From the Beginning to Plato

142 PYTHAGOREANS AND ELEATICS
But if each [of the many things] is, then it is necessary that it has some size
and bulk, and that one part of it is at a distance from another. The same
account applies to the part in front: for that too will have size and a part of
it will be in front. Now, it is alike to say this once and to keep saying it all
the time: for no such part of it will be the endmost, nor will it be that [any
such part] is not one part next to another. Thus if there are many things, it
must be that they are both small and large: so small as to have no size, so
large as to be infinite.
(DK 29 B 1, Simplicius Physics 141.2–8)
One axiom used is that anything having size contains at least two parts
themselves having size. This clearly generates an unending series of parts having
size. Less clear is the final step from ‘having infinitely many parts with size’ to
‘infinite (in size)’, which apparently was taken with no further argument. There
is some analogy with the ‘Stadium’ and ‘Achilles’ (see (c) below): just as the
runner’s supposedly finite track turns out to contain an infinite series of
substretches, each of positive length, so here the object with supposedly finite
size turns out to contain an infinite series of parts, each having size. If we try to
recompose the original thing out of the parts, we shall never finish, but always be
adding to its size; and this, Zeno might plausibly claim, is just what is meant
when we say something is infinite in size. 36
(f)
Methods and assumptions
In the light of the arguments themselves as preserved, the question of their aims
and methods can be taken up again.
It is evident that some of the assumptions used by Zeno in these arguments are
not due to simple ‘common sense’. Common sense does not make postulates
about the divisibility ad infinitum of things having size; nor suppose that ‘to be is
to be something having a quantity’; nor insist on a single correct way of counting
things. Hence Zeno’s arguments are not directed against unreflecting ‘common
sense’. In fact, these are the kind of assumptions that are naturally and plausibly
made, when one sets about theorizing, in an abstract and mathematical spirit,
about the physical world.
The methods and the style of proof are also mathematical. Note-worthy are the
constructions of progressions ad infinitum, and the remark when one is
constructed: ‘it is alike to say this once and to keep saying it all the time’.
However many times the operation is repeated, that is, it will always turn out
possible to make precisely the same step yet again. 37