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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

#41

Post by vardump »

blurbleblurble wrote: post #13
thaumasiotes wrote: post #10

> Hou and Luo’s work was suggestive, but not a true proof. 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 — meaning that the solution might be very accurate, but it’s still an approximation. I feel certain that if you run a process that approaches infinity using ordinary floating-point numbers, you will actually…

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

You can combine two doubles to double precision, to 107 bits.

https://en.wikipedia.org/wiki/Quadruple-precision_floating-p...

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

#42

Post by mjburgess »

chmod775 wrote: post #38
mjburgess wrote: post #31

Earlier quoted context omitted.

Mathematics is constrained by properties of abstract objects that the symbols are about . Here, for example, the mathematician has to imagine a scenario to describe with mathematics (two couter-flow fluids, etc.). The notation gains its meaning from this imagined scenario. Rules for manipulating symbols are therefore insufficient. The proof has to follow from the scenario , which the machine is unable to represent.

> which the machine is unable to represent. If you believe that a computer will eventually be able to accurately simulate a human brain, you might as well give up right now. Since if a computer with all its constraints is able to simulate a human (brain), but cannot do this, then a human can't do it either. Conversely if a human can do this but a computer can't, then a computer can never simulate a human. Don't tell…

Yes, I don't believe a concrete computer can "simulate" anything in the relevant sense, let alone a human brain.

Computer, as defined abstractly, is just any abstract function from {0,1}^N->{0,1}^M.

Any realisation of that, eg., by providing each {0,1} as an electrical switch, realises physical properties associated with electrical switches only.

The reason that electrical computers are useful has vastly more to do with the electrical part than the computer part. The "computational properties" of the electrical devices we call computers are relatively trivial.

But in any case, no system in virute of being an implementation of a discete function thereby acquires physical properties. A woodern "comptuer" is useless precisely because you can't play video games on it.

Likewise, even if the brain can be described by a discrete function -- (which is so implausible as to be a bit mad and certainly purely an act of faith) -- then it still requires the relevant physical properties to implement. These properties are extremely unlikely to be those of electrical switches.

The "computational work" done by biochemical signalling alone should probably be regarded as "infinite", saying much about the limitations of discrete conceptions of information.

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

#43

Post by raverbashing »

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…

Because infinity is a mathematical trick, to say where something is going but it is not calculable (it is not a number in itself)

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

#44

Post by mpweiher »

AnonCoward42 wrote: post #22
mpweiher wrote: post #11

Earlier quoted context omitted.

The proof is not a social construct. The truth is. The proof is a mechanism to reach that consensus, by convincing other mathematicians of a specific truth. That is all it is. There is a naive idea that a proof is a purely mechanical series of steps that provides access to truth. Last I checked, this isn't so for the vast majority of proofs in math. Such a proof would be way too tedious to construct or check by mathe…

> 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. It does not at all imply arbitrariness or that anything goes.

> however the truth is bound in a narrow set of assumptions.

Yes, it is. Again, something being a social construct does not make it a free-for-all. More the opposite, because the constraints are socially enforced.

> And while there is an epistemologic level to reality, you would dismiss reality entirely by calling it a social construct.

Mathematics ≠ Reality. Science is about reality, but scientific truth is also a social construct (see Popper), and highly constrained by reality (ibid).

[1] https://www.amazon.de/Mathematical-Experience-Phillip-J-Davi...

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

#45

Post by tigerlily »

krcz wrote: post #37
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…

It's a social construct in the the same sense anything not directly verifiable using senses is. Is there an Eiffel tower in Paris? Most people haven't seen it, so they can only accept the social consensus that it is there. If one can afford it, they can travel to Paris and check themselves. The same with mathematical truth: if one has means (time, intelligence, access to training), they can check the proof themselves…

To illustrate the point further, once you get to Paris how can you be sure it's an Eiffel tower? I guess you have to ask the man in the street. See the truth of it is a social construct. And whether you accept this as truth is a social construct, and so on. QED.

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

#46

Post by mjburgess »

MarcelOlsz wrote: post #40
mjburgess wrote: post #24

Earlier quoted context omitted.

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

What level of math do I need to be at to "compute" this comment?

Very little.

A computer is an abstract mathematical description (eg., like "prime") of a certain mathematical object, a function.

A computer is a way of specifying a discrete function (ie., one which maps a finite number of bits to a finite number of bits), in terms of a sequence of mathematical transitions.

It's an "algorithmic" way of specifying the domain and codomain of a discrete function.

Electrical digital computers aren't actually computers in this sense, and are extremely aproximately described by them. Inasmuch as the shape of the earth is aproximately "spherical".

In any case, pretty much all of physics does not use discrete functions (indeed, I can't think of a single case). In every way physics describes reality, ie., parameterised on space and time, functions are continuous.

They map an infinite amount of spatio-temporal information to an infinite amount of spatio-temporal information.

And there is yet no reason whatsoever, other than the AI PR machine, to suppose that all of physics is wrong in this regard, and the universe is describable by anything else.

This is relevant here, since the problem that cannot be represented to the machine uses ordinary equations of physics, none of which are computable.

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

#47

Post by moffkalast »

marginalia_nu wrote: post #35
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…

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.

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

#48

Post by ben_w »

MarcelOlsz wrote: post #40
mjburgess wrote: post #24

Earlier quoted context omitted.

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

What level of math do I need to be at to "compute" this comment?

> function from {0,1}^N -> {0,1}^M

"{0,1}": the set containing the values 0 and 1

"{0,1}^N": a discrete n-dimensional space, where the possible values in each dimension are 0 or 1

So they're saying a computer takes a length N binary sequence input and produces a length M binary output.

(As for "what level is this", I didn't cover any of this in my double A-level[0] in maths/further maths, but I am covering it in brilliant.org and some popular maths books, so my best guess is it's first or second year degree level?)

[0] https://en.wikipedia.org/wiki/A-Level

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

#49

Post by jules »

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…

There are (short) computer programs where you input a mathematical proposition and a proof in a kind of proof programming language, and the program will then check if it's a valid proof. Saying that mathematical truth is a social construct is technically true but misses the point entirely.

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

#50

Post by aaron695 »

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, genocide being bad are just social constructs. And why exactly should we follow social constructs? Lets all go and start the next FTX because everything is just a social construct so who cares?

And we've just rediscovered nihilism like the other 120 billion teens did.

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