A Strange Wilderness

by Amir D. Aczel

  In the economically depressed city of Halle—later, part of East Germany—Cantor was a respected professor. But every time he published a paper, there was a ruckus in Berlin. He took it badly. Between his lack of progress on the continuum hypothesis and the constant attacks by Kronecker and other “lions and hyenas,” Cantor’s mental health deteriorated.

  Cantor’s bitter rival and former teacher Leopold Kronecker, left, called Cantor a scientific charlatan and a corrupter of youth.

  Cantor seemed to view infinity in an almost religious sense. He deeply believed that infinity was the realm of God. Infinity included the transfinite numbers—aleph-zero, aleph-one, and any others—and, beyond the transfinite numbers, there was an unreachable level of infinity that he named the Absolute. According to Cantor, the Absolute was God himself. As Cantor continued to explore the idea of infinity and the levels he had found infinity to have, his conflict with Kronecker intensified and became even more personal.

  The bitter rivals did make one attempt at reconciliation, however. Cantor liked to vacation in Germany’s Harz Mountains, a small mountain chain west of Halle whose highest peaks do not exceed three thousand feet. The villages, forests, and streams of the Harz Mountains offered a restful environment in which to relax and discuss mathematics. Indeed, Cantor had met a number of mathematicians in these mountains, and during one of his stays, he gathered his courage and wrote a letter to his former professor offering a meeting in one of the mountain villages. To his surprise Kronecker accepted the invitation. The two men then met and spent a few days discussing mathematics, and it seemed that they might have reached some kind of understanding. But soon after they departed—Cantor to Halle and Kronecker to Berlin—the confrontation between them flared up again.

  On June 29, 1877, Cantor wrote a letter to Dedekind in which he exclaimed that he had discovered something so bizarre that he found it too shocking to comprehend—even if he had proved it on paper. “I see it,” Cantor wrote, “but I don’t believe it!” What Cantor discovered was that a central concept in mathematics and everyday life—dimension—is completely irrelevant when it comes to the powerful concept of infinity.

  Cantor drew a square and a line segment on a Cartesian coordinate system and asked himself: “Are there more points on the square than on the line segment?” The intuitive answer is “of course,” but Cantor analyzed the situation using his customary machinery of searching for a one-to-one correspondence between the points on a square and the points on a line segment. To his great surprise, he succeeded in drawing such a correspondence.

  Cantor chose for his line segment the interval of numbers between zero and one, and for each side of his square also an interval of real numbers between zero and one. Now, each point on the square is given by its two Cartesian coordinates, and since each coordinate is between zero and one, a point is given by the pair (0.a1a2a3a4a5a6 …, 0.b1b2b3b4b5b6 …). Cantor defined the transformation from the square to the line segment as alternating the decimal expansion of each of the two coordinates. Thus a number on the line segment would be of the form 0.a1b1a2b2a3b3 … This establishes a one-to-one correspondence between every number on the square and every number on the line segment. Here, the line segment between zero and one is a proper subset of the square with side being the zero-to-one interval, which proves that dimension doesn’t matter when it comes to the level of infinity of a set of numbers: a two-dimensional object (the square), has the same cardinality (i.e., the same order of infinity) as the one-dimensional object, the line segment.

  Cantor tried to get his paper on the irrelevance of dimension published in Crelle’s Journal, but Kronecker caught wind of the submission and actively sought to prevent its publication. After hearing only silence from the journal’s editor for several months, Cantor became suspicious. When he inquired, he found out that, indeed, his enemy was behind the stonewalling. Kronecker had told the editor that Cantor’s paper dealt with empty concepts that did not really exist and that the mathematical public had to be protected from meaningless concepts like infinity and irrational numbers. Crelle’s Journal eventually did publish Cantor’s paper a year later, but the mathematician decided to send his future papers to other publications that he hoped might be immune to the influence of his foe.

  Cantor befriended a Swedish mathematician named Gosta Mittag-Leffler, who edited the mathematics journal Acta Mathematica, and this journal then became the home of Cantor’s later papers. Although Cantor’s work was warmly appreciated by an editor who understood his work, Kronecker was not ready to give up his relentless assaults. After seeing Cantor’s papers in Acta Mathematica, Kronecker tried to befriend Mittag-Leffler himself by pretending that he had a paper he wanted to publish in the journal. As it turned out, there was no such paper, but Cantor became very angry with Mittag-Leffler once he suspected that Kronecker was successful in turning his friend and supporter against him. This reaction almost lost him a key ally.

  Stress was taking its toll on Cantor’s health, and in May 1884 he had the first of his many nervous breakdowns. The attack lasted two months, during which he was completely unable to work. At that time, Cantor had been working on what we now know is an impossible problem: the continuum hypothesis.

  CANTOR’S ELDEST DAUGHTER, Else, was nine years old at the time of his first nervous breakdown. She and other members of the family were so shocked by the sudden change in her father’s behavior that she remembered it vividly years later, as she recounted to Cantor’s first biographer, Arthur Schoenflies. At the onset of the problem, Cantor became very agitated and could not communicate with people. He stayed in bed for weeks, during which time he neglected mathematics altogether and read Shakespeare. Eventually, he became convinced that he had come up with a new finding: that Shakespeare’s plays were actually written by Francis Bacon. It later came out that he had come upon this idea through a book he had found at an antiquarian bookshop in Leipzig, which described Bacon as a great poet rather than a scientist, and Cantor had assumed that the book he had found was unknown. Psychologist Nathalie Charraud hypothesizes in her book Infini et Inconscient: Essai sur Georg Cantor (Infinity and the Unconscious: An Essay on Georg Cantor) that Cantor may have seen himself as a character in a Shakespearean tragedy. In a haze of mental illness, rage, and hurt from the onslaught against him from Berlin, Cantor was living in an unreal world. He decided to leave mathematics altogether and asked the university’s administration to allow him to transfer to the philosophy department, but his request was turned down.

  After publishing two pamphlets at his own expense, both of which argued that Bacon was the author of Shakespeare’s plays, Cantor returned to mathematics and to his unsuccessful attempts to prove the continuum hypothesis. In 1899, after a concentrated effort at proving this impossible hypothesis, Cantor suffered another mental breakdown and was taken to the Halle Nervenklinik, a mental care facility in the city, for treatment.

  Today we recognize that Cantor probably suffered from bipolar disorder, in which periods of depression alternate with periods of elation. During the “high” periods he would work as if in a frenzy, and during the “low” periods he would be completely immobilized. Today there are effective medicines for treating this problem, but in Cantor’s time treatment consisted of making the patient take long, hot baths. Cantor had a pleasant private room in the facility, with high windows that let the sun in—he was a respected professor at the University of Halle and was treated as such. He often walked the wooded grounds of the hospital, enjoying nature. And he rested. After a few months Cantor was released, but he did not feel up to resuming his work and wrote lengthy letters to the Ministry of Education asking to be relieved of his duties as professor and given a post as librarian—at the same salary, he insisted. Apparently, the ministry ignored these requests.

  There ensued a cycle of intense periods of work on the continuum hypothesis and frequent hospitalizations. Unfortunately, Cantor was at the Nervenklinik at the time of the 1900 Congress of Mathematicians, held i
n Paris, where the renowned German mathematician David Hilbert presented a set of ten problems—later expanded to twenty-three—that he considered the most important in mathematics and hoped would be solved in the coming century. The first problem on the list was Cantor’s continuum hypothesis.

  Cantor was well enough in 1904 to attend the Third International Congress of Mathematicians, held in Heidelberg, Germany. He came accompanied by his two daughters, Else and Anna-Marie. His youngest son, Rudolph—a gifted musician—had died at age thirteen after years of ill health, and Cantor had also lost his mother not long before the congress. At the congress the Hungarian mathematician Jules C. Koenig presented a paper claiming that the second order of infinity was not any of Cantor’s alephs. Sitting in the audience, Cantor became enraged and started ranting against the speaker, causing a great commotion in the audience before his daughters could calm him down.

  Nevertheless, Cantor was such an astute mathematician that—once he was again able to think clearly—he saw immediately that Koenig had made improper use of one of the lemmas (preliminary mathematical results) in his work. Shortly afterward the German mathematician Ernst Zermelo, who, along with Cantor and Abraham Fraenkel, would become recognized as one of the founders of modern set theory, proved that Koenig’s work was indeed flawed because of his misuse of that lemma.

  Over the next decade and a half, Cantor went through cycles of hospitalization and recuperation, and in June 1917 he was admitted to the clinic for the last time. On January 6, 1918, his emaciated body was found in his room in the Nervenklinik. Apparently, food shortages resulting from World War I had affected the hospital’s supplies, and Cantor died of starvation.

  Cantor’s pioneering work opened the door to our modern understanding of infinity, but his goal of solving the continuum problem could not be achieved—not by him and not by anyone else. In 1937 the Austrian logician Kurt Gödel proved a result that, once completed by Paul Cohen of Stanford in 1963, established that Cantor’s continuum hypothesis cannot be proved or disproved within our system of mathematics (called Zermelo-Fraenkel set theory). Incidentally, Gödel, too, suffered from mental problems throughout his life: he died from self-starvation.

  London’s Burlington House is depicted in this engraving as it appeared in 1854, when it became the headquarters for the Royal Society of London for Improving Natural Knowledge, founded in 1660. The building serves as a symbol, perhaps, of the intellectual establishment to which iconoclastic mathematicians like Srinivasa Ramanujan—who, in fact, was elected a Fellow of the Royal Society in 1918—sought to belong.

  FOURTEEN

  UNLIKELY HEROES

  By the turn of the twentieth century, humanity had made tremendous leaps in industry and communication, but prejudice was still an ever-present reality. Whereas earlier mathematicians struggled amid religious and political upheaval or against internal demons, two mathematicians born in the late nineteenth century fought the constraints of poverty and sexism in their quests to be heard as mathematicians. They came from unusual backgrounds and lived atypical lives in a world where, despite a proliferation of egalitarian principles and social ideals, the sciences were still dominated by wealthy white men. These trailblazers were the Indian genius S. R. Ramanujan and the brilliant German mathematician Emmy Noether.

  RAMANUJAN

  The mathematician Srinivasa Ramanujan (1887–1920) was born in the village of Erode, south of Madras in Tamil Nadu, southern India, to a poor family. His father worked in a small store, and his mother sang at a temple. A younger brother died from a childhood illness at just three months of age, and two other siblings also died as infants. Ramanujan himself contracted smallpox when he was two but recovered from it. At age five he enrolled in school in Kumbakonam, a town nearer to Madras, where his family lived. He did not enjoy school.

  At a young age Ramanujan exhausted the mathematical knowledge of his teachers at school and independently read books on mathematics. Although he had no formal training in mathematics, he had such an amazing aptitude for it that by age twelve he had worked out new solutions to problems in number theory and analysis. Astonishingly, he seemed to be able to come up with mathematical facts and ideas in a complete intellectual vacuum. India had a long tradition, going back to the early Middle Ages, of producing important mathematical results without proof. And like some other Indian mathematicians, Ramanujan cared little about formal proof. He simply derived beautiful mathematics as if out of thin air—most of them identities and equations. In 1902 he learned the method the Italian mathematicians had found in the sixteenth century for solving cubic equations, and he derived on his own a method of solution for quartic equations. He was not aware of the impossibility of solving quintic equations by radicals, so he spent time, in vain, trying to derive a formula for it.

  At his high-school graduation in 1904, Ramanujan was awarded the Rao prize for outstanding achievement in mathematics, having obtained grades that were higher than the maximum possible at the school. At the Government College in Kumbakonam, where he studied on scholarship, Ramanujan also performed amazingly well in mathematics but showed no aptitude for anything else. He therefore lost the scholarship and moved to another town by himself, later going to Pachaiyappa College in Madras to study. Because of his poor performance in other areas of study and some health problems, he failed to graduate. Nevertheless, his independent study of mathematics produced many results.

  Ramanujan, whose likeness appears on this 1962 commemorative stamp, said in an interview shortly before his death, “As a child, I was considered slow-minded, as my verbal abilities did not come into play until I was three years old.”

  But leaving college without a degree made Ramanujan depressed. Through an Indian tradition allowing such unions, he married a ten-year-old girl, Janaki Amal, with the stipulation that the marriage would not be consummated until she came of age. They did not live together, and Ramanujan continued to live in abject poverty. He began looking for some kind of job to support himself, hoping to find employment as a clerk. Ramanujan finally obtained the needed references for employment when he showed his mathematical work to several mathematics professors at local universities. His work was so astonishingly novel that some of these professors at first doubted that it was his own. Once he showed them how he derived his equations, however, they understood that he was not a fraud, and their enthusiastic letters of reference even enabled him to receive some financial support to work on mathematics.

  Ramanujan published in the Journal of the Indian Mathematical Society a problem that he challenged other mathematicians to solve. His riddle was to find the answer to the infinite sum of these nested square roots:

  Six months passed, and no one had come forward offering a solution, so Ramanujan revealed the answer: 3.

  In 1912 Ramanujan finally obtained a position as a clerk in the Madras Port Trust. He performed his job so efficiently that he had time left over to do more research in mathematics and publish papers. Seeing how brilliant he was, Ramanujan’s friends and associates showed his work to English mathematicians to try to gain their support for the struggling young man. Unfortunately, these attempts were completely unsuccessful.

  In January 1913 Ramanujan wrote a letter to the prominent British mathematician G. H. Hardy (1877–1947) at Cambridge University, including nine pages of his original mathematical work. Hardy looked at the paper and initially thought it was fraudulent; he thought someone must have copied the work of some mathematician from a journal without citation. He recognized some of the results as mathematical derivations that had been obtained by others and known in the West. Others made no sense to him. But he was intrigued. When he read the pages again, he realized that one result he didn’t understand was obtained from work on hypergeometric series, previously studied by Euler and Gauss. He was so impressed and stunned by the theorems that he later said, “They defeated me completely; I had never seen anything in the least like them before.”1 These theorems had to be true, he concluded, because “i
f they weren’t, nobody would have had the imagination to invent them.”

  Hardy showed the papers to his colleagues, and they were equally stunned. Then he wrote back to Ramanujan, expressing interest in his work and asking for proofs of some of the theorems. Ramanujan was elated to receive a response and wrote to Hardy, “I have found a friend in you, who views my labors sympathetically.” Eventually, Hardy invited him to Cambridge University. When the invitation arrived, the local education board decided to give Ramanujan a grant to work at the University of Madras, in hopes of keeping him in India. His parents apparently objected to Ramanujan’s proposed move to England, and he sadly turned Hardy down. Hardy was disappointed, and their relationship cooled somewhat, but Hardy tried again. This time around, Ramanujan was ready—his mother had had a dream in which the family deity told her to allow her son to leave.2

  British mathematician G. H. Hardy, seen in this ca. 1927 photograph, was an early supporter of Ramanujan’s, and remained his friend and advocate until the younger man’s premature death at the age of thirty-two.

  On March 17, 1914, Ramanujan boarded the Nevasa at Madras, arriving in London almost exactly a month later. He moved to an address very close to Hardy’s rooms at Cambridge, and the two men met daily to go over Ramanujan’s amazing theorems. Hardy had already received in letters from Ramanujan more than one hundred theorems, and Ramanujan had brought many more with him. Looking at the theorems, Hardy could see that some were already known and others were false, but many of them were new breakthroughs.

  In later years Hardy would say that his greatest achievement in mathematics was discovering Ramanujan. He considered him a mathematician of Euler’s caliber. But Hardy was also an excellent mathematician himself. In his famous book, A Mathematician’s Apology, he wrote about the life of a pure mathematician. Hardy had never been interested in applications, concerning himself instead with bringing rigor and abstract beauty to British mathematics. Despite his tremendous ability, he was generally shy and reserved. For some reason, he hated to look at himself in a mirror. It was reported that when he traveled, he would cover the mirrors in the hotel rooms where he stayed. Hardy never married, but he maintained several close relationships with people in his life. His friendship with Ramanujan was prime among them.