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The logic and topology of Kant’s temporal continuum. (English) Zbl 1395.01018

Authors’ abstract: In this paper we provide a mathematical model of Kant’s temporal continuum that yields formal correlates for Kant’s informal treatment of this concept in the Critique of Pure Reason and in other works of his critical period. We show that the formal model satisfies Kant’s synthetic a priori principles for time (whose consistence is not obvious) and that it even illuminates what “faculties and functions” must be in place, as “conditions for the possibility of experience”, for time to satisfy such principles. We then present a mathematically precise account of Kant’s transcendental theory of time – the most precise account to date.
Moreover, we show that the Kantian continuum which we obtain has some affinities with the Brouwerian continuum but that it also has “infinitesimal intervals” consisting of nilpotent infinitesimals; these allow us to capture Kant’s theory of rest and motion in the Metaphysical Foundations of Natural Science.
While our focus is on Kant’s theory of time the material in this paper is more generally relevant for the problem of developing a rigorous theory of the phenomenological continuum, in the tradition of Whitehead, Russell, and Weyl among others.
Reviewer’s remarks: In Section 1, the authors note (quotation): “The work in this paper grew out of the formalization of Kant’s transcendental logic in [T. Achourioti and the second author, ibid. 4, No. 2, 254–289 (2011; Zbl 1252.03007)]; the technical development is mostly based on the first author’s dissertation [The logic of Kant’s temporal continuum. Amsterdam: University of Amsterdam, Institute for Logic, Language and Computation (ILLC) (PhD Thesis) (2017), http://hdl.handle.net/11245.1/74c2979b-559d-47fc-9d0e-53acd23169b8]; to which we shall often refer for omitted proofs and for extended discussions that, for reasons of space, cannot be induced here.”
As reviewer, I can only admit that there is nevertheless a lot to learn from the contents of the paper. Indeed, there are omitted comments and clarifications, to be found and existent in the first author’s dissertation. Anyway, a brightful work, the result of several years of research.

MSC:

01A50 History of mathematics in the 18th century
03-03 History of mathematical logic and foundations
03B44 Temporal logic
03F99 Proof theory and constructive mathematics
06F35 BCK-algebras, BCI-algebras
54E55 Bitopologies
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