Was this reclusive mathematician a genius whose ideas could transform AI, or a madman?

In September 2014, in a village on the slopes of the Pyrenees, Jean-Claude, a gardener in his late 50s, was surprised by the appearance of a neighbor at his gate. They had not spoken for almost 15 years, after an argument over a climbing rose that Jean-Claude wanted to prune. The old man, 86, lived in complete seclusion, tending his garden in his ever-present djellaba and spending his nights writing, oblivious to everyone. But this time, the old man, with his long beard and the air of a man seeking the truth, seemed troubled.

“Can you do me a favor?” he asked Jean-Claude.

“If I can.”

“Can you buy me a revolver?”

Jean-Claude refused. Then he noticed the hermit, almost deaf and almost blind, moving unsteadily around the garden, and called his children. Even they had not spoken to their father for almost 25 years. When they arrived in the village of Lasseur, the hermit again asked for a revolver to commit suicide. His house was in terrible condition, the corridors were filled with shelves with flasks of musty liquids, overgrown plants were growing out of pots everywhere, and thousands of pages of mysterious notes were lying in boxes. But his ailments deprived him of the ability to conduct research, and he no longer saw the point of life. On November 13, he died, exhausted and alone, in a hospital in the nearby town of Saint-Lizier.

Alexander Grothendieck: Genius and Hermit

The man’s name was Alexander Grothendieck. Born in 1928, he arrived in France in 1939 as a refugee from Germany and soon revolutionized postwar mathematics, much as Einstein had transformed physics a generation earlier. Grothendieck transcended individual disciplines such as geometry, algebra, and topology and sought to find a universal language that would unite them all. At the heart of his work was a new conception of space that freed it from the Euclidean tyranny of fixed points and ushered it into a world of relativity and probability. The flood of concepts and tools he introduced in the 1950s and ’60s was admired by his peers.

But in 1970, a turning point occurred: Grothendieck resigned from the French Institute for Higher Scientific Research (IHES) in protest at the funding it received from the Ministry of Defense, ending his career in higher mathematics. He held a series of minor teaching positions until 1991, when he left his home at the foot of Mont Ventoux and disappeared. No one – not friends, not family, not colleagues, not the loved ones who knew him as “Shurik” (his childhood nickname, a Russian diminutive of Alexander) – knew where he was.

Solitude and philosophical reflections

In Lasserre, he lived almost entirely alone, without television, radio, telephone, or Internet. Only a few followers traveled to the village upon learning of his whereabouts; he politely declined to receive most of them. When he did speak, he sometimes mentioned his true friends: plants. Grothendieck believed that wood was conscious. He told Michel Camilleri, a local bookbinder who helped compile his writings, that his kitchen table “knew more about you than you will ever know about yourself.” But these strange preoccupations led him into dark corners of his mind: he told one visitor that there were beings in his house that could harm him.

Legacy and Interest in Grothendieck's Works

Grothendieck tried to erase the traces of his fame, but his genius remained indelible. He appears in one of Cormac McCarthy’s last novels, Stella Maris, as the eminence grise guiding his mentally unstable mathematician hero. The publication of his extensive memoir, Harvesting and Sowing, in 2022 has revived interest in his work. Moreover, there is growing academic and corporate interest in the possible technological applications of his ideas. Chinese telecoms giant Huawei believes his topos concept could be key to creating the next generation of AI, and has hired Fields Medalist Laurent Lafforgue to research the topic. But as his former colleague Pierre Cartier has noted, Grothendieck’s motives were far from mundane:

“Even in the mathematical environment he was not quite at home,” writes Cartier. “He had a monologue, or rather a dialogue, with mathematics and God, which for him were one and the same.”

Secrets of the last years

Were his final 70,000 pages of writing the scribbles of a madman? Or was the hermit from Lasserre making a final attempt to penetrate the secrets of the universe? And what would this man, who rejected the scientific community and modern society, say about his intellectual property now being used by tech corporations?

In a famous passage from Gatherings and Sowings, Grothendieck writes that most mathematicians work within an existing framework:

“They are like the heirs of a large and beautiful house, ready for occupancy, with all its parlors, kitchens and workshops, with all the tools and utensils needed for cooking and crafts.” But he considered himself a rarer type: builders, “whose instinctive vocation and joy is the building of new houses.”

Legacy and reflections of a son

Now his son, Mathieu Grothendieck, is wondering what to do with his father’s house. Lassere sits on a hilltop 35km from the Spanish border, in the remote Ariège department, a haven for misfits, vagabonds and utopians. On a cold morning in January 2023, I climbed up to the village, shrouded in mist, amid oak and beech forests, with red kites soaring over the fields. Grothendieck’s house, the only two-storey house in Lassere, sits on the southern edge of the village.

Mathieu, dressed in a dressing gown, opens the door with the air of a man who has just woken from hibernation. He is 57 years old, his face is lined, his nose resembles the keel of a ship. The legacy of the house in which his father suffered such mental anguish weighs on him.

“This place has a history that's bigger than me,” he says, his voice softened by smoking. “And because I don't have the means to fix it up, I feel bad. I feel like I'm still living in my father's house.”

A former ceramicist, Mathieu now works part-time as a musician. In the kitchen, a long scroll of Chinese calligraphy sits next to photographs of a Buddha statue and portraits of his mother, Mireille Dufour, whom Grothendieck left in 1970. (Mathieu is her youngest son; he has a sister, Jeanne, and a brother, Alexandre. Grothendieck also had two other sons, Serge and John, by two other women.) Above Mathieu’s bed hangs a vivid portrait of his paternal grandfather, Alexander Shapiro, a Ukrainian Jewish anarchist who lost an arm escaping a czarist prison and later fought in the Spanish Civil War.

A Legacy of Pain and Genius

Even with his great wisdom and insight, there was always a sense of excess in my father. He was always eager to put himself in dangerous positions. Schapiro and his companion, the German writer Johanna Grothendieck, left five-year-old Alexander with a foster family in Hamburg when they fled Nazi Germany in 1933 to fight for socialist ideals in Europe. He was reunited with his mother in 1939 and spent the rest of the war either in an internment camp in France or in hiding. His Jewish father, detained separately, was sent to Auschwitz, where he was murdered upon arrival in 1942. It was this legacy of abandonment, poverty, and violence that, in Mathieu’s view, made Grothendieck a genius, but also ultimately destroyed him. “Artists and geniuses compensate for their shortcomings and wounds. The wound that made Shurik a genius caught up with him at the end of his life,” Mathieu says.

Mathieu takes me into a huge, ramshackle barn behind the house. On the dirt floor lies a pile of glass flasks, encased in wicker baskets: they contain the remains of the mathematician’s herbal infusions, which required thousands of litres of alcohol. Far from traditional mathematics, Grothendieck’s final research focused on the question of why there is evil in the world. His last written work was a notebook with the names of deportees in one of his father’s convoys, in August 1942. Mathieu believes that his father’s infusions were connected to an attempt to explain the nature of evil: a kind of alchemy through which he tried to identify the properties of different species, their resistance to adversity and aggression.

“It's hard to understand,” admits Mathieu. “All I know is that these tinctures were not meant to be drunk.”

After our conversation, Mathieu agreed to show me his father’s manuscripts stored at Lasserre, which his daughter had scanned onto a hard drive. At the beginning of 2023, the family was still negotiating the transfer of these documents to the French National Library. They have now been accepted and will eventually be made available to researchers. However, serious scholarly work is needed to assess their significance in mathematics, philosophy and literature. I am clearly not in a position to judge the first of these areas.

Opening to the first page I came across, I saw legible but nervous handwriting, the occasional colorful topological diagram, references to past thinkers, often physicists—Maxwell, Planck, Einstein—and frequent references to Satan and “this damned world.” His children, too, are trying to make sense of this grand legacy.

“It's mystical, but at the same time down to earth. He talks about life with a certain moralism. It's completely out of touch with the times,” says Mathieu. “But I think there are gems in it. He was the king of formulating ideas.”

After a couple of hours of reading, my head was spinning and I felt like the abyss was looking at me. One can only imagine what it must have been like for Grothendieck himself. According to Mathieu, a friend once asked his father what his greatest dream was. The mathematician replied: “So that this hellish circle of thoughts would finally end.”

Paths to Mysticism

The southern slope of Mont Ventoux in France was shrouded in cloud shadows in April as cyclists rounded the summit. In the Vaucluse department of Provence, this is where Alexander Grothendieck began his first steps into mysticism. His other son, Alexander, now lives in the area. I walk along a bumpy road to meet the 62-year-old, who emerges from an oak forest with a smile. In a threadbare sweater, dark trousers and slippers, Alexander looks slimmer than his brother, his cheeks wind-beaten.

He takes me into the huge hangar where he lives. Amplifiers and musical instruments are piled up here; in the back is a workshop where he makes kalimbas, African keyboard instruments. In 1980, his father moved a few kilometers west, to a house outside the village of Mormoiron. In the years that followed, Grothendieck's thoughts became increasingly mystical. “Even with all his wisdom and depth of understanding, there was always a sense of excess in my father,” says Alexander.

Grothendieck left the commune he had been part of since 1973 for a village north of Montpellier, where he continued to teach at the university. From 1970 onwards, he became one of the first radical ecologists in France and became increasingly interested in meditation. In 1979, he spent a year poring over his parents’ letters, which finally dispelled any romantic illusions about them.

“The myth of their great love collapsed for Shurik – it was a pure illusion,” says Zhanna Grothendieck, the granddaughter of the writer Johanna. “He was able to decipher all the traumatic elements of his childhood. He understood that he was simply abandoned by his own mother.”

This preoccupation with the past intensified in the mid-1980s, when Grothendieck was working on the manuscript of “Harvests and Sowings.” It was a meditation on his mathematical career, filled with stunning aphoristic insights, like the metaphor of home. But the work, replete with David Foster Wallace-style footnotes, was relentless and overwhelming, permeated with a sense of betrayal by his former colleagues. After his revelations about his parents, this feeling became a kind of guiding principle.

“It was systematic with my dad – to put someone on a pedestal to see their flaws. And then – bam! – they would go up in flames,” Alexander says.

During this period, Alexander Grothendieck, although still working in mathematics, became increasingly interested in mysticism. He began to view his dreams not as simple products of his own mind, but as divine messages from a being he called “the Dreamer.” This being was for him a synonym for God, a kind of cosmic mother.

“Like the mother's breast, the 'great dream' offers us thick and rich milk, good for nourishing and strengthening the soul,” he later wrote in a treatise on the subject called The Key of Dreams.

Pierre Deligne, his brilliant student whom Grothendieck accused of betrayal in his memoirs, “Gatherings and Sowings,” believed that his old teacher had lost his clarity of thought. “He was no longer the Grothendieck I admired,” he said by telephone from the Institute for Advanced Study in Princeton.

Complete isolation

By this time, Grothendieck was completely isolated. He had severed ties with nature and everyone around him. By the summer of 1989, his prophetic dreams had intensified and evolved into daily “audiences” with an angel he called either Flora or Lucifer, depending on whether she was benevolent or tormenting. This angel instructed him in a new cosmology that centered on suffering and evil in God's design. Grothendieck, for example, believed that the speed of light, close to but not exactly 300,000 km/s, was evidence of Satan's intervention.

“He was in a state of mystical delirium,” says Jean Malgoire, another former student and now a professor at Montpellier. “That's also a form of mental illness. It would have been good if he had been examined by a psychiatrist at the time.”

In real life, Grothendieck became increasingly distant and fearful. His son Matthieu spent two months in Mormoiron working on his father's house, but during that time Grothendieck invited him inside only once. Matthieu lost his temper and snapped: “He had lost interest in other people. I no longer sensed any real or sincere empathy in him.” Despite this, Grothendieck remained interested in the souls of people. On January 26, 1990, he sent a messianic seven-page letter to 250 of his acquaintances, including his children, called the “Good News Letter.” In it, he announced the date of October 14, 1996, as “Liberation Day,” when evil on Earth would cease to exist, and said that they had been chosen to help usher in this new era.

In June 1990, as if to strengthen his spiritual commitment, Grothendieck fasted for 45 days (he wanted to surpass Christ's 40-day fast), cooling himself in the summer heat in a wine barrel filled with water. Seeing his father shrink to the emaciated state of Nazi concentration camp prisoners, Alexander Jr. realized that perhaps his father was imitating someone else:

“In a sense, he was reuniting with his father.”

Grothendieck almost died. He only broke his fast after Johanna’s partner convinced him to eat again. She believes that the fast caused irreversible damage to her father’s brain at the cellular level, causing him to lose his ability to think rationally. Shortly after, Grothendieck summoned Malgoire to Mormoiron to hand over 28,000 pages of mathematical manuscripts (now available online). He showed his student a metal barrel full of ashes: the remains of a large number of personal papers, including letters from his parents, which he had burned. The past had lost its meaning for him, and now Grothendieck could only look forward. A year later, without warning, he left his home, choosing a path known only to him.

The last days

A round black sandstone slab made by Johanna and now covered with wild roses marks Grothendieck's burial place in the Lasserre cemetery. It is almost hidden behind a telegraph pole. The mathematician was alone at the time of his death in the hospital; after a few weeks with his children, he rejected them again, accepting help only from intermediaries.

The presence of his family seemed to evoke unbearable feelings in him. In his letters, he judged the people in his life by how much they were under Satan's influence. But as Alexander points out, this was also a reflection of his own troubled subconscious:

“He didn't like what he saw in the mirror we showed him.”

Towards the end of his life, Alexander Grothendieck, with a long gray beard and glasses, lived in seclusion in Lasserre, France. In one of the last photographs taken by his son Mathieu, Grothendieck lies in bed, a knitted cap on his head, with a portrait of his father hanging on the wall behind him. The photograph captures a man who in his final years was increasingly withdrawing into himself and cutting off ties with the outside world.

Grothendieck’s residence in Lasserre was discovered by chance in the late 1990s when his son Alexandre signed up for car insurance and discovered that the insurance company already had an address in his father’s name. He tried to contact Grothendieck and saw him one day at a market in the town of Saint-Girons, south of Lasserre. “He saw me and smiled widely, he was very happy,” Alexandre says. But a minute later, Grothendieck changed, his smile disappeared, and he clearly decided that he should have nothing to do with his son. That moment was “an ice shower” for Alexandre. He did not see his father again until his death in 2014.

Until the early 2000s, Grothendieck continued to work at an intense pace, often spending nights at his kitchen table writing down his “meditations.” However, by the mid-2000s, his writing activity began to wane. According to his son Mathieu, in his later writings, Grothendieck meticulously recorded his every action, as if the smallest details of his life had profound significance. These texts were so painful for Mathieu that he decided not to donate them to the National Library.

In his later years, Grothendieck suffered from an “uncontrollable aversion” to his work, which he attributed to malign forces but which may have been linked to depression. In 1997, he wrote, “The most terrible thing about the victims is that Satan has dominion over their thoughts and feelings.” He even considered suicide, but ultimately decided to continue living as a self-proclaimed victim.

Grothendieck also tried to get rid of his house in Lasserre, offering it for free to his bookbinder, Michel Camilleri, on the condition that he take care of his plants. When Camilleri refused, Grothendieck was outraged, seeing in this again “the hand of Satan.” In 2001, the house almost burned down when the chimney of an uncleaned stove caught fire. Some witnesses claim that Grothendieck tried to prevent firefighters from entering the property, although Mathieu does not believe this.

As his condition worsened, Grothendieck became increasingly withdrawn. Attempts to engage him in the local community ended in biblical quotations and accusations. By the mid-2000s, Grothendieck was completely lost in the labyrinths of his own mind.

Grothendieck's Legacy

In mid-April, elegant Parisians emerge from the polished lobby of a renovated hotel in the 7th arrondissement, heading out for lunch. The building once broadcast France’s first television programs; now Huawei is seeking to make a similar move in artificial intelligence. It has created the Centre Lagrange, a cutting-edge mathematics research institute, on the site, and has hired leading French mathematicians, including Laurent Lafforgue, to work there. An aura of secrecy surrounds their work, raising suspicions in the West about Chinese technology. Huawei initially declined to answer any questions, later agreeing to a few email responses.

Of particular interest to Huawei is the concept of topos, developed by Alexander Grothendieck in the 1960s. Among his concepts, topos was the furthest step in his quest to identify deeper algebraic values ​​at the heart of mathematical space and thus create a geometry without fixed points. Grothendieck described toposes as a “vast and calm river” from which fundamental mathematical truths can be extracted. Olivia Caramello sees them more as “bridges” that can facilitate the transfer of information between different fields. Lafforgue confirms that Huawei is exploring the application of topos in a number of fields, including telecommunications and artificial intelligence.

Caramello describes topoi as a mathematical embodiment of the idea of ​​vision; an integration of all possible perspectives on a given mathematical situation that reveals its most essential features. Applying topoi to AI could allow computers to go beyond the data associated with, say, an apple; the geometric coordinates of its image or metadata. AI could then begin to recognize objects the way we do, through a deeper “semantic” understanding of what an apple is. However, Lafforgue says practical applications for creating the next generation of “thinking” AI are still some way off.

There’s a broader question about whether Grothendieck himself would have wanted this. In 1972, during his ecological period, he expressed concern that capitalist society was leading humanity to ruin, giving a lecture at CERN titled “Can We Continue Scientific Research?” He didn’t know about AI, but was already against the merger of academia and corporate industry. Given his pacifist values, he would likely also have been against Huawei using his work; Huawei’s CEO, Ren Zhengfei, is a former member of the People’s Liberation Army’s engineering corps. The U.S. Department of Defense and some independent researchers believe Huawei is controlled by the Chinese military.

Huawei insists it is a private company owned by its employees and founder Ren Zhengfei, and that it is “not owned, controlled or affiliated with any government or third party.”

Lafforgue notes that France's IHES, where Grothendieck and later himself worked, was funded by industry — and considers Huawei's interest legitimate. Caramello, founder and president of the Grothendieck Institute, a research organization, says he would like to see his concepts systematically studied for implementation.

“Topos theory itself is a kind of machine that can expand our imagination,” she says. “So you see, Grothendieck was not against using machines. He was against blind machines or brute force.”

What is alarming, however, is the level of opacity around Huawei’s AI goals and collaborations, including its links to the Grothendieck Institute, where Lafforgue sits on the scientific board. But Caramello stresses that it is a completely independent organization, focused on theoretical rather than applied research, and makes its findings publicly available. She says the institute does not research AI, and that Lafforgue’s involvement is purely due to his expertise in Grothendieck mathematics.

Matthieu Grothendieck is certain that his father would not have approved of Huawei or any other corporation using his work: “No. I’m not even asking. I know.” There is no doubt that the mathematician considered modern science to be morally degenerate, and the Lasserre papers attempted to reconcile it with metaphysics and moral philosophy. Unlike Grothendieck’s mysticism of the 1980s, there is structure and intent here. They begin with just under 5,000 pages devoted to the Schematics of Elementary Geometry and the Structure of the Psyche. According to mathematician Georges Maltsinotis, who has examined this section, these parts contain mathematics in “the proper and correct form.” Grothendieck then embarks on the Problem of Evil, which stretches over 14,000 pages written during the 1990s.

Judging from the 200 or so pages I am trying to decipher, Grothendieck put a Herculean effort into his new cosmology. He seems to have tried to understand the workings of evil at the level of matter and energy. He argues with Einstein, James Clerk Maxwell, and Darwin, especially about the role of chance in what he saw as a divinely created universe. There are numerological musings on the significance of lunar and solar cycles, the nine-month gestation period. He renames the months in the new calendar: January becomes Roma, August Songa.

How much of this work is significant, and how much is it an idle mania? According to Pierre Delig, Grothendieck became fatally lost in his solitude. He says he had little interest in reading Lasserre's writings “because he had little contact with other mathematicians. He was limited to his own ideas rather than using those of others.” But for others the situation is less clear-cut, including Caravello. In her view, this combination of mathematics and metaphysics accurately reflects his borderline intelligence and can lead to unexpected insights: she points to the use of tools Hopf fibrations to explain psychological phenomena in the “Structure of the Psyche”.

“We are at the very beginning of a huge study of these manuscripts. And there will certainly be miracles in them,” she says.

Grothendieck remained obsessed with evil until the very end. Perhaps broken by his traumas, he could not allow himself to forgive and imagine a kinder world. But his children, despite the long estrangement, did not become the same. Matthieu rejects the idea that his father repeated on them the abandonment he himself had experienced as a child: “We were adults, so it was nothing compared to what he went through. He coped much better than his parents.”

The rejection of his children hurt Joanna, but she understands that there was something fundamentally broken about her father. “In his mind, I don’t think he abandoned us. We existed in a parallel reality for him. The fact that he burned his parents’ letters was extremely telling: he didn’t feel part of the family chain.” Remarkably, the trio do not judge their father and openly discuss his ordeal.

“We accept it,” Alexander says. “It was the challenge he was looking for.”

All this and much more — TG “Mathematics is not for everyone”