Physics   

nothing outside it is complete and whole. For thus we define the
whole-that from which nothing is wanting, as a whole man or a whole
box. What is true of each particular is true of the whole as such-the
whole is that of which nothing is outside. On the other hand that from
which something is absent and outside, however small that may be, is
not 'all'. 'Whole' and 'complete' are either quite identical or
closely akin. Nothing is complete (teleion) which has no end (telos);
and the end is a limit.
Hence Parmenides must be thought to have spoken better than Melissus.
The latter says that the whole is infinite, but the former describes
it as limited, 'equally balanced from the middle'. For to connect the
infinite with the all and the whole is not like joining two pieces of
string; for it is from this they get the dignity they ascribe to the
infinite-its containing all things and holding the all in itself-from
its having a certain similarity to the whole. It is in fact the matter
of the completeness which belongs to size, and what is potentially a
whole, though not in the full sense. It is divisible both in the
direction of reduction and of the inverse addition. It is a whole and
limited; not, however, in virtue of its own nature, but in virtue of
what is other than it. It does not contain, but, in so far as it is
infinite, is contained. Consequently, also, it is unknowable, qua
infinite; for the matter has no form. (Hence it is plain that the
infinite stands in the relation of part rather than of whole. For the
matter is part of the whole, as the bronze is of the bronze statue.)
If it contains in the case of sensible things, in the case of
intelligible things the great and the small ought to contain them. But
it is absurd and impossible to suppose that the unknowable and
indeterminate should contain and determine.
Part 7
It is reasonable that there should not be held to be an infinite in
respect of addition such as to surpass every magnitude, but that there
should be thought to be such an infinite in the direction of division.
For the matter and the infinite are contained inside what contains
them, while it is the form which contains. It is natural too to
suppose that in number there is a limit in the direction of the
minimum, and that in the other direction every assigned number is
surpassed. In magnitude, on the contrary, every assigned magnitude is
surpassed in the direction of smallness, while in the other direction
there is no infinite magnitude. The reason is that what is one is
indivisible whatever it may be, e.g. a man is one man, not many.
Number on the other hand is a plurality of 'ones' and a certain
quantity of them. Hence number must stop at the indivisible: for 'two'
and 'three' are merely derivative terms, and so with each of the other
numbers. But in the direction of largeness it is always possible to
think of a larger number: for the number of times a magnitude can be
bisected is infinite. Hence this infinite is potential, never actual:
the number of parts that can be taken always surpasses any assigned
number. But this number is not separable from the process of
bisection, and its infinity is not a permanent actuality but consists
in a process of coming to be, like time and the number of time.
With magnitudes the contrary holds. What is continuous is divided ad
infinitum, but there is no infinite in the direction of increase. For
the size which it can potentially be, it can also actually be. Hence
since no sensible magnitude is infinite, it is impossible to exceed
every assigned magnitude; for if it were possible there would be
something bigger than the heavens.
The infinite is not the same in magnitude and movement and time, in
the sense of a single nature, but its secondary sense depends on its
primary sense, i.e. movement is called infinite in virtue of the
magnitude covered by the movement (or alteration or growth), and time
because of the movement. (I use these terms for the moment. Later I
shall explain what each of them means, and also why every magnitude is
divisible into magnitudes.)
Our account does not rob the mathematicians of their science, by