The genius logician Kurt Gödel gave his name to his famous Incompleteness Theorems, which in the 1930s helped define the limits of both logic and mathematics. It might be thought that the justification for another biography of Gödel is that previous biographies were in some ways incomplete—or, to put it another way, that a new work should add substantially to what we already know. Does Stephen Budiansky’s “Journey to the Edge of Reason” pass this test? The author, whose most recent works include a biography of Oliver Wendell Holmes Jr. and a history of the National Security Agency, writes vividly, and the book overflows with fascinating detail. Although it mostly steers clear of math and logic, it does a good enough job to convince general readers that they have understood some of the problems with which Gödel grappled. Plus, there is some fresh material to draw upon, including Gödel’s diary, which covers two years before the outbreak of World War II. Much of the story has already been told in biographies by John Dawson and Rebecca Goldstein. Born in 1906, Gödel was raised in a German-speaking family in Brünn (now Brno), when it was part of the Austro-Hungarian Empire. Although Gödel was sometimes mistaken for being Jewish by his contemporaries, his family was, in fact, Lutheran. His father worked in the textile business, and the Gödels were comfortably off. Little Kurt was inquisitive and, because he wouldn’t stop asking questions, he acquired the nickname Herr Warum, “Mr. Why.” Following the collapse of the polyglot Austro-Hungarian Empire and the rise of Czech nationalism, Gödel joined the throng of young, Czech Germans moving to Vienna—he arrived in 1924, becoming a student at the prestigious University of Vienna. Shortly afterward he was invited to attend the Vienna Circle, a discussion group made up of mathematically and scientifically literate philosophers, led by professor Moritz Schlick. For a decade or so, their philosophical approach, logical empiricism, became the most fashionable in the world. Crudely put, the Circle maintained that for a statement to be meaningful it had to be either testable (“water boils at 100 degrees centigrade”) or true by virtue of the meaning of its terms (“all bachelors are unmarried” or other tautologies). Many statements about God, ethics and aesthetics were therefore meaningless. Math posed a problem for the Circle. Was 2+2=4 an empirical claim? Did we discover its truth by adding two apples to another two apples and counting four apples? This didn’t seem right. We could surely work out that 2+2=4 without the aid of fruit or any other material prop. Inspired in particular by Ludwig Wittgenstein, the Circle argued that we should treat mathematical truths as tautologies. Gödel was mostly silent during Vienna Circle discussions, but he passionately disagreed. His instincts were Platonist; that is to say, he believed that mathematical truths weren’t invented but existed somewhere “out there,” independent of the human mind, and that it was the task of mathematicians to discover these truths. Gödel’s reputation and fame rest principally on a proof that received its first public airing in September 1930, at a scientific gathering in Königsberg. Gödel—only 24 years old—demonstrated to the assembled delegates that there were limits to what could be proved in mathematics; that whatever axioms were postulated as the basic blocks of mathematics, there would inevitably be some truths within mathematics that could not be proved. By all accounts, the delegates at the conference were a bit flummoxed; the significance of this discovery took a few days to sink in. Then news of Gödel’s first Incompleteness Theorem (it would be followed by a second), spread rapidly around the world. “A scientific achievement of the first order” was the rather understated verdict of Gödel’s supervisor, Hans Hahn, when Gödel submitted the proof for his thesis. The work is now widely accepted as a seminal development in the history of logic. From 1933, Gödel began an on-off-on-off-on association with the recently formed Institute for Advanced Studies in Princeton, N.J., of which Einstein was a founding member. After Vienna, Gödel found Princeton a little parochial, though he liked the scenery. “All in all the country looks much like a park; only the true Alpine forest is missing.” Gödel had already started to show signs of mental instability. In 1934 he spent time at a sanatorium outside Vienna. He became obsessed with the stomachaches he suffered and began to develop paranoid delusions about being poisoned by his enemies. Schlick appealed to a Viennese psychiatrist, Otto Pötzl, for help: “If Dr. Gödel does not regain his health, it would be a loss of immeasurable consequence for our university and for science throughout the world.” Vienna’s political, social and cultural milieu forms the backdrop to the first half of Gödel’s life and has a central place in Mr. Budiansky’s narrative. Many, if not most, of Gödel’s friends and associates were of Jewish origin. The Austrian fascists and Nazis would always exaggerate Jewish power, and Mr. Budiansky unfortunately overstates the size of Vienna’s Jewish population (it peaked at 200,000). Gödel had just been invited back to Princeton when German troops marched into Austria. The Viennese were top-division anti-Semites, and the Anschluss unleashed a wave of sickening violence and sadism. Nonetheless, and shockingly, Gödel (along with most of his compatriots) voted in the subsequent plebiscite in favor of his country’s absorption by the German state. He later claimed he had done so only to secure a passport, though according to Mr. Budiansky Gödel became “wracked with guilt.” He was always politically naive. Toward the end of 1938, when he was briefly in the U.S. before returning to Vienna, he met a Jewish-Austrian philosopher for lunch. “And what brings you to America?” Gödel asked. Gödel faced various fraught struggles with the Nazi authorities before he was able to settle back in Princeton for good. By this stage, he had married Adele, despite vehement objections from members of his family. Adele was domineering, was older than he, was situated several rungs lower on the class ladder, and was a divorcée. She was the butt of snobbery both in Austria and in the U.S. (“a Viennese washerwoman type,” according to Gödel’s friend, Oskar Morgenstern, one of the founders of game theory). Albert Einstein and Kurt Gödel, ca. 1948. Photo: Oksar Morgenstern Estate At Princeton, the serious and shy Gödel developed an unlikely friendship with the more genial and gregarious Albert Einstein. They would walk to and from the institute. “I go to my office just to have the privilege of being able to walk home with Kurt Gödel,” Einstein once joked. No book on Gödel would be complete without one particular story that reads like something out of a Tom Stoppard play. Once the war was over, Gödel applied for U.S. citizenship and set about preparing for his citizenship test with more zeal than was warranted. He read books and books about U.S. history and laws—and during the course of his study discovered what he thought was an inner contradiction in the U.S. constitution. His two sponsors were Einstein and Morgenstern, and as they drove together to the court in Trenton, N.J., tried to convince him not to bring this up. But at some stage during the proceedings the judge claimed that the constitution in the U.S., unlike in Austria, would never permit a dictatorship. “Oh, yes,” said Gödel, and “I can prove it.” Fortunately, Einstein and the others soon managed to shut him up, and Kurt Gödel became an American. What contradiction he spotted is still a matter of debate. Periodically, over the next few decades, Gödel’s bouts of paranoia returned. If sent a wrapped present he would refuse to open it, suspecting it might contain poison. When Adele went into the hospital in 1977, he retreated into isolation and stopped eating. His imagined enemies now included Nazis, family members, doctors, tax inspectors and evil spirits. He died on Jan. 14, 1978: The death certificate gave the cause as “malnutrition and inanition, secondary to personality disturbance.” It was a distressing end to an emotionally troubled life. Mr. Budiansky’s account of this life does little to shift our perspective on Gödel, nor does it tell us much of factual importance about him that has not already been revealed in other books. On the other hand, almost every episode in the life of Gödel’s friend Einstein has been accorded its own monograph; if that degree of biographical scrutiny is justified for the 20th century’s most important scientist, then there is surely room in the world for an enthralling book about its most important logician. —Mr. Edmonds is the author of “The Murder of Professor Schlick: The Rise and Fall of the Vienna Circle.”