Mathematics is out there

Sergiu Klainerman spent years proving that black holes won’t fly apart; and arguing that maths is not a human invention

A computer simulation of supermassive black holes only 40 orbits from merging, rotating a still version of the simulation through 360 degrees as viewed from the plane of the disk. Courtesy NASA’s Goddard Space Flight Center/Scott Noble; simulation data, d’Ascoli et al, 2018

is a freelance science writer living in Cambridge, Massachusetts. His articles have appeared in numerous magazines, including Quanta, Discover and Astronomy. He is the co-author, most recently, of The Gravity of Math: How Geometry Rules the Universe (2024).

Edited byPam Weintraub

The equations that govern black holes were true before there were black holes. That claim is hotly contested, and cuts through one of the deepest fault lines in the philosophy of mathematics. On one side are those who hold that mathematical structures, including well-established principles and basic geometric shapes like the tetrahedron, exist independently of human thought – not as a language we invented to describe reality, but rather as the substrate of reality itself. On the other side of the debate are those who argue that mathematics is the product of human labours, imposed on a world that would be wholly indifferent to it were we not here.

Sergiu Klainerman, professor of mathematics at Princeton University in New Jersey, stands resolutely in the first camp, affirming that mathematical truth precedes us, and that our job is simply to unearth it. His work includes landmark proofs that empty space is stable and that black holes – collapsed stars so dense that nothing inside can escape their gravitational pull – do not disintegrate when perturbed. Theorems like this that he has proved, and others he has built upon, do not represent human creations, he says, but instead stand as discoveries. While the tools used to uncover mathematical truths may be invented, the truths themselves are things to be found. Conflating the two is a category error, Klainerman believes, with consequences that obscure our understanding of what mathematics is and why it works.

Klainerman, 76 this month, was drawn to mathematics as a youth, partly because he was good at it. But he had a more compelling reason. He was born and raised in Romania under the country’s repressive Communist regime, and when he began his undergraduate studies at the University of Bucharest in 1969, mathematics was one of the few subjects not ideologically controlled. It offered ‘a sense of purity’, he said, an escape from the propaganda that saturated every other corner of public life. In mathematics, which he regarded as ‘a field of uncompromising truth and abstract beauty’, he found both intellectual challenge and something even rarer: a welcome refuge. This was one realm over which Nicolae Ceaușescu’s regime had absolutely no control. Mathematical theorems, once proven with rigour, do not bend to political pressure. Unlike diamonds, they truly last forever.

Klainerman didn’t arrive at his philosophy of mathematics in a seminar room. He had early inklings of it as a young man in Bucharest who had watched a state try to control what was true, and had found in one discipline a truth that the government could not touch. That conviction had not come easily, nor was it the result of a linear process. The education waiting for him at the University of Bucharest was, in its own way, almost as resistant to genuine understanding as the regime that surrounded it.

His main source of frustration was with the mathematics department’s reigning pedagogical doctrine: students were expected to keep studying and studying, assimilating information and acquiring technical skills without gaining real understanding. The prevailing attitude was: ‘Climb the mountain first and the true meaning of the subject will be revealed at the top.’ But no one ever seemed to reach the top. Consequently, they never obtained a clear picture of what they were being taught, nor did they gain a sense of what the most interesting questions were.

Disappointed with the formal instruction available to him, Klainerman, along with a few other students, joined an informal study group led by the budding mathematician Otto Liess, who was a few years older than him. There they began working on partial differential equations (PDEs), a way of describing change across space and time (or other variables), from the spread of heat throughout a solid object to the evolution of a gravitational field in space. Much of fundamental physics is written in this language, and almost everything Klainerman would later devote his research to – the study of shock waves, the stability of empty space, black holes – would rest upon such equations. Although Klainerman appreciated the enthusiasm of his fellow students, this extracurricular experience still left him unsatisfied. He could follow the arguments step by step, yet still had no sense of what they were for or where they were meant to lead. It was the same frustration he had encountered in the classroom. And yet those sessions proved decisive. They drew him, ineluctably, into PDEs and, with them, into a field of extraordinary depth and range. By chance, he had found the subject that would shape the rest of his career.

But life in Ceaușescu’s Romania came with strict limits. So, while Klainerman was learning everything he could about these equations, he was also trying to find a way out. In addition to the restrictions on free thought imposed on everyone in the country, he faced another obstacle: he was Jewish. That fact had already shaped the fortunes of his family, hindering the advancement of his father, a well-known cardiologist, and it sharply constrained Klainerman’s future prospects too. To pursue a PhD in mathematics, he would have had to join the Communist Party, something he was loath to do. And even then, his chances of becoming a professor were slim to none.

Professor Sergiu Klainerman. Courtesy Princeton University

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After earning his bachelor’s degree and a year of graduate school in Bucharest in 1974, Klainerman found a way out of Romania (via Israel), landing in the United States as a graduate student at the Courant Institute of Mathematical Sciences at New York University – itself a haven built by refugees. The Institute had been founded in 1935 by Richard Courant, a Jewish mathematician forced out of Germany by the Nazis, and it had grown into one of the world’s leading mathematical research centres, in part by offering a home to talented Jewish mathematicians shunned elsewhere. There, Klainerman’s education was transformed.

For the first time, he encountered teachers who did not treat mathematics as a mountain to be climbed blindly, betraying no hints as to what may lie above, but rather as a field whose inner logic and motivation could be grasped from within. And his specific area of focus, PDEs, was intimately connected with........

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