Peter Krautzberger · on the web

Waiting for the Polymath revolution -- thoughts from a bystander

Tim Gowers has hinted at a revival of the fifth Polymath project. Which brings something back from the bottom of my draft folder.

Let's talk about Polymath

If you haven't heard of the Polymath project, then, hm, well... anyway, here's the beginning of its Wikipedia entry:

The Polymath Project is a collaboration among mathematicians to solve important and difficult mathematical problems by coordinating many mathematicians to communicate with each other on finding the best route to the solution. The project began in January 2009 on Tim Gowers' blog when he posted a problem and asked his readers to post partial ideas and partial progress toward a solution. This experiment resulted in a new answer to a difficult problem, and since then the Polymath Project has grown to describe a particular process of using an online collaboration to solve any math problem.

(If you've really never heard of the Polymath project, then you might want to go through the references on Wikipedia.)

I think nobody who stumbled upon Polymath in 2009 and 2010 could escape the deep fascination and exhilaration of its early successes and the positive spirit it created in the online math community. It was an exciting time. Personally, I was not actively involved in any Polymath project, have neither contributed nor seriously tried to follow the intricate but vast amount of information that Polymath projects have left in their wake -- it was simply too far from my own research interests. Like many mathematicians, however, I have followed it as an interested party.

I write, therefore, as a bystander, a simple mathematician who has an ongoing interested in the future of the field and its community. And as such, I've begun to worry about the impact of the Polymath project.

To return to the above quote, what continues to bug me is the last half-sentence.

[...] since then the Polymath Project has grown to describe a particular process of using an online collaboration to solve any math problem.

O RLY?
O RLY? by shianux, on Flickr

Isn't Polymath dead?

Now that's just silly, of course it isn't. I just told you Tim Gowers will have another one, didn't I? In fact, I'm quite certain that a revival led by Tim Gowers will get another paper out of it (I mean, he got the dHJ paper into Annals for crying out loud).

But that doesn't change my feeling that Polymath is dead -- or rather, that it is a mirage.

What is Polymath, actually?

Polymath is certainly not dead if you think of it as a research project of Tim Gowers, Terry Tao, Michael Nielsen and some of their friends. However, this is not how most people think about it. Instead, Polymath has left a much larger impression in- and outside the mathematical community thanks to numerous mentions on all kinds of news outlets. And then there's the position Polymath takes in Michael Nielsen's writing, in particular in Reinventing Discovery which has lead to every science journalist hearing about this "great revolution" of how we do (mathematical) research.

If you think that that's what Polymath is -- a revolutionary new way of doing research -- then, unfortunately, Polymath is either dead or, more appropriately, has never existed in the first place, has only been an illusion.

Would the real revolutionary please stand up?

Wikipedia tells us

A revolution (from the Latin revolutio, "a turn around") is a fundamental change in power or organizational structures that takes place in a relatively short period of time.

Hm...

Maybe it's still too early. It's only been three years.

Maybe it isn't and instead it's time to think about what's keeping this revolution from happening.

Compared to open notebook science, citizen science, tricki.org or Wikipedia, Polymath fails to be revolutionary in its output. It does not add to the accepted notions of "research activity". It produces more papers, in traditional journals. Perfectly fine mathematics, just not exactly revolutionary.

Six Polymath projects have been formally started so far. Only two seem to have come to the desired conclusion. These two were headed by Gowers and Tao. Two Fields Medalists leading a research project, even a cool one like Polymath, well, it's not "fundamental change in power or organizational structures".

Besides the fact that only Gowers and Tao seem to be able to bring a Polymath project to a conclusion, there's a much bigger issue: Nobody else is even trying. The projects can be traced back to a close circle of bloggers around Gowers, Nielsen and Tao -- Kalai, Lipton, Ellenberg etc. Here's an (unfair) comparison: the arXiv did not revolutionize the dissemination of preprints because Paul Ginsparg put his own and his bff's papers up on the web. Instead, a continued effort reached more and more of the community and established the arXiv as the standard for pre-print releases and self-archival, making it a corner stone of many scientific communities today. (Fun fact: according to Michael Nielsen in Reinventing Discovery, a physicist once described Ginsparg as wasting his talents, collecting "garbage". Would anybody say anything like this about Polymath? In other words, is a Polymath project in any sense risky?)

The few Polymath projects that have been attempted have all had a reasonably high level of complexity. Additionally, the tendency seems to be towards more complicated research questions rather than simpler ones. Maybe this is MathOverflow's fault for taking care of so many "easy" questions, i.e., questions somebody else simply seems to have the answer to. Additionally, people on MO often spend a considerable amount of work on solving questions -- an effort that could just easily be considered a micro-contribution if we had a platform for these.

There's a lot of talk about how awesomesauce !eleventy!1! Polymath projects are but there seems to be no effort towards understanding the successes and failures. Of course, this is not surprising since there aren't that many examples to consider. But there is nevertheless a lot of data. Analyzing the available data, the process, what works and what doesn't could help immensely. In particular, if the "failed" projects could be turned into productive contributions, we might actually get a new form of research activity that benefits the greater part of the community. Failed atttempts are the mathematical analogue of negative data in the sciences and there's similar lack of dissemination.

Evolution or how could Polymath become something meaningful for the entire community?

It may seem that I have some kind of beef with Polymath, but that's not the case at all. Polymath was quite simply amazing. I did wish, however, it (or something like it) would work for more people and would actually turn into a (much needed) revolution.

Mathematics has the greatest potential for "doing research online". There's no physical entity needed and our primary standard of scientific communication is the written word. There's nothing in mathematical research that cannot be digitalized. We will never face the problem that somebody on the other side on the net would have to actually look at our specimen, our antibody staining, our test subjects. The web works perfectly for us.

Hence, mathematicians could be at the forefront of experimenting with new research activities that use the connectivity the web can offer in new and imaginative ways. Polymath was one experiment and it worked to a certain point. Even if Polymath5 can be revived in its current form, this most likely won't help the community as a whole (except in lazy bragging rights). Just like anywhere else on the web, the experiments continue -- 20 years into the invention of this mind boggling creation, we don't have a clue what the future has in store for us.

The goal could be to find a way to do Polymath-esque research on a large scale, involving large parts of the mathematical community (or at least, the online community). But maybe we need something completely different. It would simply be a shame if the Polymath projects ended up "a fun project for a few top mathematicians". Maybe we need another Ginsparg, ready to endure the ridicule of "wasting" a research career. But I don't like heroic sacrifice.

I would rather hope for a group effort, maybe led by an innovative department or a group of college faculties coming together or some grant agency or academice society supporting a significant grant for the development of new ideas, looking to other successful projects like MathOverflow, wikipedia, the n-lab, blogs, activity on social networks etc. But more likely we simply need a crazy group of young researchers fighting to revolutionize the community regardless of the consequences for themselves, ready to kick the hornets' nest of old white dudes established researchers.

Just keep experimenting. Polymath doesn't work for most people, that's ok, let's try something else, change it, revamp it, do the complete opposite.

But, please, for a change, let's not ask Tim Gowers to do everything for us!

Aufklärung ist der Ausgang des Menschen aus seiner selbstverschuldeten Unmündigkeit.

Coda.

Do you expect a project led by Fields Medalists to create a revolution? Overthrowing what exactly?

But there might be a more fundamental problem: do we have the right people in the first place? Michael Nielsen wrote in WSJ:

The tools are a way of connecting the right people to the right problems at the right time, activating what would otherwise be latent expertise.

Throughout Reinventing Discovery, there's something that Michael Nielsen does not discuss: the social impact. New ways of doing research will have a huge impact on the actual people involved and will require them to be willing and able incorporate these changes. The "right people" does not just mean "the researchers who know the right thing" but also "the researchers who can work with these tools".

It could be that the groups of mathematicians that influence the conversations and developments within the mathematical community, e.g., tenured faculty at the top math departments, consist entirely of the "wrong" people, unable to be in the right place at the right time. Simply because it's not why they got the job -- they are where they are because they are excellent at doing research in the current model, the steampunk approach of "papers in journals".

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