As a result of comments on my post Is Massively Collaborative Mathematics Possible? and also as a result of thinking about the proposal a little further I have a few extra remarks to make, and a slight redrafting of the procedural rules. (As I said before, these rules are just my first guess about what would work, and if a consensus emerges that they should change then they can of course be changed.)
Michael Nielsen, in this interesting response to my original post, felt that the rule I stated there as rule 10 was too restrictive. The rule said that one should keep posts focused and not change the subject. Now that I’ve thought of a mechanism, albeit a crude one, for starting new threads to the discussion, I’m inclined to agree and to want to be more flexible. So now if someone says something like, “The discussion has given me an idea for a different but related problem that we might try to tackle,” then as long as there is a genuine connection and people are interested in the related project, I see no harm in starting a new thread for discussing it.
I said earlier that I would create all these threads in advance as empty posts. But now that Terry has pointed out to me that if I just put everything related to this project into one category then it will be easy for people to collect the relevant posts together, I’ve decided that I’ll just create new-thread posts as the need arises.
I’ve dropped the use of the word “stupid” in the rules (see rules 3-5 of the original post). That’s because I don’t want to encourage comments that are stupid in a tiresome way, as opposed to “intelligently stupid”. I hope the new rules 3-4 express what I meant slightly better. (I’m afraid the numbers have had to change. Let us say that if you refer to rule n then you mean rule n in this post unless you specify otherwise.)
The newly formulated rules.
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.
2. Comments should be as easy to understand as you can possibly make them. 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.
3. When you do research, you are more likely to succeed if you try out lots of ideas, even if it is often the case that soon afterwards you can see why they never had a chance of working. Similarly, you are encouraged to express your mathematical ideas here, even if you have not spent time checking whether simple arguments show that they don’t work.
4. If they don’t work for an obvious reason, then almost certainly (if a lot of people are working together) someone will point that reason out very quickly. That is more efficient than you spending time in advance checking whether your ideas are good ones. Similarly, if you can’t immediately see how to make your idea precise, it may be more efficient to give it in a vague form and wait for somebody else to bring it into focus.
5. The ideal outcome would be a solution of the problem with no single individual having to think all that hard. The hard thought would be done by a sort of super-mathematician whose brain is distributed amongst bits of the brains of lots of interlinked people. So try to resist the temptation to go away and think about something and come back with carefully polished thoughts: just give quick reactions to what you read and hope that the conversation will develop in good directions. A good general rule to apply is this: don’t try to write a comment that requires you to think with a piece of paper (or a blackboard).
6. If you are convinced that you could answer a question but that it would just need a couple of weeks to go away and try a few things out, then still resist the temptation to do that. Instead, explain briefly, but as precisely as you can, why you think it is feasible to answer the question and see if the collective approach gets to the answer more quickly. (The hope is that every big idea can be broken down into a sequence of small ideas. The job of any individual collaborator is to have these small ideas until the big idea becomes obvious — and therefore just a small addition to what has gone before.) Only go off on your own if there is a general consensus that that is what you should do.
7. Similarly, suppose that somebody has an imprecise idea and you think that you can write out a fully precise version. This could be extremely valuable to the project, but don’t rush ahead and do it. First, announce in a comment what you think you can do. If the responses to your comment suggest that others would welcome a fully detailed proof of some substatement, then write a further comment with a fully motivated explanation of what it is you can prove, and give a link to a pdf file that contains the proof.
8. Actual technical work, as described in 7, will mainly be of use if it can be treated as a module. That is, one would ideally like the result to be a short statement that others can use without understanding its proof.
9. One can summarize much of the above by saying that each comment should represent a “quantum of progress”. That is, the discussion should have been advanced in some small way that is not obviously complex or divisible.
10. Rule 9 is a lower bound as well as an upper bound. A comment such as “I don’t think that idea will work” does not really advance the discussion. But if you say “I don’t think that idea will work because you would need a very strong analogue of such-and-such a result and nobody has any idea how to do that” then it does, because it suggests possible ways for the conversation to continue. (People could then attempt to back up or alleviate your worries.)
11. Sometimes, comments such as “What you said in comment 32 sounded interesting, but I don’t quite see what you are driving at, and I think others probably don’t either,” advance the discussion. Again, the reason is that such a comment clearly demands a response, and the understanding of the problem is likely to be slightly greater after the response than it was before.
12. If at some point a clearly defined subdiscussion starts, then a new post should be written to summarize what has been said so far, and the subdiscussion should continue as a series of comments on that post.
13. Suppose the experiment actually results in something publishable. Even if only a very small number of people contribute the lion’s share of the ideas, the paper will still be submitted under a collective pseudonym with a link to the entire online discussion.
14. If it becomes clear that the discussion has run out of steam, then anything that is worth writing up will be written up (this may well be a collaborative process) and submitted to the arXiv, for use by anybody who wishes to use it.
15. Comments on the collaborative procedure should be carefully kept apart from the mathematical comments. Procedural comments should be attached to this post, and mathematical comments should be attached to the relevant mathematical posts.
A few more words about what is expected.
I am not 100% confident that this experiment will work, but I am very confident that something like this could in principle work. It is clear from the responses to my original post that many people share this confidence, probably because we have all read similar popular science books about artificial intelligence, large networks, and so on. I also have a more personal reason for this confidence, which is that I feel as though the kind of conversation I am advocating is very similar to the kind of conversation I have with myself when I am doing research on my own. (Although different people go about research in different ways, I would guess that many others do something similar to what I am about to describe.) For as long as possible I try to avoid doing any technical calculations: if the temptation arises, I try instead to look for heuristic arguments that will allow me to predict what the results of the calculations will be, or else for reasons to expect them not to be useful after all. Only if it has become very clear that actually doing a calculation is likely to lead to an insight that I can’t see how to obtain without doing it will I go ahead and do it (or try to do it). The aim of all this is to build up a plausible sketch of an entire argument, reducing the original problem from one big mystery to a series of exercises. In practice, it rarely works as smoothly as that, because very often I find a step plausible that I later discover to be wrong, sometimes after having used it as the foundation for quite a lot else. So the hard work of doing calculations, when I eventually get down to it, keeps feeding into the main sketch and forcing me to change it, sometimes quite drastically. But that just means that the result of a calculation can be to throw me back to the searching-for-a-plausible-sketch stage.
Now the precise calculations might not be all that easy to do collaboratively (though I could be wrong about this — perhaps there could be agreement that a certain lemma needs to be proved, and there could be some dialogue about how best to present the proof, after which the proof would more or less write itself, with people suggesting the next line, revising earlier lines, etc.). But everything else — the search for a sketch, backed up by heuristic arguments — does seem very naturally suited to a big collaboration. And I think it is fairly obvious from the description above that a big collaboration could potentially carry out a search of this kind much more efficiently than a single person. In particular, each heuristic argument would be subjected to quick scrutiny from a lot of people, so it if survived then it would have to be pretty good. So the experience I have often had, of doing detailed work on a sketch that was doomed to fail, would be much less likely.
I’ve written these last two paragraphs just to try to explain in a different way what I hope the contributions to this project will be like. The way I’ve presented my initial thoughts on the density Hales-Jewett theorem gives examples of the kind of thing I have in mind.