I'm not so sure that that assumption is truly necessary for classical mechanics. From the mathematical side, at least, much of what we do with the reals can be done as well (though with greater effort) with the rationals alone. This is the idea behind finitism. I won't claim to be well versed in this interesting but rather obscure field, but my point is that a lot of the mathematical results we think require that our variables be continuous actually remain valid (possibly with minor adaptations) without that requirement.

But I'm actually more interested in the physical side of the equation. What makes you think that classical mechanics implies or assumes an infinitely divisible world, from a physical point of view? Could you give a couple of specific examples?

Well, we currently think we know better...

I said almost the same a while ago here in the forum, in a different context. Still, perhaps we shouldn't be too surprised. You analyse a phenomenon carefully, and you attempt to identify its most important features. If you then use those features as axioms to construct a model for the phenomenon, there's a good chance your model will provide a fair approximation to the real thing.

Although the atom isn't really made up of little particles called electrons revolving around a nucleus in circular orbits, matter is nevertheless atomic to first approximation, and atoms do contain a nucleus with protons and neutrons, and much smaller electrons dispersed around the atom. So, while Rutherford's model of the atom is strictly speaking false, reality is close enough that it can't help behaving a little bit like the model. Perhaps this is simply a logical necessity.

Remember that Galileo, who founded the scientific method, also said that the book of the world was written in mathematical language.

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