Hostname: page-component-7479d7b7d-pfhbr Total loading time: 0 Render date: 2024-07-12T10:18:17.767Z Has data issue: false hasContentIssue false
  • English
  • Français

What Do Symmetries Tell Us about Structure?

Published online by Cambridge University Press:  01 January 2022

Thomas William Barrett
Get access Rights & Permissions [Opens in a new window]

Abstract

Mathematicians, physicists, and philosophers of physics often look to the symmetries of an object for insight into the structure and constitution of the object. My aim in this article is to explain why this practice is successful. In order to do so, I present a collection of results that are closely related to (and, in a sense, generalizations of) Beth’s and Svenonius’s theorems.

Type
Articles
Information
Philosophy of Science , Volume 85 , Issue 4 , October 2018 , pp. 617 - 639
Copyright
Copyright © The Philosophy of Science Association

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

I am especially grateful to Hans Halvorson for many discussions about this material. Thanks to Neil Dewar, John Dougherty, Ben Feintzeig, Phillip Kremer, Alex Meehan, J. B. Manchak, and Jim Weatherall for comments and discussion on earlier versions of this article. Thanks also to two anonymous referees for their helpful comments and feedback.

References

Andréka, H., Madarász, J. X., and Németi, I. 2005. “Mutual Definability Does Not Imply Definitional Equivalence: A Simple Example.” Mathematical Logic Quarterly 51 (6): 591–97..CrossRefGoogle Scholar
Baez, J., Bartels, T., Dolan, J., and Corfield, D. 2006. “Property, Structure and Stuff.” http://math.ucr.edu/home/baez/qg-spring2004/discussion.html.Google Scholar
Barrett, T. W. 2015a. “On the Structure of Classical Mechanics.” British Journal for the Philosophy of Science 66 (4): 801–28..CrossRefGoogle Scholar
Barrett, T. W. 2015b. “Spacetime Structure.” Studies in History and Philosophy of Science B 51:37–43.CrossRefGoogle Scholar
Barrett, T. W. 2018. “Equivalent and Inequivalent Formulations of Classical Mechanics.” British Journal for the Philosophy of Science. doi:10.1093/bjps/axy017.CrossRefGoogle Scholar
Barrett, T. W., and Halvorson, H. 2016a. “Glymour and Quine on Theoretical Equivalence.” Journal of Philosophical Logic 45 (5): 467–83..CrossRefGoogle Scholar
Barrett, T. W., and Halvorson, H. 2016b. “Morita Equivalence.” Review of Symbolic Logic 9 (3): 556–82..CrossRefGoogle Scholar
Barrett, T. W., and Halvorson, H. 2017a. “From Geometry to Conceptual Relativity.” Erkenntnis 82 (5): 1043–63..CrossRefGoogle Scholar
Barrett, T. W., and Halvorson, H. 2017b. “Quine’s Conjecture on Many-Sorted Logic.” Synthese 194 (9): 3563–82..CrossRefGoogle Scholar
Borceux, F. 1994. Handbook of Categorical Algebra. Vol. 1. Cambridge: Cambridge University Press.Google Scholar
Curiel, E. 2014. “Classical Mechanics Is Lagrangian: It Is Not Hamiltonian.” British Journal for the Philosophy of Science 65 (2): 269–321..CrossRefGoogle Scholar
da Costa, N. C. A., and Rodrigues, A. A. M. 2007. “Definability and Invariance.” Studia Logica 86 (1): 1–30..CrossRefGoogle Scholar
Dasgupta, S. 2015. “Substantivalism vs Relationalism about Space in Classical Physics.” Philosophy Compass 10 (9): 601–24..Google Scholar
Dasgupta, S. 2016. “Symmetry as an Epistemic Notion (Twice Over).” British Journal for the Philosophy of Science 67 (3): 837–78..CrossRefGoogle Scholar
de Bouvére, K. L. 1965. “Synonymous Theories.” In Symposium on the Theory of Models, 402–6. Amsterdam: North-Holland.Google Scholar
Earman, J. 1989. World Enough and Spacetime: Absolute versus Relational Theories of Space and Time. Cambridge, MA: MIT Press.Google Scholar
Feintzeig, B. H. 2017. “Deduction and Definability in Infinite Statistical Systems.” Synthese. doi:10.1007/s11229-017-1497-6.CrossRefGoogle Scholar
Friedman, H. M., and Visser, A. 2014. “When Bi-interpretability Implies Synonymy.” Logic Group Preprint Series 320:1–19.Google Scholar
Geroch, R. 1978. General Relativity from A to B. Chicago: University of Chicago Press.Google Scholar
Glymour, C. 1971. “Theoretical Realism and Theoretical Equivalence.” In PSA 1970: Proceedings of the 1970 Biennial Meeting of the Philosophy of Science Association, 275–88. Boston: Springer.Google Scholar
Glymour, C. 1977. “The Epistemology of Geometry.” Noûs 11:227–51.CrossRefGoogle Scholar
Glymour, C. 1980. Theory and Evidence. Princeton, NJ: Princeton University Press.Google Scholar
Halvorson, H. 2016. “Scientific Theories.” In The Oxford Handbook of Philosophy of Science, ed. Humphreys, P., 585–608. Oxford: Oxford University Press.Google Scholar
Hodges, W. 2008. Model Theory. Cambridge: Cambridge University Press.Google Scholar
Hudetz, L. 2015. “Linear Structures, Causal Sets and Topology.” Studies in History and Philosophy of Modern Physics B 52:294–308.CrossRefGoogle Scholar
Hudetz, L. Forthcoming. “Definable Categorical Equivalence: Towards an Adequate Criterion of Theoretical Intertranslatability.” Philosophy of Science.Google Scholar
Kanger, S. 1968. “Equivalent Theories.” Theoria 34 (1): 1–6..CrossRefGoogle Scholar
Knox, E. 2014. “Newtonian Spacetime Structure in Light of the Equivalence Principle.” British Journal for the Philosophy of Science 65 (4): 863–80..CrossRefGoogle Scholar
Korbmacher, J., and Schiemer, G. 2017. “What Are Structural Properties?” Philosophia Mathematica. doi:10.1093/philmat/nkx011.CrossRefGoogle Scholar
Mac Lane, S. 1971. Categories for the Working Mathematician. New York: Springer.CrossRefGoogle Scholar
Malament, D. B. 1977. “Causal Theories of Time and the Conventionality of Simultaneity.” Noûs 11:293–300.CrossRefGoogle Scholar
Maudlin, T. 2012. Philosophy of Physics: Space and Time. Princeton, NJ: Princeton University Press.Google Scholar
Narens, L. 2002. Theories of Meaningfulness. Mahwah, NJ: Erlbaum.Google Scholar
Nguyen, J., Teh, N. J., and Wells, L. 2018. “Why Surplus Structure Is Not Superfluous.” British Journal for the Philosophy of Science. doi:10.1093/bjps/axy026.CrossRefGoogle Scholar
North, J. 2009. “The ‘Structure’ of Physics: A Case Study.” Journal of Philosophy 106:57–88.CrossRefGoogle Scholar
Pelletier, F. J., and Urquhart, A. 2003. “Synonymous Logics.” Journal of Philosophical Logic 32 (3): 259–85..CrossRefGoogle Scholar
Pinter, C. C. 1978. “Properties Preserved under Definitional Equivalence and Interpretations.” Mathematical Logic Quarterly 24 (31–36): 481–88.CrossRefGoogle Scholar
Rosenstock, S., Barrett, T. W., and Weatherall, J. O. 2015. “On Einstein Algebras and Relativistic Spacetimes.” Studies in History and Philosophy of Science B 52:309–16.Google Scholar
Rosenstock, S., and Weatherall, J. O. 2016. “A Categorical Equivalence between Generalized Holonomy Maps on a Connected Manifold and Principal Connections on Bundles over That Manifold.” Journal of Mathematical Physics 57 (10). arXiv:1504.02401 [math-ph].CrossRefGoogle Scholar
Swanson, N., and Halvorson, H. 2012. “On North’s ‘The Structure of Physics.’” Unpublished manuscript. http://philsci-archive.pitt.edu/9314/.Google Scholar
Weatherall, J. O. 2016a. “Are Newtonian Gravitation and Geometrized Newtonian Gravitation Theoretically Equivalent?” Erkenntnis 81 (5): 1073–91..CrossRefGoogle Scholar
Weatherall, J. O. 2016b. “Understanding Gauge.” Philosophy of Science 83 (5): 1039–49..CrossRefGoogle Scholar
Weatherall, J. O. 2018a. “Category Theory and the Foundations of Classical Field Theories.” In Categories for the Working Philosopher, ed. Landry, E. Oxford: Oxford University Press.Google Scholar
Landry, E. 2018b. “Regarding the ‘Hole Argument.’” British Journal for the Philosophy of Science 69 (1): 329–50..Google Scholar
Weyl, H. 1952. Symmetry. Princeton, NJ: Princeton University Press.CrossRefGoogle Scholar
Winnie, J. 1986. “Invariants and Objectivity: A Theory with Applications to Relativity and Geometry.” In From Quarks to Quasars, ed. Colodny, R. G., 71–180. Pittsburgh: Pittsburgh University Press.Google Scholar