It is obvious that mathematics needs both sorts of mathematicians [theory-builders and problem-solvers] . . . It is equally obvious that different branches of mathematics require different aptitudes. In some, such as algebraic number theory, or the cluster of subjects now known simply as Geometry, it seems . . . to be important for many reasons to build up a considerable expertise and knowledge of the work that other mathematicians are doing, as progress is often the result of clever combinations of a wide range of existing results. Moreover, if one selects a problem, works on it in isolation for a few years and finally solves it, there is a danger, unless the problem is very famous, that it will no longer be regarded as all that significant.

At the other end of the spectrum is, for example, graph theory, where the basic object, a graph, can be immediately comprehended. One will not get anywhere in graph theory by sitting in an armchair and trying to understand graphs better. Neither is it particularly necessary to read much of the literature before tackling a problem: it is of course helpful to be aware of some of the most important techniques, but the interesting problems tend to be open precisely because the established techniques cannot easily be applied.

*The two cultures of mathematics*

JOC/EFR April 2011

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