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

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

#121

Post by naasking » Sat, Nov 19, 2022, 3:12 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…

> So cat math likely wouldn't have integers as we know them, but would have some kind of size-based analogue.

Sorry, but no. Any species capable of actually creating some kind of math will have some mathematical structure isomorphic to the integers. If cats can't count, then that just says that cats are not capable of creating some kind of math.

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

#122

Post by zaphar » Sat, Nov 19, 2022, 3:12 PM UTC

mola wrote: post #83

bheadmaster wrote: post #65

Earlier quoted context omitted.

The fact that mathematics gives us power to predict events in the real world makes it independent of social consensus. If everyone in the world believes that 2+2=5, that doesn't make it less true 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. I hate this sociologist view that everything depends on the social consensus. G…

You got it backwards. 1984 is possible because it's fundemental that some truths are social constructs.

I would argue that those aren't truths. They need some other term to describe them but truth doesn't really fit the bill. Conflating them with truth leads to weird outcomes in your reasoning.

A more accurate term might be stories rather than truth.

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

#123

Post by SideburnsOfDoom » Sat, Nov 19, 2022, 3:12 PM UTC

bheadmaster wrote: post #113

SideburnsOfDoom wrote: post #109

Earlier quoted context omitted.

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.

" In terms of their behavior under particular circumstances" is very specific. "The maths is useful under particular circumstances" is not the same thing as "numbers are real"

Bricks are in the category of "made objects" not natural objects, which generally means that they are _designed_ to come off a production line as similar as humanly possible to the other products. "My iPhone is physically interchangeable to yours" is a statement about the huge efforts of industrial manufacturing to standardise matter, not about the natural world.

Bricks too have quality thresholds that they have to meet or exceed. That alone should tell you that treating them like integers is a convenient abstraction, nothing more. The sibling comment has it right: counting bricks is a great model, but confusing your model for reality is still an old error.

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

#124

Post by postalrat » Sat, Nov 19, 2022, 3:13 PM UTC

bheadmaster wrote: post #65

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…

The fact that mathematics gives us power to predict events in the real world makes it independent of social consensus. If everyone in the world believes that 2+2=5, that doesn't make it less true 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. I hate this sociologist view that everything depends on the social consensus. G…

2+2=4 stops being true when you use a different method of counting.

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

#125

Post by naasking » Sat, Nov 19, 2022, 3:15 PM UTC

mpweiher wrote: post #44

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…

[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.

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

#126

Post by musingsole » Sat, Nov 19, 2022, 3:17 PM UTC

bheadmaster wrote: post #113

SideburnsOfDoom wrote: post #109

Earlier quoted context omitted.

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.

> Does it matter that two bricks are not "the same piece of matter" if our predictions work the same for both of them

No, because of a dense, interconnected web of other social truths (the rest of the arithmetic model), the relative error of this one truth/model is negligible.

However, confusing your model for reality is a fallacy perhaps older than time.

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

#127

Post by ookdatnog » Sat, Nov 19, 2022, 3:18 PM UTC

marcusverus wrote: post #111

TheOtherHobbes wrote: post #39

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. 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…

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 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.

If aliens then visit who can't eat the apple and aren't interested in botany, they may simply choose not to distinguish between apples and pears, or apples and any other fruit, or even apples and any other form of organic material. The same clump of molecules is there, but the categories it belongs to have changed.

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

#128

Post by musingsole » Sat, Nov 19, 2022, 3:21 PM UTC

naasking wrote: post #120

dorchadas wrote: post #68

Earlier quoted context omitted.

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 str…

> 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.

The definition of "consistent" seems completely entangled with a given social group's ideas of "rational". You might imply that our word for it hints at a Platonic ideal of "consistent", but if that's true, then you're caught in an infinite cascade of which nuances of meaning between the Platonic Ideal and our concrete reality are actually reflections of the truth or corruption of it.

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

#129

Post by roywiggins » Sat, Nov 19, 2022, 3:24 PM UTC

jojobas wrote: post #26

planck01 wrote: post #6

Earlier quoted context omitted.

There are such reachable truths, but every mathematical system has a non empty set of axioms - or assumptions- which are 'given'.

These axioms are not given in course of socialization and generally are observations of nature rather than human.

There are multiple set theories using different axioms. Is the Axiom of Choice based on an observation of nature or do mathematicians keep it around because it's useful? It's a statement about infinities that absolutely have no physical reality. You can do mathematics without it, and the question of whether to do math relying on it is a matter of opinion.

(Yes, proofs relying on AC are arguably true even if you don't accept AC, but as a social reality some sets of axioms are considered valid bases for work and some aren't, you can keep adding stronger axioms to ZFC to prove more things more easily, but how far you go with that before it stops being interesting is a matter of opinion)

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

#130

Post by nuclearnice1 » Sat, Nov 19, 2022, 3:26 PM 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…

What a wonderfully informative and educational comment. Thank you.

Would you also be able to shed some light on what a singularity is? It was not intuitive to me that incompressiblity should lead to a singularity.

The article dances around the term:

> At that point, the Euler equations are said to give rise to a “singularity” — or, more dramatically, to “blow up.”

> Once they hit that singularity, the equations will no longer be able to compute the fluid’s flow.