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apart from the substances (e.g. a 'mobile' or a pale'), pale is
prior to the pale man in definition, but not in substantiality. For it
cannot exist separately, but is always along with the concrete
thing; and by the concrete thing I mean the pale man. Therefore it
is plain that neither is the result of abstraction prior nor that
which is produced by adding determinants posterior; for it is by
adding a determinant to pale that we speak of the pale man.
It has, then, been sufficiently pointed out that the objects of
mathematics are not substances in a higher degree than bodies are, and
that they are not prior to sensibles in being, but only in definition,
and that they cannot exist somewhere apart. But since it was not
possible for them to exist in sensibles either, it is plain that
they either do not exist at all or exist in a special sense and
therefore do not 'exist' without qualification. For 'exist' has many
senses.
3
For just as the universal propositions of mathematics deal not
with objects which exist separately, apart from extended magnitudes
and from numbers, but with magnitudes and numbers, not however qua
such as to have magnitude or to be divisible, clearly it is possible
that there should also be both propositions and demonstrations about
sensible magnitudes, not however qua sensible but qua possessed of
certain definite qualities. For as there are many propositions about
things merely considered as in motion, apart from what each such thing
is and from their accidents, and as it is not therefore necessary that
there should be either a mobile separate from sensibles, or a distinct
mobile entity in the sensibles, so too in the case of mobiles there
will be propositions and sciences, which treat them however not qua
mobile but only qua bodies, or again only qua planes, or only qua
lines, or qua divisibles, or qua indivisibles having position, or only
qua indivisibles. Thus since it is true to say without qualification
that not only things which are separable but also things which are
inseparable exist (for instance, that mobiles exist), it is true
also to say without qualification that the objects of mathematics
exist, and with the character ascribed to them by mathematicians.
And as it is true to say of the other sciences too, without
qualification, that they deal with such and such a subject-not with
what is accidental to it (e.g. not with the pale, if the healthy thing
is pale, and the science has the healthy as its subject), but with
that which is the subject of each science-with the healthy if it
treats its object qua healthy, with man if qua man:-so too is it
with geometry; if its subjects happen to be sensible, though it does
not treat them qua sensible, the mathematical sciences will not for
that reason be sciences of sensibles-nor, on the other hand, of
other things separate from sensibles. Many properties attach to things
in virtue of their own nature as possessed of each such character;
e.g. there are attributes peculiar to the animal qua female or qua
male (yet there is no 'female' nor 'male' separate from animals); so
that there are also attributes which belong to things merely as
lengths or as planes. And in proportion as we are dealing with
things which are prior in definition and simpler, our knowledge has
more accuracy, i.e. simplicity. Therefore a science which abstracts
from spatial magnitude is more precise than one which takes it into
account; and a science is most precise if it abstracts from
movement, but if it takes account of movement, it is most precise if
it deals with the primary movement, for this is the simplest; and of
this again uniform movement is the simplest form.
The same account may be given of harmonics and optics; for neither
considers its objects qua sight or qua voice, but qua lines and
numbers; but the latter are attributes proper to the former. And
mechanics too proceeds in the same way. Therefore if we suppose
attributes separated from their fellow attributes and make any inquiry
concerning them as such, we shall not for this reason be in error, any
more than when one draws a line on the ground and calls it a foot long
when it is not; for the error is not included in the premisses.
Each question will be best investigated in this way-by setting
up by an act of separation what is not separate, as the
arithmetician and the geometer do. For a man qua man is one
indivisible thing; and the arithmetician supposed one indivisible
thing, and then considered whether any attribute belongs to a man
qua indivisible. 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 [ Pobierz całość w formacie PDF ]