In this paper the claim that Zeno's paradoxes have been solved is contested.
Although no one has ever touched Zeno without refuting him (Whitehead), it will
be our aim to show that, whatever it was that was refuted, it was certainly not
Zeno. The paper is organised in two parts. In the first part we will
demonstrate that upon direct analysis of the Greek sources, an underlying
structure common to both the Paradoxes of Plurality and the Paradoxes of Motion
can be exposed. This structure bears on a correct - Zenonian - interpretation
of the concept of division through and through. The key feature, generally
overlooked but essential to a correct understanding of all his arguments, is
that they do not presuppose time. Division takes place simultaneously. This
holds true for both PP and PM. In the second part a mathematical representation
will be set up that catches this common structure, hence the essence of all
Zeno's arguments, however without refuting them. Its central tenet is an
aequivalence proof for Zeno's procedure and Cantor's Continuum Hypothesis. Some
number theoretic and geometric implications will be shortly discussed.
Furthermore, it will be shown how the Received View on the motion-arguments can
easely be derived by the introduction of time as a (non-Zenonian) premiss, thus
causing their collapse into arguments which can be approached and refuted by
Aristotle's limit-like concept of the potentially infinite, which remained -
though in different disguises - at the core of the refutational strategies that
have been in use up to the present. Finally, an interesting link to Newtonian
mechanics via Cremona geometry can be established.Comment: 41 pages, 7 figure
