Hard analysis, soft analysis

“Hard analysis” and “soft analysis” are technical terms. They don’t necessarily mean “difficult” and “easy.”

Soft analysis often asks qualitative questions where hard analysis asks more quantitative questions. For example, soft analysis might ask whether a series converges, when hard analysis asks at what rate a series converges. Soft analysis is often more abstract, asking questions about classes of functions, where hard analysis asks questions about specific functions.

The difference between soft and hard analysis reminds me of Freeman Dyson’s classification of mathematicians into birds and frogs:

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.

I was trained as a bird, but I’ve become more of a frog over time. Of course these categories are not exclusive. Frogs sometimes act as birds and vice versa. The categories of hard and soft analysis are not exclusive either. See Terry Tao’s article on relating hard and soft analysis.

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