The Regressus ad infinitum in Zenos of Elea argumentation for the simplicityof what is

In the present paper we are presentingthe interpretation of the one Zenos of Elea arguments in favour of the position that a plurality ofbeings is impossible or unthinkable. This is the a contrario proof, in which the regressus ad infinitumis used. The proof is based on the logically exhaustive dilemma: elements of the initial plural being (orthe whole) either have their own elements (i.e. they are divisible), or they have not ones (i.e. they areindivisible). If they have the elements, than the dilemma is repeated with respect to those last elements.Thus, either we have regressus ad infinitum- if we are attempting to think the being as plural orcomplex, - or we reach a element that is absolutely simple, but, nevertheless, is connected with otherelements, so that to obtain the whole. In the first case Zeno affirms that it is impossible to think such ainfinitely complex being. For to think it would mean to complete the last act in an infinite sequence ofacts - i.e. in one which contains acts of the thought distinguishing of parts from the whole. The lastclaim seems to be as much unrealizable one as the claim to point out the last member of an infiniteseries. It is the claim that the completion of an infinite sequence of discrete acts is impossible, as G.Vlastos formulates this. We pay some attention to similarity of this argument to the argument from theDichotomy (or the Race Course), which is the argument against possibility to think a locomotion. Alsoit is necessary to notice that Zeno is using here the principle «the whole is not identical with any of itsparts». In the second case Zeno affirms that the being, which is connected with something, also isconnected with the connection, by means of which it is connected with something. In other words, thebeing, if it possesses some property, also possesses the possession this property . Hence, such abeing isn't absolutely simple one (or isn't indivisible one). But this issue contradicts the selectedalternative. Eo ipso, Zeno accepts here the assumption: any absolutely simple being can't be a part ofsomething, can't belong to something, can't be connected with something, can't have any properties. Intrying to illustrate this assumption, Zeno gives the example of the line and points, which are located onthe line: those points - as simple objects, which have zero magnitudes, - cannot compose in the sum anon-zero magnitude line, even if they are infinite in number. This assumption is, in our view, can beseen as somehow obvious and intuitively acceptable one, unlike the assumptions used by somehistorians of philosophy to justify that multiple things cannot be composed of absolutely simpleelements. For example, S. Makin suggest that Zeno have used the assumption: part inherits all theproperties of the whole. So, if the whole is divided, then each its part is divided, and so on adinfinitum. Other assumption is the W.E. Abrahams one: all what is has a magnitude and is divisibleeverywhere. We provide examples of the so-called "Zeno objects", which are discussed by manycontemporary philosophers. The interest to those objects is generated by the paradoxes, which they arecausing. Thus we show that at least some of Zenos paradoxes are by now far from being resolved,despite the fact that this paradoxes are the subject of a huge number of publications (see the works ofdistinguished scholars, e.g. J. Benardete, P. Benacerraf, A. Grunbaum).

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