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Computer proof ‘blows up’ centuries-old fluid equations

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

#101

Post by wwalker3 »

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 unphysical kind of fluid. Incompressibility means that the speed of wave propagation in such a fluid is infinite, which means that normal causality is not respected. Effects in such a fluid happen simultaneously with their causes.

For example, if you apply a force to one end of a pipe full of Euler fluid, the fluid instantly starts coming out of the other end of the pipe, with no time taken for this effect to propagate from one end of the pipe to the other. You could use a long pipe full of Euler fluid as a superluminal communication device!

Intuitively, it seems reasonable that in such an unphysical fluid, it would be possible to form a singularity even from smooth initial conditions. The difficulty, of course, is proving that intuition, which is what the paper is trying to do.

1) https://arxiv.org/pdf/2210.07191.pdf "Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data", Jiajie Chen and Thomas Y. Hou.

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

#102

Post by yazaddaruvala »

mik1998 wrote: post #77
yazaddaruvala wrote: post #18

Earlier quoted context omitted.

1+1=2 is a social construct. It’s simple, repeatable and therefore programable - but still a social construct.

It's not. It's a fact derived a priori from the definition of addition and the axioms of the field of real numbers. There's nothing special about the claim that 1 + 1 = 2. You can also define addition such that 1 + 1 = 0 or anything else.

“a priori from the definition of addition” is effectively saying “a social construct”

And I hate to break it to you but all “axioms” are social constructs that we use to create hopefully useful models.

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

#103

Post by jxdxbx »

bheadmaster wrote: post #91
jfengel wrote: post #70

Earlier quoted context omitted.

That's actually an example of what OP was talking about. You have defined + as the operator that mimics what piles of rocks do, and defined numbers as counting rocks. That's only a tiny fraction of what math does. An interesting and useful one, and mathematicians have put a lot of work into studying basic arithmetic. They have expanded out into numerous other forms, some of which turn out to have correspondence to th…

> But the hard part is convincing other mathematicians to care. My point is that whether other mathematicians care or not is completely irrelevant and doesn't subtract from mathematics' power of predicting phenomena in the real world. Each and every mathematical theory has to be consistent with basic rules of reality - if nothing else, symbolic manipulation relies on basic arithmetic and set theory. Without symbolic…

You can construct a formal system with any axioms you choose. It is not arbitrary that some of these systems turn out to be useful in modeling the world. But there are other systems that are only dry exercises in symbol manipulation that may be of little use to physics or engineering. Or of course they might end up being super important 100 years later. But in the meantime mathematicians might be interested in them anyway.

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

#104

Post by gilbetron »

thaumasiotes wrote: post #17
blurbleblurble wrote: post #13

Earlier quoted context omitted.

Double precision floats have a maximum value of 1.7976931348623158 E + 308

That is not correct; they have a maximum value of positive infinity. See what you get when you square 1.7e+307.

1.7e+307 squared is 2.89e+614, not infinity even though a "computer" will say +inf

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

#105

Post by AnonCoward42 »

TheOtherHobbes wrote: post #39
AnonCoward42 wrote: post #22

Earlier quoted context omitted.

> The proof is not a social construct. > The truth is. Yeah, that is how it feels like nowadays, however the truth is bound in a narrow set of assumptions. These assumptions are bound in reality even in mathematics (One apple is one apple, you add another one, you have two). And while there is an epistemologic level to reality, you would dismiss reality entirely by calling it a social construct. The details of how a…

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. To a non-human consciousness those features may be uninteresting, irrelevant, or incomprehensible, so they might not see apples at all. But they could see " "s, which we don't even have a concept for, never mind a word. And which we either ignore or possibly…

> 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.

It was clearly not about the apple, but about distinct entities with similar features. Now for another being these might not be similar, but something else probably is, disregard of different dimensions, different senses or the likes. As the observed reality for an alien species or actually any other species on this earth is different of course.

> You cannot create an absolute internally consistent mathematics, because foundational axioms depend on subjective experience, not on objective logic.

> And you can't prove a subjective experience objectively.

I think the misconception comes from the fact that you need basic assumptions to build an abstraction. One of the most basic assumptions is that something like a shared reality exists and we're not for example in a virtual world or a dream.

You can happily deny this shared reality, however I would not necessarily encourage you to touch fire (literally and figuratively speaking).

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

#106

Post by [deleted] »

jxdxbx wrote: post #103
bheadmaster wrote: post #91

Earlier quoted context omitted.

> But the hard part is convincing other mathematicians to care. My point is that whether other mathematicians care or not is completely irrelevant and doesn't subtract from mathematics' power of predicting phenomena in the real world. Each and every mathematical theory has to be consistent with basic rules of reality - if nothing else, symbolic manipulation relies on basic arithmetic and set theory. Without symbolic…

You can construct a formal system with any axioms you choose. It is not arbitrary that some of these systems turn out to be useful in modeling the world. But there are other systems that are only dry exercises in symbol manipulation that may be of little use to physics or engineering. Or of course they might end up being super important 100 years later. But in the meantime mathematicians might be interested in them a…

[deleted]

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

#107

Post by Maursault »

mjburgess wrote: post #24
somat wrote: post #21

"That’s because it’s impossible for a computer to calculate infinite values. It can get very close to seeing a singularity, but it can’t actually reach it" Why not? Is it impossible to calculate infinite values in general? I suspect not, My understanding is that a lot of calculus is in fact on how to calculate infinite values. And a computer is a universal machine, this means that while it can not calculate everythin…

Almost all functions aren't computable, as they aren't discrete. A "computer" is just a function from {0,1}^N -> {0,1}^M

A function is merely a rule, and rules do not do anything other than define relationships, but this is not quite right either, because who is defining? Taking into account Newton's Second Law of Motion, which the practical application of cause and effect, a computer is always always a person: one who computes. Consider that no matter how complicated they become, pencils do not calculate, cars do not drive, and guns do not shoot. You tell everybody. Listen to me! You've gotta tell 'em! Computers are people! We gotta stop them! Somehow!

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

#108

Post by [deleted] »

bheadmaster wrote: post #91
jfengel wrote: post #70

Earlier quoted context omitted.

That's actually an example of what OP was talking about. You have defined + as the operator that mimics what piles of rocks do, and defined numbers as counting rocks. That's only a tiny fraction of what math does. An interesting and useful one, and mathematicians have put a lot of work into studying basic arithmetic. They have expanded out into numerous other forms, some of which turn out to have correspondence to th…

> But the hard part is convincing other mathematicians to care. My point is that whether other mathematicians care or not is completely irrelevant and doesn't subtract from mathematics' power of predicting phenomena in the real world. Each and every mathematical theory has to be consistent with basic rules of reality - if nothing else, symbolic manipulation relies on basic arithmetic and set theory. Without symbolic…

[deleted]

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

#109

Post by SideburnsOfDoom »

bheadmaster wrote: post #99
SideburnsOfDoom wrote: post #90

Earlier quoted context omitted.

> that 2+2=4 - in the sense that I know for sure, if I take throw two rocks on a pile of two rocks, I'll get a pile of four rocks, not five rocks. That depends on if one of the rocks breaks in half as you throw it onto the rock-pile or not. And also if the resulting piece knocked off is large enough to pass your fuzzy and contextual distinction between "rock" and "pebble". But IMHO, arithmetic such as counting number…

> tl;dr the natural world is not fungible, but behaving as if it is, is a convenient abstraction for mathematics and commerce, not a property of the natural objects. If it wasn't a property of natural objects (in some way), then how could our predictions work so well in the real world?

What prediction is that? Are you arguing that two rocks or apples or people are actually "can't tell them part" identical?

It works for electrons. But when was the last time that you interacted knowingly with a single electron?

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

#110

Post by thaumasiotes »

mjburgess wrote: post #84
thaumasiotes wrote: post #81

Earlier quoted context omitted.

> The issue is to demonstrate that a given function, say f, for given real-valued inputs, say x, has an output y which is not real-valued and goes to infinity. Why would the output need to be not real? There's no difficulty with saying a real-valued function has a singularity. The issue is to demonstrate that this function has a singularity at some point, yes. Simulation is a bad way to do that, though conceivably yo…

The issue with bit-patterns are, at least, they're discrete. And so cannot, eg., represent pi. This project is about real-valued functions which are taken to describe physical reality. Almost all of physical reality has no closed-form analytical description that "traditional mathematics" can operate on. So there arent any relevant symbolic rules of inference yet invented to resolve this problem. If you want to progra…

Computers work with perfect representations of pi all the time. The TI-89 will do it routinely.

All of your objections continue to apply just as strongly to human mathematicians as they do to computers. But you apparently believe there is a difference between what the mathematicians can do and what the computers can do. This is false. Any problem that occurs in computers' representations of values will also occur in human representations of values.

Using your example, a system that is sensitive to differences so fine that they cannot be held in any realistic amount of memory is quite possible. But humans will have just as much trouble using it as computers do. If it is easy to show that x=pi in particular causes trouble, computers will find that easy too, using the same tools -- symbolic computation on pi -- that humans do.

The fact that computers have discrete internal representations is not relevant to anything. All human mathematics is also performed using exclusively discrete representations.

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