Benjamín Labatut

he was not so much interested in the world as he was invaded by its many forms.

the man who pulled bread from air

He spent nine months there, consumed by hatred, humiliated by the conditions the victorious parties had imposed on his adopted country, and feeling betrayed by the cowardice of the generals, who had surrendered rather than fighting to the last man.

Cantor was born and raised in Russia, a nation whose inhabitants have become famous for their depth of feeling, the intensity of their religious and political beliefs, and a certain inclination for all things tragic,

Meanwhile, Nelly said, artists had already fully embraced it; she believed that the rediscovery of the irrational was the driving force behind all vanguard movements, movements that, even to a lay observer, were evidently suffused with a Faustian, boundless energy, a haste, a tragic fall in which everything was permitted.

Cantor could apparently demonstrate that there were as many points in a one-inch line as there were in all of space. He had taken a giant leap into the unknown and found something unique, something that nobody had ever considered before him, but his critics, who were many and varied, argued that he had simply gone too far. His infinities, while undoubtedly interesting, could never be considered objects for serious mathematical study. However, Cantor was armed with a proof that seemed completely airtight. “I see it, but I don’t believe it!” 79he wrote to a close friend when he had finished it, and his biggest problem, from then on, was that so many others were equally unable to accept this new and confounding article of faith.

” Trying to silence his opponents by making his theory complete, Cantor developed an entire hierarchy of infinities, but he struggled with increasingly strong episodes of uncontrollable mania, attacks that were followed by deep melancholy and the darkest possible depression. These spells became so regular that he was unable to work on 81mathematics; as a substitute, he devoted his boundless manic energy to try to prove that Shakespeare’s plays had actually been written by the English philosopher Francis Bacon, and that Christ was the natural son of Joseph of Arimathea, views that only helped to give more credence to the arguments of those who said that he was slowly going insane. In May 1884, he had a massive mental breakdown and had to be institutionalized at a sanatorium in Halle.

In 1901, Bertrand Russell, one of Europe’s foremost logicians, discovered a fatal paradox in set theory, and it became a veritable obsession for him. It would not let him rest, even when he was sound asleep, because he would dream of it, again and again. To try to excise it, Russell and his colleague Alfred North Whitehead wrote a massive treatise intended to reduce all of mathematics to logic. They did not use axioms as Hilbert and von Neumann did, but an extreme form of logicism: to them, the foundation of mathematics had to be logical, and so they went about it, building mathematics from the ground up. This was not an easy task by any measure: the first seven hundred and sixty-two pages of their gargantuan treatise—Principia Mathematica—were dedicated solely to proving that one plus one equals two, at which point the authors dryly note, “The above proposition is occasionally useful.” Russell’s attempt to establish all of mathematics on logic also failed, and his paradox dreams were replaced by a new and recurring nightmare, which expressed the insecurities he felt regarding the value of his own work: In his reverie, Russell would stride along the halls of an endless library, with spiral staircases winding down into the abyss, and a high vaulted ceiling that rose up to meet the heavens. From where he was standing, he could see a young, gaunt librarian pacing the rows of books with a metal pail, such as one would use to draw water from a well, hanging from his arm, an undying fire burning within it. One by one, he would pick up the volumes from the shelves, open their dust-covered jackets, and flip through their pages, placing them back or tossing them into the bucket, to be consumed by the flames. Russell would watch him advance, knowing, with that certainty that we can only fully 83experience in dreams, that the young man was edging toward the last extant copy of his Principia Mathematica.

When you combined Gödel’s and von Neumann’s ideas, the outcome defied logic itself: from here to eternity, mathematicians would have to choose between accepting terrible paradoxes and contradictions, or work with unverifiable truths. It was an almost intolerable dilemma, but there appeared to be no way around it. Gödel’s logic, however mysterious, was airtight.

Jancsi, as obsessive as he could be, was never one to get stuck on a problem like that; he was not one of those scientists 97who forget to brush their teeth or change their clothes. Quite the contrary, it was always a joy for him to work out his ideas. His intelligence was playful, not tortured, and his insights were usually immediate, practically instantaneous, not labored. But Gödel had broken something in him, so he locked himself up and Mariette would hear him scream in six different languages. When he finally emerged in late November, sporting a patchy beard that she would later make fun of whenever she wanted to humiliate him, he walked straight to the post office to send a letter to Gödel, informing him that he had developed an even more remarkable corollary to his already outstanding theorem: “Using the methods you employed so successfully … I achieved a result that seems to me to be remarkable; namely, I was able to show that the consistency of mathematics is unprovable.” Jancsi had basically turned Gödel’s argument on its head. According to the Austrian, if a system was consistent—free from contradictions—then it would be incomplete, because it would contain verities that could not be proven. Janos, meanwhile, had demonstrated the opposite: if a system was complete—if you could use it to prove every true statement—then it could never be free of contradictions, and so it would remain inconsistent! An incomplete system was not satisfactory, for obvious reasons, but an inconsistent one was much worse, because with it you could prove anything you liked