Commentary Aristotle Physics
Lecture 11. The definition of the infinite
The infinite turns out to be the contrary of what it is said to be. It is not what has nothing outside it that is infinite, but what always has something outside it.
This is indicated by the fact that rings that have no bezel are described as infinite, because it is always possible to take a part that is outside a given part.
The description depends on a certain similarity, but it is not true in the full sense of the word. This condition alone is not sufficient; it is necessary also that the next part that is taken should never be the same. In the circle, the latter condition is not satisfied: it is only the adjacent part from which the new part is different.
Our definition, then, is as follows: a quantity is infinite if it is such that we can always take a part outside of what has been already taken.
On the other hand, what has nothing outside it is perfect and whole. For thus we define the whole as that to which nothing is lacking, like a whole man or a whole box.
What is true of each particular is true of the whole as such—the whole is that of which nothing is outside. On the other hand, that from which something is absent and outside, however small that may be, is not “all.” “Whole” and “perfect” are either quite identical or closely akin. Nothing is perfect that has no end, and the end is a limit.
Hence, Parmenides must be thought to have spoken better than Melissus. The latter says that the whole is infinite, but the former describes it as limited, “striving equally from the middle.” For to connect the infinite with the all and the whole is not continuous as thread follows thread.
For it is from this they get the dignity they ascribe to the infinite—its containing all things and holding the all in itself—from its having a certain similarity to the whole.
It is, in fact, the matter of the completeness that belongs to size, and what is potentially a whole, though not in the full sense. It is divisible both in the direction of reduction and of the inverse addition.
It is a whole and limited, though not according to itself, but according to something other. It does not contain, but is contained insofar as it is infinite.
Consequently, it is also unknowable as infinite, for the matter has no species. (Hence, it is plain that the infinite stands in the relation of part rather than of whole. For the matter is part of the whole, as the bronze is of the bronze statue.)
If it contains in the case of sensible things, then, in the case of intelligible things, the great and the small ought to contain them. But it is absurd and impossible to suppose that the unknowable and indeterminate should contain and determine.
382. After showing how the infinite exists, the Philosopher now explains what it is.
About this, he does three things:
first, he shows what the infinite is;
second, from this he assigns the reason for the things said of the infinite, at “it is reasonable” (207a32; [390]);
third, he solves the difficulties mentioned earlier, at “it remains to dispose” (208a5; [400]).
In regard to the first, he does two things:
first, he shows what the infinite is and rejects the false definition of some;
second, he rejects a certain false opinion that follows from the above false definition, at “for it is from this” (207a18; [387]).
About the first, he does three things:
first, he proposes what he intends;
second, he explains the proposition, at “this is indicated” (207a2; [384]);
third, he draws a conclusion, at “hence Parmenides” (207a15; [386]).
383. He says, therefore (206b33), that the infinite must be defined in a manner contrary to the way some have defined it. For some have said that infinite is what has nothing outside it, when, to the contrary, it should be defined as what always has something outside it.
384. Then, at “this is indicated” (207a2) he explains his proposition.
First, he shows that his description is good;
second, that the description of the earlier philosophers is incompetent, at “on the other hand” (207a8; [385]).
Thus, he first shows by an example that the infinite is what always has something outside it. For some people say that a ring is infinite, since one can always take part after part because it has a circular direction.
But this is to speak analogously and not properly, because to be infinite requires this, that, beyond whatever part is taken, there be some other part, though nevertheless in such a way that one never take again a part taken previously. But this does not happen in a circle, because the part that is counted after another happens to be different from that immediately before it but not from all the parts previously counted, since one part can be counted any number of times, as is evident in a circular motion. Therefore, if rings are called infinite according to this analogy, it follows that that which is truly infinite is something that always has something beyond, if one were to measure its quantity. For it is impossible to measure the quantity of the infinite; but, if someone should desire to reckon it, he would take part after part to infinity, as said above.
385. Then, at “on the other hand” (207a8), he proves that the definition of the earlier philosophers is incompetent, since what has nothing outside it is a definition of the perfect and a whole thing. Here is his proof. Every whole is defined as that to which nothing is lacking—as we speak of a whole man and of a whole box if they lack nothing that they ought to have. And just as we speak thus in regard to some individual whole, as in the case of this or that particular, so too this account holds in regard to what is truly and perfectly whole, namely, that outside of which there is absolutely nothing. But, when something is lacking through the absence of something intrinsic, then such a thing is not a whole.
So, it is evident that this is the definition of a whole: the whole is that of which nothing is outside. But a whole thing and a perfect thing are either entirely the same or of a proximate nature. He says this because whole is not found in simple things that have no parts—in which things, nevertheless, we use the word perfect. This shows that the perfect is that of which nothing is outside. But nothing that lacks an end is perfect, because the end is the perfection of each thing. For the end is the term of that of which it is the end. Therefore, nothing infinite and unterminated is perfect. Hence the definition of the perfect as that of which nothing is outside does not apply to the infinite.
386. Then, at “hence, Parmenides” (207a15), he draws a conclusion from the foregoing. Since the definition of a whole does not apply to the infinite, it is clear that the position of Parmenides is better than that of Melissus.
For Melissus said that the whole universe was infinite. Parmenides said the whole is terminated by what is striving equally from the middle, by which he designated the body of the universe as spherical. For in a spherical figure, lines from the center to the term (that is, the circumference) are drawn according to equality, as though striving equally with each other.
And it is rightly stated that the whole universe is finite, for to be a whole and to be infinite are not reciprocally connected, that is, not continuous as thread follows thread in spinning. For there was a proverb that things that follow one upon the other should be said to be continuous as thread following thread.
387. Then, at “for it is from this” (207a18), he rejects a false opinion that arose from the given definition:
first, in a general way, with regard to everything;
second, according to the opinion of Plato, at “if it contains” (207a28; [389]).
He says first that, because some thought that “whole” and “infinite” were mutually connected, they consequently took it as a dignity of the infinite, as something self-evident, that the infinite contains all things and that it has all things in itself. This was due to the fact that the infinite has a likeness to a whole, as what is in potency has a likeness to act. For the infinite, inasmuch as it is in potency, is as matter in respect to the perfection of magnitude, and it is as a whole in potency, not as a whole in act. This is proved by the fact that the infinite is based on the possibility of dividing things into what is smaller and of making continual additions by a contrasting division, as was said above.
Consequently, the infinite in itself, according to its proper account, is a whole in potency only, and it is something imperfect, comparable to matter, not having perfection. For it is not whole and infinite according to itself—that is, according to the proper account by which it is infinite—but according to something other, that is, according to end and whole, to which it is in potency. For division, which is possible to infinity, is called “perfect” insofar as it is completed, whereas the division that goes on to infinity is imperfect. And it is clear, since it is the whole that contains but matter that is contained, that the infinite as such does not contain but is contained. This is true insofar as whatever is in act of the infinite is always contained by something greater, according as it is possible to take something beyond.
388. Now, from the fact that the infinite is as a being in potency, not only does it follow that the infinite is contained and does not contain, but two other conclusions also follow.
One is that the infinite as such is unknown, because it is as matter without species—that is, without form—and matter is not known except through form.
The other conclusion, which has the same source, is that the infinite has more the account of a part than that of a whole, since matter is compared to the whole as a part. And it is not a surprise that the infinite conducts itself as a part, inasmuch as only a part of it is ever actual.