Why Beauty is Truth Read online

  To save on printing costs, Niels distilled his ideas down to their essentials, and the printed version ran a mere six pages. This was a lot less than Ruffini’s 500 pages, but there are occasions in mathematics where brevity can make ideas more obscure. Many of the logical details—which in this area were crucial—had to be left out. The paper was a sketch, not a proof.

  Niels introduced it by writing, “Mathematicians have generally been occupied with the problem of finding a general method for solving algebraic equations, and several have made attempts to prove the impossibility of it. I dare to hope, therefore, that mathematicians will receive favorably this article which has for its purpose to fill this lacuna in the theory of equations.” It was a faint hope. Though he succeeded in visiting some mathematicians in Paris and getting them to agree to look at his paper, its reasoning was so compressed that most of them probably found it incomprehensible. Gauss filed his copy but never read it—when it was found after his death, the pages were still uncut.

  Later, perhaps realizing his mistake, Abel produced two longer versions of his proof, giving more of the details. Having by this time heard of Ruffini, he wrote in these versions, “The first to attempt a proof of the impossibility of an algebraic solution of the general equation was the mathematician Ruffini; but his memoir is so complicated that it is difficult to judge the correctness of the argument. It appears to me that his reasoning is not always satisfactory.” But like everyone else, he didn’t say why.

  Ruffini and Abel wrote their arguments in the formal mathematical language of the time, which was not well suited to the style of thinking required. Mathematics then was mainly concerned with specific, concrete ideas, whereas the key to the theory of equations is to think in rather general terms—about structures and processes rather than specific things. Thus their ideas were difficult for their contemporaries to grasp for reasons that went beyond language. But even for modern mathematicians, using the terminology of the period would make comprehension difficult.

  Fortunately, we can grasp the essential features of their analysis by employing an architectural metaphor. One way to think about Ruffini’s almost-proof, and of Abel’s complete proof, is to imagine building a tower.

  This tower has a single room on each floor, with a ladder that connects it to the room above. Each room contains a large sack. If you open the sack, millions of algebraic formulas spill out across the floor. At first sight, these formulas have no special structure and appear to have been harvested at random from the pages of algebra texts. Some are short, some long; some are simple, some extraordinarily complicated. A closer look, however, reveals family resemblances. The formulas in a given sack have lots of common features. The formulas in the sack in the room above have different common features. The higher we climb the tower, the more complicated the formulas in the sacks become.

  The sack on the first floor, at ground level, contains all of the formulas that you can build by taking the coefficients of the equation and then adding them together, subtracting, multiplying, and dividing them—over and over, as many times as you like. In the world of algebraic formulas, once you have the coefficients, all of these “harmless” combinations come along pretty much free of charge.

  To climb the ladder to the floor above, you must take some formula out of the sack, and use it to form a radical. It might be a square root, a cube root, a fifth root, whatever. But the formula whose root you are taking must have come from that sack. You can always take it to be a pth root where p is prime, because more complex roots can be built from prime ones, and this simple observation is surprisingly helpful.

  Whichever root you decide to take, when you arrive on the second floor, you find a second sack, whose contents are initially identical to those of the sack of the first floor. But you open the sack, and throw in your new radical.

  Formulas breed. When Noah landed his ark on Mount Ararat, he told all the creatures inside it to go forth and multiply. The formulas in the sack do more than that: they go forth and multiply, add, subtract, and divide. After a few seconds of frenzied activity, the sack on the second floor is bulging with all possible “harmless” combinations of the coefficients of the equation and your new radical. Compared to the sack on the first floor, there are many new formulas—but they all resemble each other; each of them includes your radical as a new component.

  You do much the same to get to the third floor. Again you pick some formula from the new sack—just one—and form a new radical by taking some (prime) root of that formula. You carry your new radical up the ladder to the third floor, toss it in the sack, and wait for the formulas to carry out their mating rituals.

  And so on. Each new floor introduces a new radical, and new formulas appear in the sack. At any stage, all of those formulas are built from the coefficients, together with any of the radicals introduced so far.

  Eventually you reach the top floor of the tower. And you complete your quest—to solve the original equation by radicals—provided that, tucked away inside the sack in the attic, you can find at least one root of that equation.

  There are many conceivable towers. They depend on which formulas you choose, and which radicals you take. Most fail dismally, and no hint of the desired root can be found. But if the quest is possible, if some formula built from successive radicals yields a solution, then the corresponding tower does indeed have a root in its attic. For the formula tells us exactly how to obtain that root by adjoining successive radicals. That is, it tells us exactly how to build the tower.

  We can reinterpret the classic solutions of the cubic, the quartic, and even the Babylonian solution of quadratics in terms of these towers. We begin with the cubic, because this is complicated enough to be typical, but simple enough to be comprehensible.

  Cardano Tower has only three floors.

  The sack on the first floor contains the coefficients and all of their combinations.

  The ladder to the second floor requires a square root. A very particular square root, that of a specific formula in the first sack. The sack on the second floor contains all combinations of this square root, together with the coefficients.

  The ladder to the third floor, the attic, requires a cube root—again, a specific one. It is the cube root of a particular formula involving the coefficients and the square root that you used to reach the floor below. Does the sack in the attic contain a root of the cubic equation? It does, and the proof is Cardano’s formula. The ascent of the tower is a success.

  Ferrari Tower is taller; it has five floors.

  The first floor, as always, has a sack that contains just the combinations formed by the coefficients. You reach the second floor by forming harmless combinations and then taking a suitable square root. You reach the third floor by forming harmless combinations and then taking a suitable cube root. You reach the fourth floor by forming harmless combinations and then taking a suitable square root. Finally, you clamber up to the fifth floor—the attic—by forming harmless combinations and then taking a suitable square root.

  Solving the quadratic, cubic, and quartic.

  And now, the sack in the attic does indeed contain what you are seeking, a root of the quartic equation. Ferrari’s formula provides the instructions for building precisely such a tower.

  The Tower of Babel, which solves the quadratic, also fits the metaphor. But it turns out to be a stumpy tower with only two floors. The sack on the first floor contains just the combinations of the coefficients. A single carefully chosen square root conducts you to the floor above, the attic. Inside that sack is a root of the quadratic—in fact, both of them. The Babylonian procedure for solving quadratics, the formula you were taught at school, tells us so.

  What about the quintic?

  Suppose that a formula to solve the quintic by radicals really does exist. We don’t know what it is, but we can infer a lot about it nonetheless. In particular, it must correspond to some tower. Let me call this hypothetical tower the Tower of Abel.

  The Tower of
Abel could contain hundreds of floors, and its ladders may involve all sorts of radicals—19th roots, 37th roots, we don’t know. All we know for sure is that the sack on the first floor contains just the harmless combinations of coefficients. We fondly imagine that up in the attic, above the clouds, is a sack containing some root of the quintic.

  We ask how to climb the tower, and the mathematics tells us that there is only one way to get to the second floor. We have to take one particular square root. There is no other way up.

  Why the quintic is unsolvable.

  Well, not quite. We could take all sorts of other roots, build a huge, tall tower. But such a tower cannot have a root in its attic unless some floor corresponds to the particular square root that I am thinking about. And none of the previous floors will help you reach the attic; building them was a waste of time and money. So any sensible builder will go for that square root right at the start.

  What do you need to climb the ladder to the third floor?

  There is no ladder to the third floor. You can reach the second floor, but then you are stuck. And if you can’t reach the third floor of the presumed tower, you certainly can’t get to the attic and find a root in the sack.

  In short, the Tower of Abel does not exist. All that exists is an abandoned attempt that peters out on the second floor; or perhaps a more elaborate structure with lots of unnecessary floors, which eventually peters out in exactly the same manner, for exactly the same reason. This is what Ruffini proved, save for one technical gap. Roughly speaking, he failed to prove that if harmless combinations of radicals live in the attic, then so do the radicals themselves.

  Ruffini’s proof and Abel’s towers have clear similarities. But by using towers, Abel improved Ruffini’s tactics and filled the gap he left. Between them, they proved that no radical tower climbs from the coefficients of the quintic to its roots. In architectural language, that tells us that there is no formula for the root of a quintic that uses nothing more elaborate than radicals. Solving the quintic by radicals is as impossible as climbing to the Moon by repeatedly standing on your own shoulders.

  As Christmas 1828 drew near, Abel arranged to stay with his old friends Catharine and Niels Treschow in Froland. He was looking forward to visiting Crelly, who lived nearby. His doctor didn’t think the trip was a good idea, because of the state of Abel’s health. In a letter to Christoffer Hansteen’s wife, Johanne, Catharine wrote, “If only you had been in town he might have been content to remain. But he tried to hide how ill he really was.” In mid-December Abel headed for Froland, bundled up against the winter cold. He arrived on 19 December wearing every scrap of clothing he had with him, including socks over his arms and hands. Despite his coughs and cold shivers, he plowed ahead with his mathematics, happy to work in the Treschows’ parlor surrounded by their children. He enjoyed the company.

  Abel was still trying to land a permanent position. Even his temporary post at Oslo was in doubt. Over Christmas he focused his main efforts on securing the job in Berlin. His friend August Crelle, busy behind the scenes, had persuaded the Department of Education to create a mathematical institute and was angling for Abel to be appointed as one of its professors. He had obtained support from the scientific giant Alexander von Humboldt, together with a recommendation from Gauss and another from Adrien-Marie Legendre, a prominent member of the French Academy. Crelle advised the education minister that Abel was willing to accept a position in Berlin, but that the authorities should move quickly because he was in demand elsewhere, notably Copenhagen.

  Abel was due to leave Froland for Oslo on 9 January, but his coughs and chills had worsened and he spent most of his time confined to his room. His intended in-laws, the Kemps, became very worried. On the morning of his planned departure he was coughing violently and spitting blood. The family doctor was immediately called to the house, and he prescribed bed rest and constant nursing. Crelly acted as nurse, and her loving attentions and various medications led to a distinct improvement. Within a few weeks Abel was allowed to sit in a chair for short periods. He had to be restrained from doing any mathematics.

  Legendre wrote to say how impressed he was with Abel’s work on elliptic functions, and urged the young man to publish his solution to the problem of deciding when an equation could be solved by radicals: “I urge you to let this new theory appear in print as quickly as you are able. It will be of great honor to you, and will universally be considered the greatest discovery which remained to be made in mathematics.” While some prominent mathematicians, actively or through neglect, were hindering the publication of Abel’s seminal works, his reputation in other quarters was growing fast.

  Toward the end of February 1829, Abel’s doctor realized that he was never going to recover, and the best he could hope for was to keep the illness at bay as long as possible. The doctor sent Abel’s former teacher Bernt Holmboe a certificate, reporting the young man’s state of health:

  . . . Shortly after his arrival at Froland Ironworks he suffered a severe attack of pneumonia with considerable expectoration of blood, which ceased after a brief period. But a chronic cough and great weakness have compelled him to rest in bed, where he must still remain; furthermore, he cannot be permitted to be exposed to the slightest variation in temperature.

  More serious, the dry cough with stinging pains in the chest makes it very probable that he suffers from hidden chest and bronchial tubercles, which easily can result in a subsequent chest phthisis, partially on account of his constitution.

  Due to this precarious state of health . . . it is most unlikely that he will be able to return to Oslo before the spring. Until then, he will be unable to discharge the duties of his office, even if the outcome of his illness should be the most desirable.

  Crelle received the bad news in Berlin, and redoubled his efforts to secure Abel a position, advising the German minister that it would be good to transfer Abel to a warmer climate.

  On 8 April, Crelle sent his protégé good news:

  The Education Department has decided to call you to Berlin for an appointment . . . In what capacity you will be appointed and how much you will be paid I cannot tell you, for I do not know myself . . . I only wanted to hurry to let you hear the main news; you may be certain that you are in good hands. For your future you need no longer have any concern; you belong to us and are secure.

  If only.

  Abel was too ill to travel. He had to stay in Froland, where despite Crelly’s nursing he became weaker and weaker, and his cough grew worse. He left his bed only to allow the sheets to be changed. When he tried to do some mathematics, he found he was unable to write. He began to dwell on the past, and his poverty, but he did not take his feelings out on the people he loved, remaining cooperative and good-natured to the very end.

  Crelly naturally found it more and more difficult to hide her distress from her fiancé. Marie or Hanna kept her company at the bedside. Abel’s worsening cough was stopping him from sleeping, and the family hired a nurse to look after him overnight so that Crelly could get some rest.

  On the morning of 6 April, after a night of severe pain, Abel died. Hanna wrote, “He endured his worst agony during the night of 5 April. Toward morning he became more quiet, and in the forenoon, at 11 o’clock, he expired his last sigh. My sister and his fiancée were with him in the last moment, and saw his quiet passing into the arms of death.”

  Five days later, Crelly wrote to Catharine Hansteen’s sister Henriette Fridrichsen, asking her to tell Catharine the sad news. “My dearest love, yes, only duty could make me demand this, for I owe your sister, Fru Hansteen, so much. I take the pen with trembling hand to ask you to inform her that she has lost a kind, devout son who loved her so infinitely.

  “My Abel is dead!. . . I have lost all on earth. Nothing, nothing have I left. Pardon me, the unfortunate can write no more. Ask her to accept the enclosed lock of my Abel’s hair. That you will prepare your sister for this in the most lenient way asks your miserable C. Kemp.”

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  THE LUCKLESS REVOLUTIONARY

  Mathematicians are never satisfied.

  Whenever a problem is solved it only raises new questions. Soon after Abel’s death, his proof that some quintics can’t be solved by radicals started to become recognized. But Abel’s work was just the start. Although all previous attempts to solve all quintics had ground to a halt, a few very clever mathematicians had proved that some quintics can be solved by radicals. Not just obvious ones, like x5 – 2 = 0, where x = but surprising ones like x5 + 15x + 12 = 0, though the solution is too complicated to state here.

  This was a puzzle. If some quintics are solvable and some not, what distinguishes one kind from the other?

  The answer to this question changed the course of mathematics, and mathematical physics. Even though the answer was given more than 170 years ago, it is still yielding important new discoveries. In retrospect, it is astonishing how far-reaching are the consequences of an innocent question about the internal structure of mathematics. Solving quintics, it appeared, had no practical use whatsoever. If some problem in engineering or astronomy involved a quintic, there were numerical methods to determine a solution to as many decimal places as were needed. The solvability—or not—of a quintic by radicals was a classic example of “pure” mathematics, of questions asked for reasons that interested no one but mathematicians.

  How wrong you can be.

  Abel had discovered an obstacle to the solution of certain quintics by radicals. He had proved that this obstacle genuinely prevented such solutions existing for at least some quintics. The next step forward, the pivot upon which our entire story revolves, was made by someone who looked the gift horse firmly in the teeth and asked the kind of question that mathematicians cannot resist when some major problem has been solved. “Yes, that’s all very nice . . . but why does it really work?”