Incompleteness Read online

  Yet the metamathematical import of the theorems, which to Gödel was their most important aspect, was disregarded. Even more paradoxically, the racier currents in the culture, hawking postmodern uncertainty and the false mythology of all absolutes, scooped his theorems up, together with Einstein’s relativity, reinterpreting them so that they precisely negated the convictions that Gödel and his fellow exile had so passionately wanted to demonstrate.

  Paradoxes, in the technical sense, are those catastrophes of reason whereby the mind is compelled by logic itself to draw contradictory conclusions. Many are of the self-referential variety; troubles arise because some linguistic item—a description, a sentence—potentially refers to itself. The most ancient of these paradoxes is known as the “liar’s paradox,” its lineage going back to the ancient Greeks.18 It is centered on the self-referential sentence: “This very sentence is false.” This sentence must be, like all sentences, either true or false. But if it is true, then it is false, since that is what it says; and if it is false, well then, it is true, since, again, that is what it says. It must, then, be both true and false, and that is a severe problem. The mind crashes.

  Paradoxes like the liar’s play a technical role in the proof that Gödel devised for his extraordinary first incompleteness theorem. Gödel was able to take the structure of self-referential paradoxicality—the sort of structure that causes our minds to crash when considering “This very sentence is false”—and turn it into an extraordinary proof for one of the most surprising results in the history of mathematics.19 This itself seems almost paradoxical. Paradoxes have always seemed specifically designed to convince us that we are simply not smart enough to take up whatever topic brought us to them. Gödel was able to twist the intelligence-mortifying material of paradox into a proof that leads us to deep insights into the nature of truth, and knowledge, and certainty. According to Gödel’s own Platonist understanding of his proof, it shows us that our minds, in knowing mathematics, are escaping the limitations of man-made systems, grasping the independent truths of abstract reality.

  The structure of Gödel’s proof, the use it makes of ancient paradox, speaks at some level, if only metaphorically, to the paradoxes in the tale that the twentieth century told itself about some of its greatest intellectual achievements—including, of course, Gödel’s incompleteness theorems. Perhaps someday a historian of ideas will explain the subjectivist turn taken by so many of the last century’s most influential thinkers, including not only philosophers but hard-core scientists, such as Heisenberg and Bohr. Such an explanation lies well beyond the scope of this book. But what I can do is to describe the effects that the revolt against objectivity had on one of the twentieth century’s greatest thinkers: how it provoked him into his proof of the incompleteness theorems and how it then reinterpreted those theorems as confirmation of itself.

  To understand the full richness—and paradox—of Gödel, his world and his work, it will be necessary to take two steps backward from the glimpse of him walking home with Einstein on a shady road in Princeton. We’ll step back first to the 1920s Vienna of his youth, the scene of so many of the young century’s intellectual and cultural assaults on tradition; and then take another retreat back to the turn of the century, when a conception of mathematics gave birth to a program for completing mathematics that would fall victim to the work of the reticent young logician with the outsized metamathematical ambitions.

  1 This is not, however, to imply that these beliefs are innate, i.e., that we are born having them. Obviously, we must first acquire the concepts, and the language for expressing them, before we can come to believe that 5 + 7 = 12. Innateness is a psychological notion, whereas aprioricity is an epistemological notion, having to do with the way in which the belief is justified, what counts as evidence both for and against it. [Note: Two types of notes will be employed in this book: footnotes to continue a thought on the page, endnotes to give citations.]

  2 As the first venture of the Institute outside the realm of purely theoretical work, it was criticized as “out of place” even by faculty members who had a high regard for the endeavor itself, according to the official account of the Institute’s School of Mathematics. After von Neumann’s death, the computer was quietly transferred to Princeton University.

  3 Many contemporaries report the “awed hush” (in the words of Helen Dukas, ibid.) that would fall over a lecture or seminar room when he entered. Princeton philosopher Paul Benacerraf, who had been a graduate student at Princeton in Einstein’s day, told me that Einstein sometimes used to attend the weekly Friday philosophy seminar, seldom speaking but still making his presence felt simply because it was his presence.

  4 Before Gödel came onto the scene, logicians were more likely to be members of a philosophy department. Simon Kochen, a logician in the mathematics department at Princeton University, remarked to me that “Gödel put logic on the mathematical map. Every mathematical department of note now has logic represented on its staff. It may only be one or two logicians, but there will, at least, be someone” (May 2002).

  5 This utopian epistemology is characteristic of the seventeenth-century rationalists—René Descartes (1596–1650), Benedictus Spinoza (1632–1677), and Gottfried Wilhelm Leibniz (1646–1716). Spinoza and Leibniz, in particular, believed it was possible to appropriate the standards and methods of the mathematicians and generalize them so that they could answer all our posed questions: scientific, ethical, even theological. Then, when theological differences of the sort that cause long and bloody wars arose, men of reason could respond: “Come, let us a priori deduce.”

  6 Gödel’s hostility to the theory of evolution becomes quite understandable the more one understands his mind. A rationalist like Gödel wishes to excise chance and randomness, whereas natural selection invokes randomness and contingency as fundamental explanatory factors. At the level of microevolution (generation-to-generation changes), the theory gives a central role to random mutation and recombination. At the level of macroevolution (patterns in the history of life), it gives a central role to historical contingency, such as the vagaries of geology and climate, or such chance events as a meteorite’s crashing to Earth, blackening out the Sun, wiping out the dinosaurs, thus allowing mouselike mammals to inhabit the vacated ecological niches. (I am indebted to Steven Pinker for this insight.)

  7 Morgenstern had also fled Nazi-occupied Austria for the Institute. Even though he was an economist, his work was sufficiently mathematical—he is one of the founders, with von Neumann, of game theory—to gain him entrance into Flexner’s Institute.

  8 Gödel produced a very original solution to the field equations of Einstein’s general relativity, and surprised Einstein with them for his seventieth birthday. In Gödel’s solution, time is cyclical. See Chapter 4.

  9 Hao Wang (1921–1995), a logician at Rockefeller University, devoted himself to understandiing Gödel’s views on everything from the nature of mathematical intuition to the transmigration of souls, and produced three books out of the material.

  10 His other two achievements dating 1929–30 were the second incompleteness theorem and the proof of the completeness of the predicate calculus.

  11 Measurements of properties like length are, according to special relativity, relative to a particular coordinate system or reference frame. But to reduce these technical terms—coordinate system, reference frame—to the idea of human points of view, is, well, nonsense. We have the choice of various coordinate systems to describe the motion of something, and, according to the theory of relativity, all coordinate systems are equal; none is privileged. In one coordinate system an “observer” (who does not even have to be a conscious entity, and hence does not have to be literally observing or even capable of observing anything) will be at rest; in another he or she or it will be moving. It’s often natural, though not determined, to choose a coordinate system with respect to which a particular observer is at rest. Thus it is often natural (though not determined) to choose the coordinate s
ystem in which the earth, for example, is at rest. The motion of all of us terrestrials, with our myriad subjective points of view would then be described relative to one coordinate system, in which the earth is at rest.

  12 In relativity theory, for example, time doesn’t flow, but rather is, as the fourth dimension, as static as space. In vivid contrast, the most dramatic (and poignant) aspect of our experience of time is its ceaseless, unidirectional motion, carrying us away from the past and toward the future.

  13 “Here I sit in order to write, at the age of 67, something like my own obituary. I am doing this not merely because Dr. Schilpp has persuaded me to do it; but because I, in fact, believe that it is a good thing to show those who are striving alongside of us how one’s striving and searching appears to one in retrospect” (Schilpp, p. 3).

  14 Contrast it, for example, with this statement of Werner Heisenberg’s: “The idea of an objective real world whose smallest parts exist objectively in the same sense as stones or trees exist, independently of whether or not we observe them . . . is impossible.”

  15 By descriptive proposition one means a proposition that is not just true (or false) by virtue of its meaning alone. Propositions whose truth-value (truth or falsity) is a function of their meaning alone are called “analytic” or, sometimes, “trivial.” So, for example, “all bilingual people speak at least two languages” is analytic. A proposition that is, on the other hand, descriptive is not true or false simply by virtue of its meaning but also by virtue of the facts of the matter. So the proposition “I am bilingual” is false by virtue of both its meaning and the facts of the matter.

  16 Interestingly, this is true even of David Hilbert, whose formalism was sharply opposed to Platonism (see chapter 2).

  17 The circumstances of the writing of Hardy’s classic are as moving as they are unusual. Hardy had lost his mathematical creativity, which tends to happen to mathematicians relatively young. (A mathematician of 40 has probably already seen his best years, which is why the most prestigious award for mathematicians [there is no Nobel Prize for mathematics], the Fields Medal, is awarded to someone 40 or younger.) Hardy attempted suicide, survived the attempt, and was persuaded by C. P. Snow to write a book explaining the life of a mathematician. The result, A Mathematician’s Apology, is incomparable. Soon after completing it, Hardy again attempted suicide, and succeeded.

  18 Here is the textual reference the paradox is derived from: “One of themselves, even a prophet of their own, said, ‘The Cretans are always liars.’ . . . This witness is true” (Titus 1:12–13).

  19 That mathematical conclusions have the ability to surprise us might itself seem paradoxical. The world might very well, and often does, confound our expectations, our experiential contact with it bringing us to rude awakenings. But how can conclusions that are arrived at through purely a priori reason do so? If a priori truths are—by definition—immune from empirical revision, then it’s not some unexpected experience of the world that delivers the punch. We ourselves must deduce the confoundment, and this seems prima facie odd. This metamathematical issue, too, is addressed by Gödel’s prolix first theorem. For Gödel, the independent reality of mathematics, of which our axioms are only incomplete descriptions, takes the surprise out of the surprisingness of mathematics.

  I

  A Platonist among the Positivists

  First Love

  Kurt Gödel 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 “Brünn.” 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 Empire’s textile industry,1 so when Rudolf proved to be no scholar at grammar school, he was enrolled, at the age of 12, at a weaver’s 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.

  Gödel’s 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 Gödels’ marriage. As so often happens in such cases, the mother’s 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 father’s 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 Gödel’s earliest years, as sparse as it is, comes by way of his older brother Rudolf, who wrote a brief “History of the Gödel Family,” as well as from Rudolf’s responses to queries from the logicians Hao Wang and John Dawson on the subject of his younger brother’s 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. Gödel’s 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 vernünftig: the world is rational. Like many gifted mathematicians, Gödel 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”2 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 Gödel 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. Gödel 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 Gödel 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 it’s reinforced by a serious illness a few years later, to derive the following additional conclus
ion: 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 don’t need to depend on them. I can figure out everything for myself. The world is thoroughly logical and so is my mind—a perfect fit.

  Quite possibly the young Gödel 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 man—even more so, perhaps, in the case of mathematical geniuses.

  At school, the K.-K. Staatsrealgymnasium mit deutscher Unterrichtssprache3 (obviously, a German-language school), Gödel 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 Klepetař, wrote to John Dawson that “from the beginning . . . Gödel kept more or less to himself and devoted most of his time to his studies.” He also reported that Gödel’s interests were “manifold,” and that “his interest in mathematics and physics [had already] manifested itself . . . at the age of 10.”