According to Schwarzschild, the most frightful thing about mass at its most extreme degree of concentration was not the way it altered the form of space, or the strange effects it exerted on time: the true horror, he said, was that the singularity was a blind spot, fundamentally unknowable. Light could never escape from it, so our eyes were incapable of seeing it. Nor could our minds grasp it, because at the singularity the laws of general relativity simply broke down. Physics no longer had any meaning.
Courant listened to him, rapt. Just before the doctors came looking for him and the young man rejoined the convoy that would take him back to Berlin, Schwarzschild asked him a question that tormented him for the rest of his life, though at the time he considered it nothing but the ravings of a dying soldier, product of a creeping madness that had overtaken Schwarzschild’s mind as weariness and despair consumed him.
If matter were prone to birthing monsters of this kind, Schwarzschild asked with a trembling voice, were there correlations with the human psyche? Could a sufficient concentration of human will—millions of people exploited for a single end with their minds compressed into the same psychic space—unleash something comparable to the singularity? Schwarzschild was convinced that such a thing was not only possible, but was actually taking place in the Fatherland. Courant tried to appease him; he said that he saw no signs of the apocalypse Schwarzschild feared, and that surely there could be nothing worse than the war they were mired in. He reminded Schwarzschild that the human soul was a greater mystery than any mathematical enigma, and that it was unwise to project the findings of physics into such far-flung realms as psychology. But Schwarzschild was inconsolable. He babbled about a black sun dawning over the horizon, capable of engulfing the entire world, and he lamented that there was nothing we could do about it. Because the singularity sent forth no warnings. The point of no return—the limit past which one fell prey to its unforgiving pull—had no sign or demarcation. Whoever crossed it was beyond hope. Their destiny was set, as all possible trajectories led irrevocably to the singularity. And if such was the nature of that threshold, Schwarzschild asked, his eyes shot through with blood, how would we know if we had already crossed it?
Courant left to return to Germany. Schwarzschild died that afternoon.
* * *
More than two decades would pass before the scientific community would accept Schwarzschild’s ideas as an inevitable consequence of the theory of relativity.
Einstein himself was the one who fought hardest to exorcise the demon Schwarzschild had invoked. In 1939, he published his article “On a Stationary System with Spherical Symmetry Consisting of Many Gravitational Masses”, which explained why singularities such as those Schwarzschild described could not exist. “The singularity does not appear for the simple reason that matter cannot be concentrated arbitrarily. And this is due to the fact that otherwise the constituting particles would reach the speed of light.” With characteristic intelligence, the German-born physicist had appealed to the internal logic of his theory to patch the rent in the fabric of space-time, protecting the universe from a catastrophic gravitational collapse.
But the calculations of the greatest mind of the twentieth century were wrong.
On September 1, 1939—the same day the Nazi tanks crossed the Polish frontier—Robert Oppenheimer and Hartland Snyder published an article in Volume 56 of the Physical Review. In it, the two physicists from North America demonstrated beyond all shadow of a doubt that “When all thermonuclear sources of energy are exhausted a sufficiently heavy star will collapse. Unless it reduces its mass due to fission, rotation or radiation, this contraction will continue indefinitely,” forming the black hole that Schwarzschild had prophesied, capable of crumpling space like a piece of paper and extinguishing time like a blown-out candle, and no natural law or physical force could avert it.
THE HEART OF THE HEART
On the morning of August 31, 2012, the Japanese mathematician Shinichi Mochizuki published four articles on his blog. Those six hundred pages contained a proof of one of the most important conjectures in number theory, known as a + b = c.
To this day, no one has managed to comprehend it.
Mochizuki had worked for years in complete isolation, developing a mathematical theory that bore no resemblance to any that had been known before.
He gave his discovery no publicity after uploading it to his blog, neither sending it to specialized publications nor presenting it at conferences. One of the first people to become aware of its existence was Akio Tamagawa, his colleague at the Research Institute for Mathematical Sciences at the University of Kyoto, who sent the articles to Ivan Fesenko, a number theorist at the University of Nottingham, attached to an email with a single question:
“Has Mochizuki solved a + b = c?”
Fesenko could hardly contain his impatience as he saved the four huge files to his computer. He spent ten minutes watching the download bar, and then shut himself away for two weeks to study the proof, ordering takeaway meals and sleeping only when exhaustion demanded it. He responded to Tamagawa with three words:
“Impossible to understand.”
In December 2013, a year after Mochizuki published his articles, several of the most prominent mathematicians in the world gathered in Oxford to study the proof. Enthusiasm reigned during the early days of the seminar. The obscure reasoning of the Japanese mathematician had begun to yield to comprehension, and, on the third night, the rumour that a major step forward was about to take place spread on internet forums and specialized websites.
On the fourth day, everything collapsed.
After a certain point, no one was capable of following the proof’s arguments any further. The greatest mathematical minds on the planet were baffled, and there was nobody who could help them, since Mochizuki himself had refused to participate in the meeting.
The new branch of mathematics that Shinichi Mochizuki had created to prove the conjecture was so bizarre, abstract and ahead of its time that a theoretician from the University of Wisconsin–Madison said that he felt as though he was studying a paper from the future: “Everybody who I’m aware of who’s come close to this stuff is quite reasonable, but afterwards they become incapable of communicating it.”
The few who have been able to follow Mochizuki’s system sufficiently to understand it in part say it consists of a series of underlying relationships between numbers that are invisible at first sight. “If researchers want to apprehend my work, they must first deactivate the thought patterns that they have installed in their brains and taken for granted for so many years,” Mochizuki wrote on his blog.
He was born in Tokyo, and from a young age was famous for his powers of concentration, which his peers have described as beyond human. As a child, he suffered attacks of muteness that became increasingly intense during his adolescence, to such an extent that hearing him talk became a rarity. Nor could he stand other people’s gazes, and would walk with his eyes pinned to the ground, a habit which gave his back a slight hump that did not, however, diminish his good looks, as his high forehead, enormous glasses and dark, slicked-back hair gave him an uncanny resemblance to Superman’s alter ego, Clark Kent.
He was sixteen when he entered Princeton, and by twenty-three he already had his doctorate. After spending two years at Harvard, he moved back to Japan, accepting a post as professor at the Research Institute for Mathematical Sciences at the University of Kyoto on condition that he be permitted to devote himself exclusively to research, with no obligation to teach classes. At the beginning of the 2000s, he stopped attending international conferences. Over the following years, his life became increasingly constricted. First he limited himself to travelling within Japan, then he no longer ventured beyond the Kyoto prefecture, and finally his range was confined to the narrow circuit between his apartment and his tiny office at the university.
From the window of that office, as orderly as the interior of a temple, there is a view of Mount Daimonji, where once a year, during th
e O-Bon festival, the monks burn a giant kanji in the form of a man with outstretched arms – – a sign which means enormous/tall/monumental, and is used to express extreme grandiloquence, similar to that employed by Mochizuki in christening his new branch of mathematics, which he called, without a trace of either modesty or irony, Inter-Universal Teichmüller Theory.
The a + b = c conjecture reaches down to the roots of mathematics. It proposes a deep and unexpected relationship between the additive and multiplicative properties of numbers. If it is proven, it will become a formidable tool capable of dispelling, as if by magic, a vast quantity of long-standing enigmas. But Mochizuki’s ambition was even greater than that; he did not stop at verifying the conjecture, but even invented a novel type of geometry, one that required mathematicians to conceive of numbers in a radically different way. According to Yuichiro Yamashita, one of the few who claims to have grasped the real scope of his Inter-Universal Theory, Mochizuki has created a complete universe, of which, for the moment, he is the sole inhabitant.
But extraordinary claims require extraordinary evidence, and Mochizuki’s refusal to give interviews, defend his results in person, or even discuss his work in a language other than Japanese, made his peers suspicious. To some it was all an elaborate hoax, while others said that he suffered from an acute psychological imbalance, and offered as proof his growing social phobia and the isolation he worked in.
Things seemed to improve in 2014, when he announced that he would travel to France in November to present his theory during a six-day seminar at the University of Montpellier. Mathematicians from all over the world fought for a seat and even lined the stairs on both sides of the lecture room, but Mochizuki stood them all up. He had arrived a week in advance, suffered the overbearing attentions of the rector of the university, and then disappeared. No one knew where he was till the day before his talks were slated to begin, when the guards ejected him from the campus after a confusing incident.
As soon as he returned to Japan, Mochizuki removed the proof from his blog and threatened legal action against anyone who tried to publish it. He was subject to a wave of attacks from his most hardened critics, while his colleagues assumed that he had discovered a fundamental flaw in the logic of his proof. Mochizuki denied this, but offered no explanations. He renounced his post at the University of Kyoto, and wrote a last entry before shutting down his blog, stating that, in mathematics, certain things should remain hidden, “for the good of all of us”. This incomprehensible and apparently capricious gesture only confirmed what many had feared: Mochizuki had succumbed to Grothendieck’s curse.
Alexander Grothendieck was one of the most important mathematicians of the twentieth century. During a creative outburst almost unparalleled in the history of science, he revolutionized our understanding of space and time, not just once, but twice. Mochizuki’s own fame dated from 1996, when he managed to prove one of Grothendieck’s conjectures, and everyone who met the Japanese mathematician while he was still a student knew that he regarded Grothendieck as his master.
Required reading for mathematicians all over the world, Grothendieck had led a team that had produced tens of thousands of pages, a colossal, intimidating oeuvre into which most undergraduates only dipped their toes to learn what was necessary to advance in their own fields, but even that could take years. Mochizuki, on the other hand, began reading the first volume of Grothendieck’s collected works as a freshman and did not stop until he had finished the last.
Minhyong Kim, Mochizuki’s roommate at Princeton, remembers having found him delirious at midnight after days without sleep or food. Exhausted and dehydrated, he babbled incoherently, his pupils as wide as an owl’s. He spoke of the “heart of the heart”, an entity Grothendieck had discovered at the very centre of mathematics, which had completely unhinged him. The next morning, when Kim asked for an explanation, Mochizuki stood and stared at him. He had no memory of the night before.
* * *
Between 1958 and 1973, Alexander Grothendieck towered over mathematics like a veritable colossus, convincing the finest minds of his generation to put aside their own research projects and ambitions and join his radical quest to unearth the structures underlying all mathematical objects.
His method of working was extraordinary. Even though he was able to solve three of the four Weil conjectures, the greatest mathematical enigmas of his time, Grothendieck was not interested in deciphering famous problems or reaching seemingly unthinkable results; his desire was to achieve an absolute understanding of the foundations of mathematics. To do so, he constructed intricate theoretical architectures around the simplest of questions, encircling them with a vast array of new concepts. Under the soft, continuous pressure of Grothendieck’s reasoning, solutions seemed to reveal themselves of their own free will, welling up to the surface, and opening, as he once said, “like a nutshell that had spent months submerged in water”.
His was the power of unbridled abstraction. He championed an approach that was based on wild generalizations, zooming further and further out and then focusing sharply, as any dilemma became clear when one viewed it from a sufficient distance. Numbers, angles, curves and equations did not interest him, nor did any other mathematical object in particular: all that he cared for was the relationship between them. “He had an extraordinary sensitivity to the harmony of things,” one of his disciples, Luc Illusie, recalled. “Not only did he introduce new techniques and prove major theorems: he changed the way we think about mathematics.”
Space was his lifelong obsession. One of his greatest strokes of genius was expanding the notion of the point. Beneath his gaze, the humble dot was no longer a dimensionless position; it swelled with a complex inner structure. Where others had seen a simple locus without depth, size or breadth, Grothendieck saw an entire universe. No one had proposed something so bold since Euclid.
For years, he devoted the whole of his energy, twelve hours a day, seven days a week, to mathematics. He did not read newspapers, watch television or go to the cinema. He liked ugly women, squalid apartments, dilapidated rooms. He worked cloistered in a cold office with flaking paint falling from the walls, his back turned to the only window, with pen and paper on his desk and only four objects as decoration: his mother’s death mask, a small wire sculpture of a goat, a jar of Spanish olives, and a charcoal portrait of his father, drawn in Le Vernet concentration camp.
Alexander Schapiro, Alexander Tanaroff, Sascha, Piotr, Sergei. His father’s real name has not come down to us, as he lived under multiple aliases while participating in the anarchist movements that rocked Europe at the beginning of the twentieth century. A Ukrainian from a Hasidic family, he was arrested by the Tsarist police at fifteen and sentenced to death along with his comrades. He was the only one to survive. For three weeks they dragged him from his cell to the execution ground, where he watched his friends die, one by one, before the firing squad. He was spared the death penalty on account of his age, condemned to life in prison, and freed ten years later, during the Russian Revolution of 1917. He dove head first into a series of conspiracies, secret plots and internal skirmishes between opposing revolutionary factions that cost him his left arm, though it remains unclear whether this was the result of combat, a frustrated assassination attempt, a failed suicide or a bomb that went off in his hand while he was still carrying it. He made his living as a street photographer. In Berlin, he met Grothendieck's mother, and the two of them moved to Paris in 1939. In 1940, he was arrested by the Vichy government and interned in Le Vernet. He was deported to Germany in 1942, and died of Zyklon B poisoning in one of the gas chambers at Auschwitz.
His son took the last name of his mother, Johanna Grothendieck, a woman who wrote her entire life, though she never managed to publish her novels or poems. When she met Grothendieck’s father, she was married and working as a journalist for a leftist daily. She abandoned her husband and joined the revolutionary struggle with her new lover. When Grothendieck was five, his mother left him in the hands of
a Protestant pastor to travel to Spain and fight with the anarchists against Franco in defence of the Second Republic. After the Republican defeat, she took refuge with her husband in France and from there sent for her son. Johanna and Grothendieck were declared “undesirables” by the French government and sent, along with other “suspect foreign elements” who had formed part of the International Brigades, and other refugees from the Spanish Civil War, to the Rieucros camp, close to Mende, where Johanna contracted tuberculosis. By the time the war ended, Grothendieck was seventeen. He survived with his mother in extreme poverty picking grapes on the outskirts of Montpellier, the city where he began his studies. Mother and son had a close, aberrant relationship. Johanna died in 1957, of a relapse of tuberculosis.
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”.
When We Cease to Understand the World Page 5