Abstract

TARANTINO, Piero. The Distinction between the "Problems" and the "Theorems" among the Mathematicians of the Academy. Nova tellus [online]. 2010, vol.28, n.1, pp.97-123. ISSN 0185-3058.

In his Commentary of the First Book of Euclid's Elements Proclus gives an account of a debate in Plato's Academy about the nature of mathematical propositions. The core of the controversy is the distinction between problems and theorems, the former pertaining to the generation and the construction of figures, the latter concerning attributes and properties of geometrical objects. According to Proclus, all mathematical propositions were called "theorems" by Speusippus and Amphinomus and "problems" by Menaechmus and the mathematicians of his school. The origin and the reasons of this remarkable debate appear obscure especially because of the lack of additional information in Proclus' commentary. My purpose is to throw light on this controversy in Plato's Academy taking into consideration a large range of views on contents and particularly methods of early ancient mathematics. In this way I attempt to contextualize the question of the status of mathematical propositions in the wider development of an axiomatic and deductive model for the systematic organization of geometrical and arithmetical contents. Consequently the alternative positions, held respectively by Speusippus and Amphinomus and by the school of Menaechmus, appear to constitute a defence of different but not conflicting stages in the mathematical work, namely the phase of the discovery and the phase of demonstration. On the one hand, Speusippus and Amphinomus support the new deductive structure of mathematical propositions, according to which theorems are derived not by proven principles. On the other hand, the school of Menaechmus aims to preserve the heuristic function of problems, whose contribution is preliminary to an axiomatic and formal exposition of discovered contents.

Keywords : problems; theorems; Academy; Proclus; Euclid.

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