Truth, Incompleteness and the Goedelian Way

Is there a more powerful modern Trinity? These reigning deities proclaim humanity's inability to thoroughly explain the world. They have been the touchstones of modernity, their presence an unwelcome burden at first, and later, in the name of postmodernism, welcome company.

Their rule has also been affirmed by their once-sworn enemy: science. Three major discoveries in the 20th century even took on their names. Albert Einstein's famous Theory (Relativity), Kurt Gödel's famous Theorem (Incompleteness) and Werner Heisenberg's famous Principle (Uncertainty) declared that, henceforth, even science would be postmodern.

Or so it has seemed. But as Rebecca Goldstein points out in her elegant new book, "Incompleteness: The Proof and Paradox of Kurt Gödel" (Atlas Books; Norton), of these three figures, only Heisenberg might have agreed with this characterization.

His uncertainty principle specified the inability to be too exact about small particles. "The idea of an objective real world whose smallest parts exist objectively," he wrote, "is impossible." Oddly, his allegiance to an absolute state, Nazi Germany, remained unquestioned even as his belief in absolute knowledge was quashed.

Einstein and Gödel had precisely the opposite perspective. Both fled the Nazis, both ended up in Princeton, N.J., at the Institute for Advanced Study, and both objected to notions of relativism and incompleteness outside their work. They fled the politically absolute, but believed in its scientific possibility.

And therein lies Ms. Goldstein's tale. From the late 1930's until Einstein's death in 1955, Einstein and Gödel, the physicist and the mathematician, would take long walks, finding companionship in each other's ideas. Late in his life, in fact, Einstein said he would go to his office just to have the "privilege" of walking with Gödel. What was their common ground? In Ms. Goldstein's interpretation, they both felt marginalized, "disaffected and dismissed in profoundly similar ways." Both thought that their work was being invoked to support unacceptable positions.

Einstein's convictions are fairly well known. He objected to quantum physics and its probabilistic clouds. God, he famously asserted, does not play dice. Also, he believed, not everything depends on the perspective of the observer. Relativity doesn't imply relativism.

The conservative beliefs of an aging revolutionary? Perhaps, but Einstein really was a kind of Platonist: He paid tribute to science's liberating ability to understand what he called the "extra-personal world."

And Gödel? Most lay readers probably know of him from Douglas R. Hofstadter's playful best-seller "Gödel, Escher, Bach," a book that is more about the powers of self-referentiality than about the limits of knowledge. But the latter is the more standard association. "If you have heard of him," Ms. Goldstein writes, perhaps too cautiously, "then there is a good chance that, through no fault of your own, you associate him with the sorts of ideas - subversively hostile to the enterprises of rationality, objectivity, truth - that he not only vehemently rejected but thought he had conclusively, mathematically, discredited."

Ms. Goldstein's interpretation differs in some respects from that of another recent book about Gödel, "A World Without Time: The Forgotten Legacy of Gödel and Einstein" by Palle Yourgrau (Basic), which sees him as more of an iconoclastic visionary. But in both he is portrayed as someone widely misunderstood, with good reason perhaps, given his work's difficulty.

Before Gödel's incompleteness theorem was published in 1931, it was believed that not only was everything proven by mathematics true, but also that within its conceptual universe everything true could be proven. Mathematics is thus complete: nothing true is beyond its reach. Gödel shattered that dream. He showed that there were true statements in certain mathematical systems that could not be proven. And he did this with astonishing sleight of hand, producing a mathematical assertion that was both true and unprovable.

It is difficult to overstate the impact of his theorem and the possibilities that opened up from Gödel's extraordinary methods, in which he discovered a way for mathematics to talk about itself. (Ms. Goldstein compares it to a painting that could also explain the principles of aesthetics.)

The theorem has generally been understood negatively because it asserts that there are limits to mathematics' powers. It shows that certain formal systems cannot accomplish what their creators hoped.

But what if the theorem is interpreted to reveal something positive: not proving a limitation but disclosing a possibility? Instead of "You can't prove everything," it would say: "This is what can be done: you can discover other kinds of truths. They may be beyond your mathematical formalisms, but they are nevertheless indubitable."

In this, Gödel was elevating the nature of the world, rather than celebrating powers of the mind. There were indeed timeless truths. The mind would discover them not by following the futile methodologies of formal systems, but by taking astonishing leaps, making unusual connections, revealing hidden meanings.

Of course, those leaps and connections could go awry. Gödel was an intermittent paranoiac, whose twisted visions often left his colleagues in dismay. He spent his later years working on a proof of the existence of God. He even died in the grip of a perverse esotericism. He feared eating, imagined elaborate plots, and literally wasted away. At his death in 1978, he weighed 65 pounds.

But he was no postmodernist. Late in his life Gödel said of mathematics: "It is given to us in its entirety and does not change, unlike the Milky Way. That part of it of which we have a perfect view seems beautiful, suggesting harmony." That beauty, he proposed, would be mirrored by the world itself. These are not exactly the views of an acolyte devoted to Relativity, Incompleteness and Uncertainty. And Einstein was his fellow dissenter.

An interview with Rebecca Goldstein at Butterflies and Wheels:

An Interview with Rebecca Goldstein

By Ophelia Benson

Rebecca Goldstein has a new book out: Incompleteness: The Proof and Paradox of Kurt Gödel.
Readers at Science Daily call Incompleteness ’Outstanding’ and ‘Superb’.
Butterflies and Wheels: Alan Sokal and Jean Bricmont call chapter 11 of their book Fashionable Nonsense: Postmodern Intellectuals’ Abuse of Science: ‘Gödel’s Theorem and Set Theory: Some Examples of Abuse.’ They give a quotation from Régis Debray as an epigraph: ‘Ever since Gödel showed that there does not exist a proof of the consistency of Peano’s arithmetic that is formalizable within this theory (1931), political scientists had the means for understanding why it was necessary to mummify Lenin…’ The chapter’s first sentence starts, ‘Gödel’s theorem is an inexhaustible source of intellectual abuses…’
Sokal and Bricmont go on to quote more such abuses, from Debray, Alain Badiou, and Michel Serres, who wrote, ‘Régis Debray applies or discovers as applicable to social groups the incompleteness theorem valid for formal systems…’
Paul Gross and Norman Levitt examine literary critic (or ‘theorist’) Katherine Hayles’ musings on Gödel in Higher Superstition: ‘Hayles then cites the Gödel incompleteness result as the deathblow to the Russell-Whitehead program…This is intended to figure the movement away from post-Enlightenment ideals of “universal” knowledge to postmodern skepticism…’
Is this a widespread view of Gödel? Is it a view held solely by people who don’t actually understand Gödel’s work? Are there any mathematicians or logicians who think Gödel is a social theorist or a postmodernist?
Rebecca Goldstein: I’m not sure that there is a “widespread view of Gödel.” While I was writing “Incompleteness” and people asked me what I was working on these days, I usually drew a blank stare when I said his name. Sometimes mentioning the title of Douglas Hofstadter’s popular book, “Gödel, Escher, Bach,” brought on a faint gleam of recognition. So, by and large, Gödel - unlike his soul-mate, Einstein - is strangely unknown, and this anonymity is in itself something I wanted to address. I say in the book that Gödel is the most famous person that you probably haven’t heard of, and that if you’ve heard of him you probably have, through no fault of your own, an entirely false impression of what it was he did to the foundations of mathematics.
Which brings me to the crux of your question. Among “humanist” intellectuals who do invoke Gödel’s name, he is often associated with the general assault on objectivity and rationality that gained such popularity in the last century. I’d often find myself pondering which would be the preferable state of affairs regarding Gödel, anonymity or misinterpretation. Which would Gödel have preferred? I’m going to indulge in “the privileged position of the biographer” to presume I know the answer to the latter question, at least: Gödel, who was so passionately committed to the truth, would have far preferred utter oblivion to the falsifications of his theorems that have given him whatever fame he has in the non-mathematical world.
And what falsifications! He had meant his incompleteness theorems to prove the philosophical position to which he was, heart and soul, committed: mathematical Platonism, which is, in short, the belief that there is a human-independent mathematical reality that grounds our mathematical truths; mathematicians are in the business of discovering, rather than inventing, mathematics. His incompleteness theorems concerned the incompleteness of our man-made formal systems, not of mathematical truth, or our knowledge of it. He believed that mathematical reality and our knowledge of mathematical reality exceed the formal rules of formal systems. So unlike the view that says there is no truth apart from the truths we create for ourselves, so that the entire concept of truth disintegrates into a plurality of points of view, Gödel believed that truth - most paradigmatically, mathematical truth - subsists independently of any human point of view. If ever there was a man committed to the objectivity of truth, and to objective standards of rationality, it was Gödel. And so the usurpation of his theorems by postmodernists is ironic. Jean Cocteau wrote in 1926 that “The worst tragedy for a poet is to be admired through being misunderstood.” For a logician, especially one with Gödel’s delicate psychology, the tragedy is perhaps even greater.
I’ll give you just one example of misinterpretation, not only because it’s quite typical, but also because it had a personal effect on me. The summer before entering college I was told I would have to read, in preparation for honors English, the then-influential book, by William Barrett, called “Irrational Man” published in 1964. Gödel’s name is linked by Barrett with thinkers like Nietzsche and Heidegger, destroyers of our illusion of objectivity. After correctly stating the first incompleteness theorem (there are in fact two theorems, the second a consequence of the first, so long as one presumes that arithmetic is free of contradictions) Barrett draws this conclusion: “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.” If you negate the conclusion that Barrett draws from Gödel’s work, you end up with precisely the conclusion that Gödel himself drew! How often does that happen? A man sets out to prove a philosophical position mathematically, so that there can be no doubt. And he does prove it, but people draw precisely the wrong conclusion from it.
So, returning to your question as to whether “it [the rejection of objective knowledge] is a view held solely by people who don’t actually understand Gödel’s work?” I would answer, unequivocally: yes.
B and W: Are there any mathematicians or logicians who think Gödel is a social theorist or a postmodernist?
Rebecca Goldstein: I don’t personally know of any, and it’s hard to imagine any either. Since mathematical logic is not the most central part of mathematics, there are mathematicians who don’t pay all that much attention to Gödel’s work and may not be terribly familiar with its details. But it’s hard to imagine - even for me, with my overworked novelist’s imagination - a mathematician who would draw the sloppy conclusions that others have regarding the incompleteness theorems.
The same, by the way, can be said about Einstein’s relativity. These very names - “incompleteness,” relativity” - have encouraged very fanciful extrapolations that stand in direct opposition to the views of the scientists connected with these important results. Einstein was as little committed to the “relativity of truth” as his good friend Gödel was committed to the view that mathematics is the result of “the human activity that mathematicians carry on.”
The two of them had, by the way, a legendary friendship. Einstein was an old man and Gödel was relatively young when they became friends in Princeton, both of them refugees from Nazified Europe. (Gödel, by the way, was not Jewish, though even Bertrand Russell made the mistake of assuming that he was.) The two of them would regularly walk home from the Institute together. In fact, toward the end of his life, Einstein confided that his own work meant little to him now, and that he went to his office primarily to “have the privilege of walking home with Gödel.” They were very different in terms of their personalities - Einstein sagacious and worldly, Gödel quite hopelessly unworldly and seriously neurotic. I interviewed people at the Institute who used to watch them making the trek home each day, wondering what it was that they spoke to one another about. In my book I speculate about this deep bond, speaking of the philosophical commitments that both men shared, commitments which were so often either dismissed or misunderstood. It’s yet another irony - the story I write is full of somewhat sad ironies - that the two intellectual titans of their age should have felt marginalized, their own work often cited as the most persuasive of reasons for making the subjectivist turn. After Einstein died, Gödel really had no one else to speak with. This isolation certainly contributed to the psychological troubles that deepened and darkened over the years.
B and W: Is your book partly intended to correct the misinterpretation of Gödel’s work?
Rebecca Goldstein:Today I got an email from a professor of English at a prestigious university saying, among other things: “By the way, I too was assigned to read William Barrett's The Irrational Man, but in my Freshman year at Saint Joseph's College (now University), and from that and other references to Godel's work over the years, I came to assume that it was a sort of proto- deconstruction of the edifice of modern math and science.”
B and W: Edward Rothstein said in the New York Times: “It is difficult to overstate the impact of his theorem and the possibilities that opened up from Gödel's extraordinary methods, in which he discovered a way for mathematics to talk about itself. (Ms. Goldstein compares it to a painting that could also explain the principles of aesthetics.)”.
Can you tell us a little about that impact?
Rebecca Goldstein:Before Gödel, logic was considered more a branch of philosophy than of mathematics, the discipline associated with Aristotle rather than, say, with Gauss. Gödel developed extraordinarily powerful tools in the course of proving his theorems which both opened up new areas of mathematical research (recursion theory, for example) and also provided the means for solving more standard problems in mathematics. Mathematical logic now, as a result, has far more mathematical respectability. As Simon Kochen, a Princeton mathematical logician, told me, “Gödel put logic on the mathematical map.” But there are many other ways in which the impact of his famous proof is felt. In the course of proving the limitations of formal systems, Gödel sharpens the very concept of a formal system, as well as a whole interrelated family of concepts: The concepts of a mechanical or an effective procedure, of recursive and computable functions, of combinatorial processes and of an algorithm: this family of concepts all pretty much come down to the same thing, centering around the idea of rules that are applied to the results of prior applications of rules, with no regard to any meanings or interpretations except for what can be captured in the rules themselves. In other words, these concepts all have to do with procedures that can be programmed into computers. There’s a sense in which Gödel’s proof, especially as it was filtered through the work of Turing, helped to invent the computer.
And then there’s the more philosophical fallout from his theorems, the light they shed not only on the nature of mathematical knowledge - the fact that it can’t be captured in a formal system - but also on the nature of the mathematical knower herself. If computers run according to formal systems and our minds provably don’t, not even in knowing arithmetic, then does this mean that our minds are provably not computers? Gödel himself, rigorous logician that he was, was reluctant to draw so conclusive a conclusion; he hedged it in logically important ways. Other important thinkers, however, have drawn precisely this conclusion. Just such an argument served as the basis, for example, of Roger Penrose’s two celebrated books, “The Emperor’s New Mind” and “Shadows of the Mind.” He used Gödel’s incompleteness theorem to argue that our minds’ activities exceed what can be programmed into computers.
B and W: We’re in something of a Golden Age of intellectual biographies of philosophers. Wittgenstein, Russell, Ayer, Kant, Hegel, Spinoza and others have had rich biographies in the past decade. What sort of work do you think biography can do? Were you inspired by any biographies in particular?
Rebecca Goldstein:I didn’t think of “Incompleteness” as a biography. The aim of the book - the aim of the entire Norton series of which this book is a part - is to fit the scientific results into a “narrative framework.” I could have chosen the biographical story as my narrative arc. That strategy was the one that my editor kept encouraging me to take. He kept urging me to begin the book with Gödel’s birth in 1906 and go on from there. But I resisted him. I wanted the intellectual passions of Gödel to supply the narrative framework. Here’s the story I wanted to tell: Gödel, like many of us, first fell in love when he was an undergraduate, and that love forever changed him. Only it wasn’t a person that Gödel fell in love with but rather an idea, a grand philosophical vision that has attracted thinkers, and most especially the mathematically inclined, since the very first Platonist in the fifth century B.C.E.. Gödel met this great love of his in a philosophy class. (So much for the claim that philosophy can have no practical results: from Plato to - by way of Gödel and then Turing - google. ) He had been a physics major until his introductory course in philosophy, but he changed his major to mathematics under the influence of his impassioned Platonism. Devoted lover that he was, he resolved to find a way of proving - mathematically proving - mathematical Platonism. This was a daunting ambition. (The dichotomy between the outward timidity of this man, prey to terrible paranoid worries, and the inner vaulting intellectual confidence is one of the most fascinating things about his personality.) And then the amazing thing was that he actually went and did it, he actually produced mathematical theorems that had the philosophical consequences he was after; and then he lived to see his ideas twisted around so that they served the very viewpoint that he had hoped to conclusively refute. The drama I wanted to create, the story I wanted to tell, was all contained in this love story, a tragic love story (as almost all gripping love stories are).
B and W: Philosophers are sometimes drawn to fiction because fiction is a kind of thought-experiment. Does this aspect of fiction interest you?
Rebecca Goldstein:Well, of course, fiction is, in a certain sense, a kind of thought-experiment, but unlike the thought-experiments we use in, say, analytic philosophy in order to tease out implications or make conceptual distinctions or provide counterexamples to theses, the thought-experiments of fiction are not deliberately put forth in order to figure something out. Sure, there’s plenty of figuring out going out, for both the reader and, even more so, for the writer, but figuring out is not the paramount aspect of the deep experience of participating in fiction. I resist the view that the pleasures of fiction derive from its purely thought-experimental aspects. And yet I do think of the narrative imagination as a cognitive faculty; but its cognitive aspects are far more complicated than “thought-experiment” suggests. I’m fascinated by the unique phenomenology of reading and, of course, writing fiction, the fact that we’re drawn into a world that we know isn’t real but that we participate in almost as if it were. I think fiction manages to tamper temporarily with the boundaries of our own personal identity - we inhabit identities not our own - and also with our sense of time - narrative time is measured out in units of significance, unlike regular time which is generally just one damned insignificant thing after another - and that this tampering puts us in the way of deep insights to which we’re not usually privy. How else to explain the fact that novelists are so much smarter when they’re writing novels than at any other time, which is why it’s often such a profound disappointment to meet a revered writer in person!
B and W: Do you agree with for instance Martha Nussbaum that fiction is one of the best ways for people to learn empathy? Do you think such a view of fiction can be in tension with aesthetic judgments? If a novel has its heart in the right place but is badly written, which do you think matters more?
Rebecca Goldstein:Yes, I do think that storytelling is the basic way that we make our way into others’ psychology, which is of course central in regarding them as people just like oneself, in all the morally relevant aspects, an observation that ushers one into the moral point of view. The narrative imaginative is not only a cognitively significant faculty but a morally significant one as well. I don’t, however, think that the moral benefits of storytelling provide us with aesthetic standards. What makes art great has little to do with its uplifting tendencies - aside from the fact that great art is intrinsically uplifting.
B and W: Did you find in writing the biography that you missed the novelist’s license to assume inside knowledge of the protagonist’s thoughts? Did you find yourself wanting to bridge gaps in the evidence with Perhapses and conditionals, or were you more interested in making clear where there was evidence and where there wasn’t?
Rebecca Goldstein:In some ways Kurt Gödel was like some of the fictional characters I’ve created. I’m thinking of, say, Noam Himmel, in my first book, "The Mind-Body Problem," or Samuel Mallach, in my last novel, "Properties of Light." I’ve always been interested in geniuses, especially of the mathematical or scientific sort. Even within this small sub-set there’s a particular type of personality that fascinates me, one that’s characterized by both the intellectual heroism of thinking one’s way where no man or woman has thought before coupled together with a marked lack of heroism in any matters removed from the intellectual high ground. It’s easy to make fun of helpless and/or lunatic geniuses; but I find the dichotomy between intellectual grandeur (and in mathematics the grandeur can seem almost superhuman) and “human-all-too-human” smallness to be touching and very telling of our uneasy human position.
I came to feel extremely close to my subject while I wrote “Incompleteness.” Of course it wasn’t that all-penetrating closeness that a writer feels with her characters, but there was something sometimes approximating it. Again, this was not a biography in the usual sense of the word; I was interested in Gödel’s life only insofar as it related to his theorems: what they meant to him as well as to others, and how the latter facts affected him. (Ludwig Wittgenstein’s hostility to Gödel’s theorems is of particular importance here.) But you can see that, given what I came to believe about the man and his most famous results, there was a great deal of pathos that I saw in his story, and - the payoff of the narrative imagination - a great deal of empathetic participation in it that then helped to further along my understanding. So I did feel quite often that I’d penetrated into the soul of the man. He was an unusually reticent person in life. Aside from those animated walks to and from the Institute with Einstein, that others watched in wonderment, he eschewed social intercourse as much as possible. He mistrusted, more and more, our ability to communicate with one another. Even when he was very young, before the historical result, and its historical misinterpretations, he remarked to one of his acquaintances that the more he considered language, the less likely it seemed to him that we ever understood one another. This is the statement of a profoundly lonely person, someone in some sense constitutionally lonely, and this, too, touched me and made me all the more eager to hear what he’d wanted to say. He had wanted to communicate through his proofs, to let his deep mathematics do the speaking for him; so again, the fact that the mathematics was heard to say the very opposite of what he’d meant by it is poignant. He did write some letters protesting others’ misinterpretations of his works, particularly Wittgenstein’s. Wittgenstein had been an enormously influential figure in the Vienna that Gödel inhabited before his move to Princeton; part of the story I reconstruct is that Gödel resented Wittgenstein’s influence, especially after Wittgenstein dismissed Gödel’s theorems as ”logische Kunststücken,” logical conjuring tricks. Gödel, being the outwardly timorous man he was, never sent these letters off, but they’re there in his literary remains, in the basement of Princeton’s Firestone Library. Those unsent resentful missives - both their content and the very fact that they were unsent - played a role in my constructing a partial model of Gödel’s psychology. But about his more terrifying demons - and unfortunately it’s very clear that he had them in abundance and, in the end, they did him and his intellectual grandeur in - I would never dare to speculate. I never deluded myself into thinking I’d arrived at the sort of access a novelist has toward her fictional characters (who, strangely, also develop something of an independent life).
B and W: Does writing a biography bring up interesting epistemological issues? Do you think people with philosophical training are more aware of such issues than, for instance, historians and journalists? Or, perhaps, aware of them in different ways? As interesting issues in themselves rather than as methodological problems?
Rebecca Goldstein:I think that anyone who tries to write a biography, even a modified biography such as mine, comes smack up against the “interesting epistemological issues.” It’s a good exercise for a biographer to consider the question of how much of her own life’s narrative, at least as she tells it to herself, could even her very best friends reproduce. I was able to read the memoirs of those who had known Gödel and to make use of their observations and speculations; and I was fortunate to have met him once, though only very briefly, during a small window of his life when he was somewhat more outgoing than usual. But in the end what I was trying to do was come up with a story that would make sense of the rather small number of external facts about his life that he left us. It was a story that made much sense to me, as I hope it will to my readers. But in the end, no story about a person can be true. We are all of us, not to speak of mathematical/philosophical geniuses, far too complicated and self-contradictory to be contained in a “narrative framework.” The biographer, as much as the mathematical logician, is keenly aware of the incompleteness necessarily inherent in her project.
Incompleteness: The Proof and Paradox of Kurt Gödel .
Rebecca Goldstein's web page is here.

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February 14, 2005

Truth, Incompleteness and the Gödelian Way

An Interview with Rebecca Goldstein

By Ophelia Benson