The Forgotten Dream of the Fields Medal

In just a few weeks, the 2018 winners of the Fields Medal—math’s most famous prize—will be announced in Brazil. I figured this was a good time to call up historian Michael J. Barany, whose tireless, startling work has transformed my conception of the Fields.

This interview has been edited for brevity and clarity, then illustrated for goofiness.

 

What is the Fields Medal?

It’s an award given by the International Congresses of Mathematics, every four years, for “outstanding achievements in mathematics.”

And what was Mr. Fields going for in starting this award?

John Charles Fields is a Canadian mathematician, one of few North Americans active in international mathematics in the first decades of the 20^th century.

He doesn’t want the medal named after him, or anyone, or anything. He doesn’t give any guidelines about what areas of study the medal should be given for. He wants it to be as open-ended as possible, and also—this is crucial for him—as non-political as possible.

He figured that if it were about assessing contributions of the past, there would always be grounds for national rivalry. So he had this famous line in his memorandum: that it should not just be for past achievements, but for future promise.

 

Interesting. He thought judging past achievements would be hyper-political, but speculating on the future wouldn’t be.

Yes, that was his idea for how to make it a place for cooperation rather than competition.

It’s this idea that evaluating mathematical achievements is not an objective thing. So you try to take away the ammunition to argue against each other.

 

Fuzzy ground no matter what; might as well pick the thing where it’s overt.

Yeah, where it’s really, deliberately fuzzy. He wanted every committee to have as much latitude as possible to decide how to celebrate international mathematics, and to make a statement about the future, to decide what the medal was going to mean.

Like any prize, the Fields appeals to the competitive, status-seeking part of us. But it seems Fields himself had a more idealistic notion.

It was this thoroughly idealistic medal. We’ve forgotten, maybe, in the decades since, just how quixotically idealistic Fields was.

It was not founded as a Nobel Prize of Mathematics. He’d seen how, in the case of the Nobels, there was this intense rivalry, the counting up of national winners. He found that distasteful; he thought it was a real shortcoming of those kinds of prizes. But he also saw the international solidarity they could inspire.

He wanted the good of the international solidarity without the bad of rivalry and competition. He thought that not naming it after anyone—any person, any place—would be the best way to signal the neutrality of the medal.

 

Was it a big deal from the beginning?

It’s common when people write retrospectively to say, “Oh, this person won a Fields Medal, therefore, they were a leader in mathematics.” Using it as an index of their importance.

But I was researching the history of mathematics in the 1940s and 1950s, and when you look at the documents circulating around that time period, the Fields medal didn’t have that cachet. It was one prize among many—like a Guggenheim Fellowship. Certainly not at the level of a National Medal of Science, or membership in a National Academy.

 

When did it become the medal we know today—with the strict age 40 limit, and the stature as the “Nobel of Mathematics”?

One of the surprising things to come out of this research was that those two things have a hidden link. Two historical phenomena come together in 1966.

One trend was already evident in 1936, 1950, 1954, 1958: the problem of narrowing down the pool of candidates. In letters that I found in the archives, you can see this tremendous difficulty: “We have all these great mathematicians we could give the award to.” The 1950 committee considered many different criteria—whether they should just look at algebraists, or people under the age of 32, or people who have become famous since the last Congress.

They took as a guideline that the medal should be a leg up for winners. If they already had a professorship at a good institution; if they already had been recognized by a lot of other prizes or awards; then that was a disqualification for the Fields.

 

What changed?

In 1966, the committee decides that all of this parsing who’s recognized, who’s not, who’s well known, who’s not—it’s getting too hard. We’d better just pick an age limit. So all of these people who would have been considered too well-known previously are suddenly under consideration.

For the first time, the winners in 1966 are people who are already famous in the mathematical community. That jump in prestige, which was an accidental effect of the age limit, helped make the “Nobel Prize” story a lot more credible.

 

So, they were trying to come up with a criterion that would be easier to apply, and they wound up picking one that was radically different.

Yes. Almost the inverse, really.

I should say, “40” doesn’t have some magical relationship to mathematical activity. That’s just the smallest round number that included all of the medalists before 1966.

 

Had they picked a younger age limit—32, 35—might the award have stayed closer to its original conception?

It’s possible. But you have some domains of research where it’s possible to get a major result by age 30, and others where you have to work for a really long time before you can make a recognizable impact. So it would still have this bias for certain topics, certain training systems, certain ways of cultivating talent.

 

Stephen Smale, one of the 1966 winners, sat down next to me at lunch in Heidelberg. I asked him: “Have you heard about this research, that people started calling the Fields ‘the Nobel of Math’ in part because they needed a rhetorical move to explain why you were traveling to Moscow in the middle of the Cold War?”

He just nodded. “I heard it. Could be right. Sounds plausible.” I don’t think he felt his memories were vivid enough to say for sure.

When I wrote that original piece on the Smale affair, and was first trying to get it published, the Notices of the AMS was the third place where I submitted it. In all three referee processes, they would always send it to mathematicians who were old enough, in principle, to remember the events, and the reaction was always, “I don’t remember it that way.”

That was a huge obstacle for getting the historical evidence to be believed. These things are so transformed by the lens of the past that it can be hard to believe what the documents are telling you.

One of the culprits in this part of the story is John Lighton Synge. In 1971, he’s an aging mathematician, and he asserts very confidently to this historian Henry Tropp that Fields was concerned about the lack of a Nobel Prize, and had intended this as a substitute—even though there’s no documentary evidence giving any kind of indication of this.

It’s on the basis of that kind of memory, that has now been colored by the events of 1966, that historians ourselves got this idea that there was this deep historical link between Fields and Nobel.

 

What are the differences between the priorities encoded by the Nobel, and those of the Fields?

The Nobel tends to be for particular breakthroughs. It’s possible in principle—and has happened in fact—for someone to win two Nobel prizes in the same field for different breakthroughs.

The Fields is a kind of promissory, a future-oriented kind of award. It’s for the promise of an individual, as evidenced by the work; if you have two major breakthroughs in math, that increases your odds of getting a Fields Medal, but you’re not going to get two Fields Medals.

 

And what is the Fields’s notion of “promise”?

A fairly traditional career path. People who come to mathematics not as talented children who progress rapidly through the educational system—they have a much harder time gaining the level of recognition they need to win by the age of 40. Taking time out to have children is a big disadvantage; this is a problem especially for women.

Some areas of study lend themselves to spectacular, public results—where you help to solve a major conjecture widely acknowledged as important. But in the post-WWII era, a lot of important mathematical work is done in environments that can’t be widely publicized, due to corporate restrictions or government secrecy.

You see certain institutional biases, too. You need people on the Fields Medal committee to know who you are. You see a lot of people who have spent some part of their career in Boston, Berkeley, Paris, or Princeton—a very small collection of institutions account for a huge share of the winners.

 

Is there any progress on expanding our notion of merit?

One thing the IMU and ICMs have been doing in recent years is to de-center the Fields Medals a bit, by creating more medals—the Nevanlinna Prize, the Chern Medal, the Gauss Prize—that spread around the spotlight. The idea is that there are a variety of ways to make a difference.

 

What do you imagine for the future of the Fields?

There will always be a drive to maintain the Fields Medal’s luster by associating it with people who are already widely agreed to be excellent mathematicians. That’s a safe approach to the Fields Medal. But my deliberately provocative idea that I closed the piece in Nature with is this: Every four years, the Fields committee has this opportunity to think about what matters to the discipline. What does the discipline lack right now? What areas of study deserve a leg up?

The committee has this built-in liberty to think more expansively about what the discipline could become, and what areas of the discipline are not sufficiently recognized or acknowledged, and to use the medal in this way that’s truer to an early conception of the award.

This is what I found out about Harald Bohr’s work as committee chair in 1950. In championing Laurent Schwartz, he was speaking up for a mathematician whose textbook had barely come out. He didn’t have a lot of major theorems to his name. What he had was an approach to mathematics that Bohr thought could unite pure and applied mathematicians, as well as a charisma and personality that Bohr thought could make him an ideal ambassador for the discipline. Bohr pushed to use the Fields Medal as a way of saying, “Here’s somebody you probably haven’t heard of, but who’s going to make a difference.”

Embracing that responsibility can, I hope, be inspiring to anybody in that position of power. There’s an opportunity to intervene. It’s wasted if you use it to affirm what people already know.

 

It’s like the end of Ratatouille. The Fields can turn away from safeguarding its own prestige, and instead stand up, and defend the animated rats of the world.

Yeah. Find talent in the rats.

 

Thanks to Michael for the interview! Any resemblance between uncelebrated mathematicians and animated rats is purely metaphorical.