Alexander Grothendieck and the search for the heart of the mathematical universe
This piece is extracted from When We Cease to Understand the World, Benjamín Labatut's new book blending fact with fiction to tell key moments in science history.
Alexander Grothendieck is considered by many to be the greatest mathematician of the 20th Century. But his upbringing was fraught, leaving his birth town of Berlin as a child to evade the Nazis and ending up in several concentration camps across France.
Grothendieck attended a college along with other refugee children, where, it is thought, his love of maths first appeared.
For much of his career he focused on algebraic geometry, and he is credited with the creation of many principles and heuristics used in mathematics today. His work helped mathematicians solve problems that had tormented them for years, including 17th Century mathematician Pierre de Fermat's Last Theorem.
Though revered, he began to withdraw from the mathematical community. He retired in 1988 and removed himself from society, living in solitude until his death in 2014.
In this extract from When We Cease to Understand the World, Labatut imagines what life was like for the great but troubled mathematician.
When Grothendieck was still an undergraduate student at the University of Montpellier, his professor, Laurent Schwartz, gave him an article he had published not long before which included fourteen major unresolved problems, and asked Grothendieck to choose one of them for his thesis. The young man, who was always bored and distracted in his classes and seemed incapable of following instructions, returned three months later. Schwartz asked him which problem he had chosen and how far along he had got. Grothendieck looked at him, baffled. What did he mean by “which one”? He had solved all of them.
His talent caught the attention of everyone who met him, but it was difficult for him to find work in France. Because of his parents’ constant displacements, Grothendieck had no nationality, and his only identity document was a Nansen passport, which classified him as “stateless”. He was physically imposing, tall, thin and athletic, with a square jaw, broad shoulders and a large, bull nose. The corners of his thick lips curved upwards, giving him a mischievous look, as though he were privy to a secret that everyone else did not even suspect. When he began to lose his hair, he shaved his head, a perfect oval. In photos, he looks like Michel Foucault’s identical twin.
A talented boxer, fanatical about Beethoven’s final quartets and Bach, he loved nature and venerated “the humble and long-lived olive tree, full of sun and life”, but above all the things of this world, including mathematics, he had a devotion to writing that bordered on the fanatical.He wrote with such fervour that, in certain parts of his manuscripts, the pencil tip would pierce the page straight through. When calculating, he wrote his equations in his notebook and then retraced them over and over, until each symbol was so thick it was no longer intelligible, in thrall to the physical pleasure he felt by scratching graphite on paper.
In 1958, the French millionaire Léon Motchane built the Institute of Advanced Scientific Studies on the outskirts of Paris to serve Grothendieck’s ambitions. There, Grothendieck, who had just turned thirty, announced a work programme that would re-establish the foundations of geometry and unify all the branches of mathematics. An entire generation of professors and students subjugated themselves to Grothendieck’s dream. He would preach aloud while they took notes, expanded his arguments and wrote out drafts for him to correct. The most devout of all his collaborators, Jean Dieudonné, would get up at five in the morning, even if the Sun had not yet risen, to review the transcripts from the previous day before Grothendieck burst into the classroom at eight on the dot, developing a fresh set of ideas that he had already begun debating with himself as he climbed the stairs of the institute. Grothendieck’s seminar produced twelve volumes, more than twenty thousand pages that manage to bring together geometry, number theory, topology and complex analysis.
Unifying mathematics is a dream that only the most ambitious minds have pursued. Descartes was among the first to show that geometric forms can be described through equations. Whoever writes x2 + y2 = 1 is describing a perfect circle. Every possible solution to this basic equation represents a circle drawn on a plane. But if one considers not only real numbers and the Cartesian plane, but also the bizarre spaces of complex numbers, there appears a series of circles of various sizes that move as if they were living creatures, growing and evolving in time. Part of Grothendieck’s brilliance was to recognise that there was something grander hidden behind every algebraic equation. He called this something a scheme. Each individual solution to an equation, each shape, was nothing but a shadow, an illusory projection that flashed forth from the general scheme, “like the contours of a rocky coast illuminated at night by the rotating lamp of a lighthouse”.
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Grothendieck was capable of creating an entire mathematical universe fit for a single equation. His topoi, for example, were seemingly infinite spaces that defied the limits of the imagination. Grothendieck compared them to “the bed of a river so vast and so deep that all the horses from all the kings of this world, and from all possible worlds, could drink from its waters together”. To think in such terms demanded a completely novel conception of space, as radical as the change brought forth by Albert Einstein, 50 years before.
He loved choosing le mot juste for the concepts he discovered, as a way to tame them and render them familiar before he could fully grasp them. His étales, for example, evoke the calm, docile waves of a low tide, a still mirror of water, the surface of a wing stretched to breaking point, or the taut, white sheets swaddling a newborn child.
He was capable of sleeping at will, as many hours as he needed, then dedicating the whole of his energy to his work. He could begin working out an idea in the morning and not move from his desk until dawn the next day, squinting under the light of an old kerosene lamp. “It was fascinating to work with a genius,” his friend Yves Ladegaillerie remembers. “I don’t care for that word, but for Grothendieck there is no other. It was fascinating but also terrifying, because this was a man who simply did not resemble other human beings.”
His capacity for abstraction seemed endless. He could make unexpected leaps to higher categories and work in orders of magnitude no one had dared to explore before. He formulated his ideas by removing one layer after the other, breaking down concepts, simplifying and abstracting until there seemed to be nothing left; there, in that apparent vacuum, he would discover the structures he had been searching for.
“My first impression on hearing him lecture was that he had been transported to our planet from an alien civilisation in some distant solar system in order to speed up our intellectual evolution,” a professor from the University of California at Santa Cruz said of him. Despite how radical they were, the mathematical landscapes Grothendieck conjured up gave no impression of artificiality. To the trained eyes of a mathematician, they revealed themselves as seemingly natural environments, for Grothendieck did not impose his will on things, preferring to let them grow and develop by themselves. The results had an organic beauty, as though each idea had budded and borne fruit following its own vital impulse.
In 1966, he won the Fields Medal, known as the Nobel of mathematics, but he refused to go to Moscow to receive it in protest against the imprisonment of the writers Yuli Daniel and Andrei Sinyavsky.
For two decades, his dominance was such that René Thom, another brilliant winner of the Fields Medal, admitted to having felt “oppressed” by Grothendieck’s overwhelming technical superiority. Frustrated by his incapacity to rival Grothendieck’s vertiginous output of groundbreaking ideas, Thom abandoned pure mathematics to develop catastrophe theory, a mathematical treatment that describes seven ways in which any dynamic system — be it a river, a tectonic fault or the fragile mind of a human being — can suddenly lose its equilibrium and collapse, falling into disorder and chaos.
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“What stimulates me is not ambition or the thirst for power. It is the acute perception of something immense and yet very delicate at the same time.” Grothendieck continued to push past the limits of abstraction. No sooner had he conquered new territory than he was preparing to expand its frontiers. The pinnacle of his investigations was the concept of motive: a ray of light capable of illuminating every conceivable incarnation of a mathematical object. “The heart of the heart” he called this strange entity located at the crux of the mathematical universe, of which we know nothing save its faintest glimmers.
Even his closest and most loyal collaborators believed he had gone too far. Grothendieck wanted to hold the Sun in the palm of his hand, uncover the secret root that could bind together countless theories that bore no apparent relation to one another. They told him his goal was unattainable, and that his project sounded more like the pipe dreams of an amateur than a legitimate programme for scientific exploration. Grothendieck did not listen. After spending so long gazing down at the foundations of mathematics, his mind had stumbled into the abyss.