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Iron, gold, silver, copper, tin, lead, phosphorus, arsenic; at the beginning of the eighteenth century, mankind was aware of only a handful of pure elements. Chemistry had not yet branched away from alchemy,
If arsenic is a patient assassin, hiding out in the most recondite of the body’s tissues and accumulating there for years, cyanide takes your breath away. In sufficient concentrations, it stimulates the carotid body’s receptors all at once, triggering a reflex that cuts off respiration. Medical literature calls this the audible gasp that precedes tachycardia, apnoea, convulsions and cardiovascular collapse.
Alan Turing, the genius mathematician and father of computing, killed himself by biting into an apple injected with cyanide.
But the apple was never examined to confirm the hypothesis of suicide (even if the seeds do contain a natural form of it, with only half a cup of them sufficient to kill a human being),
morning of Thursday, April 22, 1915, the soldiers saw an enormous greenish cloud creeping towards them across no-man’s-land. Twice as high as a man and as dense as winter fog, it stretched from one end of the horizon to the other, as far as the eye could see. The leaves withered on the trees as it passed, birds fell dead from the sky; it tinged the pastureland a sickly metallic colour. A scent like pineapple and bleach filled the throats of the soldiers when the gas reacted with the mucus in their lungs, forming hydrochloric acid. As the cloud pooled in the trenches, hundreds of men fell to
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The horses, still in the stables, cows, chickens, everything, all were dead. Everything, even the insects were dead.”
The Haber–Bosch process is the most important chemical discovery of the twentieth century. By doubling the amount of disposable nitrogen, it provoked the demographic explosion that took the human population from 1.6 to 7 billion in fewer than one hundred years. Today, nearly fifty per cent of the nitrogen atoms in our bodies are artificially created, and more than half the world population depends on foodstuffs fertilized thanks to Haber’s invention.
Schwarzschild’s Singularity
an artillery unit on the Russian front, nor because of his friend’s cryptic warnings about a coming catastrophe, but due to what was written on the back of the page: in a minuscule hand that Einstein could only decipher with a magnifying glass, Schwarzschild had sent him the first exact solution to the equations of general relativity.
Schwarzschild’s results. They worked flawlessly for an ordinary star, around which space curved softly, just as Einstein predicted, and the body of the star remained suspended in the centre of a depression, like a pair of children resting inside a cloth hammock. The problem arose when too much mass was concentrated in a very small area, as occurs when a giant star exhausts its fuel and begins to collapse. According to Schwarzschild’s calculations, in such a case, space-time would not simply bend; it would tear apart. The star would go on compressing and its density would increase till the
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Initially, even Schwarzschild cast this result aside as a mathematical anomaly. After all, physics is rife with infinities that are nothing more than numbers on paper, abstractions that do not represent real-world objects, or that simply indicate calculating errors. The singularity in his metrics was undoubtedly one of these: a mistake, an oddity, a metaphysical delirium.
For Schwarzschild, that such a thing could exist in the universe was inconceivable. Not only did it defy common sense and cast doubts on general relativity, it threatened the very foundations of physics, as, within the singularity, the notions of space and time themselves became meaningless. Schwarzschild attempted to find a logical solution to the paradox he had created. Did the fault lie in his conceit? Had he simply been too clever for
His singularity, he told himself, was nothing but an imaginary monster. A paper tiger, a Chinese dragon.
His imagination had fallen prey to the pull of his discovery: with alarm, he realized that if his singularity were ever to exist, it would endure until the end of the universe. Its ideal conditions made it an eternal object that would neither grow nor diminish, but remain eternally as it was. Unlike all other things, it was immune to becoming and doubly inescapable: in the strange spatial geometry it generated, the singularity was located at both ends of time: one could flee from it into the remotest past or escape to the furthest future only to encounter it once more. In the last letter he
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Schwarzschild tried to explain to them that the nature of the moon had not changed; it was an optical illusion, caused by a thin layer of clouds crossing in front of it, making it seem larger and faster than it was. He spoke to them with the same tenderness he would have used with his children, but he did not manage to convince them. Nor could he himself shake off the feeling that everything had begun to move more swiftly since the start of the war, as though the world was slipping off a precipice.
Before witnessing the cruel reality of combat and suffering the horror of modern war in his own flesh, Schwarzschild had found the camaraderie of the army rejuvenating. When his unit was first deployed, he discovered a system for perfecting the sights on the tanks—without anyone having requested that he do so—tinkering with them in his free time with the same eagerness with which he had built his first telescope, as if the drills and simulations of his months of training had rekindled in him the boundless curiosity he had known during childhood.
He was convinced that mathematics, physics and astronomy constituted a single body of knowledge and believed that Germany was capable of exercising a civilizing force comparable to that of ancient Greece. To do so, however, its science must be raised to the heights already achieved by its philosophy and art, for “only a vision of the whole, like that of a saint, a madman or a mystic, will permit us to decipher the true organizing principles of the universe.”
In 1905, he travelled to Algeria to observe a total eclipse, but he failed to heed the maximum exposure time and damaged the cornea of his left eye. When they removed the patch he had been forced to wear for weeks, he noticed a shadow the size of a two-mark coin in his visual field, which was apparent even when his eyes were closed.
“Is there anything that is truly at rest, something stationary around which the universe revolves, or is there nothing at all to hold on to amid this endless chain of movements in which every single thing seems bound? Just imagine how far we have fallen into uncertainty if the human imagination cannot find a single place to lay its anchor, if not a single stone in the world has the right to be considered immobile!”
ouroboros:
In his diary, and in the letters he sent to his wife, his patriotic zeal gives way to bitter complaints about the meaninglessness of the war and a growing contempt for the stupidity of his fellow officers that would only increase as his calculations came closer to the singularity. When he finally reached it, he could think of nothing more: he became so immersed that he failed to take cover during an enemy attack, and a mortar exploded a few metres away from his head. No one understood how he had survived.
Schwarzschild deduced that any object could generate a singularity if its matter were compressed into a sufficiently restricted space: for the sun, three kilometres, for the earth, eight millimetres, and 0.000000000000000000000001 centimetres for the mass of an average human body.
This distortion altered the law of causality; Schwarzschild deduced that if a hypothetical traveller were capable of surviving a journey through this rarefied zone, he would receive light and information from the future, which would allow him to see events that had not yet occurred. If he could reach the centre of the abyss without gravity tearing him apart, he would distinguish two superimposed images projected at once in a small circle over his head, like those that are visible through a kaleidoscope: in one, he would perceive the entire future evolution of the universe at an inconceivable
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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.
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.
“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.
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.
oeuvre
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
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
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“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.
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”
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.
To his wife’s despair, he founded a commune at home, where vagabonds, professors, hippies, pacifists, thieves, nuns and prostitutes dwelt side by side.
“Doing mathematics is like making love,” wrote Grothendieck, whose sexual impulses rivalled his spiritual inclinations.
guards discovered a book of matches in the man’s hand and a small jar of lighter fluid in his pocket, but they did not call the police. They expelled him from the campus, imagining he was insane or suffered from a mental handicap of some kind, because he would not look up from the floor and insisted repeatedly—yet always in the softest of voices—that they must let him go, as he had a very important seminar to present in the Department of Mathematics.
while
Looking at the waves scudding outwards and getting lost on the horizon, he could not help but recall the words of his mentor, the Danish physicist Niels Bohr, who had once told him that a part of eternity lies in reach of those capable of staring, unblinking, at the sea’s deranging expanses.
The physicist—like the poet—should not describe the facts of the world, but rather generate metaphors and mental connections.
trembled in the most innocuous circumstances: pigeons terrified him, he was afraid of cats and dogs, and the sound of our father’s shoes climbing the stairs could send him into a panic.”
“For more than a century, we have divided earthly phenomena into two fields: atoms and particles of solid matter on the one hand, and the intangible waves of light, propagated through the sea of the luminous ether, on the other. But these two systems cannot remain separate; we must bring them together in a single theory that explains their multiple interactions.
grand and capacious,
His answer arrived two days later, and meant the immediate consecration of de Broglie, whose work Einstein saw as the beginning of a new way forward for physics: “He has lifted a corner of the great veil. This is the first weak beam of light to penetrate the dilemma of the quantum world, the most terrible of our generation.”
His teachers predicted a glorious future for him, on account of his irrepressible curiosity and his evident talent in the exact sciences, but several years after graduating, The Schrödinger found himself nothing more than a run-of-the-mill physicist. None of his articles had made a significant contribution to the field. He had no siblings, nor could he have children with Anny, and if he died, his family name would be lost forever. His biological and intellectual sterility made him consider divorce: perhaps he should give up everything, perhaps he should stop drinking and chasing after every
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undulated.
modicum
eclectic
sanatorium,
used to calculate wave mechanics in the sea or the dispersion of sound through the atmosphere; but to apply it to the inner functioning of the atom, to the movement of electrons, Schrödinger had needed to include an imaginary number in his formula: the square root of minus one. In practical terms, this meant that a part of the wave his equation described escaped the three dimensions of space. Its crests and troughs travelled through multiple dimensions in a highly abstract realm that could only be described by pure mathematics. Beautiful as they were, Schrödinger’s waves were not a part of
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