Best known for his Incompleteness Theorem, Kurt Gödel (1906-1978) is considered one of the most important mathematicians and logicians of the 20th century. By showing that the establishment of a set of axioms encompassing all of mathematics would never succeed, he revolutionized the world of mathematics, logic, and philosophy.
Janna Levin is a Professor of Physics and Astronomy at Barnard College of Columbia University. Her scientific research concerns the Early Universe, Chaos, and Black Holes. Her second book—a novel, A Madman Dreams of Turing Machines—won the PEN/Bingham Fellowship for Writers that “honors an exceptionally talented fiction writer whose debut work…represents distinguished literary achievement…”
Simply Charly: Can you begin by telling us what the difference is between Gödel’s first and second Incompleteness Theorems?
Janna Levin: It’s a subtle thing, actually, that what Gödel did, when he devised his Incompleteness Theorems, was he really showed that there were some truths that could never be formally proven within the context of arithmetic.
And it was a very formal notion: it means to start with axioms, take the transformation rules to come up with theorems, and the idea of proof was very tied to that very specific kind of mechanistic idea. And he was able to show that there were some facts about numbers, really arithmetic—nothing could seem somehow more elementary or natural than arithmetic and mathematics—that could not be proven within the context of these axiomatic systems.
Now, what he really, really did, subtly, in the first theorem, was show that you can’t simultaneously prove something as complete and consistent; “complete” meaning that all facts could be proven. You could either show that it was consistent and incomplete, or complete and inconsistent. And by “inconsistent,” you mean something really bad, like, that two contradictory statements could simultaneously be proven. That’s much more serious, in a way than incompleteness, which just says that there are some facts that are true or false, but can’t be reached by this kind of approach in mathematics. His second theorem was to show that the consistency of mathematics itself was one of those undecidable statements; that you could never prove that arithmetic was consistent. That was really shocking to people. But he believed it was consistent, because he really believed in the sanctity of mathematics, that it wasn’t ever going to be the case that something was simultaneously true and false; that it was going to be consistent, but it wasn’t something that ever could be proven to be the case.
So to clarify, his second theorem is really the statement that the consistency of mathematics cannot actually be proven, or the consistency, more specifically, of arithmetic, is one of the undecidable statements in arithmetic.
SC: David Hilbert believed that eventually all mathematical things could be or would be defined. Gödel’s theorem disproves that. Are there any other commonly held beliefs or theories that the Incompleteness Theorem puts into doubt?
JL: It’s interesting. Gödel’s Incompleteness Theorem has a kind of far-reaching, philosophical implication that’s very strange. One of the strangest things about Gödel’s construction is that he built, effectively, a mathematical sentence that amounted to the claim that this statement is unprovable. So the statement is making a claim about itself.
He translates that completely into numbers. He codes it, in a sense, like you would do a binary code now, he did a kind of prime number code so that it became a relationship among numbers. The claim that this statement is unprovable becomes a claim about prime numbers being factorized in certain ways, and so it really becomes a totally arithmetic statement. In that way, he was able to put it back into arithmetic and show that, in fact, it could never be proven in the context of arithmetic.
However, its claims about itself can’t be proven. There’s this little glitch—it’s not a glitch—but this little subtlety that I seem to be able to recognize intuitively that it’s a true claim. It has somehow made a true claim about itself, but it’s not a claim I can prove, by starting with axioms, going through transformation rules, and combining theorems.
And so, something strange has been separated in notions of truth and provability, and intuitive notions. And I think that one of the far-reaching implications is about how the mind works: what does it mean to try to model the mind as a mathematical system? When we step outside of our mathematical system, we move in hierarchies of systems, we look outside of mathematics and see a truth that we can’t prove. That’s really amazing and subtle, and it was one of the most curious aspects, once people got over the shock, of the blow to Hilbert’s program. I think it was one of the most curious aspects of his theorem.
So, when people think about designing in artificial intelligence, for instance, it calls into question the idea I could simply program an intelligence by setting up axioms and transformation laws, because that system can’t recognize the truth of a claim that I can recognize by stepping outside of that axiomatic system. So it has to be, somehow, more hierarchical and more complex than I can code. And there’s been a lot of work about complexity of codes and complexity of theorems that’s been very largely inspired by that first discovery of Gödel’s. So I think it’s going to keep reaching into the future when we talk about things like designing an artificial intelligence, and that will go on, long into the future.
SC: What were Gödel’s peer’s reactions when the discovery was announced? It would have doomed to failure the efforts of all the world’s greatest mathematicians, so tell us about their reaction.
JL: It’s interesting. There wasn’t much reaction, at all. I think it was such a subtle thing that he had done, that people did not immediately recognize the significance, the implications. And he, in fact, went to a conference in Hilbert’s hometown and spoke about this at this conference. And when they summarized the big results of the conference, he wasn’t even mentioned.I just don’t think people fully grasped what he had just said. One person who did was Von Neumann, an absolutely brilliant mathematician who was located at the Institute for Advanced Study in Princeton. And that really changed the direction of Gödel’s life, because he ran up to Gödel, after his lecture, and was very engaged in trying to clarify what had just happened. He understood the significance of this accomplishment before, I think, anyone else. And so, he brought Gödel to the Institute for Advanced Study, where he eventually emigrated and spent the rest of his life. But it took twenty years for the most significant mathematician since Aristotle to get a proper status at his own institution.
SC: What are the implications of this discovery on the field of, let’s say, mathematics and logic?
JL: It’s interesting that, I would say, certain things like physics, which are based on mathematics and logic, still seem kind of unaffected. It’s actually something that I’m interested in—what can incompleteness say about the program for finding a Theory of Everything in physics?
If there can’t be a Theory of Everything in mathematics, can there be one in physics? And it’s possible that there could, but it seems like we haven’t fully understood the implications of his ideas for physics. In mathematics, a lot of sub-branches are complete. It’s not an issue. Nobody would come along and would say, “This problem’s been really hard to solve. Maybe it’s undecidable.” You would really have to prove undecidability based on very strict criteria before you would ever abandon the program to find a mathematical theorem or proof.
So, basically, mathematics goes on, kind of unaffected. It’s consistent, so we believe, and then we can’t prove it. Although in some sub-branches you can prove consistency of completeness, but that means it’s still a sound theory, or it’s still a sound way of approaching, understanding the world, and so, we’re never going to have a result that’s contradictory. We’re never going to come into a true paradox, because Gödel’s theorem isn’t really a paradox. But we might occasionally encounter undecidable propositions.
And so, it’s possible that as a mathematician approaches certain problems, certain very, very unusual problems that are highly self-referential and have certain strange characteristics, that self-referential aspect is absolutely key—then you might get into a Gödelian type of tangle. But basically, mathematics proceeds, and the attempt to formalize mathematics can proceed in a modified form. And it’s still incredibly powerful and a miracle that we can understand the world this way. It’s really quite a miracle.
We mine our own mind, and that’s how we uncover these seemingly external truths. Gödel only believed in the mind part. He wasn’t so sure about the external reality part, and he often said things like—I think I even saw it on the Simply Gödel website—that “I don’t believe in natural science.” And I think what he meant by that is he just wasn’t sure about external reality. He was really doubtful of it, which is so strange. But he really believed that mathematical concepts existed, they were real, and that the mind migrated to this pure, platonic reality, and migrated over reincarnations. He really believed in the transmigration of the soul. So he was pretty out there.
SC: To what other areas, outside of mathematics and logic, such as physics, theology, or philosophy, did Gödel contribute?
JL: He most directly, himself, personally contributed to physics. Obviously, his ideas had big implications that inspired other people to carry on, in probably all fields like theology, philosophy, physics, metaphysics, artificial intelligence, computer science—to everything he really was very influential. But he, himself, contributed to physics in a very interesting way, and it was kind of a similar idea to throwing a wrench in the works a bit.
He became very close to Einstein when he was at the Institute. And Einstein once said, “I only go to the Institute for the pleasure of Gödel’s company.” I think people have probably heard that quote before. In fact, he was such a recluse, Einstein might have been one of the only people he was speaking to, for months at a time, on these walks. And he became interested in Einstein’s ideas about the relativity of space and time.
Einstein’s theory of a curved space-time is a theory that you and I, traveling at different speeds or in a curved space-time, or somebody near a black hole, or somebody else, might measure, literally, the passage of time differently. We might age differently from each other. I won’t notice a difference. But you might look at me, near a black hole, and think, I’m aging very slowly, like thirty years have elapsed for you and only one year has elapsed for me. So there’s this very strange but very real relativity of space and time in Einstein’s theory.
So what Gödel did, he was able to construct a very unusual space-time, one which was rotating, which is not what our universe is doing, but just a hypothetical—imagine a universe, which is rotating in some specific way—and he was able to show that he could find specific observers in that world who could travel back in time.
And this was really wild. Einstein did not want to accept this, either. Again, here comes Gödel, saying, actually, you can, in some very unusual circumstances, time travel. And so, there was a lot of battling about it, that maybe you could never build that space-time. And since then, there’s been a handful of space-times we can construct, where we can move back in time, where you can find a handful of observers who can move back in time. If you can move back in time, does that mean that the laws of physics are going to become absurd? So it becomes a kind of subtle and crazy business.
SC: Did he not also come up with some ontological arguments on the existence of God?
JL: I think many times in Gödel’s life, he was interested in spiritual things. One was the transmigration of the soul, one was this kind of platonic reality, the existence of this pure, mathematical reality. And another was proving the existence of God. And he did struggle with trying to write these proofs of the existence of God.
I don’t think he ever got to a stage where he was particularly impressed with the outcome, and it seems that he had often kind of abandoned it. But later in his life, he did come back to it. And one of the things that bothered him so much—not just about the existence of God, but also about free will and the spirit and the soul—was this idea that he inspired that thought could kind of be mechanized, in a certain way. This is something that Alan Turing, the great British codebreaker who cracked the German Enigma code in World War II—it’s something that Alan Turing really brought forward.
He said, “Actually, I can mechanize this whole process of thinking about mathematics,” and he was totally inspired by Gödel. He was using Gödel coding and was very influenced by his results. And he basically invents the idea of the computer in the process. I build a machine. The machine mechanistically goes through certain operations, given certain input. And it performs mathematical operations. It thinks.
And Turing really believed he was going to build a machine that could think as well as a human being. And a lot of Gödel’s ideas get wrapped up in this. Now, Gödel could not argue with the correctness of Turing’s formalism and his mathematical proofs, but he was very disturbed by the idea that the human mind could be reduced to this mechanistic approach, even though he, himself, had sort of influenced this idea. And I think a lot of the trying to prove the existence of God was wrapped up in trying to reject Turing’s claim that we were reducible to machines.
Turing went further, because he said, “Not only can I build a machine that thinks we are machines that think. That’s all we are, we are machines that think. We are made out of biological matter, but we are basically machines that follow this mechanistic process and think.” And he abandoned the idea of the soul, does the opposite. He was a religious kind of adolescent, who becomes a kind of materialist, in the sense of being a naturalist, while Gödel really rails against it.
SC: Is anyone today working to further or bring a new dimension to Gödel’s Incompleteness Theorem?
JL: Gregory Chaitin is a great mathematician, incredibly clever guy, who just did, I think, beautiful things with Gödel’s and Turing’s ideas, and modernized it completely. And so there are probably people like Gregory Chaitin, out there, who are going to suddenly break through with some lovely new way of thinking about it.
What Gregory Chaitin really did is, he started thinking about real numbers versus the kinds of numbers that we use every day. And what that means is—this is something that Turing also showed, inspired by Gödel—that there are an infinite number of numbers, about which we can know, essentially, nothing. They have an infinitely long sequence of digits and it’s a toss of the coin what those digits are, and every statement about them is sort of undecidable and unprovable, in the way that Gödel originally conceived.
And Chaitin formalized these ideas in terms of computer science thinking and in terms of notions of complexity. And that was just brilliant. I think that that’s very advantageous, in a modern way of thinking about it. So for instance, I’ve been interested in this guy, Seth Lloyd at MIT, who’s been thinking about the universe as a computer, and thinking about the universe as processing, in its mechanistic way, that both Gödel and Turing thought about, of facts about the world.
And so, you start to apply Chaitin’s thinking to that, to the universe as a computer system, where the laws of physics are like the software, the code, in some sense. And you can combine Gödel’s and Turing’s and Chaitin’s idea all in one. So I’m sure there are people out there doing these weird, little things, trying to figure things out. Connect it to quantum mechanics. But I don’t know of a global program.
SC: Take a few moments, if you would, and tell us about Gödel’s life, specifically, the very, very tragic end to his life.
JL: I had known about Gödel’s work for a long time. It’s just one of the things you know, that there are these incompleteness theorems. You go to graduate school in physics or mathematics, it’s just something that you learn. It was many, many years before I heard about how he died. I was having lunch with a friend, who tells me that he starved himself to death. And I just, I could not even believe the description of the story.
He was so afraid of being poisoned, because he was really quite paranoid, that he refused to eat food, to the point of starving himself to death. It was like he had these two struggling desires; one to live and one to die. It was like he was simultaneously a hypochondriac and suicidal. And somehow he balanced his tension for seventy-some years. But he eventually died of self-starvation. It’s just shocking. And I became very interested in his life. Just who is this person?
I think there’s something that moves me about physics and mathematics, which has to do with the objectivity, which has to do with the fact that it doesn’t matter who discovers something; it’s just true or not true. If it hadn’t been Gödel, it would have been somebody else.
At the same time, there’s a harshness to that, which I just don’t think is honest. I think that it’s still a human endeavor. And it was Gödel, it was Einstein, it was Turing. And I think that we do long to know about those stories: those narratives are part of what we value and part of what we try to understand. And I became very interested in his personal story.
He was a really unusual, extreme, totally extreme person. I think people want to imagine all mathematicians are like this, but this was an absolutely off scale, extreme person. He was a hypochondriac, constantly—even in mid-summer,-in layers of wool and clothes and coats. He didn’t talk to people for days. He would call the person in the office next to him, rather than look them in the eye, and speak to them, face-to-face.
And yet, he was kind of charming. There was something sophisticated about him, and funny, and kind of charming, and he was liked. He had nervous breakdowns, probably several. It’s unclear how to document them, because he would often commit himself to these—I guess we would call them sanatoria, but they might have been health spa retreats, basically to recoup mentally and sometimes they were outright breakdowns, where doctors were saying that his mental condition was perilous.
He left Princeton several times by boat to go back to Europe in a state of total hysteria and despair. And his wife, who was several years his senior, would kind of nurse him back to sanity and health because he was continually dropping to anorexic weights: 65 pounds, 85 pounds. I think when he died he was 65 pounds.
And she would literally—she was kind of a more working class. People accused her of being a washerwoman in an unkind context. They were a little bit elitist towards her. And she would nurse him back to health. She would cook for him heavy, German fare and kind of spoon-feed him, literally morsel by morsel, until his weight was back up and he was more stable.
Without her, it’s clear he would not have survived for as long as he did. But when he did die, it was because she had taken so ill that she couldn’t care for him. And she died shortly after. But when Adele was too ill to care for him, he was lost.
And so, I think he’s a fascinating person, precisely because of the contrast. The great stories are the stories about the tragic hero, the hero in the sense of what makes them great is also their downfall. That is the combination that I think is so gripping—that it was his mental strength that made him extraordinary and that’s why we’re still talking about him today.
And yet, that kind of intense devotion to logic also led him astray, led him away from reality, unable to brush some of the fuzzy things under the rug that we do every day about reality and what’s true and what’s not true. He just couldn’t. He couldn’t let go.
And I wouldn’t say that it led him to be insane, exactly, because he still wasn’t illogical. It was a kind of consistent logic to the end. He wasn’t seeing purple elephants flying around the sky. It was consistent in its own strange way, and yet it just constantly led him in these two directions of living and dying, living and dying, until I just think it was all lost.
SC: So, what prompted you to write a book about Turing and Gödel, and especially to write it as a novel?
JL: I knew I wanted to write a fictionalized account. I tried not to. I tried to write non-fiction. I think there are some great books that describe the theorems, like Nagel and Newman’s book, which is not as well-known. It’s just a beautiful, beautiful approach to the ideas. And there are books on their biographies. I just couldn’t write this non-fiction book. It was missing everything.
I think what I loved about the idea of writing fiction was the idea that you could structure the entire book on their theorems, in a sense. That there was a self-referential character, the way Gödel’s theorems were self-referential. That it toyed with the idea of what we could know to be true, and the limits of what we could ever know, both about mathematics and about their lives. That somehow all these things were based on the ancient Liar’s Paradox, and in the book the narrator is a self-professed liar, who’s trying, yet, to get to the truth. And all of those ideas I just thought were so much more complex, and that in fiction, and with overwhelming hyperbolic language, I could do more to create an impact of that idea, in the sense of that, the feeling that hits you in your solar plexus of what that means to know something’s true, but can’t prove it to be true. And to only be able to approach truth, and yet for it to be your obsession. But I could do that better in fiction than I could in non-fiction. And it was just really about a freedom of language and structure that I could have done it.
And so, while it was my initial hope to write fiction, I tried not to. And I eventually realized that it was the only way the book could actually be written.