The concept of infinity has been fraught with paradox since antiquity. For this reason, Aristotle sought to banish it from his physics, claiming that there were no actual infinities in nature—only potential infinities. Over a millennium later, medieval scholars offered the following example when asked why infinity was forbidden.

Imagine two concentric circles. Each circle contains infinitely many points along its circumference, but since the outer circle has a greater circumference, it has more points than the inner circle. Now take any point A on the outer circle, and draw a line from A to the circle’s center. This line must intersect some point B on the circumference of the inner circle. Hence, for every point A on the outer circle, there is a corresponding point B on the inner circle, and vice versa. Therefore, both circles must have the same number of points, despite the fact that the outer circle appears to have more points than the inner circle. More »