Question

Why is there a consistent theory of continuum mechanics in which one just consider things like differential elements and apply Newtons laws? Is there a deeper reason for it. Is it the nature of newtonian framework that makes it happen or is it somehow related to nature of bodies (topological spaces with borel measure etc)?

Answer 1

What evidence do you have that there is a consistent theory of continuum mechanics? Certainly, when looked at through a macroscope, the universe looks like it behaves according to continuum mechanics, but this completely breaks down on the microscopic level. So you can't justify a consistent theory of continuum mechanics by using the universe. There's no reason that good approximations to our universe have to be consistent theories of physics; physicists currently believe that QED is a very accurate approximation to the electromagnetic force, but that it cannot be made consistent at the smallest scales without adding additional physics. And there are papers showing that if you just define your physics using, say, the wave equation without putting some kind of restrictions on the initial conditions, very funny things can happen. Also, Newton's laws of gravitation with point particles have some very unpleasant consequences. So I would say that, unless you're very clever about how you specify it, a theory of continuum mechanics would very likely not be consistent.

If you are very clever, you might be able to make a consistent theory of continuum mechanics, but I don't know of anybody who has actually done this.