When Samuel Hansen said in his interview “You’re not a pure mathematician” I agreed without thinking, but later the statement bothered me a little. I know what he meant: considering the two categories of pure math and applied math, you’d put yourself in the latter category. Which is true.
But the term “pure” math can be misleading, as if everyone else does impure math. Applied math is not an alternative to theoretical math. Applied mathematicians prove theorems etc. We work on applications in addition to doing what is expected of pure mathematicians. The difference between pure and applied math is motivation, not content. Applied math is motivated by direct application to non-mathematical problems. Pure math seeks to advance math for its own sake. Both are important.
Statistics uses the terms “theoretical” and “applied” rather than “pure” and “applied.” Math doesn’t use “theoretical” as an antithesis to “applied” because applied math is theoretical. But unlike math, being “applied” in statistics does mean you’re often (too often?) excused from proving theorems. The first time I was a coauthor on a statistics paper I was surprised to find out you could publish with just simulation results and no theorems. This happens in applied math as well, but not nearly as often as it does in applied statistics.
On the other hand, when I hear the term “applied statistics” I want to ask “Is there any other kind?” Statistics is applied (and theoretical!) though some statisticians work more directly on applications than others. As Andrew Gelman quips, the difference between theoretical and applied statisticians is that
The theoretical statistician uses x, the applied statistician uses y (because we reserve x for predictors).
I assume that statement wasn’t meant to be taken literally, but I agree with the sentiment that the distinction between theoretical and applied statistics can be exaggerated. I’d say the same applies to pure and applied math.