Launchpad or Purpose of the Research? : Plato and Aristotle on Truth in Geometric Proof

It is widely acknowledged that Aristotle somehow diverges from Plato’s thesis on geometry. Besides, this dissimilarity between the two has been mostly evaluated only through the general aristotelic overcoming of Plato's instances and the overall ontological critique. In Rep. 510b-511d, Plato introduces geometry to explain the defective capacity of the διάνοια compared with the νόησις. Geometers accept some common knowledges of their science a priori (e.g. the idea of straight line, the demonstration of triangle) as hypotheses and they found their method on them. For this reason, their science is limited because it can apply a representation of ideas to reality only through a conductive means (the εἴκονες): consequently, Plato defines this knowledge as διά-νοια. Consequently, he believes the geometric study worthy of the Guardians’ education, because they gradually understand how to think for themselves the ideas without any hypotheses but through dialectics (νόησις). Moreover, in a crucial passage (Rep. 510d 23), Plato emphasizes that geometrical proofs are conducted starting from visible shapes (τοῖς ὁρωμένοις εἴδεσι), but the geometer then argues about pure figures separated from the concrete ones. Aristotle, for his part, pays close attention to this point: several remarks (APr. 49b-50a, APo. 76b-77a, Met. 1078a) attest to the prominence of the topic. Aristotle clearly points out that the terms' exposition (ἐκτίθεσθαι) in geometric proof doesn't get into conflict with the sillogistic demonstration procedure. The geometer draws a figure with merely didactic purposes that have nothing to do with the proof's validity and truth. In addition, this consideration about the geometrical objects leads to pivotal differences from platonic ontology (Met. 1089a). Nevertheless, both the interpretation of the ἔκθεσις and the objections to Plato's conception of geometry have still not gained a common ground for discussion among scholars. It is of substantial interest to deepen how these issues deal with Aristotle's epistemology and conception of truth.