99.99% of the mathematics of motion consists of continuous functions with continuous derivatives of all orders. Infinitesimals are just fine for that.
There were logical contradictions even in Newtons and Leibniz works, far before anyone considered continuous functions without derivatives etc., they were basically making decisions about when a given operation or transformation can be applied based on intuition alone and not any logical deductions, and it wasn't rare they arrived at incorrect conclusions. Also, Newton himself did epsilon-delta reasonings, he just did not notice their generality:
Again, you are confusing the work of Cauchy and Weierstrass and the epsilon-delta stuff with all the latter even more formal approaches to treat more complicated functions, which by the way were developed in response to Fourier examining heat transfer and trying to describe it mathematically (so it's actually rooted in physics).
People were saying the same things you are just saying about limits about the geometry of Euclid, in fact Newton at one time was of the opinion all the formal development of geometry is useless. He reconsidered after obtaining nonsense geometrical results a few times...
The formalization of the calculus, in contrast, was a step backwards in creative terms. Although necessary, it is of interest mostly to pure mathematicians.
I wish you luck doing quantum mechanics with Newton-style calculus.
I took courses in graduate level QM and Newton-style calculus sufficed. The only unusual artifact that stands out in my memory was the Dirac delta function.
Perhaps physicists (e.g., your example of Newton above) somehow intuitively step over the "holes" in the underlying analytical frameworks that mathematicians fall into with regularly. Mathematicians are wont to build "manholes" to cover those holes and physicists have no desire to stop them, seeing that it is honest labour and keeps the mathematicians busy.
There were logical contradictions even in Newtons and Leibniz works, far before anyone considered continuous functions without derivatives etc., they were basically making decisions about when a given operation or transformation can be applied based on intuition alone and not any logical deductions, and it wasn't rare they arrived at incorrect conclusions. Also, Newton himself did epsilon-delta reasonings, he just did not notice their generality:
http://www.sciencedirect.com/science/article/pii/S0315086000...
Again, you are confusing the work of Cauchy and Weierstrass and the epsilon-delta stuff with all the latter even more formal approaches to treat more complicated functions, which by the way were developed in response to Fourier examining heat transfer and trying to describe it mathematically (so it's actually rooted in physics).
People were saying the same things you are just saying about limits about the geometry of Euclid, in fact Newton at one time was of the opinion all the formal development of geometry is useless. He reconsidered after obtaining nonsense geometrical results a few times...
The formalization of the calculus, in contrast, was a step backwards in creative terms. Although necessary, it is of interest mostly to pure mathematicians.
I wish you luck doing quantum mechanics with Newton-style calculus.