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 recognize 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”.
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, fifty 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 civilization 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.
“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.
At the tail end of the Sixties, he travelled for two months to Romania, Algeria and Vietnam to give a series of seminars. One of the colleges at which he taught in Vietnam was later bombed by American troops; two professors and dozens of students died. When he returned to France, he was a changed man. Influenced by the clamour of the May ’68 protests all around him, he called on more than a hundred students during a masterclass at the University of Paris in Orsay to renounce “the vile and dangerous practice of mathematics” in light of the hazards humanity was facing. It was not politicians who would destroy the planet, he told them, but scientists like them who were “marching like sleepwalkers towards the apocalypse”.
From that day forward, he refused to participate in any maths conference that would not allow him to devote equal time to ecology and pacifism. During his talks, he gave away apples and figs grown in his garden and warned about the destructive power of science: “The atoms that tore Hiroshima and Nagasaki apart were split not by the greasy fingers of a g
eneral, but by a group of physicists armed with a fistful of equations.” Grothendieck could not stop fretting over the possible effects that his own ideas could have on the world. What new horrors would spring forth from the total comprehension that he sought? What would mankind do if it could reach the heart of the heart?
In 1970, at the high point of his renown, creativity and influence, he resigned from the Institute of Advanced Scientific Studies after learning it accepted funds from the French Ministry of Defence.
In the following years, he abandoned his family, disavowed his friends, repudiated his colleagues, and fled the rest of the world.
“The great turning point” was the term Grothendieck used to describe the change in the direction of his life during his forties. All at once, he found himself swept up by the spirit of the age: he became obsessed by ecology, the military-industrial complex and nuclear proliferation. To his wife’s despair, he founded a commune at home, where vagabonds, professors, hippies, pacifists, thieves, nuns and prostitutes dwelt side by side.
He became intolerant of all the comforts of bourgeois life; he tore up the carpets from the floors of his house, considering them superfluous adornments, and began to make his own clothing: sandals from recycled tires, trousers sewn from old burlap sacks. He stopped using his bed, instead sleeping on a door he had torn from its hinges. He only felt at ease among the poor, the young and the marginalized. The stateless, those without a homeland.
He was generous with his possessions and gave them away without a second thought. He was generous with those of others as well. One day, one of his friends, the Chilean Christian Mallol, arrived home with his wife after a dinner out to find his front door ajar, his windows open, the fireplace raging and the furnace blasting. Grothendieck was naked and asleep in his bath. Two months later, Mallol received a cheque for three thousand francs from Grothendieck, to cover the cost.
Although in general he was caring and kind, he could suffer sudden outbreaks of violence. During a pacifist protest in Avignon, he ran towards the containment barrier and knocked out two policemen who were attempting to impede the march, before being beaten to a pulp by dozens of officers who dragged him unconscious to the police station. At home, his wife heard him rant and rave while locked alone in his study, long monologues in German which degenerated into screams so loud that they shook the windows, followed by episodes of mutism that could last for days.
“Doing mathematics is like making love,” wrote Grothendieck, whose sexual impulses rivalled his spiritual inclinations. Throughout his life he seduced both men and women. He had three children with his wife, Mireille Dufour, and two more outside his marriage.
He founded the group Survive and Live, devoting all of his resources and energy to it. He published a journal with a group of friends (though he wrote it practically on his own) to promote his ideas on self-sufficiency and care for the environment. He tried to involve those who had followed him in his mathematical undertakings, but no one shared his sense of urgency or could tolerate his extremism now that the object of his obsession was no longer the abstract world of numbers, but the concrete problems of society—which Grothendieck confronted with a degree of naivety that bordered on idiocy.
He was convinced that the environment had its own consciousness and that it was his duty to protect it. He would gather even the smallest shoots that grew between the cracks in the pavement outside his house to replant and care for them.
He began fasting once a week, then twice, and sometimes he would stop eating completely. Self-mortification became second nature to him. During a trip to Canada, he refused to wear shoes and walked through the snow in his sandals, like some Middle Eastern prophet spreading his gospel through a frozen desert. After a severe motorcycle accident, he declined anaesthesia, and agreed only to acupuncture during his surgery, as he had grown almost indifferent to physical pain. Behaviours of this kind fed the rumours that his critics spread to discredit him (and to defend themselves against the increasingly virulent charges Grothendieck levelled against them), the most outrageous of them all being that Grothendieck, in his zeal to reduce his impact on the planet, would shit in a bucket and then walk through the farms surrounding his home to spread his own excrement as fertilizer.
In 1973, the commune he had founded as a place open to all descended into pure lawlessness. First the police came to arrest two Japanese monks from the Order of the Marvellous Lotus Sutra who had overstayed their visas, and charged Grothendieck with harbouring illegal aliens. That same week, a girl he often spent the night with tried to hang herself with the curtains in his room. When he returned with her from the hospital, Grothendieck found the members of the commune dancing around an enormous bonfire they had built in the middle of the courtyard, feeding it with pages from his manuscripts. Grothendieck disbanded the community and retired to Villecun, a village of just a dozen tiny cottages.
In Villecun he lived without electricity or drinking water in a flea-infested cabin, but he was happier than he had ever been. He purchased a dilapidated hearse to get around, and, when the engine failed, he bought a second, even ricketier automobile with holes in the floorboard that opened onto the road, and drove it at the highest possible speed, though he had neither licence nor registration.
For five years, in almost total isolation, he devoted himself to manual labour, undertaking no major projects. His children did not visit him, he had no lovers, and he ignored all his neighbours with the exception of a twelve-year-old girl he helped with her arithmetic homework. When he had exhausted his savings, he began teaching mathematics at the University of Montpellier to cover the costs of his spartan existence. His undergraduate students could not have imagined that the man who received them dressed in rags like a vagrant, and whom they would find asleep on the classroom floor if they arrived early, was a living legend.
In Villecun, he focused his immense analytical powers on his own mind. The result was a change even more radical than the one that had drawn him away from mathematical research, a complete metamorphosis that he tried to encapsulate in a cryptic list tracing out the stations of his spiritual journey, which had taken him further and further away from common sense.
May 1933: longing for death
27–30 December 1933: birth of the wolf
summer (?) 1936: the Gravedigger
March 1944: existence of God the creator
June–Dec. 1957: call and betrayal
1970: the stripping away—entry into the mission
1–7 Apr. 1974: moment of truth, entry into the spiritual path
7 Apr. 1974: meeting with Nihonzan Myohoji, entry of the divine
July–Aug. 1974: insufficiency of the Law. I leave the paternal Universe
June–July 1976: awakening of the Ying
15–16 Nov. 1976: collapse of the image, discovery of meditation
18 Nov. 1976: re-encounter with my soul, entry of the Dreamer
August 1979–February 1980: I manage to meet my parents (imposture)
March 1979: discovery of the wolf
August 1980: meeting with the Dreamer—recovery of childhood
Feb. 1983–Jan. 1984: the new style (“In Pursuit of the Fields”)
Feb. 1984–May 1986: Reaping and Sowing
25 Dec. 1986: the “sacrifice” of the ReS
*NB 25.12.1986: first erotico-mystic dreams
28 Dec. 1986: death and rebirth
1–2 Jan. 1987: mystico-erotic “abduction”
27 Dec. 1986–21 March 1987: metaphysical dreams, intelligence of dreams
8.1, 24.1, 26.2, 15.3 (1987): prophetic dreams
28.3.1987: nostalgia for God
30.4.1987— … The Key to Dreams
Between 1983 and 1986, he wrote Reaping and Sowing: Reflections and Testimonies from a Mathematician’s Past, an outlandish work that no one in France dared to publish. Over thousands of pages, replete with what one colleague described as “mathematical phantasmagorias”, Grothendieck delves into his own psyche in an a
ttempt to understand everything, revealing a vast and terrifying intellect stripped increasingly bare, precariously balanced between enlightenment and paranoia.
The ideas of Reaping and Sowing turn in circles. The author returns to the same arguments over and over again, aspiring to total precision. He examines what he has just written in order to discard or affirm it with redoubled force, attempting to fix his words in a definitive form, which they naturally resist. A single page contains brusque changes of perspective, theme and tone, the product of a mind struggling against the limits of meaning and attempting, for once, to bring all things into its purview: “A perspective is by nature limited. It offers us one single vision of a landscape. Only when complementary views of the same reality combine are we capable of achieving fuller access to the knowledge of things. The more complex the object we are attempting to apprehend, the more important it is to have different sets of eyes, so that these rays of light converge and we can see the One through the many. That is the nature of true vision: it brings together already known points of view and shows others hitherto unknown, allowing us to understand that all are, in actuality, part of the same thing.”
He lived like a hermit, reading, meditating and writing. In 1988, he nearly died of starvation. He had come to identify completely with the French mystic Marthe Robin, who was afflicted with stigmata and survived for five decades eating nothing but consecrated wafers. Grothendieck tried to surpass the forty days of fasting Christ spent in the desert, and for months nourished himself on soup from the dandelions he picked in his front garden and around his home. His neighbours, used to seeing him wandering in the street picking these flowers, saved him from death, visiting him with cakes and homemade dishes, and not leaving until he agreed to eat them.
When We Cease to Understand the World Page 6