By Ernest Davis
Journey to the Edge of Reason: The Life of Kurt Gödel. By Stephen Budiansky. W.W. Norton & Company, New York, NY, May 2021. 368 pages, $30.00.
Journey to the Edge of Reason: The Life of Kurt Gödel. By Stephen Budiansky. Courtesy of W.W. Norton & Company.
Kurt Gödel’s incompleteness theorem, which demonstrates the existence of mathematical propositions about natural numbers that one can neither prove nor disprove, is probably the best-known mathematical theorem of the 20th century. This theorem, together with Gödel’s other seminal results—the inference system of first-order logic is complete, the axioms of arithmetic cannot be proved consistent, the axiom of choice is consistent with the Zermelo-Frankel axioms of set theory, and the continuum hypothesis is consistent with the ZFC axioms—firmly establish his position at the pinnacle of logicians.
Stephen Budiansky’s new biography, Journey to the Edge of Reason: The Life of Kurt Gödel, is the sixth English-language biography of Gödel; among 20th-century mathematicians, only Alan Turing, Srinivasa Ramanujan, Emmy Noether, Bertrand Russell, and Norbert Wiener have more. Previous works about Gödel include biographical accounts; philosophical disquisitions; and expositions of Gödel’s mathematical results, achievements in mathematical logic, and relation to the field’s history.1
Gödel’s life fell sharply into three parts: his childhood in Brünn, Austria-Hungary (1906-1924); his young adulthood at the University of Vienna (1924-1940); and his later years at the Institute of Advanced Studies (IAS) in Princeton, NJ (1940-1978). Biographies of Gödel naturally tend to follow this breakdown. But the first part of Budiansky’s book only occasionally describes Gödel or his family; rather, it provides an entertaining historical and cultural portrait of the Austro-Hungarian Empire at the turn of the 20th century. Oddly, Budiansky barely mentions World War I — a reader who does not know what happened in Europe between 1914 and 1918 would hardly guess it from his account.
Gödel conducted most of his important mathematical work during his years in Vienna. Budiansky’s narrative of these years includes an extended description of the Vienna Circle — a philosophical enterprise with the ambitious goal of formulating science as the logical analysis of sense data, inspired by the model of mathematics’ construction from logic in Principia Mathematica by Alfred North Whitehead and Bertrand Russell. Gödel’s doctoral advisor, Hans Hahn, was a founder of the Vienna Circle, and Gödel was one of very few students admitted as a regular member. He frequently attended gatherings, befriended some of the younger participants, and even presented his incompleteness theorem at a group meeting. However, the group did not contribute to Gödel’s mathematics and he rejected its philosophy.
Budiansky’s book comprises careful, readable accounts of Gödel’s most important results, particularly the incompleteness theorem and the consistency of the continuum hypothesis. He includes an appendix with a proof of the incompleteness theorem, though like most such presentations he omits the full demonstration that one can express the predicate “Provable(p)” in the language of arithmetic. Budiansky’s coverage of other mathematical topics is sketchier. In a book for general readership, omitting some of Gödel’s more technical accomplishments—such as his proof of the compactness theorem and contributions to intuitionistic logic—is certainly reasonable. It is less satisfactory for important colleagues and interactions to go unmentioned. For example, Alonzo Church—a leading logician in the U.S. during Gödel’s time—and his many brilliant doctoral students were at Princeton University, only a few blocks away from the IAS. Budiansky barely mentions Church, even though Gödel had significant interactions with Church and his students Stephen Cole Kleene, John Barkley Rosser, and Dana Scott.2
Gödel visited the U.S. three times in the 1930s, but he was in Vienna when World War II broke out in September 1939. After Gödel obtained permission to leave Vienna, John von Neumann arranged for him to receive a visa to the U.S. and a position at the IAS. Gödel and his wife Adele traveled east, taking the Trans-Siberian Railway to Vladivostok, a steamer to Yokahama, an American ship to San Francisco via Honolulu, and a train to Princeton. He never again left the east coast of the U.S. Gödel spent the rest of his life at the IAS, working primarily on mathematical philosophy and publishing little. He had a small number of close friends there on whom he relied, especially Albert Einstein (the two walked to work together every morning) and mathematician and economist Oskar Morgenstern. Gödel wrote to his mother in Vienna warmly and frequently; they remained very close, and she and his older brother visited Princeton in 1958.
In general, Budiansky’s book portrays mathematicians quite positively. The importance of Gödel’s results—especially the incompleteness theorem—and the quality of his work were immediately and almost universally acknowledged, barring a handful of exceptions like Ernst Zermelo and Ludwig Wittgenstein. Throughout his life, Gödel’s colleagues served as supportive and protective friends.
Though quiet, Godel was friendly, sociable, and even charming in his younger years, with a whimsical sense of humor. He was always immaculately dressed and groomed; according to fellow student Olga Taussky-Todd, Gödel was even “a bit ostentatious about his success in attracting the opposite sex.” But he was not at all egotistical. In fact, when Morgenstern met Gödel’s mother in Vienna after World War II, he was surprised to learn that she had no idea of her son’s great successes. In 1960, when Paul Cohen completed the proof that the continuum hypothesis is independent of Zermelo-Fraenkel set theory (which Gödel had been working on for decades), Gödel was nothing but excited and enthusiastic in his response. He was not an elitist either culturally or socially; his artistic tastes were mostly lowbrow and his wife Adele—to the horror of his family and many of his friends—was an uneducated dancer.
Concurrent with Gödel’s success was turmoil in both the outside world and within his own psyche. Budiansky’s description of the rise of fascism in Austria is dramatic and harrowing. The government, led by Engelbert Dollfuss and Kurt Schuschnigg, grew steadily more authoritarian and anti-Semitic, until it was swept away by Hitler’s Anschluss. The University of Vienna offered no refuge. “In 1923, the German Student Union demanded that all books in the library written by Jews be marked with a Star of David,” Budiansky writes. “Vienna’s Neue Freie Presse reported that on a stroll around the main building of the university, a visitor would encounter little but anti-Semitic posters and hate literature.” In 1933, Gödel’s colleague and friend Karl Menger wrote to Oswald Veblen about the circumstances. “[T]he situation at the university is as unpleasant as possible,” he said. “Whereas I still don’t believe that Austria has more than 45 percent Nazis, the percentage at the university is certainly 75 percent, and among the mathematicians I have to deal with, not far from 100 percent.” Moritz Schlick, the chair of the Vienna Circle, was killed by a former student in 1936; though Schlick was neither Jewish nor politically active, the Nazis defended his murder as a justified reaction to his perverse philosophy. It must be noted, however, that there is no trace of either anti-Semitism or Nazi sympathy in Gödel’s record.
Gödel himself dealt increasingly with obsessive-compulsive disorder, severe hypochondria, and an obsession with conspiracy theories. He experienced complete mental collapses in 1936 and 1970 that were marked by extreme paranoia and a fear of being poisoned; he only ate food that Adele spoon-fed him. In the end, Gödel effectively starved himself to death — he weighed just 65 pounds when he died.
As many scholars have noted, a straight line runs from some of Gödel’s intellectual characteristics—even his intellectual virtues—to other aspects of his mental state. He was always obsessed with determining the reasons for things and philosophically believed that everything has a reason; this mindset can clearly slide into conspiracy theories. Gödel was also known for being exceptionally careful and wanting to get everything exactly right before publishing or announcing any results. In the context of his own health, this same caution turned into hypochondria. Yet his paranoia, which ultimately killed him, was much less of a constant in his psychological makeup and does not seem to connect in the same way.
Nevertheless, when we think of Kurt Gödel, let us remember the young man—handsome, charming, impressively brilliant—who saw more deeply into the nature of mathematical proofs than anyone had before and realized that a mathematical proposition could be encoded as a number — that an entire proof could be encoded as a number. He understood that “being provable” was just an arithmetic property like “being prime” and that, through an ingenious twist of legerdemain, he could thereby construct a proposition that mocked the hopes of mathematicians with its paradoxical assertion: “This sentence is not provable.”
1 Existing biographies of Gödel include Reflections on Kurt Gödel by Hao Wang (1987), Logical Dilemmas: The Life and Work of Kurt Gödel by John W. Dawson, Jr. (1997), Gödel: A Life of Logic by John L. Casti and Werner DePauli (2000), Incompleteness: The Proof and Paradox of Kurt Gödel by Rebecca Goldstein (2005), and Simply Gödel by Richard Tieszen (2017).
2 Thanks to Martin Davis for helpful information on this point.
Ernest Davis is a professor of computer science at New York University’s Courant Institute of Mathematical Sciences.