In an earlier blog post on new modes of mathematical collaboration, I offered my critical views on Michael Nielsen’s ideas about making mathematics more collaborative using the Internet. Around the time, Timothy Gowers, a prominent mathematician, was inspired by Michael Nielsen’s post, to muse in this blog post about whether massively collaborated mathematics is possible. The post was later critiqued by Michael Nielsen.
Since then, Gowers decided to actually experiment with solving a problem using collaborative methods. The project is called the “polymath” project. “Polymath” means a person with extensive knowledge of a wide range of subjects. Gowers was arguably punning on the word, with the idea being that when many people do math together, it is like a “polymath”.
Gowers, who is more of a problem-solver than a theory-builder, naturally chose solving a problem as the testing ground for collaborative mathematics. Further, he chose a combinatorial problem (the density Hales-Jewett theorem) that had already been solved, albeit by methods that were not directly combinatorial, and defined his goal as trying to get to a combinatorial solution for the problem. Gowers wrote a background post about the problem and a post about the procedure, where he incorporated feedback from Michael Nielsen and others. These rules stipulated, among other things, that those participating in the collaborative project must not try to think too much about the problem away from the computer, and must not do any technical calculations away from the computer. Rather, they should share their insights. The idea was to see whether sharing and pooling insights led to discovery faster than working on them alone. I may have misunderstood Gowers’ words, so I’ll quote them here:
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.
In the next post, Gowers listed his ideas broken down into thirty-eight points. He also clarified the circumstances under which the project could be declared finished. In Gowers’ words:
It is not the case that the aim of the project is to find a combinatorial proof of the density Hales-Jewett theorem when k=3. I would love it if that was the result, but the actual aim is more modest: it is either to prove that a certain approach to that theorem (which I shall soon explain) works, or to give a very convincing argument that that approach cannot work. (I shall have a few remarks later about what such a convincing argument might conceivably look like.)
In the next post, Gowers explained the rationale for selecting this particular problem. He explained that, first, he wanted to select a serious problem, the kind whose solution would be considered important for researchers in the field. Second, he didn’t want to select a problem that was parallelizable in a natural sense — rather, he believes that the solution to every problem does parallelize at some stage, and how this parallelization is to occur can itself be determined.
By this time, Gowers’ blog was receiving hundreds of comments, mostly comments by Gowers himself, but also including comments from distinguished mathematicians such as Terence Tao. Tao has his own blog, and he published a post giving the background of the Hales-Jewett theorem and a later post with some of his own ideas about the problem.
A few days later, Gowers announced at the end of this post that there was a wiki on the enterprise of solving the density Hales-Jewett theorem. In the same post, Gowers also summarized all the proof strategies that had come up thanks to the comments. Since then, there have been no more blog posts about the problem.
A look at the wiki
It’s still early days to know the eventual shape that the Polymath1 wiki will take. One thing that seems to be conspicuous by its absence is a copyright notice. This could create problems, particularly considering that this is a collaboratively edited website aimed at solving a problem.
There are some other things that I think need to be decided.
Is the wiki intended only to provide leads or reference points to ideas elaborated elsewhere, or is it intended to provide the structure, substance and background material as well? If the former is the case, then the wiki can be designed in a problem-centric fashion. However, if the wiki is designed this way (i.e., only to provide leads), its generic comprehensibility is going to be poor. Moreover, the “cross-fertilization” of ideas with other problems is going to be minimal if the organization is centered completely around the density Hales-Jewett theorem. On the other hand, if the wiki provides too much of background information, it would be better to organize it according to the background information. This would make it lose its problem-specific focus. I think there is a trade-off here.
Style of pages: The pages currently have a very conversational style. This may be because, currently, the pages are adaptations of material put up in blog posts and blog comments. But this conversational style makes it hard to use the pages as a handy reference or lookup point.
Classifying page types: There needs to be some sort of separation between definition pages, pages about known theorems, pages about speculation and conjectures, and pages describing conjectures and thoughts. As of now, such a separation or classification is not available.
Interfacing with other reference sources: If (and this goes back to the first point) it is decided that the wiki will not provide too much background information and will focus on a style suited to the problem focus, then some decisions will need to be made on how to link up to outside reference sources.
Linking mechanisms between pages: A person who reads about one idea, definition, theorem, or conjecture, should have a way of knowing what else is most closely related to that. Robust linking mechanisms need to be decided for this.
To give an illustration of this, consider the current page on Line (permalink to current version). This page introduces definitions for three kinds of “lines” talked about in combinatorics — combinatorial lines, algebraic lines, and geometric lines. Some of the things I’d recommend for this page are:
Create separate pages for combinatorial line, algebraic line, geometric line.
In each page, create a “definition” section with a precise definition, and perhaps an “examples” section with examples, as well as links to the other two pages, explaining the differences.
For the page on combinatorial line, link to generalizations such as combinatorial subspace.
For the page on combinatorial line, provide a reverse link to pages that use this concept, or link to expository articles/blog entries that explain how and why the concept of combinatorial line is important.
Here are some suggestions on the theorem pages.
Create a separate section in the theorem page giving a precise statement of the theorem.
For each theorem, have sections of the page devoted to listing/discussing stronger and weaker theorems, generalizations and special cases. For instance, the coloring Hales-Jewett theorem is “weaker” than the density Hales-Jewett theorem as well as the Graham-Rothschild theorem.
Another suggestion I’d have would be to use the power of tools such as Semantic MediaWiki to store the relationships between theorems in useful ways.
I’ll post more comments as things progress.