The Basic Works of Aristotle (Modern Library Classics) (151 page)

3
     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,
¹⁰
(15)
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. 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,
¹¹
(20)
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,
(25)
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,
(30)
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,
(35)
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;
[1078a]
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,
(5)
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,
(10)
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.
(15)
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.
(20)

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.
(25)
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.
¹²
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.
(30)

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
¹³
are in error. For these sciences say and prove a great deal about them; if they do not expressly mention them,
(35)
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.
[1078b]
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.
(5)
But we shall speak more plainly elsewhere
¹⁴
about these matters.

4
     So much then for the objects of mathematics; we have said that they exist and in what sense they exist,
¹⁵
and in what sense they are prior and in what sense not prior.
¹⁶
Now, regarding the Ideas,
(10)
we must first examine the ideal theory itself, not connecting it in any way with the nature of numbers, but treating it in the form in which it was originally understood by those who first maintained the existence of the Ideas. The supporters of the ideal theory were led to it because on the question about the truth of things they accepted the Heraclitean sayings which describe all sensible things as ever passing away,
(15)
so that if knowledge or thought is to have an object, there must be some other and permanent entities, apart from those which are sensible; for there could be no knowledge of things which were in a state of flux. But when Socrates was occupying himself with the excellences of character, and in connexion with them became the first to raise the problem of universal definition (for of the physicists Democritus only touched on the subject to a small extent,
(20)
and defined, after a fashion, the hot and the cold; while the Pythagoreans had before this treated of a few things, whose definitions—e. g. those of opportunity, justice, or marriage—they connected with numbers; but it was natural that Socrates should be seeking the essence, for he was seeking to syllogize, and ‘what a thing is’ is the starting-point of syllogisms; for there was as yet none of the dialectical power which enables people even without knowledge of the essence to speculate about contraries and inquire whether the same science deals with contraries; for two things may be fairly ascribed to Socrates—inductive arguments and universal definition,
(25)
both of which are concerned with the starting-point of science):—but Socrates did not make the universals or the definitions exist apart;
they
,
(30)
however, gave them separate existence, and this was the kind of thing they called Ideas. Therefore it followed for them, almost by the same argument, that there must be Ideas of all things that are spoken of universally, and it was almost as if a. man wished to count certain things,
(35)
and while they were few thought he would not be able to count them, but made more of them and then counted them; for the Forms are, one may say, more numerous than the particular sensible things, yet it was in seeking the causes of these that they proceeded from them to the Forms.
[1079a]
For to each thing there answers an entity which has the same
name and exists apart from the substances, and so also in the case of all other groups there is a one over many, whether these be of this world or eternal.

Again, of the ways in which it is proved that the Forms exist, none is convincing; for from some no inference necessarily follows,
(5)
and from some arise Forms even of things of which they think there are no Forms. For according to the arguments from the sciences there will be Forms of all things of which there are sciences, and according to the argument of the ‘one over many’ there will be Forms even of negations, and according to the argument that thought has an object when the individual object has perished,
(10)
there will be Forms of perishable things; for we have an image of these. Again, of the most accurate arguments, some lead to Ideas of relations, of which they say there is no independent class, and others introduce the ‘third man’.
¹⁷

And in general the arguments for the Forms destroy things for whose existence the believers in Forms are more zealous than for the existence of the Ideas; for it follows that not the dyad but number is first,
(15)
and that prior to number is the relative, and that this is prior to the absolute
¹⁸
—besides all the other points on which certain people, by following out the opinions held about the Forms, came into conflict with the principles of the theory.

Again, according to the assumption on which the belief in the Ideas rests, there will be Forms not only of substances but also of many other things; for the concept is single not only in the case of substances,
(20)
but also in that of non-substances, and there are sciences of other things than substance; and a thousand other such difficulties confront them. But according to the necessities of the case and the opinions about the Forms,
(25)
if they can be shared in there must be Ideas of substances only. For they are not shared in incidentally, but each Form must be shared in as something not predicated of a subject. (By ‘being shared in incidentally’ I mean that if a thing shares in ‘double itself’, it shares also in ‘eternal’, but incidentally; for ‘the double’ happens to be eternal.) Therefore the Forms will be substance.
(30)
But the same names indicate substance in this and in the ideal world (or what will be the meaning of saying that there is something apart from the particulars—the one over many?). And if the Ideas and the things that share in them have the same form, there will be
something common: for why should ‘2’ be one and the same in the perishable 2’s,
(35)
or in the 2’s which are many but eternal, and not the same in the ‘2 itself’ as in the individual 2? But if they have not the same form, they will have only the name in common, and it is as if one were to call both Callias and a piece of wood a ‘man’, without observing any community between them.
¹⁹
[1079b]

But if we are to suppose that in other respects the common definitions apply to the Forms, e. g. that ‘plane figure’ and the other parts of the definition apply to the circle-itself,
(5)
but ‘what really is’ has to be added, we must inquire whether this is not absolutely meaningless. For to what is this to be added? To ‘centre’ or to ‘plane’ or to all the parts of the definition? For all the elements in the essence are Ideas, e. g. ‘animal’ and ‘two-footed’.
²⁰
Further,
(10)
there must be some Idea answering to ‘plane’ above, some nature which will be present in all the Forms as their genus.

5
     Above all one might discuss the question what in the world the Forms contribute to sensible things, either to those that are eternal or to those that come into being and cease to be; for they cause neither movement nor any change in them.
(15)
But again they help in no wise either towards the knowledge of other things (for they are not even the substance of these, else they would have been in them), or towards their being, if they are not
in
the individuals which share in them; though if they were, they might be thought to be causes, as white causes whiteness in a white object by entering into its composition.
(20)
But this argument, which was used first by Anaxagoras, and later by Eudoxus in his discussion of difficulties and by certain others, is very easily upset; for it is easy to collect many and insuperable objections to such a view.

But, further, all other things cannot come from the Forms in any of the usual senses of ‘from’.
(25)
And to say that they are patterns and the other things share in them is to use empty words and poetical metaphors. For what is it that works, looking to the Ideas? And any thing can both be and come into being without being copied from something else, so that, whether Socrates exists or not,
(30)
a man like Socrates might come to be. And evidently this might be so even if Socrates were eternal. And there will be several patterns of the same thing, and therefore several Forms; e. g. ‘animal’ and ‘two-footed’, and also ‘man-himself’, will be Forms of man. Again, the Forms are patterns not only of sensible things, but of Forms themselves
also; i. e. the genus is the pattern of the various forms-of-a-genus; therefore the same thing will be pattern and copy.

Again, it would seem impossible that substance and that whose substance it is should exist apart; how,
(35)
therefore, could the Ideas, being the substances of things, exist apart?

In the
Phaedo
²¹
the case is stated in this way—that the Forms are causes both of being and of becoming.
[1080a]
Yet though the Forms exist, still things do not come into being, unless there is something to originate movement; and many other things come into being (e. g. a house or a ring) of which they say there are no Forms.
(5)
Clearly therefore even the things of which they say there are Ideas can both be and come into being owing to such causes as produce the things just mentioned,
²²
and not owing to the Forms. But regarding the Ideas it is possible, both in this way and by more abstract and accurate arguments,
(10)
to collect many objections like those we have considered.