Highly cited theorems

Some theorems are cited far more often than others. These are not the most striking theorems, not the most advanced or most elegant, but ones that are extraordinarily useful.

I first noticed this when taking complex analysis where the Cauchy integral formula comes up over and over. When I first saw the formula I thought it was surprising, but certainly didn’t think “I bet we’re going to use this all the time.” The Cauchy integral formula was discovered after many of the results that textbooks now prove using it. Mathematicians realized over time that they could organize a class in complex variables more efficiently by proving the Cauchy integral formula as early as possible, then use it to prove much of the rest of the syllabus.

In functional analysis, it’s the Hahn-Banach theorem. This initially unimpressive theorem turns out to be the workhorse of functional analysis. Reading through a book on functional analysis you’ll see “By the Hahn-Banach theorem …” so often that you start to think “Really, that again? What does it have to do here?”

In category theory, it’s the Yoneda lemma. The most common four-word phrase in category theory must be “by the Yoneda lemma.” Not only is it the most cited theorem in category theory, it may be the only highly cited theorem in category theory.

The most cited theorem in machine learning is probably Bayes’ theorem, but I’m not sure Bayes’ theorem looms as large in ML the previous theorems do in their fields.

Every area of math has theorems that come up more often than other, such as the central limit theorem in probability and the dominated convergence theorem in real analysis, but I can’t think of any theorems that come up as frequently as Hahn-Banach and Yoneda do in their areas.

As with people, there are theorems that attract attention and theorems that get the job done. These categories may overlap, but often they don’t.