The Basic Works of Aristotle (Modern Library Classics) (150 page)

¹
This is an implication of the ordinary type of judgement, ‘
x
is not white’.

²
The Platonists.

³
The three views appear to have been held respectively by Plato, Xenocrates, and Speusippus.

⁴
sc
. an undifferentiated unity.

⁵
i. e. the principles which are elements and those which are not.

⁶
i. e. the efficient cause is identical with the formal.

⁷
i. e. the causes of substance are the causes of all things.

⁸
i. e. the division into potency and actuality stands in a definite relation to the previous division into matter, form, and privation.

⁹
e. g. the proximate causes of a child are the individual father (who on Aristotle’s view is the efficient and contains the formal cause) and the germ contained in the individual mother (which is the material cause).

¹⁰
In l. 17.

¹¹
In 1070
^b
17.

¹²
Cf. 1069
^a
30.

¹³
Anaxagoras.

¹⁴
Cf.
De Caelo
, iii. 300
^b
8.

¹⁵
Cf.
Timaeus
, 30
A
.

¹⁶
Cf.
Phaedrus
, 245
C
;
Laws
, 894
E
.

¹⁷
Cf.
Timaeus
, 34
B
.

¹⁸
Cf. 1071
^b
22–26.

¹⁹
i. e. the sphere of the fixed stars.

²⁰
i. e. the sun. Cf.
De Gen. et Corr.
ii. 336
^a
23 ff.

²¹
i. e. the outer sphere of the universe, that in which the fixed stars are set.

²²
If it had any movement, it would have the first. But it produces this and therefore cannot share in it; for if it did, we should have to look for something that is prior to the first mover and imparts this motion to it.

²³
sc
. because they are activities or actualities.

²⁴
Cf. 1075
^a
36.

²⁵
Cf. vii. 1028
^b
21, xiv. 1091
^a
34, 1092
^a
11.

²⁶
i. e. the animal or plant is more beautiful and perfect than the seed.

²⁷
i. e. impossible without.

²⁸
The reference is to Plato (Cf.
Phys
. 206
^b
32).

²⁹
Cf.
Phys
. viii. 8, 9;
De Caelo
, i. 2, ii. 3–8.

³⁰
This is to be understood as a general term including both fixed stars and planets.

³¹
Cf. ll. 5–11.

³²
i. e. inwards from, the universe being thought of as a system of concentric spheres encircling the earth.

³³
In 1073
^b
35, 38–1074
^a
4.

³⁴
sc
. in order that higher forms of being may be produced by new combinations of the elements.

³⁵
i. e. the substratum.

³⁶
The reference is to Platonists.

³⁷
The reference is to the Pythagoreans and Speusippus; Cf. xii. 1072
^b
31.

³⁸
Cf. i. 985
^a
4.

³⁹
The special reference is to Plato; Cf.
Rep
. 477.

⁴⁰
Since contraries must contain matter, and matter implies potentiality and contingency.

⁴¹
Cf. 1071
^b
19, 20.

⁴²
Speusippus is meant; Cf. vii. 1028
^b
21, xiv. 1090
^b
13–20.

⁴³
Cf.
Iliad
, ii. 204.

1
     We have stated what is the substance of sensible things, dealing in the treatise on physics
¹
with matter, and later
²
with the substance which has actual existence.
(10)
Now since our inquiry is whether there is or is not besides the sensible substances any which is immovable and eternal, and, if there is, what it is, we must first consider what is said by others, so that, if there is anything which they say wrongly, we may not be liable to the same objections, while, if there is any opinion common to them and us, we shall have no private grievance against ourselves on that account; for one must be content to state some points better than one’s predecessors,
(15)
and others no worse.

Two opinions are held on this subject; it is said that the objects of mathematics—i. e. numbers and lines and the like—are substances, and again that the Ideas are substances.
(20)
And since (1) some recognize these as two different classes—the Ideas and the mathematical numbers, and (2) some recognize both as having one nature, while (3) some others say that the mathematical substances are the only substances,
³
we must consider first
⁴
the objects of mathematics, not qualifying them by any other characteristic—not asking, for instance, whether they are in fact Ideas or not, or whether they are the principles and substances of existing things or not,
(25)
but only whether as objects of mathematics they exist or not, and if they exist, how they exist. Then after this we must separately consider
⁵
the Ideas
themselves in a general way, and only as far as the accepted mode of treatment demands; for most of the points have been repeatedly made even by the discussions outside our school, and, further, the greater part of our account must finish by throwing light on that inquiry,
(30)
viz. when we examine
⁶
whether the substances and the principles of existing things are numbers and Ideas; for after the discussion of the Ideas this remains as a third inquiry.

If the objects of mathematics exist, they must exist either in sensible objects, as some say, or separate from sensible objects (and this also is said by some); or if they exist in neither of these ways,
(35)
either they do not exist, or they exist only in some special sense. So that the subject of our discussion will be not whether they exist but how they exist.

2
     That it is impossible for mathematical objects to exist
in
sensible things, and at the same time that the doctrine in question is an artificial one, has been said already in our discussion of difficulties
⁷
; we have pointed out that it is impossible for two solids to be in the same place, and also that according to the same argument the other powers and characteristics also
⁸
should exist in sensible things and none of them separately.
[1076b]
This we have said already. But, further, it is obvious that on this theory it is impossible for any body whatever to be divided; for it would have to be divided at a plane,
(5)
and the plane at a line, and the line at a point, so that if the point cannot be divided, neither can the line, and if the line cannot, neither can the plane nor the solid. What difference, then, does it make whether sensible things are such indivisible entities, or, without being so themselves, have indivisible entities in them? The result will be the same; if the sensible entities are divided the others will be divided too,
(10)
or else not even the sensible entities can be divided.

But, again, it is not possible that such entities should exist
separately
. For if besides the sensible solids there are to be other solids which are separate from them and prior to the sensible solids, it is plain that besides the planes also there must be other and separate planes and points and lines; for consistency requires this.
(15)
But if these exist, again besides the planes and lines and points of the mathematical solid there must be others which are separate. (For incomposites are prior to compounds; and if there are, prior to the sensible bodies, bodies which are not sensible, by the same argument the
planes which exist by themselves must be prior to those which are in the motionless solids.
(20)
Therefore these will be planes and lines other than those that exist along with the mathematical solids to which these thinkers assign separate existence; for the latter exist along with the mathematical solids, while the others are prior to the mathematical solids.)
(25)
Again, therefore, there will be, belonging to these planes, lines, and prior to them there will have to be, by the same argument, other lines and points; and prior to these points in the prior lines there will have to be other points, though there will be no others prior to these. Now (1) the accumulation becomes absurd; for we find ourselves with one set of solids apart from the sensible solids; three sets of planes apart from the sensible planes—those which exist apart from the sensible planes,
(30)
and those in the mathematical solids, and those which exist apart from those in the mathematical solids; four sets of lines, and five sets of points. With which of these, then, will the mathematical sciences deal? Certainly not with the planes and lines and points in the motionless solid; for science always deals with what is prior.
(35)
And (2) the same account will apply also to numbers; for there will be a different set of units apart from each set of points, and also apart from each set of realities, from the objects of sense and again from those of thought; so that there will be various classes of mathematical numbers.

Again, how is it possible to solve the questions which we have already enumerated in our discussion of difficulties
⁹
? For the objects of astronomy will exist apart from sensible things just as the objects of geometry will; but how is it possible that a heaven and its parts—or anything else which has movement—should exist apart? Similarly also the objects of optics and of harmonics will exist apart; for there will be both voice and sight besides the sensible or individual voices and sights.
[1077a]
Therefore it is plain that the other senses as well, and the other objects of sense, will exist apart; for why should one set of them do so and another not? And if this is so,
(5)
there will also be animals existing apart, since there will be senses.

Again, there are certain mathematical theorems that are universal,
(10)
extending beyond these substances. Here then we shall have another intermediate substance separate both from the Ideas and from the intermediates—a substance which is neither number nor points nor spatial magnitude nor time. And if this is impossible, plainly it is also impossible that the
former
entities should exist separate from sensible things.

And, in general, conclusions contrary alike to the truth and to the usual views follow, if one is to suppose the objects of mathematics to exist thus as separate entities.
(15)
For because they exist thus they must be prior to sensible spatial magnitudes, but in truth they must be posterior; for the incomplete spatial magnitude is in the order of generation prior, but in the order of substance posterior, as the lifeless is to the living.

Again, by virtue of what, and when,
(20)
will mathematical magnitudes be one? For things in our perceptible world are one in virtue of soul, or of a part of soul, or of something else that is reasonable enough; when these are not present, the thing is a plurality, and splits up into parts. But in the case of the subjects of mathematics, which are divisible and are quantities, what is the cause of their being one and holding together?.

Again, the modes of generation of the objects of mathematics show that we are right. For the dimension first generated is length, then comes breadth, lastly depth, and the process is complete. If, then.
(25)
that which is posterior in the order of generation is prior in the order of substantiality, the solid will be prior to the plane and the line. And in
this
way also it is both more complete and more whole, because it can become animate. How, on the other hand, could a line or a plane be animate? The supposition passes the power of our senses.
(30)

Again, the solid is a sort of substance; for it already has in a sense completeness. But how can lines be substances? Neither as a form or shape, as the soul perhaps is, nor as matter, like the solid; for we have no experience of anything that can be put together out of lines or planes or points, while if these had been a sort of material substance,
(35)
we should have observed things which could be put together out of them.

Grant, then, that they are prior in definition.
[1077b]
Still not all things that are prior in definition are also prior in substantiality. For those things are prior in substantiality which when separated from other things surpass them in the power of independent existence, but things are prior in definition to those whose definitions are compounded out of their definitions; and these two properties are not co-extensive. For if attributes do not exist apart from their substances (e. g.
(5)
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.
(10)