In this lecture, Freeman Dyson surveys the history of progress in mathematics (and related fields of physics) and concludes that there is no One Best Way. Here's the opening:
Some mathematicians are birds, others
are frogs. Birds fly high in the air and
survey broad vistas of mathematics out
to the far horizon. They delight in concepts
that unify our thinking and bring
together diverse problems from different parts of
the landscape. Frogs live in the mud below and see
only the flowers that grow nearby. They delight
in the details of particular objects, and they solve
problems one at a time. I happen to be a frog, but
many of my best friends are birds. The main theme
of my talk tonight is this. Mathematics needs both
birds and frogs. Mathematics is rich and beautiful
because birds give it broad visions and frogs give it
intricate details. Mathematics is both great art and
important science, because it combines generality
of concepts with depth of structures. It is stupid
to claim that birds are better than frogs because
they see farther, or that frogs are better than birds
because they see deeper. The world of mathematics
is both broad and deep, and we need birds and
frogs working together to explore it.
Dyson is always worth reading, and this is a particularly rich piece, full of personal reminiscences as well as insights into the way mathematics and science work. And it demonstrates a characteristic that is far too rare in both great minds and ordinary ones: the ability to appreciate the importance of genuine difference, rather than to argue that one's own approach to problems (of whatever type) is the only correct--or the better--one. |
