> So, if classical mechanics is deterministic, and describes nature, nature must be continuous at arbitary depth.
Isn't that the classical assumption though? That nature is analogue. To avoid problems with infinities, you can just say let's assume it's continuous up to Graham's number or something.
Or, .. it's continuous but "unknowable" in the sense of becoming increasingly "out of focus" wrt measurement due to the uncertainty principle at and below the order of Planck's constant.
Not every dynamic system is chaotic, and not system with chaotic elements is chaotic everywhere ... but ...
Even quite simple systems can have chaotic regions .. and within those regions two 'particles' (or phase space initial conditions) can startout arbitrarily close (within a fuzzy out of focus impossible to measure Planck distance) and end up nowhere near each other .. (ie not continuous).
This is a mathematical result of dynamic systems and can be arrived either by Lorentz's reasoning ( "the butterfly effect" ) or via Smale's Horshoe Map (taffy folding to infinity!).
Isn't that the classical assumption though? That nature is analogue. To avoid problems with infinities, you can just say let's assume it's continuous up to Graham's number or something.