Mathematician Kurt Gödel, right, and physicist Albert Einstein, left, taking a walk in Princeton, 1954

Kurt Gödel and the romance of logic

The great theory of this emaciated genius of philosophy defeated the finest minds of the 20th century—and rescued the idea that there are truths that humans can never prove
December 13, 2018

We are sometimes inclined to make celebrities out of intellectuals despite—or perhaps precisely because of—their producing work we can never hope to understand. Bertrand Russell’s oddly old-fashioned dress sense and aristocratic bearing remain familiar features on the cultural landscape, as are Albert Einstein’s friendly face and shock of white hair. Indeed, such was the popularity of the aging Einstein that he was, decades after coming up with relativity theory, offered the (largely ceremonial) presidency of Israel. The elder Russell, meanwhile, was invited on to radio and television to give his opinion on everything from communism to what kind of lipstick women should wear. The reason he was invited on to the media was not, of course, that he was an authority on these subjects, but that he had, in his younger days, written abstruse things on mathematical logic and the philosophy of mathematics. The most notable of these, Principia Mathematica—in which he and his co-author Alfred North Whitehead put forward an axiomatic system of logic upon which they hoped to build, first arithmetic and then the whole of mathematics—is considered formidably difficult even by experts in the field.

The logician and philosopher Kurt Gödel passes the Einstein/Russell test, in doing work whose importance is beyond argument, but can also seem beyond comprehension as well. If you wanted to make the case that he should join them as a celebrated public figure, you could point out that among his work is an important contribution to the interpretation of Einstein’s relativity theory, and that he pulled the rug out from under the project on which Russell spilled most sweat. And Gödel’s advocate would start out with more name-recognition to build on than you might imagine. In 1999, when Time magazine conducted a survey of its readers to determine the 20 most influential thinkers of the 20th century (a poll topped by Einstein), Gödel, who has often been described as “the most important logician since Aristotle,” came ninth—ahead of Keynes, Watson and Crick of DNA fame and world wide web pioneer Tim Berners-Lee.

And yet, in truth, in popular culture he remains miles away from Russellian let alone Einsteinian status. Almost none of us, I’d bet, has any idea what he sounded like. Be honest, would you recognise the face on the right overleaf if it wasn’t alongside an article about Gödel? Just possibly you might, if you had previously come across him in a biography of Einstein and saw the picture of the two of them together at the Institute for Advanced Study in Princeton, New Jersey: Einstein looking bucolic and genial with his braces and pot belly, and Gödel in a white jacket, looking, by comparison, formal, distant and emaciated. But, apart from the Einstein link, what do we really know about Gödel? Even for most of those who have heard of him, he is not so much a recognisable figure or a rounded character, but rathera name.

The reason why this name—at least—lives on is mainly because of the fundamental importance of his “First Incompleteness Theorem.” It establishes that axiomatic systems like Russell and Whitehead’s Principia Mathematica cannot prove every arithmetical truth. Such systems will always be incomplete. The much less discussed “Second Incompleteness Theorem” says that the consistency of a theory of arithmetic cannot be proved within that theory. I will concentrate on the First, and try to explain later how Gödel managed to prove it and why it is considered important.

“The general idea that there are truths that can’t be proved has an appeal far beyond logic”

For now, it is worth noting that the general idea that there are truths that cannot be proved has an appeal far beyond logic, which has led the American mathematician Jordan Ellenberg to describe Gödel as “the romantic’s favourite mathematician.”

Thus, while the man may not be much known, his work has inspired all sorts of things—being cited in discussions of philosophy, literature, science and even theology; he has also been celebrated in poetry (“Homage to Gödel” by Hans Magnus Enzensberger) and music (Hans Werner Henze’s Second Violin Concerto contains a setting of Enzensberger’s poem). His famous theorem has been popularised many times, most notably in Douglas Hofstadter’s 1979 book Gödel, Escher, Bach: An Eternal Golden Braid, which won a Pulitzer and became an unlikely bestseller.

Mr Why

Though many people who knew Gödel, including Russell (who met him in Princeton in 1943), assumed he was Jewish, he was not. He was born into a German-speaking Lutheran family in 1906 in Brünn, Moravia, which was then in the Austro-Hungarian empire, but now, renamed Brno, is the second city of the Czech Republic. Unlike other German speakers in the area, Gödel did not speak Czech (although he was an able linguist who mastered English and French) and, even after the establishment of Czechoslovakia in 1918, he regarded himself an Austrian.

He was a shy, nervous boy of delicate health, but his school records show that he was an exceptional student. He was so curious about everything that he was nicknamed “Herr Warum” (Mr Why). Only once did he fail to receive the highest mark, and that, oddly enough, was in mathematics.

In 1924, he enrolled as a physics student at the University of Vienna, switching to mathematics in 1926. At about the same time, he was invited by the mathematician Hans Hahn to attend the meetings of the Vienna Circle, a group of philosophers led by Moritz Schlick that espoused “logical positivism,” a philosophy that rejected mysticism, religion and metaphysics in favour of an emphasis on the use of mathematical logic to address philosophical issues and a commitment to what they called the “scientific outlook.”

Gödel agreed with their view on the relevance of mathematical logic to philosophy, but, from his student days right up until his death, maintained two beliefs that were anything but positivistic. The first was that he believed in God. Towards the end of his life, he revealed to friends that he thought he had found a proof of God’s existence using modal logic. A sketch of the proof was found among his Nachlass (papers), and reveals it to be a variant of Leibniz’s version of the “Ontological Argument,” which argues that God’s existence follows, with necessity, from His properties. Gödel also believed in the afterlife. The second departure from positivism, arguably more important in shaping his intellectual outlook, is that he believed—one is inclined to say “passionately”—in mathematical Platonism.

Platonism stands in contrast with Constructivism. The issue between the two can be expressed as the difference between regarding the mathematician as an inventor and regarding him or her as a discoverer. The Constructivist believes mathematics to be the invention of the human mind; the Platonist believes it to be a set of truths discovered by human reason. The difficulty with the Constructivist view is that it turns mathematics into a branch of fiction; the difficulty with the Platonist view is that it appears to commit us to the existence of mathematical (and logical) objects. We can see five sheep, but we can’t see the number five. Yet Platonists insist that, in some sense, the number five exists. And, just as the planet Uranus was there in space, waiting to be discovered by Herschel, so, Platonists believe, numbers—five, zero, minus-five, the square root of minus-one—are there, together with all the relations between them, waiting to be discovered by mathematicians.

Platonism was dismissed by most members of the Vienna Circle as the kind of metaphysics that belonged to a pre-scientific worldview. For Gödel, on the other hand, it was the very foundation of his thinking. Rebecca Goldstein, in her absorbing intellectual biography Incompleteness: The Proof and Paradox of Kurt Gödel, writes that as an undergraduate, “Gödel fell in love with Platonism.” (She also emphasises, as Gödel himself did, the connections between his commitment to Platonism and his “Incompleteness Theorem”).

In 1930, Gödel received his doctorate for a dissertation that (in a revised version) was published in the same year as “The Completeness of the Axioms of the Functional Calculus of Logic.” What it showed is that what is now called “first-order logic,” but was then called “functional calculus,” is complete. That is to say, every logical truth expressible using the language of first-order predicate logic (a system of rules and symbols that modern logicians use to analyse the relations between subject-predicate propositions) can be proved within an axiomatic system. This would not, however, include any statements of arithmetic, since arithmetic is not expressible in first-order predicate logic. For arithmetic, you need a more powerful kind of logic, such as the one used by Russell and Whitehead in Principia Mathematica.

Immediately after receiving his PhD, Gödel began the work that resulted in his celebrated theorem. It took only a few months for him to realise what his work might lead to, and by the autumn of 1930 he was ready to announce his incompleteness result. He did so almost casually on the final day of the Königsberg Conference on Epistemology of the Exact Sciences on 7th September. The logician Jaakko Hintikka has written, “It is a measure of Gödel’s status that the most important moment of his career is the most important moment in the history of 20th-century logic, maybe in logic in general.” It took a little while, however, for the importance of Gödel’s announcement to be recognised. At the conference itself, it was greeted with silence. The one person present who showed any realisation of its importance was the Hungarian John von Neumann, a polymath mathematician who would lay important foundations in fields from game theory to fluid dynamics, and—more pertinently for Gödel’s story—had just been appointed to the Institute for Advanced Study in Princeton. But by the time the proof was published just a few months later, in January 1931, it was highly anticipated and Gödel himself was in great demand. In 1933-4 he was invited to lecture in Princeton on his result. Notes of this lecture provided the first English publication of the theorem.

Reality and a shadow

Gödel remained in Vienna throughout the 1930s, lecturing at the university and continuing his work in mathematical logic. His work at the university was interrupted several times by ill health and mental disturbance, some of which was attributed by friends, family and Gödel himself to the fallout caused by his love for Adele Nimbursky, of whom his parents, particularly his father, disapproved because she was six years older, a divorcee and a dancer. He married her in 1938 anyway, and the two of them moved to Princeton, where he remained until his death in 1978. In 1947, he applied for—and received—US citizenship, but the citizenship hearing was the one exam in his life that he nearly blew. He announced that he had studied the US constitution in detail, and—no doubt, forensically examining its propositions one at a time and perhaps testing it against thought experiments against wild possible futures in which the president was allowed to get out of control—he had discovered how the US could legally be turned into a dictatorship.

“In solving Einstein’s field equations, Gödel modelled a universe where it is possible to travel back in time”

Nonetheless, he was and remained safely installed in Princeton, where his interests became more overtly philosophical, and, though he wrote a great deal, his perfectionism ensured that he published very little. The papers he did publish, including most notably “Russell’s Mathematical Logic” (1944), and “What is Cantor’s continuum problem?” (1947), present his thoroughgoing Platonist view of mathematics. In “A Remark About the Relationship Between Relativity Theory and Idealistic Philosophy” (1949), his last published philosophical paper, Gödel offers a novel solution to the Einstein field equations of general relativity. There are many possible solutions to Einstein field equations, each one of which describes a different model of spacetime. Gödel’s solution (which Einstein himself considered a major contribution to theoretical physics) models a universe which is rotating and in which it is possible to travel back in time.

Gödel used this to argue that time is not objective. Plato had been right: the objects of mathematics are more real than the spatio-temporal world of everyday experience. We do after all, and as Plato’s most famous analogy had it, live in a cave where the things we can have empirical contact with are but shadows of reality. So much for positivism.

In his last years, Gödel’s interests became almost solely focused on philosophy and in 1959 he began an intensive study of the work of both Leibniz and Husserl. But by now he had not published in a decade, and he would publish nothing else before his death nearly 20 years later. Most of what we know of Gödel’s later philosophy comes from the conversations recorded by his friend Hao Wang, and the papers he left in his Nachlass, which have now been published in the five volumes of Gödel’s Collected Works. In 1974—with Gödel’s permission—Wang published extracts from conversations he had had with the logician in From Mathematics to Philosophy. Wang continued after Gödel’s death with Reflections on Kurt Gödel (1987) and again in 1996 with A Logical Journey: From Gödel to Philosophy.

Having always been prone to both mental and physical illnesses, in his last 20 years Gödel became increasingly frail, thin and disturbed. He was very concerned about his health, but also distrustful of almost everybody except his wife. He would make appointments with doctors and not attend them, or, having attended them, refuse to take the medicine provided. In 1976, he retired from the Institute. A year later, Adele underwent major surgery and was hospitalised. As she was the only person Gödel trusted to prepare food for him, this was disastrous for his health. He refused to eat, even after he too had been admitted to hospital. He died on 14th January 1978. His death certificate recorded that he died of “malnutrition and inanition caused by personality disturbance.”

Loops of logic

The year after his death came Hofstadter’s Gödel, Escher, Bach, and with it a dash of posthumous fame. Hofstadter’s book is not an easy read: a dense near-800 pages discussing logic, mathematics, philosophy, biology, psychology, physics and linguistics. But at its heart is a clever connection that Hofstadter makes between the music of Bach, the drawings of the Dutch graphic artist MC Escher and Gödel’s “First Incompleteness Theorem” via the notion of a “strange loop,” which turns out to be a great way of conveying the flavour of Gödel’s proof of his most influential work.

“The First Incompleteness Theorem” states the following. Given any consistent axiomatic system of arithmetic, there will be at least one statement about numbers which cannot be proven in that system, and neither can its negation. Moreover (though this is not actually part of the theorem, but of Gödel’s reflections upon it), that statement has to be true. In other words, the Russell-Whitehead project was doomed, and so too were the hopes of anyone else who wanted to derive the whole of arithmetic from a system of logic. There is, and can be, no such thing as a consistent and complete axiomatisation of arithmetic. Moreover, that there cannot be such a system is itself a truth that can be inferred from a theorem of mathematical logic.

To see how this can possibly be so, Hofstadter’s notion of a “strange loop” is helpful. His first example of such a thing is Bach’s “neverending” or “endlessly rising” canon, the “Canon per Tonos.” In this piece, with every repetition of the canon, the key rises a tone until it returns to its original key of C minor. The strange loop in question, then, is that it appears to have achieved the impossible feat of rising up and up and then, somehow, finding itself back where it started. In the work of Escher, a similar effect is achieved pictorially. In his famous lithograph, Waterfall (1961), for example, we see water falling further and further down in six steps only for it to return to its original position. In Drawing Hands (1948), Escher has created what Hofstadter calls a “two-step Strange Loop”: one hand is shown drawing another, which is in turn drawing the first.

Gödel’s own “strange loop” is a self-referential arithmetical statement: a statement of arithmetic that somehow manages to say something about itself. We have been used to self-referential statements in ordinary language for a long time, one famous example being the statement by Epimenides the Cretan, “All Cretans are liars.” Was he telling the truth or not? If he was, he can’t have been. More succinctly, think of the statement “This sentence is false.” If it is true, then it is false, and, if it is false, then it is true. This is what Hofstadter calls a “one-step Strange Loop.”

You start to see how self-reference can take us into a strange and paradox-laden corner of logic. But, still, it is not easy to grasp how a statement of arithmetic could be about itself. A statement of arithmetic is about numbers: the statement “7 + 5 = 12” is about the relationships between the numbers 7, 5 and 12. It is not about the statement that “7 + 5 = 12.” Statements about arithmetical statements are what logicians and mathematicians call “meta-mathematical statements,” an example of which would be: “The statement ‘7 + 5 = 12’ is provable in the axiomatic system that Russell and Whitehead created in Principia Mathematica.”

The “strange loop” at the heart of Gödel’s proof of his famous theorem is a kind of arithmetical analogue of the Epimenides Paradox, though it is important to realise that the result is not a paradox, but a theorem—a theorem about the necessary incompleteness of consistent axiomatisations of arithmetic, that is, a theorem that implies that for every consistent system of arithmetic, there will be at least one arithmetic truth which that system cannot prove.

In order to prove that theorem, Gödel showed how it was possible to construct a statement about numbers, an arithmetical statement, that is alsoabout itself. He did this by an ingenious method now called “gödel numbering,” which assigns a unique number to each symbol of the system (+, -, 1, 2, etc), each sequence of symbols (this would include every proposition, such as “2 + 2 = 4”), and each sequence of sequences of symbols (this would include every proof). Tallied up in this way, any arithmetical statement could be represented by a unique number, which in turn allows arithmetical statements (statements about numbers) to be about arithmetical statements.

Thus, he showed how it was possible for an arithmetical statement about two numbers to be also a meta-mathematical statement about whether a particular arithmetical statement can or cannot be proven in the system of—say—Principia Mathematica. This allowed him to create the self-reference needed for a “strange loop” and to create an arithmetical statement that in effect says, in an echo of Epimenides, “I am not provable in the Principia Mathematica system.”

If the system of Principia Mathematica is consistent, then this statement must be true. Why? Because, if it were false, then it would be provable in the Principia system, and, as it says that it is not provable, this would introduce an inconsistency into the system. So, if the system isconsistent, then the statement must be true. But, if it is true, and (this is the crucial bit) it expresses an arithmetical statement, then it expresses an arithmetical truth—an arithmetical truth that cannot, by virtue of its own truth be proven in Principia Mathematica. In sum, if the system of arithmetic is consistent, then it is necessarily incomplete.

Discovery and a lifetime

But why should anyone who is not a logician be concerned? Other minds that are both especially mathematical and especially creative have suggested some sweeping—and self-evidently important—applications. The theoretical physicist and mathematician Roger Penrose, for example, has argued that Gödel’s theorem shows that “Strong AI” is false: our minds cannot be computers, and that by extension the intelligence of computers will never fully replicate them. But there is another, more general, way of grasping the importance—and that starts with the broader question of Platonism. Not even Gödel thought that his incompleteness result proved Platonism to be true, but he did think it lent it some support. How? By seeming to undermine one important element of many anti-Platonist perspectives—the view that, in mathematics, truth and provability are the same thing.

This idea that the two are equivalent is associated with the image of the mathematician as an inventor rather than a discoverer. If—as apparently instead established by the “Incompleteness Theorem”—it is possible, in any given system, for there to be an arithmetical statement that is true in that system but not -provable in that system, then one might infer that truth and provability cannot be the same thing. Until and unless the anti-Platonist can find something wrong with Gödel’s proof, he or she seems to be left with the task of finding some way of interpreting it that does not involve accepting the idea that the truth of an arithmetical statement is independent from its proof, a notion that seems awfully close to the Platonic conception of arithmetical truths being (in some sense) “out there.”

Whether this challenge can be met is still—nearly 80 years after the Königsberg Conference—a matter of controversy. I think it is fair to say that many philosophers of mathematics simply do not know what to say in the face of it. Both Russell and Wittgenstein, two of my biographical subjects who were among the cleverest people of their generation, failed to understand it. So, if you find Gödel’s work and its consequences hard to make sense of, don’t worry: you are in good company. Nevertheless, it seems likely that the time will come when this emaciated genius is given his full place in the cultural spotlight alongside his close friend Einstein and his brief acquaintance Russell. Both provided the inspiration and the context for his peculiarly impenetrable, philosophically baffling, and arguably fundamentally important contribution to intellectual life.