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

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

#111

Post by marcusverus » Sat, Nov 19, 2022, 2:45 PM UTC

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.

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 change the apple one bit.

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

#112

Post by [deleted] » Sat, Nov 19, 2022, 2:52 PM UTC

qazxcvbnm wrote: post #61

prmph wrote: post #51

Earlier quoted context omitted.

> Apart from that, with the help of computers, it can be made absolutely precise and clear which statements follow from which axioms, and in that sense it is not a social construct at all. I think you are mistaken. The idea that math proofs are a social construct relates to, in my view, much deeper ideas than you seem to think [1]. It is not just that convincing other mathematicians that a proof is correct is a socia…

> It is not just that convincing other mathematicians that a proof is correct is a social process, but also that the reasoning on which any proof relies, even if it seems unassailable, even if built into an automated checker, is still a product of the human mind. Usually there is a level of logic that can challenge even what seems so basic as to be fool-proof. I think you are mistaken. I used to be very troubled by t…

[deleted]

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

#113

Post by bheadmaster » Sat, Nov 19, 2022, 2:53 PM UTC

SideburnsOfDoom wrote: post #109

bheadmaster wrote: post #99

Earlier quoted context omitted.

> 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?

> What prediction is that?

For example, all the predictions that make us capable of building skyscrapers that don't fall down for centuries. Does it matter that two bricks are not "the same piece of matter" if our predictions work the same for both of them? In terms of their behavior under particular circumstances, they are the same.

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

#114

Post by majewsky » Sat, Nov 19, 2022, 2:55 PM UTC

moffkalast wrote: post #47

marginalia_nu wrote: post #35

Earlier quoted context omitted.

An ideal computer is an universal machine. A real computer has real limitations on what is calculable, even among the things that are theoretically calculable.

Sounds like we need to make a virtual universal machine and account for the limitations when emulating it.

The problem with this is whether the emulation ever terminates. Simply put, you cannot emulate an infinitely powerful machine in finite time.

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

#115

Post by photochemsyn » Sat, Nov 19, 2022, 2:56 PM UTC

> "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 principle, if you know the location and velocity of each particle in a fluid, the Euler equations should be able to predict how the fluid will evolve for all time. But mathematicians want to know if that’s actually the case."

At some scale in a real fluid, quantum effects become important, and the notion that it's possible to know both position and momentum falls apart. To quote Peter Atkins:

> "The trouble is when you're dealing with operators, is it turns out you can't always extract explicit information about all of them simultaneously, and this leads to Heisenberg's uncertainty principle, which most people think of as a great confuser of the world and denier of information. I like to think of it as a great clarifier because old-fashioned people, like Newton and Einstein, and Lagrange, and all the people who developed classical mechanics, took it as certain that to specify the state of a particle, you had to specify where it was and how fast it was going."

That's why there are no perfect ideal fluids. Atkins expands:

> "What quantum theory did, through the uncertainty principle, was to clarify - what it said, was discuss the world if you like in terms of positions, or if you prefer, discuss the world in terms of linear momenta, don't try both at once. You get very simple descriptions in terms of positions, you get very simple descriptions in terms of linear momenta, it's only when you - like trying to start a sentence in English and ending in Latin or something, trying to mix the two together, that you get into confusion. So think of the Heisenberg principle as a clarifier - try to think one way, or try to think the other way, but don't try to think like Newton thought."

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

#116

Post by mik1998 » Sat, Nov 19, 2022, 2:59 PM UTC

yazaddaruvala wrote: post #102

mik1998 wrote: post #77

Earlier quoted context omitted.

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.

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 from it. Perhaps the only "social" thing here is what axioms we generally deem interesting enough to investigate - this does not change the facts derived from them.

> that we use to create hopefully useful models.

I think you underestimate pure mathematics :)

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

#117

Post by majewsky » Sat, Nov 19, 2022, 2:59 PM UTC

thaumasiotes wrote: post #54

Karliss wrote: post #32

Earlier quoted context omitted.

Sure calculating infinity is easy as long as you redefine "infinity" to be something which isn't actual infinity. But it's useless for many mathematical proofs. Having overflow of some finite floating point calculations labelled as "infinity" is useful for calculations of some practical problems, but it shouldn't be confused with the actual mathematical concept of infinity just because both use the same word.

Floating point infinity is "actual infinity". It has all the correct mathematical properties. If you want to slam a special IEEE constant, you should slam 0, which has different properties from mathematical zero.

According to my Javascript console (using IEEE754 double precision), 1e308+1e308 equals Infinity. That's not "actual infinity".

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

#118

Post by q-big » Sat, Nov 19, 2022, 3:06 PM UTC

ThereIsNoWorry wrote: post #20

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…

I don't know why you're being downvoted, but that is a perfectly valid statement. Everyone thinks proofs are this holy grail and totally rigorous, and they are on a certain level. But the idea is floating around that Mathematicians are infallible when in fact lots of proofs in highly complex areas of mathematics are NOT 100% perfectly rigorous. They contain a lot of skipping, because "it's trivial" and consensus. Thi…

> Doing actual rigorous proofs a computer can verify is enormously tedious and many Mathematicians dislike it for that reason, because the inherently subjective "elegance" and "beauty" gets lost in translation.

Couldn't we also interpret this fact that computerized proofs are currently often very unelegant as strong evidence that not a lot is understood about this topic and thus doing such "ugly" computerized proofs is the best we can (in most cases) currently do?

Science at the boundary of human knowledge is often quite ugly; as our understanding of it grows, it often becomes more beautiful and elegant.

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

#119

Post by gilbert_vanova » Sat, Nov 19, 2022, 3:08 PM UTC

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…

> I doubt there is a single (professional) engineer that would say something like that.

I suspect an engineering professor would be even more compelled to highlight how social consensus is the foundation of everything else that follows. You only have to read the surface layers of this or that performance debate to see how it is unfortunately the case. "Performance" is a fluid term that doesn't mean much of anything on its own -- and people arguing that they've juiced another drop of performance juice from this or that application's fruit are often just talking past each other in terms of priorities or perspective.

The symbolic manipulation at the heart of mathematics is a byproduct of language -- another system beholden to social consensus. It's inescapable.

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

#120

Post by naasking » Sat, Nov 19, 2022, 3:08 PM UTC

dorchadas wrote: post #68

qazxcvbnm wrote: post #61

Earlier quoted context omitted.

> It is not just that convincing other mathematicians that a proof is correct is a social process, but also that the reasoning on which any proof relies, even if it seems unassailable, even if built into an automated checker, is still a product of the human mind. Usually there is a level of logic that can challenge even what seems so basic as to be fool-proof. I think you are mistaken. I used to be very troubled by t…

But what this doesn't get at is that the very system of logic we use to make proofs is a social construct. Other cultures have had other systems of logic, and called valid arguments that we wouldn't today precisely because they were using a different system of logic. So the very foundations of mathematics, the logic we use behind our proofs, is inherently a social construct that arose out of Greek philosophy as it wa…

> But what this doesn't get at is that the very system of logic we use to make proofs is a social construct. Other cultures have had other systems of logic, and called valid arguments that we wouldn't today precisely because they were using a different system of logic.

I think you need to separate the process of developing mathematics, from the self-consistency and validity of the logical argument or mathematical structure itself. The process is social but the validity and mathematical structure itself is not.

Computer science has led to an explosion in different logics. We've never had more logics/formal systems than we do today, but whether any given formal system actually is consistent is not dependent on consensus, it is a fact, either true or false, completely independent of consensus.