Polymath


Finally a new mathematics post!.  

I have been holding out on commenting on the fascinating polymath project for a while, even though it touches on my central topics of maths and communication.  Now with its preliminary success feels like a good time to do so.  

Update 26/3/9: For those who want to know more about the problem Jason Dyer has a beautifully simple explanation up at the Number Warrior. This is exactly the sort of work that I find most exciting in the polymath project and heartily commend Jason.

A few months ago Tim Gowers put forward the challenge of whether massively collaborative mathematics was possible.  He also came up with a suitable problem and started it as a wiki.  As well as the wiki and articles on Gowers weblog progress was covered by various people including Terry Tao, Gil Kalai and Michael Nielsen.  

The actual work in proving this result seems not quite to have achieved the goal of massive collaboration.  In fact in this case:

the number [of contributors] settled down to a handful, all of whom I knew personally. 

Tim Gowers

So in this case the collaboration might be seen as an evolution of the small problem driven research meeting.  However even if this is all that it is, it is still a significant evolution. The web version has three key advantages.  Firstly it is open, so the group involved in the project is more self-selecting, allowing for a different collection of people than might be assembled for a meeting.  Secondly the web allows the research to take place as part of ordinary life.  This leads to the third benefit that the process can take place at a more natural speed with time to digest the ideas.   

The more open grouping of individuals leads to the problem that many mathematical questions can be asked in more than one language.  The polymath collaboration provides a solution to this.  An important part of the effort can be in translating between areas:

To give one example, Randall McCutcheon made some very useful comments, but they were in the language of ergodic theory, which I understand only in a very limited way. But Terence Tao is a master at translating concepts back and forth between combinatorics and ergodic theory, so I was able to benefit from Randall’s contributions indirectly.

Tim Gowers

I would now like to take a little time out to rant.  Perhaps one of the reasons that more people did not get involved in the project (and those that did were established enough to be recognised by Tim Gowers) is that the pressure on mathematicians, especially at the start of their careers, is to prove their own results.  This is a different statement of the classic problem of paper numbers.  Lets face it, it is far easier to get a new result published than a simplification of a far more significant result.  One consequence of this is that many important results are only studied in detail if there is a feeling that they can be used to attack a new problem.  Something related to this is the process of unnecessary generalisation, creating a result that seems new yet deals with no new interesting cases.   This emphasis decreases the overall understanding of mathematics in order to produce many marginal results.

As a personal example, my work studying aperiodic order naturally considers aperiodic sets of tiles.  These are sets of shapes that can tile the plane but do not admit any periodic tilings.  The most significant result in this area is that all substitution tilings can generate sets of aperiodic tilings.  This is a beautiful and significant result to me, yet it was only last autumn that I was able to find the time (between two weeks and a month) that I needed to really get to grips with the proof.  (The general case was proved by Chaim Goodman-Strauss).  However this understanding is not directly relevant to anything that I am currently turning into a paper, and thus of little benefit to my CV.

So we are faced with a situation where new results are granted more significance than understanding.  This is a tragedy as for mathematics clear exposition has made far more impact than deep results.  As a first example consider the beautiful language of arithmetic that we all take for granted: the arabic numerals.  Imagine having to do multiplication, even addition in Roman numerals, and it is not hard to see the massive leap forward that these provide.  Yet for hundreds of years that is what people did, so our current system is far from trivial or obvious.  In fact Leonardo of Pisa had to do a lot of work and lobbying to change the system (he is better know for the number sequence that uses his other name: Fibonacci).  

Another example of the importance of language comes from the famous dispute over calculus.  Whatever the actual chain of events that lead to the discovery, Liebnitz clearly trumps Newton in one regard.  He had a better notation.  In fact in can be argued that the insistence on Newton’s notation severly damaged British mathematics for hundreds of years (but that would need more in-depth study).  

The aspect of communication and language is addressed in Gower’s write up of the project:

 next time I think we may have to have some policy such as writing up all useful insights on the corresponding wiki before we allow ourselves a new comment thread, so that anybody who wants to join the discussion can read about the progress in a condensed and organized form.

Tim Gowers

My plea is that this idea be emphasised, and that writing up the results should be consider not just something to facilitate the smooth running of the project, but as one of the goals.  This could in fact increase the idea of a massive collaboration as many more people are capable of finding a better interpretation of an idea of Tao or Gowers than actually creating ideas.  Yet clearer explanations can be of benefit to all, even the giants themselves.  This is certainly something that seems to comes naturally to such projects as:

Better still, it looks very much as though the argument here will generalize straightforwardly to give the full density Hales-Jewett theorem…Better even than that, it seems that the resulting proof will be the simplest known proof of Szemerédi’s theorem. 

Tim Gowers

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