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  3. Mathematical Structuralism
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    • Publisher:
      Cambridge University Press
      Publication date:
      06 December 2018
      29 November 2018
      ISBN:
      9781108582933
      9781108456432
      DOI:
      https://doi.org/10.1017/9781108582933
      Dimensions:
      Weight & Pages:
      Dimensions:
      (229 x 152 mm)
      Weight & Pages:
      0.2kg, 100 Pages
      • Subjects:
      Mathematics (general), Philosophy of Science, Philosophy
      Series:
      Elements in the Philosophy of Mathematics
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    Subjects:
    Mathematics (general), Philosophy of Science, Philosophy
    Series:
    Elements in the Philosophy of Mathematics
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    Book description

    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.

    References

    References

    Assadian, B. [2017]. “The Semantic Plights of the Ante-Rem Structuralist.” Philosophical Studies, https://doi.org/10.1007/s11098-017–1001-7.
    Awodey, S. [1996]. “Structure in Mathematics and Logic: A Categorical Perspective.” Philosophia Mathematica, 4(3): pp. 209–237.
    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.
    Bell, J. L. [1986]. “From Absolute to Local Mathematics.” Synthese, 69(3): pp. 409–426.
    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.]
    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.]
    Beltrami, E. [1902]. Opere matematiche. Hoepli, Milan.
    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.
    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.
    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.
    Bolzano, B. [1950]. Paradoxes of the Infinite. Steele, D. A. (trans.), Routledge & Kegan Paul, London.
    Boolos, G. [1971]. “The Iterative Conception of Set.” In Boolos, G., Logic, Logic, and Logic, Harvard University Press, 1998, pp. 13–29.
    Burgess, J. P. [1999]. “Review of Shapiro [1997].” Notre Dame Journal of Formal Logic, 40(2): pp. 283–291.
    Burgess, J. P. and Rosen, G. [1997]. A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics. Oxford University Press.
    Burgess, J. P., Hazen, A., and Lewis, D. [1991]. “Appendix on Pairing.” In Lewis, D., Parts of Classes, Blackwell, Oxford, pp. 121–149.
    Cantor, G. [1932]. Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, Zermelo, E. (ed.), Springer, Berlin.
    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.
    Coffa, A. [1991]. The Semantic Tradition from Kant to Carnap. Cambridge University Press, Cambridge.
    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.
    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.
    Dedekind, R. [1932]. Gesammelte mathematische Werke 3, Fricke, R., Noether, E., and Ore, O. (eds.), Vieweg, Brunswick.
    Demopoulos, W. [1994]. “Frege, Hilbert, and the Conceptual Structure of Model Theory.” History and Philosophy of Logic,” 15(2): pp. 211–225.
    Drake, F. R. [1974]. Set Theory: An Introduction to Large Cardinals. North Holland.
    Dummett, M. [1991]. Frege: Philosophy of Mathematics. Harvard University Press, Cambridge, MA.
    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.
    Feferman, S. and Hellman, G. [1995]. “Predicative Foundations of Arithmetic.” Journal of Philosophical Logic, 24(1): pp. 1–17.
    Frege, G. [1879]. “Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens,” translated in van Heijenoort [1967], pp. 1–82.
    Frege, G. [1884]. The Foundations of Arithmetic. J. L. Austin (trans.), 2nd Edition. Harper, New York, 1960.
    Frege, G. [1903a]. Grundgesetze der Arithmetik 2. Olms, Hildescheim.
    Frege, G. [1903b]. “Über die Grundlagen der Geometrie.” Jahresbericht der Mathematiker-Vereinigung, 12, pp. 319–324, 368–375.
    Frege, G. [1906]. “Über die Grundlagen der Geometrie.” Jahresbericht der Mathematiker-Vereinigung, 15, pp. 293–309, 377–403, 423–430.
    Frege, G. [1967]. Kleine Schriften. Darmstadt, Wissenschaftlicher Buchgesellschaft (with I. Angelelli).
    Frege, G. [1971]. On the Foundations of Geometry and Formal Theories of Arithmetic. E.-H. W. Kluge (trans.), Yale University Press, New Haven, Connecticut.
    Frege, G. [1976]. Wissenschaftlicher Briefwechsel. Gabriel, G., Hermes, H., Kambartel, F., and Thiel, C. (eds.), Felix Meiner, Hamburg.
    Frege, G. [1980]. Philosophical and Mathematical Correspondence. Basil Blackwell, Oxford.
    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.
    Goldblatt, R. [2006]. Topoi: The Categorial Analysis of Logic (Revised Edition). Dover Publications.
    Goldfarb, W. D. [1979]. “Logic in the Twenties: The Nature of the Quantifier.” Journal of Symbolic Logic, 44(3): pp. 351–368.
    Goodman, N. [1977]. The Structure of Appearance. 3rd Edition. D. Reidel.
    Grassmann, H. [1972]. Gessammelte mathematische und physicalische Werke 1. Engels, F. (ed.), Johnson Reprint Corporation, New York.
    Hale, B. [1996]. “Structuralism’s Unpaid Epistemological Debts.” Philosophia Mathematica, (3)4: pp. 124–147.
    Hallett, M. [1990]. “Physicalism, Reductionism and Hilbert.” In Irvine, A. D. (ed.), Physicalism in Mathematics, Kluwer Academic Publishers, Dordrecht, Netherlands, pp. 183–257.
    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.
    Hellman, G. [1989]. Mathematics without Numbers: Towards a Modal-Structural Interpretation. Oxford University Press, Oxford.
    Hellman, G. [1996]. “Structuralism without Structures.” Philosophia Mathematica, (3)4: pp. 100–123.
    Hellman, G. [2003]. “Does Category Theory Provide a Framework for Mathematical Structuralism?” Philosophia Mathematica, 11(2): pp. 129–157.
    Hellman, G.. [2005]. “Structuralism.” In Shapiro, S. (ed.), The Oxford Handbook of Philosophy of Mathematics and Logic, Oxford University Press, Oxford, pp. 536–562.
    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.
    Hellman, G. [forthcoming]. “Extending the Iterative Conception: A Height-Potentialist Perspective.”
    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.
    Hilbert, D. [1899]. Grundlagen der Geometrie. Leipzig, Teubner; Foundations of Geometry, E. Townsend (trans.), Open Court, La Salle, Illinois, 1959.
    Hilbert, D. [1900]. “Mathematische Probleme.” Bulletin of the American Mathematical Society 8 (1902), pp. 437–479.
    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.
    Hilbert, D. [1935]. Gesammelte Abhandlungen, Dritter Band. Julius Springer, Berlin.
    Keränen, J. [2001]. “The Identity Problem for Realist Structuralism.” Philosophia Mathematica, 9(3): pp. 308–330.
    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.
    Klein, F. [1921]. Gesammelte mathematische Abhandlungen 1, Springer, Berlin.
    Lawvere, F. W. [1964]. “An Elementary Theory of the Category of Sets.” Proceedings of the National Academy of Sciences 52: pp. 1506–1511.
    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.
    Linnebo, Ø. [2013]. “The Potential Hierarchy of Sets.” Review of Symbolic Logic, 6(2): pp. 205–228.
    Linnebo, Ø. [forthcoming]. “Putnam on Mathematics as Modal Logic.” In Cook, R. and Hellman, G. (eds.), Putnam on Mathematics and Logic, Springer Verlag.
    Linnebo, Ø. and Pettigrew, R. [2011]. “Category Theory as an Autonomous Foundation.” Philosophia Mathematica, 19(3): pp. 227–254.
    Mac Lane, S. [1986]. Mathematics: Form and Function. Springer, Berlin.
    Mayberry, J. P. [2000]. The Foundations of Mathematics in the Theory of Sets. Cambridge University Press, Cambridge.
    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.
    McLarty, C. [1991]. “Axiomatizing a Category of Categories.” The Journal of Symbolic Logic, 56(4): pp. 1243–1260.
    McLarty, C. [2004]. “Exploring Categorical Structuralism.” Philosophia Mathematica, 12(1): pp. 37–53.
    Nagel, E. [1939]. “The Formation of Modern Conceptions of Formal Logic in the Development of Geometry.” Osiris, Vol. 7, pp. 142–224.
    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.
    Parsons, C. [1990]. “The Structuralist View of Mathematical Objects.” Synthese, 84(3): pp. 303–346.
    Pasch, M. [1926]. Vorlesungen über neuere Geometrie (Zweite Auflage). Springer, Berlin.
    Pettigrew, R. [2008]. “Platonism and Aristotelianism in Mathematics.” Philosophia Mathematica, 16(3): pp. 310–332.
    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.
    Poincaré, H. [1899]. “Des Fondements de la Géométrie.” Revue de Métaphysique et de Morale, 7, pp. 251–279.
    Poincaré, H. [1900]. “Sur les Principes de la Géométrie?” Revue de Métaphysique et de Morale, 8, pp. 72–86.
    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.
    Poncelet, J. V. [1862]. Applications d’analyse dt de geometrie, Mallett-Bachelier, Paris.
    Quine, W. V. O. [1986]. Philosophy of Logic (Second Edition). Harvard University Press, Cambridge, MA.
    Resnik, M. D. [1980]. Frege and the Philosophy of Mathematics. Cornell University Press, Ithaca, NY.
    Resnik, M. D. [1997]. Mathematics as a Science of Patterns. Oxford University Press, Oxford.
    Russell, B. [1903]. The Principles of Mathematics. Allen and Unwin, London.
    Russell, B. (1919)[1993]. Introduction to Mathematical Philosophy. Reprint by Dover, New York.
    Scanlan, M. J. [1988]. “Beltrami’s Model and the Independence of the Parallel Postulate.” History and Philosophy of Logic, 9(1), pp. 13–34.
    Shapiro, S. [1997]. Philosophy of Mathematics: Structure and Ontology. Oxford University Press, New York.
    Shapiro, S. [2006a]. “Structure and Identity.” In MacBride, F. (ed.), Identity and Modality, Oxford University Press, Oxford, pp. 109–145.
    Shapiro, S. [2006b]. “The Governance of Identity.” In MacBride, F. (ed.), Identity and Modality, Oxford University Press, Oxford, pp. 164–173.
    Shapiro, S. [2008]. “Identity, Indiscernibility, and ante rem Structuralism: The Tale of i and –i.” Philosophia Mathematica, 16(3): pp. 285–309.
    Shapiro, S. [2012]. “An ‘i’ for an i: Singular Terms, Uniqueness, and Reference.” Review of Symbolic Logic, 5(3): pp. 380–415.
    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.
    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.
    Tait, W. [1986]. “Truth and Proof: The Platonism of Mathematics.” Synthese, 69(3): pp. 341–370.
    van Heijenoort, J. [1967a]. From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Harvard University Press, Cambridge, MA.
    van Heijenoort, J. [1967b]. “Logic as Calculus and Logic as Language.” Synthese, 17(3): pp. 324–330.
    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.
    Staudt, Von, Georg Christian, Karl. [1856–60]. Beitrage zur Geometric der Lage. F. Korn, Nürnberg.
    Weyl, H. [1949]. Philosophy of Mathematics and Natural Science. Princeton University Press, Princeton (Revised and Augmented Edition, Athenaeum Press, New York, 1963).
    Whitehead, A. N., and Russell, B. [1910]. Principia Mathematica 1. Cambridge University Press, Cambridge.
    Wilson, M. [1992]. “Frege: The Royal Road from Geometry.” Noûs, 26(2): pp. 149–180.
    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.

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