Well, the proof is perfectly valid, as on some abstract level a computer == Turing machine == something you can reason about. If you can prove that your program works in the required way, then your proof holds.
However, brute force solutions are ugly and shed little (if any) new light on the problem, and in the words of G.H.Hardy - "There is no permanent place in the world for ugly mathematics". With things like this I just can't help but think that an elegant and enlightening proof will unexpectedly pop up some time in the distant future when we're all off researching some sort of new-age hypermaths or something...
I'd agree with your dislike of ugly solutions. I guess the benefit of this being accepted as a proof is that all the conjectures that begin 'Assuming that x is true...' can now be converted to theorems saying 'x is true, therefore...', which might tidy up a lot of loose ends.
There are proofs (by humans) that are too large for all but a rare few to check. Wiles' proof of "Fermat's Last Theorum" was 100 pages. Courtesy of Wikipedia:
"Because Wiles had incorporated the work of so many other specialists, it had been suggested in 1994 that only a small number of people were capable of fully understanding at that time all the details of what Wiles has done." [http://en.wikipedia.org/wiki/Wiles%27_proof_of_Fermat%27s_La...]
No one can argue that an unconfirmed computer proof is more trustworthy than an unconfirmed proof from a human.
I think you're overlooking the real thrust of the situation. Wiles' proof was relatively lengthy and involved, but it can be examined and studied by humans in an extremely reasonable amount of time. It only took three days for Wiles to present his original proof. There aren't many people in the world who can understand it, and there are fewer who are knowledgeable enough to confirm its validity, but they exist.
Conversely, the proof described in the NYTimes article is of such length that no single mathematician could confirm its validity -- rather than deducing the non-existence of the object in question by a logical argument, it examined a huge number of possible cases. In that respect, it is far more similar to the proof of the four color map theorem. The issue is not so much whether or not we trust the computer's result, but moreso what it means for mathematics to proceed with results that we do not, in a traditional sense, understand.
Cray 1S? Isn't that about the power of today's desk top? The Cray could do 160 to 250 MFLOPS depending on the problem (Wikipedia). That seems pretty slow. I answered my own question http://forums.techpowerup.com/showthread.php?t=94721. There are processors doing up to 60 GFLOPS. Must be an error in the story.
EDIT: I have to start paying attention to bylines. There are archeologists among us.
For brute force proofs, the proof basically nothing to do with the output of the program. I'll explain:
A mathematical proof is just some utterance or depiction that will convince your audience that you have proven something. The actual utterance that might 'count' is totally dependent upon who is in the audience. If you are a topologist and you are talking to another topologist working on very similar stuff, your proofs would be incomprehensible to even other mathematics researchers, who could not determine if you were giving a valid proof or not, while the other topologist could easily tell.
If your audience is a really smart physics professor, that same topology proof would have to suck in many other facts, and explicitly quote accepted theorems, or you would risk the physicist jumping up and saying "not so fast" every time he see's a break in the reasoning. At least the physicist will be comfortable with the existence of theorems.
Now, a proof of the same thing, given to an alien with an IQ of 5000 but who's race never thought up the concept of a theorem would be exquisitely painful. You might have to actually give the proof of each theorem you use, and each theorem each of those theorems use. Otherwise, he would say "your logic is flawless, but you assume facts you have not proven, so this proof is incomplete."
If people were way smarter, proofs would be way shorter, since nobody writes down things that are really obvious. Nobody says something like "since the real numbers are commutative under multiplication..." because there is no need to say it. If people were 50 times smarter or something, way more complex things would just 'go without saying'.
So, what does some kind of brute force proof really consist of? It's a program that runs on a computer generally. So the proof is really the source code. If people are convinced the program does what you claim, and then you run it, that is certainly a proof.
If you can't count that, you also can't count any other non-constructive proof. I would say that they are basically pointless, since you can use the result of the proof in other proofs, but you can't build on it very much.
no, this is really begging the question. youve reduced "the proposition is true" to "the algorithm is correct, and the algorithm returns 'true'". once you prove the former you still have to do the hard part of actually _proving_ the latter. i suppose this amounts to mathematically proving, among other things, that your pentium 4 or whatever is incapable of executing a program incorrectly.
reminds me of when the four color theorem was proved using a computer program. They had to convince everyone the proof was valid. There were lots of questions about the validity of the program.
I cant help thinking Douglas Adams brought this exact problem up ages ago when Deep Thought found the Answer to Life, The Universe and Everything.
As Adams pointed out the answer is sometimes useless because the process (or the question) was incomprehensible to the "beings" who sought the answer.
Then later when Earth is destroyed before the Answer is calculated he makes an even stronger point: It doesnt matter
The answer gave occupation to the philosophers for years before it was discovered, it provided millions of years of purpose for the "mice" in finding the answer. And Ultimately the lack of the answer didnt matter anyway.
Perhaps we can just accept the answer (knowing that it could be innacurate and keeping an eye out for indicators) and make use of it? Surely that is more worthwhile - it's something of a leap to completely trust the computer, but it seems more dangerous to refuse to trust it at all :)
[+] [-] J_McQuade|17 years ago|reply
However, brute force solutions are ugly and shed little (if any) new light on the problem, and in the words of G.H.Hardy - "There is no permanent place in the world for ugly mathematics". With things like this I just can't help but think that an elegant and enlightening proof will unexpectedly pop up some time in the distant future when we're all off researching some sort of new-age hypermaths or something...
... well, I suppose a man's got to dream, right?
[+] [-] andrew1|17 years ago|reply
[+] [-] shib71|17 years ago|reply
"Because Wiles had incorporated the work of so many other specialists, it had been suggested in 1994 that only a small number of people were capable of fully understanding at that time all the details of what Wiles has done." [http://en.wikipedia.org/wiki/Wiles%27_proof_of_Fermat%27s_La...]
No one can argue that an unconfirmed computer proof is more trustworthy than an unconfirmed proof from a human.
[+] [-] telegraph|17 years ago|reply
Conversely, the proof described in the NYTimes article is of such length that no single mathematician could confirm its validity -- rather than deducing the non-existence of the object in question by a logical argument, it examined a huge number of possible cases. In that respect, it is far more similar to the proof of the four color map theorem. The issue is not so much whether or not we trust the computer's result, but moreso what it means for mathematics to proceed with results that we do not, in a traditional sense, understand.
[+] [-] enum|17 years ago|reply
[+] [-] russell|17 years ago|reply
EDIT: I have to start paying attention to bylines. There are archeologists among us.
[+] [-] goodside|17 years ago|reply
[+] [-] justin_vanw|17 years ago|reply
A mathematical proof is just some utterance or depiction that will convince your audience that you have proven something. The actual utterance that might 'count' is totally dependent upon who is in the audience. If you are a topologist and you are talking to another topologist working on very similar stuff, your proofs would be incomprehensible to even other mathematics researchers, who could not determine if you were giving a valid proof or not, while the other topologist could easily tell.
If your audience is a really smart physics professor, that same topology proof would have to suck in many other facts, and explicitly quote accepted theorems, or you would risk the physicist jumping up and saying "not so fast" every time he see's a break in the reasoning. At least the physicist will be comfortable with the existence of theorems.
Now, a proof of the same thing, given to an alien with an IQ of 5000 but who's race never thought up the concept of a theorem would be exquisitely painful. You might have to actually give the proof of each theorem you use, and each theorem each of those theorems use. Otherwise, he would say "your logic is flawless, but you assume facts you have not proven, so this proof is incomplete."
If people were way smarter, proofs would be way shorter, since nobody writes down things that are really obvious. Nobody says something like "since the real numbers are commutative under multiplication..." because there is no need to say it. If people were 50 times smarter or something, way more complex things would just 'go without saying'.
So, what does some kind of brute force proof really consist of? It's a program that runs on a computer generally. So the proof is really the source code. If people are convinced the program does what you claim, and then you run it, that is certainly a proof.
If you can't count that, you also can't count any other non-constructive proof. I would say that they are basically pointless, since you can use the result of the proof in other proofs, but you can't build on it very much.
[+] [-] eru|17 years ago|reply
[+] [-] hc|17 years ago|reply
[+] [-] bitdiddle|17 years ago|reply
[+] [-] ErrantX|17 years ago|reply
As Adams pointed out the answer is sometimes useless because the process (or the question) was incomprehensible to the "beings" who sought the answer.
Then later when Earth is destroyed before the Answer is calculated he makes an even stronger point: It doesnt matter
The answer gave occupation to the philosophers for years before it was discovered, it provided millions of years of purpose for the "mice" in finding the answer. And Ultimately the lack of the answer didnt matter anyway.
Perhaps we can just accept the answer (knowing that it could be innacurate and keeping an eye out for indicators) and make use of it? Surely that is more worthwhile - it's something of a leap to completely trust the computer, but it seems more dangerous to refuse to trust it at all :)
[+] [-] KaiP|17 years ago|reply
http://www.cecm.sfu.ca/organics/papers/lam/paper/html/paper....
[+] [-] sireat|17 years ago|reply
The resulting perfect play for some endgames is incomprehensible to a human GM and can last for more than 100 moves.
http://www.xs4all.nl/~timkr/chess/perfect.htm
[+] [-] pj|17 years ago|reply
[+] [-] xenophanes|17 years ago|reply
[+] [-] hexis|17 years ago|reply