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New Axiomatic Method (Instead of Conclusion)

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Axiomatic Method and Category Theory

Part of the book series: Synthese Library ((SYLI,volume 364))

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Abstract

In the following long promised presentation of the New Axiomatic Method I shall use as a guide Lawvere’s description of Axiomatic Method as “unification and concentration” (Lawvere 2003, p. 213) and generalize upon some examples of axiomatic thinking due to Lawvere and Voevodsky. I begin with the unification, then turn to the concentration and, finally, discuss the place and the special character of logic in the New Axiomatic Method.

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Notes

  1. 1.

    Although Voevodsky calls the association of L (Matrin-Löf’s type theory) to G (homotopy theory) the “direct formalization” (of G by L) it is clear that L does not represent the logical form of G in the usual sense of the word. Let x,y be individual variables and R be a binary predicate variable. The expression xRy is said to represent the logical form of linguistic expression Mary loves John in the sense that this latter expression obtains from xRy when variables x,y,R are given the appropriate semantic values; by giving these variables some other semantic values one may obtain other contentual expressions like Peter hates Paul or 1 > 0, which have the same logical form but different contents. Hilbert’s formal theory of Euclidean geometry represents the logical form of the traditional intuitive Euclidean geometry in the same traditional sense of “logical form” (Sect. 3.1). However homotopy theory is not just one contentual intuitive interpretation of Matrin-Löf’s type theory among a bunch of other such interpretations. Matrin-Löf’s type theory and the appropriately adjusted homotopy theory are rather two different legs of the same theory called homotopy type theory. The basic relationship between these two legs is not that of form and content.

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    Andrei Rodin

  2. Department of Liberal Arts, State University of Saint-Petersburg, Saint-Petersburg, Russia

    Andrei Rodin

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Rodin, A. (2014). New Axiomatic Method (Instead of Conclusion). In: Axiomatic Method and Category Theory. Synthese Library, vol 364. Springer, Cham. https://doi.org/10.1007/978-3-319-00404-4_10

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