This is a great parable. I’m often mildly reluctant to talk about some of my pre-formal ideas in case it gets finished up proper by others and I counterfactually lose social credit. I usually do it anyway, especially for stuff I don’t plan on “finishing up”. But I can see how this reluctance is like heavy molasses poured all over a research community, and it makes us much less effective.
In my experience, the “finishing stage” of making an idea precise enough to be presented is not where the germs of generality are—the parts of ideas that can be used to build other ideas with in a compounding fashion.[1] If I’m just researching or working on something in order to build up a repertoire of tools in order to personally use them for other problems, then I don’t need to go through the expensive “finishing” stage of making the infrastructure for all the middle steps legible to others.
There’s an essay by fields medalist William Thurston[2] with several related points, but it’s worth reading in its entirety.
“First I will discuss briefly the theory of foliations, which was my first subject, starting when I was a graduate student. (It doesn’t matter here whether you know what foliations are.)
At that time, foliations had become a big center of attention among geometric topologists, dynamical systems people, and differential geometers. I fairly rapidly proved some dramatic theorems. I proved a classification theorem for foliations, giving a necessary and sufficient condition for a manifold to admit a foliation. I proved a number of other significant theorems. I wrote respectable papers and published at least the most important theorems. It was hard to find the time to write to keep up with what I could prove, and I built up a backlog.
An interesting phenomenon occurred. Within a couple of years, a dramatic evacuation of the field started to take place. I heard from a number of mathematicians that they were giving or receiving advice not to go into foliations—they were saying that Thurston was cleaning it out. People told me (not as a complaint, but as a compliment) that I was killing the field. Graduate students stopped studying foliations, and fairly soon, I turned to other interests as well.
… When I started working on foliations, I had the conception that what people wanted was to know the answers. I thought that what they sought was a collection of powerful proven theorems that might be applied to answer further mathematical questions. But that’s only one part of the story. More than the knowledge, people want personal understanding. And in our credit-driven system, they also want and need theorem-credits.
… I’ll skip ahead a few years, to the subject that Jaffe and Quinn alluded to, when I began studying 3-dimensional manifolds and their relationship to hyperbolic geometry.
… In reaction to my experience with foliations and in response to social pressures, I concentrated most of my attention on developing and presenting the infrastructure in what I wrote and in what I talked to people about
… There has been and there continues to be a great deal of thriving mathematical activity. By concentrating on building the infrastructure and explaining and publishing definitions and ways of thinking but being slow in stating or in publishing proofs of all the “theorems” I knew how to prove, I left room for many other people to pick up credit. There has been room for people to discover and publish other proofs of the geometrization theorem.
In this episode (which still continues) I think I have managed to avoid the two worst possible outcomes: either for me not to let on that I discovered what I discovered and proved what I proved, keeping it to myself (perhaps with the hope of proving the Poincare conjecture), or for me to present an unassailable and hard-to-learn theory with no practitioners to keep it alive and to make it grow.
(...) I think that what I have done has not maximized my “credits”. I have been in a position not to feel a strong need to compete for more credits. Indeed, I began to feel strong challenges from other things besides proving new theorems. I do think that my actions have done well in stimulating mathematics.”
And in the spirit of this post, I should HT Chris Olah for linking to this essay. It’s important to maintain a culture for remembering what hat-tips are due.
This is a great parable. I’m often mildly reluctant to talk about some of my pre-formal ideas in case it gets finished up proper by others and I counterfactually lose social credit. I usually do it anyway, especially for stuff I don’t plan on “finishing up”. But I can see how this reluctance is like heavy molasses poured all over a research community, and it makes us much less effective.
In my experience, the “finishing stage” of making an idea precise enough to be presented is not where the germs of generality are—the parts of ideas that can be used to build other ideas with in a compounding fashion.[1] If I’m just researching or working on something in order to build up a repertoire of tools in order to personally use them for other problems, then I don’t need to go through the expensive “finishing” stage of making the infrastructure for all the middle steps legible to others.
There’s an essay by fields medalist William Thurston[2] with several related points, but it’s worth reading in its entirety.
Thurston was a Togo.
“The art of doing mathematics consists in finding that special case which contains all the germs of generality.”
And in the spirit of this post, I should HT Chris Olah for linking to this essay. It’s important to maintain a culture for remembering what hat-tips are due.