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Hilbert's sixth problem: derivation of fluid equations via Boltzmann's theory

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Re: Hilbert's sixth problem: derivation of fluid equations via Boltzmann's theory

#91

Post by gnubison »

killjoywashere wrote: post #7
itsthecourier wrote: post #3

Earlier quoted context omitted.

may you please elaborate on why it is important, why hasn't been solved before and what new applications may you imagine with it, please?

David Hilbert was one of the greatest mathematicians of all time. Many of the leaders of the Manhattan Project learned the mathematics of physics from him. But he was famous long before then. In 1900 he gave an invited lecture where he listed several outstanding problems in mathematics the solution of any one of which would change not only the career of the person who solved the problem, but possibly life on Earth. M…

> the solution of any one of which would change not only the career of the person who solved the problem, but possibly life on Earth. Many have stood like mountains in the distance, rising above the clouds, for generations.

Whether or not this is AI, this comment is not true. An axiomatic derivation of a formula doesn’t change how it’s used. We knew the formulas were experimentally correct, it’s just that now mathematicians can rest easy about whether they were theoretically correct. Although it’s interesting, it doesn’t change or create any new applications.

Re: Hilbert's sixth problem: derivation of fluid equations via Boltzmann's theory

#92

Post by kgwgk »

Xmd5a wrote: post #71
kgwgk wrote: post #68

Earlier quoted context omitted.

> [Quantum Cognition] is a real field with dozen of collaborators and even a textbook. Flat Earth is also a real field, with conferences with hundreds of attendees.

Have you visited Busemeyer's website ? Here's the textbook he wrote, 2nd edition https://www.cambridge.org/us/universitypress/subjects/psycho... Looks totally respectable. Why do you feel the need to ridicule it?

https://arxiv.org/pdf/1309.5673

> Decision making – When people are given a chance to play a particular gamble twice, if they think they won the first play, or alternatively if they think they lost the first play, then the majority chooses to play again on the second round. Given these preferences, they should also play the second round even if they don’t think about the outcome of the first round. Yet people do just the opposite in the latter case (Tversky & Shafir, 1992). This finding violates the law of total probability, yet it can be explained as a quantum interference effect

I wouldn't say that "it can be explained as a quantum interference effect" for any respectable definition of "explained" but your mileage may vary.

> The Contextual Nature of Concepts and their Combinations – When quantum entities become entangled, they form a new entity with properties different from either constituent, and one cannot manipulate one constituent without simultaneously affecting the other. The mathematics of entanglement has been used to model the nonmonotonic relations observed among concepts when they are combined to form a new concepts such as STONE LION

https://scispace.com/pdf/the-emergence-and-evolution-of-inte...

> Different possible ways of looking at this combination exist, and a positive answer to both questions ‘is STONE LION a LION’ and ‘is STONE LION a STONE’ make sense. Hence a better approach is to consider both as entities, and use Fock space for a two entity situation, and forgo the more simple model of considering one of them as a context.

Vanilla Quantum Models of Cognition and Decision are not enough. One really needs Quantum Field Theory Models of Cognition and Decision. "The genuine structure of quantum field theory is needed to match predictions with experimental data." https://arxiv.org/pdf/0705.1740

Re: Hilbert's sixth problem: derivation of fluid equations via Boltzmann's theory

#93

Post by _zoltan_ »

baxtr wrote: post #69
IdealeZahlen wrote: post #4

Sabine Hossenfelder's video on this: https://youtu.be/mxWJJl44UEQ

In my perception Sabine’s quality degraded over the last year or so. Maybe it’s also the topics she covers. I’m not sure why she is getting into fantasies of AGI for example. I liked the skeptical version of her better.

I don't know if it's just the persona she plays in these videos, but it's so so so creepy and cringe.

Re: Hilbert's sixth problem: derivation of fluid equations via Boltzmann's theory

#94

Post by whatshisface »

bubblyworld wrote: post #59
LudwigNagasena wrote: post #18

So where and how does a jump from nice symmetric reversible equations to turbulent irreversibility happen?

I've been puzzling about this as well. The best answer I have (as an interested maths geek, not a physicist, caveat lector) is that it sneaks in under the assumption of "molecular chaos", i.e. that interactions of particles are statistically independent of any of their prior interactions. That basically defines an arrow of time right from the get-go, since "prior" is just a choice of direction. It also means that the…

It comes about when the deterministic collision process is integrated over all the indistinguishable initial states that could lead into an equivalence class of indistinguishable final states. If you set the collision probability to zero it's time reversible even with molecular chaos, and if the particles are highly correlated (like in a polymer) there can still arise an arrow of time when the integral is performed.

Re: Hilbert's sixth problem: derivation of fluid equations via Boltzmann's theory

#95

Post by Alive-in-2025 »

naasking wrote: post #87
baxtr wrote: post #69

Earlier quoted context omitted.

In my perception Sabine’s quality degraded over the last year or so. Maybe it’s also the topics she covers. I’m not sure why she is getting into fantasies of AGI for example. I liked the skeptical version of her better.

As far as I've seen, her position is only that AGI is pretty much inevitable. What's so fantastical about that?

I think plenty of people don't think it's inevitable. I'm no ai researcher, just another software engineer (so no real expertise). I think it will keep getting better but the end point is unclear.

Re: Hilbert's sixth problem: derivation of fluid equations via Boltzmann's theory

#96

Post by naasking »

Alive-in-2025 wrote: post #95
naasking wrote: post #87

Earlier quoted context omitted.

As far as I've seen, her position is only that AGI is pretty much inevitable. What's so fantastical about that?

I think plenty of people don't think it's inevitable. I'm no ai researcher, just another software engineer (so no real expertise). I think it will keep getting better but the end point is unclear.

The reason it's inevitable is because because it follows from physics principles. The Bekenstein Bound proves that all physical systems of finite volume contain finite information, humans are a finite volume, ergo a human contains finite information. Finite information can be fully captured by a finite computer, ergo computers can in principle perfectly simulate a human person.

This + continued technological development entails that AGI is inevitable.

Re: Hilbert's sixth problem: derivation of fluid equations via Boltzmann's theory

#97

Post by Xmd5a »

kgwgk wrote: post #92
Xmd5a wrote: post #71

Earlier quoted context omitted.

Have you visited Busemeyer's website ? Here's the textbook he wrote, 2nd edition https://www.cambridge.org/us/universitypress/subjects/psycho... Looks totally respectable. Why do you feel the need to ridicule it?

https://arxiv.org/pdf/1309.5673 > Decision making – When people are given a chance to play a particular gamble twice, if they think they won the first play, or alternatively if they think they lost the first play, then the majority chooses to play again on the second round. Given these preferences, they should also play the second round even if they don’t think about the outcome of the first round. Yet people do just…

I don't think this cursory look at the abstracts and first pages of these papers is achieving the effect you intended.

The first quote summarize the findings (page 4) of Tversky & Shafir, 1992. Google indicates this paper has 978 citations.

>is STONE LION a STONE

This is called a typicality test.

Jurafsky, Lecun & two more fellows dropped this paper recently: https://arxiv.org/pdf/2505.17117 (emphasis mine)

>This work aims to bridge this gap by integrating cognitive psychology, information theory, and modern NLP. We pose three central research questions to guide our investigation: [RQ1]: To what extent do concepts emergent in LLMs align with human-defined conceptual categories? [RQ2]: Do LLMs and humans exhibit similar internal geometric structures within these concepts, ESPECIALLY CONCERNING ITEM TYPICALITY? RQ3]: How do humans and LLMs differ in their strategies for balancing representational compression with the preservation of semantic fidelity when forming concepts?

It would be interesting to try to reproduce Hampton's original experiment on typicality with LLMs and run the same analysis as https://arxiv.org/pdf/1208.2362.

Anyway thanks for the exchange. Running these thoughts once more in my head allowed me to reconsider some stuff I found about Zipf distributions that might tie into the tradeoff between compression and meaning Lecun is talking about.

The quantum zipf stuff is here: https://arxiv.org/abs/1909.06845

It's here too: https://link.springer.com/article/10.1134/S1061920806030071

>quantum field theory

Chomsky is working on Hopf Algebra.

https://magazine.caltech.edu/post/math-language-marcolli-noa...

Re: Hilbert's sixth problem: derivation of fluid equations via Boltzmann's theory

#98

Post by kgwgk »

Xmd5a wrote: post #97
kgwgk wrote: post #92

Earlier quoted context omitted.

https://arxiv.org/pdf/1309.5673 > Decision making – When people are given a chance to play a particular gamble twice, if they think they won the first play, or alternatively if they think they lost the first play, then the majority chooses to play again on the second round. Given these preferences, they should also play the second round even if they don’t think about the outcome of the first round. Yet people do just…

I don't think this cursory look at the abstracts and first pages of these papers is achieving the effect you intended. The first quote summarize the findings (page 4) of Tversky & Shafir, 1992. Google indicates this paper has 978 citations. >is STONE LION a STONE This is called a typicality test. Jurafsky, Lecun & two more fellows dropped this paper recently: https://arxiv.org/pdf/2505.17117 (emphasis mine) >This wor…

> I don't think this cursory look at the abstracts and first pages of these papers is achieving the effect you intended.

The only effect intended was my own amusement. I doubted anyone would ever read the comment.

> Jurafsky, Lecun & two more fellows dropped this paper recently

Somehow they missed that quantum field theory formalism is essential to explain such things.

Re: Hilbert's sixth problem: derivation of fluid equations via Boltzmann's theory

#99

Post by baxtr »

naasking wrote: post #96
Alive-in-2025 wrote: post #95

Earlier quoted context omitted.

I think plenty of people don't think it's inevitable. I'm no ai researcher, just another software engineer (so no real expertise). I think it will keep getting better but the end point is unclear.

The reason it's inevitable is because because it follows from physics principles. The Bekenstein Bound proves that all physical systems of finite volume contain finite information, humans are a finite volume, ergo a human contains finite information. Finite information can be fully captured by a finite computer, ergo computers can in principle perfectly simulate a human person. This + continued technological developm…

Although the reasoning is clear, you (and her) jump from "possible in principle" to "inevitable in practice".

Just because something is physically possible doesn't make it "inevitable". That's why it's just a fantasy at this point.

Re: Hilbert's sixth problem: derivation of fluid equations via Boltzmann's theory

#100

Post by bubblyworld »

whatshisface wrote: post #94
bubblyworld wrote: post #59

Earlier quoted context omitted.

I've been puzzling about this as well. The best answer I have (as an interested maths geek, not a physicist, caveat lector) is that it sneaks in under the assumption of "molecular chaos", i.e. that interactions of particles are statistically independent of any of their prior interactions. That basically defines an arrow of time right from the get-go, since "prior" is just a choice of direction. It also means that the…

It comes about when the deterministic collision process is integrated over all the indistinguishable initial states that could lead into an equivalence class of indistinguishable final states. If you set the collision probability to zero it's time reversible even with molecular chaos, and if the particles are highly correlated (like in a polymer) there can still arise an arrow of time when the integral is performed.

Interesting, so if I understand right you are saying that coarse-graining your states can produce an arrow of time on its own? Given some fixed coarse-graining, I can see that entropy would initially increase, since your coarse-graining hides information from you. The longer you evolve the system under this coarse graining the less certain you will be about the micro-states.

But I would expect this to eventually reach an equilibrium where you are at "maximum uncertainty" with respect to your coarse graining. Does that sound right at all? And if so, then there must be something else responsible for the global arrow of time, right?

> If you set the collision probability to zero it's time reversible even with molecular chaos

Is this true for boring reasons? If nothing interacts then you just have a bunch of independent particles in free motion, which is obviously time-reversible. And also obviously satisfies molecular chaos because there are no correlations whatsoever. Maybe I misunderstand the terminology.

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