ro. (4) If number is finite, how far does the series go? They should tell us both how far it goes, and why it goes just so far. If it stops at 10, then (a) the Forms will soon run short. E.g. the kinds of animal will exceed the numbers up to ro.
18. (8) If the number 3 is the Idea of man, then all the other threes will be Ideas of man, or at any rate men ; thus there will be an infinite number of men.
21. (y) If the smaller number is a part of the greater when composed of the addible units contained in a single number, then if the. horse is 4 and man is 2, man will be a part of the horse.
25. (6) It is absurd that there should be an Idea of ro and not of rr.
27. (e) There are, and come to be, things of which there are not Forms. The Forms, then, are not the causal agencies.
2g. (¢) It is absurd if the number-series up to ro is more of an entity than 10, though never generated asa unity. Yet they treat the series up to ro as a complete number. At least they generate the
succeeding entities, the void, proportion, the odd, &c., within the series up to 10, assigning some to the first principles, others to the numbers. Further, they identify spatial magnitudes (lines, &c.) with numbers short of ro.
(3) What ts the nature of the One?
» 2g. If number is self-subsistent, is 1, or 2, 3, &c., prior? Inasmuch as the number is composite, the one is prior ; inasmuch as the universal or form is prior, the number is prior, being to the units as form to matter.
47. The right angle is in a sense prior to the acute, viz. in definition and because it is determinate ; the acute angle is in a sense prior, because it is a part of the right angle. The acute angle is prior as matter; the right angle which is the union of form and matter is prior because nearer to the form.
13. How, then, is the One a first principle? ‘ Because it is indivi- sible.’ But both the universal and the particular or element are indivisible. ‘They are first principles in different senses, however; the former is first in definition, the latter in time. :
18. They make the One a first principle in both ways. But this is impossible; it cannot have the primariness both of form and of ‘matter. Both the number and the unit are in a sense one, but if the number is actually one (not a mere aggregate), the units exist only potentially. .
23. The cause of the mistake is that they were inquiring from two points of view, that of mathematics and that of general definitions ; from the former they treated the One (i.e. the first principle) as a point, merely divested of position, a minimal material part analogous to the atoms, while from the latter they treated the unity which is predicated of each number as a formal element in the number. These characters, however, cannot belong to the same thing.
32. But if the One must only be without position, differing from the unit merely by being a first principle, and the number 2 is divisible while the unit is not, the unit is liker the One than the number 2 is, and therefore each unit in 2 is prior to 2. But they generate the number 2 first.
1085* 1. Further, if the number 2 is one thing and the number 3 is one thing, they make, together, a 2. What, then, is the origin of this 2?
3. Does the number 2, or one of the units in it, come next after 1?
444 Commentary
(4) Défficulties about the first principles of geometrical objects.
7. Similar difficulties arise about the genera posterior to number— the line, the plane, the solid. (a) Some derive them from the kinds of great and small, lines from the long and short;splanes from the broad and narrow, solids from the deep and shallow. There is a difference of opinion about the formal principle answering to the One.
‘14. These views involve many impossible results. (i) Lines, planes, and solids are cut off from one another, unless their first principles go together so that the broad and narrow is also long and short (in which case the plane would be a line and the solid a plane).
20. (ii) The same difficulty arises as with regard to number ; long, short, &c., are atirzbudes of spatial magnitude, not the ma/ter of it, any more than straight and curved are.
(23. We may put the same difficulty that arises with regard to species when we posit the existence of universals, viz. whether it is ‘animal’ itself or some other ‘animal’ that is present in a particular kind of animal. Similarly if the One and the numbers are self- subsistent, is the unit which we recognize in a number the One itself ?)
gi. (4) Others derive magnitudes from the point (which is akin to the One) and an element akin to plurality. The same difficulties follow.
35. For (i) if the matter is-one, line, plane, and solid will be the same ; (ii) if it is different, the matters either go together or not, so that the plane either will not contain a line or will be a line.
(5) Lhe difficulty of generating numbers and spatial magnitudes ~ as the Platonists generate them.
b4. The difficulties which attend the great and small as material principles of number, attend plurality also if this be taken as the. material principle (Speusippus). One thinker generates number from the plurality which is universally predicated, the other generates it from a particular plurality, viz. the first (the dyad). (@) In both cases we may ask whether the elements are united by mixture, position, fusion, generation, &c.
12. (4) Each unit must be composed of the One and either plurality or a part of plurality. Now the unit, being indivisible, cannot be a plurality, while if its material element be a part of plurality, (a) each of the parts must be indivisible, and it is not, as they say, plurality, but
a part of it, that is the material principle ; (8) number is being derived from a plurality of indivisibles, i.e. from another number.
23. (c) We may inquire with regard to these thinkers too, whether that number is infinite or finite. There was a finite plurality from which the finite units were derived, and there is another ‘ plurality itself’ which is infinite plurality. Which kind. of plurality is the first principle?
27. (d) Similarly (cf. * 32-34) a point cannot be derived from the ‘point itself’ and an interval, nor from the ‘ point itself’ and an indi- visible part of an interval ; for spatial magnitudes are not, like number, composed of indivisibles.
SUMMING UP OF CRITICISM OF IDEAL NUMBERS ° (ch. 9. 1085 3410868 18).
1085” 34. These objections show that number and spatial magni- tudes are not self-subsistent, as is shown also by the diversity of views about numbers, (1) Those who believed only in the objects of mathe- matics (Speusippus) did so because they saw the difficulties about the Ideas.
1086? 5. (2) Those who thought of the Ideas as numbers, and did not see how, if the first principles are what they suppose them to be, mathematical number could exist apart from ideal number (Xeno- crates), made them the same in name, but really did away with mathematical number.
11. (3) The first thinker who held that Forms existed and were numbers, and that mathematical objects existed (Plato), naturally separated them.
13. All are partly right and (as their mutual contradictions show) partly wrong. Their error springs from the wrongness of their assumptions.
1083” 23—1085? 34. So far Aristotle has distinguished the various modes of conceiving numbers as substantial entities, and has criticized them separately. Now he attacks the general view which was common to the Pythagoreans and the Platonists, no longer drawing the distinc- tions drawn in 1080 15—1083? 19. His criticisms from now onwards may best be classified not according to the thinkers attacked but according to the subjects on which he attacks the whole of the two schools. These fall, as Bz. has pointed out, into five groups.
(1) 1083>23-36. How are the numbers produced from the material principle ?
(2) 1083” 36—1084" 2. How many ideal numbers are there ?
446 Commentary
(3) 1084 2—-10852 7. What is the nature of the One?
(4) 1085° 7—- 4. On the principles of geometrical objects.
(5) 1085» 4-34. On the difficulty of generating numbers from unity and multitude, and spatial magnitudes from similar principles.
23. Aristotle begins with a difficulty arising out of Plato’s (cf. 1081 24) description of the units in the ideal two as produced through the equalization of the great and small by the One. Aristotle appears, as Bz. points out, to have misconceived the nature of the Platonic material principle. It was no doubt conceived as a principle which was both great and small; i.e., it was indeterminate quantity. But Aristotle habitually speaks of the great and /he small as if they were two distinct principles, and his argument here turns entirely on this point. We may note a significant looseness in Aristotle’s way of referring to the principle. Sometimes (and, we must suppose, more correctly) it is 7d peya Kal puxpor, e.g. B. 998 10, M. 1083> 32, 10852 12, N. 1087» 8. At other times it is 76 peya Kal 76 palkpor, @.g. A. 987» 20, 988 26, M. 1083? 27, 10852 9, N. 1087 11, 14, 16, and this is what the argument here requires.
28-30. ‘ Further, what account can the Platonists give of the units in the ideal Three? One of them is an odd unit and cannot be assigned to the great or to the small (since these produce only one unit each); which is perhaps why they make the ideal One the middle unit in odd numbers.’ For the fact that they did so cf. Diels, Vorsokr.’ 270. 18. And why should they not? we might ask. Aristotle’s answer would doubtless be that if they make the One a purely formative principle in the case of even numbers, they have no right to make it one of the material elements of odd numbers. We can hardly suppose, however, that they did this ; it is more likely that they represented the One as a sort of arbiter (cf. Al. 767. 17 tv THs povddos peciteiav) between the tendencies to excess and to defect.
30-32. ‘If each of the units in the ideal Two comes from both the great and the small, these being equalized, how will the ideal Two be a single entity composed of the great and the small? Or how will it differ from one of its units?’, sc. if each of the units turns out to be what they describe the ideal Two as being, viz. what is produced by the co-operation of the One and _ the-great-and-the-small (which dvorows jv |. 36).
36. ddpiotos Suds. Robin quotes this as one of the few passages definitely relating to Plato in which this term is used. The other passages he cites are N. 1088815, 10915, besides various places in later writers (Robin, 643-5).
37. xwptotov yap morodor. Aristotle holds that if you regard number as a separately existing substance, you have to say that it actually is finite or that it actually is infinite, and both alternatives are impossible ; he believes himself to escape the difficulty by holding that number does not exist as something given for all time, but only in the process of counting, and that it is potentially infinite in the sense that, however high a number has been counted, a higher can be counted. Acdrerau
M. 8. 10835 23 — 1084? Io 447
Suvdper eivar TO dreipov Phys, 206% 18,133 ovrws orl 7d dzreipov, TO det GAXOo Kat GAA AapBaverGar 206% 27: it is not a rode Tu like a man or a house but like a day or a contest, whose being is not that of a substance but is always in course of destruction or generation.
10842 4-7. The peculiar words wimrew, éumirrew (not elsewhere found in Aristotle, nor, perhaps, in other authors, in this connexion) are probably Academic terms to express the mode of generation of numbers. ‘There are three cases:
(1) By addition (dt pev) of 1 to an even number an odd number is produced.
(2) By multiplication (dt dé) (z) of 1 by 2 a power of 2 is pro- duced,
(4) of an even number by an odd number, an even number not a power of 2 is produced.