Logoi arcaici. Le matematiche antiche fra esattezza e approssimazione

In Greek pre-Euclidean geometry exactness, truth, deductive rigor don’t exclude approximation, procedural flexibility, research without an end of an unachievable result. The sources regarding the pre-Euclidean (i.e. pre-Eudoxian; cf. Eucl. Elem. V deff. 3-6) definition and conception of logos hint to the “Euclidean algorithm” in the form of an “archaic” conception of logos: the antanairesis of two magnitudes (cf. Arist. Top. VIII 3, 158 b 33-35). Particularly relevant is its application to incommensurable magnitudes, which implies the infinite production of two series of ordered couples of integer-numbers (logoi), that converge to a smaller and smaller value, without achieving it. This can be considered a failure of mathematics in the “skeptical” perspective represented by Protagoras and challenged by the Anonymous supporter (Xenocrates?) of the “indivisible lines” (80 B 7 Diels-Kranz; ps. Arist. De lin. insec., 968 b 5-22). On the other hand, mathematical procedures of approximation of contraries – one and many, smaller and bigger, defect and excess –, aiming at balancing them, offer to Plato and Aristotle a good model of “rational calculation” applied to human life, that is both rational and irrational at the same time (e.g. Plato. Resp.VII 522 e-524 c; Arist. Eth. Nic. II 5,1106 a 26-b 28).