A Good Lemma Is Worth a Thousand Theorems (2007) (sites.math.rutgers.edu)

(2007)

coyoneda lemma has been helping me out (in prod - at FAANG even!) for over a decade

I completely agree. Lemmas are simple, obvious, and yet they completely change how you think.

How many proofs cite the Axiom of Choice? How many cite Zorn's Lemma? Which is more important, the axiom that we're relying on, or the restatement of the axiom that allows us to actually prove things?

It is not just mathematics that values simple and obvious ideas that change how you think. We all should value that. Like Stein's Law, "Things that can't go on forever, don't." And Davies' Corollary, "Things that can't go on forever, can go on longer than you think they can."

I guess homological algebra must be worth at least a million theorems, then?

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Favorite lemmata:

Johnson–Lindenstrauss lemma

https://en.wikipedia.org/wiki/Johnson%E2%80%93Lindenstrauss_...

Isolation lemma

https://en.wikipedia.org/wiki/Isolation_lemma

Schwartz–Zippel lemma

https://en.wikipedia.org/wiki/Schwartz%E2%80%93Zippel_lemma

Lovász local lemma

https://en.wikipedia.org/wiki/Lov%C3%A1sz_local_lemma

> Even more important than lemmas are observations, but that is another story.

In my book about abstraction logic (http://abstractionlogic.com) I have definitions, theorems, lemmas, and even observations :-) Just did a count of the frequency. Of course, not sure what those frequencies say about the relative importance.

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Definitions 78

Theorems 20

Lemmas 76

Observations 41

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Lemmas ~ library code, theorems ~ application code.