Advertisement

A Silly Answer to a Psillos Question

  • Elaine Landry
Chapter
Part of the The Western Ontario Series in Philosophy of Science book series (WONS, volume 75)

Abstract

In this paper I offer an answer to a question raised in (Psillos, 2006): How can one speak of structures without objects? Specifically, I use category theory to show that, mathematically speaking, structures do not need objects. Next, I argue that, scientifically speaking, this category-theoretic answer is silly because it does not speak to the scientific structuralist’s appeal to the appropriate kind of morphism to make precise the concept of shared structure. Against French et al.’s approach,1 I note that to account for the scientific structuralist’s uses of shared structure we do not need to formally frame either the structure of a scientific theory or the concept of shared structure. Here I restate my (Landry, 2007) claim that the concept of shared structure can be made precise by appealing to a kind of morphism, but, in science, it is methodological contexts (and not any category or set-theoretic framework) that determine the appropriate kind. Returning to my aim, I reconsider French’s example of the role of group theory in quantum mechanics to show that French already has an answer to Psillos’ question but this answer is not found in either his set-theoretic formal framework or his ontic structural realism. The answer to Psillos is found both by recognizing that it is the context that determines what the appropriate kind of morphism is and, as Psillos himself suggests,2 by adopting a methodological approach to scientific structuralism.

Keywords

Category Theory Abstract System Shared Structure Ontic Structural Realism Scientific Structuralism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Bibliography

  1. Achinstein, P. (1968). Concepts of Science: A Philosophical Analysis. Johns Hopkins University Press, Baltimore MD.Google Scholar
  2. Bell, J. L. (1981). Category theory and the foundations of mathematics. British Journal for the Philosophy of Science, 32:349–358.CrossRefGoogle Scholar
  3. Bell, J. L. (2006). Abstract and variable sets in category theory. In What is Category Theory, pages 9–16. Polimetrica Monza.Google Scholar
  4. Benacerraf, P. (1991). What numbers could not be. In Benacerraf, P. and Putnam, H., editors, Philosophy of Mathematics, pages 272–294. Cambridge University Press, New York, NY, 2nd edition. Originally published 1965.Google Scholar
  5. Brading, K. and Landry, E. (2007). Scientific structuralism: Presentation and representation. Philosophy of Science, 73:571–581.CrossRefGoogle Scholar
  6. Cartwright, N., Shomar, T., and Suárez, M. (1995). The toolbox of science. In Herfel, W., Krajewski, W., Niiniluoto, I., and Wójcicki, R., editors, Theories and Models of Scientific Progress, pages 137–149. Rodopoi, Amsterdam.Google Scholar
  7. Castellani, E. (1993). Quantum mechanics, objects and objectivity. In Garola, C. and Rossi, A., editors, The Foundations of Quantum Mechanics — Historical Analysis and Open Questions, pages 105–114. Kluwer, Dordrecht.Google Scholar
  8. da Costa, N. C. A. and French, S. (1990). The model-theoretic approach in the philosophy of science. Philosophy of Science, 57:248–265.CrossRefGoogle Scholar
  9. da Costa, N. C. A., Bueno, O., and French, S. (1997). Suppes predicates for space-time. Synthese, 112:271–279.CrossRefGoogle Scholar
  10. Dummett, M. (1991). Frege and Other Philosophers. Oxford University Press, Oxford.Google Scholar
  11. Feferman, S. (1977). Categorical foundations and foundations of category theory. In Butts, R. and Hintikka, J., editors, Foundations of Mathematics and Computability Theory. Reidel, Dordrecht.Google Scholar
  12. French, S. (1999a). Models and Mathematics in Physics: The Role of Group Theory. Cambridge University Press, Cambridge.Google Scholar
  13. French, S. (1999b). Theories, models and structures: Thirty years on. Philosophy of Science, Supplement: Proceedings of PSA 1998.Google Scholar
  14. French, S. (2000). The reasonable effectiveness of mathematics: Partial structures and the application of group theory to physics. Synthese, 125:103–120.CrossRefGoogle Scholar
  15. French, S. (2006). Structure as a weapon of the realist. Proceedings of the Aristotelian Society, 106(2):1–19.Google Scholar
  16. Hale, B. (1996). Structuralism’s unpaid epistemological debts. Philosophia Mathematica, 4: 124–143.Google Scholar
  17. Hellman, G. (1996). Structuralism without structures. Philosophia Mathematica, 4:100–123.Google Scholar
  18. Hellman, G. (2001). Three varieties of mathematical structuralism. Philosophia Mathematica, 9:184–211.Google Scholar
  19. Hellman, G. (2003). Does category theory provide a framework for mathematical structuralism? Philosophia Mathematica, 11:129–157.Google Scholar
  20. Hesse, M. (1963). Models and Analogies in Science. Oxford University Press, Oxford.Google Scholar
  21. Ladyman, J. (1998). What is structural realism? Studies in the History and Philosophy of Science, 29:409–424.CrossRefGoogle Scholar
  22. Landry, E. (1999). Category theory: The language of mathematics. Philosophy of Science, 66: S14–S27.CrossRefGoogle Scholar
  23. Landry, E. (2006). Category Theory as a Framework for an In Re Interpretation of Mathematical Structuralism. Kluwer, Dordrecht.Google Scholar
  24. Landry, E. (2007). Shared structure need not be shared set-structure. Synthese, 158:1–17.CrossRefGoogle Scholar
  25. Landry, E. and Marquis, J. P. (2005). Categories in context: Historical, foundational and philosophical. Philosophia Mathematica, 13:1–43.CrossRefGoogle Scholar
  26. Mayberry, J. (1994). What is required of a foundation for mathematics? Philosophia Mathematica, 2:16–35.CrossRefGoogle Scholar
  27. Parsons, C. (1990). The structuralist view of mathematical objects. Synthese, 84:303–346.CrossRefGoogle Scholar
  28. Psillos, S. (1995). Is structural realism the best of both worlds? Dialectica, 49:15–46.CrossRefGoogle Scholar
  29. Psillos, S. (2001). Is structural realism possible? Philosophy of Science, Supplement: Proceedings of PSA 2000, pages 13–24.Google Scholar
  30. Psillos, S. (2006). The structure, the whole structure and nothing but the structure? Philosophy of Science, 73:560–570.CrossRefGoogle Scholar
  31. Quine, W. (1948). On what there is. Review of Metaphysics, 2:21–38.Google Scholar
  32. Redhead, M. (1975). Symmetry in inter-theory relations. Synthese, 32:77–112.CrossRefGoogle Scholar
  33. Redhead, M. (1980). Models in physics. British Journal for the Philosophy of Science, 31: 145–163.CrossRefGoogle Scholar
  34. Redhead, M. (1995). From Physics to Metaphysics. Cambridge University Press, Cambridge.Google Scholar
  35. Shapiro, S. (1997). Philosophy of Mathematics: Structure and Ontology. Oxford University Press, Oxford.Google Scholar
  36. Suárez, M. (2003). Scientific representation: Against similarity and isomorphism. International Studies in the Philosophy of Science, 17:225–244.CrossRefGoogle Scholar
  37. Suárez, M. (2006). An inferential conception of scientific representation. Philosophy of Science, Supplement: Proceedings of PSA 2004.Google Scholar
  38. Suppe, F. (1977). Alternatives to the received view. In Suppe, F., editor, The Structure of Scientific Theories, pages 119–232. University of Illinois Press, Chicago IL.Google Scholar
  39. Suppes, P. (1962). Models of data. In Nagel, E., Suppes, P., and Tarski, A., editors, Logic, Methodology and Philosophy of Science, pages 252–261. Stanford University Press, Stanford.Google Scholar
  40. Suppes, P. (1967). Set theoretical structures in science. Technical report, Stanford. Mimeograph.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.University of CaliforniaDavisUSA

Personalised recommendations