R Hansen says: >Not only irrational, but transcendental (not algebraic). I am talking about the actual positions of the molecules and atoms, although they are moving, but you know what I mean. The argument is not much different than what you said about the improbability (impossibility) of a ray of light ever hitting a particular number. I am saying that said ray will not even hit any algebraic number. We know the ray of light will hit a number, I am saying that number will always be transcendental.
I think if you model a light ray as a 1 dimensional line, that its just an approximation good for some purposes and not for others. Likewise, number lines and points and infinitely sharp darts makes a nice mental image, but the absurd result that says the dart does hit a particular point, but that it has zero % chance of hitting any point in particular, tells you something is not quite right with that picture. Probability is an area of mathematics where the difference between models, physics, and math is often murky, though I don't think it has to be.
As far as the distance between molecules, if you believe Heisenberg then surely you know the uncertainty is finite > 0 and greater than the tail end of any transcendental number. But the math of physics still uses the real number system, so one could argue the exact center of that uncertain area was exactly some transcendental number. Now I'm not even sure what that means, but to me it looks like faith at work (if one believes such things,) not anything that can be verified or proved.
R Hansen says: >In fact, the ray of light would never hit any "particular" (constructed?) number because there are infinitely more non particular numbers for it to hit.
I find that insight interesting. It seems to touch on the delightful "paradoxes" G. Chaitin like to point out in "How Real are Real Numbers" and his book "Meta Math". I would suggest that by the very fact that the dart hits some number (one of the anonymous, invisible "reals") then by that very fact, that number instantly becomes very particular, and therefore, unhittable.