The Complete Works of Aristotle

by Barnes, Jonathan, Aristotle

  Again, since the ideal 1 is first, and then there is a 1 which is first among the others and next after the ideal 1, and again a third which is next after the second [30] and next but one after the first 1, the units must be prior to the numbers by which they are named in counting, e.g. there will be a third unit in 2 before 3 exists, and a fourth and a fifth in 3 before the numbers 4 and 5 exist.—None of these thinkers [35] has said the units are non-comparable in this way, but according to their principles even this way is reasonable, though in truth it is impossible. For it is reasonable that [1081b1] the units should have priority and posteriority if there is a first unit and a first 1, and the 2’s also if there is a first 2; for after the first it is reasonable and necessary that there should be a second, and if a second, a third, and so with the others [5] successively. (And to say both at the same time, that a unit is first and another unit is second after the ideal 1, and that a 2 is first after it, is impossible.) But they make a first unit and 1, but not a second and a third, and a first 2, but not a second and a third.

  [10] Clearly, also, it is not possible, if all the units are non-comparable, that there should be an ideal 2 and 3; and similarly in the case of the other numbers. For whether the units are undifferentiated or each differs from each, number must be [15] counted by addition, e.g. 2 by adding another one to the one, 3 by adding another one to the two, and 4 similarly. This being so, numbers cannot be generated, as they generate them, from the dyad and the 1; for 2 becomes part of 3, and 3 of 4, and the [20] same happens in the case of the succeeding numbers, but for them 4 came from the first 2 and the indefinite 2,—which makes it two 2’s other than the ideal 2; if not, the ideal 2 will be a part of 4 and one other 2 will be added. And similarly 2 will [25] consist of the ideal 1 and another 1; but if this is so, the other element cannot be an indefinite 2; for it generates a unit, but not a definite 2. Again, besides the ideal 3 and the ideal 2 how can there be other 3’s and 2’s? And how do they consist of prior [30] and posterior units? All these doctrines are absurd and fiction, and there cannot be a first 2 and then an ideal 3. Yet there must, if the 1 and the indefinite dyad are to be the elements. But if the results are impossible, it is also impossible that these are the principles.

  If the units, then, are differentiated, each from each, these results and others [35] similar to these follow of necessity. But if those in different numbers are differentiated, but those in the same number are alone undifferentiated from one [1082a1] another, even so the difficulties that follow are no less. E.g. in the ideal 10 there are ten units, and the 10 is composed both of them and of two 5’s. But since the ideal 10 is not any chance number nor composed of any chance 5’s—or, for that matter, [5] units—the units in this 10 must differ. For if they do not differ, neither will the 5’s of which the 10 consists differ; but since they differ, the units also will differ. But if they differ, will there be no other 5’s in the 10 but only these two, or will there be [10] others? If there are not, this is paradoxical; and if there are, what sort of 10 will consist of them? For there is no other 10 in the 10 but itself. But it is also necessary that the 4 should not consist of any chance 2’s; for the indefinite 2, as they say, took the definite 2 and made two 2’s; for its nature was to double what it took.

  [15] Again, as to the 2 being a thing apart from the two units, and the 3 a thing apart from the three units, how is this possible? Either by one’s sharing in the other, as white man is different from white and man (for it shares in these), or when one is a differentia of the other, as man is different from animal and two-footed. Again, [20] some things are one by contact, some by intermixture, some by position; none of which relations can belong to the units of which the 2 or the 3 consists; but as two men are not a unity apart from both, so must it be with the units. And their being [25] indivisible will make no difference to them; for points are indivisible, but yet a pair of them is nothing apart from the two.

  But this consequence also we must not forget, that it follows that there are prior and posterior 2’s, and similarly with the other numbers. For let the 2’s in the 4 [30] be simultaneous; yet these are prior to those in the 8, and as the 2 generated them, they generated the 4’s in the ideal 8. Therefore if the first 2 is an Idea, these 2’s also will be Ideas. And the same account applies to the units; for the units in the first 2 generate the four in 4, so that all the units come to be Ideas and an Idea will be [35] composed of Ideas. Clearly therefore those things also, of which these are Ideas, will be composite, e.g. one might say that animals are composed of animals, if there are [1082b1] Ideas of them.

  In general, to differentiate the units in any way is an absurd fiction; and by a fiction I mean that which is brought in forcibly to suit a hypothesis. For neither in quantity nor in quality do we see unit differing from unit, and number must be [5] either equal or unequal—all number but especially that which consists of abstract units—so that if one number is neither greater nor less than another, it is equal; but what is equal and in no wise differentiated we take to be the same when we are speaking of numbers. If not, even the 2’s in the ideal 10 will be differentiated though they are equal; for what reason will the man who says they are not differentiated be [10] able to allege?

  Again, if every unit plus another unit makes two, a unit from the ideal 2 and one from the ideal 3 will make a 2. Now this consists of differentiated units; and will it be prior to the 3 or posterior? It rather seems that it must be prior; for one of the [15] units is simultaneous with the 3, and the other is simultaneous with the 2. And we, for our part, suppose that in general 1 and 1, whether the things are equal or unequal, is 2, e.g. the good and the bad, or a man and a horse; but those who hold these views say that not even two units are 2.

  If the number of the ideal 3 is not greater than that of the 2, this is surprising; [20] and if it is, clearly there is a number in it equal to the 2, so that this is not different from the ideal 2. But this is not possible, if there is a first and a second number. Nor will the Ideas be numbers. For in this particular point they are right who claim that the units must be different, if there are to be Ideas, as has been said before. For the [25] Form is unique; but if the units are not different, the 2’s and the 3’s also will not be different. Therefore they must say that when we count thus—‘1, 2,’ we do not add to the previous number; for if we do, neither will the numbers be generated from the [30] indefinite dyad, nor can a number be an Idea; for one Idea will be in another, and all the Forms will be parts of one Form. Therefore with a view to their hypothesis they are right, but absolutely they are wrong; for their view is very destructive, since they will admit that this question itself affords some difficulty—whether, when we count and say ‘1, 2, 3,’ we count by addition or by partitions. But we do both; therefore it is [35] absurd to refer this to so great a difference of substance.

  8 · First of all it is well to determine what is the differentia of a [1083a1] number—and of a unit, if it has a differentia. Units must differ either in quantity or in quality; and neither of these seems to be possible. But number qua number differs in quantity. And if the units also differed in quantity, number would differ from [5] number, though equal in number of units. Again, are the first units greater or smaller, and do the later ones increase or diminish? All these are irrational suppositions. But neither can they differ in quality. For no attribute can attach to them; for even to numbers quality is said to belong after quantity. Again, quality [10] could not come to them either from the 1 or from the dyad; for the former has no quality, and the latter gives quantity; for its nature is to cause things to be many. If [15] the facts are really otherwise, they should above all state this at the beginning and determine if possible, regarding the differentia of the unit, why it must exist; otherwise, what do they mean by it?

  Evidently then, if the Ideas are numbers, the units cannot all be comparable, [20] nor can they be non-comparable in either of the two ways. But neither is the way in which some others speak about numbers correct. These are those w
ho do not think there are Ideas, either without qualification or as identified with certain numbers, but think the objects of mathematics exist and the numbers are the first of real [25] things, and the ideal 1 is the starting-point of them. It is paradoxical that there should be a 1 which is first of 1‘s, as they say, but not a 2 which is first of 2’s, nor a 3 of 3’s; for the same reasoning applies to all. If, then, the facts with regard to number are so, and one supposes mathematical number alone to exist, the 1 is not the [30] starting point. For this sort of 1 must differ from the other units; and if this is so, there must also be a 2 which is first of 2’s, and similarly with the other successive numbers. But if the 1 is the starting-point, the truth about the numbers must rather [35] be what Plato used to say, and there must be a first 2 and 3, and the numbers must not be comparable with one another. But if on the other hand one supposes this, many impossible results, as we have said, follow. But either this or the other must be [1083b1] the case, so that if neither is, number cannot exist separately.

  It is evident from this that the third view is the worst,—that ideal and mathematical number is the same. For two mistakes evidently meet in the one [5] opinion. (1) Mathematical number cannot be of this sort, but the holder of this view has to spin it out by making suppositions peculiar to himself. And (2) he must also admit all the consequences that confront those who speak of numbers as Forms.

  The doctrine of the Pythagoreans in one way affords fewer difficulties than those before named, but in another way has others peculiar to itself. For not [10] thinking of number as capable of existing separately removes many of the impossible consequences; but that bodies should be composed of numbers, and that this should be mathematical number, is impossible. For it is not true to speak of indivisible magnitudes; and however much there might be magnitudes of this sort, [15] units at least have no magnitude; and how can a magnitude be composed of indivisibles? But arithmetical number, at least, consists of abstract units, while these thinkers identify number with real things; at any rate they apply their propositions to bodies as if they consisted of those numbers.

  [20] If then it is necessary, if number is a self-subsistent real thing, that it should be conceived in one of these ways which have been mentioned, and if it cannot be conceived in any of these, evidently number has no such nature as those who make it separable construct for it.

  Again, does each unit come from the great and the small, equalized, or one [25] from the small, another from the great? If the latter, neither does each thing contain all the elements, nor are the units without difference; for in one there is the great and in another the small, which is contrary in its nature to the great. Again, how is it with the units in the ideal 3? There is one over. But perhaps it is for this reason that they give the ideal 1 the middle place in odd numbers. But if each of the two units consists of both the great and the small, equalized, how will the 2, which is [30] one thing, consist of the great and the small? Or how will it differ from the unit? Again, the unit is prior to the 2; for when it is destroyed the 2 is destroyed. It must, then, be the Idea of an Idea since it is prior to an Idea, and it must have come into being before it. From what, then? Not from the indefinite dyad, for its function was [35] to double.

  Again, number must be either infinite or finite; for these thinkers think of number as capable of existing separately, so that it is not possible that neither of [1084a1] those alternatives should be true. Clearly it cannot be infinite; for infinite number is neither odd nor even, but the generation of numbers is always the generation either of an odd or of an even number,—when 1 operates in one way on an even number, an odd number is produced, and when 2 (or an odd number) operates in the other [5] way, the numbers got from 1 by doubling (or the other even numbers) are produced. Again, if every Idea is an Idea of something, and the numbers are Ideas, infinite number will be an Idea of something, either of some sensible thing or of something else. Yet this is not possible in view of their hypothesis any more than it is reasonable in itself, if they conceive of the Ideas as they do. [10]

  But if number is finite, how far does it go? With regard to this not only the fact but the reason should be stated. But if number goes only up to 10, as some say, firstly the Forms will soon run short; e.g. if 3 is man-in-himself, what number will be the horse-in-itself? The numbers which are Ideas of the several things go up to 10. It [15] must, then, be one of the numbers within these limits; for it is these that are substances and Ideas. Yet they will run short; for the various kinds of animal will exceed them. At the same time it is clear that if in this way the 3 is the Idea of man, the other 3’s are so also (for those in the same number are similar), so that there will [20] be an infinite number of men, and if each 3 is an Idea, each of the men will be man-in-himself, and if not, they will at least be men. And if the smaller number is part of the greater (being number of such a sort that the units in the same number are comparable), then if the ideal 4 is an Idea of something, e.g. of horse or of white, man will be a part of horse, if man is 2. It is paradoxical also that there should be an [25] Idea of 10, but not of 11, nor of the succeeding numbers. Again, there both are and come to be certain things of which there are no Forms; why, then, are there not Forms of them also? We infer that the Forms are not causes. Again, it is paradoxical if the number-series up to 10 is more of a real thing and a Form than 10 [30] itself. There is no generation of the former as one thing, and there is of the latter. But they try to form a theory on the assumption that the series of numbers up to 10 is a complete series. At least they generate other things—the void, proportion, the odd, and the others of this kind—within the 10. For some things, e.g. movement, rest, good, bad, they assign to the principles, and the others to the numbers. This is [35] why they identify 1 with the odd; for if the odd implied 3, how would 5 be odd? Again, magnitudes and all such things are explained without going beyond a definite number, e.g. the first indivisible line, then the 2, then the others up to [1084b1] 10.

  Again, if number can exist separately, one might ask which is prior—1, or 2 or 3? Inasmuch as the number is composite, 1 is prior, but inasmuch as the universal [5] and the form is prior, the number is prior; for each of the units is part of the number as its matter, and the number acts as form. And in a sense the right angle is prior to the acute, because it is definite and in virtue of its formula; but in a sense the acute is prior, because it is a part and the right angle is divided into acute angles. As matter, then, the acute angle and the element and the unit are prior, but as regards [10] the form and the substance (in the sense of the formula), the right angle, and the whole consisting of the matter and the form, are prior; for the compound thing is nearer the form and the object of the formula, but in generation it is later. How then is 1 the starting-point? Because it is not divisible, they say. But both the universal, [15] and the particular or the element, are indivisible; but in different ways, one in formula and the other in time. In which way then is 1 the starting-point? As has been said, the right angle is thought to be prior to the acute, and the acute to the right, and each is one. They make 1 the starting-point in both ways. But this is impossible. For one kind of starting-point is the form or substance, the other the [20] part or matter. For each is in a way one—in truth, each unit exists potentially (at least if the number is a unity and not like a heap, i.e. if different numbers consist of different units, as they say), but not actually.

  The cause of the mistake they fell into is that they conducted their inquiry at the same time from the standpoint of mathematics and from that of universal [25] formulae, so that from the former standpoint they treated unity, their first principle, as a point; for the unit is a point without position. They put things together out of the smallest parts, as some others have done. Therefore the unit becomes the matter of numbers and at the same time prior to 2; and again posterior, [30] 2 being treated as a whole, a unity, and a form. But because their inquiry was universal they treated the unity which can be predicated of a number, as in this sense also a part of the
number. But these characteristics cannot belong at the same time to the same thing.

  If the ideal 1 must be merely without position5 (for it differs in nothing from other 1’s except that it is the starting-point), and the 2 is divisible but the unit is not, [35] the unit must be more like the ideal 1. But if so, it must be more like the unit than the 2; therefore each of the units must be prior to the 2. But they deny this; at least [1085a1] they generate the 2 first. Again, if the ideal 2 is a unity and the ideal 3 is one also, both form a 2. From what, then, is this 2 produced?

  9 · Since there is not contact in numbers, but the units between which there [5] is nothing, e.g. those in 2 or in 3, are successive, one might ask whether they succeed the ideal 1 or not, and whether, of the terms that succeed it, 2 or either of the units in 2 is prior.

  Similar difficulties occur with regard to the classes of things posterior to number,—the line, the plane, and body. For some construct these out of the forms [10] of great and small; e.g. lines from long and short, planes from broad and narrow; masses from deep and shallow; which are forms of great and small. And the principle of these which answers to the 1 different men describe in different ways. And in these also the impossibilities, the fictions, and the contradictions of all [15] probability are seen to be innumerable. For they are severed from one another, unless the principles of these imply one another in such a way that the broad and narrow is also long and short; but if this is so, the plane will be a line and the solid a plane. Again, how will angles and figures and such things be explained? And the [20] same happens as in regard to number; for these things are attributes of magnitude, but magnitude does not consist of these, any more than the line consists of straight and curved, or solids of smooth and rough.