Since its very beginning mathematics was deeply related to logic and ontology. Greek mathematicians consciously applied the contradiction principle and had a clear idea of the soundness of modus ponens and of the implicational transitivity of deduction. When Pythagoras (or the Pythagoreans) demonstrated the irrationality of √2 by applying the method of reductio ad absurdum, Greek mathematics was already quite developed. It must be signalled that in this first clash between mathematics and logic, nobody thought that the culprit was logic. Greek mathematicians never thought that it was logic and not mathematics that had to be readjusted. This spontaneous attitude among the ancients, has prevailed up to the present times¹. When a strange or paradoxical result was obtained through mathematical reasoning nobody thought that logic had to be readjusted or even radically changed. Without this conception of logic (naive but based on very strong intuitions), the creation and development of set theory would have been impossible and, consequently, the project of finding a trustable foundation for classical mathematics (or, perhaps, it would have taken place many years later).