the sun completes thirty. Thus, the number of Saturn’s orbits is smaller than the number of the sun’s orbits. What is less than another is finite, and whatever exceeds a finite quantity by a finite amount is finite. Therefore, both numbers are finite.”

Al-Urmawī said:

“One might say: multiplying one without limit yields less than multiplying two in the same way, although both are infinite.”

I say:

What al-Urmawī has offered is a counterargument: it neither challenges any premise of the proof nor shows where its reasoning breaks down.

The proponent of the proof might then say:

Numerical sequences differ from the numbers of orbits in two respects.

First, sequences of abstract numbers have no existence in external reality; the mind merely conceives them, just as it conceives abstract shapes. It conceives a circular shape, then a shape larger than it, then another larger still, and so on. These shapes conceived by the mind have no existence in external reality. Likewise, abstract numbers have no existence in external reality. Thus, continuous or discrete quantity, when abstracted from that which possesses it, exists only in the mind. The same applies to a mathematical body—that is, length, breadth, and depth conceived in abstraction from that which possesses them.

This being so, the possibility of conceiving something in the mind does not entail the possibility of its existence in external reality. For the mind can entertain impossibilities, such as the coexistence of contradictories or contraries. It can conceive something as both existent and nonexistent, or as both moving and stationary; and it can also conceive