I recently saw this fun video about the tension between clarity and mystery and the role of this tension in creating good design. It made me think about giving a math talk. I think the element of mystery is often neglected when one prepares a math class or just a math talk.
It is understood that unlike, say, in fiction, mystery is undesirable in mathematics. While it is good to have a bunch of unsolved problems to work on, we would very much like to know the answers to them. All mathematicians are glad that we now know the proof of Fermat’s big conjecture and the Poincare conjecture, and most mathematicians would sell their soul to know the proofs of the Riemann hypothesis and solution to the Navier-Stokes.
To some extent, this revulsion toward mystery extends to math talks. Unlike in a detective novel where the entire plot is shrouded in mystery and the audience only gets to know the resolution at the very end, in a math talk the result is on the board in the first few minutes. Nobody ever wonders whether something is true or false (unless it is not yet known), there is no doubt, there is no suspense about the content.
And yet, I think it is wrong to say that there is no element of mystery in a good math talk. I think a reason why a lot of math talks end up being boring is that they lack this element of mystery. But the mystery is not about facts, it is not about truths. If it is not about facts, then what is it about? Isn’t math just a collection of facts and truths?
First, let me say that this element of mystery only exists in a math talk and not in a math paper. Math papers are written in the most boring fashion because they are not meant to be read linearly. Most math papers are never fully read. A math paper is not unlike a dictionary or an encyclopedia, you just read the section that you are interested in.
But a math talk is different, you cannot fiddle back and forth or skip a chapter you do not like.
You are forced to listen to the talk as it happens in front of you.
Because of this, there is a human element to the talk, an almost artistic character to it,
which is where the element of suspense comes from.
The suspense in a math talk is not about the facts, but about the presenter’s interpretation of the facts, it is about how they got to uncover or understand an abstruse phenomenon.
This, I think, is fundamental to giving a good math talk. You have to tell a story, and a story needs its twists and plots, its mysteries and its revelations. But these twists cannot be about facts, as facts in math are immutable. Instead, they are about the interactions between the human world and the math world.
Of course, this is an overgeneralization and an oversimplification. Giving a good math talk is a skill, and like any other skill, it needs to be practiced to get good at. But it is not a mindless, automatic skill like driving. Instead, I think it is more akin to conducting an orchestra. As with an orchestra, the conductor does not really have a choice when it comes to the music, but they do have a choice when it comes to the interpretation.
Clarity is of utmost importance too and cannot be sacrificed. But it can be easily overdone.
A good math talk is not about the content but rather the interpretation, it is made interesting by the constant tension between the mathematical clarity and the human mystery, and in that, it is expressive and extremely personal and is often riddled with mysteries and adventures.