Oldest and youngest

It could have been a brilliant career

“He had a fine mind as well as an arithmetically prodigious brain and it was assumed that he would make his way to Trinity and contribute something to the field of pure mathematics before he was thirty or whatever age it is that marks the Anno Domini of mathematicians”

“I believe they’re pretty much over the hill by twenty-six these days”

- Stephen Fry, The Liar (1991)

I’ve been thinking a lot about career progressions and age. As someone soon heading into his fourth decade of doing maths research, is it all over for me? As Fry suggests, the stereotype of a mathematician is often that of a young genius burned out by their mid-twenties, but is that really the case?

The Stars of Track and Field

I was prompted to this question from an unexpected direction, by watching Court of Gold on Netflix. I’m generally a sucker for these kinds of programmes, and this one was no exception. It tells the story of the mens’ Olympic basketball tournament of 2024, focusing on four teams. The theme is that international basketball is becoming ever more competitive. It argues that while in the past non-American teams may have had single stars like Steve Nash and Dirk Nowitzki, they generally lacked strength in depth. However, now many countries can pick a world-class star backed up by a range of solid NBA players. It’s no longer the case that the USA can pick a bunch of stars, play together for a bit and expect to win gold by right.

To be honest, part of the fun is the characters and the national stereotypes (for a show with the Obamas as executive producers, I’m not sure if this was deliberate). France have a chef to provide great cuisine, and fall out over the existential philosophy of how to play the game. They also have Wemby, a man who answers the question of what would happen if you grafted Mr Tickle’s arms onto Mr Tall. Once Serbia’s tournament is over they get blind drunk. It takes a while for them to get knocked out because they have three-time NBA MVP Nikola Jokić, who somehow looks like the most Eastern European man ever, but also has the vision and delicacy to flick a basketball behind his back to pick out a teammate nobody else even knows is there.

Against that, the USA have a roster of galacticos. But as PSG showed by failing to win the Champions League despite having Messi, Neymar and Suarez up front, we know that’s not enough. Part of what keeps the team together is the calming guidance of nine-time NBA winner Steve Kerr, who Last Dance viewers will remember being shaped by the murder of his father by Islamic Jihad in Beirut. But equally, down the stretch, when Serbia and then France push them close, they have Kevin Durant, Steph Curry and LeBron James, all-time stars in their mid-to-late 30s, to bail them out. (Spoilers, sorry!)

And it was really LeBron that got me thinking about career trajectories. He now holds one of those records you have to think about when you first hear it: he’s the oldest and youngest man to score 40 points in an NBA game. He did it in his teens, and he did it in his forties. It’s quite likely that this week he’ll be the first player to reach 50,000 NBA points (regular season and playoffs), when only two players (Kareem Abdul Jabbar and Karl Malone) have even made 40,000.

It’s an astonishing track record of longevity. It made me think that while we often talk about prodigies, and sometimes about late bloomers, there’s also another even more special category of people who keep knocking it out throughout a long career.

Like Dylan in the Movies

I think part of the problem is what statisticians call censored data. That is, you don’t see the end of peoples’ careers until they are over. So in the sixties and seventies, when that type of music was in its infancy, Hendrix, Jones, Joplin, Morrison and the rest of the Twenty-Seven Club might have led us to think that rock ‘n’ roll careers inevitably end in an early death (and to be fair hard drugs and heavy living don’t always help). But it’s only now, when we can buy tickets to see all four original members of Black Sabbath performing together in their mid-seventies, that we know what other trajectories can look like.

With A Complete Unknown tearing up the Oscar nominations, it’s tempting to focus on the period of Bob Dylan’s life when he “burst on the scene already a legend, the unwashed phenomenon, the original vagabond” (as Joan Baez would later put it). But equally, we shouldn’t forget how nobody was really prepared for the idea that Dylan would now still be going strong at 83.

We’re talking about someone who has been an elder statesman longer than most people have had a career. It’s nearly 37 years since the source of the Traveling Wilburys meme, 32 years since his triumphant career celebration at Madison Square Garden, and 27 years since we all assumed he was meditating on his imminent death in Not Dark Yet (“Don't even hear the murmur of a prayer. It's not dark yet, but it's gettin' there”).

For those of us who aspire to a long lifetime of being cantankerous, of going endlessly on tour but not giving people the greatest hits set they’d like, it’s kind of inspiring.

3.. 6.. 9 Seconds of Light

So, who is the LeBron James or Bob Dylan of mathematics? Not a household name perhaps, but I’d have to argue for Jean-Pierre Serre. Somewhat like LeBron, he’s both the youngest and oldest Fields medallist, winning it at 27, and still going strong at 98. And when I say “still going strong”, his most recent arxiv paper revision is dated November 2024. Having won the Fields Medal in 1954, while Dylan was still in high school, Serre is still giving his own version of the Never Ending Tour, with talks as late as 2023 (and possibly later). I’m not qualified to describe his contributions, but it’s clear that this is an incredible track record.

Why then is the stereotype that mathematicians burn out early? Some of it is based on the old stories of Galois and Abel, neither of whom even became eligible to join the Twenty-Seven Club. A lot of it is the fault of G.H. Hardy in my view, and of all the well-intentioned teachers who set their pupils reading that:

I had better say something here about this question of age, since it is particularly important for mathematicians. No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game.

In the phrase I’ve italicized, you’ll notice that there’s no mention of women of any age. It’s perhaps to be expected from a man writing at a time when his university didn’t even give degrees to female students. But if we can recognise Hardy’s views on gender as that of a dinosaur, why are we clinging to them when it comes to age?

We ended up with a situation where Fields Medals are only given to those under 40 more or less by accident, meaning that Andrew Wiles’ ground-breaking proof of Fermat’s Last Theorem could not be recognised in this way. And for every genius who died young or burned out early, there’s at least as many who had a very different trajectory. Leonhard Euler published his equations for fluid flow when he was 50, and was 65 when he discovered a prime number which would be the largest known for nearly a century.

For subscribers who weren’t here eighteen months ago, I wrote about one particular genius:

And again, Andrey Kolmogorov showed no sign of burning out. He was in his mid-fifties when he made a major contribution to a gold riband Hilbert problem and sixty when he first published his ideas on algorithmic complexity of numbers.

By Mike Murphy, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=295635

I sometimes think about mathematical progress as humanity working together to climb an enormous rock face - something like El Capitan except much harder. It can be the work of a lifetime for someone to even reach up one handhold higher than anybody got to before (and most of us don’t even manage that). But when they do, they can hammer in a piton, hang down a rope, and the next generation can reach that point more easily. But even that “more easily” is a relative thing. So fundamentally, it’s not a surprise that maths careers look different to how they used to. It would take years in graduate school to master Serre’s work in number theory, let alone to start to build on it - the days when Fermat could be a lawyer by day and prove theorems as a hobby are long gone.

Of course, if you look at career trajectories in more detail, people’s productivity will often slow down over time. There’s an element of regression towards the mean here. Maybe Dylan never wrote anything as good as Like a Rolling Stone again, but then who did? Mathematicians only get hired to permanent jobs with a strong track record in their 20s, so we aren’t seeing a proper random sample of what could be possible - with the story of Yitang Zhang only giving a hint. As time goes on, senior peoples’ productivity can be constrained at least as much by spending more time serving on committees, reviewing grants and doing other jobs than any fundamental barrier to productivity.

Most of my own best papers (so far!) came out in my forties, and if there’s no law of nature which says that mathematicians will definitely burn out, I’m kind of curious to see how long that can go on. Even if I can’t keep publishing, then at least I hope to watch Terence Tao (Maths Olympiad gold medallist aged 13, full professor at 24) keeping pushing back the boundaries after he turns 50 this year. And at the very least, nobody who follows a less conventional career trajectory should be put off by the words of Fry and of Hardy.