Massively Collaborative Mathematics
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Massively Collaborative Mathematics
Keep an eye on this, when Gowers gets his problem up I have a feeling there are a number of people here who will want to participate!
Massively Collaborative Mathematics
Edit: spelled his name wrong.
Massively Collaborative Mathematics
Edit: spelled his name wrong.
Last edited by GreedyAlgorithm on Thu Jan 29, 2009 7:00 pm UTC, edited 1 time in total.
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 skeptical scientist
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Re: Massively Collaborative Mathematics
That sounds like a smashing idea. I've sort of been hoping something similar might happen spontaneously on this forum, but never actually thought about it enough to go to the effort of trying to set it up. We came close in the thread on the values of pi in different metric spaces (which, not coincidentally, is one of my favorite threads ever, and actually became a "pizza seminar" talk.) If only someone hadn't posted a link to a paper with the full solution, I'm certain we would have gotten there on our own before too much longer. But working with unsolved problems would be even more fun, and remove the risk of someone spoiling the answer.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
"With math, all things are possible." —Rebecca Watson
Re: Massively Collaborative Mathematics
I vaguely remember a blog attempting to do something similar with the Goldbach conjecture. I don't think it ever really got off the ground because of a lack of sufficiently educated people being involved. That being said, we have some highly educated people on this forum already, so I guess getting over that beginning hump is a bit easier for us.
 skeptical scientist
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Re: Massively Collaborative Mathematics
That's probably because the Goldbach conjecture is exactly the wrong problem to try this method on.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
"With math, all things are possible." —Rebecca Watson
 qinwamascot
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Re: Massively Collaborative Mathematics
I read this on his blog. I'm excited to see what kind of work can be done. If this is a massive failure, at least we'll know more about why it fails. And if it works, we could come out with a handy and useful result that would otherwise be difficult or impossible to prove simply because such a proof would take too much time. So long as the amount of time spent on the problem is small, it's a winwin situation.
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 imatrendytotebag
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Re: Massively Collaborative Mathematics
What about: Counting the number of possible filledin nxn sudoku boards. Or how many tetrahedra with side length one can be packed in a sphere with radius 1, given that each tetrahedron has a vertex at the center of the sphere? Or... or... I don't know. Those are my first two suggestions.
Hey baby, I'm proving love at nth sight by induction and you're my base case.
 qinwamascot
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Re: Massively Collaborative Mathematics
Gowers is a prominent combinatorialist (he won the fields medal) and he mentions that the problem will require a modest (or more) amount of knowledge in combinatorics. Speculating as to what the problem will be is probably foolish given that he will release it pretty soon, but if we're going to, then that would be the best direction to attempt. I'd guess that it's probably neither of those, as a computer can do the first one for small n, while for large n there doesn't seem to be a lot of possibility of division of labor. The second would be more likely from a geometer as opposed to a combinatorialist.
My guess is that it's a relatively unknown problem, but one that he has tried to solve, but came to a point where it was difficult to continue. Otherwise, there's too much risk that people will be unwilling to cooperate on one hand, or someone will solve it in a relatively easy way on the other. Being that I only have 1 semester of combinatorics, i'd assume that I have no chance to guess the problem.
My guess is that it's a relatively unknown problem, but one that he has tried to solve, but came to a point where it was difficult to continue. Otherwise, there's too much risk that people will be unwilling to cooperate on one hand, or someone will solve it in a relatively easy way on the other. Being that I only have 1 semester of combinatorics, i'd assume that I have no chance to guess the problem.
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 Cosmologicon
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Re: Massively Collaborative Mathematics
When I started reading that blog entry, I thought that there's no way that this could possibly work. But then I got to the sentence "The lemma you suggested trying to prove is known to be false," and changed my mind. That's a perfect example of how one person can help another solve a problem without knowing how to do the whole thing.
It makes me wonder if the best way to do this might be as a database of lemmas and dependencies between them. They could be marked true, false, or open, and have importance automatically assigned based on what other lemmas are known to depend on them. This is probably way too restrictive a format, though. Not everyone likes to state their ideas in terms of lemmas.
Anyway, I'm still highly skeptical, but I'm starting to see how it might work.
It makes me wonder if the best way to do this might be as a database of lemmas and dependencies between them. They could be marked true, false, or open, and have importance automatically assigned based on what other lemmas are known to depend on them. This is probably way too restrictive a format, though. Not everyone likes to state their ideas in terms of lemmas.
Anyway, I'm still highly skeptical, but I'm starting to see how it might work.
Re: Massively Collaborative Mathematics
It's the way of the future, man! Don't doubt!
Re: Massively Collaborative Mathematics
If I were to hazard a guess, I'd say it might be on computational complexity (but not something ridiculous like P != NP)
This is on the basis that:
The problems in this area are often understandable without PhDlevel expertise.
He's about to lecture a nonexaminable graduate course on computational complexity.
At Cambridge that usually means the lecturer wants to exhibit some interesting area of research as inspiration for PhD students.
I personally think this is a great idea, I've seen several cases of problems being solved collaboratively here, although not of the level that would constitute a research paper.
The irony is that Gowers himself almost never collaborates  I think he's only ever cowritten one paper.
This is on the basis that:
The problems in this area are often understandable without PhDlevel expertise.
He's about to lecture a nonexaminable graduate course on computational complexity.
At Cambridge that usually means the lecturer wants to exhibit some interesting area of research as inspiration for PhD students.
I personally think this is a great idea, I've seen several cases of problems being solved collaboratively here, although not of the level that would constitute a research paper.
The irony is that Gowers himself almost never collaborates  I think he's only ever cowritten one paper.
 imatrendytotebag
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Re: Massively Collaborative Mathematics
qinwamascot wrote:I'd guess that it's probably neither of those, as a computer can do the first one for small n, while for large n there doesn't seem to be a lot of possibility of division of labor. The second would be more likely from a geometer as opposed to a combinatorialist.
I meant, we could try those problems. This project concept isn't limited to the problem he poses.
Hey baby, I'm proving love at nth sight by induction and you're my base case.
 qinwamascot
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Re: Massively Collaborative Mathematics
I misread your post. I thought you were guessing what he would pose. But we could try those problems. Either might fit the format. Unfortunately, I don't know enough about them to be able to say.
It'd be difficult to do, but possible. Obviously, the lemmas would need to be written in a mathematically unambiguous way to be searchable. But this would be too dense for humans to read easily. What's more, there are multiple ways to formalize a given statement. So it would have to be able to find things that are equivalent or can easilly be proven to do so. At this point, we're already getting into one of the most powerful proofengines in the world, and my guess is that it wouldn't be sufficient right now. Maybe in 20 years, when we can all have supercomputers for laptops.
Perhaps, rather than searching, then lemmas could be categorized. For example, you start at a main page. Then you might navigate to, for example, abstract algebra. You go one more step down to group theory, and another to specifically abelian groups. If the categorization was indepth and unambiguous, it could be a usable system. Plus, it categorizes the lemmas for you in a way that will naturally be useful together. The problem with this is that it'd be difficult to find something that is out of place, and placements could get ambiguous when multiple categories are involved.
The third option that I see is a sort of universal forum, where mathematicians go to ask these kinds of questions. Anyone knowing about the literature in a given field can help reference it to others. The problem here is that it consumes a lot of time on the part of the helpful mathematicians.
It makes me wonder if the best way to do this might be as a database of lemmas and dependencies between them. They could be marked true, false, or open, and have importance automatically assigned based on what other lemmas are known to depend on them. This is probably way too restrictive a format, though. Not everyone likes to state their ideas in terms of lemmas.
It'd be difficult to do, but possible. Obviously, the lemmas would need to be written in a mathematically unambiguous way to be searchable. But this would be too dense for humans to read easily. What's more, there are multiple ways to formalize a given statement. So it would have to be able to find things that are equivalent or can easilly be proven to do so. At this point, we're already getting into one of the most powerful proofengines in the world, and my guess is that it wouldn't be sufficient right now. Maybe in 20 years, when we can all have supercomputers for laptops.
Perhaps, rather than searching, then lemmas could be categorized. For example, you start at a main page. Then you might navigate to, for example, abstract algebra. You go one more step down to group theory, and another to specifically abelian groups. If the categorization was indepth and unambiguous, it could be a usable system. Plus, it categorizes the lemmas for you in a way that will naturally be useful together. The problem with this is that it'd be difficult to find something that is out of place, and placements could get ambiguous when multiple categories are involved.
The third option that I see is a sort of universal forum, where mathematicians go to ask these kinds of questions. Anyone knowing about the literature in a given field can help reference it to others. The problem here is that it consumes a lot of time on the part of the helpful mathematicians.
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Re: Massively Collaborative Mathematics
skeptical scientist wrote:That's probably because the Goldbach conjecture is exactly the wrong problem to try this method on.
The failure wasn't too surprising to me. The site did seem more like an amateur blog than one by a mathematician/researcher.
I am, however, curious as to why you say the Goldbach conjecture is exactly the wrong type of problem for this method. I would think the multitude of approaches (due to the multitude of theorems involving primes) would lend itself to a massive collaboration. Where I am mistaken?
 Talith
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Re: Massively Collaborative Mathematics
qinwamascot wrote:I misread your post. I thought you were guessing what he would pose. But we could try those problems. Either might fit the format. Unfortunately, I don't know enough about them to be able to say.It makes me wonder if the best way to do this might be as a database of lemmas and dependencies between them. They could be marked true, false, or open, and have importance automatically assigned based on what other lemmas are known to depend on them. This is probably way too restrictive a format, though. Not everyone likes to state their ideas in terms of lemmas.
It'd be difficult to do, but possible. Obviously, the lemmas would need to be written in a mathematically unambiguous way to be searchable. But this would be too dense for humans to read easily. What's more, there are multiple ways to formalize a given statement. So it would have to be able to find things that are equivalent or can easilly be proven to do so. At this point, we're already getting into one of the most powerful proofengines in the world, and my guess is that it wouldn't be sufficient right now. Maybe in 20 years, when we can all have supercomputers for laptops.
Perhaps, rather than searching, then lemmas could be categorized. For example, you start at a main page. Then you might navigate to, for example, abstract algebra. You go one more step down to group theory, and another to specifically abelian groups. If the categorization was indepth and unambiguous, it could be a usable system. Plus, it categorizes the lemmas for you in a way that will naturally be useful together. The problem with this is that it'd be difficult to find something that is out of place, and placements could get ambiguous when multiple categories are involved.
The third option that I see is a sort of universal forum, where mathematicians go to ask these kinds of questions. Anyone knowing about the literature in a given field can help reference it to others. The problem here is that it consumes a lot of time on the part of the helpful mathematicians.
It would seem to me that a wiki is well equipped to provide some kind of database of lemmas in a way which can be easily navigated. Ofcouse, wikis have their own awkward points but until we can have, as you say, supercomputers as laptops, it would be an avenue that looks attractive at the moment.
After a few google searches I found this site which looks like a promising place for this kind of project.
Re: Massively Collaborative Mathematics
Collaborative intellectual works and what tools are needed for them is a question that keeps tickling the back of my head since I read xkcd and see threads, of otherwise good quality, derail into a spiral of offtopicness. It seems clear that a flat Internet forum is not what it takes even with very active moderators.
Can wikis work ? A wiki works best as a documentation tool : when there is a verifiable way of checking that what is written there is correct. A research discussion, however, is more opened than that. You will either have to authorize any content that anyone thinks could have an interest in the proof (and quickly enough it will be filled by offtopic statements) or you can just authorize facts that have a proved connection with the problem at hand, in which case many insightful remarks will probably be excluded.
What about a threaded forum ? There some users can digress on a subproblem while letting other people talk about another facet of the problem. But there you still have the problem of many remarks of lesser interest appearing.
I am a big fan of the slashcodestyle automoderation (mostly known through the Slashdot website). It has a very good noise filter but it lacks the iterative building one can find on wikis.
I think there are many tools that need to be merged into something new and that this something will witness the same success that wikis did.
Can wikis work ? A wiki works best as a documentation tool : when there is a verifiable way of checking that what is written there is correct. A research discussion, however, is more opened than that. You will either have to authorize any content that anyone thinks could have an interest in the proof (and quickly enough it will be filled by offtopic statements) or you can just authorize facts that have a proved connection with the problem at hand, in which case many insightful remarks will probably be excluded.
What about a threaded forum ? There some users can digress on a subproblem while letting other people talk about another facet of the problem. But there you still have the problem of many remarks of lesser interest appearing.
I am a big fan of the slashcodestyle automoderation (mostly known through the Slashdot website). It has a very good noise filter but it lacks the iterative building one can find on wikis.
Talith wrote:It would seem to me that a wiki is well equipped to provide some kind of database of lemmas in a way which can be easily navigated. Ofcouse, wikis have their own awkward points but until we can have, as you say, supercomputers as laptops, it would be an avenue that looks attractive at the moment.
Cosmologicon wrote:It makes me wonder if the best way to do this might be as a database of lemmas and dependencies between them. They could be marked true, false, or open, and have importance automatically assigned based on what other lemmas are known to depend on them.
I think there are many tools that need to be merged into something new and that this something will witness the same success that wikis did.
 qinwamascot
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Re: Massively Collaborative Mathematics
I am, however, curious as to why you say the Goldbach conjecture is exactly the wrong type of problem for this method. I would think the multitude of approaches (due to the multitude of theorems involving primes) would lend itself to a massive collaboration. Where I am mistaken?
There are several reasons why Goldbach's conjecture is not a good problem for this. First, no one would want to share work. It's a very prominent problem, and even a minor discovery relating to it would attract a lot of attention. People would rather publish their own papers on the matter. Secondly, it's difficult to subdivide into different subproblems. I'm not saying it's impossible, but it's not easy either. The optimum problem would be equivalent to the combination of several smaller problems, which are more managable to make progress on even for just a single individual. Finally, there are a large number of specialists who are trying to prove Goldbach's conjecture, like any other major problem. It's unlikely that a large mass of people would beat out specialists in the area.
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 Talith
 Proved the Goldbach Conjecture
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Re: Massively Collaborative Mathematics
Maybe we should try and find a problem that would be suitable to us on this forum. Something which would be hard to take off on a tangent or otherwise diverge from the primary problem (as alot of problems that appear here seem to do) and one that can be split up into alot of smaller parts that can be taken on by different people that may have more knowledge on that part of the problem.
I suppose a good place to start would be a list of possible areas of maths that we can explore. Is anyone willing to try and come up with a list?
I suppose a good place to start would be a list of possible areas of maths that we can explore. Is anyone willing to try and come up with a list?
Re: Massively Collaborative Mathematics
qinwamascot wrote:I am, however, curious as to why you say the Goldbach conjecture is exactly the wrong type of problem for this method. I would think the multitude of approaches (due to the multitude of theorems involving primes) would lend itself to a massive collaboration. Where I am mistaken?
There are several reasons why Goldbach's conjecture is not a good problem for this. First, no one would want to share work. It's a very prominent problem, and even a minor discovery relating to it would attract a lot of attention. People would rather publish their own papers on the matter. Secondly, it's difficult to subdivide into different subproblems. I'm not saying it's impossible, but it's not easy either. The optimum problem would be equivalent to the combination of several smaller problems, which are more managable to make progress on even for just a single individual. Finally, there are a large number of specialists who are trying to prove Goldbach's conjecture, like any other major problem. It's unlikely that a large mass of people would beat out specialists in the area.
Ok, thanks. I get the first two points. With the third point, however, I was assuming that it was a large group of people consisting of a number of specialists.
 qinwamascot
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Re: Massively Collaborative Mathematics
Perhaps, but it's unlikely (just due to numbers) that a lot of specialists will join. There are only so many of them for any one field or problem. If we're talking about the kind of collaboration that Gowers is proposing, it would be doable by nonspecialists (or at least, the subparts would be). I suppose it's possible to conceive of a problem tackled this way by specialists, but it seems unlikely that many would join.
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Re: Massively Collaborative Mathematics
qinwamascot wrote:Perhaps, but it's unlikely (just due to numbers) that a lot of specialists will join. There are only so many of them for any one field or problem. If we're talking about the kind of collaboration that Gowers is proposing, it would be doable by nonspecialists (or at least, the subparts would be). I suppose it's possible to conceive of a problem tackled this way by specialists, but it seems unlikely that many would join.
That's true. I guess I was living in some kind of fantasy world where mathematicians have some kind of hive mindlike ability to share ideas. Anyway, I'm sorry for sidetracking the thread into the Goldbach conjecture. My knowledge of combinatorics is limited, but hopefully I'll be able to contribute something when this problem finally is announced.
 qinwamascot
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Re: Massively Collaborative Mathematics
It's already been announced. You can read it here.
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