What do I mean by that?
Is there a way to make research mathematics a more collaborative, asynchronous and open endeavor? Is it possible to open it up to the public, to scale it up in some sense? So that, at least in principle, all mathematicians around the world could participate on a given problem, maybe even aspiring young mathematicians and hobbyists?
In the past, there has been one successful approach to this question (that I know of) that is called Polymath Projects. These were (up to now) 16 attempts to solve research projects in public, under the guidance of the proposing professional mathematician. As far as I can see, many of the proposing mathematicians used their personal blog to coordinate the progress on the projects. They would post a problem definition together with their current knowledge of the area and then everybody would be able to comment on the blog and contribute their ideas and knowledge. Every now and then, the blog owner would update everybody with status blog posts (and thereby also partition the comment history into more readable threads).
According to the Polymath Projects Wiki, it all started with this blog post by Timothy Gowers, in which he stated the following vision:
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 — in whatever form it took — would be that comments would mostly be kept short.https://gowers.wordpress.com/2009/01/27/is-massively-collaborative-mathematics-possible/
I have always liked the fail-fast-learn-fast attitude. the generosity of ideas and the collaborative aspect of this approach. Also, Gowers’ original blog post seems to be lowering the entry barrier to participating in such problems, thereby making it more fun to engage.
The title picture of this post, by the way, is from a study called “The polymath project: lessons from a successful online collaboration in mathematics” by Justin Cranshaw, A. Kittur. It was published in 2011 in Proceedings of the SIGCHI Conference on Human Factors in Computing Systems.
Recently, I wondered if this model could somehow be standardized and made more scalable? As it was done previously, one person would have to organize the progress centrally. What if it was done unsupervised(-ly?) and asynchronously?
And also: would it be possible to make the process of solving problems more diverse? For instance, some people might be interested in taking over expository tasks, like summarizing the state of a problem. Other people might be good at performing numerical computations if needed. Yet other people are good at using theorem proving assistants or formal computer tools to contribute something. In general, making the process more open, collaborative and diverse would be something I would very much look forward to.
What is there already?
I have recently found and written about a list of open math problems in this blog post. But to solve such problems, one would need a sort of central website that would contain not only a list of open problems but furthermore some kind of scorecard for each problem and a forum for people to discuss their ideas and upload content. As an open platform, people should be able to enter new problems and add references and the state of the art to that problem. Everybody could then go and read the resume of the problem and try to make progress on a problem her or she is interested in.
While researching the list of lists of open problems, I stumbled upon a website called SciLag: https://www.scilag.net/. This seems, at first glance, to be a promising first step towards an implementation of such a standardization. They even have a “Problem Mining” section which is supposed to mine papers on the arXiv for potential open problems or conjectures (wonder how well that works yet). The existing open problems in their database are tagged with areas of mathematics but unfortunately, these tags are not clickable as of now.
A similar website is called http://www.openproblemgarden.org/, which seems to have a similar functionality, except for the problem mining part.
Which problems do I see with this?
While this approach sounds very promising and fun to me, I can imagine (because I am not a professional mathematician) that there might be some inertia in the system and that the amount of onboarding to be done by people who want to join a problem might be large.
Resistance from the establishment
After all, most of the knowledge that I would love to see open-sourced is currently guarded by a group of professional mathematicians. They share their knowledge openly in conferences and (open and paywalled) publications. But they do still (mostly) share it at a very suboptimal state of the problem, namely when they think it has been solved. In some sense, this is too late and, more often than not, progress might have been faster (albeit less heroic) had they collaborated more.
As Gowers writes:
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’t there a danger that somebody would manage to use the idea to solve the problem and rush to (individual) publication?
I can imagine that some people might be hesitant to share their ideas and progress on a problem in public, partly because of a fear of looking stupid or losing a race to credit. I am currently convinced that sharing ideas earlier would be better for mathematical progress overall, but maybe someone can convince that this is false. I fear that some people want to work only with people who they know and trust in order for them to be able to publish regular papers that give them reputation in the regular currency and not some fancy new (blockchain-ish) online blog/polymath reputation currency.
Too high of an entry barrier for non-professionals to make any progress
The other question that comes to my mind is whether or not it would even be worth the effort. Does it even make sense to invest the time to post a problem and its state of the art on a platform given that very few people in the world are even remotely able to contribute anything useful. And those few people might be familiar with the problem and its progress already anyway. So why bother?
As mentioned by Gowers in his original post, there is probably a class of problems for which it is worthwhile to invest the time. But I feel that there should be more than 16 such problems. And as mentioned before, there could also be a diversification of tasks in the sense of exposition, library search, idea generation, computation, etc. that might fit many people’s needs.
As a last remark, let me comment on the following quote from Gowers’ original article:
Now I don’t 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.
I don’t quite understand why that would be the case. I have heard (or read) at least one mathematician say that they would love to see a solution to the Riemann hypothesis in their lifetime or first thing after getting off from a time machine in some distant future. I think it would be to the benefit of many people to share even vague ideas on such prestigious problems openly. For almost all people it would be better to know the solution in their lifetime than to have a huge price for a single person attached to it. Most people would hopefully agree that they won’t be the single person who solves the problem anyway.