A Strange Wilderness

by Amir D. Aczel

  On the night before his death in 1832, Galois wrote down many of his mathematical theorems in a letter to his friend Auguste Chevalier; in the letter’s last page, above, Galois writes: “I hope some men will find it profitable to sort out this mess.”

  On May 29, the night before the duel, Galois did not sleep. In a hotel room he wrote down some of his mathematical theories, leaving out many details (“I have no time”), and sent it in a letter to his friend Auguste Chevalier. In other letters written that night to Republican comrades, the tormented genius wrote that he knew he would not survive the confrontation: “I am dying, the victim of an infamous coquette,” he wrote.

  At dawn on May 30, 1832, Galois went to a deserted field to meet his adversary. We can only guess how he must have felt. Apparently he did not have a second (an assistant), as duelers usually did, and he probably was not versed in the rules of this barbaric practice—by then illegal in France for two centuries. Around noon that day, a peasant found him lying at the side of a road outside Paris, mortally wounded in the stomach. His challenger and second had left the wounded Galois to die. The peasant took Galois to the Cochin Hospital, where he died the next morning at 10:00 a.m., apparently from peritonitis caused by the wound. His last words were to his brother, who was sitting at his hospital bedside: “Don’t cry, Alfred,” he said. “I need all my courage to die at twenty.”

  Galois was buried in an unmarked grave in the Montparnasse Cemetery in Paris. In response to his untimely death, his friends organized a demonstration on June 5, which led to the massacre of the Saint Merri cloister—an event that Victor Hugo described in his tragic novel Les Misérables.

  But what really happened to Galois? Why would someone—who may never have even fired a gun—get involved in a duel when he knew he had no chance of winning it and had no strong conviction or passion about wanting to duel? The mystery of Galois’s death has dogged historians of mathematics for almost two centuries, and there have been a number of theories about what actually happened. One of them is that the Royalists saw in Galois a dangerous Republican leader and wanted to kill him. According to this theory, they ensnared him with Stephanie and then challenged him to a duel for her “lost honor,” knowing that he was inexperienced in duels and would likely die. Another theory was that he challenged the other man to the duel because he was smitten with Stephanie. In support of this theory we have the speculations that Galois’s adversary was not a Royalist, but rather one of his Republican friends. Some have surmised that the adversary was a fellow Republican prisoner kept at the halfway house who had competed with him for Stephanie’s affection. Another hypothesis, more outlandish and less credible than the others, is that Galois wanted to die in the duel so that his death would become a rallying point for the Republicans. We don’t know what really happened, and it is likely that the true cause of Galois’s death will forever remain a mystery.

  GALOIS’S BROTHER ALFRED published the tragic young mathematician’s papers and letters. Thanks to the work of Galois and Abel, we know that an equation of the fifth order or higher has no solution by radicals (i.e., through a formula using arithmetic operations, including the extraction of roots, on the coefficients of the equation). Galois’s deep and comprehensive analysis led to yet another important theory in algebra: Galois theory, which refers to a special study of groups and fields in the context of solving equations. Galois theory is so powerful (and beautiful) that it settled problems that had occupied the minds of mathematicians for millennia.

  It took two decades from Galois’s death in 1832 before mathematicians finally understood the theory, which explains why all three classical problems of antiquity—the squaring of the circle, the doubling of the cube, and the trisecting of an arbitrary angle, all using only a straight-edge and a compass—are absolutely impossible to solve. Galois theory is important in many areas of mathematics, including number theory, and helped in the proof of Fermat’s Last Theorem in the 1990s by Andrew Wiles. It also underlies the entire theory of particle physics through the use of Galois’s idea of groups and symmetries. It is hard to overestimate the influence of the work of this tormented young genius on modern mathematics.

  In 1846 the French mathematician Joseph Liouville (1809–82) edited several papers written by Galois and published them together with the letter and manuscript Galois had sent to his friend Chevalier the night before the fatal duel. Galois’s work was based on results obtained by Gauss that relate equations to properties of prime numbers, and on work by Lagrange on the permutations of the roots of a polynomial equation. Galois then used the term group for the first time in reference to mathematical groups of permutations of the roots of polynomials. When these groups had certain desirable properties of symmetry, then the polynomial equation from which they were derived had solutions obtained as radicals through arithmetic operations on the equation’s coefficients. The quintic equation does not have such properties, and thus Galois showed definitively what Abel and Ruffini had been able to show in a more restricted and incomplete way. Today the theory of groups that emerged from Galois’s findings is a very important part of modern abstract algebra.

  The tragic death of Galois marks the end of the prominence of France in mathematics until the end of the nineteenth century. Laplace, Fourier, and Legendre all died between 1827 and 1833, and only Cauchy would continue to write papers until his death in 1857. With the deaths of most of the best French mathematicians, the leadership in mathematics was passed to England and Germany.

  ALGEBRAISTS OF BRITAIN

  Interestingly, it would be the British who would take over leadership in algebra, the field of Galois’s fertile investigations. George Boole (1815–64) was born to a family of tradesmen in Lincoln, England. He studied Greek and Latin early in his education, and then started reading the works of Laplace and Lagrange before embarking on a career in mathematics. In 1847 he published a book called The Mathematical Analysis of Logic, which introduced mathematical analysis into logic, the systematic study of forms of valid deductive argument. Seven years later Investigation of the Laws of Thought, which extended his theory, followed. Boole introduced what is now known as Boolean algebra—the algebra of logic that every computer programmer knows by heart. This is the algebra of true/false statements. For example, if x and y each represent something that can either be true (1) or false (0), then one can determine the truth value of their conjunction (xy). One only needs to grasp the basic rules of multiplication to know that, if either x or y is false, then xy must be false. Our world of computers would not be the same without Boolean algebra.

  Other developers of abstract algebra include George Peacock (1791–1858), who, along with Augustus De Morgan (1806–71), attended Trinity College, Cambridge. De Morgan was born in India to British parents, and after moving to England, he tried to generalize algebraic notation to virtues and vices.5 Important contributions to algebra were also made by Arthur Cayley (1821–95), who invented the theory of matrices, a concept that is absolutely essential not only in mathematics but also in all of applied science. In 1858 Cayley wrote a paper on the theory of linear transformations. Trying to impose order on the transformations, Cayley invented the matrix. He also studied the determinant, which we think of as naturally associated with a matrix.

  The British mathematician William Kingdon Clifford is known for his writings on the geometrical properties of gravity. Like fellow mathematician and Oxford alumnus C. L. Dodgson, a.k.a. Lewis Carroll, Clifford also wrote children’s stories.

  Cayley’s close lifelong friend, James Joseph Sylvester (1814–97), was born to a Jewish family named Joseph, but he later changed his name to Sylvester (a name associated with the papacy). Sylvester also studied matrices and devised a method, called Sylvester’s criterion, of eliminating an unknown from two polynomial equations. Together, the two mathematicians worked on developing the theory of forms. Sylvester moved to the United States and, for many years, was one of the greatest mathematicians working at Johns Hopkins University before
he returned to England.

  Benjamin Peirce (1809–80) of Harvard did important work on linear associative algebras, and is famous for having said, “Mathematics is the science which draws necessary conclusions.”6 His ideas were further pursued in England by William Kingdon Clifford (1845–79), who gave us Clifford algebras, which include concepts and tools that are useful in mathematics and theoretical physics. Clifford enjoyed entertaining young children with stories, much like his fellow Oxford mathematician C. L. Dodgson (1832–98). We all know Dodgson by his pen name, Lewis Carroll, the author of Alice in Wonderland.

  WILLIAM ROWAN HAMILTON

  At Göttingen, Germany, a Norwegian mathematician named Sophus Lie (1842–99) extended Galois’s work to continuous groups. In a discrete group, such as the permutation groups studied by Galois, there are discrete symmetries. For example, rotating a triangle one third of the way around leaves the triangle the same, and such transformations are modeled by a discrete group of rotations. These discrete groups aid in the study of solutions of polynomial equations. Lie derived the new, continuous groups, such as the group of all possible rotations of a circle, in order to study differential equations. Unlike discrete polynomial equations, differential equations are “continuous,” in that taking the derivative is a continuous operation; hence, Lie “invented” these groups to try to perform for differential equations what Galois had done for polynomial equations. Lie groups are one of the most powerful and most important mathematical tools in modern theoretical physics.

  Algebraic operations are commutative—i.e., the order of the operations is unimportant (5 × 3 = 3 × 5 = 15)—but there are other types of systems in which the order of operations is important. For example, we know that matrix multiplication is generally noncommutative. The study of a particular noncommutative algebra was carried out by the illustrious Irish mathematician and physicist Sir William Rowan Hamilton (1805–65).

  Hamilton’s father and mother were intellectually inclined, but they died young. Even before they died, however, Archibald and Sarah Hamilton sent their young son to live with his uncle, Reverend James Hamilton, so that he would receive a good education at the hands of this very knowledgeable and scholarly man. With the encouragement of his uncle, who was a gifted linguist, William learned to read Greek, Latin, and Hebrew by the age of five. By the age of ten, he knew six languages, including Persian and Hindustani.

  As a young man, Hamilton became friendly with the poets William Wordsworth and Samuel Coleridge. He even wrote his own poems. However, when he matriculated at Trinity College, Dublin, he turned his attention to mathematics, studying Laplace’s classic Mécanique Céleste and other works of the great French mathematicians who had advanced mathematical methods in physics and astronomy.

  While still a student, at the age of twenty-two, Hamilton was appointed the Royal Astronomer of Ireland and director of the Dunsink Observatory, as well as professor of astronomy. In 1833 Hamilton presented to the Royal Irish Academy a paper in which he introduced an algebra of pairs of real numbers and defined multiplication thus: (a, b)(c, d) = (ac – bd, ad + bc). This is actually the law of multiplication of complex numbers, which Gauss understood, but for the first time a mathematician had made the law explicit in algebraic terms. Hamilton tried hard to extend this idea further, but he couldn’t do it.

  The mathematician John C. Baez provides a fitting metaphor for the relations among number systems, which was central to Hamilton’s work:

  There are exactly four normed division algebras: the real numbers, complex numbers, quaternions, and octonions. The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are the slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative.7

  The Dunsink Observatory in Dublin, Ireland, was established in 1785. Between 1880 and 1916, when Ireland was decreed to be in the same time zone as England, the observatory recorded Ireland’s official time, then called Dublin Mean Time. There are several references to “Dunsink time” in James Joyce’s Ulysses.

  The story of Hamilton’s discovery of the quaternions is amazing. Hamilton was fascinated by the idea of a relationship between the complex numbers as an algebra and as a two-dimensional geometry, where multiplication by i is rotation in the plane by 90 degrees counterclockwise. He tried to go one step further and look at three-dimensional geometry as a home for an extension of the complex numbers, studying this problem at length from 1835 until 1843. By then he had been knighted by the British king.

  William Rowan Hamilton, the Irish mathematician famous for the algebraic graffito he inscribed on Dublin’s Brougham Bridge, could read six languages by the time he was ten years old.

  In a letter to one of his sons, describing the events of October 1843, Hamilton later wrote, “Every morning in the early part of the above-cited month, on my coming down to breakfast, your (then) little brother William Edwin, and yourself, used to ask me: ‘Well, Papa, can you multiply triplets?’ Whereto I was always obliged to reply, with a sad shake of the head: ‘No, I can only add and subtract them.’”8 The system Hamilton was looking for simply did not exist. What he was really looking for, algebraically, was an algebra modeled by a four-dimensional geometry, not a three-dimensional one.

  On October 16, 1843, Hamilton was walking with his wife along Dublin’s Royal Canal, heading for a meeting at the Royal Irish Academy. As he and his wife crossed the Brougham Bridge over the canal, he suddenly had a great “aha” moment. As he described it in the letter to his son, “That is to say, I then and there felt the galvanic circuit of thought close; and the sparks which fell from it were the fundamental equations between i, j, k; exactly such as I have used them ever since.”

  At that moment, Hamilton stopped dead in his tracks, right at the top of the bridge. He wanted to write it all down, but he had no pencil or paper, so he picked up a nail from the ground. In what has been called the most famous act of mathematical vandalism in history, he carved into the stone surface of the bridge the following equation: i2 = j2 = k2 = i j k = –1. (This is an extension of the special multiplication law for complex numbers, i2 = –1.) The original carving is now gone, but every year his act and mathematical achievement is commemorated by mathematicians from around the world who, on October 16, replicate Hamilton’s walk, culminating at the top of the bridge, where a plaque describing his discovery now stands.

  The algebraic system Hamilton discovered is called the quaternions. The (unit) quaternions form a Lie group, which exhibits key symmetries found in theoretical physics. Thus all these developments that originated from the work of a young genius named Galois are not only important in pure mathematics, but also form key elements of modern theoretical physics and other fields.

  PART VI

  TOWARD A NEW MATHEMATICS

  Sunlight filters through a dense forest of spruce trees in the Harz Mountains of Germany, where mathematician Georg Cantor sought rest and relaxation. The image aptly conveys the obstacles—be they foes, doubters, or internal demons—that many mathematical geniuses face during the arduous quest toward truth.

  THIRTEEN

  INFINITY AND

  MENTAL ILLNESS

  The political upheaval of France in the mid-1800s had slowed the progress of mathematics there, so the center for mathematical research in the world moved to Germany. Major mathematicians worked in Berlin and Göttingen, including Richard Dedekind (1831–1916) of the Brunswick Polytechnic, who made progress in our understanding of mathematical analysis—the theoretical basis of calculus.

  A key mathematician in Berlin was Karl Weierstrass (1815–97), a high-school teacher who, at age forty, published a mathematical paper so profound that he was offered a full professorship in Berlin. There, he nurtured many students, including
a young Russian woman named Sofia Kovalevskaya (1850–91). This was unusual because women were not allowed to matriculate at universities, nor did they have their own passports. She had written him from her native Russia and managed to travel to Berlin (with her parents’ permission) before he agreed to take her on as a private student. Kovalevskaya received a Ph.D. from the University of Göttingen and continued to study under Weierstrass, but struggled to find employment. Eventually, however, she gained a reputation as an important mathematician, lecturing at the University of Stockholm before dying from the flu at age forty.

  Another mathematician in Berlin, Leopold Kronecker (1823–91), was the spoiled son of a business magnate. He was a good mathematician—the Kronecker delta function, useful in many areas of mathematics, is named after him—but he believed that irrational numbers (already known to the ancient Greeks) did not really exist. He was intolerant of anyone who did not hold his views about mathematics … or anything else.