일반성과 특수성

The mathematical universe is inhabited not only by important species but also by interesting individuals.—C.L. Siegel 수학의 우주에는 중요한 종들뿐만 아니라 흥미로운 개체들이 살고 있다.

from Magnus, Vignette of a cultural episode

Freeman Dyson

Birds and Frogs

Birds vs. Frogs, or Descartes vs. Bacon

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.

쪼개는 사람과 합치는 사람