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I've long dreamt of an occasion where hundreds or thousands of mathematicians work on a single problem in a truly coordinated way. Well, now that I'm nearing my doctoral defense in mathematics, I've decided to consider making this dream a reality.

But I can't do it alone....

What are some of your thoughts on how best to implement such a scheme? What are some practical pitfalls preventing such an effort taking place? How could an organizing body avoid them? Would it be feasible to try a test run on Math.SE of some coordinated work, perhaps a simple but unsolved problem in an area of mathematics with various connections to subject areas? What would be the most appropriate level so that most could contribute?

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@Pete The way to do it is to collect ideas from every 1000, then check every idea. I dont think there is much of a gradual progress in unsolved problems, you have an idea and either it works or not. All you need is 1 revolutionary idea, and someone with technical skills to pursue it. –  user1708 Mar 3 '11 at 20:37
    
(I deleted my previous comment because it seemed to be mostly negative. Which is not to say that I have really changed my mind, but I try to keep negative thoughts and comments to myself as much as I can.) –  Pete L. Clark Mar 4 '11 at 5:22
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Maybe this response will seem oblique bordering on irrelevant, but...Your question reminds me of the classic Sandman story Dream of A Thousand Cats. In it, a cat learns that reality could be completely restructured (or rather, re-restructured...) if only (any) one thousand cats would simultaneously believe and dream in unison of a different reality. When asked what he thinks of this, one of the characters -- a cat -- says "I would like to see anyone -- prophet, king or god -- persuade a thousand cats to do anything at the same time."

I feel like herding mathematicians is only a little easier than herding cats. What grand project would convince a thousand of us to get involved in a coordinated way?

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A Clay Prize problem. 1 mill /1000 ppl = 1000$ for each. Fame, Fun & Fortune. –  user1708 Mar 3 '11 at 20:10
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+1 for your point. +1000 for Sandman reference. –  Willie Wong Mar 4 '11 at 1:18
    
From Dream of a Thousand Cats: "Thank you for coming to listen to me; for your willingness to hear my message ... and I hope that when I have finished, some of you may share my dream." How appropriate. ;) –  user02138 Mar 4 '11 at 7:41
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There is already a similar scheme the Polymath projects which you can find at the blog and wiki

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Yes, it is a similar idea, but how successful is Polymath? At a first glance, it doesn't seem particularly active. –  user02138 Mar 3 '11 at 14:38
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@user02138: not particularly active? This is what a typical polymath dicussion look like. And successful? In the two years since its inception, the social experiment has already produced three papers. –  Willie Wong Mar 3 '11 at 14:55
    
@Willie: I mean on the scale of hundreds or thousands of participants. –  user02138 Mar 3 '11 at 15:05
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@user02138: oh. I see. Somehow I think perhaps "active" is not the right word. You should take a look at this article about the polymath project. While there were initially more than a hundred participants in polymath1, it quickly dwindled to around 27 active people. Some analysis was given about why that is the case and perhaps gives some perspective addressing some of your listed questions above. –  Willie Wong Mar 3 '11 at 16:44
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It's near impossible to expect thousands of participants unless you cut the problem into parts that require essentially no specialized knowledge. But if you can cut a problem into such small parts, you can likely teach a computer to solve the problem for you. –  Ryan Budney Mar 3 '11 at 20:32
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I don't think that there is a way such that hundreds or thousands of people could collaborate on a problem without subdivision into smaller groups with certain synchronization points. Large software projects with thousands of developers are usually broken down to groups with 2-20 members. Imagine that you'd have to talk to each member in your group for 30 minutes per week, that would have to talking for 10 hours in the latter case, every week, just to synchronize with what the others are doing (this is sometimes called the combinatorial explosion of communication).

In this sense I'd say that Wikipedia is very successful in collecting and explaining a lot of mathematical knowledge and pointing out open problems. If you supplement this wiki ansatz with links to more specialized wikis where subgroups document their work about certain open problems, you'd come close to a large scale collaboration. The nLab is an example of this, but it has not attracted a large crowd yet. The nLab has an accompanying forum, where people talk about their thoughts before there is an agreement about what should be written on the wiki.

You can write about your work and where you are stuck on the wiki, explaining all the background and ask on the forum for help. Others can chime in either by supplementing what you wrote on the wiki, or by joining the discussion on the forum. I don't see why this could not work with a lot more participants, as long as they form subgroups and avoid a situation where every participant has to talk and listen to everyone else.

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