the reply is: “Yes, but why did you say that what this expression denotes can exist?”

This is like someone saying, “I conceive of the infinite in my mind and express it in words.” Nothing in this implies that it can exist in external reality, just as one can mentally conceive and verbally describe infinitely many bodies, dimensions, and shapes. The two cases are alike. This is one way the proponent of the argument can answer the counterargument based on successive numerical magnitudes.

This distinction—albeit as we have presented it—has been mentioned by several thinkers who distinguish between numbers and motions, including Muslim theologians and others. They explain this distinction, familiar to those who agree with the proponent of the argument on this refutation, as follows: doubling numbers is not something that actually exists, but something posited, unlike motions that have occurred.

Likewise, those who distinguish between the past and the future do so on the grounds that the past has existed, whereas the future has not. What is impossible is the existence of the infinite, not the supposition of the infinite.

Those who agree with the objector say: “The past, too, has ceased to exist, so its individual instances do not exist together. What is inadmissible is the existence of infinitely many things assembled together—more precisely, assembled together and ordered in relation to one another, so that they have a natural or positional ordering.”

This is the distinction drawn by Ibn Sīnā and his followers among the philosophers. Ibn Rushd, however,