1There is a city by the cold eastern sea called Königsberg. You have been there before in these books. It was the city of Kant, who walked the same street at the same hour every day and asked what a mind can know.
2In the thirtieth year of the century the masters of number and of thought gathered there for three days, to speak about the ground on which all sure knowledge stands.
3They were in good heart. For a great work was under way, and its captain was old Hilbert of Göttingen, the same who had fought for Emmy Noether. Hilbert's hope was this: to take the whole science of number and set it on a few plain rules, written out like the rules of a game, and then to prove two things about those rules for ever. First, that they could never lead to a contradiction. Second, that every true saying about numbers could be reached from them, step by step, by anyone with patience.
4If that were done, there would be one house of knowledge, at least, with no dark room in it. Nothing in it would be out of reach. Every question would have its answer waiting, and a sure road to it.
5On the third day, when the speeches were over and the masters sat round a table to talk, a thin young man from Vienna asked leave to say something. He was twenty-four. He wore round spectacles and a good suit, and spoke so softly that those at the far end leaned forward.
6He said, in one careful sentence, that he could show there were true sayings about numbers which could never be proved from the rules, whatever rules one chose.
7Then he stopped. Nobody answered. The talk moved on to other things. The account of the meeting that was printed afterward did not speak of it.
8Only one person in the room understood what had happened. He was a Hungarian named von Neumann, the quickest mind of that age. He took the young man by the sleeve afterward and drew him into a corner, and questioned him, and went away pale.
9The young man's name was Kurt Gödel.
10And see how the Weave will sometimes arrange a thing better than a storyteller. The very next day, in that same city, old Hilbert stood up before a great hall and spoke of his faith. No question is beyond us, he said. There is nothing that we shall never know. He ended with six words, and they were sent out over the land by the new wireless, and they are cut now upon his grave:
11"We must know. We will know."
12He did not yet know that the roof had been taken off his house the afternoon before, by a boy speaking under his breath.
13Now hear what the boy had found. It can be told to a child, though it took a strange and patient mind to make it firm.
14There is an old riddle about a man who says, "I am lying." If he is lying, then he tells the truth; and if he tells the truth, he is lying. People had laughed over it for two thousand years and called it a trick of words.
15Gödel saw that it was a key. He found a way to make numbers speak about numbers: to give every saying a number of its own, as every house in a street has one, so that a sum could be made to talk about sums. And with great labour he built, inside the plain science of counting, a saying that said of itself:
16I cannot be proved from these rules.
17Think about that saying. Suppose the rules could prove it. Then they would have proved a false thing, for it says it cannot be proved; and rules that prove false things are worthless. So, if the rules are sound, they cannot prove it. But then what it says is so. It is true. It is true, and it stands outside the fence for ever.
18You may add it to the rules by hand, as a new rule. Then there will be another such saying outside the new fence. And another after that. There is no end to it.
19And he showed a second thing, which was worse for Hilbert. No such set of rules can prove of itself that it is free from contradiction. To be sure of the house, you must stand outside the house.
20So there is no book that holds all truth. It is not that we have been too lazy to write it. It cannot be written. Any net of rules that is fine enough to catch the numbers has a hole in it, and a true fish swims through.
21Some heard this as a sorrow. Be slower than they were. Ask what the other world would have been, the one Hilbert wanted: a finished house, where every answer could be ground out by turning a handle. In that world there would be nothing left to find that a mill could not find. What Gödel showed is that truth is larger than any way of proving it. The mind will always have somewhere further to go. The last door of that house can never be shut from inside.
22A few years after, a runner of long roads, an Englishman of twenty-three named Alan Turing, lay down in a meadow by a river near Cambridge to get his breath. And there he asked the question from the other side: what, exactly, is turning a handle?
23He pictured the simplest clerk there could be: a machine with a long ribbon of paper marked in squares, which could read one square, write one sign, and step left or right, by a short list of orders. He showed that such a machine, given the right list, can do anything that can be done by rule at all. He had dreamed, on paper, the pattern of every reckoning-engine that would ever be built.
24Then he showed what it could not do. No machine can be made that will look at any other machine and tell you surely whether its work will ever come to an end. Some questions have no handle. It was Gödel's fish again, swimming through a different net.
25Remember that young man in the meadow. You will meet him at the end of this book.
26As for Gödel, hear what kind of man had done this, and what it cost to bring him out alive.
27He had been a child who asked so many questions that his family called him Mister Why. As a small boy he had a fever of the joints, and read in a book that it can harm the heart; and from that day he believed his heart was weak, though no doctor ever found it so. He was afraid of bad air, of cold, of food he had not seen cooked. His mind, that trusted nothing it could not prove, could not prove that the world was safe.
28In Vienna there was a woman named Adele. She had been a dancer in a night-café. She was six years older than he was, she had been married before, and she had a red mark on her cheek from birth. His family thought her far beneath him. She had no learning at all. She had what he had not: she was afraid of nothing. When his fears came on him and he would not eat, she sat by him and tasted every spoonful first.
29The city grew dark around them. His teacher died. Then came a morning in June when the wise and gentle Schlick, who had gathered the thinkers of Vienna about him every Thursday evening, was going up the great stairs of the university to give his lesson. A former pupil stepped out with a pistol and shot him dead.
30The newspapers of the shouting man's party did not mourn. They wrote that such teachers had brought it on themselves. And when, two years later, the shouting man's army marched into Vienna and the crowds threw flowers, the killer was let out of prison.
31Gödel was more than once in a house for the sick in mind in those years. Adele brought him back each time. They were married quietly in the autumn after the armies came.
32He was not a Jew. But he had sat with Jews, and learned from Jews, and the new masters took his right to teach away and told him to beg for it again. Then they called him before the army doctors, the man who thought his heart would stop if he climbed a stair, and wrote on his paper: fit for service.
33One evening that November he and Adele were walking down the long stone steps near the university. A band of young toughs of the shouting man's party came up toward them. They saw a thin man in spectacles with a book under his arm, and that was enough. They shouted the word for Jew. One of them struck him across the face, and his spectacles flew off and broke on the stone.
34He stood where he was, with his hands at his sides. He could have proved to them, without one flaw, that they were in error. It would not have helped him.
35Adele did not prove anything. She had an umbrella. She went at them with it, swinging it with both arms and shouting such words as she had learned in the night-café, and those brave young men gave back down the steps before a dancer of forty with a furled umbrella. Then she found his spectacles, and took his arm, and walked him home.
36"We are going to America," she said. "Write to them."
37A place was waiting for him in Princeton, in the same house of thinkers where the clerk of Bern now worked. But it was winter, and the second great war had begun. Between Vienna and America lay the ocean, and the ocean was full of iron fish that sank ships without warning. A German who was taken off a ship by the English would be held as an enemy to the end of the war.
38So they went the other way round the world.
39They left Vienna in the dead of January with what they could carry. They went north by train through the Baltic lands. They crossed the Russian border in the snow. At Moscow they boarded the long train that runs to the far side of Asia, and for nine days and nights they rode it: six thousand miles of white forest and frozen river, with the stove at the end of the wagon and the glasses of tea rattling, and soldiers walking through at every halt to look at papers.
40He was sure he would be taken off. At each halt he sat with his coat buttoned to the chin and their papers in order on his knee. She fed him bread she had cut herself.
41They came out at the edge of the eastern sea. They crossed to Japan, and reached the harbour to find that their ship had gone. More than two weeks they waited in a strange city, counting their money. Then a ship took them across the widest water on earth. They saw the hills of San Francisco in the first days of March, and rode yet another train, three thousand miles, to a small town with white wooden houses.
42They had gone three quarters of the way round the earth to travel from one room of thinkers to another. He had carried in his head, the whole way, a truth that no rule could reach. And she had carried him.
43Mark that, when you count who the prophets are. The books name him. They seldom name her. He found that no house of rules can prove itself sound from within; it takes one who stands outside. And he lived by the help of one who stood outside all his reasoning, and held an umbrella.
44In Princeton an old man with white hair was waiting, who had himself come there as an exile seven years before. The two became friends: the one loud in his laughter and careless in his dress, the other silent and buttoned up in a heavy coat on the warmest day. Every afternoon they walked home together, slowly, under the trees, talking in their own tongue.
45The old man told a friend, near the end, that his own work no longer meant much to him. He came to his desk, he said, only to have the privilege of walking home with Gödel.
46What did they speak of? No one wrote it down. But think who was walking there. One of them had shown that there is no clock over the world. One had shown that there is no book under it. Between them, with a pencil each, they had taken the lid off time and the floor out of certainty. And neither of them was afraid, for both believed that what was left, when the false roofs were gone, was the true sky.
47But in those same years, on both sides of the ocean, other men were learning a different thing from the new knowledge. They had heard that the atom has rooms, and a fire in its rooms. And a woman who had once sat at Boltzmann's feet had just seen, on a walk in the snow, how the door could be forced.