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CATEGORICAL HARMONY AND PATH INDUCTION

Published online by Cambridge University Press:  14 March 2017

PATRICK WALSH
PATRICK WALSH*
Affiliation:
Department of Philosophy, Carnegie Mellon University
*
*DEPARTMENT OF PHILOSOPHY CARNEGIE MELLON UNIVERSITY PITTSBURGH, PA 15213, USA E-mail: [email protected]
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Abstract

This paper responds to recent work in the philosophy of Homotopy Type Theory by James Ladyman and Stuart Presnell. They consider one of the rules for identity, path induction, and justify it along ‘pre-mathematical’ lines. I give an alternate justification based on the philosophical framework of inferentialism. Accordingly, I construct a notion of harmony that allows the inferentialist to say when a connective or concept is meaning-bearing and this conception unifies most of the prominent conceptions of harmony through category theory. This categorical harmony is stated in terms of adjoints and says that any concept definable by iterated adjoints from general categorical operations is harmonious. Moreover, it has been shown that identity in a categorical setting is determined by an adjoint in the relevant way. Furthermore, path induction as a rule comes from this definition. Thus we arrive at an account of how path induction, as a rule of inference governing identity, can be justified on mathematically motivated grounds.

Keywords

03A05philosophy of logichomotopy type theorycategory theoryharmony
Type
Research Article
Information
The Review of Symbolic Logic , Volume 10 , Issue 2 , June 2017 , pp. 301 - 321
Copyright
Copyright © Association for Symbolic Logic 2017 

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References

BIBLIOGRAPHY

Awodey, S. (2009). Introduction to Categorical Logic. Available at: https://www.andrew.cmu.edu/user/awodey/catlog/notes/catlog2B.pdf (accessed October 31, 2009).Google Scholar
Awodey, S. (2010). Category Theory (second edition). Oxford, UK: Clarendon Press.Google Scholar
Belnap, N. D. (1962). Tonk, plonk and plink. Analysis, 22(6), 130–134.Google Scholar
Bernays, P. (1922). Über hilberts gedanken zur grundlegung der arithmetik. Jahresbericht der Deutschen Mathematiker-Vereinigung, 31, 10–19.Google Scholar
Coquand, T. (2011). Equality and Dependent Type Theory. Available at: http://www.cse.chalmers.se/∼coquand/equality.pdf (accessed October 5, 2015).Google Scholar
Coquand, T. (2014). A Remark on Singleton Types. Available at: http://www.cse.chalmers.se/∼coquand/singl.pdf (accessed March 6, 2017).Google Scholar
Dummett, M. (1981). Frege: Philosophy of Language. Cambridge, MA: Harvard University Press.Google Scholar
Dummett, M. (1991). The Logical Basis of Metaphysics. Cambridge, MA: Harvard University Press.Google Scholar
Schreiber, U. (2009). ‘Philosophical’ meaning of the yoneda lemma, MathOverflow. Available at: http://mathoverflow.net/users/381/urs schreiber.Google Scholar
Jacobs, B. (1999). Categorical Logic and Type Theory. Amsterdam: Elsevier.Google Scholar
Koslow, A. (2005). A Structuralist Theory of Logic. New York: Cambridge University Press.Google Scholar
Ladyman, J. & Presnell, S. (2014). Does homotopy type theory provide a foundation for mathematics. The British Journal for the Philosophy of Science, forthcoming.Google Scholar
Ladyman, J. & Presnell, S. (2015). Identity in homotopy type theory, part I: The justification of path induction. Philosophia Mathematica, 23(3), 386–406.CrossRefGoogle Scholar
Lawvere, F. W. (1969). Adjointness in foundations. Dialectica, 23(3–4), 281–296.Google Scholar
Lawvere, F. W. (1970). Equality in hyperdoctrines and comprehension schema as an adjoint functor. Applications of Categorical Algebra, 17, 1–14.Google Scholar
Martin-Löf, P. (1995). Verificationism then and now. In van der Schaar, M., editor. The Foundational Debate: Complexity and Constructivity in Mathematics and Physics. Vienna Circle Institute Yearbook, Vol. 3. Dordrecht: Springer Netherlands, pp. 3–14.Google Scholar
Martin-Löf, P. (1996). On the meanings of the logical constants and the justifications of the logical laws. Nordic Journal of Philosophical Logic, 1(1), 11–60.Google Scholar
Maruyama, Y. (2016). Categorical harmony and paradoxes in proof-theoretic semantics. In Piecha, T. and Schroeder-Heister, P., editors. Advances in Proof-Theoretic Semantics. Trends in Logic, Vol. 43. Heidelberg: Springer, pp. 95–114.Google Scholar
Mayberry, J. (1994). What is required of a foundation for mathematics? Philosophia Mathematica, 2(1), 16–35.Google Scholar
Morehouse, E. (2013). An Adjunction-Theoretic Foundation for Proof Search in Intuitionistic First-Order Categorical Logic Programming, Ph.D. Thesis, Wesleyan University.Google Scholar
Peregrin, J. (2012). What is inferentialism? In Gurova, L., editor. Inference, Consequence, and Meaning: Perspectives on Inferentialism. Newcastle upon Tyne, UK: Cambridge Scholars Publishing, pp. 3–16.Google Scholar
Peregrin, J. (2014). Inferentialism: Why Rules Matter. New York: Palgrave Macmillan.Google Scholar
Pfenning, F. & Davies, R. (2001). A judgemental reconstruction of modal logic. Mathematical Structures in Computer Science, 11(4), 511–540.Google Scholar
Prawitz, D. (1965). Natural Deduction: a Proof-Theoretical Study. Stockholm: Almqvist & Wiksell.Google Scholar
Prior, A. N. (1960). The runabout inference-ticket. Analysis, 21(2), 38–39.Google Scholar
Schroeder-Heister, P. (2007). Generalized definitional reflection and the inversion principle. Logica Universalis, 1(2), 355–376.Google Scholar
Steinberger, F. (2009). Harmony and Logical Inferentialism. Ph.D. Thesis, University of Cambridge.Google Scholar
Steinberger, F. (2009). Not so stable. Analysis, 69(4), 655–661.Google Scholar
Stevenson, J. T. (1961). Roundabout the runabout inference-ticket. Analysis, 21(6), 124–128.Google Scholar
Tennant, N. (1978). Natural Logic. Edinburgh: Edinburgh University Press.Google Scholar
Tennant, N. (1987). Anti-realism and Logic: Truth as Eternal. Oxford: Clarendon Press.Google Scholar
Tennant, N. (2007). Inferentialism, logicism, harmony, and a counterpoint. Essays for Crispin Wright: Logic, Language, and Mathematics, 2, 105–132.Google Scholar
Tennant, N. (2010). Inferential semantics for first-order logic: Motivating rules of inference from rules of evaluation. In Smiley, T. J., Lear, J., and Oliver, A., editors. The Force of Argument: Essays in Honor of Timothy Smiley. London: Routledge, pp. 223–257.Google Scholar
The Univalent Foundations Program (2013). Homotopy Type Theory: Univalent Foundations of Mathematics. Institute for Advanced Study. Available at: http://homotopytypetheory.org/book.Google Scholar
Walsh, P. (2015). Justifying Path Induction: An Inferentialist Analysis of Identity Elimination in Homotopy Type Theory. Master’s Thesis, Carnegie Mellon University.Google Scholar