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

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

#151

Post by uoaei » Sat, Nov 19, 2022, 5:09 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…

Math doesn't do that, models (including language: a model of communication around which we have already reached consensus) do.

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

#152

Post by pencilguin » Sat, Nov 19, 2022, 5:12 PM UTC

mpweiher wrote: post #7

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…

A proof is a rhetorical device to convince others of the truth of a proposition.

If you can't get anybody to read your proof, does it demonstrate anything?

Fred Moxley has (what seems to me like, but what do I know?) a nice proof of the Riemann conjecture that he got by quantizing the problem. But nobody will read it, because mathematicians don't like that method. It might be right or not, but it anyway doesn't tell you anything surprising about prime numbers, so nobody can be bothered.

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

#153

Post by WalterBright » Sat, Nov 19, 2022, 5:16 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…

Best comment I ever read on HN.

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

#154

Post by pencilguin » Sat, Nov 19, 2022, 5:16 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.

... of what people can and also choose to observe about nature.

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

#155

Post by naasking » Sat, Nov 19, 2022, 5:18 PM UTC

gmfawcett wrote: post #140

naasking wrote: post #121

Earlier quoted context omitted.

> 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.

IDK, it seems easy to imagine an alien mathematics based only upon continuous values? There's nothing obviously universal about discretizing things.

Firstly, the reals contain the integers, so there is an isomorphism as I said.

Secondly, discretization absolutely is universal. It's literally in the laws of physics for one (particles are discrete, energy levels are discrete, etc.). For another, are you suggesting a physical alien species will have a continuous number of appendages, or organs, or that their population will somehow be continuous? I frankly don't see how you can possibly escape aliens capable of math developing a notion of basic counting.

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

#156

Post by dist1ll » Sat, Nov 19, 2022, 5:20 PM UTC

aaron695 wrote: post #50

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…

> that truth in mathematics was a social construct. This is garbage. Everything under this definition is a social construct and as such why would you only relate it to mathematics? The rock I'm holding is a social construct. Deep. If they want to get stoned and talk about the meaning of life cool, but it's beneath a math professor (Who's not at home getting stoned) Following it logically you quickly find murder, rape…

Just because something is a social construct doesn't mean it's worthless, impure or required to be rejected.

Going from self-reflection to nihilism is a pretty big overreaction.

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

#157

Post by paulpauper » Sat, Nov 19, 2022, 5:20 PM UTC

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.

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

#158

Post by ChadNauseam » Sat, Nov 19, 2022, 5:25 PM UTC

auggierose wrote: post #87

soVeryTired wrote: post #75

Earlier quoted context omitted.

> I can sit down and prove something using a tool like Isabelle, and I will be as sure of its "truth" as I can possibly be, and it really doesn't matter what other people, mathematicians or not, think about it. That's the beauty of it. But I guess the point is that almost no-one does this. I would guess that if someone tried to formally verify every published paper out there (or even every textbook), they would uncov…

Not many do this currently, that is true. But this will change. In a hundred years every mathematician will do this. I think it will reach "mainstream" much much earlier than this, probably around 2030.

Probably by 2030 and by 2040 at the latest, you will be able to give a proposition to a machine learning model and the model will output a machine-verifiable proof of its truth or falsity at least as often as a human can.

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

#159

Post by version_five » Sat, Nov 19, 2022, 5:28 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.

If Winston and O'Brian both believe O'Brian is levitating, that still doesn't make it actually true. "Sanity is not statistical" as Winston says in the book

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

#160

Post by auggierose » Sat, Nov 19, 2022, 5:46 PM UTC

prmph wrote: post #147

musingsole wrote: post #128

Earlier quoted context omitted.

> 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 n…

Exactly the point I was making. Many do not seems to see the fundamental issue at play here, and another way to think of them is what you have hinted at: the role of language. There is no objective way to nail down the meaning of words, like "consistent", "proof", "equal", etc. Suppose one wanted a maximally rigorous definition of "equal". Does it mean two things that cause people to think of the same thing when they…

Equality is actually quite easy to axiomatise in most logics, here in my favourite logic:

1) x = x

2) x = y => P[x] => P[y]

In my opinion, there is a mathematical reality, which is shared by everyone, even by those who don't believe in it :-) For example, a logical system exists in that reality, and you can either derive a theorem in that system or not in this reality. I don't think it is possible that there is a third possibility. I don't think it is possible that I have a different reality from you in that respect. This reality is not socially constructed, it just is. Intuitionism will tell you that because you don't know if a certain theorem is derivable, it is in some sort of hybrid state until we know for sure via a intuitionistic proof or a counter example. I think that is bullocks. Either there is a proof or not. Either there is a counter example, or not.

Beyond that, extending this mathematical reality, there is a wider, not as easily accessible reality. We can try to understand that reality by modelling it via certain assumptions, and then applying our mathematical reality to those assumptions. I believe the mathematical conclusions we draw from this will be real to the extent that the assumptions are true; but of course you cannot ever be sure about those assumptions, and so you cannot be sure about the conclusions. But if you notice that your conclusions do not hold, you need to challenge your assumptions, not your mathematics.