Like many others, I like to read about the process of making and discovering science and mathematics. I enjoy reading biographies, and especially autobiographies, and especially whatever Feynmann and Atiyah have written about their own work and process of discovery.
Many histories of science and mathematics, however, and many papers in which people argue how others should do science and mathematics, are premised on a view that bothers me a lot: that at any given time there are a lot of people chugging along doing incremental work, and then once in a while a Great Person (usually, a Great Man) comes along and starts a revolution. Then people start chugging along (or “beavering away”, in Atiyah’s wonderful turn of phrase) in the direction dictated by the revolution.
Needless to say, in this view it is the Great Men who are the main principal actors, and the rest are just extras, if not nearly superfluous.
I think that reality is quite different. It is true that our understanding and knowledge of math and science proceeds quite discontinuously. This is quite obvious in mathematical, where a statement can go, in one day, from being an open question for which there seems to be no promising approach to being a solved problem with a new proof that opens up several exciting new directions.
But these discrete advances are not made by geniuses who magically produce an entirely new discovery. Sometimes, a major advance is genuinely ahead of its time (in theoretical computer science, I think that the creation of the modern foundations of cryptography in 1981-82 was such an example). Usually, however, a major advance happens because the time is ripe. Because the work of all the people “beavering away” has completely clarified a number of tools that were previously less well understood, because a large collection of “incremental” results and “observations” has been made, and all the pieces are there, ready to be put together. In fact, major discoveries are usually made independently and around the same time, with a synchronicity that is often stunning.
This is not to detract from the people who actually make great contributions. If finding the right pieces and putting them together was that easy, then we would have revolutions every day. And if it was just dumb luck, then every great discoverer would be a one-hit wonder, while we tend to see the same people making great contributions again and again.
Indeed, realizing that the time is ripe for an advance, and that all the pieces are there, and that they can be put together, is an amazing skill, which is pretty much what people mean by “genius.” Great works of art are not pulled out of nowhere, they are expressions of the sensibility and culture of a given period; seeing a great work of art (be it a movie, a novel, a painting, an installation) makes us think something like “now I better know how I feel.” Similarly, a great leader is not someone who makes other people do what he wants. It is someone who makes people do what they want, who realizes that there is potential for something to be done if only it is articulated in a way that people can identify in.
So, back to mathematics, an extreme version of the Great Man view is something like Lee Smolin’s The Trouble with Physics, in which he proposes a distinction between “seers” and “craftspeople.” Even in the choice of words, it is clear that “seers” are supposed to make their advances by being “visionary,” and it is also clear whose role is more important.
I have seen few discussions on the process of making science and mathematics emphasizing the importance of collective work and the inevitability of major discoveries once the time is right.
Those thoughts, which I had for a long time, and certainly ever since reading Smolin’s book two years ago, resurfaced when I read about Tim Gowers’s wonderful polymath experiment.
The idea is to try out Massively Collaborative Mathematics, in which an open question and a direction of investigation are posed, and a large number of people make “atomic” contributions, such as asking questions, answering questions, proposing ideas, counterexamples, and any kind of remarks or partial steps towards a proof.
Indeed, if any significant mathematical contribution comes from some kind of combination of pieces all of them well known to some, then the reason why the discovery was not made before was that nobody already knew all the pieces, or at least not from the right perspective. Hence a large part of any mathematical discovery is to find out the lore of some adjacent research area, which is relevant but often not known to the people working on the question itself. If many people join together in this way, however, their collective knowledge is huge, and this process of (re)discovery can be remarkably accelerated.
The “genius” that I defined before as an ability to put together the zeitgeist, could just be in the union of many minds, each doing nothing more than saying what is obvious to them.
Gowers’s inspiration for this project was this post by Michael Nielsen, whose blog makes for very interesting reading. Nielsen’s inspiration, in turn, is open source software, and indeed he talks about open science.
Mathematics, however, is a form of creative endeavor more similar to art than to engineering, and I think that if Massively Collaborative Mathematics works out, its philosophical, and not just practical, consequences (in terms of how we think of creativity, genius, and “paradigm shifts”) would be remarkable.