Readit NewsThis seems like a mathematical trick -- pretending like perfectly smooth functions exist in reality -- which is then extrapolated from in bizarre and unilluminating ways.
The only reasonable reading of Newton's laws (and descriptions of e.g. physically-constructed curves like this dome) is that they are true up to some small epsilon length scale. No matter how small the epsilon, as long as it is not literally zero, this doesn't work.
(for instance if things aren't perfectly smooth then there is some small force proportional to the discrepancy ~O(e^2) or O(sin e) or whatever, which moves the ball off the dome)
If I was teaching physics from sceatch, I would state on day one: physics is the practice of building models whose low-order approximations give correct predictions about reality. There is no such thing as a perfect model to infinite decimal places.
(That one 9 decimal place calculation from QFT doesn't invalidate this: given a model, the calculations may be perfectly accurate! But the model is still fuzzy because of fuzzy inputs. In that case, it has been possible to make it very very not fuzzy.)
You may be interested to know that this exact objection has been made in the philosophical literature. See "Causal Fundamentalism in Physics" by Zinkernagel (2010). Available here: https://philsci-archive.pitt.edu/4690/1/CausalFundam.pdf
At the end, the author notes (as you do) that if you consider a finite difference equation with small time steps, there are no pathological solutions. He also mentions that Newton takes this difference equation approach when solving problems in his Principia.
See also "The Norton Dome and the Nineteenth Century Foundations of Determinism" by van Strien:
>> Abstract. The recent discovery of an indeterministic system in classical mechanics, the Norton dome, has shown that answering the question whether classical mechanics is deterministic can be a complicated matter. In this paper I show that indeterministic systems similar to the Norton dome were already known in the nineteenth century: I discuss four nineteenth century authors who wrote about such systems, namely Poisson, Duhamel, Boussinesq and Bertrand. However, I argue that their discussion of such systems was very different from the contemporary discussion about the Norton dome, because physicists in the nineteenth century conceived of determinism in essentially different ways: whereas in the contemporary literature on determinism in classical physics, determinism is usually taken to be a property of the equations of physics, in the nineteenth century determinism was primarily taken to be a presupposition of theories in physics, and as such it was not necessarily affected by the possible existence of systems such as the Norton dome.
It does take some asking around to discover the optimal Julia workflow with Revise.jl, PkgTemplates.jl, VSCode settings/debugger, Pluto.jl, but now it's probably my best development experience. Julia 1.10 improves much of this as well.