This paper re-constructs Plato's ‘philosophy of geometry’ by arguing that he uses a geometrical method of hypothesis in his account of the cosmos’ generation in the Timaeus. Commentators on Plato's philosophy of mathematics often start from Aristotle's report in the Metaphysics that Plato admitted the existence of mathematical objects in-between ( metaxu) Forms and sensible particulars ( Meta. 1.6, 987b14–18). I argue, however, that Plato's interest in mathematics was centred on its methodological usefulness for philosophical inquiry, rather than on questions (...) of mathematical ontology. My key passage of interest is Timaeus’ account of the generation of the primary bodies in the cosmos, i.e. fire, air, water and earth ( Tim. 48b–c, 53b–56c). Timaeus explains the primary bodies’ origin by hypothesising two right-angled triangles as their starting-point ( arkhê) and describing their individual geometrical constitution. This hypothetical operation recalls the hypothetical method which Socrates introduces in the Meno (86e–87b), as well as the use of hypotheses by mathematicians which is described in the Republic (510b–c). Throughout the passage, Timaeus is focussed on explicating the bodies in terms of their formal structure, without however considering the ontological status of the triangles in relation to the physical world. (shrink)

This paper aims to gain multiple perspectives on Plato’s ontology through the analysis of the Intermediates. Plato refers to them to provide his cosmos with a dynamic structure, in which every part relates to a whole. Nevertheless, this definition could reverse in an opposite: the Intermediate justifies the sensible world to reach the ideal stability, concerning a mixture of being and nothing that gets purification from the becoming by degrees. In this interpretation, the sensibility absorbs life from the Forms, living (...) as their parasite. According to J. N. Findlay, the metaxy state is parasitical because of his logical exemplification of the Ideas, lacking an ontological substance. On the contrary, the void of the half-nothing takes a different role, necessary to establish a symbolic betweenness. (shrink)