The first clear distinction between the heuristic conjecture and the deductive proof in the ancient mathematics

Abstract

The mathematics of the ancient river-valley cultures was purely empirical, while the classical Greek mathematics was entirely deductive without any written sign of the heuristic arguments. In the forthcoming Hellenistic period there were significant changes. One of them is that in spite of the rigorous (deductive) proofs some heuristic arguments appeared in separate treatises. We show a nice example due to Archimedes. "We have learned from the very pioneers of this science not to have regard to mere plausible imaginings when it is a question of the reasonings to be included in our geometrical doctrine." – Proclus