Incompleteness Page 2
Some thinkers have seen in Gödel’s theorems high-grade grist for the postmodern mill, pulverizing the old absolutist ways of thinking about truth and certainty, objectivity and rationality. One writer expressed the postmodern sentiment in lively eschatological terms: “He [Gödel] is the devil, for math. After Gödel, the idea that mathematics was not just a language of God but language we could decode to understand the universe and understand everything—that just doesn’t work any more. It’s part of the great postmodern uncertainty that we live in.” The necessary incompleteness of even our formal systems of thought demonstrates that there is no nonshifting foundation on which any system rests. All truths—even those that had seemed so certain as to be immune to the very possibility of revision—are essentially manufactured. Indeed the very notion of the objectively true is a socially constructed myth. Our knowing minds are not embedded in truth. Rather the entire notion of truth is embedded in our minds, which are themselves the unwitting lackeys of organizational forms of influence. Epistemology is nothing more than the sociology of power. So goes, more or less, the postmodern version of Gödel.
Other thinkers have argued that, in regard to the nature of the human mind, the implications of Gödel’s theorems point in an entirely different direction. For example, Roger Penrose in his two bestselling books, The Emperor’s New Mind and Shadows of the Mind, has made the incompleteness theorems central to his argument that our minds, whatever they are, cannot be digital computers. What Gödel’s theorems prove, he argues, is that even in our most technical, rule-bound thinking—that is, mathematics—we are engaging in truth-discovering processes that can’t be reduced to the mechanical procedures programmed into computers. Notice that Penrose’s argument, in direct opposition to the postmodern interpretation of the previous paragraph, understands Gödel’s results to have left our mathematical knowledge largely intact. Gödel’s theorems don’t demonstrate the limits of the human mind, but rather the limits of computational models of the human mind (basically, models that reduce all thinking to rule-following). They don’t leave us stranded in postmodern uncertainty but rather negate a particular reductive theory of the mind.
Gödel’s theorems, then, appear to be that rarest of rare creatures: mathematical truths that also address themselves—however ambiguously and controversially—to the central question of the humanities: what is involved in our being human? They are the most prolix theorems in the history of mathematics. Though there is disagreement about precisely how much, and precisely what, they say, there is no doubt that they say an awful lot and that what they say extends beyond mathematics, certainly into metamathematics and perhaps even beyond. In fact, the metamathematical nature of the theorems is intimately linked with the fact that the Encyclopedia of Philosophy stated them in (more or less) plain English. The concepts of “formal system,” “undecidable,” and “consistency” might be semi-technical and require explication (which is why the reader should not worry if the succinct statement of the theorems yielded little understanding); but they are metamathematical concepts whose explication (which will eventually come) is not rendered in the language of mathematics. Gödel’s conclusions are mathematical theorems that manage to escape mere mathematics. They speak from both inside and outside mathematics. This is yet another facet of their distinct fascination, the facet seized upon in yet another popular book, Douglas Hofstadter’s Pulitzer-prize-winning Gödel, Escher, Bach: An Eternal Golden Braid.
The prefix meta comes from the Greek, and it means “after,” “beyond,” suggesting the view from outside, as it were. The metaview of a cognitive area poses such questions as: how is it possible for this area of knowledge to be doing what it is doing? Mathematics, just because it is sui generis—the severest of disciplines—using a priori methods to establish its often astounding, though incorrigible, results, has always forcefully presented theorists of knowledge (known as “epistemologists”) with metaquestions, most specifically the question of how it is possible for it to be doing what it is doing. The certainty of mathematics, the godlike infallibility it seems to bestow on its knowers, has been seen as presenting both a paradigm to be emulated—if we can do it there, let’s do it everywhere5—and also a riddle to be pondered: how can we do it, there or anywhere? How can the likes of us, thrown up out of the blindfolded thrashings of evolution, attain any sort of infallibility? To grasp this riddle it might be helpful to recall a famous remark of Groucho Marx’s, to the effect that he would not belong to any country club that would accept the likes of him. Similarly, some have fretted that if mathematics is really so certain then how can it be known by the likes of us? How can we have gained entry into so restricted a cognitive club?
Metaquestions about a field, say about science or mathematics or the law, are not normally questions that are contained in the field itself; they are not, respectively, scientific or mathematical or legal. Rather they are categorized as philosophical questions, residing, respectively, in the philosophy of science, of mathematics, of law. Gödel’s theorems are spectacular exceptions to this general rule. They are at once mathematical and metamathematical. They have all the rigor of something that is a priori proved, and yet they establish a metaconclusion. It is as if someone has painted a picture that manages to answer the basic questions of aesthetics; a landscape or portrait that represents the general nature of beauty and perhaps even explains why it moves us the way it does. It is extraordinary that a mathematical result should have anything at all to say about the nature of mathematical truth in general.
Gödel’s two theorems address themselves to the very issue that has always singled out mathematics: the certainty, the incorrigibility, the aprioricity. Do the theorems cast us out of the most exclusive cognitive club in epistemology, undermining our claim of being able to attain, in the area of mathematics at least, perfect certitude? Or do the theorems leave us members in good standing? Gödel himself, as we shall see, held strong convictions on this metaquestion, sharply at odds with interpretations that are commonly linked with his work.
For both Gödel and Einstein, metaquestions of how, respectively, physics and mathematics are to be interpreted—what it is that these powerful forms of knowledge actually do and how they do it—are central to their technical work. Einstein, too, had extremely strong metaconvictions regarding physics. More specifically, Einstein’s and Gödel’s metaconvictions were addressed to the question of whether their respective fields are descriptions of an objective reality—existing independent of our thinking of it—or, rather, are subjective human projections, socially shared intellectual constructs.
The emphasis that each placed on these metaquestions was, in itself, enough to separate them from most practitioners in their respective fields. Not only were both men centrally interested in the metalevel, but, even more unusually, they also wanted their technical work to shed metalight. Gödel, in fact, had acquired the ambition, while still an undergraduate at the University of Vienna, of devoting himself only to the sort of mathematics that would have broader philosophical implications. This is a truly daunting goal, in some sense historically ambitious, and one of the most astounding aspects of his story is that he managed to achieve it. This daunting ambition, which he preserved throughout his life, may have limited how much he did, but it also determined that whatever he did was profound. Einstein, though not quite so strict with himself as Gödel, still shared the conviction that truly good science always keeps the larger philosophical questions in view: “Science without epistemology is—insofar as it is thinkable at all—primitive and muddled.”
The friendship between Einstein and Gödel is still the stuff of both legend and speculation. Every day the two men made the trek back and forth from the Institute, and others watched them with curiosity and wondered that they had so much to say to one another. For example, Ernst Gabor Straus wrote:
No story of Einstein in Princeton would be complete without mentioning his really warm and very close friendship with Kurt Gödel. They were very, very
dissimilar people, but for some reason they understood each other well and appreciated each other enormously. Einstein often mentioned that he felt that he should not become a mathematician because the wealth of interesting and attractive problems was so great that you could get lost in it without ever coming up with anything of genuine importance. In physics, he could see what the important problems were and could, by strength of character and stubbornness, pursue them. But he told me once, “Now that I’ve met Gödel, I know that the same thing does exist in mathematics.” Of course, Gödel had an interesting axiom by which he looked at the world; namely, that nothing that happens in it is due to accident or stupidity. If you really take that axiom seriously all the strange theories that Gödel believed in become absolutely necessary. I tried several times to challenge him, but there was no out. I mean, from Gödel’s axioms they all followed. Einstein did not really mind it, in fact thought it quite amusing. Except the last time we saw him in 1953, he said, “You know, Gödel has really gone completely crazy.” And so I said, “Well, what worse could he have done?” And Einstein said, “He voted for Eisenhower.”
Straus’s language indicates a certain puzzlement as to what the two men saw in one another; in particular, what the sagacious physicist could have seen in the neurotic logician. Einstein, wrote Straus, was “gregarious, happy, full of laughter and common sense.” Gödel, on the other hand was “extremely solemn, very serious, quite solitary and distrustful of common sense as a means of arriving at the truth.”
The Einstein of legend—with his wild hair and absent-mindedness, his quixotic embrace of one-world politics and other lost causes—is not usually portrayed as a savvy, worldly sort; but, compared to Gödel, he was. Most in Princeton, even his mathematical colleagues, found Gödel, with his “interesting axiom” exponentially complicating every discussion and practical decision, all but impossible to speak with. As the mathematician Armand Borel wrote in his history of the Institute’s School of Mathematics, he and the others sometimes “found the logic of Aristotle’s successor. . . quite baffling.” Eventually, the mathematicians solved their Gödel problem by banishing him from their meetings, making him a department of one: the sole decision-maker on anything having strictly to do with logic.
Though Princeton’s population is well accustomed to eccentricity, trained not to look askance at rumpled specimens staring vacantly (or seemingly vacantly) off into space-time, Kurt Gödel struck almost everyone as seriously strange, presenting a formidable challenge to conversational exchange. A reticent person, Gödel, when he did speak, was more than likely to say something to which no possible response seemed forthcoming:
John Bahcall was a promising young astrophysicist when he was introduced to Gödel at a small Institute dinner. He identified himself as a physicist, to which Gödel’s curt response was “I don’t believe in natural science.”
The philosopher Thomas Nagel recalled also being seated next to Gödel at a small gathering for dinner at the Institute and discussing the mind-body problem with him, a philosophical chestnut that both men had tried to crack. Nagel pointed out to Gödel that Gödel’s extreme dualist view (according to which souls and bodies have quite separate existences, linking up with one another at birth to conjoin in a sort of partnership that is severed upon death) seems hard to reconcile with the theory of evolution. Gödel professed himself a nonbeliever in evolution and topped this off by pointing out, as if this were additional corroboration for his own rejection of Darwinism: “You know Stalin didn’t believe in evolution either, and he was a very intelligent man.”
“After that,” Nagel told me with a small laugh, “I just gave up.”6
The linguist Noam Chomsky, too, reported being stopped dead in his linguistic tracks by the logician. Chomsky asked him what he was currently working on, and received an answer that probably nobody since the seventeenth-century’s Leibniz had given: “I am trying to prove that the laws of nature are a priori.”
Three magnificent minds, as at home in the world of pure ideas as anyone on this planet, yet they (and there are more) reported hitting an insurmountable impasse in discussing ideas with Gödel.
Einstein, too, was presented time and again, on their daily walks to and from the Institute, with examples of Gödel’s strange intuitions, his profound “anti-empiricism.” Nevertheless Einstein consistently sought out the logician’s company. In fact, economist Oskar Morgenstern,7 who had known Gödel back in Vienna, confided in a letter: “Einstein had often told me that in the late years of his life he has continually sought Gödel’s company, in order to have discussions with him. Once he said to me that his own work no longer meant much, that he came to the Institute merely um das Privileg zu haben, mit Gödel zu Fuss nach Hause gehen zu dürfen,” that is, in order to have the privilege of walking home with Gödel. Even given their shared interest in the metalevel of their respective fields, Einstein’s avowal of devotion strikes one as extravagant.
For his part, Gödel’s letters to his mother, Marianne, who remained behind in Europe (a correspondence that gives us some knowledge of his life until her death in 1966), are filled with references to Einstein. If Einstein, in his last years, went to the Institute merely for the privilege of walking home with Gödel, for Gödel there was simply nobody else in all the world with whom to talk, at least not in the way in which he could talk to Einstein (an exclusivity made all the more poignant when one considers that Gödel had a wife). So that, for example, on 4 July 1947, he wrote to his mother that Einstein had been ordered by his doctor to take a rest cure. “So I am now quite lonesome and speak scarcely with anybody in private.”
It was, and remains, a minor mystery to those who observed their powerful friendship. “I used to see them walking across the path from Fuld Hall to Olden Farm every day,” the Swiss-born Armand Borel, who came to the Institute a little after Gödel, told me as I sat in his office at the Institute. “I do not know what it was they spoke about. It was most probably physics, because Gödel, too, was interested in physics, you know.8 They didn’t want to speak to anybody else. They only wanted to speak to each other,” he concluded with a shrug.
It is important in understanding the relationship between Einstein and Gödel, in trying to peer behind Straus’s bemused “somehow they understood each other very well,” not simply to stop short at the easy explanation that these two were uniquely each other’s intellectual peer, that they constituted, in the logician Hao Wang’s words, a “two-membered ‘natural kind’ consisting of the leading ‘natural philosophers’ of the century.”9 There is much more, even beyond membership in so exclusive a set, to be said for what bound the two together.
There are the surface similarities, of course. There is the fact, for example, that they had both done their most important work in Central Europe, in German-speaking lands, from which they had been forced to flee. But in this respect at least, Einstein and Gödel were hardly unique in the Princeton of their day. Scholar after scholar had had to flee Vienna and Göttingen and Budapest for places like Pasadena and Princeton. The fact that they were political exiles, who spoke the same native tongue and found themselves strolling the improbable landscape of suburban New Jersey, certainly does not begin to explain the special bond between them, which mystified even their fellow refugees.
There are other striking similarities between the two. There is, for example, the fact that both of them had done their most important work when quite young men. Einstein had been 26 in 1905, his annus mirabilis, when, as an obscure patent clerk in Bern, Switzerland, he had published his articles on (special) relativity, the light quantum, and Brownian motion, as well as completed his Ph.D. dissertation. Gödel’s results (which also were three in number, though it is the first incompleteness theorem that far outshines all else10) had been accomplished three years before reaching the comparable age.
More important than this shared autobiographical detail is the fact that each man had toyed, at an even earlier age, with the idea of entering the field that th
e other had chosen. Gödel had entered the University of Vienna intending to study physics. Einstein had first thought of becoming a mathematician. There is a sense in which each saw in the other a realization of what he might have become had he opted otherwise, and there was undoubtedly a certain fascination in this.
Still there is far more that bound the two of them together. I would like to propose that the reason for the profound understanding and appreciation that held between these two “very, very dissimilar people” lay on the deepest level of their revolutionary ideas. They were comrades in the most profound sense in which thinkers can be comrades. Both men were committed to an understanding of reality, and of their own work in relation to that reality, that placed them painfully at odds with the international community of thinkers.
One might have thought, with each having presented results so uniquely transformative that their respective fields had been forced to remake themselves to contain these results at their center, that the last thing that Einstein and Gödel would have felt is marginalized. Feelings of alienation, disaffection, dismissal, isolation are for the noninfluential and the failed. But disaffected and even dismissed they felt, and, moreover, disaffected and dismissed in profoundly similar ways, at the metalevel of their fields, the level at which you interpret what it all means.
There is a sense, then, at least as I have tried to penetrate to the core of a friendship that mystified onlookers, in which Einstein and Gödel were fellow exiles within a larger exile, and it is a sense that goes far beyond the geopolitical conditions that caused them to seek safety in Princeton, New Jersey. I believe that they were fellow exiles in the deepest sense in which it is possible for a thinker to be an exile. Strange as it might seem for men so celebrated for their contributions, they were intellectual exiles.