Three of Zeno's objections to motion are answered by utilizing a version of nonstandard analysis, internal set theory, interpreted within an empirical context. Two of the objections are without force because they rely upon infinite sets, which always contain nonstandard real numbers. These numbers are devoid of numerical meaning, and thus one cannot render the judgment that an object is, in fact, located at a point in spacetime for which they would serve as coordinates. The third objection, an arrow never (...) appears to be moving, is answered by showing that it only applies to a finite number of instants of time. A theory of motion is also advanced; it consists of a finite series of contiguous infinitesimal steps. The theory is immune to Zeno's first two objections because the number of steps is finite and each lies outside the domain of observation. Present motion is hypothesized to be an unobservable process taking place within each step. The fact of motion is apparent through a summing (Riemann integration) of the steps. (shrink)

After presenting the rules of Eleatic antilogic, i.e., dialectic, I argue that Zeno was a practitioner, and, on the basis of key passages from Plato’s Parmenides, that his paradoxes of divisibility and movement were notreductio ad absurdum, but simple derivation of impossibilities meant to ridicule Parmenides’ adversaries. Thus, Zeno did not try to prove that there is no motion, but simply derived this consequence from premises held by his opponents. I argue further that these paradoxes were devised, in accordance with (...) Eleatic antilogic, following a scheme that included hypotheses and their contradictories, within which the subject is to be treated both “in relation to itself,” and “in relation to other things”. (shrink)

Dans la première partie duParménide, Socrate présente une théorie des Formes qui explique la comprésence d’opposés dans les choses ordinaires et soutient que les Formes ne peuvent avoir des caractéristiques opposées. Dans la deuxième partie, Parménide s’appuie sur les propos de Socrate; il en dérive des conséquences inacceptables — que la Forme de l’Un n’existe pas, et ainsi, que rien n’existe. Cette conclusion est indéniablement fausse. Pour éviter ceci, Socrate doit abandonner la thèse exposée dans la première partie et trouver (...) une manière de préserver la fonction explicative des Formes. Cet article expose la structure de l’exercice qui occupe la deuxième partie du dialogue. (shrink)

_ Source: _Volume 10, Issue 1, pp 65 - 94 In his highly influential 1928 article ‘The _Parmenides_ of Plato and the Origin of the Neoplatonic “One”,’ E.R. Dodds argued, _inter alia_, that among the so-called Neoplatonists Plotinus was the first to interpret Plato’s _Parmenides_ in terms of the distinctive three ‘hypostases’, One, Intellect, and Soul. Dodds argued that this interpretation was embraced and extensively developed by Proclus, among others. In this paper, I argue that although Plotinus took _Parmenides_ to (...) contain a sort of outline of the true metaphysical principles, he understood the One of the first hypothesis of the second part of the dialogue in a way importantly different from the way that Proclus understood it. The characterization of this One, especially its identity with the Idea of the Good of _Republic_, has significant ramification for Plotinus’ philosophy that set it apart from Proclus’ philosophy in ways hitherto infrequently noted. The widely accepted reasons for rejecting Proclus’ interpretation do not apply to the interpretation of Plotinus. The two different interpretations help explain why Proclus’ notorious proliferation of entities in the intelligible realm is not found in Plotinus. (shrink)

This article shows how Plato’s Parmenides explores the relationship between place and the geometric forms that are inscribed in it, independently of sensation, becoming and causality. The analysis concurs with essential points of what is said in the Timaeus about khôra, this receptacle deprived of any intrinsic qualities, on which every sensitive reality is drawn.

This is a study of the correspondence between Forms and particulars in Plato. The aim is to determine whether they exhibit an ontological symmetry, in other words, whether there is always one where there is the other. This points to two questions, one on the existence of things that do not have corresponding Forms, the other on the existence of Forms that do not have corresponding things. Both questions have come up before. But the answers have not been sufficiently sensitive (...) to the intricacies of the questions. Nor have they been adequately resourceful with what little evidence there is in the original sources. The intention here is to make up for that deficiency, not just with better answers but also with better insight into the questions. (shrink)

A discussão nos livros dos Tópicos giram em torno dos debates dialéticos e seus elementos. Aristóteles discorre sobre os gêneros, as propriedades e os chamados acidentes e suas relações predicativas em categorias. Interessa-nos, sobretudo, compreender o papel desempenhado pela Definição e qual sua relação com os demais instrumentos para a dialética.