Rebecca Goldstein - Incompleteness: The Proof and Paradox of Kurt Gödel

If there is any aspect of being the perfect literary agent that Tina Bennett lacks, I have yet to discover it. The current project only served to reveal new aspects of Tinas ways of unstintingly supporting her writers.

I am extremely grateful to the following people who shared their recollections of Kurt Gdel with me: John Bahcall, Paul Benacerraf, Armand Borel, Thomas Nagel, Morton White. Each one was stintless with his time. Simon Kochen not only spoke long hours with me but also generously read over my completed manuscript, catching some technical errors, for which I am profoundly grateful, and answering further queries by e-mail. Berel Lang also read the manuscript and his comments, too, were insightful, substantive, and helpful.

I thank John Dawson, not only for the Herculean work he did as Gdels archivist, which made the job of all scholars who follow possible, but also his prompt response to any question that arose.

As always, Sheldon Goldstein, physicist-philosopher, had insights that were invaluable. He, more than anyone, helped me to ease my way back into mathematical logic. There is not a man on Earth, Id wager, quite equal to him for reminding one of the beauty and elegance of abstract thought. Steven Pinker generously read some early inchoate chapters, when I was feeling my way toward popular technical writing, and his comments and encouragement were sustaining. And when I threw up my hands, Yael Goldstein calmly placed them back on the keyboard, offering the sort of sage advice and substantive criticism and guidance without which this book, quite literally, would not have been written.

Accordingly, I dedicate the book to her, with gratitude, love, and stupefied admiration.

The Mind-Body Problem

The Late-Summer Passion of a Woman of Mind

The Dark Sister

Strange Attractors

Mazel

Properties of Light

Incompleteness

First Love

K urt Gdel was 18 when he arrived in Vienna to begin his studies at the university. Though he had been born in Moravia, in what is now the Czech Republic but was then part of the Hapsburg Empire, his arrival in Vienna must have felt like something of a homecoming. He had considered himself an exile even in the land of his birth.

He was born on 28 April 1906 in Brno, or what the Germans and Austrians still call Brnn. His parents, Rudolf and Marianne, were of German rather than Czech origin, and associated exclusively with the other Sudeten Germans who dominated in Brno. The city was the center of the Hapsburg Empires textile industry, so when Rudolf proved to be no scholar at grammar school, he was enrolled, at the age of 12, at a weavers school, where he found his calling. He completed his studies there with distinction and was given a job in the textile factory of Friedrich Redlich, where he worked until his death. He rose swiftly through the ranks, eventually becoming a director and joint partner. Consequently, the family lived comfortably, eventually acquiring a villa in a fashionable neighborhood.

Gdels mother, Marianne, was far more educated and cultured than his father, which was not unusual among the bourgeoisie of the Empire. It was also common for marriage choices to be forged out of practical concerns rather than romantic inclinations, and this, too, seemed to be the case in the Gdels marriage. As so often happens in such cases, the mothers strongest emotional ties were supplied by her children, in her case Rudolf, born a year after her marriage, and then four years later Kurt, who was baptized Kurt Friedrich, the middle name honoring his fathers employer, who served as godfather. For some reason, the logician dropped his middle name when he became a U.S. citizen in 1948.

Almost all of our knowledge of Kurt Gdels earliest years, as sparse as it is, comes by way of his older brother Rudolf, who wrote a brief History of the Gdel Family, as well as from Rudolfs responses to queries from the logicians Hao Wang and John Dawson on the subject of his younger brothers childhood. (Rudolf was a physician who never married and remained in Austria. He died in 1992, at the age of 90.)

We learn from Rudolf that Kurt asked so many questions that his nickname was der Herr Warum , or Mr. Why. Little children, as anyone who has spent any significant amount of time with them knows, tend to push the why questions pretty hard. We are born into a sort of ontological wonder ( thaulamazein ) that passes into oblivion as we get used to the lay of the land. Gdels intense childhood thaulamazein persisted throughout his adult life, so that the child who was called Herr Warum grew into the man who began the 14 principles of his private credo with Die Welt is vernnftig : the world is rational. Like many gifted mathematicians, Gdel reached a certain level of precocious maturity while still a young child; then, having arrived at this level, he remained there. The picture en famille we have of the future successor to Aristotle at age four shows a cherubic little man, seriously staring straight into the camera, his hand precisely poised before him, the little forward hunch giving the suggestion of solemn contemplation.

The Gdel family, ca. 1910: Marianne, Kurt, father Rudolf, brother Rudolf.

We also learn from Rudolf, in a letter to the logician Hao Wang, that at about the age of five, the younger brother suffered a mild anxiety neurosis ( leichte Angst Neurose ) , and at the age of eight he suffered a severe bout of joint rheumatism, with high fever. The patient did research on his illness and, learning that the illness could cause possible permanent heart damage, he inferred that precisely this outcome had occurred in his particular case. Gdel held to the conviction of an injured heart throughout his life, despite the absence of any evidence. The conclusion he reached as an eight-year-old child, entirely on his own, was to contribute to his lifelong hypochondria.

When the random permutations of genetic blending produce an offspring whose intelligence far outstrips that of his parents that child faces a special sort of predicament: he both recognizes his utter dependence, being after all only a child; and he also clearly perceives the severe limits of his own parents understanding. Most people come to the latter recognition only during adolescence, when the normal reaction is an explosive mixture of hubris, contempt, and outrage (how can they be so dumb?). But the reaction of a young child is more likely to be blind terror (how can they be trusted to take care of me?). The leichte Angst Neurose is some indication that the precocious Gdel grasped the limits of parental omniscience at about the age of five. It would be comforting, in the presence of such a shattering conclusion, especially when its reinforced by a serious illness a few years later, to derive the following additional conclusion: There are always logical explanations and I am exactly the sort of person who can discover such explanations. The grownups around me may be a sorry lot, but luckily I dont need to depend on them. I can figure out everything for myself. The world is thoroughly logical and so is my minda perfect fit.

Quite possibly the young Gdel had some such thoughts to quell the terror of discovering at too young an age that he was far more intelligent than his parents. It would explain much about the man he would become. The child is father to the maneven more so, perhaps, in the case of mathematical geniuses.

At school, the K.-K. Staatsrealgymnasium mit deutscher Unterrichtssprach e (obviously, a German-language school), Gdel excelled in all his studies and began to show the marked aloofness and solemnity that would characterize him throughout his life. A fellow schoolmate, Harry Klepeta, wrote to John Dawson that from the beginning... Gdel kept more or less to himself and devoted most of his time to his studies. He also reported that Gdels interests were manifold, and that his interest in mathematics and physics [had already] manifested itself... at the age of 10.