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Definable Categorical Equivalence

Published online by Cambridge University Press:  01 January 2022

Laurenz Hudetz
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Abstract

This article proposes to explicate theoretical equivalence by supplementing formal equivalence criteria with preservation conditions concerning interpretation. I argue that both the internal structure of models and choices of morphisms are aspects of formalisms that are relevant when it comes to their interpretation. Hence, a formal criterion suitable for being supplemented with preservation conditions concerning interpretation should take these two aspects into account. The two currently most important criteria—generalized definitional equivalence (Morita equivalence) and categorical equivalence—are not optimal in this respect. I put forward a criterion that takes both aspects into account: the criterion of definable categorical equivalence.

Type
Articles
Information
Philosophy of Science , Volume 86 , Issue 1 , January 2019 , pp. 47 - 75
Copyright
Copyright © The Philosophy of Science Association

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Footnotes

†

I am indebted to Charlotte Werndl for her invaluable support and advice. I also thank Hajnal Andréka, Thomas Barrett, Alexander Bors, Jeremy Butterfield, Adam Caulton, Erik Curiel, Marton Gömöri, Sam Fletcher, Hannes Leitgeb, Sebastian Lutz, István Németi, Karl-Georg Niebergall, Miklos Redei, Bryan Roberts, and Jim Weatherall as well as two anonymous referees for very helpful feedback and discussions.

References

Ahlbrandt, G., and Ziegler, M. 1986. “Quasi-Finitely Axiomatizable Totally Categorical Theories.” Annals of Pure and Applied Logic 30:63–83.CrossRefGoogle Scholar
Andréka, H., Madarász, J., and Németi, I. 2008. “Defining New Universes in Many-Sorted Logic.” Unpublished manuscript, Renyi Institute of Mathematics, Hungarian Academy of Sciences.Google Scholar
Andréka, H., and Németi, I. 2014. “Comparing Theories: The Dynamics of Changing Vocabulary.” In Johan van Benthem on Logic and Information Dynamics, ed. Smets, S. and Baltag, A., 143–72. Cham: Springer.Google Scholar
Awodey, S. 1997. “Logic in Topoi: Functorial Semantics for Higher-Order Logic.” PhD diss., University of Chicago.Google Scholar
Awodey, S. 2006. Category Theory. Oxford: Oxford University Press.CrossRefGoogle Scholar
Barrett, T. 2017. “Equivalent and Inequivalent Formulations of Classical Mechanics.” British Journal for the Philosophy of Science, forthcoming.Google Scholar
Barrett, T. 2018. “What Do Symmetries Tell Us about Structure?” Philosophy of Science 85 (4): 617–39.CrossRefGoogle Scholar
Barrett, T., and Halvorson, H. 2016. “Morita Equivalence.” Review of Symbolic Logic 9 (3): 556–82.CrossRefGoogle Scholar
Barrett, T., and Halvorson, H. 2017. “From Geometry to Conceptual Relativity.” Erkenntnis 82:1043–63.CrossRefGoogle Scholar
Barrett, T., Rosenstock, S., and Weatherall, J. 2015. “On Einstein Algebras and Relativistic Spacetimes.” Studies in History and Philosophy of Modern Physics B 52:309–16.Google Scholar
Coffey, K. 2014. “Theoretical Equivalence as Interpretive Equivalence.” British Journal for the Philosophy of Science 65 (4): 821–44.CrossRefGoogle Scholar
Feintzeig, B. 2017. “Deduction and Definability in Infinite Statistical Systems.” Synthese, forthcoming.CrossRefGoogle Scholar
Gelfand, I., and Naimark, M. 1943. “On the Imbedding of Normed Rings into the Ring of Operators in Hilbert Space.” Matematiceskij sbornik 54 (2): 197–217.Google Scholar
Halvorson, H. 2016. “Scientific Theories.” In Oxford Handbook of the Philosophy of Science. Oxford: Oxford University Press.Google Scholar
Halvorson, H., and Tsementzis, D. 2017. “Categories of Scientific Theories.” In Categories for the Working Philosopher, ed. Landry, E. Oxford: Oxford University Press.Google Scholar
Hodges, W. 1993. Model Theory. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Hudetz, L. 2015. “Linear Structures, Causal Sets and Topology.” Studies in History and Philosophy of Modern Physics 52:294–308.CrossRefGoogle Scholar
Hudetz, L. 2017. “The Semantic View of Theories and Higher-Order Languages.” Synthese, forthcoming.CrossRefGoogle Scholar
Ismael, J., and van Fraassen, B. C. 2003. “Symmetry as a Guide to Superfluous Theoretical Structure.” In Symmetries in Physics: Philosophical Reflections, ed. Brading, K. and Castellani, E., 371–92. Cambridge: Cambridge University Press.Google Scholar
Johnstone, P. 2002. Sketches of an Elephant: A Topos Theory Compendium. Vols. 1 and 2. Oxford: Oxford University Press.Google Scholar
Landry, E. 2007. “Shared Structure Need Not Be Shared Set-Structure.” Synthese 158 (1): 1–17.CrossRefGoogle Scholar
Lefever, K., and Székely, G. 2018. “Comparing Classical and Relativistic Kinematics in First-Order Logic.” Logique et Analyse 61 (241): 57–117.Google Scholar
MacLane, S. 1998. Categories for the Working Mathematician. 2nd ed. New York: Springer.Google Scholar
Malament, D. 2009. “Geometry and Spacetime.” Lecture notes, University of California, Irvine.Google Scholar
Naber, G. L. 2012. The Geometry of Minkowski Spacetime: An Introduction to the Mathematics of the Special Theory of Relativity. New York: Springer.CrossRefGoogle Scholar
Nguyen, J., Teh, N. J., and Wells, L. 2017. “Why Surplus Structure Is Not Superfluous.” British Journal for the Philosophy of Science, forthcoming.Google Scholar
Osterwalder, K., and Schrader, R. 1973. “Axioms for Euclidean Green’s Functions.” Communications in Mathematical Physics 31 (2): 83–112.CrossRefGoogle Scholar
Osterwalder, K., and Schrader, R. 1975. “Axioms for Euclidean Green’s Functions.” Pt. 2. Communications in Mathematical Physics 42 (3): 281–305.CrossRefGoogle Scholar
Putnam, H. 1983. “Equivalence.” In Philosophical Papers, Vol. 3, Realism and Reason, ed. Putnam, H., 26–45. Cambridge: Cambridge University Press.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 57:102902.CrossRefGoogle Scholar
Strocchi, F. 2005. An Introduction to the Mathematical Structure of Quantum Mechanics: A Short Course for Mathematicians. Singapore: World Scientific.CrossRefGoogle Scholar
Teh, N., and Tsementzis, D. 2017. “Theoretical Equivalence in Classical Mechanics and Its Relationship to Duality.” Studies in History and Philosophy of Science B 59:44–54.CrossRefGoogle Scholar
Tsementzis, D. 2017. “A Syntactic Characterization of Morita Equivalence.” Journal of Symbolic Logic 82 (4): 1181–98.CrossRefGoogle Scholar
Wallace, D. 2016. “Who’s Afraid of Coordinate Systems? An Essay on the Representation of Spacetime Structure.” Studies in the History and Philosophy of Modern Physics, forthcoming.Google Scholar
Weatherall, J. O. 2016. “Are Newtonian Gravitation and Geometrized Newtonian Gravitation Theoretically Equivalent?” Erkenntnis 81 (5): 1073–91.CrossRefGoogle Scholar
Weatherall, J. O. 2017. “Categories and the Foundations of Classical Field Theories.” In Categories for the Working Philosopher, ed. Landry, E. Oxford: Oxford University Press.Google Scholar
Wightman, A. S. 1956. “Quantum Field Theory in Terms of Vacuum Expectation Values.” Physical Review 101:860–66.CrossRefGoogle Scholar