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."
speak only for yourself :o) , they are the reason we have:
lemma now, dilemma laterBut the fact that most lemmas are like this, does not mean that all are. Whether we call it a lemma, or something else, the more important ideas are the ones that result in thinking differently. And something like Zorn's lemma, makes us think differently.
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
So it needs judicious care when used along with algorithm s that work with inner-products
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