Actual and Potential Infinity

In mathematics one often encounters the assertion of the existence of an actual infinity. It is apparent that this statement rules out any form of finitism, but what is more subtle, and in my opinion much more interesting, is the third option that the word actual contradicts. Actual here originates from the Aristotelian distinction between actual and potential existence. This, according to Aristotle, is one fundamental division of existence.

So, what is the distinction between an actual infinity and a potential one, according to Aristotle?

In the following, we assume an Aristotelian ontology of mathematics, at its core that mathematics is an abstraction of one aspect of physical, natural things. We discuss the infinite strictly with regard to quantity, ruling out any non-quantifiable infinites, such as God.

From chapters 4-8 of Physics book 3, corresponding to lectiones 7-13 of Aquinas' commentary on the work, it appears that Aristotle means actually infinite in a very literal way: something is actually infinite if all of its (infinitely many) parts are physically present together, at the same time. Take the set of all natural numbers, ℕ, as an example. F or ℕ to be an actually infinite set per Aristotle, you need to have all the elements together in one place at the same time, somehow instantiated (perhaps written down). That’s clearly impossible, as Aristotle argues.

Despite denying the existence of actual infinity, Aristotle rejects finitism, instead maintaining that infinity exists in potency. In chapter 6, 206a18-25, Aristotle clarifies what exactly is the potential existence of the infinite:

But the phrase “potential existence” is ambiguous. When we speak of the potential existence of a statue we mean that there will be an actual statue. It is not so with the infinite. There will not be an actual infinite. The word “is” has many senses, and we say that the infinite “is” in the sense in which we say “it is day” or “it is the games,” because one thing after another is always coming into existence."

Aquinas explains, “The infinite is not said to be in potency in the sense that at some later time the whole is in act.” It is in potency “in the sense that at some later time it comes to be in act, not at once as a whole, but successively.” From 207a18-28, we see that the (potential) infinite has is similar to a whole. “It is a whole and limited; not, however, in virtue of its own nature, but in virtue of what is other than it.”

Returning to modern math, it seems to me that pinning down the set of all natural numbers is a lot like talking about a line segment as a whole–both contain infinitely many points, but those points are not all actually instantiated. Jacob Klein’s book is illuminating about the merging of magnitude and number, which are separate concepts for Aristotle, in the modern conception of number. The way we talk about infinities in math today reminds me of Aristotle’s treatment of magnitude as continuum.

Under Aristotle’s definition of potential, nothing prevents us from talking about a potential infinity as a whole in many respects. After all, Aristotle does indicate that the infinite is a whole, and thus limited, in virtue of what is other than it. Aquinas too grants that the infinite in itself is a whole in potency (§387).

Furthermore, given the meaning of an actually infinite set for Aristotle (that is, one that has all its elements physically instantiated), it seems that an actually infinite set is absurd. Unless we understand mathematical objects Platonically as existing in a separate realm, it doesn’t make much sense to talk about whether all the natural numbers “exist simultaneously.” This indicates that when mathematicians assert the existence of an actual infinity these days, they almost never mean actually as Aristotle did.

Take what the Stanford Encyclopedia of Philosophy says about Aristotle’s view on infinity regarding time, in the article “Aristotle and Mathematics”:

Since Aristotle believes that the universe has no beginning and is eternal, it follows that in the past there have been an infinite number of days. Hence, his rejection of the actual infinite in the case of magnitude does not seem to extend to the concept of time.

But Aristotle illustrates what potential infinity means precisely with the example of time in Physics chapter 10! This is clearly a misunderstanding. For time, or days, to be an actual infinity for Aristotle, all those infinite number of days must exist together at the same time.

It seems that a common objection to understanding infinity as existing in potency is that such an infinity is incomplete (see Wikipedia, which surprising does not have a page on potential infinity), in that you never get the “whole thing”, and consequently you can’t talk about everything in such an infinite set as really existing. However, Aristotle didn’t say that infinity doesn’t exist; he didn’t say that there are only arbitrarily big finite things. He didn’t say that the infinite is completely unlimited or cannot be considered as a whole, only that it’s not whole and limited in virtue of its own nature. In fact, the reader might remember that Aristotle’s solution to Zeno’s paradox is precisely to look at the infinity of a magnitude as a whole, which is only made possible by Aristotle’s observation that the infinity in question is a potential one. It seems that there is a prevalent misunderstanding of the Aristotelian concept of potentiality. Once again, Aristotle illuminates the third way between the false dichotomy of “actual infinity” and finitism.