What is it like to have an understanding of very advanced mathematics?

You can answer many seemingly difficult questions quickly. But you are not very impressed by what can look like magic, because you know the trick. The trick is that your brain can quickly decide if question is answerable by one of a small number of powerful general purpose “machines” (e.g. continuity arguments, combinatorial arguments, correspondence between geometric and algebraic objects, linear algebra, compactness arguments that reduce the infinite to the finite, dynamical systems, etc.).

One of a long list.

It’s hard to convince those that don’t “speak” math how necessary it is, rather than being forced to explain things in a much less precise language (be it English or something else) that the audience understands.

via @seanmcarroll