The rationals are also infinitely dividable, it's just countable. But I think I take your meaning, and it's an important one-- that if we designate in advance the accuracy goal we desire, we can apply a form of mathematics that only uses a finite number of operations on the natural numbers to achieve that accuracy, and continuous calculus is just a more elegant form for obtaining estimates within that same accuracy target. In that view, we don't use continuous calculus because it is "closer to the reality", or even more accurate, but simply because it is easier.

But the problem is not that you have to use continuous calculus to do classical mechanics, because the equations could be translated into finite versions. The problem is that we cannot arbitrarily tighten our accuracy goal-- at some point the classical predictions no longer work. If we ask, at what time does Achilles catch the tortoise, we can only get the answer right to a certain accuracy, after which the question no longer has meaning, even though we tend to imagine that the time that comes out of the classical calculation actually means something other than just a continuous estimate of an arbitrarily chosen finite process. Classical mechanics does not tell you that limitation, you have to apply it externally to avoid false predictions. Yet classical mechanics works fine at lower levels of accuracy, completely oblivious to this catastrophic failure at higher accuracies. That this is not actually a catastrophe, but simply the way we have become accustomed to using physics axioms, is very much the point I'm making-- that we were never doing anything different, we just became aware of the pretense.
The dynamical equations are differential equations, so they presume divisability of space and time. I agree they could be written in a finite (and very clunky) form, but the choice of the intervals, and the loss of accuracy that results, would be entirely arbitrary. There would be nothing in the theory itself that suggested it needed to be written that way, we would simply be imposing it because we know that otherwise it makes assumptions that are unverifiable. We do it the way we do because we prefer inconsistency to arbitrariness, it's just that simple. Maybe also because the math is easier.

Yes, I agree-- things get less surprising when we step away from the idea that we are "unmasking reality" and just accept that everything we do is an effective theory. Somehow we have become masters of modeling a fake version of reality that matches the real one, in many cases, astonishingly well.

But it still doesn't seem likely that this would even be possible-- like the difficulties we have predicting weather, for example. Why is it that we can isolate the behavior of particles enough to ignore "the weather"? If we couldn't do that, we'd be dead in the water. Perhaps there's some anthropic way of looking at that, whereby a universe that is not understandable by intelligence does not evolve intelligence. In fact, that's pithy enough for a signature!
I really don't know-- it seems pretty mysterious. It's ironic, most people find it weird that classical mechanics breaks down in the atom, but I find it weird that it doesn't do worse.
Yes, famous physicists have a very uneasy marriage with mathematics-- the same ones will be found to make statements like that, then turn around and poke fun at abstract thinking!

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