# GODEL AND THE NATURE OF MATHEMATICAL TRUTH [1]

GÖDEL AND THE NATURE OF MATHEMATICAL TRUTH

EDGE: You seem to have a strange collection of interests: mathematics and physics and philosophy, on the one hand, and fiction, on the other. Why would a novelist teach philosophy of science and have enough of an interest in mathematics to write a book on Gödel's incompleteness theorems?

REBECCA GOLDSTEIN: To me the affinities are natural. It's a matter of different forms of beauty. Mathematicians and physicists are just as guided by principles of elegance and beauty as novelists and musicians are. Einstein told the philosopher of science Hans Reichenbach that he'd known even before the solar eclipse of 1918 supported his general theory of relativity that the theory must be true because it was so beautiful. And Hermann Weyl, who worked on both relativity theory and quantum mechanics, said "My work always tried to unite the true with the beautiful, but when I had to choose one or the other, I usually chose the beautiful." I would say the same thing about writing novels. The question comes up, when you're using ideas in math or physics or philosophy in a work of fiction, just how far can you distort the idea to make it work in the novel, work as a metaphor. I try to keep as close to the truth as possible, but when I have to choose, then I choose Weyl-ly.

Mathematics seems to be the one place where you don't have to choose, where truth and beauty are always united. One of my all-time favorite books is A Mathematicians' Apology. G.H. Hardy tries to demonstrate to a general audience that mathematics is intimately about beauty. He gives as examples two proofs, one showing that the square root of 2 is irrational, the other showing that there's no largest prime number. Simple, easily graspable proofs, that stir the soul with wonder. I read G.H. Hardy's book the summer after graduating college, right before going on to graduate school. It was the same summer that I read Newman and Nagel's lovely little book, Gödel's Proof. It was great to read them at the same time. Nothing could have convinced me more of Hardy's point about mathematics and beauty than reading at the same time about Gödel's proof.

Hardy's book is not only intellectually engaging but also moving, even elegiac, because he was mourning his loss of mathematical creativity. He was in his fifties, and, as he wrote, mathematics is a young man's game. He wrote the book after his first suicide attempt and before his second—and successful—suicide attempt. C.P. Snow talked him into writing a book that would describe the special joys of mathematical creativity to those who had never experienced it. The book had a big impact on me, impressing me with the hollowness of bifurcating the intellect and the passions. The intellect is passionate.

And of course it was Snow, too, who coined the phrase that you've one-upped, the two cultures, warning that practitioners of the mathematical sciences, on the one hand, and the arts and humanities, on the other, are losing the ability to understand each other, to the impoverishment of all. Your idea of bridging the two cultures, creating a third culture, approaches the bridge primarily from the scientific side. A lot of your Edge scientists engage themselves with the kinds of questions that have traditionally been addressed by humanists, questions that have to do with what it means to be human. But there's movement from the other direction as well. There are other other narrative artists —I'm thinking of the novelists Richard Powers, Alan Lightman, and Dan Lloyd, and the playwrights Michael Frayn and Paul Parnell (who wrote QED about Richard Feynmann) —who are integrating mathematical and scientific ideas into their work. It's a hopeful spot in the culture.

I like to think that the shallower aspects of the intellectual scene of the last century have played themselves out. I mean in particular the assaults on objectivity and rationality, which often take the form of attacks on science. There's nothing less exhilarating than reducing everything to social constructs and to our piddly human points of view. The pleasure of thinking is in trying to get outside of ourselves—this is as true in the arts and the humanities as in math and the sciences. There's something heroic in the idea of objective knowledge; the farther away knowledge takes you from your own individual point of view, the more heroic it is. Maybe the new ideas that are going to revitalize the arts and humanities are going to be allied with the sciences. It's not, of course, that novels will all address scientific themes—that would be ridiculously restrictive. But I hope that the spirit of expansiveness that's associated with the pursuit of scientific truth can get infused into the arts and humanities.

EDGE: How do these connections between the sciences and humanities relate to Gödel, if they do?

GOLDSTEIN: One of the strange things that happened in the twentieth century was that results from mathematics and physics got co-opted into the assault on objectivity and rationality. I'm thinking primarily of relativity theory and Gödel's incompleteness theorems.

The summer before entering college I had to read a book that was popular back then, by an NYU philosopher, William Barrett, called Irrational Man. It was, vaguely existentialist and it argued pretty strenuously that man constructs all truths. It spoke a lot about Nietzsche and Heidegger, but there were a few pages on relativity theory and the incompleteness theorems, arguing that the upshot of these results was that even in physics and mathematics there's no objective truth and rationality: everything is relative to man's point of view, and that the proofs of mathematics are incomplete because there's no foundation for mathematical knowledge. Everything is infected with man's subjectivity, leaving us no grounds for distinguishing between rational and irrational. I read this right before entering college and it took the wind out of my sails. I had been excited about learning the important things but now I was reading that the one important thing to learn is that there aren't any important things, at least none that we haven't made up, which seemed to undermine their importance. I liked making up things as well as anyone; after all, I was a future novelist. Still, the thought that this making-up business penetrated even to mathematics deflated me.

And the irony is that both Einstein and Gödel—who had a legendary friendship when they were at the Institute for Advanced Study—could not have been more committed to the idea of objective truth. Both were super-realists when it came to their fields, Einstein in physics, Gödel in mathematics. The irony is sharpened in Gödel's case since not only was he a mathematical realist, believing that mathematical truth is grounded in reality, but, even more ironically, it was this meta-mathematical conviction that actually motivated his famous proofs.

Gödel was a mathematical realist, a Platonist. He believed that what makes mathematics true is that it's descriptive—not of empirical reality, of course, but of an abstract reality. Mathematical intuition is something analogous to a kind of sense perception. In his essay "What Is Cantor's Continuum Hypothesis?", Gödel wrote that we're not seeing things that just happen to be true, we're seeing things that must be true. The world of abstract entities is a necessary world—that's why we can deduce our descriptions of it through pure reason.

One of the interesting things about Gödel is that he became enraptured with Platonism when he was a student, an undergraduate at the University of Vienna. He took a course in philosophy with Heinrich Gomperz. When I read Gödel's papers in the basement of the Firestone Library at Princeton, I discovered that later in life he was sent a questionnaire asking about his philosophical influences. The sociologists had listed a bunch of weighty philosophers, and Gödel disregarded almost all of them—(though he said that Kant was a little bit influential). According to Gödel, the greatest influence on his life was Professor Gomperz, who introduced him to philosophical position, Platonism. Gödel's response was strong. He switched his major from physics to mathematics, specializing first in number theory, since he thought that it was there that he would find results closest to his Platonist heart. That shows you the philosophical orientation motivated his work. It seemed that Gödel hatched an audacious ambition while still a young student: to produce a mathematical result that would have meta-mathematical implications implications, or at least suggestions, about the nature of mathematics itself. It's as if a painter produces a picture that has something to say about the nature of beauty, perhaps even something to say about why beauty moves us. Mathematics forcefully raises meta- questions, since it is a priori, immune from empirical revision, necessary. How can we have knowledge of this sort? What's it about? The truths we learn about the spatio-temporal realm are all ultimately empirical; and they're contingent. They're not immune to empirical revision, which is why physics requires expensive equipment for testing its predictions against the world. Mathematicians are cheap; they are thus cost-effective for universities —which is another way of saying that mathematics is a priori. But this aprioricity and necessity present problems. What can necessary, a priori truths be about? Maybe they're about nothing at all, other than the formal systems we construct mere consequences of manipulating symbols according to rules, as in chess. Platonism rejects this answer. It claims that mathematics is descriptive of abstract entities, of numbers and sets, that exist separately from our attempt to understand them through our mathematical systems

Platonism has always had a great appeal for mathematicians, because it grounds their sense that they're discovering rather than inventing truths. When Gödel fell in love with Platonism, it became, I think, the core of his life. He happened to have been married, but the real love of his life was Platonism, and he fell in love, like so many of us, when he was an undergraduate.

Platonism was an unpopular position in his day. Most mathematicians, such as David Hilbert, the towering figure of the previous generation of mathematicians, and still alive when Gödel was a young man, were formalists. To say that something is mathematically true is to say that it's provable in a formal system. Hilbert's Program was to formalize all branches of mathematics. Hilbert himself had already formalized geometry, contingent on arithmetic's being formalized. And what Gödel's famous proof shows is that arithmetic can't be formalized. Any formal system of arithmetic is either going to be inconsistent or incomplete.

On October 7, 1931, when he was 24 years old, he announced his result, a proof that showed that any formal system that is rich enough to express arithmetic will have a proposition which is true and unprovable. He actually showed how to construct, in each consistent formal system, a true arithmetical proposition that can't be proved. It sounds paradoxical, because if he's showing that it's true, hasn't he proved that it's true? But it's not paradoxical. The proof skirts the edge of paradox.

Part of the immediate background of Gödel's Proof is not only Hilbert's Program, but the Vienna of the late '20s and early '30s. When he was a student, Gödel had been invited by Hans Hahn, one of his professors, to attend the legendary meetings of the logical positivists, what came to be know as the Vienna Circle. Sometimes Gödel is categorized as a logical positivist because of this early association. And it's true that Gödel didn't argue with them while he attended their meetings, held in a dismal room in the basement of the University of Vienna. But just because he chose not to argue doesn't mean he didn't vehemently disagree with them. A passionate Platonist must be profoundly at odds with logical positivists.

Gödel mistrusted our ability to communicate. Natural language, he thought, was imprecise, and we usually don't understand each other. Gödel wanted to prove a mathematical theorem that would have all the precision of mathematics—the only language with any claims to precision—but with the sweep of philosophy. He wanted a mathematical theorem that would speak to the issues of meta-mathematics. And two extraordinary things happened. One is that he actually did produce such a theorem. The other is that it was interpreted by the jazzier parts of the intellectual culture as saying, philosophically exactly the opposite of what he had been intending to say with it. Gödel had intended to show that our knowledge of mathematics exceeds our formal proofs. He hadn't meant to subvert the notion that we have objective mathematical knowledge or claim that there is no mathematical proof—quite the contrary. He believed that we do have access to an independent mathematical reality. Our formal systems are incomplete because there's more to mathematical reality than can be contained in any of our formal systems. More precisely, what he showed is that all of our formal systems strong enough for arithmetic are either inconsistent or incomplete. Now an inconsistent system is completely worthless since inconsistent systems allow you to derive contradictions. And once you have a contradiction then you can prove anything at all.

EDGE: Do you think that Godel's proof reveals something about the relationship between language and reality?

GOLDSTEIN: Gödel's did not see language as constructive of reality. Language rather is subordinate to reality. But that doesn't mean that language isn't important in the proof, that there isn't something fascinating going on in the languages spoken, so to speak, within the proof. In fact, the proof is a layering of different kinds of language, and the way in which the proof links these layers is the essence of the proof's strategy.

There's the purely mathematical language, and then there is the meta-language that's describing the formal systems themselves, the rules of the formal systems. The cunning is that he gets sentences which say something straightforwardly arithmetical to also say something about themselves. These sentences manage to speak on two levels, and this double-speak is accomplished through what we now call Gödel numbering. Each of the elements in the system has a number, and you can also assign numbers to the well-formed formulas composed of those element, and to the sequences of well-formed formulas, which are what proofs are—by combinatorial rules. Given any string of symbols you can derive the unique number that goes with that string, and vice-versa. Because of the Gödel numbering those propositions are saying something straightforwardly arithmetical but they're also saying something about themselves, something about their own formal properties. This is the way in which self-referentiality—gets utilized in the proof.

Self-referentiality, which produces many devilish logical problems—the logician Raymond Smullyan has written particularly well about them in his entertaining books—goes back to the time of ancient Greece, when Epimenides, the Cretan, said that all Cretans were liars. This is a paradox. There's nothing, strictly speaking, paradoxical in Epimenides' statement, but it does lead to the following sentence, which is, famously, paradoxical: "This sentence is false". What Epimenides was saying was: "I'm a Cretan, everything that Cretans say is false; this very thing that I'm saying is false". And this last statement is indeed paradoxical. Because if it's true then it's false and if it's false then it's true. And the mind crashes.

Gödel appropriated this ancient form of paradox in order to produce a proposition which we can see is true precisely because we can see it's unprovable. This proposition has a purely straightforward mathematical meaning but it's also a proposition that speaks about itself. : The proposition is, in effect: "This very proposition is unprovable". Is it true, or is it false? If it's false, then its negation is true. Its negation says that the proposition is provable. So, assuming the system to be consistent, if this problematic double-speaking proposition is false, its negation is true, which would mean the problematic proposition itself is thus provable. So if it's false it can't be false. If it's false it's true. Therefore it has to be true. But unprovable!

That's how he does it. That's the proposition that's both true and unprovable. And remember that it has a strictly arithmetical meaning as well. That's accomplished through the Gödel numbering. So he's shown that in any consistent formal system of arithmetic there will be true but unprovable arithmetical propositions. A formal system of arithmetic is either going to be inconsistent or incomplete.

The second incompleteness theorem, which follows pretty straightforwardly from the first, proves that one of the things that you can't prove in a formal system of arithmetic is the consistency of that very system. So while you're working in a system you can't prove within that system that it's consistent. And of course an inconsistent system is worthless because you can prove anything in an inconsistent system.

EDGE: Could you say something about the milieu in which Gödel lived at the time when he produced these theorems?

GOLDSTEIN: Between the two world wars, Vienna was a place of intellectual ferment. There was disappointment and disillusionment with the old ways of doing things. The horrors of World War I were still a current memory and there was an attempt to throw off the old ways, to rethink things, in many areas. So we see psychoanalysis starting there, and the modernist architecture of Adolf Loos, and Arnold Schoenberg with his atonal music. There was a lot of intercultural, interdisciplinary dialogue. The logical positivists were very much part of this. They tried to rethink the foundations of knowledge, to rethink the foundations of language. They claimed that if we purify language we'll be able to purify knowledge.

As the logical positivist would have it, so much of the horror that had resulted in the Great War had come from confused thinking. People claimed to know things they couldn't possibly know. The political concerns gave a fervor to the movement. If we were more modest in our claims to knowledge, perhaps we'd avoid some of the tragedy that our species is prone to. A lot of them—Neurath and Carnap certainly—had left-leaning politics as well. They toned this down when they got to America. But in Vienna, when Gödel was there, there was a fervor in trying to rethink language and the limits of what we can say.

Wittgenstein had an enormous influence on the Vienna Circle of logical positivists. He had written the Tractatus Logico-Philosophicus in the trenches of World War I. In that book, published in 1922, he tried to delineate the outer reaches of language and show that language has a border around it. There are rules that allow us to say what's sayable, and there's a great deal that lies on the other side. Most of the important things, he notes, can't be said.

But there he disagreed with the positivists. The positivists fought about the other side of the divide; outside of the sayable there was nothing at all. Beyond that which we can say there's nothing. But Wittgenstein in fact believed that the most important thing, what he referred to as "the mystical," is merely unsayable, not that it doesn't exist at all. If we try to say it we will utter nonsense. But it's important nonsense. Thus, Wittgenstein was not really a positivist. But the Vienna Circle understood him to be a positivist and they admired him tremendously. They undertook to study the Tractatus and they studied it for an entire year. And it's a slim book! They met on Thursday evenings and studied the Tractatus sentence by sentence by sentence. It has the appearance of great clarity, but in fact it's rather obscure. It's quite beautiful, quite poetic, more artistic than scientific, as in fact the logician Frege wrote to Wittgenstein, in one of those seemingly flattering letters that an author probably would rather not get. Though the logical positives were inspired by Wittgenstein, he had an ambivalent attitude towards them. There were a few that he spoke to: Schlick and Friedrich Waissman, an acolyte, who worshiped Wittgenstein. Wittgenstein was a powerful personality, a man of great charisma.

Wittgenstein was in his early 40s around that time. He had gone to Cambridge before the war as an undergraduate, and had galvanized Bertrand Russell. Russell, together with Alfred North Whitehead, had written Principia Mathematica, trying to reduce arithmetic to logic and set theory. An overwhelming problem was how to defend your system of logic and set theory from paradoxical sets. Russell himself had discovered one of these atrocities of a priori reason: the set of all sets that aren't members of themselves. This leads to a paradox—you see this when you ask whether this set is a member of itself: if it is then it isn't and if it isn't then it is. In other words, there can't be such a set since it leads to paradox. Therefore, your formal system has got to block the formation of this sort of set, and of others that lead to paradoxes. Russell and Whitehead's Principia Mathematica had a set of special rules—they called it the theory of types—that would keep certain problematic sets—the kind that yield paradoxes—out of set theory. But it was completely ad hoc. They didn't have a theory as to what was going on there, and they issued an invitation for mathematicians and logicians to come up with a better explanation— a theory of when language is clean and pure, and when is it tainted by paradox. How can we purify our mathematical logical language so that it can't form paradoxes? That lured Wittgenstein to Cambridge.

Wittgenstein had an enormous effect on Russell. First Russell thought Wittgenstein had a new kind of sensibility. He gave Russell the sense that he really knew something incommunicable, that there was something he was trying to get at that Russell was not seeing it. As a result of these interactions, Russell actually gave up mathematical logic. Wittgenstein convinced him that his old ways of doing things were wrong. Russell said that he couldn't quite understand what Wittgenstein was saying, but he felt in his bones that he must be right. That's the kind of effect that Wittgenstein had on people.

Then he went off into the trenches wrote the Tractatus. It had an enormous effect on those thinkers in Vienna who were trying to rethink the foundations of all knowledge and all language. And as I mentioned, he always disavowed them. He said they never understood him. One of the ways of really understanding this is that last proposition of the Tractatus: " Of what we cannot speak thereof we must be silent." It's ambiguous. It could mean that all facts can be said and they can be said clearly, or it could mean that there are facts that are out there, but our language is not adequate for expressing them: that our language is leaving out chunks of reality. If we try to express the unsayable in language, we'll violate the rules of language and commit nonsense.

Wittgenstein seemed to be saying the latter; that there are aspects of reality that exceed our ability to express them. Positivists understood it in the other way, the former way, that the criterion for meaningfulness exhausts all facts. Everything that can be said can be said clearly, and there's nothing else out there.

Interesting, isn't it, that here are philosophers obsessed with trying to say things precisely, with giving the rules for precision, and what they're saying about precision isn't precise enough for them to understand one another. You can understand Gödel's saying, as he's quoted saying to the mathematician Menger one night when they were walking home together from one of the meetings of the positivists, something like: "The more I think about language the less possible it seems to me that we ever understand one another."

I believe that Gödel was taking the measure of his elders. His views about mathematics, about meaningfulness, about what we can know, about how important language is to shaping reality, were out of sync with those of the positivists.

When I was a graduate student in philosophy, there were still some professors, Cambridge philosophers, who had been in the inner Wittgensteinian circle, and sometimes they'd come to Princeton and I'd get tolisten to them. They still retained some of Wittgenstein's mannerisms. He had had various tics; he would make certain faces and guttural sounds when he was thinking, and they made these faces and sounds, too. He had a mesmerizing effect on people. But not on Gödel.

Here's what I think. Gödel was irked by Wittgenstein. He not only held meta-mathematical views that were deeply at odds with Wittgenstein's—and though Wittgenstein wasn't a positivist, his views on the foundations of mathematics, especially in the Tractatus, were in the positivist vein—but he was irked, too, I think, by the fuss that those around him, the positivists of the Vienna Circle, made about Wittgenstein. Maybe he was even irked by the fuss that Wittgenstein made about Wittgenstein. We only let people get away with that sort of stuff if we think they're worthy. And by Gödel's lights, Wittgenstein wasn't.

In any case, these Wittgenstein-dominated conversations were the discussions he was frequenting when he was incubating his own ideas on the foundations of mathematics, not only incubating but, for all we know, actually working out the intuitions that would lead to his incompleteness theorems.

Gödel was a reticent man, an opaque man. He doesn't give one a lot with which to try to imagine the inner man. A novelist is trained in the art of inhabiting characters, both real and imagined. A lot of the novelist's skill resides in trying to insinuate oneself into others' inner lives. Gödel is a hard one to penetrate. I'm fairly confident that there was some strong emotion connected with Wittgenstein; I can construct a fairly convincing story to this effect. But in the end it might be a made-up story. It's compelling to me, for what that's worth, and it makes sense, psychological sense. And there's even some written evidence.

Gödel had harsh things to say about Wittgenstein later in his life. Never, of course, face to face, usually not even to other people, but in letters. Most of them he never sent, and I came upon them in Firestone among the literary remains, the Nachlass, of Gödel. Wittgenstein never accepted Gödel's result; he said in The Foundations of Mathematics, posthumously published, that his task is not to discuss Gödel, but rather to bypass Gödel. He also called Gödel's results the tricks of a logical conjurer, logical artifices. Kunststücken. Someone told Gödel about this and it was then that he let vent some of his annoyance about Wittgenstein, annoyance that was, if my psychological speculations are right, decades old, hatched long ago while Gödel listened to the positivists extolling Wittgenstein's views, understanding him to vindicate their condemnation of all metaphysical views, including Platonism.

Of course it wasn't only Wittgenstein who dismissed Gödel's theorems. There are mathematicians who still argue with the incompleteness results, sometimes on constructivist grounds, namely strict scruples about what can and, more importantly, can't be appealed to in proofs. Then there are those who accept that Gödel mathematically proved his results about incompleteness, but reject the meta-mathematical view, mathematical realism, that Gödel thought was strongly suggested by his results. There are certainly mathematical logicians who are formalists, even in the light of the incompleteness theorems. Gödel's Platonism may have psychologically motivated his search for incompleteness, for helping him to drive what he saw as a wedge between the concepts of mathematical truth and provability; but that doesn't mean that his theorems logically disprove formalism. Gödel's Platonist heart may have rejoiced in his results, as they seemed to have vindicated his belief that mathematical reality exists independent of formal systems. But Platonism isn't implied by the incompleteness results. Platonism isn't a mathematical theorem at all.

Of course, Gödel made it harder not to be a Platonist. He proved that there are true but unprovable propositions of arithmetic. That sounds at least close to Platonism. That sounds close to the claim that arithmetical truths are independent of any human activity. Philosophers of mathematics can certainly avoid the Platonist conclusion but, so long as they don't just "bypass Gödel," they have to do fancy footwork. Even Wittgenstein, who said his task wasn't to address Gödel's theorems, couldn't help returning to them again and again. He argued about them in his class with Alan Turing. And of course Turing's own work, his demonstration that we can't solve the halting problem (roughly, knowing whether a given computer program will produce a result given an input or will grind away forever), itself entails Gödel's first incompleteness theorem.

There are mathematicians who say that what Gödel did is just irrelevant to their working lives as mathematicians, that they never have to think about incompleteness, or even know what exactly it is, in doing mathematical work. So you can do your mathematics and stay out of the meta-discussion. This is probably a pretty common attitude among mathematicians. And in some sense it's a natural attitude. When you're working within the discipline you're doing what can be done within that discipline. The fish doesn't have to be an expert on the nature of water.

That's true in other field, too, say in physics. Physicists who disagree radically on the interpretation of physical theories—some thinking they're descriptive of an objective physical reality, others thinking theories are just instruments for predictions—can collaborate qua physicists, can employ the same physical theories to get out scientific results, whether theoretical or applied. Your day-to-day work as a physicist isn't necessarily going to be changed one way or the other because of your meta-view of what physics is; and your day-to-day life as a mathematician isn't necessarily going to be changed by your meta-view of what mathematics is. You don't even have to have a meta-view. Gödel's theorems only matter if you're interested in those meta- questions and to be a mathematician you don't necessarily have to be interested in those questions.

EDGE: You connect this interest in the philosophical foundations of mathematics and physics with Gödel's and Einstein's famous friendship, don't you?

GOLDSTEIN: Many of those who watched the two of them walking back home together every day from the Institute for Advanced Study deep in conversation—acquaintances of theirs told me they only wanted to speak to one another—wondered how two such different people could be so bonded. But what bonded them was that, first of all, they were so keenly interested in the meta-questions of their respective fields, those interpretive questions regarding what is it that these fields are really doing and how is it that they manage to do it.

Both of them saw their work in a certain philosophical context. They were both strong realists—Einstein in physics, and obviously Gödel in mathematics. That philosophical perspective put them at odds with many of their scientific peers. So it's interesting that these two figures were very celebrated yet felt themselves to be marginalized, and marginalized in similar ways. This may explain something of the bond which was otherwise baffling to their acquaintances. And again the two of them are joined in that same ironic twist which which I began our conversation , that their work was swept up into the general assault on objectivity and rationality. Again I think back to that summer before I entered college and read that both relativity incompleteness had shown, with the full authority of physics and mathematics, that there are no objective criteria for truth and rationality. They were both exiled from Nazified Europe, but they were also—strange to say—intellectual exiles, and that's astounding, given how central their work is usually taken to be.

Both of them had effected revolutions in their particular fields, but the way they saw their own work, the philosophical light they thought that their work was shedding, could not have been more at odds with most of their contemporaries.

Einstein and Gödel were true allies, and after Einstein died, Gödel's natural solitariness deepened, along with his paranoid tendencies. He came to a sad end. His mistrust of language was, in a sense, , vindicated. Mistrusting language he had tried to make his mathematics speak his meta-mathematical convictions, but others often interpreted his theorems to be saying precisely the opposite of what he'd meant them to say. Who wouldn't become even more distrustful of human communication? Who wouldn't retreat even more into isolation? I'm not justifying his paranoia of course. But there's some shadow of a rational response in the irrationality, too.

I'm saddened by the sense of his isolation, by how profound it must have been. It's chilling to consider that he felt the world to be so hostile that he believed his food was being poisoned and so stopped eating and so starved to death. I've spent a long time imagining what that must have felt like for such a man. And I contrast that dark and cold place in which he lived many long years and in which he ended his life with the sense of bright wonderment that I experienced that summer before graduate school, when I first understood Gödel's masterpiece of reason. He gave that experience to countless people, and we're grateful.