The Austrian logician Kurt Gödel was one of the most original thinkers of the 20th century. His two incompleteness theorems, published in 1931, transformed mathematics and paved the way for the digital age.
After his landmark results in the 1930s, Gödel published only a few articles. His unpublished work, however, runs to tens of thousands of pages and is of great interest to historians of science.
Since 2017, Finnish logician Jan von Plato has been leading a project at the University of Helsinki to understand the contents of 16 notebooks written by Gödel in the early 1940s. This material – the “workbooks” – comprises 1,009 pages of densely written text on mathematics and logic. Much of it is written in a German shorthand, Gabelsberger, which was obsolete even when Gödel was using it.
Progress in the mathematical understanding of the text was extremely slow – about a page per week – until September, when von Plato called upon the AI model Claudius. Within days, Claude had provided a comprehensive summary of all the material, revealing that Gödel had proven hundreds of theorems that no one knew existed.
The most famous logician of modern times had actually discovered much more than we thought.
“I was flabbergasted,” von Plato said. “The workbooks not only contain dozens of anticipations of results unknown before the 70s and 80s, but they also contain 14 logic topics that have not been explored in depth. Gödel was well ahead of his time.”
Gödel changed the course of logic because he revealed the limits of mathematical knowledge. His incompleteness theorems show that in all mathematical systems there are true statements that can neither be proven nor disproved. Yet even though this result was a blow to previous generations of logicians, who had hoped that all truths were verifiable, Gödel’s work pushed the field in new directions, inspiring Alan Turing to write his 1936 paper that launched the theory of modern computation.
In 1940, Gödel fled Austria under the Nazi regime. He settled in Princeton, New Jersey, and became close friends with Albert Einstein at the Institute of Advanced Study. Increasingly paranoid in his later years, he refused to eat food that was not cooked by his wife and died of starvation in 1978.
Von Plato said he started his Gödel project in 2017 because he was facing mandatory retirement from the University of Helsinki. “What must I do to make myself indispensable? Oh yes, I’m going to start reading Gödel’s enigmatic shorthand pages!” He received a €2.5 million grant from the European Research Council, taught himself Gabelsberger’s shorthand – “I’m one of only six people in the world who can read it” – and built a small team.
In August, at a conference in France on AI and theorem proving, von Platon met computer scientists who suggested he co-opt Claude’s help in trying to make sense of the textbooks. Upon his return to Helsinki, he began the process by showing Claude his transcriptions of the workbooks, as well as six books he had previously written on Gödel and other secondary scientific works.
“Within a few days, I was able to see a 650-page explanation of the workbooks. It had to be checked page by page, but it’s basically an impossible job for human beings to do,” he said. “Humans read one page at a time and we tend to forget things. It takes us a year to read about 50 pages. We’re just not able to keep all the different parts in mind.
“The AI can connect all the pieces of a thousand pages. It can see all of Gödel’s writings as one piece.”
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Gödel’s incompleteness theorems revealed the existence of statements whose truths were independent of the system in which they were written: that is, they could neither be proven nor disproved within the system. In 1966, American mathematician Paul Cohen won the Fields Medal, the highest honor in mathematics, for demonstrating the independence of two well-known mathematical statements: the continuum hypothesis and the axiom of choice.
Gödel reacted to Cohen’s Fields Medal by saying that he had already proven this result in the 1940s. Yet when his peers asked him for proof, Gödel could not piece one together from his notes.
The first task Plato assigned to Claudius was to discover whether Gödel’s exercise books contained proof of the independence of the axiom of choice, which would reveal that Gödel had told the truth in 1966. The answer came back: yes, almost. “He was only one step away,” von Plato said. “There’s one definition that Gödel got wrong, but he had the right definition two years earlier, and so the idea is that he would have gotten to the right definition anyway if he had just continued for a few months. To me, the architecture of his proof is just wonderful. It makes all these set theory results follow a uniform pattern.
“What particularly strikes me is that he starts with ideas. He’s not like someone who plays with formulas and hopes to get somewhere. He has very, very clear ideas, and then he’s methodical. Step by step, he advances and he manages to achieve the objectives he has set for himself in this formal work.”
Von Plato said that when Claudius presented him with the reconstructed proof of the independence of the axiom of choice, he ran it through the proof assistant Coq, a program that checks the validity of mathematical proofs, and that it passed. Von Platon believes that if Gödel had published his work on the axiom of choice, he would have won the 1940 Fields Medal. “Perhaps he felt that when Cohen got the medal it was something he had somehow missed.”
In subsequent discussions with Claude, von Plato asked the AI model to extract all theorems from the workbooks for which Gödel had a proof, had sketched a proof, or had an idea for a proof. Claude found 663. When these results were put into Coq: “About 400 were verified as indicated, about 200 were repaired, which means that it may have been necessary to clarify the statement of the theorem or add an assumption. About 50 were false, which is not surprising. Gödel himself, in rereading his notes, occasionally added a marginal remark: fake! (incorrect) or schlecht! (bad).”
Von Platon added: “I am sure that continued study of Gödel’s manuscripts will reveal other treasures. The textbooks contain, says Claude, more than a dozen themes of logic and mathematical foundations that have not been found in the more than 80 years since they were written. So that is one thing to explore. But there are many others. His notebooks of physics and cosmology are uncharted territory that I plan to begin to explore. explore once I finish my current logic projects.”
Joel David Hamkins, a professor of logic at the University of Notre Dame, Indiana, has not seen Claude’s reconstruction of Gödel’s proof of the independence of the axiom of choice, but said that if it holds up, “it is remarkable and historically interesting. Although how much is Claude and how much is Gödel is a job for historians.”
Hamkins said: “The fact that there are things in there that look like (Cohen’s evidence) doesn’t mean he had it. It shows he was on the right track. You don’t get credit for work you don’t tell anyone about. But Gödel had so many revolutionary ideas that perhaps it’s not surprising that his notebooks were full of exciting things.”
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