In this paper we introduce the concept of "archimedian leap" as a non deductive argumentation process in mathematics which produces an epistemological rupture in the direction of giving objectivity to certain mathematical facts. We also discuss the qualitative character of archimedian leap emphasizing its hermeneutical function in mathematical thought and its relation with an aesthetical knowledge in mathematics. Some examples of epistemological ruptures produced by an archimedian leap, among others, are the following: (1) the archimedian property of the real line, which structures the Euclidean line through the real number system; (2) the complete induction principle of the theory of natural numbers, which models our intuition about recursive processes; (3) the passage of potential infinite to actual infinite, which permits giving an objective content to the theory of infinite sets and cardinalities.
Archimedian leap; Archimedian property; Epistemological rupture; Mathematical myths; Mathematical aesthetics