rxwine
rxwine
Joined: Feb 28, 2010
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November 22nd, 2015 at 2:26:51 PM permalink
The setup:
Quote:

A pair of mathematicians, Alexei Lisitsa and Boris Konev of the University of Liverpool, U.K., have come up with an interesting problem—if a computer produces a proof of a math problem that is too big to study, can it be judged as true anyway? In a paper they've uploaded to the preprint server arXiv, the two describe how they set a computer program to proving a small part of what's known as "Erdős discrepancy problem"—the proof produced a data file that was 13-gigabytes in size—far too large for any human to check, leading to questions as to whether the proof can be taken as a real proof.



Related to the question:
Quote:

It wasn't long, however, before some began to realize that at some point, the proofs spit out by the computer would be too long, complicated or both for a human reader to fully comprehend



The question: Will the future begin to be answers to long held questions, without human comprehension as to how it was solved?

Can you develop AI for instance by another computer without knowing how it is actually done because the process becomes to complex to analyze?




(Sunday is funday.)

http://phys.org/news/2014-02-math-proof-large-humans.html#jCp
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Wizard
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Wizard
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November 22nd, 2015 at 3:32:20 PM permalink
Weren't Fermat and the four-color map theorems proven by computer? I don't know the details. It seems unsatisfying to me to trust that a computer solved a theorem by brute force.
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EvenBob
EvenBob
Joined: Jul 18, 2010
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November 22nd, 2015 at 3:43:16 PM permalink
It's ironic that we invent computers to
solve things and then it bothers us
when they do. It's like inventing dynamite
and being so bothered that it's used
to make bombs, you set up a prize
foundation to make amends. What
did Nobel think they would do with
dynamite, just blow stumps out of
the ground?
"It's not enough to succeed, your friends must fail." Gore Vidal
Wizard
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Wizard
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November 22nd, 2015 at 3:51:04 PM permalink
Quote: EvenBob

It's ironic that we invent computers to
solve things and then it bothers us
when they do.



I'm not bothered by computes solving problems. My Odds site is full of computer analysis of casino games I could have never done without a computer. However, I do find it unsatisfying when somebody says, "A computer solved Fermat but you have to have a PhD in computer science to understand the proof. Otherwise, you'll just have to have faith in the programmer." Its probably not unlike how a kid feels when he asks a why question and is told, "You're too young to understand the answer."
It's not whether you win or lose; it's whether or not you had a good bet.
teliot
teliot
Joined: Oct 19, 2009
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November 22nd, 2015 at 4:28:22 PM permalink
Quote: Wizard

I'm not bothered by computes solving problems. My Odds site is full of computer analysis of casino games I could have never done without a computer. However, I do find it unsatisfying when somebody says, "A computer solved Fermat but you have to have a PhD in computer science to understand the proof. Otherwise, you'll just have to have faith in the programmer." Its probably not unlike how a kid feels when he asks a why question and is told, "You're too young to understand the answer."

Hi Mike, this is actually a very interesting topic.

One can actually prove that there are theorems with "arbitrarily long" proofs, in a technical sense. That is, one can prove that there must be a Theorem that has a proof that requires more electrons than there are in the universe to express, so in a practical sense, it can never be written down. And of course, one can't formally identify that Theorem.

However, that differs from Goedel's incompleteness theorem, which asserts that in any theory at least as rich as the integers, there is a true statement which is not provably true. Statements like the twin prime conjecture often feel like they are of this type.
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