Logicity and non-logicity of the axiomatic (Axiomatization in mathematics)

It is intended to investigate how far contemporary mathematical axiomatization refers to logic, and in what measure it brings, to the contrary, beyond any logical perspective. Pursuing such a goal, one analyses first axiomatics in the light of logical formulation: taking into account, among other sources, the so-called Curry-Howard correspondence, one concludes that extremality is the main property of the logically conceived axiom. One comes then to evoke the actual ways of axiomatization in mathematics, which leads to the distinction between conceptual, non-foundational, with the program of classification connected axiomatization, and pure axiomatization, bearing the value of some kind of institutional act introducing the object, as it happens for the sets or an universe of Set Theory. Finally, one argues that "pure axiomatization" comes with some special kind of intuition, which appears to have many similarities with what Kant called pure intuition(of the forms of sensibility).