Modes to present some Zeno Of Elea’s arguments “against plurality”

In this article we try to formalize, summarize and classify some Zeno of Elea’s arguments “against any plurality of a being”. We show that these arguments can be interpreted as revealing the difficulties in describing the relation of the whole and its parts. In the first group of these arguments the whole is considered as “non-holistic” one, and a contradiction is derived from this, and in the second group the whole is considered as “holistic” one, which is about, on the one hand, some difficulties in identifying the parts of this whole, and on the other hand it is about an infinite regress. It is shown that there is a connection of these arguments’ meaning and structure with some cases of Plato’s and Aristotle’s identifying mereological difficulties (some Plato’s and Aristotle’s texts belong to the first group, and other - to the second one). In addition to the properly mereological difficulties, these arguments are associated with several discussions in Plato's “Parmenides”, “Theaetetus” and others dialogues. Those discussions are aimed to demonstrate the impossibility to constitute a bound object (and in this sense the “whole” one), if such a connection or relationship as “participation” (in “Parmenides”) is used. Also those discussions are aimed to demonstrate proposition inability for one proposition to be a “part” of the other proposition (in “Theaetetus”). Also, these arguments related to the discussion of the impossibility of the formation of a complex proposition from simple ones in Sextus Empiricus. We show that Sextus’ arguments on the incomprehensibility of the intentional object of one cognitive power by other cognitive power, on unknowability external world’s objects, on the impossibility to grasp anything through its designation, on a regress that arises in searching criterion of truth or justification, too, are all of Zeno’s type arguments.The same type of argument applies to a famous problem ofNeoplatonic metaphysics, which was clearly expounded by Proclus: how the First Cause can be completely independent of their effects, and? At the same time, to be related with them as their cause? However, in this case, there are difficulties with the identification and individuation of ta eide in Nous. Those difficulties are similar to the difficulties which arise in trying to distinguish between interdependent propositions - if in this propositions’ distinguishing we use the criterion of G. Frege.If these arguments generalize and formalize, they can be incorporated into a modern discussion of holistic systems (W.V.O. Quine gave the impulse to this discussions): the assumptions on which the description of holistic systems is based, lead to serious difficulties; but to give up these assumptions is in many situations more difficult than to agree with them - as can be seen from the work of M. Harrell and M. Esfeld. In addition, we show how the arguments that use an infinite regress, can be presented in convincing form to meet the strict criteria of C. Gratton, and how the “viciousness” of the regress under consideration associated with paradoxes identified by B. Russell at the beginning of the XX entury.

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