"Non si tratta di voltare una conchiglia". Platone e la svolta delle matematiche

In Resp. VII Plato submits to a radical reform the mathematical disciplines of the future quadrivium: the four “hard sciences” (arithmetic, geometry, astronomy, music) that together with the three “soft arts” of the trivium (grammar, rhetoric, dialectic) were to represent Western “liberal culture” – the basic culture of the free members of Western society from late antiquity onwards. Socrates and his fine respondent Glaucon insist on the fact that reformed mathematical disciplines must exercise a certain power over the human soul. This power is described in their dialogue through metaphors and similes, that suggest how arithmetic and logistiké (art of calculation), plane and solid geometry, can force the soul to approximate and balance the two opposite dimensions of sense and intelligence, sensible and intelligible world, becoming and being. In particular, the application of logistiké to incommensurable magnitudes provides a concrete exemplification of the apparently absurd “confusion of contraries” implied by mathematics, that has the power to “tow” the soul up to the “Good as such”. The result of the reform of mathematics achieved by Plato in Republic is no doubt the first version of the “mathematical Platonism” we will meet in the history of philosophy, but its meaning and aim relates to problematical and even ridiculous aspects of mathematical procedures, that alone have the power to awaken and “turn” our thought to the Good as absolute value.