Those, then, who posit the Ideas and identify them with numbers, by their assumption (in accordance with their method of abstracting each general term from its several concrete examples) that every general term is a unity, make some attempt to explain why number exists. Footnote 1I have followed Ross’s text and interpretation of this sentence. For the meaning cf. Aristot. Met. 14.2.20. Since, however, their arguments are neither necessarily true nor indeed possible, there is no justification on this ground for maintaining the existence of number.
The Pythagoreans, on the other hand, observing that many attributes of numbers apply to sensible bodies, assumed that real things are numbers; not that numbers exist separately, but that real things are composed of numbers. Footnote 2See Introduction. But why? Because the attributes of numbers are to be found in a musical scale, in the heavens, and in many other connections. Footnote 3Cf. Aristot. Met. 14.6.5.
As for those who hold that mathematical number alone exists, Footnote 4Cf. Aristot. Met. 14.2.21. they cannot allege anything of this kind Footnote 5i.e., that things are composed of numbers. consistently with their hypotheses; what they did say was that the sciences could not have sensible things as their objects. But we maintain that they can; as we have said before. And clearly the objects of mathematics do not exist in separation; for if they did their attributes would not be present in corporeal things.
Thus in this respect the Pythagoreans are immune from criticism; but in so far as they construct natural bodies, which have lightness and weight, out of numbers which have no weight or lightness, they appear to be treating of another universe and other bodies, not of sensible ones. Footnote 6See Introduction.
But those who treat number as separable assume that it exists and is separable because the axioms will not apply to sensible objects; whereas the statements of mathematics are true and appeal to the soul. Footnote 7The statements of mathematics appeal so strongly to our intelligence that they must be true; therefore if they are not true of sensible things, there must be some class of objects of which they are true. The same applies to mathematical extended magnitudes.
It is clear, then, both that the contrary theory Footnote 8The Pythagorean theory, which maintains that numbers not only are present in sensible things but actually compose them, is in itself an argument against the Speusippean view, which in separating numbers from sensible things has to face the question why sensible things exhibit numerical attributes. can make out a case for the contrary view, and that those who hold this theory must find a solution for the difficulty which was recently raised Footnote 9sect. 3. —why it is that while numbers are in no way present in sensible things, their attributes are present in sensible things.
There are some Footnote 10Probably Pythagoreans. Cf. Aristot. Met. 7.2.2, Aristot. Met. 3.5.3. who think that, because the point is the limit and extreme of the line, and the line of the plane, and the plane of the solid, there must be entities of this kind.
We must, then, examine this argument also, and see whether it is not exceptionally weak. For (1.) extremes are not substances; rather all such things are merely limits. Even walking, and motion in general, has some limit; so on the view which we are criticizing this will be an individual thing, and a kind of substance. But this is absurd. And moreover (2.) even if they are substances, they will all be substances of particular sensible things, since it was to these that the argument applied. Why, then, should they be separable?
Again, we may, if we are not unduly acquiescent, further object with regard to all number and mathematical objects that they contribute nothing to each other, the prior to the posterior. For if number does not exist, none the less spatial magnitudes will exist for those who maintain that only the objects of mathematics exist; and if the latter do not exist, the soul and sensible bodies will exist. Footnote 11That the criticism is directed against Speusippus is clear from Aristot. Met. 7.2.4. Cf. Aristot. Met. 12.10.14.
But it does not appear, to judge from the observed facts, that the natural system lacks cohesion, like a poorly constructed drama. Those Footnote 12Xenocrates (that the reference is not to Plato is clear from sect. 11). who posit the Ideas escape this difficulty, because they construct spatial magnitudes out of matter and a number—2 in the case of lines, and 3, presumably, in that of planes, and 4 in that of solids; or out of other numbers, for it makes no difference.
But are we to regard these magnitudes as Ideas, or what is their mode of existence? and what contribution do they make to reality? They contribute nothing; just as the objects of mathematics contribute nothing. Moreover, no mathematical theorem applies to them, unless one chooses to interfere with the principles of mathematics and invent peculiar theories Footnote 13e.g. that of indivisible lines.
of one’s own. But it is not difficult to take any chance hypotheses and enlarge upon them and draw out a long string of conclusions.
These thinkers, then, are quite wrong in thus striving to connect the objects of mathematics with the Ideas. But those who first recognized two kinds of number, the Ideal and the mathematical as well, neither have explained nor can explain in any way how mathematical number will exist and of what it will be composed; for they make it intermediate between Ideal and sensible number.
For if it is composed of the Great and Small, it will be the same as the former, i.e. Ideal, number. But of what other Great and Small can it be composed? for Plato makes spatial magnitudes out of a Great and Small. Footnote 14This interpretation (Ross’s second alternative, reading τίνος for τινος) seems to be the most satisfactory. For the objection cf. Aristot. Met. 3.4.34. And if he speaks of some other component, he will be maintaining too many elements; while if some one thing is the first principle of each kind of number, unity will be something common to these several kinds.
We must inquire how it is that unity is these many things, when at the same time number, according to him, cannot be derived otherwise than from unity and an indeterminate dyad. Footnote 15The argument may be summarized thus. If mathematical number cannot be derived from the Great-and-Small or a species of the Great-and-Small, either it has a different material principle (which is not economical) or its formal principle is in some sense distinct from that of the Ideal numbers. But this implies that unity is a kind of plurality, and number or plurality can only be referred to the dyad or material principle.
All these views are irrational; they conflict both with one another and with sound logic, and it seems that in them we have a case of Simonides’ long story Footnote 16The exact reference is uncertain, but Aristotle probably means Simonides of Ceos. Cf. Simonides Fr. 189 (Bergk).
; for men have recourse to the long story,
such as slaves tell, when they have nothing satisfactory to say.
The very elements too, the Great and Small, seem to protest at being dragged in; for they cannot possibly generate numbers except rising powers of 2. Footnote 17Assuming that the Great-and-Small, or indeterminate dyad, is duplicative (Aristot. Met. 13.7.18).
It is absurd also, or rather it is one of the impossibilities of this theory, to introduce generation of things which are eternal.
There is no reason to doubt whether the Pythagoreans do or do not introduce it; for they clearly state that when the One had been constituted—whether out of planes or superficies or seed or out of something that they cannot explain—immediately the nearest part of the Infinite began to be drawn in and limited by the Limit. Footnote 18Cf. Aristot. Physics 3.4, Aristot. Physics 4.6, and Burnet, E.G.P. sect. 53.
However, since they are here explaining the construction of the universe and meaning to speak in terms of physics, although we may somewhat criticize their physical theories, it is only fair to exempt them from the present inquiry; for it is the first principles in unchangeable things that we are investigating, and therefore we have to consider the generation of this kind of numbers.