Skip to main content Accessibility help
×
×
Home
Mathematical Structuralism

Mathematical Structuralism

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.

  • Copyright

  • COPYRIGHT: © Geoffrey Hellman and Stewart Shapiro 2019

References

Hide all
Assadian, B. Assadian, B. [2017]. “The Semantic Plights of the Ante-Rem Structuralist.” Philosophical Studies, https://doi.org/10.1007/s11098-017–1001-7. CrossRef | Google Scholar
Awodey, S. Awodey, S. [1996]. “Structure in Mathematics and Logic: A Categorical Perspective.” Philosophia Mathematica, 4(3): pp. 209–237. CrossRef | Google Scholar
Awodey, S. 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. CrossRef | Google Scholar
Bell, J. L. Bell, J. L. [1986]. “From Absolute to Local Mathematics.” Synthese, 69(3): pp. 409–426. CrossRef | Google Scholar
Beltrami, E. 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. 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.] CrossRef | Google Scholar
Beltrami, E. Beltrami, E. [1902]. Opere matematiche. Hoepli, Milan. Google Scholar
Benacerraf, P. Benacerraf, P. Putnam, H. Benacerraf, P. [1965]. “What Numbers Could Not Be,” reprinted in P. Benacerraf and H. Putnam (eds.), Philosophy of Mathematics: Selected Readings ( Second Edition), Cambridge University Press, 1983, pp. 272–294. Google Scholar
Benacerraf, P. Benacerraf, P. Putnam, H. Benacerraf, P. [1965]. “What Numbers Could Not Be,” reprinted in P. Benacerraf and H. Putnam (eds.), Philosophy of Mathematics: Selected Readings ( Second Edition), Cambridge University Press, 1983, pp. 272–294. Google Scholar
Bernays, P. Edwards, P. Bernays, P. [1967]. “Hilbert, David,” in P. Edwards (ed.), The Encyclopedia of Philosophy, Volume 3, Macmillan Publishing Company and The Free Press, New York, pp. 496–504. Google Scholar
Bolzano, B. Steele, D. A. Bolzano, B. [1950]. Paradoxes of the Infinite. D. A. Steele (trans.), Routledge & Kegan Paul, London. Google Scholar
Boolos, G. Boolos, G. Boolos, G. [1971]. “The Iterative Conception of Set.” In G. Boolos , Logic, Logic, and Logic, Harvard University Press, 1998, pp. 13–29. Google Scholar
Burgess, J. P. Burgess, J. P. [1999]. “Review of Shapiro [1997].” Notre Dame Journal of Formal Logic, 40(2): pp. 283–291. CrossRef | Google Scholar
Burgess, J. P. Rosen, G. 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. Lewis, D. Lewis, D. Burgess, J. P. , Hazen, A. , and Lewis, D. [1991]. “Appendix on Pairing.” In D. Lewis , Parts of Classes, Blackwell, Oxford, pp. 121–149. Google Scholar
Cantor, G. Zermelo, E. 87 Cantor, G. [1932]. Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, E. Zermelo (ed.), Springer, Berlin. Google Scholar
Coffa, A. Colodny, R. G. Coffa, A. [1986]. “From Geometry to Tolerance: Sources of Conventionalism in Nineteenth-Century Geometry.” In R. G. Colodny (ed.), From Quarks to Quasars: Philosophical Problems of Modern Physics, Pittsburgh University Press, Pittsburgh, pp. 3–70. Google Scholar
Coffa, A. Coffa, A. [1991]. The Semantic Tradition from Kant to Carnap. Cambridge University Press, Cambridge. CrossRef | Google Scholar
Dedekind, R. Beman, W. W. Dedekind, R. [1872]. “Stetigkeit und irrationale Zahlen,” translated as “Continuity and Irrational Numbers.” In W. W. Beman (ed.), Essays on the Theory of Numbers, Dover Press, New York, 1963, pp. 1–27. Google Scholar
Dedekind, R. Beman, W. W. Dedekind, R. [1888]. “Was sind und was sollen die Zahlen?,” translated as “The Nature and Meaning of Numbers.” In W. W. Beman (ed.), Essays on the Theory of Numbers, Dover Press, New York, 1963, pp. 31–115. Google Scholar
Dedekind, R. Fricke, R. Noether, E. Ore, O. Dedekind, R. [1932]. Gesammelte mathematische Werke 3, R. Fricke , E. Noether , and O. Ore (eds.), Vieweg, Brunswick. Google Scholar
Demopoulos, W. Demopoulos, W. [1994]. “Frege, Hilbert, and the Conceptual Structure of Model Theory.” History and Philosophy of Logic,” 15(2): pp. 211–225. CrossRef | Google Scholar
Drake, F. R. Drake, F. R. [1974]. Set Theory: An Introduction to Large Cardinals. North Holland. Google Scholar
Dummett, M. Dummett, M. [1991]. Frege: Philosophy of Mathematics. Harvard University Press, Cambridge, MA. Google Scholar
Feferman, S. Butts, R. E. Hintikka, J. Feferman, S. [1977]. “Categorical Foundations and Foundations of Category Theory.” In R. E. Butts and J. Hintikka (eds.), Logic, Foundations of Mathematics, and Computability Theory, D. Reidel, Dordrecht, pp. 149–169. CrossRef | Google Scholar
Feferman, S. Hellman, G. Feferman, S. and Hellman, G. [1995]. “Predicative Foundations of Arithmetic.” Journal of Philosophical Logic, 24(1): pp. 1–17. CrossRef | Google Scholar
Frege, G. Frege, G. [1879]. “Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens,” translated in van Heijenoort [1967], pp. 1–82. Google Scholar
Frege, G. Frege, G. [1884]. The Foundations of Arithmetic. J. L. Austin (trans.), 2nd Edition. Harper, New York, 1960. Google Scholar
Frege, G. Frege, G. [1903a]. Grundgesetze der Arithmetik 2. Olms, Hildescheim. Google Scholar
Frege, G. Frege, G. [1903b]. “Über die Grundlagen der Geometrie.” Jahresbericht der Mathematiker-Vereinigung, 12, pp. 319–324, 368–375. Google Scholar
Frege, G. Frege, G. [1906]. “Über die Grundlagen der Geometrie.” Jahresbericht der Mathematiker-Vereinigung, 15, pp. 293–309, 377–403, 423–430. Google Scholar
88Frege, G. 88Frege, G. [1967]. Kleine Schriften. Darmstadt, Wissenschaftlicher Buchgesellschaft (with I. Angelelli). Google Scholar
Frege, G. 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. Gabriel, G. Hermes, H. Kambartel, F. Thiel, C. Frege, G. [1976]. Wissenschaftlicher Briefwechsel. G. Gabriel , H. Hermes , F. Kambartel , and C. Thiel (eds.), Felix Meiner, Hamburg. Google Scholar
Frege, G. Frege, G. [1980]. Philosophical and Mathematical Correspondence. Basil Blackwell, Oxford. Google Scholar
Freudenthal, H. Nagel, E. Suppes, P. Tarski, A. Freudenthal, H. [1962]. “The Main Trends in the Foundations of Geometry in the 19th Century.” In E. Nagel , P. Suppes , and A. Tarski (eds.), Logic, Methodology and Philosophy of Science: Proceedings of the 1960 Congress, Stanford University Press, Stanford, pp. 613–621. Google Scholar
Goldblatt, R. Goldblatt, R. [2006]. Topoi: The Categorial Analysis of Logic (Revised Edition). Dover Publications. Google Scholar
Goldfarb, W. D. Goldfarb, W. D. [1979]. “Logic in the Twenties: The Nature of the Quantifier.” Journal of Symbolic Logic, 44(3): pp. 351–368. CrossRef | Google Scholar
Goodman, N. Goodman, N. [1977]. The Structure of Appearance. 3rd Edition. D. Reidel. CrossRef | Google Scholar
Grassmann, H. Engels, F. Grassmann, H. [1972]. Gessammelte mathematische und physicalische Werke 1. F. Engels (ed.), Johnson Reprint Corporation, New York. Google Scholar
Hale, B. Hale, B. [1996]. “Structuralism’s Unpaid Epistemological Debts.” Philosophia Mathematica, (3)4: pp. 124–147. CrossRef | Google Scholar
Hallett, M. Irvine, A. D. Hallett, M. [1990]. “Physicalism, Reductionism and Hilbert.” In A. D. Irvine (ed.), Physicalism in Mathematics, Kluwer Academic Publishers, Dordrecht, Netherlands, pp. 183–257. CrossRef | Google Scholar
Hallett, M. George, A. Hallett, M. [1994]. “Hilbert’s Axiomatic Method and the Laws of Thought.” In A. George (ed.), Mathematics and Mind, Oxford University Press, Oxford, pp. 158–200. Google Scholar
Hellman, G. Hellman, G. [1989]. Mathematics without Numbers: Towards a Modal-Structural Interpretation. Oxford University Press, Oxford. Google Scholar
Hellman, G. Hellman, G. [1996]. “Structuralism without Structures.” Philosophia Mathematica, (3)4: pp. 100–123. Google Scholar
Hellman, G. Hellman, G. [2003]. “Does Category Theory Provide a Framework for Mathematical Structuralism?” Philosophia Mathematica, 11(2): pp. 129–157. CrossRef | Google Scholar
Hellman, G. Shapiro, S. Hellman, G. . [2005]. “Structuralism.” In S. Shapiro (ed.), The Oxford Handbook of Philosophy of Mathematics and Logic, Oxford University Press, Oxford, pp. 536–562. CrossRef | Google Scholar
Hellman, G. van Benthem, J. Heinzmann, G. Rebuschi, M. Visser, H. Hellman, G. [2006]. “What Is Categorical Structuralism?” In J. van Benthem , G. Heinzmann , M. Rebuschi , and H. Visser (eds.), The Age of Alternative Logics: Assessing Philosophy of Logic and Mathematics Today, Springer Netherlands, Dordrecht, pp. 151–161. CrossRef | Google Scholar
89Hellman, G. 89Hellman, G. [forthcoming]. “Extending the Iterative Conception: A Height-Potentialist Perspective.” Google Scholar
Hellman, G. Bell, J. L. Kellert, S. H. Longino, H. E. Waters, C. K. Hellman, G. and Bell, J. L. [2006]. “Pluralism and the Foundations of Mathematics.” In S. H. Kellert , H. E. Longino , and C. K. Waters (eds.), Scientific Pluralism, Minnesota Studies in the Philosophy of Science, Vol. XIX, University of Minnesota Press, Minneapolis, pp. 64–79. Google Scholar
Hilbert, D. Hilbert, D. [1899]. Grundlagen der Geometrie . Leipzig, Teubner; Foundations of Geometry, E. Townsend (trans.), Open Court, La Salle, Illinois, 1959. Google Scholar
Hilbert, D. Hilbert, D. [1900]. “Mathematische Probleme.” Bulletin of the American Mathematical Society 8 (1902), pp. 437–479. CrossRef | Google Scholar
Hilbert, D. 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
Hilbert, D. Hilbert, D. [1935]. Gesammelte Abhandlungen, Dritter Band. Julius Springer, Berlin. Google Scholar
Keränen, J. Keränen, J. [2001]. “The Identity Problem for Realist Structuralism.” Philosophia Mathematica, 9(3): pp. 308–330. CrossRef | Google Scholar
Kitcher, P. Haaparanta, L. Hintikka, J. Reidel, D. Kitcher, P. [1986]. “Frege, Dedekind, and the Philosophy of Mathematics.” In L. Haaparanta and J. Hintikka (eds.), Frege Synthesized, D. Reidel , Dordrecht, Holland, pp. 299–343. CrossRef | Google Scholar
Klein, F. Klein, F. [1921]. Gesammelte mathematische Abhandlungen 1, Springer, Berlin. CrossRef | Google Scholar
Lawvere, F. W. Lawvere, F. W. [1964]. “An Elementary Theory of the Category of Sets.” Proceedings of the National Academy of Sciences 52 : pp. 1506–1511. CrossRef | Google Scholar
Lawvere, F. W. Eilenberg, S. Lawvere, F. W. [1966] “The Category of Categories as a Foundation for Mathematics.” In S. Eilenberg , et al. (eds.), Proceedings of the Conference on Categorical Algebra: La Jolla 1965, Springer, Berlin, pp. 1–20. Google Scholar
Linnebo, Ø. Linnebo, Ø. [2013]. “The Potential Hierarchy of Sets.” Review of Symbolic Logic, 6(2): pp. 205–228. CrossRef | Google Scholar
Linnebo, Ø. Cook, R. Hellman, G. Linnebo, Ø. [forthcoming]. “Putnam on Mathematics as Modal Logic.” In R. Cook and G. Hellman (eds.), Putnam on Mathematics and Logic, Springer Verlag. Google Scholar
Linnebo, Ø. Pettigrew, R. Linnebo, Ø. and Pettigrew, R. [2011]. “Category Theory as an Autonomous Foundation.” Philosophia Mathematica, 19(3): pp. 227–254. CrossRef | Google Scholar
Mac Lane, S. Mac Lane, S. [1986]. Mathematics: Form and Function. Springer, Berlin. CrossRef | Google Scholar
Mayberry, J. P. Mayberry, J. P. [2000]. The Foundations of Mathematics in the Theory of Sets. Cambridge University Press, Cambridge. Google Scholar
McCarty, D. C. Hintikka, J. 90 McCarty, D. C. [1995]. “The Mysteries of Richard Dedekind.” In J. Hintikka (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. CrossRef | Google Scholar
McLarty, C. McLarty, C. [1991]. “Axiomatizing a Category of Categories.” The Journal of Symbolic Logic, 56(4): pp. 1243–1260. CrossRef | Google Scholar
McLarty, C. McLarty, C. [2004]. “Exploring Categorical Structuralism.” Philosophia Mathematica, 12(1): pp. 37–53. CrossRef | Google Scholar
Nagel, E. Nagel, E. [1939]. “The Formation of Modern Conceptions of Formal Logic in the Development of Geometry.” Osiris, Vol. 7, pp. 142–224. CrossRef | Google Scholar
Nagel, E. Nagel, E. Nagel, E. [1979]. “Impossible Numbers: A Chapter in the History of Modern Logic.” In E. Nagel (ed.), Teleology Revisited and Other Essays in the Philosophy and History of Science, Columbia University Press, New York, pp. 166–194. Google Scholar
Parsons, C. Parsons, C. [1990]. “The Structuralist View of Mathematical Objects.” Synthese, 84(3): pp. 303–346. CrossRef | Google Scholar
Pasch, M. Pasch, M. [1926]. Vorlesungen über neuere Geometrie (Zweite Auflage). Springer, Berlin. Google Scholar
Pettigrew, R. Pettigrew, R. [2008]. “Platonism and Aristotelianism in Mathematics.” Philosophia Mathematica, 16(3): pp. 310–332. CrossRef | Google Scholar
Plücker, J. 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. Poincaré, H. [1899]. “Des Fondements de la Géométrie.” Revue de Métaphysique et de Morale, 7, pp. 251–279. Google Scholar
Poincaré, H. Poincaré, H. [1900]. “Sur les Principes de la Géométrie?” Revue de Métaphysique et de Morale, 8, pp. 72–86. Google Scholar
Poincaré, H. 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. Poncelet, J. V. [1862]. Applications d’analyse dt de geometrie, Mallett-Bachelier, Paris. Google Scholar
Quine, W. V. O. Quine, W. V. O. [1986]. Philosophy of Logic ( Second Edition). Harvard University Press, Cambridge, MA. Google Scholar
Resnik, M. D. Resnik, M. D. [1980]. Frege and the Philosophy of Mathematics. Cornell University Press, Ithaca, NY. Google Scholar
Resnik, M. D. Resnik, M. D. [1997]. Mathematics as a Science of Patterns. Oxford University Press, Oxford. Google Scholar
Russell, B. Russell, B. [1903]. The Principles of Mathematics. Allen and Unwin, London. Google Scholar
Russell, B. Russell, B. (1919)[1993]. Introduction to Mathematical Philosophy. Reprint by Dover, New York. Google Scholar
Scanlan, M. J. 91 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. Shapiro, S. [1997]. Philosophy of Mathematics: Structure and Ontology. Oxford University Press, New York. Google Scholar
Shapiro, S. MacBride, F. Shapiro, S. [2006a]. “Structure and Identity.” In F. MacBride (ed.), Identity and Modality, Oxford University Press, Oxford, pp. 109–145. Google Scholar
Shapiro, S. MacBride, F. Shapiro, S. [2006b]. “The Governance of Identity.” In F. MacBride (ed.), Identity and Modality, Oxford University Press, Oxford, pp. 164–173. Google Scholar
Shapiro, S. Shapiro, S. [2008]. “Identity, Indiscernibility, and ante rem Structuralism: The Tale of i and –i .” Philosophia Mathematica, 16(3): pp. 285–309. CrossRef | Google Scholar
Shapiro, S. Shapiro, S. [2012]. “An ‘i’ for an i: Singular Terms, Uniqueness, and Reference.” Review of Symbolic Logic, 5(3): pp. 380–415. CrossRef | Google Scholar
Shapiro, S. Wright, C. Rayo, A. Uzquiano, G. Shapiro, S. and Wright, C. [2006]. “All Things Indefinitely Extensible.” In A. Rayo and G. Uzquiano (eds.), Absolute Generality, Oxford University Press, Oxford, pp. 255–304. Google Scholar
Stein, H. Aspray, W. Kitcher, P. Stein, H. [1988]. “Logos, Logic, and Logistiké: Some Philosophical Remarks on the Nineteenth-Century Transformation of Mathematics.” In W. Aspray and P. Kitcher (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. Tait, W. [1986]. “Truth and Proof: The Platonism of Mathematics.” Synthese, 69(3): pp. 341–370. CrossRef | Google Scholar
van Heijenoort, J. 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. van Heijenoort, J. [1967b]. “Logic as Calculus and Logic as Language.” Synthese, 17(3): pp. 324–330. CrossRef | Google Scholar
von Neumann, J. van Heijenoort, J. von Neumann, J. [1925]. “An Axiomatization of Set Theory.” In J. van Heijenoort (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 Von Staudt , Karl Georg Christian . [1856–60]. Beitrage zur Geometric der Lage. F. Korn, Nürnberg. Google Scholar
Weyl, H. Weyl, H. [1949]. Philosophy of Mathematics and Natural Science. Princeton University Press, Princeton (Revised and Augmented Edition, Athenaeum Press, New York, 1963). CrossRef | Google Scholar
Whitehead, A. N. Russell, B. Whitehead, A. N. , and Russell, B. [1910]. Principia Mathematica 1. Cambridge University Press, Cambridge. Google Scholar
Wilson, M. Wilson, M. [1992]. “Frege: The Royal Road from Geometry.” Noûs, 26(2): pp. 149–180. CrossRef | Google Scholar
Zermelo, E. 92 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. CrossRef | Google Scholar
Hellman, Geoffrey Shapiro, Stewart Mathematic Structuralism Geoffrey Hellman and Stewart Shapiro Google Scholar

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed