On Constructive Axiomatic Method

Andrei Rodin

Abstract


The received notion of axiomatic method stemming from Hilbert  is not fully adequate to the recent successful practice of axiomatizing mathematical theories. The axiomatic architecture of Homotopy type theory (HoTT) does not fit the pattern of formal axiomatic theory in the standard sense of the word.  However this theory falls under a more general and in some respects more traditional notion of axiomatic theory, which I call after Hilbert and Bernays constructive and demonstrate using the Classical example of the First Book of Euclid's  Elements. I also argue that HoTT is not unique in the respect but represents a wider trend in today's mathematics, which also includes Topos theory and some other developments. On the basis of these modern and ancient examples I claim that the received semantic-oriented formal axiomatic method defended recently by Hintikka is not self-sustained but requires a support of  constructive method. Finally I provide an epistemological argument showing that the constructive axiomatic method is more apt to present scientific theories than the received axiomatic method.

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