Comment by ajkjk | Hacker News Reader

Comment by ajkjk | Hacker News Reader

This 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.)

> This seems like a mathematical trick -- pretending like perfectly smooth functions exist in reality -- which is then extrapolated from in bizarre and unilluminating ways.

Given that it is common to make certain claims about the model of NM (e.g. determinism and reversibility) a mathematical trick that demonstrates this is not true is a valid refutation of claims about the model.

You can use NM to model other physically impossible systems (such as objects with arbitrarily high rigidity or arbitrarily low friction), but they are not internal contradictions within the NM model, so are less valid for criticism of NM.

This example is not practically that different from chaotic systems e.g. the double pendulum, which end up in states that can't be time-reversed back to the initial state without infinite precision. In the case of a double-pendulum pointing straight up at t=0, the same argument holds: it will eventually move in an indeterminate direction. In both cases, reversing time to recover the initial state is impossible without infinite precision.

I don't think anybody is losing sleep over the non-determinacy of a double-pendulum. It's just understood as evidence that your model is only as deterministic as your ability to say that something happens at t=0 or x=0 instead of |t| < epsilon or |x| < epsilon.

> This example is not practically that different from chaotic systems...

This is wildly different from chaotic systems in that it cannot be time reversed even with infinite precision.

For cryptography, compare to a one-time-pad. There are plenty of practical attacks on one-time-pads, but they are theoretically unbreakable. If someone were to show an actual weakness in OTP in theory it would be a big deal, even if (in practice) OTPs are seldom used due to practical considerations.

Well I agree that it's not "theoretically" different from chaotic systems, that's why I said "practically". Practically, infinite precision doesn't exist.

Just saying. Doesn't really affect anything, though.

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.

Somewhat interested, but tbh, the reason I thought of it is the same reason that they thought of it -- that it's obvious to anyone who studies physics from a philosophical angle that it really has to be that way. They just put it in a lot of fancy words so that it has academic rigor.

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