. (ed.), Mathematics and Mind, Oxford University Press, Oxford, pp. 158–200.10.1093/oso/9780195079296.003.0008CrossRefGoogle Scholar
Hellman, G. [1989]. Mathematics without Numbers: Towards a Modal-Structural Interpretation. Oxford University Press, Oxford.Google Scholar
Hellman, G. [1996]. “Structuralism without Structures.” Philosophia Mathematica, (3)4: pp. 100–123.10.1093/philmat/4.2.100CrossRefGoogle Scholar
Hellman, G. [2003]. “Does Category Theory Provide a Framework for Mathematical Structuralism?” Philosophia Mathematica, 11(2): pp. 129–157.10.1093/philmat/11.2.129CrossRefGoogle Scholar
Hellman, G.. [2005]. “Structuralism.” In Shapiro, S. (ed.), The Oxford Handbook of Philosophy of Mathematics and Logic, Oxford University Press, Oxford, pp. 536–562.10.1093/0195148770.003.0017CrossRefGoogle Scholar
Hellman, G. [2006]. “What Is Categorical Structuralism?” In van Benthem, J., Heinzmann, G., Rebuschi, M., and Visser, H. (eds.), The Age of Alternative Logics: Assessing Philosophy of Logic and Mathematics Today, Springer Netherlands, Dordrecht, pp. 151–161.10.1007/978-1-4020-5012-7_11CrossRefGoogle Scholar
Hellman, G. [forthcoming]. “Extending the Iterative Conception: A Height-Potentialist Perspective.”Google Scholar
Hellman, G. and Bell, J. L. [2006]. “Pluralism and the Foundations of Mathematics.” In Kellert, S. H., Longino, H. E., and Waters, C. K. (eds.), Scientific Pluralism, Minnesota Studies in the Philosophy of Science, Vol. XIX, University of Minnesota Press, Minneapolis, pp. 64–79.Google Scholar
Hilbert, D. [1899]. Grundlagen der Geometrie. Leipzig, Teubner; Foundations of Geometry, E. Townsend (trans.), Open Court, La Salle, Illinois, 1959.Google Scholar
Hilbert, D. [1900]. “Mathematische Probleme.” Bulletin of the American Mathematical Society 8 (1902), pp. 437–479.10.1090/S0002-9904-1902-00923-3CrossRefGoogle Scholar
Hilbert, D. [1905]. “Über der Grundlagen der Logik und der Arithmetik,” Verhandlungen des dritten internationalen Mathematiker-Kongresses in Heidelberg vom 8 bis 13 August 1904, Leipzig, Teubner, pp. 174–185; translated as “On the Foundations of Logic and Arithmetic,’‘ in van Heijenoort [1967], pp. 129–138.Google Scholar
Keränen, J. [2001]. “The Identity Problem for Realist Structuralism.” Philosophia Mathematica, 9(3): pp. 308–330.10.1093/philmat/9.3.308CrossRefGoogle Scholar
Kitcher, P. [1986]. “Frege, Dedekind, and the Philosophy of Mathematics.” In Haaparanta, L. and Hintikka, J. (eds.), Frege Synthesized, Reidel, D., Dordrecht, Holland, pp. 299–343.10.1007/978-94-009-4552-4_11CrossRefGoogle Scholar
Klein, F. [1921]. Gesammelte mathematische Abhandlungen 1, Springer, Berlin.10.1007/978-3-642-51960-4CrossRefGoogle Scholar
Lawvere, F. W. [1964]. “An Elementary Theory of the Category of Sets.” Proceedings of the National Academy of Sciences 52: pp. 1506–1511.10.1073/pnas.52.6.1506CrossRefGoogle ScholarPubMed
Lawvere, F. W. [1966] “The Category of Categories as a Foundation for Mathematics.” In Eilenberg, S., et al. (eds.), Proceedings of the Conference on Categorical Algebra: La Jolla 1965, Springer, Berlin, pp. 1–20.Google Scholar
Linnebo, Ø. [2013]. “The Potential Hierarchy of Sets.” Review of Symbolic Logic, 6(2): pp. 205–228.CrossRefGoogle Scholar
Linnebo, Ø. [forthcoming]. “Putnam on Mathematics as Modal Logic.” In Cook, R. and Hellman, G. (eds.), Putnam on Mathematics and Logic, Springer Verlag.Google Scholar
Linnebo, Ø. and Pettigrew, R. [2011]. “Category Theory as an Autonomous Foundation.” Philosophia Mathematica, 19(3): pp. 227–254.10.1093/philmat/nkr024CrossRefGoogle Scholar
Mac Lane, S. [1986]. Mathematics: Form and Function. Springer, Berlin.10.1007/978-1-4612-4872-9CrossRefGoogle Scholar
Mayberry, J. P. [2000]. The Foundations of Mathematics in the Theory of Sets. Cambridge University Press, Cambridge.Google Scholar
McCarty, D. C. [1995]. “The Mysteries of Richard Dedekind.” In Hintikka, J. (ed.), From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics, Synthese Library Series 251, Kluwer Academic Publishers, Dordrecht, Netherlands, pp. 53–96.10.1007/978-94-015-8478-4_4CrossRefGoogle Scholar
McLarty, C. [1991]. “Axiomatizing a Category of Categories.” The Journal of Symbolic Logic, 56(4): pp. 1243–1260.CrossRefGoogle Scholar
McLarty, C. [2004]. “Exploring Categorical Structuralism.” Philosophia Mathematica, 12(1): pp. 37–53.10.1093/philmat/12.1.37CrossRefGoogle Scholar
Nagel, E. [1939]. “The Formation of Modern Conceptions of Formal Logic in the Development of Geometry.” Osiris, Vol. 7, pp. 142–224.10.1086/368504CrossRefGoogle Scholar
Nagel, E. [1979]. “Impossible Numbers: A Chapter in the History of Modern Logic.” In Nagel, E. (ed.), Teleology Revisited and Other Essays in the Philosophy and History of Science, Columbia University Press, New York, pp. 166–194.10.7312/nage93038-010CrossRefGoogle Scholar
Parsons, C. [1990]. “The Structuralist View of Mathematical Objects.” Synthese, 84(3): pp. 303–346.10.1007/BF00485186CrossRefGoogle Scholar
Pasch, M. [1926]. Vorlesungen über neuere Geometrie (Zweite Auflage). Springer, Berlin.Google Scholar
Pettigrew, R. [2008]. “Platonism and Aristotelianism in Mathematics.” Philosophia Mathematica, 16(3): pp. 310–332.10.1093/philmat/nkm035CrossRefGoogle Scholar
Plücker, J. [1846]. System der Geometrie des Raumes in neuer analytischer Behandluungsweise, insbesondere die Theorie der Flächen zweiter Ordnung und Classe enthaltend. W. H. Scheller, Düsseldorf.Google Scholar
Poincaré, H. [1899]. “Des Fondements de la Géométrie.” Revue de Métaphysique et de Morale, 7, pp. 251–279.Google Scholar
Poincaré, H. [1900]. “Sur les Principes de la Géométrie?” Revue de Métaphysique et de Morale, 8, pp. 72–86.Google Scholar
Poincaré, H. [1908]. The Foundations of Science: Science and Hypothesis, The Value of Science, Science and Method. G. Halsted (trans.), The Science Press, New York, 1921, pp. 359–546.Google Scholar
Poncelet, J. V. [1862]. Applications d’analyse dt de geometrie, Mallett-Bachelier, Paris.Google Scholar
Quine, W. V. O. [1986]. Philosophy of Logic (Second Edition). Harvard University Press, Cambridge, MA.10.4159/9780674042445CrossRefGoogle Scholar
Resnik, M. D. [1980]. Frege and the Philosophy of Mathematics. Cornell University Press, Ithaca, NY.Google Scholar
Resnik, M. D. [1997]. Mathematics as a Science of Patterns. Oxford University Press, Oxford.Google Scholar
Russell, B. (1919)[1993]. Introduction to Mathematical Philosophy. Reprint by Dover, New York.Google Scholar
Scanlan, M. J. [1988]. “Beltrami’s Model and the Independence of the Parallel Postulate.” History and Philosophy of Logic, 9(1), pp. 13–34.Google Scholar
Shapiro, S. [1997]. Philosophy of Mathematics: Structure and Ontology. Oxford University Press, New York.Google Scholar
Shapiro, S. [2006a]. “Structure and Identity.” In MacBride, F. (ed.), Identity and Modality, Oxford University Press, Oxford, pp. 109–145.10.1093/oso/9780199285747.003.0006CrossRefGoogle Scholar
Shapiro, S. [2006b]. “The Governance of Identity.” In MacBride, F. (ed.), Identity and Modality, Oxford University Press, Oxford, pp. 164–173.10.1093/oso/9780199285747.003.0008CrossRefGoogle Scholar
Shapiro, S. [2008]. “Identity, Indiscernibility, and ante rem Structuralism: The Tale of i and –i.” Philosophia Mathematica, 16(3): pp. 285–309.10.1093/philmat/nkm042CrossRefGoogle Scholar
Shapiro, S. [2012]. “An ‘i’ for an i: Singular Terms, Uniqueness, and Reference.” Review of Symbolic Logic, 5(3): pp. 380–415.10.1017/S1755020311000347CrossRefGoogle Scholar
Shapiro, S. and Wright, C. [2006]. “All Things Indefinitely Extensible.” In Rayo, A. and Uzquiano, G. (eds.), Absolute Generality, Oxford University Press, Oxford, pp. 255–304.10.1093/oso/9780199276424.003.0010CrossRefGoogle Scholar
Stein, H. [1988]. “Logos, Logic, and Logistiké: Some Philosophical Remarks on the Nineteenth-Century Transformation of Mathematics.” In Aspray, W. and Kitcher, P. (eds.), History and Philosophy of Modern Mathematics, Minnesota Studies in the Philosophy of Science, Vol. XI, University of Minnesota Press, Minneapolis, pp. 238–259.Google Scholar
Tait, W. [1986]. “Truth and Proof: The Platonism of Mathematics.” Synthese, 69(3): pp. 341–370.10.1007/BF00413978CrossRefGoogle Scholar
van Heijenoort, J. [1967a]. From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Harvard University Press, Cambridge, MA.Google Scholar
van Heijenoort, J. [1967b]. “Logic as Calculus and Logic as Language.” Synthese, 17(3): pp. 324–330.10.1007/BF00485036CrossRefGoogle Scholar
von Neumann, J. [1925]. “An Axiomatization of Set Theory.” In van Heijenoort, J. (ed.), From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, Cambridge, MA, pp. 394–413.Google Scholar
Staudt, Von, Georg Christian, Karl. [1856–60]. Beitrage zur Geometric der Lage. F. Korn, Nürnberg.Google Scholar
Weyl, H. [1949]. Philosophy of Mathematics and Natural Science. Princeton University Press, Princeton (Revised and Augmented Edition, Athenaeum Press, New York, 1963).Google Scholar
Whitehead, A. N., and Russell, B. [1910]. Principia Mathematica 1. Cambridge University Press, Cambridge.Google Scholar
Wilson, M. [1992]. “Frege: The Royal Road from Geometry.” Noûs, 26(2): pp. 149–180.10.2307/2215733CrossRefGoogle Scholar
Zermelo, E. [1930]. “Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre.” Fundamenta Mathematicae, 16, pp. 29–47; translated as “On Boundary Numbers and Domains of Sets: New Investigations in the Foundations of Set Theory,” in W. Ewald (ed.), From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Volume 2, Oxford University Press, Oxford, 1996, pp. 1219–1233.10.4064/fm-16-1-29-47CrossRefGoogle Scholar
Accessibility standard: Unknown
Why this information is here
This section outlines the accessibility features of this content - including support for screen readers, full keyboard navigation and high-contrast display options. This may not be relevant for you.Accessibility Information
Accessibility compliance for the HTML of this element is currently unknown
and may be updated in the future.
- 23
- Cited by