The so-called platonism in the modern philosophy of mathematics rests on two fundamental contentions : [p1]mathematical objects, which are nonphysical, exist independently of human cognitive ability, and[p2]the truth of axioms is intuitively evident. [p1]can be legitimately ascribed to Plato himself, but how about[p2]? It is currently believed that[p2]was Plato's own conviction and led him to[p1]so that geometry thus conceived as a hierarchical structure of truths served as the model of his philosophical dialectic. I wish to criticize this prevailing interpretation and help to understand Plato's philosophy of mathematics better. Now in the Euthydemus, which I assume to be one of the early dialogues, mathematicians are depicted as specialists who discover mathematical reality and describe it correctly. This picture and also the famous doctrine of Recollection(anamnesis)seem to give support to our opponents. However, the middle books of the Republic warn us against such an optimistic view of the nature of mathematics, because there dianoia is definitely distinguished from noesis (the equivalent of anamnesis) and emphasis is laid on the consistency(homologia)in mathematics rather than on its truth. I assume that Plato was well aware of the fervent disputes concerning axiomatization of geometry being carried on among Euclid's forerunners, where the Zenonian paradoxical arguments seem to have played some role, and that he was thus hesitant to place unlimited faith in the truth of mathematical axioms. As to the relationship between mathematics and dialectic, it is very often supposed that inadequacies or defects of existing mathematics can be made good by dialectics. The task of dialecticians is to derive the basic propositions of mathematics by some kind of deductive process from a single logico-mathematical principle(e.g., the existence of a One or the definition of the Good, or a proposition or set of propositions about it), in order to complete the whole system of mathematical truths. Interpreters who support this view usually have recourse to a difficult phrase "kaitoi noeton onton meta arches"(511d2)in the Republic and take it to mean that mathematical objects become intelligible when linked up with the first principle. However, such an interpretation of Plato's text is quite alien to the genuine Platonic way of thinking. Plato simply observes the fact that mathematical subjects form a family(511b ; adelphais technais, 537c ; oikeiotetos), and it is noteworthy that dialectic is supposed to confirm itself and not mathematics (533d). The phrase at 511d2 is open to another interpretation : all that is hinted is that mathematical objects belong to the topos noetos together with (i.e. as well as or just like)the Good, and are not intermediates between sensible things and Forms. In my opinion, Plato suggests that : 1)axiomatization may make mathematical theories clear and stable to a certain extent, but they are doomed to include basic statements the truth of which is not guaranteed ; 2)though mathematical activity cannot fully grasp reality, it aims at doing so (527b ; heneka), and therefore can serve the prisoners of the Cave as a "thread of Ariadne" ; and 3)dialectics leaves mathematics as it is, but it does not follow that mathematics is value-free, since the positing of axioms, which is not a result of deductive reasoning, but of the "upward" process, involves choice (cf. Phaedo 10ld). I hope that the above consideration discourages us from assimilating Plato to a 'full-blooded' platonist or a Neoplatonist.
