Alternative interpretations of classical physics (2019) | Hacker News Reader

Alternative interpretations of classical physics (2019)

journals.aps.org

94 pointsJPLeRouzic39 comments

Classical Mechanics is a well-defined mathematical framework. Consequently, the word 'determinism' when applied to classical mechanical systems has a sharp meaning. For example, the dynamics of a finite set of point masses with a smooth potential function is 'deterministic' in the sense that there are mathematical theorems proving that initial conditions can be uniquely evolved forward for some finite amount of time.

The paper wants to argue for general non-determinism, but can only do that by changing the definition of determinism to something more fuzzy. It then proposes a new set of mathematical axioms to incorporate this definition, but the question that the paper does not answer is whether all this buys us anything or not. I think that any reasonable physicist would say that there is nothing foundational left to do for classical mechanics purely in itself, so it must be that possible applications can only lie in the search for a more fundamental theory of nature.

But then I am very pessimistic. We understand quite well how classical mechanics arises as a limit from quantum mechanics, which in turn emerges from relativistic quantum field theories. Together with general relativity, it is the latter that are the most fundamental (experimentally confirmed) theories. Therefore, trying to uproot the mathematics of classical mechanics feels like starting completely at the wrong end to me.

In the abstract, the authors say

""" Building on recent information-theoretic arguments showing that the principle of infinite precision (which translates into the attribution of a physical meaning to mathematical real numbers) leads to unphysical consequences, we consider possible alternative indeterministic interpretations of classical physics

"""

which seems to be the jumping off point to disregard strict determinism in that sense. So I guess it'd buy you out of the unphysical consequences.

Don't have aps access though so I can't say more (probably couldn't even with aps access :D)

Agreed. One of the first things I learned early in Physics was to choose your model wisely. There are many models, some will be accurate for the problem you are solving, while others will not (or will not be tractable). Often a simplification (simple harmonic oscillation, thermodynamics, etc) gives excellent insight and results, and in others a change of perspective (lagrangian dynamics, state-space) are helpful. Although there are cases strikingly close to Classical (photon optics, charge quantization, etc) where quantum effects become significant and useful, they are rare. It is mostly in the realm of (sub)atomic and interstellar where obviously quantum and relativistic effects (other than magnetism) become necessary to consider.

As much as I dream of understanding the non-local modifications necessary for quantum gravity, it doesn't seem like inserting non-determinism into classical physics has many useful effects (hah, showing I'm an engineer at heart!). I know that a lot of pop-sci goes into understanding "reality" and wave-function collapse, but I find Scott Aaronson's take on the difficulty of over-interpretation more seriously.

https://www.scottaaronson.com/blog/?p=5359

I've been always taught that classical (Newtonian) physics doesn't have to be interpreted as deterministic. However, the reason was simpler than the lack of the "infinite precision". Basically, some classical systems can have several solutions, e.g. the famous Norton's dome https://en.wikipedia.org/wiki/Norton%27s_dome .

I find this write-up enlightening : https://blog.gruffdavies.com/2017/12/24/newtonian-physics-is...

It seems the equations given by Norton really are unphysical, so actually it's not a proof on indeterminism, but simply incompleteness - newtonian physics cannot model some really really fucky situations - its incomplete.

I can recommend taking this blog post with a grain of salt. I'm a physics masters student and after working through the math myself I believe the Lipschitz continuity violation that Gruff rejects as a red herring is actually the real source of the nondeterminism, and is not just some mathematical fluff.

The first law and stitching arguments he makes appear to both be flawed. Having non-zero derivatives of force in combination with zero velocity and zero force is perfectly in accordance with Newton's first law. And in his frictionless ball counterexample, his equation is incorrect because it violates Newton's second law, not because two solutions are stitched together.

Lipschitz continuity is required for guaranteed uniqueness of differential equation solutions, and non-uniqueness can appear as nondeterminism or incompleteness.

I think he reaches the right conclusion but his reasoning is flawed.

That’s a brilliant analysis of the Norton’s dome; thank you.

Thanks. It's a great piece of writing!

Why is virtual work not taught anymore?

Who says it’s not? I suspect most courses in advanced classical mechanics still cover it, although probably briefly.

Norton’s dome is intriguing; other departures from determinism in classical physics include “space invaders”: in some many-particle systems a particle can develop an infinite velocity, which sends it “out of world”; then time reversibility means it can enter the world unpredictably. I mention both of these in an article about unsolved problems in classical physics: http://arstechnica.com/science/2014/08/the-never-ending-conu...

Doesn't conservation of energy preclude this? If all particles start with finite energy, and only finite energy is added to the system, won't the total energy stay finite? Or are you assuming an infinite potential somewhere, e.g. gravity from a point "planet"?

The examples I know of all involve gravitational or similar potentials, so there is unbounded negative energy available from 1/r. But you can still get singularities without collisions: Xia, Annals of Math. 135 411–468 (1992).

At least a few of these seem to be cases where the concept of a point particle breaks down.

There are similar problems in E&M at caustics that are solved by remembering that E&M waves have non-zero wavelength.

>which sends it “out of world”

Wouldn't any system have to include this "out of world" particle? Doesn't its inclusion mean "the world" simply grows at that infinite rate?

> Take a simple bottle of milk like the ones pictured here, with a cylindrical section below a tapering segment.

Those bottles are rectangular!

They are rectangular cylinders. The shape works for the problem under discussion.

I thought you where making up words, saying “rectangular cylinder”, but apparently it’s a thing:

https://www.quora.com/Geometry-What-is-a-rectangular-cylinde...

A cylinder can have any cross section.

Norton doesn't understand the principle of virtual work and should go back and read D'Alembert and Lanczos.

Link to actual explanation from Norton: http://www.pitt.edu/~jdnorton/Goodies/Dome/

Are these kinds of math tricks less likely to be a lack of modeling than nondeterminism, ie, does the fact that one mathematical model produces a non-deterministic result mean that the actual situation is non-deterministic?

No, it doesn’t, as you suspect. The article that sudosysgen links to up above treats this question nicely.

Published in 2019.

Requires subscription, preprint is here: https://arxiv.org/abs/1909.03697

Seems this paper was cited by this[1] Quanta article which was discussed here[2].

[1]: https://www.quantamagazine.org/does-time-really-flow-new-clu...

[2]: https://news.ycombinator.com/item?id=22848766

This seems to overlap with that Wolfram thread from yesterday https://news.ycombinator.com/item?id=26544651

One interesting thing about the universe is that it tends to hide infinities.

Black holes are wrapped in event horizons. The digits of pi would require all the energy of the universe to calculate. Calculating the brain with sufficient accuracy (using our current computational framework) would require a computer so large it would generate a black hole...

> Black holes are wrapped in event horizons.

Not really - the event horizon is the effect of the black hole, not something separate from it. Also, from one perspective there is nothing mysterious about it - an observer passing through it would not notice any kind of border there (though an observer who is not passing through it would see it as some kind of barrier emitting Hawking radiation).

Also note that physicists do not believe that there is any infinity at the center of a black hole. With a theory of quantum gravity we may even be able to describe the structure of the core of a black hole (though it would remain forever an un-testable model). Just like classical mechanics allows a particle to gain unbounded speed, GR allows a mass to gain unbounded gravity, but that is likely incorrect, we just don't know how to describe the limit yet.

> The digits of pi would require all the energy of the universe to calculate.

This makes no sense - pi has an infinite number of digits, so all of the energy in the universe can't possibly be enough to compute it.

> Calculating the brain with sufficient accuracy (using our current computational framework) would require a computer so large it would generate a black hole...

I find this hard to believe. Regardless, there is no infinity in the human brain.

Note that in general infinity is an inherently non-scientific concept - there is no way to experimentally distinguish an infinity from a quantity larger than the largest possible experiment.

> Also note that physicists do not believe that there is any infinity at the center of a black hole.

This is my impression as well, but do you have a good source for it? If I'm talking with people about black holes, it would be nice to have something to point to besides my vague impression from watching physics lectures.

This blog post from a theoretical physicist should be a decent source:

http://backreaction.blogspot.com/2020/05/a-brief-history-of-...

Any physics equations have limits of applicability. If a singularity appears in the solution to the equations, that is a sign that you have found one of these limits, not that there is an actual infinity there.

>pi has an infinite number of digits, so all of the energy in the universe can't possibly be enough to compute it.

If the universe is also infinite, then you could calculate Pi with X% of the universe's energy, where X is an arbitrarily small positive number.

If the universe is infinite, you could compute all of the digits of Pi in finite time by using an infinite amount of parallel computers, true. But computing a percentage of an infinite number (total energy in the universe) doesn't make sense.

Just calculate digits of pi in base pi, it's 1.0 exactly.

Then calculate the digits of the area of a square in base pi :)

What I dislike about clickbaity paper titles like this is the following:

Using a general term like "determinism" in the title of a paper might be fine for an expert audience: They will know that the paper deals with a specific definition inside a specific discussion.

But the general public will pick up on this and incorporate it into their superstitions. In the end this could (indirectly, through a funnel of regurgitations) strengthen some persons believe in pseudoscience resulting in them not vaccinating their kids or treating cancer with homeopathy because "everything is connected on the quantum level".

Scientists should be more responsible with the choice of their titles.

I think adding a discriminatory suffix to terms like "determinism" would provide a solution. Something like "Physics without infinite precision determinism: [...]" is much less likely to be misinterpreted.

> Physics without determinism

What does this even mean? The whole universe and life itself are possible precisely because there is determinism, which is captured in what we call logic, (and arithmetic).

Sects, fucking sects everywhere.

The turn of phrase is probably bad, but determinism can exist even in a fundamentally undeterministic universe. All that is required for the universe to be undetermimistic is for SOME events to lack a cause. For example, a universe in which once every billion years a particle appears out of thin air somewhere in the universe and flies off in a random direction is, on the whole, non-deterministic, even if every other interaction works according to simple mechanistic rules.

The universe and biology are stochastic.

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