From the Beginning to Plato

FROM THE BEGINNING TO PLATO 141
are leads to a definite result. But a definite result implies finitely many things: if
there were infinitely many, counting them would lead to no result at all.
The second limb invokes the relation ‘between’ (metaxu). Any two distinct
things are spatially separate (the converse of Parmenides’ argument for the
oneness of reality from its undividedness). But what separates them must itself
be something that is, and distinct from either. From this principle, an infinite
progression of new entities is constructed.
Though this involves an appeal to spatial properties, it might easily be
rephrased in terms of logical ones. The principle would be: for any two distinct
things, there must be some third thing different from either which distinguishes
them from one another; and so on.
(e)
The argument by ‘sizeless’ and ‘of infinite size’
Again Simplicius is our source. He quotes two chunks of the text, and enough
information to recover the rest in outline.
The first limb claimed that ‘if there are many things, they are so small as to
have no size’. The argument proceeded, according to Simplicius, ‘from the fact
that each of the many things is the same as itself and one’ (Physics 139, 18–19).
It is not difficult to make a plausible reconstruction here. First, to speak of a
‘many’ implies, as in (d), a correct way of counting. The many must be made up
of securely unified ones. Then consider each of these units. The line may have
been (compare Melissus DK B 9): what has size has parts; what has parts is not
one. Hence each of the units must be without size.
The second limb contradicted this in successively stronger ways. First, it
claimed to show that, in a plurality, what is must have size. Suppose something
does not have size, then it cannot be:
For if it were added to another thing that is, it would make it no larger: for
if something is no size, and is added, it is not possible that there should be
any increase in size. This already shows that what is added would be
nothing. But if when it is taken away the other thing will be no smaller, and
again when it is added [the other thing] will not increase, it is clear that
what was added was nothing, and so was what was taken away.
(DK 29 B 2, Simplicius Physics 139.11–15)
This argument in terms of adding and taking away obviously makes essential use
of the assumption ‘there are many things’; it could not, therefore, have been
turned against Parmenides. It also needs some principle such as ‘to be is to be
(something having) a quantity’: not a ‘commonsense’ axiom, but one likely to be
held by most mathematizing theorists of the time. 35
The next and final step proceeds from size to infinite size: