{"id":550,"date":"2009-01-30T11:20:44","date_gmt":"2009-01-30T15:20:44","guid":{"rendered":"http:\/\/michaelnielsen.org\/blog\/?p=550"},"modified":"2009-01-30T13:54:14","modified_gmt":"2009-01-30T17:54:14","slug":"is-massively-collaborative-mathematics-possible","status":"publish","type":"post","link":"https:\/\/michaelnielsen.org\/blog\/is-massively-collaborative-mathematics-possible\/","title":{"rendered":"Is massively collaborative mathematics possible?"},"content":{"rendered":"

This is the title of a thought-provoking essay by Tim Gowers<\/a>, which seems to have been stimulated in part by my recent essay<\/a> on doing science online. What follows are some excerpts from Gowers’ essay, and some thoughts by me: <\/p>\n

Of course, one might say, there are certain kinds of problems that lend themselves to huge collaborations. One has only to think of the proof of the classification of finite simple groups, or of a rather different kind of example such as a search for a new largest prime carried out during the downtime of thousands of PCs around the world. But my question is a different one. What about the solving of a problem that does not naturally split up into a vast number of subtasks? Are such problems best tackled by n people for some n that belongs to the set \\{1,2,3\\}? (Examples of famous papers with four authors do not count as an interesting answer to this question.)<\/p>\n

It seems to me that, at least in theory, a different model could work: different, that is, from the usual model of people working in isolation or collaborating with one or two others. Suppose one had a forum (in the non-technical sense, but quite possibly in the technical sense as well) for the online discussion of a particular problem. The idea would be that anybody who had anything whatsoever to say about the problem could chip in. And the ethos of the forum \u00e2\u20ac\u201d in whatever form it took \u00e2\u20ac\u201d would be that comments would mostly be kept short. In other words, what you would not tend to do, at least if you wanted to keep within the spirit of things, is spend a month thinking hard about the problem and then come back and write ten pages about it. Rather, you would contribute ideas even if they were undeveloped and\/or likely to be wrong. <\/p><\/blockquote>\n

A similar approach is used in the open-source software community – essentially, a dynamic division of labour that is not planned entirely in advance, but rather arises in response to the exigencies of the problem at hand. This dynamic division of labour is typically co-ordinated through one or more online forums. Examples close to this spirit, and also somewhat close in spirit to modern mathematics include Kasparov versus the World<\/a> and the Matlab programming competition<\/a>.<\/p>\n

On the subject of the desirable size of contributions, in open source the most frequent contributions change just a single line of code. The second most frequent contributions change two lines of code, and so on. One study<\/a> suggests the number of contributions [tex]n[\/tex] scales as [tex]n(l) \\propto l^{-1.13}[\/tex], where [tex]l[\/tex] is the number of lines of code changed or added (“committed”) in a single contribution.<\/p>\n

It’s notable that with this distribution the total line count is still dominated by the larger contributions. Despite this, my guess is that the smaller contributions are still very significant for maintaining momentum and morale, which are so important in creative projects. In this regard, it’s a little like a good creative conversation – not all contributions to the conversation need to be world-shaking, some are simply needed to keep the conversation moving.<\/p>\n

This suggestion raises several questions immediately. First of all, what would be the advantage of proceeding in this way? My answer is that I don\u00e2\u20ac\u2122t know for sure that there would be an advantage. However, I can see the following potential advantages.<\/p>\n

(i) Sometimes luck is needed to have the idea that solves a problem. If lots of people think about a problem, then just on probabilistic grounds there is more chance that one of them will have that bit of luck.<\/p>\n

(ii) Furthermore, we don\u00e2\u20ac\u2122t have to confine ourselves to a purely probabilistic argument: different people know different things, so the knowledge that a large group can bring to bear on a problem is significantly greater than the knowledge that one or two individuals will have. This is not just knowledge of different areas of mathematics, but also the rather harder to describe knowledge of particular little tricks that work well for certain types of subproblem, or the kind of expertise that might enable someone to say, “That idea that you thought was a bit speculative is rather similar to a technique used to solve such-and-such a problem, so it might well have a chance of working,” or “The lemma you suggested trying to prove is known to be false,” and so on\u00e2\u20ac\u201dthe type of thing that one can take weeks or months to discover if one is working on one\u00e2\u20ac\u2122s own. <\/p><\/blockquote>\n

I think of this as the “annoying little conjecture” problem: many conjectures that arise in the course of research are often essentially routine to prove or disprove, but it can take days or weeks to determine which it’s going to be. If you talk to just the right person, they can often cut that down to minutes or hours. Ordinarily, though, finding that right person is often just as laborious (and may be less enlightening) than solvig the problem yourself. Having a mechanism to find the right person<\/a>, even if it’s essentially just broadcast search, would be enormously beneficial.<\/p>\n

The next obvious question is this. Why would anyone agree to share their ideas? Surely we work on problems in order to be able to publish solutions and get credit for them. And what if the big collaboration resulted in a very good idea? Isn\u00e2\u20ac\u2122t there a danger that somebody would manage to use the idea to solve the problem and rush to (individual) publication?<\/p>\n

Here is where the beauty of blogs, wikis, forums etc. comes in: they are completely public, as is their entire history. To see what effect this might have, imagine that a problem was being solved via comments on a blog post. Suppose that the blog was pretty active and that the post was getting several interesting comments. And suppose that you had an idea that you thought might be a good one. Instead of the usual reaction of being afraid to share it in case someone else beat you to the solution, you would be afraid not to share it in case someone beat you to that particular idea. And if the problem eventually got solved, and published under some pseudonym like Polymath, say, with a footnote linking to the blog and explaining how the problem had been solved, then anybody could go to the blog and look at all the comments. And there they would find your idea and would know precisely what you had contributed. There might be arguments about which ideas had proved to be most important to the solution, but at least all the evidence would be there for everybody to look at. <\/p><\/blockquote>\n

The open source world demonstrates this in action. You can see every single contribution a person has made to a project – the code, the conversations in online forums, and so on. There’s even beautiful visualizations<\/a> that let you see different people’s contributions to a project. As a result, it’s very difficult to fool people about the extent of your contributions. I’m sure people are sometimes dishonest about this, but I’ll bet they’re a lot more honest than some scientists are about what they contributed to some papers.<\/p>\n

True, it might be quite hard to say on your CV, “I had an idea that proved essential to Polymath\u00e2\u20ac\u2122s solution of the *** problem,” but if you made significant contributions to several collaborative projects of this kind, then you might well start to earn a reputation amongst people who read mathematical blogs, and that is likely to count for something. (Even if it doesn\u00e2\u20ac\u2122t count for all that much now, it is likely to become increasingly important.) And it might not be as hard as all that to put it on your CV: you could think of yourself as a joint author, with the added advantage that people could find out exactly what you had contributed.<\/p>\n

And what about the person who tries to cut and run when the project is 85 [percent] finished? Well, it might happen, but everyone would know that they had done it. The referee of the paper would, one hopes, say, “Erm, should you not credit Polymath for your crucial Lemma 13?” And that would be rather an embarrassing thing to have to do.<\/p>\n

Now I don\u00e2\u20ac\u2122t believe that this approach to problem solving is likely to be good for everything. For example, it seems highly unlikely that one could persuade lots of people to share good ideas about the Riemann hypothesis. <\/p><\/blockquote>\n

At present, this is undoubtedly true. However, if this sort of approach takes off and comes to be seen as a legitimate and orthodox way of making a contribution to mathematics, the kind of thing valued by (for example) hiring committees, then I think it may eventually be possible to do this for some of the more famous problems. There’s no intrinsic difference between sharing your ideas in a paper, or in a blog comment: a good idea is a good idea. The difference at present is mainly social: one is seen as legitimate, while the other is questionable.<\/p>\n

At the other end of the scale, it seems unlikely that anybody would bother to contribute to the solution of a very minor and specialized problem. Nevertheless, I think there is a middle ground that might well be worth exploring, so as an experiment I am going to suggest a problem and see what happens.<\/p>\n

I think it is important to do more than just say what the problem is. In order to try to get something started, I shall describe a very preliminary idea I once had for solving a problem that interests me (and several other people) greatly, but that isn\u00e2\u20ac\u2122t the holy grail of my area. Like many mathematical ideas, mine runs up against a brick wall fairly quickly. However, like many brick walls, this one doesn\u00e2\u20ac\u2122t quite prove that the approach is completely hopeless\u00e2\u20ac\u201djust that it definitely needs a new idea.<\/p>\n

It may be that somebody will almost instantly be able to persuade me that the idea is completely hopeless. But that would be great\u00e2\u20ac\u201dI could stop thinking about it. And if that happens I\u00e2\u20ac\u2122ll dig out another idea for a different problem and try that instead. <\/p><\/blockquote>\n

I’ve been toying for quite some time with doing something similar, though with problems from theoretical physics. I’ll be utterly fascinated to see the result of this experiment, and will certainly follow along. Not sure I’ll have much of mathematical interest to contribute, though – combinatorics is a long way from my expertise.<\/p>\n

It\u00e2\u20ac\u2122s probably best to keep this post separate from the actual mathematics, so that comments about collaborative problem-solving in general don\u00e2\u20ac\u2122t get mixed up with mathematical thoughts about the particular problem I have in mind. So I\u00e2\u20ac\u2122ll describe the project in my next post. Actually, make that my next post but one. The next post will say what the problem is and give enough background information about it to make it possible for anybody with a modest knowledge of combinatorics (or more than a modest knowledge) to think about it and understand my preliminary idea. The following post will explain what that preliminary idea is, and where it runs into difficulties. Then it will be over to you, or rather over to us. I\u00e2\u20ac\u2122ve already written the background-information post, but will hold it back for a few days in case the responses to this post affect how I decide to do things.<\/p>\n

The blog medium is almost certainly not optimal for this purpose, so if a serious discussion starts with lots of worthwhile contributions, then I\u00e2\u20ac\u2122ll look into the possibility of migrating it over to some purpose-built site. If anyone has any suggestions for this (apart from the obvious one of using the Tricki \u00e2\u20ac\u201d I\u00e2\u20ac\u2122m not sure that\u00e2\u20ac\u2122s appropriate just yet though) then I\u00e2\u20ac\u2122d be delighted to receive them. My feelings at the moment are that blogs are too linear\u00e2\u20ac\u201dit would be quite hard to see which comments relate to which, which ones are most worth reading, and so on. A wiki, on the other hand, seems not to be linear enough\u00e2\u20ac\u201dit would be quite hard to see what order the comments come in. So my guess is that the ideal forum would probably be a forum: if someone knows an easy way to set up a mathematical forum, I might even do that. But if the discussion is on this blog, then I might from time to time try to assess where it has got to and create new posts if I feel that genuine progress has been made that can be summarized and then built on.<\/p>\n

I\u00e2\u20ac\u2122ve been thinking of doing this for a long time. The reason I\u00e2\u20ac\u2122ve suddenly decided to go ahead is that I followed a couple of links from this post on Michael Nielsen\u00e2\u20ac\u2122s blog, and discovered that, unsurprisingly, others have had similar ideas, and some people are already doing research in public. But the idea still seems pretty new, particularly when applied to one single mathematics problem, so I wanted to try it out when it was still fresh. (I would distinguish what I am proposing from what goes on at the n-category caf\u00c3\u00a9, which is an excellent example of collaborative mathematics, but focused on an entire research programme rather than just one problem.)<\/p>\n

To finish, here is a set of ground rules that I hope it will be possible to abide by. At this stage I\u00e2\u20ac\u2122m just guessing what will work, so these rules are subject to change. If you can see obvious flaws let me know.<\/p>\n

1. The aim will be to produce a proof in a top-down manner. Thus, at least to start with, comments should be short and not too technical: they would be more like feasibility studies of various ideas.<\/p>\n

2. Comments should be as easy to understand as is humanly possible. For a truly collaborative project it is not enough to have a good idea: you have to express it in such a way that others can build on it. <\/p><\/blockquote>\n

Points 3-5 all concern norms of behaviour, and the problem of maintaining a civil tone: <\/p>\n

3. When you do research, you are more likely to succeed if you try out lots of stupid ideas. Similarly, stupid comments are welcome here. (In the sense in which I am using “stupid”, it means something completely different from “unintelligent”. It just means not fully thought through.)<\/p>\n

4. If you can see why somebody else’s comment is stupid, point it out in a polite way. And if someone points out that your comment is stupid, do not take offence: better to have had five stupid ideas than no ideas at all. And if somebody wrongly points out that your idea is stupid, it is even more important not to take offence: just explain gently why their dismissal of your idea is itself stupid.<\/p>\n

5. Don\u00e2\u20ac\u2122t actually use the word “stupid”, except perhaps of yourself. <\/p><\/blockquote>\n

Clay Shirky has pointed out<\/a> that this problem – the problem of maintaining healthy conduct in online communities – has been around for decades, yet because no-one has synthesized all that is known, the same mistakes keep being made over and over (and over and over) again. The closest thing I know is a short blog post(!)<\/a> from Theresa Nielsen Hayden, which is nice, but hardly comprehensive. Two suggestions:<\/p>\n