The conceptualisation of numbers is culturally bound. This may seem like a counterintuitive claim, but one illustration thereof is the limitations of the resemblance of the ancient Greek concept of number to that in modern mathematics.

Despite the fact that the theory of Forms is regarded as the hallmark of Plato’s philosophy, it has remained remarkably elusive, because it is more hinted at than explained in his dialogues. Given the uncertainty concerning the nature and extension of the Forms, this article makes no pretense to coming up with solutions to all problems that have occupied scholars since antiquity. It aims to elucidate only two aspects of that theory: the indication in certain dialogues that the Forms are (...) what in modern parlance are called functions or purposes, and the indication in other dialogues that such functions rely on harmonious structures. (shrink)

On Malcolm Schofield’s highly influential reading of the Similarity Regress in Part I of the Parmenides, the problem that the Regress poses is explanatory. Socrates posited the Similarity Form in order to explain why similar things are similar: similar things are similar because they participate in the Form Similarity as copies of the same original. Yet, the Similarity Regress generates an infinite series of Similarity Forms such that explanation is deferred ad infinitum. Schofield provides a philosophical incentive for adopting his (...) reading. He argues that the treatment of similarity in Part II of the dialogue yields a complete explanation of similarity. If we adopt this account, we can avoid the Similarity Regress altogether since a Form of Similarity is not needed in order to explain why similar things are similar. Thus, his interpretation has a hugely important philosophical pay-off. However, there is a different way to read the argument. Socrates claims that each Form is only one. Yet, the Similarity Regress is an argument that generates an infinite series of Similarity Forms. This results in a violation of the principle of non-contradiction: there is both only one Similarity Form and infinitely many Similarity Forms. Yet, anything that is incompatible with the principle of non-contradiction is surely absurd. Nobody, as far as I am aware, has explored whether this reading also has philosophical pay-off if we look at it together with similarity in Part II. However, should this interpretation pay off, it would be a viable alternative to Schofield’s. In this paper, I explore both views in the context of the treatment of similarity in Part II of the Parmenides. I argue for an account of similarity that differs from Schofield’s. Although my account is not wholly dismissive of Schofield’s, it renders the pay-off of Schofield’s account less appealing than he suggests. Furthermore, I show that the account of similarity in Part II also yields important lessons for the proponent of the alternative reading of the Similarity Regress: similarity as treated in Part II simply leads to further infinite regresses, thereby pushing us to consider rejecting the account of similarity in Part II too and to look for some other account intead. (shrink)

Resumo O argumento do Terceiro Homem presente no “Parmênides” constitui uma formidável objeção à teoria das Ideias, à qual Platão não apresentou resposta explícita. A despeito disso, cremos ser possível haurir da filosofia platônica uma solução para a referida objeção, solução esta que estaria verossimilmente à disposição do filósofo. Sendo assim, compreende-se que Platão não tenha sido forçado a renunciar à teoria das Ideias, ao menos no que se refere às dificuldades que o Terceiro Homem lhe opõe.The Third Man Argument (...) presented in the “Parmenides” is a formidable objection to the Theory of Forms, one which Plato did not provide an explicit answer to. In spite of that we believe it is possible to draw from Plato’s philosophy a solution to the referred objection, and one which likely would be available to the philosopher. Thus, it becomes clear that Plato was not forced to renounce his Theory of Forms, at least with regard to the difficulties that the Third Man Argument raises to it. (shrink)

What does Plato mean in saying that, for all forms, “F-ness is F”? In such claims, I argue, ‘F’ is being used metonymically to refer to the property of being productive of F-ness rather than to the property of being F, in a way consistent with univocity and the rejection of a genuine Self-Predication Assumption. I explain and defend this productive metonymy reading and show how it can resolve the troubling argument at Phaedo 74b7-c6.