2.5.7 Potential Infinity
Aristotle presented several criticisms of Zeno, but his most notable contribution was his concept of “potential infinity”. He says that he agrees that it is impossible for a thing to pass over an actually infinite number of things in a finite time, but that it is possible for a thing to pass over a “potentially infinite” number of things in a finite time: “For motion, although what is continuous contains an infinite number of halves, they are not actual but potential halves.” Barnes (2014).
Therefore, according to Aristotle, in the Achilles and Dichotomy paradoxes, Zeno made the mistake of discounting the fact that the physical long path from start to finish is finite and exists undivided, but it is the mathematician who is -by estimation- dividing it infinitely and demanding the physical completion of such an infinite process.
Similarly, using the same idea of time being divided only into a potential infinity of intervals, Aristotle tried to refute the Arrow Paradox by saying that “time is not composed of indivisible moments any more than any other magnitude is composed of indivisibles.” Barnes (2014).
The idea of “potential infinity” was a very clever idea by Aristotle, which convinced philosophers for many centuries, including George Berkeley, Immanuel Kant, Carl Friedrich Gauss, and Henri Poincaré who were influential defenders of potential infinity. Leibniz and other mathematicians and physicists, on the other hand, did not accept this distinction between actual and potential infinities. Bernard Bolzano and Georg Cantor view a potentially infinite set as a variable quantity that is dependent on being abstracted from a pre-existing actually infinite set, and they argued that “each potential infinite ... presupposes an actual infinite” Tannery (1885).