Doing maths on the internet
Right now it might feel like a lot of us have been doing pretty much everything on the internet – even if your job doesn’t involve hours of brain-sapping video calls and online meetings, you might still find that a lot of events, conferences, evening classes, comedy shows and music gigs have been moved online during these difficult times. But while it can feel like a slog sometimes, the internet does provide some fantastic opportunities for people to communicate, network and collaborate – including on mathematics research.
Websites like StackExchange and MathOverflow are often home to rich and fruitful discussions on mathematical topics – as well as being a useful resource for anyone looking for alternative explanations or different approaches to a problem. While the many tangled threads can be difficult to navigate, maybe there could be a way to harness the potential of the many mathematical thinkers sitting behind their keyboards, and get them to all focus on the same problem?
In January 2009, Fields Medalist Tim Gowers wrote a post on his mathematics blog which posed the question, ‘Is massively collaborative mathematics possible?’. It interrogated this question, along with the notion that while some maths research is done by individuals or small groups of people, working on things for months before sharing any thoughts, this might not be the only model for effective maths research. Could a forum with a large number of members, each chipping in with quick, undeveloped (and potentially wrong!) thoughts and suggestions, lead to more fruitful collaboration?
Gowers suggested an experiment: to take a current open problem and subject it to this kind of online collaboration. He proposed, and later refined, a series of rules such a community could follow. The goal was for people to share ideas and approach the proof in a top-down way – thinking generally about approaches, throwing in ideas that could be quickly debunked if necessary, and not relying on any one individual to shoulder too much of the hard thinking. (As you might expect of mathematicians, there are 15 rules – all carefully worded and clearly define the space in which they’re working).
The proposed first problem to tackle was a question from combinatorics: the Hales-Jewett Theorem, which is a result from an area of mathematics called Ramsey Theory, concerned with the structures of large abstract objects (like graphs, which I’ve talked about here before). The theorem has a precise statement, but generally concerns how random large, high-dimensional structures are allowed to be – if you make them big enough and high-dimensional enough, certain patterns always emerge.
A nice analogy for the Hales-Jewett Theorem concerns what would happen if you played a large, higher dimensional game of multi-player tic-tac-toe. Imagine instead of a 3 × 3 board, you’re playing on an n × n × n × n × … × n board (in some large number of dimensions), and you need to get n in a row to win. Then, no matter how big you choose the size of your board (n) to be, and no matter how many players you have, the game cannot end in a draw – as long as you play in sufficiently many dimensions.
In particular, Gowers was interested in the ‘Density Hales-Jewett Theorem’, posed by Furstenberg and Katznelson in 1989. While Hales-Jewett itself was already proven, this ‘density’ version (in our analogy) roughly means that you’d expect to be able to find where such a winning line might be within the grid, not just to show that it exists. The problem itself has a strict mathematical statement, and blog posts from Gowers outlined the problem and his suggested starting approaches, along with a justification of why this would be a good problem to use for an online collaboration.
The main forum for discussion that ended up housing this conversation was the comments thread under the blog post itself. Commenters took on the task of finding a combinatorial proof, as Gowers set out, then a separate conversation branched off in the comments of another Fields Medalist’s blog. Terence Tao, who had posted about the project on his blog, found himself hosting a parallel discussion about, among other things, the density of Hales-Jewett numbers (these are the values of the minimum number of dimensions needed to guarantee no draws for a particular board size and number of players).
The project continued, with over 40 people continuing to contribute to the discussion for over three months. On 10th March, Gowers posted that the problem was (probably) solved, and while the conversation continued, papers were published announcing results from both discussion threads. This became known as the ‘Polymath Project’ – and in particular, this discussion was named ‘Polymath1’. The results were published under the pseudonym D.H.J. Polymath (where the initials stand for ‘Density Hales-Jewett’).
Since then, the Polymath project has continued – Polymath5 contributed to Tao’s 2015 proof of the Erdös Discrepancy problem, and Polymath12, proposed by Timothy Chow of MIT, worked to resolve Rota’s basis conjecture (a result about bases of vector spaces). All the Polymath projects are now collated on a website at polymathprojects.org, and new topics to tackle are being proposed all the time.
The project has been so successful that discussion has also focused on how this kind of working might change maths research – Gowers himself wrote an article in Nature called ‘Massively Collaborative Mathematics’, reflecting on what this teaches us about the concept open-sourcing science, and others have studied the project to determine what lessons can be learned from it.
The Polymath project has also resulted in opportunities for young people to get involved in maths research – the maths education organisation Art of Problem Solving has collaborated with MIT’s after-school research program MIT PRIMES to coordinate annual CrowdMath projects, giving high-school students the chance to collaborate and tackle real maths research problems so they can experience it at a younger age. And if we’re hoping to inspire the next generation of mathematics researchers, what better way than letting them discover some new maths themselves?