A Proportional World: the Mathematical and Philosophical Outlook of the Proportion of the Platonic Solids

Plato is the first philosopher to provide a robust account of the mathematical structure of the world; in this framework, the idea that the elements are proportionally arranged plays a crucial role. However, scholars agree nowadays that the proportional relation of the elements is not to be read at face value since no mathematical criterion has been found for which the proportion is appropriate. By contrast, this paper aims, on the one hand, to explain the mathematical reasons allowing Plato to state that the elements, each of which is to be associated with a perfect solid, are in proportion to one another (Ti. 31b4–32c5) and, on the other, to highlight the philosophical importance of ascribing to Plato the awareness of these reasons. More specifically, we show that Plato can arrange the solids in proportion ‘as far as it is possible’ by considering as a parameter the radius of the sphere into which each body is inscribed. This parameter establishes the proportion with a very low deviation, whose presence and tolerability can be explained from a technical point of view and exploited from a philosophical point of view. Thus, our solution not only ensures the overall consistency of Plato’s account of elementary bodies but also justifies the fundamental role he ascribes to the spherical shape of the world and, more generally, the geometrical foundation of the goodness of the world’s structure.