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Computer proof ‘blows up’ centuries-old fluid equations

quantamagazine.org

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

#91

Post by bheadmaster »

jfengel wrote: post #70
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…

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 manipulation, you can't even express all those "abstract" mathematics - to say that "abstract" mathematics can not have correspondence to the real world is completely false, because of this basic connection.

Now that I think about it - claiming that "mathematics is a social consensus" is exactly what I'd expect from a mathematics professor - a person whose whole life is isolated from reality, limited to the rigid structure of academia, and whose whole existence depends on other people caring. I doubt there is a single (professional) engineer that would say something like that.

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

#92

Post by ethanwillis »

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…

What's a rock?

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

#93

Post by ethanwillis »

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…

No post body was provided.

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

#94

Post by bheadmaster »

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…

Computers can, in fact, calculate infinite values in style of calculus, but they must use symbolic methods. Computer Algebra Systems often implement such methods.

I believe this article is talking about numerical methods, which are always bound to finite values, because of finite memory.

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

#95

Post by version_five »

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…

Reading the discussion, I think you're both right. There is a common sense notion of mathematics that exists independent of what people want to believe or agree upon.

There are also ways to philosophize about the nature of things in order to frame math as a human construct. I think both views can be simultaneously correct

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

#96

Post by Agingcoder »

karmakurtisaani wrote: post #4
rapsey wrote: post #3

Earlier quoted context omitted.

Huh? Mathemathical proof is not a social construct. This makes no sense.

I suppose they mean what's commonly accepted as true and can be referred to as truths. No one can read all the proofs, so they have to trust others who have. There was that one example where a mathematician "proved" something terribly complicated using his own methods and terminology developed over several years. The truth value of that kind of proof is very much a social construct.

Yes, mochizuki and the abc conjecture. It was an interesting conundrum : he was a really good mathematician, not a crank so his funky proof couldn't be dismissed. However, people were wary of approaching the proof since it was risky career wise (it takes time, etc). You end up with a weird situation where something is probably true, but you won't know that until a trusted group of mathematicians have read it and said so.

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

#97

Post by bheadmaster »

ethanwillis wrote: post #92
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…

What's a rock?

What's a "what's"?

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

#98

Post by mpweiher »

feanaro wrote: post #73
mpweiher wrote: post #44

Earlier quoted context omitted.

[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…

Saying Mathematics ≠ Reality fails to capture a large portion of the story, since a subset of mathematics is clearly necessary to be able to encode science, and confirmed by science, and is in that sense a part of reality.

It's not necessary. It is empirically useful.

Mathematics is about describing possible worlds. Given these assumptions (including the rules of the game), what follows?

Science is about figuring out the real world. The real world has no obligation to be describable by mathematics. That it is so describable is fortuitous.

"The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning."

https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...

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

#99

Post by bheadmaster »

SideburnsOfDoom wrote: post #90
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…

> 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?

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