5 ms·

Is a Proof a Proof If No One Can Check It?

32 points by kqr2 17y ago

pj 17y ago
As the singularity trots backward in time toward us, these types of computer found solutions that humans can't comprehend are going to increase.
- xenophanes 17y ago
  Huh? Singularities do not trot or time travel.
  - jderick 17y ago
    Try using your imagination
    - xenophanes 17y ago
      Try being nice :(
      - pj 17y ago
        It was a metaphor, here I'll explain it. The singularity is something in the future. Some believe the singularity will be an entity smarter than humans, that can do things, including move, perhaps even move itself in time. Who knows? Thus the trotting use as an anthropomorphic description of singularity behavior. Furthermore, as technology progresses, the date at which the singularity will arrive is getting closer and closer, thus the singularity is moving backward in time, from the future to the present. So humanity and the singularity, in a plane beyond the space time continuum are moving toward each other, humanity moving forward through time, and the singularity moving backward through time. It's just poetry. That's why jderick said to use your imagination.
enum 17y ago
If you're willing to understand the program behind the proof, I think it's more acceptable than a handwritten proof.
bitdiddle 17y ago
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.
shib71 17y ago
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_Last_Theorem 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 17y ago
  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.
  - skermes 17y ago
    This question came up in some of my more abstract classes in college. A few professors in my department were working on slightly different problems in the same general domain of automated mathematical problem solving and proof construction. The general consensus as I remember it was that the simplest way to get around the problem of no human being able to survey these proofs was to do something like this: 1. Define a machine read- and writable logic that can express your theorem and the steps you think it'll take to get there. 2. Write an automated proof-checker that can verify that proofs in this language are correct. 3. Prove the correctness of the checker. 4. Write a program that starts with your axioms and writes out a proof that ends in your theorem. 5. Verify it with the checker. Now the only proof that needs to be human-surveyable for us to be certain that everything is correct is the one in step 3. The proof created by step 4 can fill up a skyscraper full of hard disks, and as long as the proof checker verifies it we know that it must be correct. Given a simple enough proof language (FOPL, for example) and a suitable programming language (the choice at the time was lisp, I believe) step 3's proof is easily short enough for a human to verify. The only hole left is the possibility of a subtle machine malfunction that causes the checker to falsely categorize a proof as correct. On modern hardware this possibility is remote enough that once a proof has been verified a number of times by independent researchers on their own hardware we can safely ignore it.
russell 17y ago
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 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.
- goodside 17y ago
  No error. The story is from 1988.
J_McQuade 17y ago
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... ... well, I suppose a man's got to dream, right?
- andrew1 17y ago
  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.
hexis 17y ago
If it's called a "proof" shouldn't it prove something to someone? If it can't be checked, it seems like it should be called something else.
KaiP 17y ago
Explanation of the proof by the person who implemented it: http://www.cecm.sfu.ca/organics/papers/lam/paper/html/paper.html http://www.cecm.sfu.ca/organics/papers/lam/paper/html/paper....
justin_vanw 17y ago
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.
- eru 17y ago
  I agree with your notion of "proof as conversation". That's the way presenting proofs verbally works in practice. (And I am a mathematician.)
  - justin_vanw 17y ago
    I got most of this from conversations with Rich Schwartz. http://www.math.brown.edu/~res/ http://www.math.brown.edu/~res/
ErrantX 17y ago
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 :)
sireat 17y ago
In some ways this kind of proof is similar to computer generating massive end-game databases for chess. 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 http://www.xs4all.nl/~timkr/chess/perfect.htm

32 points by kqr2 17y ago

pj 17y ago

As the singularity trots backward in time toward us, these types of computer found solutions that humans can't comprehend are going to increase.

xenophanes 17y ago
Huh? Singularities do not trot or time travel.
- jderick 17y ago
  Try using your imagination
  - xenophanes 17y ago
    Try being nice :(
    - pj 17y ago
      It was a metaphor, here I'll explain it. The singularity is something in the future. Some believe the singularity will be an entity smarter than humans, that can do things, including move, perhaps even move itself in time. Who knows? Thus the trotting use as an anthropomorphic description of singularity behavior. Furthermore, as technology progresses, the date at which the singularity will arrive is getting closer and closer, thus the singularity is moving backward in time, from the future to the present. So humanity and the singularity, in a plane beyond the space time continuum are moving toward each other, humanity moving forward through time, and the singularity moving backward through time. It's just poetry. That's why jderick said to use your imagination.

enum 17y ago

If you're willing to understand the program behind the proof, I think it's more acceptable than a handwritten proof.

bitdiddle 17y ago

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.

shib71 17y ago

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_Last_Theorem 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 17y ago
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.
- skermes 17y ago
  This question came up in some of my more abstract classes in college. A few professors in my department were working on slightly different problems in the same general domain of automated mathematical problem solving and proof construction. The general consensus as I remember it was that the simplest way to get around the problem of no human being able to survey these proofs was to do something like this: 1. Define a machine read- and writable logic that can express your theorem and the steps you think it'll take to get there. 2. Write an automated proof-checker that can verify that proofs in this language are correct. 3. Prove the correctness of the checker. 4. Write a program that starts with your axioms and writes out a proof that ends in your theorem. 5. Verify it with the checker. Now the only proof that needs to be human-surveyable for us to be certain that everything is correct is the one in step 3. The proof created by step 4 can fill up a skyscraper full of hard disks, and as long as the proof checker verifies it we know that it must be correct. Given a simple enough proof language (FOPL, for example) and a suitable programming language (the choice at the time was lisp, I believe) step 3's proof is easily short enough for a human to verify. The only hole left is the possibility of a subtle machine malfunction that causes the checker to falsely categorize a proof as correct. On modern hardware this possibility is remote enough that once a proof has been verified a number of times by independent researchers on their own hardware we can safely ignore it.

russell 17y ago

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 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.

goodside 17y ago
No error. The story is from 1988.

J_McQuade 17y ago

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... ... well, I suppose a man's got to dream, right?

andrew1 17y ago
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.

hexis 17y ago

If it's called a "proof" shouldn't it prove something to someone? If it can't be checked, it seems like it should be called something else.

KaiP 17y ago

Explanation of the proof by the person who implemented it: http://www.cecm.sfu.ca/organics/papers/lam/paper/html/paper.html http://www.cecm.sfu.ca/organics/papers/lam/paper/html/paper....

justin_vanw 17y ago

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.

eru 17y ago
I agree with your notion of "proof as conversation". That's the way presenting proofs verbally works in practice. (And I am a mathematician.)
- justin_vanw 17y ago
  I got most of this from conversations with Rich Schwartz. http://www.math.brown.edu/~res/ http://www.math.brown.edu/~res/

ErrantX 17y ago

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 :)

sireat 17y ago

In some ways this kind of proof is similar to computer generating massive end-game databases for chess. 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 http://www.xs4all.nl/~timkr/chess/perfect.htm