### The Duties of a Mathematician

#### Posted by John Baez

What are the ethical responsibilities of a mathematician? I can think of many, some of which I even try to fulfill, but this document raises one that I have mixed feelings about:

- Ivan Fesenko, Remarks on aspects of pioneering modern mathematics research.

Namely:

The ethical responsibility of mathematicians includes a certain duty, never precisely stated in any formal way, but of course felt by and known to serious researchers: to dedicate an appropriate amount of time to study each new groundbreaking theory or proof in one’s general area. Truly groundbreaking theories are rare, and this duty is not too cumbersome. This duty is especially applicable to researchers who are in the most active research period of their mathematical life and have already senior academic positions. In real life this informal duty can be taken to mean that a reasonable number of mathematicians in each major mathematical country studies such groundbreaking theories.

My first reaction to this claimed duty was quite personal: namely, that I couldn’t possibly meet it. My research is too thinly spread over too many fields to “study each new groundbreaking theory or proof” in my general area. While Fesenko says that “truly groundbreaking theories are rare, and this duty is not too cumbersome”, I feel the opposite. I’d really love to learn more about the Langlands program, and the amplitudohedron, and Connes’ work on the Riemann Hypothesis, and Lurie’s work on $(\infty,1)$-topoi, and homotopy type theory, and Monstrous Moonshine, and new developments in machine learning, and … many other things. But there’s not enough time!

More importantly, while it’s undeniably *good* to know what’s going on, that doesn’t make it a “duty”. I believe mathematicians should be free to study what they’re interested in.

But perhaps Fesenko has a specific kind of mathematician in mind, without mentioning it: not the larks who fly free, but the solid, established “gatekeepers” and “empire-builders”. These are the people who master a specific field, gain academic power, and strongly influence the field’s development, often by making pronouncements about what’s important and what’s not.

For such people to ignore promising developments in their self-proclaimed realm of expertise can indeed be damaging. Perhaps these people have a duty to spend a certain amount of time studying each new ground-breaking theory in their ambit. But I’m fundamentally suspicious of these people in the first place! So, I’m not eager to figure out their duties.

What do you think about “the duties of a mathematician”?

Of course I would be remiss not to mention the obvious, namely that Fesenko is complaining about the reception of Mochizuki’s work on inter-universal Teichmüller theory. If you read his whole article, that will be completely clear. But this is a controversial subject, and “hard cases make bad law”—so while it makes a fascinating read, I’d rather talk about the duties of a mathematician more generally. If you want to discuss what Fesenko has to say about inter-universal Teichmüller theory, Peter Woit’s blog might be a better place, since he’s jumped right into the middle of that conversation:

- Peter Woit, abc News, 17 July 2018.

As for me, my *joy* is to learn new mathematics, figure things out, explain things, and talk to people about math. My *duties* include helping students who are having trouble, trying to make mathematics open-access, and coaxing mathematicians to turn their skills toward saving the planet. The difference is that joy makes me do things spontaneously, while duty taps me on the shoulder and says “don’t forget….”

## Re: The Duties of a Mathematician

Many people tell the story of arriving at IAS late in the evening and having some variation on the following conversation when picking up their keys from the porter.

Porter: Let me guess. You are a geometer.

Geometer: How did you know!?

Porter: You seem far too nice to be a number theorist.

I suspect that old porter would have some insight into l’affaire Mochizuki.