CHAPTER 8
Mathematics: Reality to Infinity
Chaos Is Defined by New Calculus
A new mathematical definition of chaos, which brings “utter confusion” for the first time under the control of man, was reported to the Fourth International Congress for Applied Mechanics here [in Cambridge, Mass.] today.
The definition is a new form of calculus. It enables scientists to predict what will happen in states of complete confusion. Practical uses are many. Examples are the solving of air turbulence which hampers airline flights and the flow of liquids in pipes.
This calculus was reported by Dr. Norbert Wiener of Massachusetts Institute of Technology. The mathematics first card indexes sample kinds of chaos. An engineer can select the sample which most nearly resembles his problem.
The steps of a drunken person illustrate the samples. The wabbly walk was one of the problems in chaos which first interested Dr. Wiener. Each step has no relation to previous steps. But calculus can show far the “drunk” is likely to go in a given time.
Problem in Planets’ Motions
“Perturbations” of the planets, their slight, irregular wanderings from their orbits, are a practical example of chaotic movement. Astronomers have been figuring them for centuries, and an explanation by Dr. Albert Einstein of one unexpected motion helped build his fame.
The new definition, Dr. Wiener explained, is entirely mathematical. It defines “pure” chaos. Practical examples of the “pure” state are difficult, he said, but closest to 100 percent chaos are the noises you here in a vacuum tube, like the radio. Electrons make them, by striking a target. But no single electron makes a sound. The racket comes from “mass,” that is, when a lot of electrons hit simultaneously.
There are three steps in the new calculus, said Dr. Wiener. The first was the “ergotic” (from the Greek word ergon, meaning “work”) theorem of Willard Gibbs, American scientist of the last century. It declared that a chaotic system would run through all possible phases of confusion.
Scientists, said Dr. Wiener, found that the systems did not run the entire gamut.
The next step was by Dr. G. D. Birkhoff of Harvard, with new mathematics by which the time when chaotic events would happen could be averaged.
Adds Factor of Space
Dr. Wiener’s word adds “space” to chaos, so as to tell not only the time but the place where a change is likely.
In airplane flights one of the “chaos” troubles is to figure out when and where the smooth flow of air over the top of a streamlined wing will bring up into eddies; and what kind of eddies. The eddies interfere with the plane’s lifting power.
Engineers by experiment measure some of these eddies. But it is impossible to measure them all. Designers have to guess on the basis of their sample eddies. The new mathematics is designed to take the guesswork out of these experiments by showing whether the samples are well chosen.
—September 14, 1938
Puzzle on Map Conjecture: Color It Solved
By TOM FERRELL
One of the best-known problems in mathematics, the so-called four-color conjecture, has at last been solved. The problem has engaged both amateur and professional attention for many years because it is so easily stated: Prove that four colors are enough to color any conceivable map, such that no two adjacent areas will be colored the same.
The four-color conjecture has long been considered probably true, because no one has ever been able to devise an imaginary map that violated it. However, such failure is not considered proof. For proof to exist, mathematicians must show that the conjecture is true for all possible maps that might ever be imagined.
The new proof has been carried out by Kenneth Appel and Wolfgang Haken of the University of Illinois, using about 1,200 hours of computer time. A brief and incomplete statement of their method, in nontechnical terms, might be this:
They took advantage of the fact that maps (divisions of a plane into regions) can be, by standard mathematical procedures, converted into graphs (points connected by lines). They found that 1,936 graphs could be used to represent all the possible configurations of maps. The computer analysis began by supposing the four-color conjecture is false; if so, a “bad map,” one that would require five colors, could exist. The computer showed for each of the 1,936 configurations that no “bad map” exists. Therefore, since the possibilities of falsehood are exhausted, the conjecture must be true.
Mathematicians are still in search of a simpler proof of the conjecture, one that will not lean so heavily on the computer but will be accessible to the techniques of more or less ordinary geometry.
There is some anxiety in mathematical circles that because the four-color conjecture has been broken at last, amateur mathematicians will now be encouraged to proceed with such projects as squaring the circle and trisecting the angle. Both, according to professionals, have been proven to be simply impossible.
—September 26, 1976
But Aren’t Truth and Beauty Supposed to Be Enough?
By JAMES GLEICK
Could the mathematicians, winners of the most prestigious awards of their discipline, please tell the audience what their work is good for?
Flush with pleasure, these four young men, carrying home three Fields Medals and a Nevanlinna Prize, were telling a lay audience what their work was about. Two had discovered astonishing facts about shapes in four-dimensional space. One had developed important insights into what makes hard problems hard. One had proved Mordell’s conjecture, the idea that a large class of equations can have only a finite number of rational solutions.
To the nearly 4,000 mathematicians who gathered here [in Berkeley, Calif.] for the International Congress of Mathematicians, which ended Monday, these were an astonishing set of breakthroughs demonstrating new vitality in the purest of sciences. But a reporter-cameraman for a local television station wanted at least one of the prizewinners to address a basic question: How would their achievements improve life for the viewers at home?
Embarrassed silence. The mathematicians suddenly seemed to have remembered pressing engagements elsewhere. They looked at one another. Gerd Faltings, a boyish, blond West German who became one of mathematics’ great men when he proved the Mordell conjecture, gave an awkward smile and flatly refused to speak.
It was Michael Freedman, a topologist in California, who rose in the end to say what all the mathematicians felt: That theirs is a way of thinking that thrives by disdaining the need for practical applications. Let the applications come later by accident—they always do. A weird, curved parody of Euclidean geometry turns out to be just the framework a physicist needs to invent the General Theory of Relativity. Notoriously unpractical techniques of number theory turn out to be just what the National Security Agency needs to make efficient, secure codes.
Usually unspoken, but always present, is the faith that doing mathematics purely, following an internal compass, seeking elegance and beauty in a strange abstract world, is the best way in the long run to serve practical science. As physics or biology progress, they will inevitably find that the way ahead has been cleared by some odd piece of pure mathematics that was thought dead and buried for many decades.
“We’re a part of a gigantic enterprise that has gone on for hundreds of years, and interacts in interesting ways with science, and operates on a very low budget,” Dr. Freedman said, “and we’ve learned that it’s hard to prophesy what piece of mathematics will have what particular applications. Mathematics has to advance as an organic whole, in ways that seem right to the people inside it.”
An older mathematician, Sir Michael Atiyah of Oxford University, who won a Fields Medal himself in 1966, offered one correction. “It’s been thousands of years,” he said. “So we’re in business on a long-term basis.”
Yet the meaning of mathematical purity is changing—has changed, many mathematicians said, even since the last Congress in Warsaw in 1983. Questions about the nature of that change, and what it might mean for the future, hovered in the air through 16 “plenary addresses,” scores of 45-minute lectures and hundreds of 10-minute “short communications” in the nine-day conference.
Mostly unrepresented was the somewhat less exalted discipline known as applied mathematics, the traditional route for mathematical ideas to filter down to engineering and other sciences. A few mathematicians could not help noticing, though, that recently physical scientists have been plucking ideas directly from the heart of pure mathematics, bypassing applied mathematics altogether. Many unexpected connections have arisen—between knot theory and genetic processes in DNA strands, for example—but the most important has been the use of geometric ideas in the theory of cosmic strings, the hottest new game in the physics of fundamental forces and particles.
The suspicion of a few mathematicians here was that biologists and chemists can no longer be relied on to be naive about the arcana of number theory or topology. That will take some getting used to.
And in the case of strings, the physics has begun feeding back into mathematics, meaning that the up-to-date pure mathematician may now have to learn some unpure science. This state of affairs was highlighted by two unusual talks by physicists, Edward Witten of Princeton University and Aleksandr Polyakov of the Soviet Union’s Landau Institute for Theoretical Physics.
Dr. Polyakov, an intense man with long sandy hair, paced nervously before his lecture, a red knapsack on his back. He was worried that his mathematician audience would be put off by having to hear a foreign language—physics, not Russian.
“I apologize if you are irritated by the reckless manner of a physicist,” he told his audience. Reckless, because the two disciplines have different standards of proof: A physicist is content to say that the earth orbits the sun; a mathematician will say only that there is convincing evidence.
Dr. Freedman, in the work that won his Fields Medal, proved that certain exotic four-dimensional spaces exist. Another medalist, Simon Donaldson of Oxford, meanwhile, used tools from physics to prove that these same spaces could not exist.
“So the conclusion a mathematician would draw,” Dr. Freedman said, “is that physics doesn’t exist.”
To some mathematicians, purity has always meant a certain degree of inscrutability. That, at least, has not changed.
Sometimes inscrutability comes with the territory—for example, when the territory has four dimensions or more. A mathematician needs to be comfortable with shapes in many dimensions, but not everyone can actually visualize more than the usual three. That is one reason geometry relies, for the sake of purity, on rigorous proofs using numbers and symbols. Visual imagination cannot be trusted.
One dimension is a line. The second dimension comes when you add a second line at right angles to the first, so that now you have east-west and north-south. The third dimension requires a new line at right angles to the others, so you must leave the flat plain and draw one up-down. To imagine a fourth dimension, it is necessary to imagine a fourth line at right angles to all the others, and this most mortals cannot do.
Yet some kind of inner vision led John Milnor of the Institute for Advanced Study in Princeton, in describing the four-dimensional discoveries of Dr. Freedman to an audience of several thousand, to start gesturing with his hands.
“The problem is,” he was saying, “when you try to embed a two-dimensional disk inside a four-dimensional manifold, it will usually intersect itself.” His hands formed loops and handles in the air, as though he were describing some new kind of suitcase.
Sometimes inscrutability is just a matter of style.
It has been said that the ideal mathematics talk has three parts. The first part should be understood by most of your audience. The second part should be understood by four or five specialists in your field. The third part should be understood by no one—because how else will people know you are serious?
Some speakers seemed to follow these guidelines, mathematicians felt. Others, perhaps to save time, skipped directly to part three.
There was one question that Fields medalists could not wait to answer, and that was whether they used computers, the unloved child of mathematics and an object whose influence was more in evidence at the congress than ever before.
“No,” Dr. Freedman said. Dr. Donaldson: “No.” Dr. Faltings said, “Perhaps it could reduce some sorts of tedious work for us, but it doesn’t do the thinking.” Personally he doesn’t use one.
Eyes turned to Leslie G. Valiant of Harvard University, winner of the recently established Nevanlinna Prize for Information Science, whose work centered on computer algorithms. “Maybe I should clarify my own position,” Dr. Valiant said. “I don’t use computers either.”
If the mathematicians were inclined toward parable and metaphor—and they most definitely are not—they might describe a vast wilderness, and in it a small society of men and women whose business it is to lay railroad track. This has become an art, and they have become artists—artists of track, lovers of track, connoisseurs of track.
Almost perversely, they ignore the landscape around them. A network of track may head to the northeast for many years and then be abandoned. An old, nearly forgotten line to the south may sprout new branches, heading toward a horizon that the tracklayers seem unable or unwilling to see.
As long as each new piece of track is carefully joined to the old, so that the progression is never broken, an odd thing happens. People come along hoping to explore this forest or that desert, and they find that a certain stretch of track takes them exactly where they need to go. The tracklayers, for their part, may have long since abandoned that place. But the track remains, and track, of course, is the stuff on which the engines of knowledge roll forward.
—August 12, 1986
At Last, Shout of “Eureka!” in Age-Old Math Mystery
By GINA KOLATA
More than 350 years ago, a French mathematician wrote a deceptively simple theorem in the margins of a book, adding that he had discovered a marvelous proof of it but lacked space to include it in the margin. He died without ever offering his proof, and mathematicians have been trying ever since to supply it.
Now, after thousands of claims of success that proved untrue, mathematicians say the daunting challenge, perhaps the most famous of unsolved mathematical problems, has at last been surmounted.
The problem is Fermat’s last theorem, and its apparent conqueror is Dr. Andrew Wiles, a 40-year-old English mathematician who works at Princeton University. Dr. Wiles announced the result yesterday at the last of three lectures given over three days at Cambridge University in England.
Within a few minutes of the conclusion of his final lecture, computer mail messages were winging around the world as mathematicians alerted each other to the startling and almost wholly unexpected result.
Dr. Leonard Adelman of the University of Southern California said he received a message about an hour after Dr. Wiles’s announcement. The frenzy is justified, he said. “It’s the most exciting thing that’s happened in—geez—maybe ever, in mathematics.”
Impossible Is Possible
Mathematicians present at the lecture said they felt “an elation,” said Dr. Kenneth Ribet of the University of California at Berkeley, in a telephone interview from Cambridge.
The theorem, an overarching statement about what solutions are possible for certain simple equations, was stated in 1637 by Pierre de Fermat, a 17th-century French mathematician and physicist. Many of the brightest minds in mathematics have struggled to find the proof ever since, and many have concluded that Fermat, contrary to his tantalizing claim, had probably failed to develop one despite his considerable mathematical ability.
With Dr. Wiles’s result, Dr. Ribet said, “the mathematical landscape has changed.” He explained: “You discover that things that seemed completely impossible are more of a reality. This changes the way you approach problems, what you think is possible.”
Dr. Barry Mazur, a Harvard University mathematician, also reached by telephone in Cambridge, said: “A lot more is proved than Fermat’s last theorem. One could envision a proof of a problem, no matter how celebrated, that had no implications. But this is just the reverse. This is the emergence of a technique that is visibly powerful. It’s going to prove a lot more.” Remember Pythagoras?
Fermat’s last theorem has to do with equations of the form xn + yn = zn. The case where n is 2 is familiar as the Pythagorean theorem, which says that the squares of the lengths of two sides of a right-angled triangle equal the square of the length of the hypotenuse. One such equation is 32 + 42 = 52, since 9 + 16 = 25.
Fermat’s last theorem states that there are no solutions to such equations when n is a whole number greater than 2. This means, for instance, that it would be impossible to find any whole numbers x, y and z such that x3 + y3 = z3. Thus 33 + 43 (27 + 64) = 91, which is not the cube of any whole number.
Mathematicians in the United States said that the stature of Dr. Wiles and the imprimatur of the experts who heard his lectures, especially Dr. Ribet and Dr. Mazur, convinced them that the new proof was very likely to be right. In addition, they said, the logic of the proof is persuasive because it is built on a carefully developed edifice of mathematics that goes back more than 30 years and is widely accepted.
Experts cautioned that Dr. Wiles could, of course, have made some subtle misstep. Dr. Harold M. Edwards, a mathematician at the Courant Institute of Mathematical Sciences in New York, said that until the proof was published in a mathematical journal, which could take a year, and until it is checked many times, there is always a chance it is wrong. The author of a book on Fermat’s last theorem, Dr. Edwards noted that “even good mathematicians have had false proofs.”
Luring the World’s “Cranks”
But even he said that Dr. Wiles’s proof sounded like the real thing and “has to be taken very seriously.”
Despite the apparent simplicity of the theorem, proving it was so hard that in 1815 and in again 1860, the French Academy of Sciences offered a gold medal and 300 francs to anyone who could solve it. In 1908, the German Academy of Sciences offered a prize of 100,000 marks for a proof that the theorem was correct. The prize, which still stands but has been reduced to 7,500 marks, about $4,400, has attracted the world’s “cranks,” Dr. Edwards said. When the Germans said the proof had to be published, “the cranks began publishing their solutions in the vanity press,” he said, yielding thousands of booklets. The Germans told him they would even award the prize for a proof that the theorem was not true, Dr. Edwards added, saying that they “would be so overjoyed that they wouldn’t have to read through these submissions.”
But it was not just amateurs whose imagination was captured by the enigma. Famous mathematicians, too, spent years on it. Others avoided it for fear of being sucked into a quagmire. One mathematical genius, David Hilbert, said in 1920 that he would not work on it because “before beginning I should put in three years of intensive study, and I haven’t that much time to spend on a probable failure.”
Mathematicians armed with computers have shown that Fermat’s theorem holds true up to very high numbers. But that falls well short of a general proof.
Tortuous Path to Proof
Dr. Ribet said that 20th-century work on the problem had begun to grow ever more divorced from Fermat’s equations. “Over the last 60 years, people in number theory have forged an incredible number of tools to deal with simple problems like this,” he said. Eventually, “people lost day-to-day contact with the old problems and were preoccupied with the objects they created,” he said.
Dr. Wiles’s proof draws on many of these mathematical tools but also “completes a chain of ideas,” said Dr. Nicholas Katz of Princeton University. The work leading to the proof began in 1954, when the late Japanese mathematician Yutaka Taniyama made a conjecture about mathematical objects called elliptic curves. That conjecture was refined by Dr. Goro Shimura of Princeton University a few years later. But for decades, Dr. Katz said, mathematicians had no idea that this had any relationship to Fermat’s last theorem. “They seemed to be on different planets,” he said.
In the mid-’80s, Dr. Gerhard Frey of the University of the Saarland in Germany “came up with a very strange, very simple connection between the Taniyama conjecture and Fermat’s last theorem,” Dr. Katz said. “It gave a sort of rough idea that if you knew Taniyama’s conjecture you would in fact know Fermat’s last theorem,” he explained. In 1987, Dr. Ribet proved the connection. Now, Dr. Wiles has shown that a form of the Taniyama conjecture is true and that this implies that Fermat’s last theorem must be true.
“One of the things that’s most remarkable about the fact that Fermat’s last theorem is proven is the incredibly roundabout path that led to it,” Dr. Katz said.
Another remarkable aspect is that such a seemingly simple problem would require such sophisticated and highly specialized mathematics for its proof. Dr. Ribet estimated that a tenth of one percent of mathematicians could understand Dr. Wiles’s work because the mathematics is so technical. “You have to know a lot about modular forms and algebraic geometry,” he said. “You have to have followed the subject very closely.”
The general idea behind Dr. Wiles’s proof was to associate an elliptic curve, which is a mathematical object that looks something like the surface of a doughnut, with an equation of Fermat’s theorem. If the theorem were false and there were indeed solutions to the Fermat equations, a peculiar curve would result. The proof hinged on showing that such a curve could not exist.
Dr. Wiles, who has told colleagues that he is reluctant to speak to the press, could not be reached yesterday. Dr. Ribet, who described Dr. Wiles as shy, said he had been asked to speak for him.
Dr. Ribet said it took Dr. Wiles seven years to solve the problem. He had a solution for a special case of the conjecture two years ago, Dr. Ribet said, but told no one. “It didn’t give him enough and he felt very discouraged by it,” he said.
Dr. Wiles presented his results this week at a small conference in Cambridge, England, his birthplace, on “P-adic Galois Representations, Iwasawa Theory and the Tamagawa Numbers of Motives.” He gave a lecture a day on Monday, Tuesday and Wednesday with the title “Modular Forms, Elliptic Curves and Galois Representations.” There was no hint in the title that Fermat’s last theorem would be discussed, Dr. Ribet said.
“As Wiles began his lectures, there was more and more speculation about what it was going to be,” Dr. Ribet said. The audience of specialists in these arcane fields swelled from about 40 on the first day to 60 yesterday. Finally, at the end of his third lecture, Dr. Wiles concluded that he had proved a general case of the Taniyama conjecture. Then, seemingly as an afterthought, he noted that that meant that Fermat’s last theorem was true. Q.E.D.
People raised their cameras and snapped pictures of this historic moment, Dr. Ribet said. Then “there was a warm round of applause, followed by a couple of questions and another warm round of applause,” he added.
“I had to give the next lecture,” Dr. Ribet said. “It was something incredibly mundane.” Since mathematicians are “a pretty well behaved bunch,” they listened politely. But, he said, it was hard to concentrate. “Most people in the room, including me, were incredibly shell-shocked,” he said.
—June 24, 1993
The Spies’ Code and How It Broke
By GEORGE JOHNSON
To the human brain, with its insatiable hunger for order, nothing is more disorienting than randomness. Soviet cryptographers knew this well when they set out to devise a code for communicating with the spy ring that included Julius and Ethel Rosenberg. With a system as simple as it was ingenious, they tried to insure that any message intercepted by United States intelligence agents would seem as meaningless as the snow on a television set tuned to an empty station.
But pure, unadulterated randomness can be extremely difficult to manufacture. As was revealed last week after decades of secrecy, in a ceremony at the Central Intelligence Agency headquarters, the Russians suffered from a lapse in quality control. They inadvertently let some pattern find its way into their scrambled codes, a loose thread that allowed American code breakers to unravel the scheme slowly.
The fine details of the Soviet encryption remain among the secrets of the National Security Agency, America’s premier decoding service. But the principle behind the system, called a “one-time pad,” has been known to cryptologists for years.
One begins with an alphabetical list of words or phrases likely to be needed in messages. These are numbered sequentially. Suppose that “Antenna,” an early code name for Julius Rosenberg (later changed to “Liberal”), was assigned the number 2222. If the next item on the list was “anti-tank,” it would be 2223, and “Anton,” the code name for the Rosenbergs’ KGB handler, Leonid Kvasnikov, would be 2224. In such a system, a message can be converted into a sequence of numbers and decoded by anyone with a copy of the translation table.
Names not on the list could be spelled out. A code number—9953, say—would tell the recipient that the following numbers stood for individual letters encrypted according to some agreed-upon scheme. Then 9954 could be a signal to stop spelling and pop back up to the dominant system in which whole words or phrases are assigned numbers.
If that is all there were to the code, even someone without the table might be able to crack it. One might profitably assume that the most frequent pattern of numbers probably represented the period or full stop. Articles like “a” and “the” would be among the next most common patterns. In many languages, the next pattern after a period would probably be a noun. With some lucky guesses, sophisticated statistical analysis and a lot of trial and error, meaning could be squeezed from the noise.
Code upon Code
Hoping to guard against this possibility, the Russian cryptographers added another layer of obfuscation. After using the table to translate the message into a string of digits, they disguised it further by adding to it a long random number. The result would also be a random number, patternless and theoretically indecipherable.
The message would be decodable by the Russians because sender and receiver each knew the random number used in the encoding scheme. If the sender wanted to encrypt “Antenna,” he would translate it into 2222, then take out a pad imprinted with the random number key and copy down the first four digits, perhaps 3913. Adding the two numbers would produce 6135. Then he would move onto the next part of the message, adding it to the next digits on the pad. Once they had been used, the random numbers would be discarded—hence the name “one-time pad.”
To decode the message, the recipient would take out his random number pad, copy the appropriate digits and subtract them from the message to recover the original number string. Proceeding like this, always carefully keeping their place on the pad, sender and receiver would be able to read dispatches that to anyone intercepting them would look like pure noise. While “Antenna” might be 3913 in one sentence, in the next it might be 4710. Since there is no structure to the key, there are none of the patterns cryptanalysts need to get a statistical foothold.
“Given a pure, perfect one-time system, you’re not going to break it,” said David Kahn, visiting historian at the NSA’s Center for Cryptologic History and author of The Codebreakers (Macmillan, 1967). Even if the message were short enough for an intelligence agent to systematically subtract from it every possible number string, the result would be meaningless. “You would simply find that you had generated every possible message in every possible language with no way of telling which one was correct,” Mr. Kahn said.
But no system is foolproof. First, generating a truly random number is harder than it sounds. Flipping a coin produces a random pattern containing an equal number of heads and tails—but only if the coin is perfectly balanced. More likely, differences in the engravings could make one side heavier than the other. The result would be deviations from randomness that might allow an observer to tell from the record of a coin toss whether it was more likely generated by a nickel or a dime. Similarly, if the code sender’s random number generator is flawed, there might be enough order in the message for cryptanalysts to reconstruct the key.
Asking for Infinity
As revealed by American cryptologists last week, the Russians’ crucial flaw was much more trivial. The problem with the one-time pad is that it depends on generating a number that is, as Mr. Kahn put it, not only absolutely random but endless. In practice, the number cannot be infinite, of course, but it must be long enough to encode every possible message that will conceivably be sent over a channel.
As traffic between Moscow and the KGB office in New York increased in volume, the Russians apparently ran out of numbers and committed the cryptographer’s cardinal sin. They repeated themselves, betraying details of Soviet espionage efforts on American soil. The spell of randomness was broken, and meaning began seeping in.
—July 16, 1995
The Mighty Mathematician You’ve Never Heard Of
By NATALIE ANGIER
Scientists are a famously anonymous lot, but few can match in the depths of her perverse and unmerited obscurity the 20th-century mathematical genius Amalie Noether.
Albert Einstein called her the most “significant” and “creative” female mathematician of all time, and others of her contemporaries were inclined to drop the modification by sex. She invented a theorem that united with magisterial concision two conceptual pillars of physics: symmetry in nature and the universal laws of conservation. Some consider Noether’s theorem, as it is now called, as important as Einstein’s theory of relativity; it undergirds much of today’s vanguard research in physics, including the hunt for the almighty Higgs boson.
Yet Noether herself remains utterly unknown, not only to the general public, but to many members of the scientific community as well.
When Dave Goldberg, a physicist at Drexel University who has written about her work, recently took a little “Noetherpoll” of several dozen colleagues, students and online followers, he was taken aback by the results. “Surprisingly few could say exactly who she was or why she was important,” he said. “A few others knew her name but couldn’t recall what she’d done, and the majority had never heard of her.”
Noether (pronounced NER-ter) was born in Erlangen, Germany, 130 years ago this month. So it’s a fine time to counter the chronic neglect and celebrate the life and work of a brilliant theorist whose unshakable number love and irrationally robust sense of humor helped her overcome severe handicaps—first, being female in Germany at a time when most German universities didn’t accept female students or hire female professors, and then being a Jewish pacifist in the midst of the Nazis’ rise to power.
Through it all, Noether was a highly prolific mathematician, publishing groundbreaking papers, sometimes under a man’s name, in rarefied fields of abstract algebra and ring theory. And when she applied her equations to the universe around her, she discovered some of its basic rules, like how time and energy are related, and why it is, as the physicist Lee Smolin of the Perimeter Institute put it, “that riding a bicycle is safe.”
Ransom Stephens, a physicist and novelist who has lectured widely on Noether, said, “You can make a strong case that her theorem is the backbone on which all of modern physics is built.”
Noether came from a mathematical family. Her father was a distinguished math professor at the universities of Heidelberg and Erlangen, and her brother Fritz won some renown as an applied mathematician. Emmy, as she was known throughout her life, started out studying English, French and piano—subjects more socially acceptable for a girl—but her interests soon turned to math. Barred from matriculating formally at the University of Erlangen, Emmy simply audited all the courses, and she ended up doing so well on her final exams that she was granted the equivalent of a bachelor’s degree.
She went on to graduate school at the University of Göttingen, where she earned her doctorate summa cum laude and met many of the leading mathematicians of the day, including David Hilbert and Felix Klein, who did for the bottle what August Ferdinand Möbius had done for the strip. Noether’s brilliance was obvious to all who worked with her, and her male mentors repeatedly took up her cause, seeking to find her a teaching position—better still, one that paid.
“I do not see that the sex of the candidate is an argument against her,” Hilbert said indignantly to the administration at Göttingen, where he sought to have Noether appointed as the equivalent of an associate professor. “After all, we are a university, not a bathhouse.” Hilbert failed to make his case, so instead brought her on staff as a more or less permanent “guest lecturer”; and Noether, fittingly enough, later took up swimming at a men-only pool.
At Göttingen, she pursued her passion for mathematical invariance, the study of numbers that can be manipulated in various ways and still remain constant. In the relationship between a star and its planet, for example, the shape and radius of the planetary orbit may change, but the gravitational attraction conjoining one to the other remains the same—and there’s your invariance.
In 1915 Einstein published his general theory of relativity. The Göttingen math department fell “head over ear” with it, in the words of one observer, and Noether began applying her invariance work to some of the complexities of the theory. That exercise eventually inspired her to formulate what is now called Noether’s theorem, an expression of the deep tie between the underlying geometry of the universe and the behavior of the mass and energy that call the universe home.
What the revolutionary theorem says, in cartoon essence, is the following: Wherever you find some sort of symmetry in nature, some predictability or homogeneity of parts, you’ll find lurking in the background a corresponding conservation—of momentum, electric charge, energy or the like. If a bicycle wheel is radially symmetric, if you can spin it on its axis and it still looks the same in all directions, well, then, that symmetric translation must yield a corresponding conservation. By applying the principles and calculations embodied in Noether’s theorem, you’ll see it is angular momentum, the Newtonian impulse that keeps bicyclists upright and on the move.
Some of the relationships to pop out of the theorem are startling, the most profound one linking time and energy. Noether’s theorem shows that a symmetry of time—like the fact that whether you throw a ball in the air tomorrow or make the same toss next week will have no effect on the ball’s trajectory—is directly related to the conservation of energy, our old homily that energy can be neither created nor destroyed but merely changes form.
The connections that Noether forged are “critical” to modern physics, said Lisa Randall, a professor of theoretical particle physics and cosmology at Harvard. “Energy, momentum and other quantities we take for granted gain meaning and even greater value when we understand how these quantities follow from symmetry in time and space.”
Dr. Randall, the author of the newly published Knocking on Heaven’s Door, recalled the moment in college when she happened to learn that the author of Noether’s theorem was a she. “It was striking and even exciting and inspirational,” Dr. Randall said, admitting, “I was surprised by my reaction.”
For her part, Noether left little record of how she felt about the difficulties she faced as a woman, or of her personal and emotional life generally. She never married, and if she had love affairs she didn’t trumpet them. After meeting the young Czech math star Olga Taussky in 1930, Noether told friends how happy she was that women were finally gaining acceptance in the field, but she herself had so few female students that her many devoted pupils were known around town as Noether’s boys.
Noether lived for math and cared nothing for housework or possessions, and if her long, unruly hair began falling from its pins as she talked excitedly about math, she let it fall. She laughed often and in photos is always smiling.
When a couple of students started showing up to class wearing Hitler’s brown shirts, she laughed at that, too. But not for long. Noether was one of the first Jewish scientists to be fired from her post and forced to flee Germany. In 1933, with the help of Einstein, she was given a job at Bryn Mawr College, where she said she felt deeply appreciated as she never had been in Germany.
That didn’t last long, either. Only 18 months after her arrival in the United States, at the age of 53, Noether was operated on for an ovarian cyst, and died within days.
—March 27, 2012
The Life of Pi, and Other Infinities
By NATALIE ANGIER
On this day that fetishizes finitude, that reminds us how rapidly our own earthly time share is shrinking, allow me to offer the modest comfort of infinities.
Yes, infinities, plural. The popular notion of infinity may be of a monolithic totality, the ultimate, unbounded big tent that goes on forever and subsumes everything in its path—time, the cosmos, your complete collection of old Playbills. Yet in the ever-evolving view of scientists, philosophers and other scholars, there really is no single, implacable entity called infinity.
Instead, there are infinities, multiplicities of the limit-free that come in a vast variety of shapes, sizes, purposes and charms. Some are tailored for mathematics, some for cosmology, others for theology; some are of such recent vintage their fontanels still feel soft. There are flat infinities, hunchback infinities, bubbling infinities, hyperboloid infinities. There are infinitely large sets of one kind of number, and even bigger, infinitely large sets of another kind of number.
There are the infinities of the everyday, as exemplified by the figure of pi, with its endless post-decimal tail of nonrepeating digits, and how about if we just round it off to 3.14159 and then serve pie on March 14 at 1:59 p.m.? Another stalwart of infinity shows up in the mathematics that gave us modernity: calculus.
“All the key concepts of calculus build on infinite processes of one form or another that take limits out to infinity,” said Steven Strogatz, author of the recent book The Joy of x: A Guided Tour of Math, From One to Infinity and a professor of applied mathematics at Cornell. In calculus, he added, “infinity is your friend.”
Yet worthy friends can come in prickly packages, and mathematicians have learned to handle infinity with care.
“Mathematicians find the concept of infinity so useful, but it can be quite subtle and quite dangerous,” said Ian Stewart, a mathematics researcher at the University of Warwick in England and the author of Visions of Infinity, the latest of many books. “If you treat infinity like a normal number, you can come up with all sorts of nonsense, like saying, infinity plus one is equal to infinity, and now we subtract infinity from each side and suddenly naught equals one. You can’t be freewheeling in your use of infinity.”
Then again, a very different sort of infinity may well be freewheeling you. Based on recent studies of the cosmic microwave afterglow of the Big Bang, with which our known universe began 13.7 billion years ago, many cosmologists now believe that this observable universe is just a tiny, if relentlessly expanding, patch of space-time embedded in a greater universal fabric that is, in a profound sense, infinite. It may be an infinitely large monoverse, or it may be an infinite bubble bath of infinitely budding and inflating multiverses, but infinite it is, and the implications of that infinity are appropriately huge.
“If you take a finite physical system and a finite set of states, and you have an infinite universe in which to sample them, to randomly explore all the possibilities, you will get duplicates,” said Anthony Aguirre, an associate professor of physics who studies theoretical cosmology at the University of California, Santa Cruz.
Not just rough copies, either. “If the universe is big enough, you can go all the way,” Dr. Aguirre said. “If I ask, will there be a planet like Earth with a person in Santa Cruz sitting at this colored desk, with every atom, every wave function exactly the same, if the universe is infinite the answer has to be yes.”
In short, your doppelgangers may be out there and many variants, too, some with much better hair who can play Bach like Glenn Gould. A far less savory thought: There could be a configuration, Dr. Aguirre said, “where the Nazis won the war.”
Given infinity’s potential for troublemaking, it’s small wonder the ancient Greeks abhorred the very notion of it.
“They viewed it with suspicion and hostility,” said A. W. Moore, professor of philosophy at Oxford University and the author of The Infinite (1990). The Greeks wildly favored tidy rational numbers that, by definition, can be defined as a ratio, or fraction—the way 0.75 equals ¾ and you’re done with it—over patternless infinitums like the square root of 2.
On Pythagoras’ Table of Opposites, “the finite” was listed along with masculinity and other good things in life, while “the infinite” topped the column of bad traits like femininity. “They saw it as a cosmic fight,” Dr. Moore said, “with the finite constantly having to subjugate the infinite.”
Aristotle helped put an end to the rampant infiniphobia by drawing a distinction between what he called “actual” infinity, something that would exist all at once, at a given moment—which he declared an impossibility—and “potential” infinity, which would unfold over time and which he deemed perfectly intelligible. As a result, Dr. Moore said, “Aristotle believed in finite space and infinite time,” and his ideas held sway for the next 2,000 years.
Newton and Leibniz began monkeying with notions of infinity when they invented calculus, which solves tricky problems of planetary motions and accelerating bodies by essentially breaking down curved orbits and changing velocities into infinite series of tiny straight lines and tiny uniform motions. “It turns out to be an incredibly powerful tool if you think of the world as being infinitely divisible,” Dr. Strogatz said.
In the late 19th century, the great German mathematician Georg Cantor took on infinity not as a means to an end, but as a subject worthy of rigorous study in itself. He demonstrated that there are many kinds of infinite sets, and some infinities are bigger than others. Hard as it may be to swallow, the set of all the possible decimal numbers between 1 and 2, being unlistable, turns out to be a bigger infinity than the set of all whole numbers from 1 to forever, which in principle can be listed.
In fact, many of Cantor’s contemporaries didn’t swallow, dismissing him as “a scientific charlatan,” “laughable” and “wrong.” Cantor died depressed and impoverished, but today his set theory is a flourishing branch of mathematics relevant to the study of large, chaotic systems like the weather, the economy and human stupidity.
With his majestic theory of relativity, Einstein knitted together time and space, quashing old Aristotelian distinctions between actual and potential infinity and ushering in the contemporary era of infinity seeking. Another advance came in the 1980s, when Alan Guth introduced the idea of cosmic inflation, a kind of vacuum energy that vastly expanded the size of the universe soon after its fiery birth.
New theories suggest that such inflation may not have been a one-shot event, but rather part of a runaway process called eternal inflation, an infinite ballooning and bubbling outward of this and possibly other universes.
Relativity and inflation theory, said Dr. Aguirre, “allow us to conceptualize things that would have seemed impossible before.” Time can be twisted, he said, “so from one point of view the universe is a finite thing that is growing into something infinite if you wait forever, but from another point of view it’s always infinite.”
Or maybe the universe is like Jorge Luis Borges’s fastidiously imagined Library of Babel, composed of interminable numbers of hexagonal galleries with polished surfaces that “feign and promise infinity.”
Or like the multiverse as envisioned in Tibetan Buddhism, “a vast system of 1059 universes, that together are called a Buddha Field,” said Jonathan C. Gold, who studies Buddhist philosophy at Princeton.
The finite is nested within the infinite, and somewhere across the glittering, howling universal sample space of Buddha Field or Babel, your doppelganger is hard at the keyboard, playing a Bach toccata.
—January 1, 2013
Don’t Expect Math to Make Sense
By MANIL SURI
Each year, March 14 is Pi Day, in honor of the mathematical constant. Saturday is the once-in-a-century event when the year, ’15, brings the full date in line with the first five digits of pi’s decimal expansion—3.1415. Typical celebrations revolve around eating pies and composing “pi-kus” (haikus with three syllables in the first line, one in the second and four in the third). But perhaps a better way to commemorate the day is by trying to grasp what pi truly is, and why it remains so significant.
Pi is irrational, meaning it cannot be expressed as the ratio of two whole numbers. There is no way to write it down exactly: Its decimals continue endlessly without ever settling into a repeating pattern. No less an authority than Pythagoras repudiated the existence of such numbers, declaring them incompatible with an intelligently designed universe.
And yet pi, being the ratio of a circle’s circumference to its diameter, is manifested all around us. For instance, the meandering length of a gently sloping river between source and mouth approaches, on average, pi times its straight-line distance. Pi reminds us that the universe is what it is, that it doesn’t subscribe to our ideas of mathematical convenience.
Early mathematicians realized pi’s usefulness in calculating areas, which is why they spent so much effort trying to dig its digits out. Archimedes used 96-sided polygons to painstakingly approximate the circle and showed that pi lay between 223/71 and 22/7. By the time Madhava (in India, around 1400) calculated pi to over 10 decimal places using his groundbreaking infinite series (which regrettably bears Leibniz’s name), it was already more than accurate enough to address all practical applications. Pursuing pi further had essentially become a mathematical challenge.
With the advent of computers, pi offered a proving ground for successively faster models. But eventually, breathless headlines about newly cracked digits became less compelling, and the big players moved on. Recent records (currently in the trillions of digits) have mostly been set on custom-built personal computers. The history of pi illustrates how far computing has progressed, and how much we now take it for granted.
So what use have all those digits been put to? Statistical tests have suggested that not only are they random, but that any string of them occurs just as often as any other of the same length. This implies that, if you coded this article, or any other, as a numerical string, you could find it somewhere in the decimal expansion of pi. Of course, that’s relatively meaningless, since you don’t know where to find the material you want. An apt metaphor for an age when we are being asphyxiated by mushrooming clouds of information.
But pi’s infinite randomness can also be seen more as richness. What amazes, then, is the possibility that such profusion can come from a rule so simple: circumference divided by diameter. This is characteristic of mathematics, whereby elementary formulas can give rise to surprisingly varied phenomena. For instance, the humble quadratic can be used to model everything from the growth of bacterial populations to the manifestation of chaos. Pi makes us wonder if our universe’s complexity emerges from similarly simple mathematical building blocks.
Pi also opens a window into a more uncharted universe, the one consisting of transcendental numbers, which exclude such common irrationals as square and cube roots. Pi is one of the few transcendentals we ever encounter. One may suspect that such numbers would be quite rare, but actually, the opposite is true. Out of the totality of numbers, almost all are transcendental. Pi reveals how limited human knowledge is, how there exist teeming realms we might never explore.
The combination of utility and mystery makes pi a perfect symbol for all of mathematics. Surely the ancients, had they understood pi better, would have deified it, just as they did the moon and the sun. They would have praised pi’s immutability: Pi = 3.14159 … is one of the few absolutes that remain.
Or is it? The ratio of circumference to diameter might not be as fixed as we think. To understand why, imagine a circle drawn on the surface of a sphere. Its diameter, as measured along the bulging surface, will be greater than if the same circle is traced out on a flat sheet of paper. This observation might have been of only academic interest except for our inability, so far, to definitively determine whether the geometry of our universe is flat. If there is even a little curvature, then the value of pi, as defined by this ratio, is not what we think.
But pi, on cue, reminds us that it is an abstraction, like all else in mathematics. The perfect flat circle is impossible to realize in practice. An area calculated using pi will never exactly match the same area measured physically. This is to be expected whenever we approximate reality using the idealizations of math.
Many decades ago, a teacher had me memorize another approximation, pi = 22/7. Except she told my class this was pi’s exact value. Perhaps she feared we’d be as traumatized as Pythagoras by the idea of a non-fractional universe; more likely, she didn’t want to confuse us. I wish we’d had Pi Day then, because pi = 22/7 was a misconception I carried all the way to college. Since then I’ve learned that it’s only when we try to stretch our minds around mathematics’ enigmas that true understanding can set in.
—March 14, 2015