Andrew Wiles

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pages: 289 words: 85,315

Fermat’s Last Theorem by Simon Singh

Albert Einstein, Andrew Wiles, Antoine Gombaud: Chevalier de Méré, Arthur Eddington, Augustin-Louis Cauchy, Fellow of the Royal Society, Georg Cantor, Henri Poincaré, Isaac Newton, John Conway, John von Neumann, kremlinology, probability theory / Blaise Pascal / Pierre de Fermat, RAND corporation, Rubik’s Cube, Simon Singh, Wolfskehl Prize

SIMON SINGH Fermat’s Last Theorem THE STORY OF A RIDDLE THAT CONFOUNDED THE WORLD’S GREATEST MINDS FOR 358 YEARS In memory of Pakhar Singh Birring CONTENTS Cover Title Page Dedication Foreword Preface 1 ‘I Think I’ll Stop Here’ 2 The Riddler 3 A Mathematical Disgrace 4 Into Abstraction 5 Proof by Contradiction 6 The Secret Calculation 7 A Slight Problem Epilogue Grand Unified Mathematics Appendices Suggestions for Further Reading Index About the Author Also by the Author Copyright About the Publisher Foreword We finally met across a room, not crowded, but large enough to hold the entire Mathematics Department at Princeton on their occasions of great celebration. On that particular afternoon, there were not so very many people around, but enough for me to be uncertain as to which one was Andrew Wiles. After a few moments I picked out a shy-looking man, listening to the conversation around him, sipping tea, and indulging in the ritual gathering of minds that mathematicians the world over engage in at around four o’clock in the afternoon. He simply guessed who I was. It was the end of an extraordinary week. I had met some of the finest mathematicians alive, and begun to gain an insight into their world. But despite every attempt to pin down Andrew Wiles, to speak to him, and to convince him to take part in a BBC Horizon documentary film on his achievement, this was our first meeting. This was the man who had recently announced that he had found the holy grail of mathematics; the man who claimed he had proved Fermat’s Last Theorem.

At the heart of his proof of Fermat, Andrew had proved an idea known as the Taniyama-Shimura Conjecture, which created a new bridge between wildly different mathematical worlds. For many, the goal of one unified mathematics is supreme, and this was a glimpse of just such a world. So in proving Fermat, Andrew Wiles had cemented some of the most important number theory of the post-war period, and had secured the base of a pyramid of conjectures that were built upon it. This was no longer simply solving the longest-standing mathematical puzzle, but was pushing the very boundaries of mathematics itself. It was as if Fermat’s simple problem, born at a time when maths was in its infancy, had been waiting for this moment. The story of Fermat had ended in the most spectacular fashion. For Andrew Wiles, it meant the end of professional isolation of a kind almost alien to maths, which is usually a collaborative activity. Ritual afternoon tea in mathematics institutes the world over is a time when ideas come together, and sharing insight before publication is the norm.

In Chapter 2 we shall find out more about the mysterious Pierre de Fermat and how his theorem came to be lost, but for the time being it is enough to know that Fermat’s Last Theorem, a problem that had captivated mathematicians for centuries, had captured the imagination of the young Andrew Wiles. Sat in Milton Road Library was a ten-year-old boy staring at the most infamous problem in mathematics. Usually half the difficulty in a mathematics problem is understanding the question, but in this case it was simple – prove that xn + yn = zn has no whole number solutions for n greater than 2. Andrew was not daunted by the knowledge that the most brilliant minds on the planet had failed to rediscover the proof. He immediately set to work using all his textbook techniques to try and recreate the proof. Perhaps he could find something that everyone else, except Fermat, had overlooked. He dreamed he could shock the world. Thirty years later Andrew Wiles was ready. Standing in the auditorium of the Isaac Newton Institute, he scribbled on the board and then, struggling to contain his glee, stared at his audience.


pages: 434 words: 135,226

The Music of the Primes by Marcus Du Sautoy

Ada Lovelace, Andrew Wiles, Arthur Eddington, Augustin-Louis Cauchy, computer age, Dava Sobel, Dmitri Mendeleev, Eratosthenes, Erdős number, Georg Cantor, German hyperinflation, global village, Henri Poincaré, Isaac Newton, Jacquard loom, lateral thinking, music of the spheres, New Journalism, P = NP, Paul Erdős, Richard Feynman, Rubik’s Cube, Search for Extraterrestrial Intelligence, Simon Singh, Solar eclipse in 1919, Stephen Hawking, Turing machine, William of Occam, Wolfskehl Prize, Y2K

Much of mathematics had found itself entangled with physics over the past few decades. Despite being a problem with its heart in the theory of numbers, the Riemann Hypothesis had for some years been showing unexpected resonances with problems in particle physics. Mathematicians were changing their travel plans to fly in to Princeton to share the moment. Memories were still fresh with the excitement of a few years earlier when an English mathematician, Andrew Wiles, had announced a proof of Fermat’s Last Theorem in a lecture delivered in Cambridge in June 1993. Wiles had proved that Fermat had been right in his claim that the equation xn + yn = zn has no solutions when n is bigger than 2. As Wiles laid down his chalk at the end of the lecture, the champagne bottles started popping and the cameras began flashing. Mathematicians knew, however, that proving the Riemann Hypothesis would be of far greater significance for the future of mathematics than knowing that Fermat’s equation has no solutions.

Fundamental particles have been given some crazy names – gluons, cascade hyperons, charmed mesons, quarks, the last of these courtesy of James Joyce’s Finnegans Wake. But ‘morons’? Surely not! Bombieri has an unrivalled reputation for appreciating the ins and outs of the Riemann Hypothesis, but those who know him personally are also aware of his wicked sense of humour. Fermat’s Last Theorem had fallen foul of an April Fool prank that emerged just after a gap had appeared in the first proof that Andrew Wiles had proposed in Cambridge. With Bombieri’s email, the mathematical community had been duped again. Eager to relive the buzz of seeing Fermat proved, they had grabbed the bait that Bombieri had thrown at them. And the delights of forwarding email meant that the first of April had disappeared from the original source as it rapidly disseminated. This, combined with the fact that the email was read in countries with no concept of April Fool’s Day, made the prank far more successful than Bombieri could have imagined.

But it was striking how, as the century drew to a close, more and more mathematicians were prepared to talk about attacking it. The proof of Fermat’s Last Theorem only helped to fuel the expectation that great problems could be solved. Mathematicians had enjoyed the attention that Wiles’s solution to Fermat had brought them as mathematicians. This feeling undoubtedly contributed to the desire to believe Bombieri. Suddenly, Andrew Wiles was being asked to model chinos for Gap. It felt good. It felt almost sexy to be a mathematician. Mathematicians spend so much time in a world that fills them with excitement and pleasure. Yet it is a pleasure they rarely have the opportunity to share with the rest of the world. Here was a chance to flaunt a trophy, to show off the treasures that their long, lonely journeys had uncovered. A proof of the Riemann Hypothesis would have been a fitting mathematical climax to the twentieth century.


pages: 389 words: 112,319

Think Like a Rocket Scientist by Ozan Varol

Affordable Care Act / Obamacare, Airbnb, airport security, Albert Einstein, Amazon Web Services, Andrew Wiles, Apple's 1984 Super Bowl advert, Arthur Eddington, autonomous vehicles, Ben Horowitz, Cal Newport, Clayton Christensen, cloud computing, Colonization of Mars, dark matter, delayed gratification, different worldview, discovery of DNA, double helix, Elon Musk, fear of failure, functional fixedness, Gary Taubes, George Santayana, Google Glasses, Google X / Alphabet X, Inbox Zero, index fund, Isaac Newton, James Dyson, Jeff Bezos, job satisfaction, Johannes Kepler, Kickstarter, knowledge worker, late fees, lateral thinking, lone genius, longitudinal study, Louis Pasteur, low earth orbit, Marc Andreessen, Mars Rover, meta analysis, meta-analysis, move fast and break things, move fast and break things, multiplanetary species, obamacare, Occam's razor, out of africa, Peter Thiel, Pluto: dwarf planet, Ralph Waldo Emerson, Richard Feynman, Richard Feynman: Challenger O-ring, Ronald Reagan, Sam Altman, Schrödinger's Cat, Search for Extraterrestrial Intelligence, self-driving car, Silicon Valley, Simon Singh, Steve Ballmer, Steve Jobs, Steven Levy, Stewart Brand, Thomas Kuhn: the structure of scientific revolutions, Thomas Malthus, Upton Sinclair, Vilfredo Pareto, We wanted flying cars, instead we got 140 characters, Whole Earth Catalog, women in the workforce, Yogi Berra

., “REM, Not Incubation, Improves Creativity by Priming Associative Networks,” Proceedings of the National Academy of Sciences 106, no. 25 (June 23, 2009): 10,130–10,134, www.pnas.org/content/106/25/10130.full. 41. Ben Orlin, “The State of Being Stuck,” Math With Bad Drawings (blog), September 20, 2017, https://mathwithbaddrawings.com/2017/09/20/the-state-of-being-stuck. 42. NOVA, “Solving Fermat: Andrew Wiles,” interview with Andrew Wiles, PBS, October 31, 2000, www.pbs.org/wgbh/nova/article/andrew-wiles-fermat. 43. Judah Pollack and Olivia Fox Cabane, Butterfly and the Net: The Art and Practice of Breakthrough Thinking (New York: Portfolio/Penguin, 2017), 44–45. 44. Cameron Prince, “Nikola Tesla Timeline,” Tesla Universe, https://teslauniverse.com/nikola-tesla/timeline/1882-tesla-has-ac-epiphany. 45. Damon Young, “Charles Darwin’s Daily Walks,” Psychology Today, January 12, 2015, www.psychologytoday.com/us/blog/how-think-about-exercise/201501/charles-darwins-daily-walks. 46.

Yuval Noah Harari, 21 Lessons for the 21st Century (New York: Spiegel & Grau, 2018). 12. The section on Fermat’s Last Theorem draws on the following sources: Stuart Firestein, Ignorance: How It Drives Science (New York: Oxford University Press, 2012); Simon Singh, Fermat’s Last Theorem: The Story of a Riddle That Confounded the World’s Greatest Minds for 358 Years (London: Fourth Estate, 1997); NOVA, “Solving Fermat: Andrew Wiles,” interview with Andrew Wiles, PBS, October 31, 2000, www.pbs.org/wgbh/nova/proof/wiles.html; Gina Kolata, “At Last, Shout of ‘Eureka!’ in Age-Old Math Mystery,” New York Times, June 24, 1993, www.nytimes.com/1993/06/24/us/at-last-shout-of-eureka-in-age-old-math-mystery.html; Gina Kolata, “A Year Later, Snag Persists in Math Proof,” New York Times, June 28, 1994, www.nytimes.com/1994/06/28/science/a-year-later-snag-persists-in-math-proof.html; John J.

“I have a truly marvelous demonstration of this proposition,” he wrote, “which this margin is too narrow to contain.” And that’s all he wrote. Fermat died before supplying the missing proof for what came to be known as Fermat’s last theorem. The teaser he left behind continued to tantalize mathematicians for centuries (and made them wish Fermat had a bigger book to write on). Generations of mathematicians tried—and failed—to prove Fermat’s last theorem. Until Andrew Wiles came along. For most ten-year-olds, the definition of a good time doesn’t include reading math books for fun. But Wiles was no ordinary ten-year-old. He would hang out at his local library in Cambridge, England, and surf the shelves for math books. One day, he spotted a book devoted entirely to Fermat’s last theorem. He was mesmerized by the mystery of a theorem that was so easy to state, yet so difficult to prove.


pages: 345 words: 84,847

The Runaway Species: How Human Creativity Remakes the World by David Eagleman, Anthony Brandt

active measures, Ada Lovelace, agricultural Revolution, Albert Einstein, Andrew Wiles, Burning Man, cloud computing, computer age, creative destruction, crowdsourcing, Dava Sobel, delayed gratification, Donald Trump, Douglas Hofstadter, en.wikipedia.org, Frank Gehry, Google Glasses, haute couture, informal economy, interchangeable parts, Isaac Newton, James Dyson, John Harrison: Longitude, John Markoff, lone genius, longitudinal study, Menlo Park, microbiome, Netflix Prize, new economy, New Journalism, pets.com, QWERTY keyboard, Ray Kurzweil, reversible computing, Richard Feynman, risk tolerance, self-driving car, Simon Singh, stem cell, Stephen Hawking, Steve Jobs, Stewart Brand, the scientific method, Watson beat the top human players on Jeopardy!, wikimedia commons, X Prize

id=10.1371/journal.pone.0071275 “Noh and Kutiyattam – Treasures of World Cultural Heritage.” The Japan-India Traditional Performing Arts Exchange Project 2004. December 26, 2004. Accessed August 21, 2015, <http://noh.manasvi.com/noh.html> Norman, Donald A. The Design of Everyday Things: Revised and Expanded Edition. New York: Basic Books, 2013. NOVA, “Andrew Wiles on Solving Fermat.” PBS. November 1, 2000. Accessed May 11, 2016. <http://www.pbs.org/wgbh/nova/physics/andrew-wiles-fermat.html> Oates, Joyce Carol. “The Myth of the Isolated Artist.” Pyschology Today 6, 1973: 74–5. O’Bannon, Ricky. “By the Numbers: Female Composers.” Baltimore Symphony Orchestra. Accessed May 11, 2016. <https://www.bsomusic.org/stories/by-the-numbers-female-composers.aspx> Oden, Maria, Yvette Mirabal, Marc Epstein, and Rebecca Richards-Kortum.

Hall, Michelangelo’s Last Judgment (Cambridge: Cambridge University Press, 2005). 6 Marcia B. Hall, Michelangelo’s Last Judgment. 7 Richard Steinitz, György Ligeti: Music of the Imagination (Boston: Northeastern University Press, 2003). 8 T.J. Pinch and Karin Bijsterveld, The Oxford Handbook of Sound Studies (New York: Oxford University Press, 2012). 9 NOVA, “Andrew Wiles on Solving Fermat,” PBS, November 1, 2000, accessed May 11, 2016, <http://www.pbs.org/wgbh/nova/physics/andrew-wiles-fermat.html> 10 Simon Singh, Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem (New York: Walker, 1997). 11 Michael J. Gelb, How to Think like Leonardo Da Vinci (New York: Dell, 2000). 12 Dean Keith Simonton, “Creative Productivity: A Predictive and Explanatory Model of Career Trajectories and Landmarks,” Psychological Review 104 no. 1 (1997): p. 66–89, <http://dx.doi.org/10.1037/0033-295X.104.1.66> 13 Yasuyuki Kowatari et al., “Neural Networks Involved in Artistic Creativity,” Human Brain Mapping 30 no. 5 (2009): pp. 1678-90, <http://dx.doi.org/10.1002/hbm.20633> 14 Suzan-Lori Parks, 365 Days/365 Plays (New York: Theater Communications Group, Inc., 2006). 11.

Hall, Michelangelo’s Last Judgment (Cambridge: Cambridge University Press, 2005). 6 Marcia B. Hall, Michelangelo’s Last Judgment. 7 Richard Steinitz, György Ligeti: Music of the Imagination (Boston: Northeastern University Press, 2003). 8 T.J. Pinch and Karin Bijsterveld, The Oxford Handbook of Sound Studies (New York: Oxford University Press, 2012). 9 NOVA, “Andrew Wiles on Solving Fermat,” PBS, November 1, 2000, accessed May 11, 2016, <http://www.pbs.org/wgbh/nova/physics/andrew-wiles-fermat.html> 10 Simon Singh, Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem (New York: Walker, 1997). 11 Michael J. Gelb, How to Think like Leonardo Da Vinci (New York: Dell, 2000). 12 Dean Keith Simonton, “Creative Productivity: A Predictive and Explanatory Model of Career Trajectories and Landmarks,” Psychological Review 104 no. 1 (1997): p. 66–89, <http://dx.doi.org/10.1037/0033-295X.104.1.66> 13 Yasuyuki Kowatari et al., “Neural Networks Involved in Artistic Creativity,” Human Brain Mapping 30 no. 5 (2009): pp. 1678-90, <http://dx.doi.org/10.1002/hbm.20633> 14 Suzan-Lori Parks, 365 Days/365 Plays (New York: Theater Communications Group, Inc., 2006). 11.


pages: 158 words: 49,168

Infinite Ascent: A Short History of Mathematics by David Berlinski

Alan Turing: On Computable Numbers, with an Application to the Entscheidungsproblem, Albert Einstein, Andrew Wiles, Benoit Mandelbrot, Douglas Hofstadter, Eratosthenes, four colour theorem, Georg Cantor, Gödel, Escher, Bach, Henri Poincaré, Isaac Newton, John von Neumann, Murray Gell-Mann, Stephen Hawking, Turing machine, William of Occam

Fermat believed that he had discovered a marvelous proof of his own conjecture, and within the margins of his own paper noted sadly that the margins were too small to contain it. Very good mathematicians were intrigued and often obsessed. Amateurs and cranks, all of them curiously aware of my e-mail address, busied themselves with crackpot proofs, some of them fiendishly ingenious. For more than three centuries the conjecture remained unyielding. And then in 1993, the English mathematician Andrew Wiles announced a proof, one retrospectively validating Fermat. The old boy had been right after all. Wiles’ proof ran to more than two hundred pages and it made use of an immense body of modern mathematics. A first version, announced in a very dramatic setting at Oxford University, contained an error. The proof required revision. But then everything came right. Although his paper addresses an old problem, it is completely an exercise in the most modern mathematics.

The coordination between certain equations and certain structures in space proceeds directly up the dimensional chain. The fourth-order equation V = x2y2 + y4/4, for example, describes an undulating surface in a three-dimensional space. Analytical geometry may be conducted in four dimensions, if need be, and although the results cannot easily be seen—let us be honest: They cannot be seen at all—the analysis is much the same. When Andrew Wiles offered his proof of Fermat’s conjecture, he used an immense array of tools, but at the very center of his proof a tingling trail led backward to Descartes, for what he had succeeded in proving was the Taniyama-Shimura conjecture, a thesis about elliptical equations and modular forms, one that in the complexity of its formulation hid that old, shrub-covered trail between the form of the discriminant and various curves in the plane.

Ten years later, algebraic geometry seemed to mathematicians simply to scintillate, Alexander Grothendieck dominating the field with what René Thom once described as crushing technical superiority. Grothendieck has since given up mathematics and is said to be resident in a cave somewhere in the south of France, where he is occupied by various ecological issues. The classification of the finite simple groups, I suppose, is next, and after that the Taniyama-Shimura conjecture and the proof of Fermat’s famous theorem, the work collectively of Andrew Wiles, Ken Ribet, Barry Mazur, and Gerhard Frey. But this list, resembling as it does various trite accounts of what is in and what is out, or what is hot and what is not, could easily be rewritten in a dozen different ways, evidence that mathematics no longer has what for so long it had, and that is a stable center. If lacking a center, the modern era in mathematics nonetheless displays certain identifiable but inconsistent tendencies, almost as if a river were suddenly to separate itself into a number of hissing streams.


pages: 262 words: 65,959

The Simpsons and Their Mathematical Secrets by Simon Singh

Albert Einstein, Andrew Wiles, Benoit Mandelbrot, cognitive dissonance, Donald Knuth, Erdős number, Georg Cantor, Grace Hopper, Isaac Newton, John Nash: game theory, Kickstarter, mandelbrot fractal, Menlo Park, Norbert Wiener, Norman Mailer, P = NP, Paul Erdős, probability theory / Blaise Pascal / Pierre de Fermat, Richard Feynman, Rubik’s Cube, Schrödinger's Cat, Simon Singh, Stephen Hawking, Wolfskehl Prize, women in the workforce

Fermat’s last theorem has also appeared in novels (The Girl Who Played with Fire by Stieg Larsson), in films (Bedazzled with Brendan Fraser and Elizabeth Hurley), and plays (Arcadia by Tom Stoppard). Perhaps the theorem’s most famous cameo is in a 1989 episode of Star Trek: The Next Generation titled “The Royale,” in which Captain Jean-Luc Picard describes Fermat’s last theorem as “a puzzle we may never solve.” However, Captain Picard was wrong and out of date, because the episode was set in the twenty-fourth century and the theorem was actually proven in 1995 by Andrew Wiles at Princeton University.5 Wiles had dreamed about tackling Fermat’s challenge ever since he was ten years old. The problem then obsessed him for three decades, which culminated in seven years of working in complete secrecy. Eventually, he delivered a proof that the equation xn + yn = zn (n > 2) has no solutions. When his proof was published, it ran to 130 dense pages of mathematics. This is interesting partly because it indicates the mammoth scale of Wiles’s achievement, and partly because his chain of logic is far too sophisticated to have been discovered in the seventeenth century.

In other words, the Doctor is tacitly acknowledging that Wiles’s proof exists, but he rightly does not accept that it is Fermat’s proof, which he considers to be the “real one.” Perhaps the Doctor went back to the seventeenth century and obtained the proof directly from Fermat. So, to summarize, in the seventeenth century, Pierre de Fermat states that he can prove that the equation xn + yn = zn (n > 2) has no whole number solutions. In 1995, Andrew Wiles discovers a new proof that verifies Fermat’s statement. In 2010, the Doctor reveals Fermat’s original proof. Everyone agrees that the equation has no solutions. Thus, in “The Wizard of Evergreen Terrace,” Homer appears to have defied the greatest minds across almost four centuries. Fermat, Wiles, and even the Doctor state that Fermat’s equation has no solutions, yet Homer’s blackboard jottings present us with a solution: 3,98712 + 4,36512 = 4,47212 You can check it yourself with a calculator.

By the time this episode aired in 1998, Wiles’s proof had been published for three years, so Cohen was well aware that Fermat’s last theorem had been conquered. He even had a personal link to the proof, because he had attended some lectures by Ken Ribet while he was a graduate student at the University of California, Berkeley, and Ribet had provided Wiles with a pivotal stepping-stone in his proof of Fermat’s last theorem. Cohen obviously knew that Fermat’s equation had no solutions, but he wanted to pay homage to Pierre de Fermat and Andrew Wiles by creating a solution that was so close to being correct that it would apparently pass the test if checked with only a simple calculator. In order to find his pseudosolution, he wrote a computer program that would scan through values of x, y, z, and n until it found numbers that almost balanced. Cohen finally settled on 3,98712 + 4,36512 = 4,47212 because the resulting margin of error is minuscule—the left side of the equation is only 0.000000002 percent larger than the right side.


Prime Obsession:: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics by John Derbyshire

Albert Einstein, Andrew Wiles, Colonization of Mars, Eratosthenes, Ernest Rutherford, four colour theorem, Georg Cantor, Henri Poincaré, Isaac Newton, John Conway, John von Neumann, Paul Erdős, Richard Feynman, Turing machine, Turing test

To Heinrich Olbers, who had urged him to compete, Gauss replied “I confess that Fermat’s Theorem … has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of.” Gauss’s indifference is in this case a minority viewpoint, it must be said. A problem that can be stated in a few plain words, yet which defies proof by the best mathematical talents for decades or—in the case of Goldbach’s Conjecture or Fermat’s Last Theorem—for centuries, has an irresistible attraction for most mathematicians. They know that they can achieve great fame by solving it, as Andrew Wiles did when he proved Fermat’s Last Theorem. They know, too, from the history of their subject, that even failed attempts can generate powerful new results and techniques. And there is, of course, the Mallory factor. When the New York Times asked George Mallory why he wanted to climb Mount Everest, Mallory replied: “Because it’s there.” V. The connection between measuring and continuity is this.

On the other hand, by the mid-1890s it had been 10 years since Stieltjes’s announcement, and a lot of people must have been entertaining doubts. Not doubts about Stieltjes’s character; it is a very common thing for a mathematician to believe he has proved a result, only to find, going over his arguments (or more commonly, having them peer-reviewed), that there is a logical flaw in them. This happened with Andrew Wiles’s first proof of Fermat’s Last Theorem in 1993. It happens somewhat more dramatically to the narrator of Philibert Schogt’s 2000 novel The Wild Numbers. Nobody would have thought the worse of Stieltjes if this had been the case, this being much too common an event in mathematical careers. But where was that proof? Both Charles de la Vallée Poussin at the University of Louvain in Belgium and Jacques Hadamard in Bordeaux took up the lesser challenge and soon got the result.

I mentioned that I was trying to think of a way to explain big oh to readers who weren’t familiar with it. “Oh,” said Peter, “You should speak to my colleague Nick” (i.e., Nicholas Katz, also a professor at Princeton, though mainly an algebraic geometer). “Nick hates big oh. Won’t use it.” I swallowed this and made a note of it, thinking I might find some place for it in this book. Then that evening I happened to be talking to Andrew Wiles, who knows Sarnak and Katz both very well. I mentioned Katz’s not liking big oh. “That’s all nonsense,” said Wiles. “They’re just teasing you. Nick uses it a lot.” Sure enough, he used it in a lecture the next day. Funny sense of humor, mathematicians. IV. So much for big oh. Now, the Möbius function. There are many ways to introduce the Möbius function. I am going to approach it by way of the Golden Key.


pages: 337 words: 103,522

The Creativity Code: How AI Is Learning to Write, Paint and Think by Marcus Du Sautoy

3D printing, Ada Lovelace, Albert Einstein, Alvin Roth, Andrew Wiles, Automated Insights, Benoit Mandelbrot, Claude Shannon: information theory, computer vision, correlation does not imply causation, crowdsourcing, data is the new oil, Donald Trump, double helix, Douglas Hofstadter, Elon Musk, Erik Brynjolfsson, Fellow of the Royal Society, Flash crash, Gödel, Escher, Bach, Henri Poincaré, Jacquard loom, John Conway, Kickstarter, Loebner Prize, mandelbrot fractal, Minecraft, music of the spheres, Narrative Science, natural language processing, Netflix Prize, PageRank, pattern recognition, Paul Erdős, Peter Thiel, random walk, Ray Kurzweil, recommendation engine, Rubik’s Cube, Second Machine Age, Silicon Valley, speech recognition, Turing test, Watson beat the top human players on Jeopardy!, wikimedia commons

Mathematics is about intuition, logical moves into the unknown that feel right even if I’m not sure quite why I have that feeling. But when DeepMind’s algorithm discovered how to do something with a very similar flavour, it triggered an existential crisis. If these algorithms can play Go, the mathematician’s game, can they play the real game: could they prove theorems? One of my crowning pinnacles as a mathematician was getting a theorem published in the Annals of Mathematics. It is the journal in which Andrew Wiles published his proof of Fermat’s Last Theorem. It is the mathematician’s Nature. So how long would it be before we might expect to see a paper in the Annals of Mathematics authored by an algorithm? In order to play a game it’s important to understand the rules. What am I challenging a computer to do? I’m not sitting at my desk doing huge calculations. If that had been the case, computers would have put me out of a job years ago.

Fermat proved why if you raise a number to the power of a prime bigger than that number and then divide the result by the prime, the remainder will be the number you started with. Euler proved why when you raise e to the power of i times pi the answer is –1. Gauss proved that every number can be written as the sum of at most three triangular numbers (writing ‘Eureka’ next to his discovery). And eventually my colleague Andrew Wiles proved that Fermat was right in his hunch that the equations xn + yn = zn don’t have solutions when n>2. These breakthroughs are representative of what it is a mathematician does. A mathematician is not a master calculator but a constructor of proofs. So here is the challenge at the heart of this book: why can’t a computer join the ranks of Fermat, Gauss and Wiles? A computer can clearly out-perform any human when it comes to calculation, but what about our ability to prove theorems?

I should know: I’ve subsequently discovered that a couple of proofs I’d published had holes in them. The gaps were pluggable, but the referees and editors had missed them. If a proof is important, scrutiny generally wheedles out any gaps or errors. That is why the Millennium Prizes are released within two years of publication: twenty-four months is regarded as enough time for a mistake to reveal itself. Take Andrew Wiles’s first proof of Fermat’s Last Theorem. Referees spotted a mistake before it ever made it to print. The miracle was that Wiles was able to repair the mistake with the help of his former student Richard Taylor. But how many incorrect proofs might be out there leading us to build our mathematical edifices on falsehoods? Some new proofs are now so complex that mathematicians fear hard-to-pick-up errors will be missed.


pages: 114 words: 30,715

The Four Horsemen by Christopher Hitchens, Richard Dawkins, Sam Harris, Daniel Dennett

3D printing, Andrew Wiles, cognitive dissonance, cosmological constant, dark matter, Desert Island Discs, en.wikipedia.org, phenotype, Richard Feynman, stem cell, Steven Pinker

DAWKINS: But what we actually do when we, who are not physicists, take on trust what physicists say, is that we have some evidence that suggests that physicists have looked into the matter – that they’ve done experiments, that they’ve peer-reviewed their papers, that they’ve criticized each other, that they’ve been subjected to massive criticism from their peers in seminars and in lectures. DENNETT: And remember the structure that’s there, too; it’s not just that there’s peer review. But it’s very important that [science is] competitive. For instance, when Fermat’s last theorem was proved by— DAWKINS: Andrew Wiles. DENNETT:—Andrew Wiles, the reason that those of us who said, ‘Forget it, I’m never going to understand that proof,’ the reason that we can be confident that it really is a proof is that— HARRIS: Nobody wanted him to get there first. [Laughter] DENNETT: —every other mathematician who was competent in the world was very well motivated to study that proof. DAWKINS: To find it, yes. DENNETT: And believe me, if they grudgingly admit that this is a proof, it’s a proof.


Elliptic Tales: Curves, Counting, and Number Theory by Avner Ash, Robert Gross

Andrew Wiles, fudge factor, Georg Cantor, P = NP

When a relationship between different kinds of Lfunctions holds, it provides a link of great interest in itself, which is also useful for proving further theorems. For example, the fact that the Lfunction of an elliptic curve is equal to the L-function of a certain modular form is the essence of the great theorem proved by the cumulative work in (Wiles, 1995; Taylor and Wiles, 1995; Breuil et al., 2001). In turn, that equality can be used to prove Fermat’s Last Theorem, as British mathematician Andrew Wiles (1953–) did. Dirichlet’s L-functions can be thought of as a generalization of the Riemann zeta-function ζ (s). In the next section, we will describe a monster generalization of ζ (s) called the Hasse–Weil zeta-function. (The German mathematician Helmut Hasse lived from 1898 to 1979. The French mathematician André Weil lived from 1906 to 1998.) This thing generally breaks up into a product of L-functions (and zeta-functions) and their reciprocals, so it can fruitfully be viewed as one source of L-functions.

Birch, Bryan John, and Henry Peter Francis Swinnerton-Dyer. 1963. Notes on elliptic curves. I, J. Reine Angew. Math., 212, 7–25. ———1965. Notes on elliptic curves. II, J. Reine Angew. Math., 218, 79–108 Breuil, Christophe, Brian Conrad, Fred Diamond, and Richard Taylor. 2001 On the modularity of elliptic curves over Q: Wild 3-adic exercises, J. Amer. Math. Soc., 14, no. 4, 843–939. Carlson, James, Arthur Jaffe, and Andrew Wiles (eds.), The Millennium Prize Problems, Clay Mathematics Institute, Cambridge, MA, 2006. Available at http://www.claymath.org/library/monographs/MPP.pdf. Conrad, Keith. 2008. The congruent number problem, Harv. Coll. Math. Rev., 2, no. 2, 58–74. Available at http://thehcmr.org/node/17. Courant, Richard, and Herbert Robbins, What is Mathematics?: An Elementary Approach to Ideas and Methods, Oxford University Press, New York, 1979.

.), 39, no. 4, 455–474. Available at http://www.ams.org/ journals/bull/2002-39-04/S0273-0979-02-00952-7/. Silverman, Joseph H., The Arithmetic of Elliptic Curves, 2nd ed., Graduate Texts in Mathematics, Vol. 106, Springer, Dordrecht, 2009. Silverman, Joseph H., and John Tate, Rational Points on Elliptic Curves, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. Taylor, Richard, and Andrew Wiles. 1995. Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2), 141, no. 3, 553–572. Thomas, Ivor (trans.), Greek Mathematical Works, revised, Vol. 1, Harvard University Press, Loeb Classical Library, Cambridge, MA, 1980. Titchmarsh, Edward Charles, The Theory of the Riemann Zeta-Function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by David Rodney Heath-Brown.


pages: 236 words: 50,763

The Golden Ticket: P, NP, and the Search for the Impossible by Lance Fortnow

Alan Turing: On Computable Numbers, with an Application to the Entscheidungsproblem, Albert Einstein, Andrew Wiles, Claude Shannon: information theory, cloud computing, complexity theory, Donald Knuth, Erdős number, four colour theorem, Gerolamo Cardano, Isaac Newton, Johannes Kepler, John von Neumann, linear programming, new economy, NP-complete, Occam's razor, P = NP, Paul Erdős, Richard Feynman, Rubik’s Cube, smart grid, Stephen Hawking, traveling salesman, Turing machine, Turing test, Watson beat the top human players on Jeopardy!, William of Occam

After Karp, computer scientists started to realize the incredible importance of the P versus NP problem, and it dramatically changed the direction of computer science research. Today, P versus NP has become a critical question not just in computer science but in many other fields, including biology, medicine, economics, and physics. The P versus NP problem has achieved the status of one of the great open problems in all of mathematics. Following the excitement of Andrew Wiles’s 1994 proof of Fermat’s Last Theorem, the Clay Mathematics Institute decided to run a contest for solutions to the most important unsolved mathematical problems. In 2000, the Clay Institute listed seven Millennium Problems and offered a $1 million bounty for each of them. 1. Birch and Swinnerton-Dyer conjecture 2. Hodge conjecture 3. Navier-Stokes equations 4. P versus NP 5. Poincaré conjecture 6.

In other words, there are no natural numbers a, b, and c all greater than zero and n greater than 2 such that an + bn = cn. Fermat never mentioned this proof again, so it is likely he never had a true solution. The problem gained great notoriety as it became the classic unsolvable math puzzle. Kids like me dreamed of being the first person to solve this famous problem. One of those kids grew up and did just that. In 1994 the Princeton mathematician Andrew Wiles, building on a long series of papers in number theory, developed a proof of Fermat’s claim and became an instant celebrity, at least as much of a celebrity as a mathematician could be. This chapter won’t show how to solve the P versus NP problem, or this would have been a very different book. Instead we explore a few of the ideas that people have tried to resolve the P versus NP problem. Alas, these ideas have not panned out to anything close to a solution to the problem.


pages: 459 words: 103,153

Adapt: Why Success Always Starts With Failure by Tim Harford

Andrew Wiles, banking crisis, Basel III, Berlin Wall, Bernie Madoff, Black Swan, car-free, carbon footprint, Cass Sunstein, charter city, Clayton Christensen, clean water, cloud computing, cognitive dissonance, complexity theory, corporate governance, correlation does not imply causation, creative destruction, credit crunch, Credit Default Swap, crowdsourcing, cuban missile crisis, Daniel Kahneman / Amos Tversky, Dava Sobel, Deep Water Horizon, Deng Xiaoping, disruptive innovation, double entry bookkeeping, Edmond Halley, en.wikipedia.org, Erik Brynjolfsson, experimental subject, Fall of the Berlin Wall, Fermat's Last Theorem, Firefox, food miles, Gerolamo Cardano, global supply chain, Intergovernmental Panel on Climate Change (IPCC), Isaac Newton, Jane Jacobs, Jarndyce and Jarndyce, Jarndyce and Jarndyce, John Harrison: Longitude, knowledge worker, loose coupling, Martin Wolf, mass immigration, Menlo Park, Mikhail Gorbachev, mutually assured destruction, Netflix Prize, New Urbanism, Nick Leeson, PageRank, Piper Alpha, profit motive, Richard Florida, Richard Thaler, rolodex, Shenzhen was a fishing village, Silicon Valley, Silicon Valley startup, South China Sea, special economic zone, spectrum auction, Steve Jobs, supply-chain management, the market place, The Wisdom of Crowds, too big to fail, trade route, Tyler Cowen: Great Stagnation, web application, X Prize, zero-sum game

Christensen, The Innovator’s Solution (Harvard Business School Press, 2003), p. 198. 243 Whole structure of Virgin Group has always been: Richard Branson, Business Stripped Bare (Virgin Books, 2008), pp. 169–214. 244 ‘I’ll be damned if I permit’: anonymous officer quoted in John Nagl, Learning to Eat Soup with a Knife (University of Chicago Press, 2005), p. 172. 8 Adapting and you 247 ‘He was not a very careful person’: Shimura is quoted in ‘Andrew Wiles and Fermat’s Last Theorem’, MarginalRevolution.com, 29 August 2010, http://www.marginalrevolution.com/marginalrevolution/2010/08/andrew-wiles-and-fermats-last-theorem.html 247 ‘Let us try for once not to be right’: Tristan Tzara214. , 1918. 247 ‘Stupefyingly clichéd and almost embarrassingly naïve’: Hedy Weiss, ‘Good music, flashy moves can’t fill emotional void’, Chicago Sun- Times, 21 July 2002. 247 ‘Crazily uneven’: Michael Phillips, ‘“Movin’ Out”? Maybe not; Broadway-bound Tharp-Joel show has to get acts together’, Chicago Tribune, 22 July 2002. 248 ‘If you stand in Twyla’s way’: Cathleen McGuigan, ‘Movin’ to Broadway: Twyla Tharp heads uptown with Billy Joel’, Newsweek,28 October 2002. 248 Oklahoma!


pages: 185 words: 43,609

Zero to One: Notes on Startups, or How to Build the Future by Peter Thiel, Blake Masters

Airbnb, Albert Einstein, Andrew Wiles, Andy Kessler, Berlin Wall, cleantech, cloud computing, crony capitalism, discounted cash flows, diversified portfolio, don't be evil, Elon Musk, eurozone crisis, income inequality, Jeff Bezos, Lean Startup, life extension, lone genius, Long Term Capital Management, Lyft, Marc Andreessen, Mark Zuckerberg, minimum viable product, Nate Silver, Network effects, new economy, paypal mafia, Peter Thiel, pets.com, profit motive, Ralph Waldo Emerson, Ray Kurzweil, self-driving car, shareholder value, Silicon Valley, Silicon Valley startup, Singularitarianism, software is eating the world, Steve Jobs, strong AI, Ted Kaczynski, Tesla Model S, uber lyft, Vilfredo Pareto, working poor

When it was exposed that Dunn arranged a series of illegal wiretaps to identify the source, the backlash was worse than the original dissension, and the board was disgraced. Having abandoned the search for technological secrets, HP obsessed over gossip. As a result, by late 2012 HP was worth just $23 billion—not much more than it was worth in 1990, adjusting for inflation. THE CASE FOR SECRETS You can’t find secrets without looking for them. Andrew Wiles demonstrated this when he proved Fermat’s Last Theorem after 358 years of fruitless inquiry by other mathematicians—the kind of sustained failure that might have suggested an inherently impossible task. Pierre de Fermat had conjectured in 1637 that no integers a, b, and c could satisfy the equation an + bn = cn for any integer n greater than 2. He claimed to have a proof, but he died without writing it down, so his conjecture long remained a major unsolved problem in mathematics.


pages: 998 words: 211,235

A Beautiful Mind by Sylvia Nasar

"Robert Solow", Al Roth, Albert Einstein, Andrew Wiles, Brownian motion, business cycle, cognitive dissonance, Columbine, experimental economics, fear of failure, Gunnar Myrdal, Henri Poincaré, invisible hand, Isaac Newton, John Conway, John Nash: game theory, John von Neumann, Kenneth Arrow, Kenneth Rogoff, linear programming, lone genius, longitudinal study, market design, medical residency, Nash equilibrium, Norbert Wiener, Paul Erdős, Paul Samuelson, prisoner's dilemma, RAND corporation, Ronald Coase, second-price auction, Silicon Valley, Simon Singh, spectrum auction, The Wealth of Nations by Adam Smith, Thorstein Veblen, upwardly mobile, zero-sum game

And he was looking forward to hearing Busemann’s seminar on the state of Soviet mathematics because everyone knew that the Russians were doing great things, but the authorities were no longer allowing even abstracts of their mathematics articles to be translated into English. The signal event of the summer institute turned out to be the surprise announcement, within a day or two of the start of the meetings, of Milnor’s proof of the existence of exotic spheres.2 For the mathematicians gathered there, it had the same electrifying effect as the announcement of a solution of Fermat’s Last Theorem by Andrew Wiles of Princeton University four decades later. It stole Nash’s thunder. Nash reacted to the news of Milnor’s triumph with a display of adolescent petulance.3 The mathematicians were all camping out in a student dormitory and eating their meals in the cafeteria. Nash protested by grabbing gigantic portions. Once he demolished a pile of bread. Another time, he threw a glass of milk at a cashier.

“Yes,” he says, “my son is in a mental hospital in Arkansas but he got a job offer!” He is laughing at the absurdity of this juxtaposition. This is too much for Alicia. “You have to be fair to Johnny,” she returns. Nash says nothing. But later in the evening he goes to some lengths to make amends. He brings an offering, maps of Mexico, that he found in books on the Borels’ shelves, to Alicia. He takes the opportunity — during a conversation about Andrew Wiles’s successful proof of Fermat’s Last Theorem — to point out that Johnny had done some “classical” number theory in graduate school. Johnny had published “one correct result, one incorrect, but the correct one was a breakthrough of sorts,” he tells the other guests. Alicia responds by paying attention, by taking in what he means. Much of the renewal of their marriage has taken place since the Nobel.

Cohen, interview, 1.5.96. 4. Stanislaw Ulam, “John von Neumann, 1903–1957,” op. cit., p. 5. 5. Hardy, op. cit. 6. Felix Browder, interview, 11.10.95. 7. Harold Kuhn, interview, 7.95. 8. Ibid. 9. John Nash, plenary lecture, op. cit. 10. Elias Stein, interview, 12.28.95. 11. Cohen, interview. 12. E. T. Bell, Men of Mathematics, op. cit. 13. Enrico Bombieri, interview, 12.6.95. 14. Bell, op. cit. 15. Andrew Wiles, professor of mathematics, Princeton University, personal communication, 6.97. 16. Lars Hörmander, interview, 2.13.97. 17. F. Browder, interview. 18. John Forbes Nash, Jr., Les Prix Nobel 1994, op. cit. 19. Bell, op. cit. 20. Ibid. 21. Ibid. 22. Jacob Schwartz, professor of computer science, Courant Institute, interview, 1.29.96. 23. Jerome Neuwirth, interview, 5.27.97. 24. Stein, interview. 25.


pages: 237 words: 50,758

Obliquity: Why Our Goals Are Best Achieved Indirectly by John Kay

Andrew Wiles, Asian financial crisis, Berlin Wall, bonus culture, British Empire, business process, Cass Sunstein, computer age, corporate raider, credit crunch, Daniel Kahneman / Amos Tversky, discounted cash flows, discovery of penicillin, diversification, Donald Trump, Fall of the Berlin Wall, financial innovation, Gordon Gekko, greed is good, invention of the telephone, invisible hand, Jane Jacobs, lateral thinking, Long Term Capital Management, Louis Pasteur, market fundamentalism, Myron Scholes, Nash equilibrium, pattern recognition, Paul Samuelson, purchasing power parity, RAND corporation, regulatory arbitrage, shareholder value, Simon Singh, Steve Jobs, Thales of Miletus, The Death and Life of Great American Cities, The Predators' Ball, The Wealth of Nations by Adam Smith, ultimatum game, urban planning, value at risk

The oblique solution complicates the problem to simplify it: The direct solution is inefficient, the oblique more direct. Invented puzzles frequently have this paradoxical character. They are a response to the everyday pleasure we take in obliquity. Most people regard arithmetic as a boring task—they do not enjoy long division or calculating the square root of a large number—but many like mathematical puzzles. The most famous such problem—Fermat’s Last Theorem—continues to intrigue. The 1994 proof by Andrew Wiles demands powerful computational tools.4 But Fermat hinted at a simpler solution that has not yet been rediscovered. Perhaps there is an oblique approach that, like Brunelleschi’s egg, or the presentation of perspective, or the Japanese assault on Singapore, is direct once thought of. If we sometimes recast problems before we begin, more often we revise our specification in the process of actually tackling them.


pages: 449 words: 123,459

The Infinity Puzzle by Frank Close

Albert Einstein, Andrew Wiles, Arthur Eddington, dark matter, El Camino Real, en.wikipedia.org, Ernest Rutherford, Isaac Newton, Murray Gell-Mann, Richard Feynman, Ronald Reagan, Simon Singh

At that time, it never occurred to me as suspicious that a child of ten had seen what generations of the ablest minds had overlooked. When the editor told me of the typo, I was astonished; not at my naïveté (I was still too naive) but that rotating a multiplication sign through 45 degrees to turn it into “+” had such profound consequences. That defeat convinced me there was no point in continuing with Fermat’s puzzle. A few years later, Andrew Wiles came across Fermat’s theorem in a book at his local library. Like me, he was then ten years old; like me, he decided to solve it; unlike me, he succeeded—though it took him more than a decade of dedicated work some thirty years later. ’t Hooft had the perseverance of Wiles, or at least more than most. Fermat’s Last Theorem had tantalized mathematicians for three centuries, whereas the problem of building a viable theory of the weak interaction had existed for only three decades.

The remarkable emergence of ’t Hooft, a mere student solving the puzzle that had defeated great masters, gained its own life. Over the years, Veltman’s role tended to be overlooked. What ’t Hooft had achieved was indeed remarkable, but no one, least of all himself, would claim that he did it alone. My comparison with Fermat’s Last Theorem is an example of the media oversimplification, not least because Andrew Wiles solved Fermat on his own; ’t Hooft’s triumph, by contrast, was the culmination of a vast effort. In our analogy of climbing Everest, one could say that Veltman had made a map of the route, prepared the equipment, and almost reached the summit before ’t Hooft started. When Veltman, nearing the summit, had been faced with an impassable crevasse, ’t Hooft found a route, but even then it was Veltman’s tools that established this to be the way and got them successfully through.


What We Cannot Know: Explorations at the Edge of Knowledge by Marcus Du Sautoy

Albert Michelson, Andrew Wiles, Antoine Gombaud: Chevalier de Méré, Arthur Eddington, banking crisis, bet made by Stephen Hawking and Kip Thorne, Black Swan, Brownian motion, clockwork universe, cosmic microwave background, cosmological constant, dark matter, Dmitri Mendeleev, Edmond Halley, Edward Lorenz: Chaos theory, Ernest Rutherford, Georg Cantor, Hans Lippershey, Harvard Computers: women astronomers, Henri Poincaré, invention of the telescope, Isaac Newton, Johannes Kepler, Magellanic Cloud, mandelbrot fractal, MITM: man-in-the-middle, Murray Gell-Mann, music of the spheres, Necker cube, Paul Erdős, Pierre-Simon Laplace, Richard Feynman, Skype, Slavoj Žižek, Solar eclipse in 1919, stem cell, Stephen Hawking, technological singularity, Thales of Miletus, Turing test, wikimedia commons

In fact, there are infinitely many solutions, and the ancient Greeks found a formula to produce them all. But it is often a much easier task to find solutions than to prove that you will never find numbers which satisfy any of Fermat’s equations. Fermat famously thought he had a solution but scribbled in the margin of his copy of Diophantus’ Arithmetica that the margin was too small for his remarkable proof. It took another 350 years before my Oxford colleague Andrew Wiles finally produced a convincing argument to explain why you will never find whole numbers that solve Fermat’s equations. Wiles’s proof runs to over one hundred pages, in addition to the thousands of pages of preceding theory that it is built on. So even a very wide margin wouldn’t have sufficed. The proof of Fermat’s Last Theorem is a tour de force. I regard it as a privilege to have been alive when the final pieces of the proof were put in place.

The importance of the unattained destination is illustrated by the strange reaction that many mathematicians have when a great mathematical theorem is finally proved. Just as there is a sense of sadness when you finish a great novel, the closure of a mathematical quest can have its own sense of melancholy. I think we were enjoying the challenge of Fermat’s equations so much that there was a sense of depression mixed with the elation that greeted Andrew Wiles’s solution of this 350-year-old enigma. It is important to recognize that we must live with uncertainty, with the unknown, the unknowable. Even if we eventually manage to produce a theory which describes the way the universe works, we will never know that there isn’t another chapter in the story, waiting for us to discover it. We can never know whether we’ve come to the end of the story. As much as we may crave certainty, to do science we must always be prepared to move on from the stories we tell now.


pages: 227 words: 62,177

Numbers Rule Your World: The Hidden Influence of Probability and Statistics on Everything You Do by Kaiser Fung

American Society of Civil Engineers: Report Card, Andrew Wiles, Bernie Madoff, Black Swan, business cycle, call centre, correlation does not imply causation, cross-subsidies, Daniel Kahneman / Amos Tversky, edge city, Emanuel Derman, facts on the ground, fixed income, Gary Taubes, John Snow's cholera map, moral hazard, p-value, pattern recognition, profit motive, Report Card for America’s Infrastructure, statistical model, the scientific method, traveling salesman

Whereas the pure scientist is chiefly concerned with “what’s new,” applied work must deal with “how high,” as in “how high would profits go?” or “how high would the polls go?” In addition to purely technical yardsticks, applied scientists have goals that are societal, as with the Minnesota highway engineers; or psychological, as with the Disney queue managers; or financial, as with hurricane insurers and loan officers. The pursuit of pure science is rarely limited by time; as an extreme example, mathematician Andrew Wiles meticulously constructed his proof of Fermat’s last theorem over seven years. Such luxury is not afforded the applied scientist, who must deliver a best effort within a finite time limit, typically in the order of weeks or months. External factors, even the life cycle of green produce or the pipeline of drug innovations, may dictate the constraint on time. What use would it be to discover the cause of an E. coli outbreak the day after the outbreak dies down?


pages: 210 words: 62,771

Turing's Vision: The Birth of Computer Science by Chris Bernhardt

Ada Lovelace, Alan Turing: On Computable Numbers, with an Application to the Entscheidungsproblem, Albert Einstein, Andrew Wiles, British Empire, cellular automata, Claude Shannon: information theory, complexity theory, Conway's Game of Life, discrete time, Douglas Hofstadter, Georg Cantor, Gödel, Escher, Bach, Henri Poincaré, Internet Archive, Jacquard loom, John Conway, John von Neumann, Joseph-Marie Jacquard, Norbert Wiener, Paul Erdős, Turing complete, Turing machine, Turing test, Von Neumann architecture

Post died of a heart attack shortly after receiving his last electroshock at the age of fifty seven. 2. In fact, there are no positive integer solutions to xn + yn = zn when n is larger than 2. This is the famous Fermat’s Last Theorem. Fermat wrote the statement in the margin of a book with the comment that he had a marvelous proof, but that the margin wasn’t large enough for him to include it. In 1994, 357 years later, Andrew Wiles finally proved it. Most mathematicians don’t believe that Fermat had a legitimate proof. The result should be more properly called Fermat’s conjecture or Wiles’s theorem. 3. All three of the problems we looked at have the property that there was an algorithm that works in the case when the answer of the decision problem was yes, but not in the case when the answer was no. Problems like this, where there is an algorithm for the Yes case, but not for the No case are sometimes called partially decidable, but not decidable (or recursively enumerable, but not recursive).


pages: 212 words: 65,900

Symmetry and the Monster by Ronan, Mark

Albert Einstein, Andrew Wiles, conceptual framework, Everything should be made as simple as possible, G4S, Henri Poincaré, John Conway, John von Neumann, Kickstarter, New Journalism, Pierre-Simon Laplace, Richard Feynman, V2 rocket

Jean-Pierre Tignol, Galois’ Theory of Algebraic Equtions, English translation, Longman, 1988, p. 274. 20 Tignol, Galois’ Theory, p. 274, translation altered slightly. 24 Toti-Rigatelli, Evariste Galois, p. 98. 25 Toti-Rigatelli, Evariste Galois. Chapter 3: Irrational Solutions 33 Fauvel and Gray, A History of Mathematics, pp. 504, 505. 35 Fauvel and Gray, A History of Mathematics, p. 503. Fauvel and Gray, A History of Mathematics, p. 503. Chapter 4: Groups 45 Quoted in Yu. I. Manin, Mathematics and Physics, Birkhäuser, 1981, p. 35. 46 A proof of Fermat’s Last Theorem was finally given by Andrew Wiles in the mid-1990s. Chapter 5: Sophus Lie 53 A. Stubhaug, The Mathematician Sophus Lie, English translation, Springer, 2002, p. 3. 54 Stubhaug, The Mathematician Sophus Lie, p. 9. 55 Stubhaug, The Mathematician Sophus Lie, p. 10. 56 Stubhaug, The Mathematician Sophus Lie, p. 12. 63 There are three degrees of freedom for the velocity (direction of motion and speed), and three for the spin (direction and amount of spin).


pages: 297 words: 77,362

The Nature of Technology by W. Brian Arthur

Andrew Wiles, business process, cognitive dissonance, computer age, creative destruction, double helix, endogenous growth, Geoffrey West, Santa Fe Institute, haute cuisine, James Watt: steam engine, joint-stock company, Joseph Schumpeter, Kenneth Arrow, Kevin Kelly, knowledge economy, locking in a profit, Mars Rover, means of production, Myron Scholes, railway mania, Silicon Valley, Simon Singh, sorting algorithm, speech recognition, technological singularity, The Wealth of Nations by Adam Smith, Thomas Kuhn: the structure of scientific revolutions

It is valid if it can be constructed under accepted logical rules from other valid components of mathematics—other theorems, definitions, and lemmas that form the available parts and assemblies in mathematics. Typically the mathematician “sees” or struggles to see one or two overarching principles: conceptual ideas that if provable provide the overall route to a solution. To be proved, these must be constructed from other accepted subprinciples or theorems. Each part moves the argument part of the way. Andrew Wiles’s proof of Fermat’s theorem uses as its base principle a conjecture by the Japanese mathematicians Taniyama and Shimura that connects two main structures he needs, modular forms and elliptic equations. To prove this conjecture and link the components of the argument, Wiles uses many subprinciples. “You turn to a page and there’s a brief appearance of some fundamental theorem by Deligne,” says mathematician Kenneth Ribet, “and then you turn to another page and in some incidental way there’s a theorem by Hellegouarch—all of these things are just called into play and used for a moment before going on to the next idea.”


pages: 284 words: 79,265

The Half-Life of Facts: Why Everything We Know Has an Expiration Date by Samuel Arbesman

Albert Einstein, Alfred Russel Wallace, Amazon Mechanical Turk, Andrew Wiles, bioinformatics, British Empire, Cesare Marchetti: Marchetti’s constant, Chelsea Manning, Clayton Christensen, cognitive bias, cognitive dissonance, conceptual framework, David Brooks, demographic transition, double entry bookkeeping, double helix, Galaxy Zoo, guest worker program, Gödel, Escher, Bach, Ignaz Semmelweis: hand washing, index fund, invention of movable type, Isaac Newton, John Harrison: Longitude, Kevin Kelly, life extension, Marc Andreessen, meta analysis, meta-analysis, Milgram experiment, Nicholas Carr, P = NP, p-value, Paul Erdős, Pluto: dwarf planet, publication bias, randomized controlled trial, Richard Feynman, Rodney Brooks, scientific worldview, social graph, social web, text mining, the scientific method, Thomas Kuhn: the structure of scientific revolutions, Thomas Malthus, Tyler Cowen: Great Stagnation

Instead, in one of the most maddening episodes in math history, he scribbled this idea in the margins of a book and wrote that he had a brilliant proof, but, alas, the margin was too small to contain it. We now think he might have been mistaken. But no one had found any numbers greater than 2 that fit the equation since he wrote this statement. So it was assumed to be true, but no one could prove it. This elegant problem in number theory had gone unproven since the seventeenth century, until Andrew Wiles completed a proof in 1995, using pages and pages of very complex math, which would most certainly not have fit in Fermat’s margin. But, crucially, along the way, mathematicians proved other, smaller, proofs in their quest to crack Fermat’s Last Theorem. When finally solved, whole new pieces of math were involved in the construction of the proof. I decided to tackle predicting the proof of one of the most famous unsolved problems, something known as P versus NP.


pages: 292 words: 88,319

The Infinite Book: A Short Guide to the Boundless, Timeless and Endless by John D. Barrow

Albert Einstein, Andrew Wiles, anthropic principle, Arthur Eddington, cosmological principle, dark matter, Edmond Halley, Fellow of the Royal Society, Georg Cantor, Isaac Newton, mutually assured destruction, Olbers’ paradox, prisoner's dilemma, Ray Kurzweil, scientific worldview, short selling, Stephen Hawking, Turing machine

If they could search systematically through all possibilities in a finite amount of our time, then they could print out ‘true’ or ‘false’ and stop. This is not as exciting to mathematicians as it might sound. Mathematicians are not only interested in whether conjectures like Goldbach’s are true or false, they are interested in the forms of reasoning needed to prove it. They want to see new types of argument. A classic example was the proof of Fermat’s Last Theorem by Andrew Wiles and Richard Taylor.18 The truth of Fermat’s conjecture emerged as a particular case of a much more general result that opened up types of proof and alternative formulations of old questions. A ‘proof ’ by direct search would provide no new insights of that sort. It would, in effect, be like looking up the answer in the back of the book. In fact, if a conjecture like Goldbach’s were shown to be true by an infinity machine, then we would feel aggrieved at being denied the insight provided by a proof.


pages: 407 words: 104,622

The Man Who Solved the Market: How Jim Simons Launched the Quant Revolution by Gregory Zuckerman

affirmative action, Affordable Care Act / Obamacare, Albert Einstein, Andrew Wiles, automated trading system, backtesting, Bayesian statistics, beat the dealer, Benoit Mandelbrot, Berlin Wall, Bernie Madoff, blockchain, Brownian motion, butter production in bangladesh, buy and hold, buy low sell high, Claude Shannon: information theory, computer age, computerized trading, Credit Default Swap, Daniel Kahneman / Amos Tversky, diversified portfolio, Donald Trump, Edward Thorp, Elon Musk, Emanuel Derman, endowment effect, Flash crash, George Gilder, Gordon Gekko, illegal immigration, index card, index fund, Isaac Newton, John Meriwether, John Nash: game theory, John von Neumann, Loma Prieta earthquake, Long Term Capital Management, loss aversion, Louis Bachelier, mandelbrot fractal, margin call, Mark Zuckerberg, More Guns, Less Crime, Myron Scholes, Naomi Klein, natural language processing, obamacare, p-value, pattern recognition, Peter Thiel, Ponzi scheme, prediction markets, quantitative hedge fund, quantitative trading / quantitative finance, random walk, Renaissance Technologies, Richard Thaler, Robert Mercer, Ronald Reagan, self-driving car, Sharpe ratio, Silicon Valley, sovereign wealth fund, speech recognition, statistical arbitrage, statistical model, Steve Jobs, stochastic process, the scientific method, Thomas Bayes, transaction costs, Turing machine

In addition to poker, he took up golf and bowling, while emerging as one of the nation’s top backgammon players. “Jim was a restless man with a restless mind,” Kochen says. Ax focused the bulk of his energies on math, a world that is more competitive than most realize. Mathematicians usually enter the field out of a love for numbers, structures, or models, but the real thrill often comes from being the first to make a discovery or advance. Andrew Wiles, the Princeton mathematician famous for proving the Fermat conjecture, describes mathematics as a journey through “a dark unexplored mansion,” with months, or even years, spent “stumbling around.” Along the way, pressures emerge. Math is considered a young person’s game—those who don’t accomplish something of significance in their twenties or early thirties can see their chances slip away.1 Even as Ax made progress in his career, anxieties and irritations built.


pages: 416 words: 112,268

Human Compatible: Artificial Intelligence and the Problem of Control by Stuart Russell

3D printing, Ada Lovelace, AI winter, Alan Turing: On Computable Numbers, with an Application to the Entscheidungsproblem, Alfred Russel Wallace, Andrew Wiles, artificial general intelligence, Asilomar, Asilomar Conference on Recombinant DNA, augmented reality, autonomous vehicles, basic income, blockchain, brain emulation, Cass Sunstein, Claude Shannon: information theory, complexity theory, computer vision, connected car, crowdsourcing, Daniel Kahneman / Amos Tversky, delayed gratification, Elon Musk, en.wikipedia.org, Erik Brynjolfsson, Ernest Rutherford, Flash crash, full employment, future of work, Gerolamo Cardano, ImageNet competition, Intergovernmental Panel on Climate Change (IPCC), Internet of things, invention of the wheel, job automation, John Maynard Keynes: Economic Possibilities for our Grandchildren, John Maynard Keynes: technological unemployment, John Nash: game theory, John von Neumann, Kenneth Arrow, Kevin Kelly, Law of Accelerating Returns, Mark Zuckerberg, Nash equilibrium, Norbert Wiener, NP-complete, openstreetmap, P = NP, Pareto efficiency, Paul Samuelson, Pierre-Simon Laplace, positional goods, probability theory / Blaise Pascal / Pierre de Fermat, profit maximization, RAND corporation, random walk, Ray Kurzweil, recommendation engine, RFID, Richard Thaler, ride hailing / ride sharing, Robert Shiller, Robert Shiller, Rodney Brooks, Second Machine Age, self-driving car, Shoshana Zuboff, Silicon Valley, smart cities, smart contracts, social intelligence, speech recognition, Stephen Hawking, Steven Pinker, superintelligent machines, Thales of Miletus, The Future of Employment, Thomas Bayes, Thorstein Veblen, transport as a service, Turing machine, Turing test, universal basic income, uranium enrichment, Von Neumann architecture, Wall-E, Watson beat the top human players on Jeopardy!, web application, zero-sum game

Mathematical Guarantees We will want, eventually, to prove theorems to the effect that a particular way of designing AI systems ensures that they will be beneficial to humans. A theorem is just a fancy name for an assertion, stated precisely enough so that its truth in any particular situation can be checked. Perhaps the most famous theorem is Fermat’s Last Theorem, which was conjectured by the French mathematician Pierre de Fermat in 1637 and finally proved by Andrew Wiles in 1994 after 357 years of effort (not all of it by Wiles).1 The theorem can be written in one line, but the proof is over one hundred pages of dense mathematics. Proofs begin from axioms, which are assertions whose truth is simply assumed. Often, the axioms are just definitions, such as the definitions of integers, addition, and exponentiation needed for Fermat’s theorem. The proof proceeds from the axioms by logically incontrovertible steps, adding new assertions until the theorem itself is established as a consequence of one of the steps.


pages: 467 words: 114,570

Pathfinders: The Golden Age of Arabic Science by Jim Al-Khalili

agricultural Revolution, Albert Einstein, Andrew Wiles, Book of Ingenious Devices, colonial rule, Commentariolus, Dmitri Mendeleev, Eratosthenes, Henri Poincaré, invention of the printing press, invention of the telescope, invention of the wheel, Isaac Newton, Islamic Golden Age, Johannes Kepler, Joseph Schumpeter, Kickstarter, liberation theology, retrograde motion, scientific worldview, Silicon Valley, Simon Singh, stem cell, Stephen Hawking, the scientific method, Thomas Malthus, trade route, William of Occam

In fact, for three and a half centuries it should have more correctly been called a ‘conjecture’ rather than a theorem. Stated mathematically, it says that there are no whole number values for x, y and z such that xn + yn = zn, when n is greater than 2. For instance, there are no whole numbers for which the sum of the cubes of two integers equals the cube of another (unless they are all equal to zero, of course). A proof was finally found by the British mathematician Andrew Wiles in 1995, and I for one have no intention of checking his method, since it runs to more than a hundred pages and took him seven years to complete. None of this should be credited to Diophantus, of course, but what it is meant to show is that his interest, like Fermat’s, was more in the properties of numbers than in the algebraic manipulation of symbols. In the seventh century the great Hindu mathematician Brahmagupta took up the challenge of tackling another Diophantine equation – what is known today as the Pell equation, which has the general form x2 − ay2 = 1.


pages: 415 words: 125,089

Against the Gods: The Remarkable Story of Risk by Peter L. Bernstein

"Robert Solow", Albert Einstein, Alvin Roth, Andrew Wiles, Antoine Gombaud: Chevalier de Méré, Bayesian statistics, Big bang: deregulation of the City of London, Bretton Woods, business cycle, buttonwood tree, buy and hold, capital asset pricing model, cognitive dissonance, computerized trading, Daniel Kahneman / Amos Tversky, diversified portfolio, double entry bookkeeping, Edmond Halley, Edward Lloyd's coffeehouse, endowment effect, experimental economics, fear of failure, Fellow of the Royal Society, Fermat's Last Theorem, financial deregulation, financial innovation, full employment, index fund, invention of movable type, Isaac Newton, John Nash: game theory, John von Neumann, Kenneth Arrow, linear programming, loss aversion, Louis Bachelier, mental accounting, moral hazard, Myron Scholes, Nash equilibrium, Norman Macrae, Paul Samuelson, Philip Mirowski, probability theory / Blaise Pascal / Pierre de Fermat, random walk, Richard Thaler, Robert Shiller, Robert Shiller, spectrum auction, statistical model, stocks for the long run, The Bell Curve by Richard Herrnstein and Charles Murray, The Wealth of Nations by Adam Smith, Thomas Bayes, trade route, transaction costs, tulip mania, Vanguard fund, zero-sum game

The problem is, in fact impossible, as by my method I am able to prove with all rigor."11 Fermat observes that Pythagoras was correct that a2 + b2 = c2, but a3 + b3 would not be equal to c3, nor would any integer higher than 2 fit the bill: the Pythagorean theorem works only for squaring. And then Fermat wrote: "I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain."12 With this simple comment he left mathematicians dumbfounded for over 350 years as they struggled to find a theoretical justification for what a great deal of empirical experimentation proved to be true. In 1993, an English mathematician named Andrew Wiles claimed that he had solved this puzzle after seven years of work in a Princeton attic. Wiles's results were published in the Annals of Mathematics in May 1995, but the mathematicians have continued to squabble over exactly what he had achieved. Fermat's Last Theorem is more of a curiosity than an insight into how the world works. But the solution that Fermat and Pascal worked out to the problem of the points has long since been paying social dividends as the cornerstone of modem insurance and other forms of risk management.


pages: 437 words: 132,041

Alex's Adventures in Numberland by Alex Bellos

Andrew Wiles, Antoine Gombaud: Chevalier de Méré, beat the dealer, Black Swan, Black-Scholes formula, Claude Shannon: information theory, computer age, Daniel Kahneman / Amos Tversky, Edward Thorp, family office, forensic accounting, game design, Georg Cantor, Henri Poincaré, Isaac Newton, Johannes Kepler, lateral thinking, Myron Scholes, pattern recognition, Paul Erdős, Pierre-Simon Laplace, probability theory / Blaise Pascal / Pierre de Fermat, random walk, Richard Feynman, Rubik’s Cube, SETI@home, Steve Jobs, The Bell Curve by Richard Herrnstein and Charles Murray, traveling salesman

Fermat never produced a proof – marvellous or otherwise – of his proposition even when unconstrained by narrow margins. His jottings in Arithmetica may have been an indication that he had a proof, or he may have believed he had a proof, or he may have been trying to be provocative. In any case, his cheeky sentence was fantastic bait to generations of mathematicians. The proposition became known as Fermat’s Last Theorem and was the most famous unsolved problem in maths until the Briton Andrew Wiles cracked it in 1995. Algebra can be very humbling in this way – ease in stating a problem has no correlation with ease in solving it. Wiles’s proof is so complicated that it is probably understood by no more than a couple of hundred people. Improvements in mathematical notation enabled the discovery of new concepts. The logarithm was a massively important invention in the early seventeenth century, thought up by the Scottish mathematician John Napier, the Laird of Merchiston, who was, in fact, much more famous in his lifetime for his work on theology.


How I Became a Quant: Insights From 25 of Wall Street's Elite by Richard R. Lindsey, Barry Schachter

Albert Einstein, algorithmic trading, Andrew Wiles, Antoine Gombaud: Chevalier de Méré, asset allocation, asset-backed security, backtesting, bank run, banking crisis, Black-Scholes formula, Bonfire of the Vanities, Bretton Woods, Brownian motion, business cycle, business process, butter production in bangladesh, buy and hold, buy low sell high, capital asset pricing model, centre right, collateralized debt obligation, commoditize, computerized markets, corporate governance, correlation coefficient, creative destruction, Credit Default Swap, credit default swaps / collateralized debt obligations, currency manipulation / currency intervention, discounted cash flows, disintermediation, diversification, Donald Knuth, Edward Thorp, Emanuel Derman, en.wikipedia.org, Eugene Fama: efficient market hypothesis, financial innovation, fixed income, full employment, George Akerlof, Gordon Gekko, hiring and firing, implied volatility, index fund, interest rate derivative, interest rate swap, John von Neumann, linear programming, Loma Prieta earthquake, Long Term Capital Management, margin call, market friction, market microstructure, martingale, merger arbitrage, Myron Scholes, Nick Leeson, P = NP, pattern recognition, Paul Samuelson, pensions crisis, performance metric, prediction markets, profit maximization, purchasing power parity, quantitative trading / quantitative finance, QWERTY keyboard, RAND corporation, random walk, Ray Kurzweil, Richard Feynman, Richard Stallman, risk-adjusted returns, risk/return, shareholder value, Sharpe ratio, short selling, Silicon Valley, six sigma, sorting algorithm, statistical arbitrage, statistical model, stem cell, Steven Levy, stochastic process, systematic trading, technology bubble, The Great Moderation, the scientific method, too big to fail, trade route, transaction costs, transfer pricing, value at risk, volatility smile, Wiener process, yield curve, young professional

Some problems are harder than others, and to assess how hard a problem is we look at how long it has remained unsolved and how many great mathematicians have failed to solve it. To my mind, the greatest mathematicians are like explorers—they search for solutions to problems that no one knows how to solve and they risk achieving nothing. A terrific example of this is the famous Fermat’s Last Theorem.3 Fermat stated his famous theorem around 1637 and great mathematicians of every generation tried to solve it until Andrew Wiles finally cracked it in the early 1990s after famously spending seven years working on the problem in secret and alone. Why did he do it? Why does any mathematician work on hard problems? Consider the stakes. In the case of Fermat’s Last Theorem, no one knew for sure whether we could solve the problem at all. The fundamental question—“Can it be solved?”—had not been resolved. Yet Wiles risked seven years of his career on a hunch that he could solve it.


The Greatest Show on Earth: The Evidence for Evolution by Richard Dawkins

Alfred Russel Wallace, Andrew Wiles, Arthur Eddington, back-to-the-land, Claude Shannon: information theory, correlation does not imply causation, Craig Reynolds: boids flock, Danny Hillis, David Attenborough, discovery of DNA, Dmitri Mendeleev, double helix, en.wikipedia.org, epigenetics, experimental subject, if you see hoof prints, think horses—not zebras, invisible hand, Louis Pasteur, out of africa, phenotype, Thomas Malthus

Fermat’s Last Theorem, like the Goldbach Conjecture, is a proposition about numbers to which nobody has found an exception. Proving it has been a kind of holy grail for mathematicians ever since 1637, when Pierre de Fermat wrote in the margin of an old mathematics book, ‘I have a truly marvellous proof . . . which this margin is too narrow to contain.’ It was finally proved by the English mathematician Andrew Wiles in 1995. Before that, some mathematicians think it should have been called a conjecture. Given the length and complication of Wiles’s successful proof, and his reliance on advanced twentieth-century methods and knowledge, most mathematicians think Fermat was (honestly) mistaken in his claim to have proved it. I tell the story only to illustrate the difference between a conjecture and a theorem.


pages: 573 words: 157,767

From Bacteria to Bach and Back: The Evolution of Minds by Daniel C. Dennett

Ada Lovelace, Alan Turing: On Computable Numbers, with an Application to the Entscheidungsproblem, Andrew Wiles, Bayesian statistics, bioinformatics, bitcoin, Build a better mousetrap, Claude Shannon: information theory, computer age, computer vision, double entry bookkeeping, double helix, Douglas Hofstadter, Elon Musk, epigenetics, experimental subject, Fermat's Last Theorem, Gödel, Escher, Bach, information asymmetry, information retrieval, invention of writing, Isaac Newton, iterative process, John von Neumann, Menlo Park, Murray Gell-Mann, Necker cube, Norbert Wiener, pattern recognition, phenotype, Richard Feynman, Rodney Brooks, self-driving car, social intelligence, sorting algorithm, speech recognition, Stephen Hawking, Steven Pinker, strong AI, The Wealth of Nations by Adam Smith, theory of mind, Thomas Bayes, trickle-down economics, Turing machine, Turing test, Watson beat the top human players on Jeopardy!, Y2K

A single person, or family, can make a simple house or canoe, and a small community can raise a barn or a stockade, but it takes hundreds of workers with dozens of different talents to build a cathedral or a clipper ship. Today peer-reviewed papers with hundreds of coauthors issue from CERN and other bastions of Big Science. Often none of the team members can claim to have more than a bird’s-eye-view comprehension of the whole endeavor, and we have reached a point where even the most brilliant solo thinkers are often clearly dependent on their colleagues for expert feedback and confirmation. Consider Andrew Wiles, the brilliant Princeton mathematician who in 1995 proved Fermat’s Last Theorem, a towering achievement in the history of mathematics. A close look at the process he went through, including the false starts and unnoticed gaps in the first version of his proof, demonstrates that this triumph was actually the work of many minds, a community of communicating experts, both collaborating and competing with each other for the glory, and without the many layers of achieved and battle-tested mathematics on which Wiles’s proof depended, it would have been impossible for Wiles or anyone else to judge that the theorem had, in fact, been proven.101 If you are a lone wolf mathematician and think you have proved Fermat’s Last Theorem, you have to consider the disjunction: Either I have just proved Fermat’s Last Theorem or I am going mad, and since history shows that many brilliant mathematicians have been deluded in thinking they had succeeded, you have to take the second alternative seriously.


pages: 492 words: 149,259

Big Bang by Simon Singh

Albert Einstein, Albert Michelson, All science is either physics or stamp collecting, Andrew Wiles, anthropic principle, Arthur Eddington, Astronomia nova, Brownian motion, carbon-based life, Cepheid variable, Chance favours the prepared mind, Commentariolus, Copley Medal, cosmic abundance, cosmic microwave background, cosmological constant, cosmological principle, dark matter, Dava Sobel, Defenestration of Prague, discovery of penicillin, Dmitri Mendeleev, Edmond Halley, Edward Charles Pickering, Eratosthenes, Ernest Rutherford, Erwin Freundlich, Fellow of the Royal Society, fudge factor, Hans Lippershey, Harlow Shapley and Heber Curtis, Harvard Computers: women astronomers, Henri Poincaré, horn antenna, if you see hoof prints, think horses—not zebras, Index librorum prohibitorum, invention of the telescope, Isaac Newton, Johannes Kepler, John von Neumann, Karl Jansky, Kickstarter, Louis Daguerre, Louis Pasteur, luminiferous ether, Magellanic Cloud, Murray Gell-Mann, music of the spheres, Olbers’ paradox, On the Revolutions of the Heavenly Spheres, Paul Erdős, retrograde motion, Richard Feynman, scientific mainstream, Simon Singh, Solar eclipse in 1919, Stephen Hawking, the scientific method, Thomas Kuhn: the structure of scientific revolutions, unbiased observer, Wilhelm Olbers, William of Occam

I am just grateful that this postscript essay has at least allowed me the opportunity to get these three curiosities out of my system. Read on Have You Read? Other titles by Simon Singh Fermat’s Last Theorem In 1963 a schoolboy browsing in his local library stumbled across the world’s greatest mathematical problem: Fermat’s Last Theorem, a puzzle that has baffled mathematicians for over 300 years. Aged just ten, Andrew Wiles dreamed that he would crack it. Wiles’s lifelong obsession with a seemingly simple challenge set by a long-dead Frenchman is an emotional tale of sacrifice and extraordinary determination. In the end Wiles was forced to work in secrecy and isolation for seven years, harnessing all the power of modern maths to achieve his childhood dream. Many before him had tried and failed, including an eighteenth-century philanderer who was killed in a duel.


pages: 626 words: 181,434

I Am a Strange Loop by Douglas R. Hofstadter

Albert Einstein, Andrew Wiles, Benoit Mandelbrot, Brownian motion, double helix, Douglas Hofstadter, Georg Cantor, Gödel, Escher, Bach, Isaac Newton, James Watt: steam engine, John Conway, John von Neumann, mandelbrot fractal, pattern recognition, Paul Erdős, place-making, probability theory / Blaise Pascal / Pierre de Fermat, publish or perish, random walk, Ronald Reagan, self-driving car, Silicon Valley, telepresence, Turing machine

And then along came a vast team of mathematicians who had set their collective bead on the “big game” of Fermat’s Last Theorem (the notorious claim, originally made by Pierre de Fermat in the middle of the seventeenth century, that no positive integers a, b, c exist such that an + bn equals c n, with the exponent n being an integer greater than 2). This great international relay team, whose final victorious lap was magnificently sprinted by Andrew Wiles (his sprint took him about eight years), was at last able to prove Fermat’s centuries-old claim by using amazing techniques that combined ideas from all over the vast map of contemporary mathematics. In the wake of this team’s revolutionary work, new paths were opened up that seemed to leave cracks in many famous old doors, including the tightly-closed door of the small but alluring Fibonacci power mystery.


Evidence-Based Technical Analysis: Applying the Scientific Method and Statistical Inference to Trading Signals by David Aronson

Albert Einstein, Andrew Wiles, asset allocation, availability heuristic, backtesting, Black Swan, butter production in bangladesh, buy and hold, capital asset pricing model, cognitive dissonance, compound rate of return, computerized trading, Daniel Kahneman / Amos Tversky, distributed generation, Elliott wave, en.wikipedia.org, feminist movement, hindsight bias, index fund, invention of the telescope, invisible hand, Long Term Capital Management, mental accounting, meta analysis, meta-analysis, p-value, pattern recognition, Paul Samuelson, Ponzi scheme, price anchoring, price stability, quantitative trading / quantitative finance, Ralph Nelson Elliott, random walk, retrograde motion, revision control, risk tolerance, risk-adjusted returns, riskless arbitrage, Robert Shiller, Robert Shiller, Sharpe ratio, short selling, source of truth, statistical model, stocks for the long run, systematic trading, the scientific method, transfer pricing, unbiased observer, yield curve, Yogi Berra

It is a dog. Therefore, has 4 legs. If a dog, then has 4 legs. It is not a dog. Therefore, not 4 legs. Denying The Consequent Fallacy: Affirming Consequent If a dog, then has 4 legs. Legs not equal to 4. Therefore not a dog. If a dog, then has 4 legs. Has 4 legs. Therefore dog. FIGURE 3.5 Conditional syllogisms: example. hinted at in the margin of a book in 1665 but not proven until 1994, by Andrew Wiles and Richard Taylor. Inductive Logic Induction is the logic of discovery. It aims to reveal new knowledge about the world by reaching beyond the knowledge contained in the premises of an inductive argument. However, this new knowledge comes with a price—uncertainty. Conclusions reached by induction are inherently uncertain. That is, they can only be true with some degree of probability. Thus, the notion of probability is intimately connected with induction.


pages: 893 words: 199,542

Structure and interpretation of computer programs by Harold Abelson, Gerald Jay Sussman, Julie Sussman

Andrew Wiles, conceptual framework, Donald Knuth, Douglas Hofstadter, Eratosthenes, Fermat's Last Theorem, Gödel, Escher, Bach, industrial robot, information retrieval, iterative process, Johannes Kepler, loose coupling, probability theory / Blaise Pascal / Pierre de Fermat, Richard Stallman, Turing machine

The most famous of Fermat's results – known as Fermat's Last Theorem – was jotted down in 1637 in his copy of the book Arithmetic (by the third-century Greek mathematician Diophantus) with the remark “I have discovered a truly remarkable proof, but this margin is too small to contain it.” Finding a proof of Fermat's Last Theorem became one of the most famous challenges in number theory. A complete solution was finally given in 1995 by Andrew Wiles of Princeton University. 46 The reduction steps in the cases where the exponent e is greater than 1 are based on the fact that, for any integers x, y, and m, we can find the remainder of x times y modulo m by computing separately the remainders of x modulo m and y modulo m, multiplying these, and then taking the remainder of the result modulo m. For instance, in the case where e is even, we compute the remainder of be/2 modulo m, square this, and take the remainder modulo m.


pages: 1,387 words: 202,295

Structure and Interpretation of Computer Programs, Second Edition by Harold Abelson, Gerald Jay Sussman, Julie Sussman

Andrew Wiles, conceptual framework, Donald Knuth, Douglas Hofstadter, Eratosthenes, Gödel, Escher, Bach, industrial robot, information retrieval, iterative process, Johannes Kepler, loose coupling, probability theory / Blaise Pascal / Pierre de Fermat, Richard Stallman, Turing machine, wikimedia commons

The most famous of Fermat’s results—known as Fermat’s Last Theorem—was jotted down in 1637 in his copy of the book Arithmetic (by the third-century Greek mathematician Diophantus) with the remark “I have discovered a truly remarkable proof, but this margin is too small to contain it.” Finding a proof of Fermat’s Last Theorem became one of the most famous challenges in number theory. A complete solution was finally given in 1995 by Andrew Wiles of Princeton University. 46 The reduction steps in the cases where the exponent is greater than 1 are based on the fact that, for any integers , , and , we can find the remainder of times modulo by computing separately the remainders of modulo and modulo , multiplying these, and then taking the remainder of the result modulo . For instance, in the case where is even, we compute the remainder of modulo , square this, and take the remainder modulo .