Written: Feb. 15, 2009
The usual names are "theory-builders" vs. "problem-solvers", but these two protoypes of mathematicians also appear under various other appellations. Some other names are strategists vs. tacticians, "monkeys that climb up the tree" vs. "monkeys that climb down", globalists vs. localists, and folowing א ח ד ה ע ם, I once called them prophets vs. priests.
The great mathematician-turned physicist-turned essayist, Freeman Dyson, has recently (Notices of the AMS, Feb. 2009) called these traditional archetypes birds vs. frogs. He then went on to describe some of his favorite birds and frogs, and emphasized that the world needs both kinds.
Another way to describe this dichotomy is "people who think globally" vs. "people who act locally". Ideally, everyone should be both a bird and a frog, and many people are! There are few people who are purely birds or purely frogs (e.g. Gian-Carlo Rota and Paul Erdös respectively). Dyson grossly exaggerated when he, quite arbitrarily, labeled von Neumann, and himself, and even more erroneously, Francis Bacon, frogs. Francis Bacon was the greatest bird of all, commanding us that we should all be frogs, and that there is no science without frogs.
I agree that both frogs and birds are crucial for the progress of science, but, even more important, for the progress of mathematics in the computer age, is the beaver, who will build the needed infrastructre of computer mathematics, that would eventually enable us to solve many outstanding open problems, and many new ones. Consequently, the developers of computer algebra systems, and creators of algorithms, are even more important than both birds and frogs.
Unfortunately, the importance of beavers is grossly underrated in the reward system of mathematics. My favorite example, that borders with a scandal, is Bruno Buchberger, who revolutionized mathematics with the Buchberger algorithm that he modestly christened the "Gröbner bases" algorithm. For example, he is conspicuously not a member of the Austrian Academy of Science. I am sure that a one-by-one comparison between any of the members of the Austrian Academy of Science and Bruno Buchberger, in google scholar, would reveal that his impact far surpasses almost all current members (perhaps all!, I was too lazy to try them all.).
But then again, wordly recognition in today's world is not as important as future impact. The best reward is having many people (and computers!) use your algorithm, and this is worth ten Fields medals.