The Universe Speaks in Numbers Page 2

  Since the mid-1970s, many talented thinkers have been drawn to fertile common ground between mathematics and physics. Nonetheless, most physicists have steered clear of this territory, preferring the conventional and more prudent approach of waiting for nature to disclose more of its secrets through experiments and observations. Nima Arkani-Hamed, one of Einstein’s successors on the faculty of the Institute for Advanced Study at Princeton, made his name by taking this orthodox approach. About a decade ago, however, after he began to study the collisions between subatomic particles, he and his colleagues repeatedly found themselves working on the same topics as some of the world’s leading mathematicians. Arkani-Hamed quickly became a zealous promoter of the usefulness of advanced mathematics to fundamental physics.

  He remains a physicist to his fingertips. ‘My number one priority will always be to help discover the most fundamental laws of nature,’ he says. ‘We must listen to the universe as carefully as we possibly can, and make use of every observation and experimental measurement that might have something to teach us. Ultimately, experiments will always be the judge of our theories.’ But his mathematical work has radically changed the way he thinks about physics research: ‘We can eavesdrop on nature not only by paying attention to experiments but also by trying to understand how their results can be explained by the deepest mathematics. You could say that the universe speaks to us in numbers.’16

  CHAPTER 1

  MATHEMATICS DRIVES AWAY THE CLOUD

  The things that so often vexed the minds of the ancient philosophers

  And fruitlessly disturb the schools with noisy debate

  We see right before our eyes, since mathematics drives away the cloud

  —EDMOND HALLEY, ODE TO NEWTON AND HIS ‘PRINCIPIA’, 1687

  Einstein was modest about his achievements. He knew his place in the history of science, however, and was aware that he was standing on giants’ shoulders, none broader than Isaac Newton’s. Two centuries after the Englishman’s death, Einstein wrote that ‘this brilliant genius’ had ‘determined the course of western thought, research, and practice, like no one else before or since’.1 Among Newton’s greatest achievements, Einstein later remarked, was that he was ‘the first creator of a comprehensive, workable system of theoretical physics’.2

  Newton never spoke of ‘physicists’ and ‘scientists’, terms that were coined more than a century after his death.3 Rather, he regarded himself as primarily a man of God and only secondarily as a mathematician and natural philosopher, attempting to understand rationally the entirety of God’s creation, using a combination of reasoning and experiment. He first publicly set out his mathematical approach to natural philosophy in 1687, when he published his Principia, a three-book volume soon to make him famous and help to establish him as one of the founders of the Enlightenment. In the preface to that edition, he made clear that he was proposing nothing less than ‘a new mode of philosophising’.4

  Newton rejected the way of working that virtually all his contemporaries regarded as the best way to proceed. They were making guesses about the mechanisms that can explain how nature works, as if it were a giant piece of clockwork that needed to be understood. Instead, Newton focused on the motion of matter, on Earth and in the cosmos—part of God’s creation that he could describe precisely using mathematics. Most significantly, he insisted that a theory must be judged solely according to the accuracy of the account it gives of the most precise observations on the real world. If they do not agree within experimental uncertainties, the theory needs to be modified or replaced by a better one. Today, all this sounds obvious, but in Newton’s day it was radical.5

  When Newton published his Principia, he was a forty-four-year-old professor living a quiet bachelor life in Trinity College, Cambridge, in rooms that now overlook the row of stores that includes Heffers Bookshop.6 Almost two decades before, the university had appointed him to its Lucasian Chair of Mathematics, although he had published nothing on the subject. Mathematics was only one of his interests—he was best known in Cambridge for designing and building a new type of telescope, which attested to his exceptional practical skills.

  A devout and stony-faced Protestant, he believed he was born to understand God’s role in creating the world, and he was determined to rid Christian teaching of corruptions by perverted priests and others who preyed on the tendency of many people to wallow in idolatry and superstition.7 To this and all his other work, Newton brought a formidable energy and a concentration so intense that he would occasionally forget to eat.8 For this prickly and suspicious scholar, life was anything but a joke—a smile would occasionally play across his face, but he was only rarely seen to laugh.9

  Newton invited only a small number of acquaintances into his chambers, and relatively few experts appreciated the extent of his talent. He was not interested in sharing his new knowledge and once remarked that he had no wish to have his ‘scribbles printed’—the relatively new print culture was not for him.10 His circle of confidants included the chemist Francis Vigani, who was disappointed to find himself cut off after telling the great thinker a ‘loose story’ about a nun.11

  Newton’s new scheme for natural philosophy did not arrive out of the blue—it emerged after decades of gestation and close study. In the opening words of the Principia, he acknowledged his debts: first to the ancient Greeks, who had focused above all on the need to understand motion, and second to recent thinkers who had ‘undertaken to reduce the phenomena of nature to mathematical laws’.12 To understand the background to Newton’s achievement, it is instructive to look briefly at these influences, beginning with the ancient Greeks, who had taught Europeans the art of thinking.

  The nearest the ancient Greeks came to doing science (from scientia, Latin for ‘knowledge’) in the modern sense was in the work of the philosopher Aristotle (384–322 BC). He believed that, underneath the messiness of the world around us, nature runs on principles that human beings can discover and that are not subject to external interference from, for example, meddlesome deities.13 Of all the ancients’ schools of philosophy, Aristotle’s paid most attention to physica—a word derived from physis, meaning nature—which included studies that ranged from astronomy to psychology. The word ‘physics’ derives from this but didn’t acquire its modern meaning until the early nineteenth century.

  The sheer breadth of Aristotle’s studies—from cosmology to zoology and from poetry to ethics—made him perhaps the most influential thinker about nature in our history. He believed that the natural world can be described by general principles that express the underlying reasons for all the types of change that can affect any matter, including changes in its shape, colour, size, and motion. His writings on science, including his book Physica, seem strange to most modern readers partly because he attempted to understand the world using pure reason, albeit supported by careful observation.

  One characteristic of his view of the world is that mathematics has no place in it. Aristotle declined, for example, to use the elements of arithmetic and geometry, whose rudiments were already thousands of years old when he began to think about science. Both branches of mathematics were grounded in human experience and had been developed by thinkers who had taken the crucial step of moving from observations of the real world to completely general abstraction. The most basic elements of arithmetic, for example, began when human beings first generalised the concept of two sticks, two wolves, two fingers, and so on, to the existence of the abstract concept of the number 2, not associated with any one concrete object. This was a profound insight, though it is not easy to say when it was first made. The beginnings of geometry—the relationships between points, lines, and angles in space—are easier to date: about 3000 BC, when people in ancient Babylonia and the ancient Indus Valley began to survey the land, sea, and sky. In Aristotle’s view, however, there was no place in science for mathematics, whose ‘method is not that of natural science’.14

  Aristotle’s rejection of mathematical thinking was antithetical to the philosophy of his teacher Plato and of another of the most famous ancients, Pythagoras, who may never have existed (his putative teachings may have been the work of others). Pythagoreans studied arithmetic, geometry, music, and astronomy and held the view that whole numbers were crucially important. Their remarkable ability to explain, for example, the relationship between musical harmonies and the properties of geometric objects led the Pythagorean school to believe that whole numbers were essential to a fundamental understanding of the way the universe works.

  Plato had believed that mathematics was fundamental to philosophy and was convinced that geometry would lead to understanding the world. For Plato, the complicated realities around us are, in a sense, shadows of perfect mathematical objects that exist quite separately, in the abstract world of mathematics. In that world, shapes and other geometric objects are perfect—points are infinitely small, lines are perfectly straight, planes are perfectly flat, and so on. So, for example, he would have regarded a roughly square table top as the ‘shadow’ of a perfect square, whose infinitely thin and perfectly straight lines all meet at precisely 90 degrees. Such a perfect mathematical object cannot exist in the real world, but it is a feature of what modern mathematicians often describe as the Platonic world, which can seem to them no less real than the world around us.

  Within a quarter of a century of Aristotle’s death, the Greek thinker Euclid introduced new standards of rigour to mathematical thinking. In his magnificent thirteen-book treatise The Elements, he set out the fundamentals of geometry clearly and comprehensively, setting new standards of logical reasoning in the subject. Although no one’s idea of easy reading, The Elements became the most influential book in the history of mathematics and exerted a powerful influence on thinkers for centuries. One of the leading physicists who later fell under its spell was Einstein, who remarked, ‘If Euclid failed to kindle your youthful enthusiasm, then you were not born to be a scientist.’15

  Mathematics was becoming practically useful, too. Archimedes was especially adept at putting mathematical ideas to work in his inventions, for example, his water-raising screw and parabolic mirror. Several of his contemporaries in Greece used geometric reasoning to measure the distance of the Sun and the Moon from the Earth, the circumference of the Earth, and the tilt of the Earth’s axis of spin, often to an impressively high degree of accuracy. The idea that regularities in the behaviour of objects that human beings observe around them on Earth could be described by mathematical laws was centuries away. However, mathematics was already enabling earth-bound human beings to transcend their senses and deploy their powers of imaginative reasoning way out into the heavens.

  Simple mathematical concepts began to be useful to many of the thinkers who were advancing science. In the Middle Ages, many of the most notable mathematical innovations arose in Islamic territories, roughly in the areas now spanned by Iran and Iraq.16 Scholars of this region made impressive mathematical advances, including the development of algebra, from the Arabic al-jabr, meaning ‘reunion of broken parts’. These innovations formed the basis of modern algebra, which uses abstract symbols, say x and y, to represent quantities that can take numerical values and be mathematically manipulated.

  By the middle of the sixteenth century, when Shakespeare was born, mathematics featured prominently in almost every branch of physical science—including astronomy, optics, and hydraulics—as well as in music. New ideas about the way mathematics relates to the world were gaining traction, calling into question the Aristotelian way of thinking that had dominated Christian and Islamic thinking for 2,000 years. One of the most important contributions was Nicolaus Copernicus’s proposal in 1543 that the centre of the universe is not the Earth but the Sun, a radical notion that marked the beginning of what became known as the Scientific Revolution. Among its leading pioneers were two astronomers who were also mathematicians: the German Johannes Kepler and the Italian Galileo Galilei. They believed that the best way to understand the world was not to focus on the superficial appearances of things but to give precise descriptions of motion. To them, it was especially important to identify mathematical regularities in measurements made on moving objects. Among Kepler’s achievements, he identified such regularities in the motion of the planets orbiting the Sun, while Galileo discovered regularities closer to home—in the paths of objects falling freely to the ground.

  For the devout Kepler, God was the ‘architect of the universe’ and had created it according to a plan that human beings could understand using geometry, a subject that Kepler regarded as divine.17 The disputatious Galileo often stressed the importance of comparing the predictions of scientific theories directly with observations made on the real world: this insistence made him ‘the father of modern science’ in Einstein’s view, though Galileo was given to exaggerating the accuracy of his experimental data.18 He was also no slouch as a mathematician and appreciated its importance to human understanding of the natural world, famously declaring in 1623 that the book of nature ‘is written in the language of mathematics’.19 His thinking was part of a cultural trend in many of the most prosperous European cities: mathematics was beginning to underpin commercial and artistic life, through new bookkeeping methods and the use of geometric perspective in art and architecture.20

  Neither Kepler nor Galileo fully grasped an idea that was to become central to science—that the natural world appears to be described by laws that apply everywhere, perhaps for all time.21 The idea, mooted by Aristotle, that there exist fundamental laws of nature, emerged most clearly in the writings of the Frenchman René Descartes, whose work was to dominate European thinking about nature for several decades from the early 1640s, an era that saw both the death of Galileo and the birth of Newton. Descartes set aside Aristotelian science and tried to account for gravity, heat, electricity, and other aspects of the real world using mechanisms that he described with impressive vividness, bearing in mind that neither he nor anyone else had any direct evidence that they were correct.22

  Descartes published his ideas in the book Principles of Philosophy, which he recommended should be read straight through like a novel (he advised his readers that most of their difficulties with the text would have disappeared by their third reading). The book used very little mathematics and gave no indication of how experimenters could test his mechanical theories, such as his idea that huge swirling vortices of matter drive each planet around the Sun. London’s most eminent experimenter, Robert Hooke, was a fulsome admirer of Descartes but was nonetheless becoming impatient with the prevailing cerebral approach to science: ‘The truth is, the Science of Nature has already been too long made only a work of the Brain and the Fancy: It is now high time that it should return to the plainness and soundness of observations on material and obvious things.’23

  When Hooke wrote those words, in 1665, the twenty-two-year-old Isaac Newton was doing breathtakingly creative work in both mathematics and natural philosophy. By then, he was familiar with the thinking of the ancient Greeks, and in one of his notebooks he had written an old scholastic tag: ‘Plato is a friend, Aristotle is a friend, but truth is a greater friend.’24 Newton was also well acquainted with the discoveries of Kepler, Galileo, and Descartes and how these thinkers and others had overturned the Aristotelian consensus. The most decisive event in Newton’s mathematical education was his reading of Descartes’s Geometry: in the words of the eminent Newton scholar David Whiteside, from the first hundred or so pages of this book, Newton’s ‘mathematical spirit took fire’.25 If he had published the mathematical discoveries he made in this period, he would have been recognised as one of the world’s leading experts in the discipline, though hardly any of his peers knew what he had done. The world found out almost a quarter of a century later, when he began to move science towards more systematic studies of the natural world, grounded in mathematics and quantitative observations. He did this in his magnum opus, one of the most important volumes in the history of human thought.

  Newton may never have written the Principia had it not been for the initiative and perseverance of the astronomer Edmond Halley, now best remembered for the observations of the comet later named after him. One of Newton’s few friends, Halley spent almost three years coaxing, assisting, and cajoling the reluctant author to deliver his masterpiece. He even offered to pay the cost of publishing it. The Principia, about five hundred pages long, went on sale in London on Saturday, 5 July 1687—a red-letter day in the history of science, though at the time it was a non-event. The publishers printed about six hundred copies, but selling them all proved to be a struggle, even after an anonymous review praised the ‘incomparable Author’ for delivering ‘a most notable instance of the powers of the Mind’ (the words were Halley’s).26 Newton had presented his scheme in a forbiddingly austere style, partly to ‘avoid being baited by little smatterers in mathematics’, as he later put it.27 As a result, the volume was virtually impenetrable for everyone except a handful of his peers: during his lifetime, fewer than a hundred people read the entire Principia, and it is certain that only a few of them had even a faint chance of understanding it.28

  Newton had subtitled his volume Mathematical Principles of Natural Philosophy, in an unsubtle swipe at Descartes’s Principles of Philosophy. The point of this was to signal that he was focusing entirely on natural philosophy—the real world—rather than on philosophy in general, and that his principles were essentially mathematical.29 Newton had studied Descartes’s error-strewn book closely and had become increasingly critical of its ‘tapestry of assumptions’.30 In a sense, the Principia was a narrowing and mathematical corrective to Descartes’s narrative philosophy of nature: Newton focused entirely on the part of the real world that could be accounted for mathematically, with both generality and precision.