Examples bring a subject to life

Steve Awodey said of category theory

Material at this level of abstraction is simply incomprehensible without the applications and examples that bring it to life.

Michael Atiyah would say this is not unique to categories.

When someone tells me a general theorem I say that I want an example that is both simple and significant. It’s very easy to give simple examples that are not very interesting or interesting examples that are very difficult. If there isn’t a simple, interesting case, forget it. … Without knowing the examples, the whole thing is pointless. It is a mistake to focus on the techniques without constantly, not only at the beginning, knowing what the good examples are, where they came from, why they’re done. The abstractions and the examples have to go hand in hand.

More from Michael Aityah here.

2 thoughts on “Examples bring a subject to life

  1. I agree completely! Some applied math journals have steered away from examples as a key part of their content, much to the detriment of the readers. They seem to have “pure” math envy where they just want to have proofs without examples of utility. I enraged one of the Associated Editors by commenting on one of these journals, “it used to be good,” but stand by the assessment. A paper on numerical analysis is close to useless without examples of the analysis actually producing meaningful results. I will note that this is my opinion, and there are those who just like reading proofs.

  2. This is exactly what’s wrong with unix man pages. If you need to learn how to use a car you want someone to tell you to put it in gear, press down on the gas, and use the wheel to steer. You don’t want a treatise on how the transmission and electronic fuel injection systems work.

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