What is the experience of mathematical motion?

(This is not so much an argument or a claim as an attempt to describe an experience.)

Imagine any simple motion of a mathematical thing: a point going across a line, a secant becoming a tangent, a rotation making a circle, etc. Question: how long does that take you? Is there a minimum time that you need for that, or can you see that any time T is always one you could better?  This seems to be a classic limit argument – we can make the time it takes less than any given time. We almost immediately get to the point where we are imagining a flicker of a speed that we understand through numbers that we can increase indefinitely.

Now so far as we are really demanding that the motion go across some distance, we cannot make it instantaneous; but so far (1) as the amount of time it takes can be made less than any given time (2) it is not meaningful to speak about the speed of the motion (3) since the motion just is the representation of a numerical value that can be increased indefinitely, the motion is instantaneous.

(I’ve wondered if we could make a similar point from the fact that there need be no time factor for taking an integral or a derivative.)

Minimally, this might show a difference between physical and mathematical motion, but this leave the question as a mere negation. It might be an insight into that mysterious reality that St. Thomas claim constitutes mathematical reality: intelligible matter.

Is the whole question simpler? Just as we can’t meaningfully ask “how long does that take you?” It seems equally meaningless to say “how long is the line you imagine? or “How large is the circle you are demonstrating  property X about?” It either has no length, or its length is the one you say it represents. Either way, this is some mysterious length!

I wonder how this might compare to the motion of the angels: which is both discrete and yet covering all points between its termini. Is mathematics the angelic intuitions that animals have? This might be another approach to hylemorphism, or a metaphysical account of number lines or the system of all numbers – which both have points only potentially, and yet in such a way that the mind passing over these points does not require the imperfection of time or local separation from a term.

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