But the geometer treats him neither qua man nor qua
indivisible, but as a solid. For evidently the properties which
would have belonged to him even if perchance he had not been
indivisible, can belong to him even apart from these attributes. Thus,
then, geometers speak correctly; they talk about existing things,
and their subjects do exist; for being has two forms-it exists not
only in complete reality but also materially.
Now since the good and the beautiful are different (for the former
always implies conduct as its subject, while the beautiful is found
also in motionless things), those who assert that the mathematical
sciences say nothing of the beautiful or the good are in error. For
these sciences say and prove a great deal about them; if they do not
expressly mention them, but prove attributes which are their results
or their definitions, it is not true to say that they tell us
nothing about them. The chief forms of beauty are order and symmetry
and definiteness, which the mathematical sciences demonstrate in a
special degree. And since these (e.g. order and definiteness) are
obviously causes of many things, evidently these sciences must treat
this sort of causative principle also (i.e. the beautiful) as in
some sense a cause. But we shall speak more plainly elsewhere about
these matters.
4
So much then for the objects of mathematics; we have said that
they exist and in what sense they exist, and in what sense they are
prior and in what sense not prior.
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