Form come from many Forms? And if the number comes not from the many
numbers themselves but from the units in them, e.g. in 10,000, how
is it with the units? If they are specifically alike, numerous
absurdities will follow, and also if they are not alike (neither the
units in one number being themselves like one another nor those in
other numbers being all like to all); for in what will they differ, as
they are without quality? This is not a plausible view, nor is it
consistent with our thought on the matter.

    Further, they must set up a second kind of number (with which
arithmetic deals), and all the objects which are called 'intermediate'
by some thinkers; and how do these exist or from what principles do
they proceed? Or why must they be intermediate between the things in
this sensible world and the things-themselves?

    Further, the units in must each come from a prior but this is
impossible.

    Further, why is a number, when taken all together, one?

    Again, besides what has been said, if the units are diverse the
Platonists should have spoken like those who say there are four, or
two, elements; for each of these thinkers gives the name of element
not to that which is common, e.g. to body, but to fire and earth,
whether there is something common to them, viz. body, or not. But in
fact the Platonists speak as if the One were homogeneous like fire
or water; and if this is so, the numbers will not be substances.
Evidently, if there is a One itself and this is a first principle,
'one' is being used in more than one sense; for otherwise the theory
is impossible.

    When we wish to reduce substances to their principles, we state
that lines come from the short and long (i.e. from a kind of small and
great), and the plane from the broad and narrow, and body from the
deep and shallow. Yet how then can either the plane contain a line, or
the solid a line or a plane? For the broad and narrow is a different
class from the deep and shallow. Therefore, just as number is not
present in these, because the many and few are different from these,
evidently no other of the higher classes will be present in the lower.
But again the broad is not a genus which includes the deep, for then
the solid would have been a species of plane. Further, from what
principle will the presence of the points in the line be derived?
Plato even used to object to this class of things as being a
geometrical fiction. He gave the name of principle of the line-and
this he often posited-to the indivisible lines. Yet these must have
a limit; therefore the argument from which the existence of the line
follows proves also the existence of the point.

    In general, though philosophy seeks the cause of perceptible
things, we have given this up (for we say nothing of the cause from
which change takes its start), but while we fancy we are stating the
substance of perceptible things, we assert the existence of a second
class of substances, while our account of the way in which they are
the substances of perceptible things is empty talk; for 'sharing',
as we said before, means nothing.

    Nor have the Forms any connexion with what we see to be the
cause in the case of the arts, that for whose sake both all mind and
the whole of nature are operative,-with this cause which we assert
to be one of the first principles; but mathematics has come to be
identical with philosophy for modern thinkers, though they say that it
should be studied for the sake of other things. Further, one might
suppose that the substance which according to them underlies as matter
is too mathematical, and is a predicate and differentia of the
substance, ie. of the matter, rather than matter itself; i.e. the
great and the small are like the rare and the dense which the physical
philosophers speak of, calling these the primary differentiae of the
substratum; for these are a kind of excess and defect. And regarding
movement, if the great and the small are to he movement, evidently the
Forms will be moved; but if they are not to be movement, whence did
movement come? The whole study of nature has been annihilated.

    And what is thought to be easy-to show that all things are
one-is not done; for what is proved by the method of setting out
instances is not that all things are one but that there is a One
itself,-if we grant all the assumptions. And not even this follows, if
we do not grant that the universal is a genus; and this in some
cases it cannot be.

    Nor can it be explained either how the lines and planes and solids
that come after the numbers exist or can exist, or what significance
they have; for these can neither be Forms (for they are not
numbers), nor the intermediates (for those are the objects of
mathematics), nor the perishable things. This is evidently a
distinct fourth class.

    In general, if we search for the elements of existing things
without distinguishing the many senses in which things are said to
exist, we cannot find them, especially if the search for the
elements of which things are made is conducted in this manner. For
it is surely impossible to discover what 'acting' or 'being acted on',
or 'the straight', is made of, but if elements can be discovered at
all, it is only the elements of substances; therefore either to seek
the elements of all existing things or to think one has them is
incorrect.

    And how could we learn the elements of all things? Evidently we
cannot start by knowing anything before. For as he who is learning
geometry, though he may know other things before, knows none of the
things with which the science deals and about which he is to learn, so