On Constructive Axiomatic Method

Abstract

The received notion of axiomatic method stemming from Hilbe rt 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. Ho wever this theory falls under a more general and in some respects more traditional no tion 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 mathem atics, which also includes Topos theory and some other developments. On the basis of the se modern and ancient examples I claim that the received semantic-oriented forma l axiomatic method defended recently by Hintikka is not self-sustained but requires a su pport of constructive method. Finally I provide an epistemological argument showing that the constructive axiomatic method is more apt to present scientific theories than the rec eived axiomatic method.