Sunday, April 5, 2015

Aristotle and the infinite

The world envisioned by Aristotle contains the infinite. Let’s consider where and how. His world fits within a heavenly sphere, so every physical line is finitely long. So is every mathematical line. Aristotle’s reason is that the mathematicians’ diagrams used in his day always could fit on a piece of papyrus. Unfortunately, he was mistaken about what mathematicians required. Euclid’s fifth postulate, the axiom of parallel lines, contains the phrase “if extended indefinitely.” Later mathematicians understood this deficiency in the Aristotelian system and tried very hard, but unsuccessfully, to deduce the fifth postulate from the others.

Aristotle used two kinds of infinity. He explicitly denied the existence of the actual infinite both in the physical world and in mathematics, but he accepted the potential infinite in both realms. Forcing Zeno to say that Achilles’ path to catch the tortoise is potentially infinite but not actually infinite is Aristotle’s clever way out of Zeno’s Paradox. Unfortunately for Aristotle there is a more fruitful treatment of the paradox.

Aristotle’s chosen labels for kinds of infinity easily lead to misunderstanding. Potential infinities do not have the potential to become actual infinities. “Potential infinity” is Aristotle’s technical term that requires a repeatable, but incomplete, process. The term “actual infinite” does not imply being actual (i.e., real). It implies there is no dependency on some process in time. When Aristotle says something is infinite, he does not mean this infinity is merely possible. He means it is real or actual, but just not an actual infinity. Actual infinities are not actual. Potential infinities are actual, not potential. Thus the confusion.

Aristotle believed the sequence of natural numbers is potentially infinite in two different ways—by adding and by dividing. Regarding addition, numbers can be abstracted from existing objects and then a unit can be repeatedly added over time to the previous number. Yet at any time, there are only finitely many natural numbers produced by this or any other process. This reasoning also works for future time; for any day there is always tomorrow, or so Aristotle believed. The reasoning does not work for cows in the field; we cannot truly say there is always another cow.

The second way that the sequence of natural numbers is potentially infinite is based on the way that continuous magnitudes are potentially infinite. Given a specific continuous magnitude such as the distance between Achilles and the tortoise, it can be repeatedly split or divided, and one can count the number of its divisions so far. Yet it always could be divided again, thereby adding one to the count. It is in this second way that natural numbers are potentially infinite, Aristotle would say. Georg Cantor (1845-1918) would disagree and claim the natural numbers are an actual infinity. Cantor re-defined the term “potentially infinite” so that any potentially infinite set of numbers is a growing subset of a pre-existing actually infinite set of numbers.

There is more to be learned from examining the details of Aristotle’s notion of potential infinity. He says that what is infinitely divisible is continuous, but he does not believe that continuous magnitudes are divisible into indivisible points. Cantor does. Aristotle also believes that lines are not composed of points, although he believes there is an infinity of points on any line.

Consider Achilles’ continuous path in pursuit of a tortoise that is crawling away from him. What is the ontological status, for Aristotle, of the points where Achilles might be? The points where he stops are real because they are the end of a line and are independent of any activity by the analyst. The points where he merely might be are not independent of the analyst. They exist only in the derivative sense of being the product of a permanent possibility of division, and they do not exist independent of this division.

Zeno and 21st century realist philosophers would say instead that the points do exist independent of this division. Those philosophers infer from the fact that Achilles might stop at a point to there being this point where he might stop. Aristotle would not accept this inference. You can chase unicorns without there being unicorns you are chasing, he might say.

Aristotle probably would not have been happy with saying Achilles is always at some point or other. What then does Aristotle mean by saying Achilles runs past a potential infinity of points? The key idea is that, for Aristotle, a continuous line can be divided anywhere but not everywhere. Aristotle would say the line is not composed of a potential infinity of points, but rather it has a structure such that the analyst theoretically can imagine Achilles stopping somewhere new even if he does not actually stop there, and in that sense the analyst can create a new point along Achilles’ path, and either of those newly created sub-paths on each side of the point can in turn be subdivided by yet another point, and so on.

However, the line is potentially infinite only because there exists a theoretical division of the line, not a practical one, since the analyst cannot live long enough nor engage in the mental effort to have Achilles arrive at a new point for every old point, Aristotle would say. At any moment in the analyst’s theoretical division, there always will be unactualized potential future divisions. Having a new analyst pick up the job left incomplete by the first analyst would not change this result. So, the potential infinity of Achilles’ path does not depend upon any unending process existing. It is enough that the theoretical division exists.

One last comment on Aristotle and infinity. He never thought of infinity as having a measure or number. Archimedes (287-212 B.C.E.) was the first person to do this. In the Archimedes Palimpsest, he argued that the number of lines inside a rectangle is equal to the number of triangles inside a prism. Unfortunately, there is no evidence that he paused to explore these numbers.

Brad Dowden
Department of Philosophy
Sacramento State


  1. Brad, thanks very much for this interesting post. How do you think Aristotle would have received Cantor's diagonalisation proof? Would it have blown his mind? I ask because in some ways at least it actually seems to meet Aristotle's constructivist requirements. Of course, given it's reliance on decimal fractions, maybe this isn't a coherent question.

  2. Randy, that’s an interesting thought about Aristotle. He was a great genius, but it is difficult to know what he’d say if he were to read my post that is based upon 2,200 years of subsequent analysis of his ideas and progress beyond them. If I had to pick two ancient Greeks who would have been most likely to appreciate the need to change their minds due to subsequent progress over the next two millennia, I would pick Aristotle and Archimedes. Aristotle would have been attracted to the constructivist feature of Cantor’s diagonalization proof that some infinite sets are smaller than others, but Archimedes probably would have been more likely than Aristotle to accept Cantor’s actual infinite sets and the calculus of Leibniz and Newton. Aristotle lived at a time when there was no coherent concept of zero or decimal. It took until the beginning of the 20th century to achieve a solid foundation to these or other notions we now believe are needed for fruitful mathematical physics--such as instantaneous speed, convergence of an infinite series, and continuity of a path. If Aristotle were to hold on to his preference for avoiding new technical terms and making do with common sense terms, then he would find today’s mathematics and physics to be repugnant.