Computer proof ‘blows up’ centuries-old fluid equations • Hacker News Board

Re: Computer proof ‘blows up’ centuries-old fluid equations

#201

Post by d2049 » Sun, Nov 20, 2022, 7:33 AM UTC

paulpauper wrote: post #157

How is it even possible to learn this much math or organize such complex ideas. Undergrad comes nowhere even close to this. I guess the leap occurs in grad school. But even though, this is way more advanced than 99.75% of math papers I have read.

In my limited few years of doing undergraduate / graduate research in this kind of field, and thus having to read research papers like this, I can say that this is BY FAR the longest paper I have ever encountered. Several times over. Forget writing a 177-page paper, how do you even begin to read this? Yes there is a helpful outline on page 6, but sheesh!

Re: Computer proof ‘blows up’ centuries-old fluid equations

#202

Post by Ferrotin » Sun, Nov 20, 2022, 7:36 AM UTC

photochemsyn wrote: post #115

> "Yet much remains unknown about the Euler equations — including whether they’re always an accurate model of ideal fluid flow." In general the word 'model' implies a mathematical construct that mimics the behavior of a real-world experimental or observational system, i.e the experimental or observational data can be generated by the model. Where's the real-world ideal fluid flow? No such systems exist. > "In princip…

No, atoms are why there aren’t perfect ideal fluids, and quantum effects have nothing to do with it.

Re: Computer proof ‘blows up’ centuries-old fluid equations

#203

Post by voidwtf » Sun, Nov 20, 2022, 7:52 AM UTC

wwalker3 wrote: post #101

The question that the referenced paper (1) is trying to answer is "do the 3D incompressible Euler equations develop a finite time singularity from smooth initial data of finite energy?" This is an important question in the theory of nonlinear partial differential equations, but is probably not as relevant to real fluid flow as a lay reader might imagine. The incompressible Euler equations model a very strange and unp…

This sounds similar in some ways to the discussion that arose between AlphaPhoenix and Veritasium YouTube channels concerning the speed of electrons along a closed loop of wire.

https://youtu.be/2Vrhk5OjBP8 With a relevant illustration at 9:41 at where I think these topics intersect conceptually.

Re: Computer proof ‘blows up’ centuries-old fluid equations

#204

Post by mik1998 » Sun, Nov 20, 2022, 10:13 AM UTC

yazaddaruvala wrote: post #176

mik1998 wrote: post #116

Earlier quoted context omitted.

No, this is not true at all. It is a certain _mathematical_ construction. It has nothing to do with society. Given the specific definition of addition in the field of real numbers, the statement 1 + 1 = 2 is true in any society. > but all “axioms” are social constructs Again, they are mathematical constructs. Axioms don't have anything to do with society. I can construct any axiom I want, and derive true statements f…

I’ll try one last time: > definition of addition What do you think any definition is other than the creation of a socially shared construct? > I can construct any axiom I want, and derive true statements from it. Let’s test this. Axiomatically I am always right! As such, I derive that it is illogical, dare I say irrational, of you to disagree with me. A truer statement has never even been written! Facts! Hmm… turns o…

> What do you think any definition is other than the creation of a socially shared construct?

A definition is simply a definition. It doesn't have to be social or shared. I can define addition in any manner, and in fact there are many other definitions of addition.

> Let’s test this. Axiomatically I am always right! As such, I derive that it is illogical, dare I say irrational, of you to disagree with me. A truer statement has never even been written! Facts!

You have a very interesting misunderstanding here. A fact is true only relative to an axiom. So yes, relative to such an axiom you are right, similarly how relative to the axioms of the field of real numbers the statement 1 + 1 = 2 is true.

You can construct any axiom and derive facts from it. I can construct axioms such that 1 + 1 = 0. That doesn't make the statement 1 + 1 = 2 in the real numbers any less false!

Re: Computer proof ‘blows up’ centuries-old fluid equations

#205

Post by bheadmaster » Sun, Nov 20, 2022, 10:14 AM UTC

edgyquant wrote: post #174

bheadmaster wrote: post #97

Earlier quoted context omitted.

What's a "what's"?

An inquiry for a description of an object. Nothing in language is difficult it’s finding the smallest abstraction to generate all the rules for all language that is. You’re asking a simple grammar question under the impression it’s an unsolved science question.

No, I'm merely pointing out the absurdity of asking "what's X" to a needless extent.

Everyone knows what a rock is. Everyone knows what I mean by "throwing two rocks on a pile of two rocks and getting a pile of four rocks" - almost everyone has done it countless times as a child. The "What's a rock" comment is just pointless philosophizing, masquerading as profoundness.

Re: Computer proof ‘blows up’ centuries-old fluid equations

#206

Post by robertlagrant » Sun, Nov 20, 2022, 11:16 AM UTC

robertlagrant wrote: post #138

Agingcoder wrote: post #2

I remember my math professor at university telling me that truth in mathematics was a social construct, and that nothing was true until a social consensus had been reached between mathematicians. This struck me at the time as a very powerful statement, yet unexpected, since very much not what most people expect from mathematics. After all, it's supposed to be a field where there is such a thing as a (most of the time…

Social constructionalism is to my understanding least surprisingly found in universities. I think the issue is if you call every type of thought and communication "social construction" then you don't end up with anything useful.

Sorry: constructionism*.

Re: Computer proof ‘blows up’ centuries-old fluid equations

#207

Post by mpweiher » Sun, Nov 20, 2022, 2:32 PM UTC

naasking wrote: post #125

mpweiher wrote: post #44

Earlier quoted context omitted.

[Mathematical truth as a social construct] > Yeah, that is how it feels like nowadays, It's always been that way. (Again, I really recommend the book[1] ). And it's hard to see how it could be otherwise. (Also depends a little about what exact mathematical truth we are talking about and whether you are a Platonist or Constructionist) That doesn't imply what either the recent proponents or the critics seem to think. I…

> Mathematics ≠ Reality Conjecture. A mathematical universe is consistent with everything we know, in which case math is literally the study of reality.

This is simply false, for at least three reasons. There are probably more.

1. Math can describe lots of universes that differ from observation. For starters, just look at all the different kinds of space-time there could be.

2. On top of that, even the math that we have that describes our current universe is...tricky: the math at the core of our two best physical theories (General Relativity and Quantum Mechanics) is inconsistent, i.e. they both cannot be true at the same time.

Lots of our greatest minds have tried to find a mathematical formulation that makes the two compatible, but so far none have succeeded. So there may not actually be a mathematical formulation.

3. And finally, nature is under no obligation to be describable mathematically. That it is so describable is a fortuitous circumstance, and one we might be finding the limits of (see (2)).

"The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning."

https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...

Re: Computer proof ‘blows up’ centuries-old fluid equations

#208

Post by wwalker3 » Sun, Nov 20, 2022, 2:56 PM UTC

acchow wrote: post #196

wwalker3 wrote: post #101

The question that the referenced paper (1) is trying to answer is "do the 3D incompressible Euler equations develop a finite time singularity from smooth initial data of finite energy?" This is an important question in the theory of nonlinear partial differential equations, but is probably not as relevant to real fluid flow as a lay reader might imagine. The incompressible Euler equations model a very strange and unp…

The infinite implied speed is of course because the Euler equations are just a rough approximation of what is happening, right? Much like Newtonian physics was an approximation? There’s a real causality bound of the speed of light

If you use the theory of nonlinear partial differential equations to analyze the behavior of compressible materials, you can find what are called the "characteristic" speeds, which are the speeds that various types of waves propagate at.

Compressible materials tend two have two different characteristic speeds, one for sound waves and one for shock waves.

The speed of sound basically works out to speed = sqrt(stiffness / density). So as a material gets stiffer, the speed of sound goes up. An infinitely stiff (i.e. incompressible) material by implication would have an infinite speed of sound, though this can't happen in any real material.

Shock waves travel faster than sound, at a speed related to the pressure difference across the shock. The greater the pressure difference, the faster the shock travels. So if you had an infinite pressure difference, you could have an infinitely fast shock wave, but again this can't happen in the real world.

However, sound and shock speeds only apply to pressure waves in a material. Other influences like gravity and electromagnetism travel at the speed of light. So for example, if you're doing fluid dynamics for plasma, then you'll have a third characteristic speed, the speed of light, because of the charged nature of the material attracting and repelling itself.

There are also more exotic characteristics, like the speed of a propagating combustion front in a flammable material. But when you get to this level you're no longer just solving one simple set of differential equations.

Re: Computer proof ‘blows up’ centuries-old fluid equations

#209

Post by naasking » Sun, Nov 20, 2022, 3:36 PM UTC

mpweiher wrote: post #207

naasking wrote: post #125

Earlier quoted context omitted.

> Mathematics ≠ Reality Conjecture. A mathematical universe is consistent with everything we know, in which case math is literally the study of reality.

This is simply false, for at least three reasons. There are probably more. 1. Math can describe lots of universes that differ from observation. For starters, just look at all the different kinds of space-time there could be. 2. On top of that, even the math that we have that describes our current universe is...tricky: the math at the core of our two best physical theories (General Relativity and Quantum Mechanics) is…

> Math can describe lots of universes that differ from observation.

This has all been hashed out already. Look up the mathematical universe hypothesis and Wolfram's Ruliad for two different but similar approaches. The basic idea is that all consistent mathematical objects exist, and we simply inhabit one of them.

The fact that QM and GR are inconsistent with each other is not an issue because they describe different universes in mathematical space that are close but not equal to our own.

> And finally, nature is under no obligation to be describable mathematically. That it is so describable is a fortuitous circumstance

That too is a conjecture. There is exactly zero evidence that nature is non-mathematical, but people who keep saying this are basically insisting that it's effectiveness at describing reality is simply a continuous chain of grand coincidences. To say this is statistically implausible bordering on impossible is the kindest way to put it.

Re: Computer proof ‘blows up’ centuries-old fluid equations

#210

Post by marcusverus » Sun, Nov 20, 2022, 3:44 PM UTC

ookdatnog wrote: post #127

marcusverus wrote: post #111

Earlier quoted context omitted.

> An apple is not an apple. An apple is a subjective construct that summarises the distinguishing features of a certain kind of object as it appears to our sense. This is utter nonsense. An apple IS an apple. If I put one on a table, then obliterate every human being that’s capable of sensing it, the apple is utterly unaffected. If aliens come down and experience the apple differently than we would have, that doesn’t…

It's not nonsense, they are exactly right. The world is a sea of particles and energy which behave according to certain patterns (both fundamental laws and emergent behavior). Some of these patterns are pertinent to us, so we name them, giving rise to a category. "Apple" is such a category. The clump of molecules on the table we denote with the term "apple" doesn't care that our brains have deemed it similar enough t…

> The clump of molecules on the table we denote with the term "apple" doesn't care that our brains have deemed it similar enough to certain other clumps of molecules to be placed in the same category. If all humans cease to exist, the clump of molecules may still be on the table, but there's no one left to consider it part of any category.

Correct. If all humans cease to exist, the human-made abstraction of "the apple" passes away. But an apple--the one on the table--is completely unaffected.

The difference here is that between an abstraction and a concrete instantiation of said abstraction. The abstraction of "the apple" is indeed a subjective construct which would pass away with humanity. But OP said "an apple". "An apple" is a part of the physical universe. It is, like you said, a clump of molecules. Molecules that can be measured objectively.

An apple is no more subjective than the orbital period of the earth or the length of the standard meter.