Incompleteness Read online

  To fully understand their sense of shared isolation, which provided the cohesive force to their famous friendship, it will be necessary to consider the metaconvictions that alienated them from their peers. How ought we to interpret, in terms of the larger philosophical questions, Einstein’s relativity theory and Gödel’s incompleteness theorems? How did the authors of these masterpieces of human thought interpret them and how did others?

  Gödel’s incompleteness theorems. Einstein’s relativity theories. Heisenberg’s uncertainty principle. The very names are tantalizingly suggestive, seeming to inject the softer human element into the hard sciences, seeming, even, to suggest that the human element prevails over those severely precise systems, mathematics and theoretical physics, smudging them over with our very own vagueness and subjectivity. The embrace of subjectivity over objectivity—of the “nothing-is-but-thinking-makes-it-so” or “man-is-the-measure-of-all-things” modes of reasoning—is a decided, even dominant, strain of thought in the twentieth-century’s intellectual and cultural life. The work of Gödel and Einstein—acknowledged by all as revolutionary and dubbed with those suggestive names—is commonly grouped, together with Heisenberg’s uncertainty principle, as among the most compelling reasons modern thought has given us to reject the “myth of objectivity.” This interpretation of the triadic grouping is itself part of the modern—or, more accurately, postmodern—mythology.

  So, for example, in the 1998 acclaimed play Copenhagen, the playwright Michael Frayn not only correctly presents the physicists Niels Bohr and Werner Heisenberg as rejecting the idea that physics is descriptive of an objective physical reality, but he also inaccurately identifies Einstein’s relativity theory as the first of modern physics’ moves in the direction of that ultimate rejection:

  Bohr: It [quantum mechanics] works, yes. But it’s more important than that. Because you see what we did in those three years, Heisenberg? Not to exaggerate but we turned the world inside out. Yes, listen, now it comes, now it comes. . . . We put man back at the center of the universe. Throughout history we keep finding ourselves displaced. We keep exiling ourselves to the periphery of things. First we turn ourselves into a mere adjunct of God’s unknowable purposes. Tiny figures kneeling in the great cathedral of creation. And no sooner have we recovered ourselves in the Renaissance, no sooner has man become, as Protagoras proclaimed him, the measure of all things, than we’re pushed aside again by the products of our reasoning! We’re dwarfed again as physicists build the great new cathedrals for us to wonder at—the laws of classical mechanics that predate us from the beginning of eternity, that will survive us to eternity’s end, that exist whether we exist or not. Until we come to the beginning of the twentieth century, and we’re suddenly forced to rise from our knees again.

  Heisenberg: It starts with Einstein.

  Bohr: It starts with Einstein. He shows that measurement—measurement, on which the whole possibility of science depends—measurement is not an impersonal event that occurs with impartial universality. It’s a human act, carried out from a specific point of view in time and space, from the one particular viewpoint of a possible observer. Then here in Copenhagen in those three years in the mid-twenties we discover that there is no precisely determinable objective universe. That the universe exists only as a series of approximations. Only within the limits determined by our relationship with it. Only through the understanding lodged inside the human head.

  Like Einstein’s relativity theory, Gödel’s incompleteness theorems have been seen as holding a prominent place in the twentieth-century’s intellectual revolt against objectivity and rationality. For example, in a popular work of philosophy written by William Barrett, Irrational Man: A Study in Existentialist Philosophy, published in 1962 while Gödel was still alive (and which I was required to read the summer before entering college), Gödel is placed alongside such thinkers as Martin Heidegger (1889–1976) and Friedrich Nietzsche (1844–1900), destroyers of our illusions of rationality and objectivity:

  Gödel’s findings seem to have even more far-reaching consequences [than Heisenberg’s Uncertainty and Bohr’s Complementarity], when one considers that in the Western tradition, from the Pythagoreans and Plato onward, mathematics as the very model of intelligibility has been the central citadel of rationalism. Now it turns out that even in his most precise science—in the province where his reason had seemed omnipotent—man cannot escape his essential finitude; every system of mathematics that he constructs is doomed to incompleteness. Gödel has shown that mathematics has insoluble problems, and hence can never be formalized in any complete system. . . . Mathematicians now know they can never reach rock bottom; in fact, there is no rock bottom, since mathematics has no self-subsistent reality independent of the human activity that mathematicians carry on.

  Barrett correctly states the (first) incompleteness theorem, that mathematics can never be formalized in any complete system. And the philosophical conclusion he draws from it is very much in sync with the most fashionable intellectual trends of the twentieth century. So it might surprise the reader to learn that Gödel himself drew no such conclusion. In fact, if we replace the “no” before “self-subsistent reality” with an “a,” we will arrive at an accurate statement of Gödel’s own metamathematical view, the view that inspired all of his mathematical work, including his famous incompleteness theorems.

  Though intellectual gurus may have interpreted Gödel as falling into step with the great revolt against objectivity and rationality that characterizes much of twentieth-century thinking, this was not the interpretation that Gödel himself held of his revolutionary results. Precisely the same thing can be truthfully said of Einstein. Both men, in fact, were staunch believers in objectivity and interpreted their own most famous work as support positive of this increasingly unpopular position. While so many of their intellectual peers might have made the subjectivist turn—citing the great achievements of relativity theory and the incompleteness theorems as signposts pointing them in that direction—Einstein and Gödel did not.

  Both Einstein and Gödel are as far from seconding the ancient Sophist’s “man is the measure of all things” as it is perhaps possible to be. For both of these men the methodology of their respective fields—the complex mixtures of reasoning, including both intuition and deduction (and, in the case of physics, which is not a priori, observation as well)—does not consist of arbitrary sets of rules that govern an elaborate made-up mind-game or language-game, which could just as well have been played by some other sets of rules entirely, leading to an altogether different construction of reality. No, for both thinkers these are the rules that lead our minds out beyond the circumscriptions of personal experience to gain access to aspects of reality that it is impossible otherwise to know.

  Einstein’s profound isolation from his scientific peers is as well known (if as little understood) as most other aspects of his celebrated life. It is often explained as stemming from his curmudgeonly refusal to accept the revolutionary advance of quantum mechanics, in particular its fundamentally stochastic nature, from which the element of pure chance cannot be excised. The familiar story told about him is that, having made his own conceptual revolution as a young man with his relativity theories, both special and general, he settled, as is the wont of older men, into a conservative mind-set unable to wrap itself around the revolutions of the next generation, even if those later revolutions were the logical extensions of his own. This telling of Einstein’s story is also part of the intellectual mythology of the twentieth century.

  Yet it is not accurate. The heart of Einstein’s scientific alienation is his rejection of the subjectivist turn that the playwright has his characters declare “all began with Einstein.” Einstein had not understood his relativity theory as pointing toward the subjectivist interpretation of physics but, rather, precisely in the opposite direction. “Relativity,” as it occurs in Einstein’s theory, means something far more technical and restricted than that measurement (and so everyt
hing) is relative to human points of view.11 For Einstein, in fact, to have followed men like Werner Heisenberg and Niels Bohr in the direction of subjectivity would have been to deny what he took to be the most fundamental meta-implications of relativity theory. Einstein interpreted his theory as representing the objective nature of space-time, so very different from our human, subjective point of view of space and time.12 Far from restoring us to the center of the universe, describing everything as relative to our experiential point of view, Einstein’s theory, expressed in terms of beautiful mathematics, offers us a glimpse of an utterly surprising physical reality, surprising precisely because it is nothing like what we are presented with in our experiential apprehension of it.

  Einstein sometimes speaks of objective reality as the “out yonder,” and in the “Autobiographical Notes” that he supplied with his typical self-mocking good humor for the Festschrift that P. A. Schilpp edited in honor of the physicist’s seventieth birthday,13 he explicitly identifies his belief in this reality as the spiritual center of his life as a scientist:

  It is quite clear to me that the religious paradise of youth, which was thus lost, was a first attempt to free myself from the chains of the “merely personal,” from an existence which is dominated by wishes, hopes, and primitive feelings. Out yonder there was this huge world, which exists independently of us human beings and which stands before us like a great, eternal riddle, at least partially accessible to our inspection and thinking. The contemplation of this world beckoned like a liberation. . . . The mental grasp of this extra-personal world within the frame of the given possibilities swam as highest aim half consciously and half unconsciously before my mind’s eye. . . . The road to this paradise was not as comfortable and alluring as the road to the religious paradise; but it has proved itself as trustworthy, and I have never regretted having chosen it.

  This is an eloquent statement of Einstein’s credo as a scientist, and it really could not be more at odds with the sentiments of almost all the other prominent physicists of his circle.14 Einstein understood the business of physics to be to discover theories that offer a glimpse of the objective nature that stands “out yonder” behind our experiences. Werner Heisenberg, together with such men as the Danish Niels Bohr and the German Max Born (who are together the leading advocates of the Copenhagen interpretation of quantum mechanics) reject this view in the name of an intellectual movement known as “positivism,” according to which any attempt to reach out beyond our experience results in arrant nonsense.

  We will have occasion to look more closely at positivism in the next chapter, when we move from Princeton, New Jersey, to Vienna, Austria, and examine the circumstances that brought forth, almost as an act of supreme intellectual rebellion against the positivists, Gödel’s two incompleteness theorems.

  Positivism, especially as it came to be espoused by the group of scientists, mathematicians, and philosophers of the famed Vienna Circle, under the strong influence of the charismatic Viennese-born philosopher Ludwig Wittgenstein, is a severe theory of meaning that makes liberal use of the word meaningless. In particular, it brands as meaningless any descriptive proposition15 that cannot in principle be verified through the contents of our experience. The meaning of a proposition is given by the means of empirically verifying it (the verificationist criterion of meaning).

  Gödel, like Einstein, is committed to the possibility of reaching out, pace the positivists, beyond our experiences to describe the world “out yonder.” Only since Gödel’s field is mathematics, the “out yonder” in which he is interested is the domain of abstract reality. His commitment to the objective existence of mathematical reality is the view known as conceptual, or mathematical, realism. It is also known as mathematical Platonism, in honor of the ancient Greek philosopher whose own metaphysics was a vehement rejection of the Sophist Protagoras’ “man is the measure of all things.”

  Platonism is the view that the truths of mathematics are independent of any human activities, such as the construction of formal systems—with their axioms, definitions, rules of inference, and proofs. The truths of mathematics are determined, according to Platonism, by the reality of mathematics, by the nature of the real, though abstract, entities (numbers, sets, etc.) that make up that reality. The structure of, say, the natural numbers (which are the regular old counting numbers: 1, 2, 3, etc.) exists independent of us, according to the mathematical realist, just as does the structure of space-time, according to the physical realist; and the properties of the numbers 4 and 25—that, for example, one is even, the other is odd and both are perfect squares—are as objective as are, according to the physical realist, the physical properties of light and gravity.

  For Gödel mathematics is a means of unveiling the features of objective mathematical reality, just as for Einstein physics is a means of unveiling aspects of objective physical reality. Gödel’s understanding of what we are doing when we are doing mathematics could be rendered in words echoing Einstein’s credo: “Out yonder there is this huge world, which exists independently of us human beings and which stands before us like a great, eternal riddle, at least partially accessible to our inspection and thinking.” Only here the “out yonder” is to be understood as at an even further remove from the subject of experience, with his distinctly human point of view. The “out yonder” is out beyond physical space-time; it is a reality of pure abstraction, of universal and necessary truths, and our faculty of a priori reason provides us—mysteriously—with the means of accessing this ultimate “out yonder,” of gaining at least partial glimpses of what might be called (in the current fashion in naming television shows: “Extreme Survival,” “Extreme Makeover,” “The Most Extreme”) “extreme reality.”

  Gödel’s mathematical Platonism was not in itself unusual. Many mathematicians have been mathematical realists; and even those who do not describe themselves as such, when they are cornered and asked pointblank about their metamathematical position, will slip unself-consciously into realism when they speak of their work as their “discoveries.”16 G. H. Hardy (1877–1947), an English mathematician of great distinction, expressed his own Platonist convictions in his classic A Mathematician’s Apology, with no apologies at all:

  I believe that mathematical reality lies outside of us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our “creations,” are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards, and I shall use the language which is natural to a man who holds it. . . .

  [T]his realistic view is much more plausible of mathematical than of physical reality, because mathematical objects are so much more what they seem. A chair or a star is not in the least like what it seems to be; the more we think of it, the fuzzier its outlines become in the haze of sensation which surrounds it; but “2” or “317” has nothing to do with sensation, and its properties stand out the more clearly the more closely we scrutinize it. It may be that modern physics fits best into some framework of idealistic philosophy—I do not believe it, but there are eminent physicists who say so. Pure mathematics, on the other hand, seems to me a rock on which all idealism founders: 317 is a prime, not because we think so, or because our minds are shaped in one way or another, but because it is so, because mathematical reality is built that way.17

  The almost three millennia since Plato have given us plenty of new and amazing mathematics, but not much more reason to believe in Platonism than the ancient Greek philosopher himself had. Mathematician after mathematician has testified, like Hardy, to their Platonist conviction that they are discovering, rather than creating, mathematical truths. However, testifying is just about all we ever got . . . until Gödel. Gödel’s audacious ambition to arrive at a mathematical conclusion that would simultaneously be a metamathematical result supporting mathematical realism was precisely what yielded his incompleteness theorems.

  Gödel’
s metamathematical view, his affirmation of the objective, independent existence of mathematical reality, constituted perhaps the essence of his life, which is to say what is undoubtedly true: that he was a strange man indeed. His philosophical outlook was not an expression of his mathematics; his mathematics were an expression of his philosophical outlook, his Platonism, which was the deepest expression, therefore, of the man himself. That his work, like Einstein’s, has been interpreted as not only consistent with the revolt against objectivity but also as among its most compelling driving forces is then more than a little ironic.

  Einstein was fortunate in his last years to have a kindred philosophical spirit, even if one as unstable and fey as Gödel, to soften the sense of exile. The words that Morgenstern quoted from Einstein, that in his last years he went to his Institute office only in order to have the privilege of walking home with Gödel, become, in the subtleties of the metalight, less surprising.

  After Einstein’s death in 1955, Gödel’s sense of intellectual exile deepened; his most profound sense of identification was with the über-rationalist Leibniz, who had been dead for almost 300 years. The explanations the logician arrived at through the rigorous application of his “interesting axiom” took on ever darker tones. The young man in the dapper white suit shriveled into an emaciated man, entombed in a heavy overcoat and scarf even in New Jersey’s hot humid summers, seeing plots everywhere. He came to believe that there was a vast conspiracy, apparently in place for centuries, to suppress the truth “and make men stupid.” Those who had discovered the full power of a priori reason, men such as the seventeenth-century’s Leibniz and the twentieth-century’s Gödel, were, he believed, marked men. His profound isolation, even alienation, from his peers provided fertile soil for that rationality run amuck which is paranoia.

  That the greatest logician since Aristotle should have followed reason so unwaveringly to such illogical conclusions has struck many people as paradoxical. But, as I hope will become ever clearer in the chapters to come, the internal paradoxes in Gödel’s personality were at least partially provoked by the world’s paradoxical responses to his famous work. His incompleteness theorems were simultaneously celebrated and ignored. Their technical content transformed the fields of logic and mathematics; the method of proof he used, the concepts he defined in the course of the proof, led to entirely new areas of research, such as recursion theory and model theory. Other central areas of research were abandoned, particularly those sanctioned by the greatest mathematician of the generation just preceding Gödel’s, David Hilbert (1862–1943), having been shown to be futile by reason of Gödel’s theorems.