Michael Nielsen, in his ongoing series on the principles of effective research, talks about problem solvers and problem-creators as two models of research:
The problem-solver: This is the person who works intensively on well-posed technical problems, often problems known (and sometimes well-known) to the entire research community in which they work. The best problem-solvers are often extremely technically proficient and hard-working. Problem-solvers often attach great social cache to the level of difficulty of the problem they solve, without necessarily worrying so much about other indicators of the importance of the problem.
The problem-creator: This is a rarer working style. Problem-creators may often write papers that are technically rather simple, but ask an interesting new question, or pose an old problem in a new way, or demonstrate a simple but fruitful connection that no-one previously realized existed.
In an interesting coincidence, Chandra Chekuri just pointed me to Tim Gowers' (the 1998 Fields Medallist) website. There, Prof. Gowers has an article titled 'The Two Cultures', where he talks essentially about the same schism in mathematics, between problem solvers, and theory builders.
It is interesting to compare the two reflections. Nielsen feels that the 'problem-creator' style is rarer, and "less popular" than the problem-solver mode. In Gowers' view, theory builders look down on problem solvers (for example, people who do combinatorics) because of a perceived "lack of depth" and other such reasons.
The two polarities are not quite the same; but read together they provide a thoughtful perspective on the kind of schisms that we all face in our research work. I can definitely attest to feeling the kinds of conflicts they talk about, between 'solving problems' and 'building theories'.
Somehow writing a blog itself seems to fly in the face of Hardy's dictum:
There is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds..
Browsing the amazing list of expository articles at Gowers' website, it is comforting to see this shibboleth too go by the wayside.
Add another one to writings on two styles -- Gian-Carlo Rota in _Indiscrete Thoughts_ discusses the difference between problem solvers and theory builders. Grassmann is offered up as an example of a quintessential theory builder, whose real contributions are not noticed in his lifetime. Worth reading just for Rota's writing style itself, but may be interesting to compare to the pieces
ReplyDeletementioned. Unfortunately my copy is in storage.
Incidentally, Rota also discusses at some length how mathematicians are remembered best for their expository work. I forget if this is seen as a good thing.
I would be wary of reading too much into these dichotomies, however. Most people I know partake of both styles. Further, it can take remarkable technical skill to state a crazy theory. :)
Oops - forgot to sign the last comment.
ReplyDelete-David Molnar (dmolnar@gmail.com)