# A Silly Answer to a Psillos Question

## 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## Bibliography

- Achinstein, P. (1968).
*Concepts of Science: A Philosophical Analysis*. Johns Hopkins University Press, Baltimore MD.Google Scholar - Bell, J. L. (1981). Category theory and the foundations of mathematics.
*British Journal for the Philosophy of Science*, 32:349–358.CrossRefGoogle Scholar - Bell, J. L. (2006). Abstract and variable sets in category theory. In
*What is Category Theory*, pages 9–16. Polimetrica Monza.Google Scholar - 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 - Brading, K. and Landry, E. (2007). Scientific structuralism: Presentation and representation.
*Philosophy of Science*, 73:571–581.CrossRefGoogle Scholar - 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 - 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 - 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 - da Costa, N. C. A., Bueno, O., and French, S. (1997). Suppes predicates for space-time.
*Synthese*, 112:271–279.CrossRefGoogle Scholar - Dummett, M. (1991).
*Frege and Other Philosophers*. Oxford University Press, Oxford.Google Scholar - 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 - French, S. (1999a).
*Models and Mathematics in Physics: The Role of Group Theory*. Cambridge University Press, Cambridge.Google Scholar - French, S. (1999b). Theories, models and structures: Thirty years on.
*Philosophy of Science, Supplement: Proceedings of PSA 1998*.Google Scholar - French, S. (2000). The reasonable effectiveness of mathematics: Partial structures and the application of group theory to physics.
*Synthese*, 125:103–120.CrossRefGoogle Scholar - French, S. (2006). Structure as a weapon of the realist.
*Proceedings of the Aristotelian Society*, 106(2):1–19.Google Scholar - Hale, B. (1996). Structuralism’s unpaid epistemological debts.
*Philosophia Mathematica*, 4: 124–143.Google Scholar - Hellman, G. (1996). Structuralism without structures.
*Philosophia Mathematica*, 4:100–123.Google Scholar - Hellman, G. (2001). Three varieties of mathematical structuralism.
*Philosophia Mathematica*, 9:184–211.Google Scholar - Hellman, G. (2003). Does category theory provide a framework for mathematical structuralism?
*Philosophia Mathematica*, 11:129–157.Google Scholar - Hesse, M. (1963).
*Models and Analogies in Science*. Oxford University Press, Oxford.Google Scholar - Ladyman, J. (1998). What is structural realism?
*Studies in the History and Philosophy of Science*, 29:409–424.CrossRefGoogle Scholar - Landry, E. (1999). Category theory: The language of mathematics.
*Philosophy of Science*, 66: S14–S27.CrossRefGoogle Scholar - Landry, E. (2006).
*Category Theory as a Framework for an In Re Interpretation of Mathematical Structuralism*. Kluwer, Dordrecht.Google Scholar - Landry, E. (2007). Shared structure need not be shared set-structure.
*Synthese*, 158:1–17.CrossRefGoogle Scholar - Landry, E. and Marquis, J. P. (2005). Categories in context: Historical, foundational and philosophical.
*Philosophia Mathematica*, 13:1–43.CrossRefGoogle Scholar - Mayberry, J. (1994). What is required of a foundation for mathematics?
*Philosophia Mathematica*, 2:16–35.CrossRefGoogle Scholar - Parsons, C. (1990). The structuralist view of mathematical objects.
*Synthese*, 84:303–346.CrossRefGoogle Scholar - Psillos, S. (1995). Is structural realism the best of both worlds?
*Dialectica*, 49:15–46.CrossRefGoogle Scholar - Psillos, S. (2001). Is structural realism possible?
*Philosophy of Science, Supplement: Proceedings of PSA 2000*, pages 13–24.Google Scholar - Psillos, S. (2006). The structure, the whole structure and nothing but the structure?
*Philosophy of Science*, 73:560–570.CrossRefGoogle Scholar - Quine, W. (1948). On what there is.
*Review of Metaphysics*, 2:21–38.Google Scholar - Redhead, M. (1975). Symmetry in inter-theory relations.
*Synthese*, 32:77–112.CrossRefGoogle Scholar - Redhead, M. (1980). Models in physics.
*British Journal for the Philosophy of Science*, 31: 145–163.CrossRefGoogle Scholar - Redhead, M. (1995).
*From Physics to Metaphysics*. Cambridge University Press, Cambridge.Google Scholar - Shapiro, S. (1997).
*Philosophy of Mathematics: Structure and Ontology*. Oxford University Press, Oxford.Google Scholar - Suárez, M. (2003). Scientific representation: Against similarity and isomorphism.
*International Studies in the Philosophy of Science*, 17:225–244.CrossRefGoogle Scholar - Suárez, M. (2006). An inferential conception of scientific representation.
*Philosophy of Science, Supplement: Proceedings of PSA 2004*.Google Scholar - 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 - 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 - Suppes, P. (1967). Set theoretical structures in science. Technical report, Stanford. Mimeograph.Google Scholar