What's your favorite math paradox

mathematics : The genius & the madness

It was a strange team that roamed the streets of the small East American town of Princeton every evening at the beginning of the 1950s: on the one hand, Albert Einstein, a good 70-year-old, mostly in a sweater and without socks, well-fed, full of mischief and common sense. Next to him is Kurt Gödel, 27 years his junior, correctly dressed, but often wrapped in a coat and scarf until summer, emaciated, unworldly and irony-free.

“I have made up my mind,” wrote Einstein once, “to bite the grass with a minimum of medical help when my hour has come, but until then to sin, as my nefarious soul tells me: smoke like a chimney, work like a horse, eat without thought or choice. "

How different Godel! He was extremely hypochondriac, measured his body temperature obsessively, swallowed tons of medication. Because he was afraid of being poisoned, his wife had to try the food first, and he adhered to a strict self-imposed diet: one eighth kilo of butter, three eggs, the beaten protein from two other eggs, milk, mashed potatoes and baby food, rarely a little meat. He was allergic to smoke, and once he disposed of a newly bought bed after a few days because he couldn't stand its smell.

The unequal strollers, both of whom had revolutionized their field, were close friends. For the shy mathematician Gödel, Einstein was the only one in whose presence he felt comfortable. And for Einstein, Gödel's scientific thinking was so profound and original that he once said that he only came to the institute "to have the privilege of being able to walk home with Gödel".

Their taste in art diverged considerably (Einstein adored Bach and Mozart, Gödel was into Schlager and Mickey Mouse), and in science they were seldom of the same opinion. This was not detrimental to their friendship: “I have often thought about it,” wrote Gödel after the death of his colleague in 1955, “why Einstein took pleasure in talking to me, and I believe that one of the reasons was that I was often of the opposite opinion and made no secret of it. ”Gödel was the only one“ who was on eye level with Einstein, ”says Freeman Dyson, a physicist who, like the two friends, was doing research at the Institute for Advanced Study in Princeton.

Godel's scientific greatness is undisputed. He has shown that there is a third category in logic in addition to “right” and “wrong”: “undecidable”. There are undecidable sentences in every meaningful mathematical system - that is what the famous Incompleteness Theorem states that Gödel came up with at the age of 23. Logicians praised the proof of the sentence as a “breathtaking intellectual symphony”. The philosopher Karl Popper compared its effect with an "earthquake". And the mathematician John von Neumann wrote: "Logic will never be the same again."

He has been called the “greatest logician since Aristotle” and a “Mozart of mathematics”. He published less than 100 pages in his life (that would not even make him assistant professor at a provincial university today), but each of his theorems established a new branch of mathematical logic. "His way of expressing himself (verbally as well as in writing) was always of the highest precision and at the same time of unsurpassable brevity", characterized his longtime friend, the Austrian mathematician Karl Menger. An artist of scarcity, then, who was only interested in the fundamental themes; an accountant of the metaphysical who combined the rigor of mathematics with the reach of philosophy. What distinguished him: his Kafkaesque proof strategies - the way in which he created seemingly surrealistic constructions with relentless logic, from which he conjured up the desired result in an adventurous way. A magician with a slide rule.

The tragedy of Kurt Gödel is that he tried to get his life under control with the same mixture of logical rigorism and bold deductions. "In the world of mathematics, everything is in balance and in perfect order," he wrote. “Shouldn't one assume the same for the world of reality?” Gödel firmly believed: “The world is reasonable”, is the first sentence of his philosophical creed that was discovered in his estate. And that is why there had to be a strictly logical explanation for every occurrence, no matter how random, and Godel always found one, however ludicrous it may sound. This ruthless rationalism drove Gödel into paranoia, and in the end he perished miserably when he logically refused all food for fear of being poisoned.

Kurt Gödel was born on April 28, 1906 in what was then Brno, Austria (now Brno, Czech Republic). Little Kurt soon got the nickname "Mr. Why" because he kept asking and researching. At the age of ten he began to be interested in mathematics, in high school he had already mastered university material, and it was rumored that he had not made a single grammatical mistake in Latin during his school days.

Perhaps the most defining event of his childhood was the rheumatic fever he suffered at the age of eight. "Mr. Why" began to read into the subject and found that the disease sometimes causes heart damage. From that moment on, despite the doctors' assurances to the contrary, he was convinced to death that he had a heart defect: the beginning of a lifelong hypochondria.

In 1924 Gödel moved to Vienna and began to study physics, in 1926 he switched to mathematics. He also dealt with philosophy, he was particularly impressed by Plato: The Greek philosopher had a passion for the objectively truth and pure reason, which is "not full of human flesh and colors and other mortal tinsel". Gödel decided to only deal with mathematical topics that were also philosophically relevant. First, at the age of 22, he tackled the most important problem in his subject: the question of the foundation of mathematics.

Mathematics has been a bastion of reason and security since ancient times. There is no such thing as relativism: A proposition is true or false, arguments are used instead of evidence, and if a mathematical truth has been proven, it will remain so for all time. But as clear as the rules of the game of this science, so unclear its basis: What is the foundation of mathematics from which it derives its theorems?

The central concept in this discussion is the axiom. Axioms are basic rules from which all theorems of a mathematical sub-area can be derived strictly logically. By definition, axioms are not provable, but they should be so “obviously true” that no one questions them.

In 1889 the Italian mathematician Giuseppe Peano proposed a system of five axioms for numerology, arithmetic. The first reads: "0 is a natural number", the second: "Every natural number has exactly one successor". As plausible as these axioms are: How could one be sure that they were the "right" ones? You can't prove it. But one thing could be shown (or so one hoped): their consistency. Axioms that contradict each other are worthless. Proof of consistency would be something like a legitimation card for a system of axioms.

The most ardent proponent of the axiom idea was the German mathematician David Hilbert (1862–1943). To put his science on a secure footing once and for all, he called on his guild to carry out a huge program. Starting from arithmetic, one area after another should be placed on the basis of axioms and its entire content logically built on them. Mathematics would thus become a formal system: a game with strict rules that determine which steps are allowed and which are not - comparable to chess.

In a keynote address at the International Congress of Mathematicians in Paris in 1900, Hilbert called on his colleagues to prove the consistency of the arithmetic system of axioms: “Conviction that every mathematical problem can be solved is a powerful incentive for us during our work; we hear the constant call within us: there is the problem, look for the solution. You can find it through pure thinking; because in mathematics there is no ignorabimus! ”(“ Ignorabimus ”means“ We will not know ”.)

Hilbert underestimated the problem. Certainly, with a clear game like chess there are no contradictions, as long as one does not introduce nonsensical rules like for example: "With every move one of the opponent's pieces has to be captured" (even with the opening move this would not be possible without violating other rules) . However, more comprehensive systems sometimes have logical inconsistencies, even if they look harmless at first glance.

In mathematics, too, contradictions soon appeared: in set theory. One tried more badly than right to bend the axioms so that the inconsistencies disappear. Hilbert found it “unbearable” that there were such “inconsistencies” in his subject, “this pattern of security and truth”. All the more vehemently he demanded proof of consistency for arithmetic, and he still had hope. He celebrated one last time at the meeting of German natural scientists and doctors on September 8, 1930 in his hometown of Königsberg: So far, an unsolvable problem has never been found in mathematics, and in his opinion, because there are “no unsolvable problems at all . Instead of foolish ignorance, on the contrary, the slogan is: We must know, we will know. "

What Hilbert could not know: The day before, on September 7, 1930, a man appeared at a mathematicians' conference in Königsberg who had just proven what Hilbert so vehemently denied: the existence of unsolvable problems. That man was Kurt Gödel.

Gödel's appearance in Königsberg was called “the most important moment in the history of logic”, and yet it was incredibly unspectacular. Typically, Gödel, the great silent man, had waited until almost the end of the conference to say a single sentence that he had probably been working on for days: "One can - assuming that classical mathematics is consistent - even give examples of sentences, which are correct in terms of content, but unprovable in the formal system of classical mathematics. "

Was it because of Godel's shy lecture? Was it because of the inappropriateness of his statement that called into question the mathematical worldview of most of those present? We do not know it. In any case, the reaction to Godel's sentence was: none. Nobody replied. The discussion continued as if nothing had happened. It took months for its discovery to spread throughout the scientific world.

What did Godel say? There are mathematical theorems that are correct, but they still cannot be proven. Correct, but not provable! And you can even specify such sentences in concrete terms. Gödel thought it possible, for example, that the so-called Goldbach conjecture was one of them: The German mathematician Christian Goldbach (1690–1764) had claimed that every even number greater than 2 can be represented as the sum of two prime numbers (4 = 2 + 2, 6 = 3 + 3, 8 = 5 + 3 ...). To this day, neither a counterexample nor proof of this conjecture has been found. (Which does not mean, however, that the theorem is unprovable.)

But Gödel's statement was by no means a mere assertion: at that point in time he already had strong evidence of it. He formulated it precisely in his famous incompleteness sentence, which he finally published in Vienna in 1931. The incompleteness theorem consists of two parts, and both are mathematical provocations of the highest order.

Outrage number one: In every consistent formal system that deals with natural numbers, there are undecidable sentences. In every meaningful mathematical system, formulas can be constituted, of which it has been proven that nobody can say whether they are right or wrong. As paradoxical as it sounds, Godel has proved unprovability strictly mathematically. It is as if he had thrown at Hilbert in mathematical language: "Yes, there is also an ignorance in mathematics!"

Outrageousness number two: The consistency of a formal system that deals with natural numbers is unprovable. It will never be possible to use mathematical means to ensure that mathematics does not contain contradictions. For David Hilbert, things could hardly have been worse: Not only was there no proof of consistency for arithmetic, there could be none at all - not just in arithmetic, but in all of mathematics.

That one could make such fundamental statements about mathematical systems at all was something completely new. Both the first and the second Gödel’s sentence fascinate with their "quasi-paradoxical self-negation" (John von Neumann): A formal-logical system emerges from itself, as it were, in order to make statements about itself.

Large parts of the mathematical world perceived Gödel's theorem as a catastrophe. Even the great English philosopher and mathematician Bertrand Russell was “confused” and asked whether “2 + 2 no longer results in 4, but 4,001”. But Gödel had by no means shown that mathematics produces inaccurate or incorrect results. Godel did not discover any contradiction in the axiom system of arithmetic, and neither he nor any other mathematician believe that such a contradiction even exists. The French mathematician André Weil summed up the situation aptly: "God exists because mathematics is free of contradictions, and the devil exists because we cannot prove it."

This is the crucial point: mathematicians have to believe in consistency, they cannot prove it. David Hilbert's program to formalize all mathematics on a secure basis had failed terribly. In mathematics, too, there is no final certainty: that is the statement of the incompleteness theorem.

Godel himself did not find this negative. He had never liked the idea of ​​mathematics as a chess game anyway: for him, the Platonist, mathematics was not a human invention. Rather, he believed in the existence of a mathematical reality independent of man; and axioms could at best serve to imitate this mathematical reality - comparable to the laws of nature with which physicists try to depict physical reality. From this point of view, the idea that mathematics can be derived purely formally from axioms is just as absurd as the idea of ​​a purely theoretical physics that dispenses with any experiments.

And now Kurt Gödel had found mathematical theorems; Sentences that existed in mathematical reality, that could even be true, but unprovable, not accessible to the formalists. In his opinion, this proved the correctness of his Platonic standpoint.

Shortly after the publication of his Independence Theorem, the young mathematician had a severe psychological crisis and had to be admitted to a sanatorium for the first time because of the risk of suicide.

The hallmark of the psychopath is not irrationality, but exaggerated rationality. Conspiracy theories are sometimes elaborate systems that may look bizarre from the outside, but are not easy to disprove. Gödel lived almost entirely in his strictly rationally constructed thought structure and was probably precisely for that reason particularly prone to paranoia. Gödel's favorite professor, Philipp Furtwängler, saw a connection between Gödel's mathematics and his paranoia: "Is his illness a consequence of the proof that it cannot be proven, or is his illness a necessary prerequisite for dealing with such questions?"

Godel was not very sensitive to signals from the outside world. For example, although he had many Jewish friends, he hardly seemed to be affected by the rise in anti-Semitism. When he stayed at the Institute for Advanced Study in Princeton for a long time in 1938 and received a visit from the Jewish German philosopher Gustav Bergmann, he innocently asked: "And what brings you to America, Mr. Bergmann?"

The Nazis - Austria is now affiliated to Hitler's Germany - revoke Gödel's license to teach at the University of Vienna because of a triviality. Now the logician, who is notorious for his meticulous interpretations of rules, feels challenged: although he could stay in America and get his wife to live with him, he returned to Vienna in mid-1939 to, as he put it, “to enforce his rights”. He almost paid with his life for this folly. But Godel doesn't seem to notice the state in which Europe is.

Two days before the outbreak of World War II, he wrote to his friend Karl Menger, who had fled to the USA: “I've been back in Vienna since the end of June and had a lot of running around in the last few weeks, so unfortunately I haven't been able to find anything for to write down the colloquium. How did the exams for my logic lectures turn out? "This letter is probably" a record of carelessness on the threshold of world historical events, "comments Menger.

But life in Vienna is gradually becoming uncomfortable for Kurt Gödel too. Completely surprisingly, when he was drafted in September 1939, he was found to be “fully fit” - despite all his assertions that he suffered from a heart defect. So now he's in Vienna, the war is raging in the east, and Godel can be called up at any time.

Then one day, near the university, Gödel was attacked by a gang of Nazi thugs who apparently thought the intellectual with the horn-rimmed glasses was a Jew (in fact, he came from a Christian family). His wife Adele can prevent worse things from happening by chasing the mob with her umbrella. The heroic deed is emblematic of the Gödels' marriage: Adele, a former nightclub dancer, was six years older than Kurt, loud and uneducated - but she grounded him, she cared for him, without her he was not viable.

Now the couple finally want to leave. But how? The Americans no longer issue tourist visas for Germans. It is only thanks to the network of Gödel's friends in the United States that he gets an American work visa and a German exit visa. And then Kurt and Adele are lucky one more time: Crossing the Atlantic is out of the question, but thanks to the Hitler-Stalin Pact, the eastern escape route is still open. The Gödels travel via German and Soviet-occupied territories to Moscow, from there with the trans-Siberian railway to Vladivostok, with the ferry to Japan and then across the Pacific to America. The journey lasted seven arduous weeks.

In Princeton, Gödel meets an old friend from Viennese times: the Jewish economist Oskar Morgenstern, who had fled to the USA in 1938. Understandably, he is eager for a situation report from Vienna, but Gödel is once again unimpressed by the external events. Morgenstern noted in his diary: “Gödel came from Vienna. When asked about Vienna: 'The coffee is pathetic. ‘(!) It is very fun, in its mixture of depth and cosmopolitanism."

Godel never set foot on European soil again. "I am so happy to have escaped beautiful Europe," he wrote to his mother. He thinks “the country and the people here are 10 times more personable”, and the authorities function “10 x 10 x ... better” than in Austria. One can well understand that Gödel had no desire to return to the, at best, half-heartedly denazified University of Vienna, where the once flourishing mathematics was paralyzed. It is less understandable that Austria did not remember his famous son: During his lifetime, Gödel, who was showered with honorary doctorates in America, never received an Austrian award, and he was never reported in the Austrian media. "Apparently they want to prove that I do not exist and never have existed," he commented bitterly.

In 1947 Kurt Gödel decided to be naturalized in the United States and he carefully prepared for this act. The day before the citizenship hearing, he came to Oskar Morgenstern excitedly: he had discovered a loophole in the American constitution that would allow America to be legally transformed into a dictatorship. Morgenstern was alarmed - not because of America, but because of Godel. Together with Einstein, he discussed what to do. The next morning a curious trio was on their way to the New Jersey State Courthouse: At the wheel of the car, Oskar Morgenstern, in the back seat, Kurt Gödel, and next to him, Albert Einstein, who told one string after the other to distract those seeking naturalization from his constitutional mind games. As soon as he got in, Einstein asked: "Well, are you ready for your penultimate test?"

Then Gödel: "What does penultimate mean?"

Einstein: "The last one comes when you step into your grave."

But the precautionary measures were of no avail. The judge opened the hearing with the statement: "Until now you had German citizenship." Gödel intervened: "Excuse me, the Austrian one." - "In any case, an evil dictatorship," the judge continued. "Fortunately, something like this is impossible in America." That was the keyword for Gödel: "On the contrary, I know exactly how it works!" And he started. But the judge was in a mild mood, and after Einstein, who knew him by his own naturalization, gave him a meaningful look, he interrupted Gödel's torrent of words: "You don't need to explain that exactly!"

Yes, Godel was a precision fanatic, obsessed with order. The order that he saw in mathematics was also felt in the world. “Because it is by no means chaotic, but, as science shows, there is the greatest regularity and order in everything.” Any alleged chaos is “just a wrong impression”. Where others saw chance at work, Gödel felt connections.

For example, in 1946, he blamed the Republicans' election victory for the fact that "the films had deteriorated significantly over the course of the year." There was no room for chance in Kurt Gödel's world. He didn't believe in the statistical chance of quantum mechanics, he didn't believe in the blind chance of evolution, and he didn't believe in chance in the lottery: “Adele has developed a real talent for guessing the numbers in games of chance. I found that with about 200 attempts. ”But he believed in ghosts, telepathy and an afterlife.

And he believed in God. “In religion,” he wrote, “there is much more reason than is usually believed.” One could “set up an exact system of postulates with such terms that are usually taken to be metaphysical: 'God', 'soul', 'ideas' . “Godel even tried - and that in the 20th century! - to establish a proof of God: a strictly mathematical version of the so-called ontological proof of God by Anselm of Canterbury (1033–1109), who had inferred the existence of God from the concept of the perfect being. Godel's proof, however, takes place in a highly abstract system; the formula for "God" is:


Gödel's contribution to Einstein's 70th birthday in 1951 was equally fantastic, scientifically but much more significant: he gave him a solution to the basic equations of general relativity. These cosmological formulas describe the structures of space and time, and Kurt Gödel had found a solution that could also have come from a science fiction author. His model described a universe with cyclical time in which the entire course of the world repeats itself endlessly. "If you go on a round trip in a spaceship on a sufficiently wide curve," says Gödel, "you can visit any region of the past, present and future in these worlds and travel back again." While Gödel researched observational data until the end of his life confirmed his model, Einstein was less enthusiastic. Godel's solution is mathematically correct, but impossible "for physical reasons". To this day, research is still not in agreement.

Einstein's death in 1955 was "a great shock" for Gödel. After losing his best friend, he did not publish a single work, never gave a lecture and became increasingly lonely. His paranoia attacks also increased. Godel was particularly afraid of refrigerators, had the heating removed from his apartment once in winter because it was said to have released poisonous gas, and when certain foreign mathematicians were at Princeton he was afraid of being murdered by them.

Kurt Gödel's last friend, Oskar Morgenstern, died in July 1977. Apparently Godel did not notice that Morgenstern was dying. 16 days before his death, the terminally ill person wrote in his diary: “Briefly asked how I was feeling and assured me that it was very clear that my cancer would not only be stopped, but would go back. Then he went on to his own problems. He assumed the doctors weren't telling him the truth, that they didn't want to deal with him, that he was an emergency, and that I should help him get to Princeton Hospital. He also assured me that about two years ago two men appeared who pretended to be doctors. They were swindlers and he had great difficulty exposing them. It is difficult to describe what such a conversation means to me: Here is one of the most brilliant men of our century, he is very fond of me and clearly suffers from a form of paranoia, expects help from me that I cannot give. By clinging to me - and he obviously has no one else, he makes the burden I carry heavier. "

For fear of poisoning Godel ate and drank less and less - for breakfast an egg, a teaspoon or two of tea, sometimes a little milk or orange juice; at lunchtime mostly beans, never meat. What kept him alive was his wife, who cared for him touchingly ("I have to hold him like a baby," she says). But Adele was sick herself, was able to support Kurt less and less and had to go to the hospital for several months in the fall of 1977. Now Godel ate nothing at all. When his wife came home at the end of December, he weighed just under 30 kilos. Gödel was admitted to hospital, where he died on January 14, 1978, with his knees drawn up and his back rolled up like a fetus in the womb. The cause of death was given as: "Malnutrition and emaciation", caused by a "personality disorder". Adele died three years later.

Now new: We give you 4 weeks of Tagesspiegel Plus! To home page