1) Those that work Problems in order to better understand Mathematics
2) Those who try to understand Mathematics better in order to better solve problems.
He felt he was of the latter variety and called his interest one in “Combinatorics“, a term he used loosely. (An example of a very simple combinatorics problem would be a Sudoku Puzzle).
I must admit that I am of the former camp. I prefer the theoretical side of Mathematics, and find combinatorial problems to be antithetical to my natural state. I am a philosopher at the core, an artist, and my interest in Mathematics is in its complexity of Beauty, its comprehensiveness, its mystery. I work problems so that I can learn more about that mystery. Sure I like the act of solving problems, but I’m not the type of mathematics student who spend his youth lashed to a rubrics cube.
Combinitorics problems are ones that generally you can start solving on day one (though it may take years to solve), but that require very little prior knowledge. The Sudoku example is right on here. The hard part about Sudoku is the solving, not the understanding of the rules. And once you’ve solved the problem, it is solved. Combinitorics is attractive in part, because of the closed loop it presents. It’s tangible.
That is far different than Real Analysis, or Algebraic Topology. These fields take years of training to even get in the door. Only then can you start on the problems, which always lead to more problems, and more, and more. (Can anyone really understand Compactness in the fullest sense? Or define a Set or Category?)
Both takes on Math are important, and neither is more important than the other. It’s like the difference between an English Teacher and a PE Teacher. They are quite different people, but we need them both (for more on the Mind Body Connection, see my fitness Blog: The Dojo).
The Buddhists have always known that life is about balance. As in all fields, Mathematics attracts certain kinds of people that then splinter off into different fields based on aptitudes and interests. The world needs both Theory Builders and Problem Solvers. And it needs all of the variations of gray that lay between them.