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The Universe Speaks in Numbers
How Modern Math Reveals Nature's Deepest Secrets
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One of the great insights of science is that the universe has an underlying order. The supreme goal of physicists is to understand this order through laws that describe the behavior of the most basic particles and the forces between them. For centuries, we have searched for these laws by studying the results of experiments.
Since the 1970s, however, experiments at the world's most powerful atom-smashers have offered few new clues. So some of the world's leading physicists have looked to a different source of insight: modern mathematics. These physicists are sometimes accused of doing 'fairy-tale physics', unrelated to the real world. But in The Universe Speaks in Numbers, award-winning science writer and biographer Farmelo argues that the physics they are doing is based squarely on the well-established principles of quantum theory and relativity, and part of a tradition dating back to Isaac Newton.
With unprecedented access to some of the world's greatest scientific minds, Farmelo offers a vivid, behind-the-scenes account of the blossoming relationship between mathematics and physics and the research that could revolutionize our understanding of reality.
A masterful account of the some of the most groundbreaking ideas in physics in the past four decades. The Universe Speaks in Numbers is essential reading for anyone interested in the quest to discover the fundamental laws of nature.
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PROLOGUE
LISTENING TO THE UNIVERSE
I hold it to be true that pure thought can grasp reality, as the Ancients dreamed.
—ALBERT EINSTEIN, ‘ON THE METHOD OF THEORETICAL PHYSICS’, 1933
‘Einstein is completely cuckoo’. That was how the cocky young Robert Oppenheimer described the world’s most famous scientist in early 1935, after visiting him in Princeton.^{1} Einstein had been trying for a decade to develop an ambitious new theory in ways that demonstrated, in the view of Oppenheimer and others, that the sage of Princeton had lost the plot. Einstein was virtually ignoring advances made in understanding matter on the smallest scale, using quantum theory. He was seeking an ambitious new theory, not in response to puzzling experimental discoveries, but as an intellectual exercise—using only his imagination, underpinned by mathematics. Although this approach was unpopular among his peers, he was pioneering a method similar to what some of his most distinguished successors are now using successfully at the frontiers of research.
Oppenheimer and many other physicists at that time can hardly be blamed for believing that Einstein’s mathematical approach was doomed: for one thing, it seemed to contradict one of the principal lessons of the past 250 years of scientific research, namely that it is unwise to try to understand the workings of nature using pure thought, as Plato and other thinkers had believed. The conventional wisdom was that physicists should listen attentively to what the universe tells them about their theories, through the results of observations and experiments done in the real world. In that way, theorists can avoid deluding themselves into believing they know more about nature than they do.
Einstein knew what he was doing, of course. From the early 1920s, he often commented that experience had taught him that a mathematical strategy was the best hope of making progress in his principal aim: to discover the most fundamental laws of nature. He told the young student Esther Salaman in 1925, ‘I want to know how God created this world. I’m not interested in this or that phenomenon, in the [properties] of this or that element. I want to know His thoughts, the rest are details.’^{2} In his view, ‘the supreme task of the physicist’ was to comprehend the order that underlies the workings of the entire cosmos—from the behaviour of the tiny particles jiggling around inside atoms to the convulsions of galaxies in outer space.^{3} The very fact that underneath the diversity and complexity of the universe is a relatively simple order was, in Einstein’s view, nothing short of a ‘miracle, or an eternal mystery’.^{4}
Mathematics has furnished an incomparably precise way of expressing this underlying order. Physicists and their predecessors have been able to discover universal laws—set out in mathematical language—that apply not only here and now on Earth but to everything everywhere, from the beginning of time to the furthest future. Theorists, including Einstein, who pursue this programme may be accused quite reasonably of overweening hubris, though not of a lack of ambition.
The potential of mathematics to help discover new laws of nature became Einstein’s obsession. He first set out his mathematical approach to physics research in the spring of 1933, when he delivered a special lecture to a public audience in Oxford. Speaking quietly and confidently, he urged theoreticians not to try to discover fundamental laws simply by responding to new experimental findings—the orthodox method—but to take their inspiration from mathematics. This approach was so radical that it probably startled the physicists in his audience, though understandably no one dared to contradict him. He told them that he was practising what he was preaching, using a mathematical approach to combine his theory of gravity with the theory of electricity and magnetism. That goal could be achieved, he believed, by trying to predict its mathematical structure—the mathematics of the two theories were the most potent clues to a theory that unified them.
As Einstein well knew, a mathematical strategy of this type would not work in most other scientific disciplines, because their theories are usually not framed in mathematical language. When Charles Darwin set out his theory of evolution by natural selection, for example, he used no mathematics at all. Similarly, in the first description of the theory of continental drift, Alfred Wegener used only words. One potential shortcoming of such theories is that words can be treacherous—vague and subject to misinterpretation—whereas mathematical concepts are precise, well-defined, and amenable to logical and creative development. Einstein believed that these qualities were a boon to theoretical physicists, who should take full advantage of them. Few of his colleagues agreed—even his most ardent admirers scoffed. His acid-tongued friend Wolfgang Pauli went so far as to accuse him of giving up physics: ‘I should congratulate you (or should I say send condolences?) that you have switched to pure mathematics.… I will not provoke you to contradict me, in order not to delay the death of [your current] theory.’^{5} Brushing such comments aside, Einstein continued on his lonely path, though he had little to show for his labours: he had become the Don Quixote of modern physics.^{6} After he died in 1955, the consensus among leading physicists was that the abject failure of his approach had vindicated his critics, but this judgement has proved premature.
Although Einstein was wrong to gloss over advances in theories of matter at the subatomic level, he was in one respect more far-sighted than his many detractors. In the mid-1970s, twenty years after he died, several prominent physicists were following in his footsteps, trying to use pure thought—bolstered by mathematics—to build on well-established but flawed theories. At that time, I was a greenhorn graduate student, wary of this cerebral strategy and pretty much convinced that it was perverse and heading nowhere. It seemed obvious to me that the best way forward for theorists was to be guided by experimental findings. That was the orthodox method, and it had worked a treat for the theorists who developed the modern theory of subatomic forces. Later known as the Standard Model of particle physics, it was a thing of wonder: based on only a few simple principles, it quickly superceded all previous attempts to describe the behaviour of subatomic particles. It accounted handsomely for the inner workings of every atom. What I did not fully appreciate at the time was how fortunate I was to be sitting in the back row of the stalls, watching an epic contemporary drama unfold.
During those years, I remember attending dozens of seminars about exotic new theories that looked impressive but agreed only roughly with experiments. Yet their champions were obviously confident that they were on to something, partly because the theories featured interesting new mathematics. To me, this seemed a peculiar way of researching physics—I thought it much better to listen to what nature was telling us, not least because it never lies.
I sensed a new wind was blowing and, as far as I could tell, it was going in an unappealingly mathematical direction. Privately, I expected the trend to peter out, but once again I was wrong. In the early 1980s, the wind gathered momentum, as the flow of new information from experiments on subatomic particles and forces slowed from a gush to a drip. For this reason, more theoreticians turned to pure reasoning, supplemented by mathematics. This led to a new approach to fundamental physics—string theory, which aspires to give a unified account of nature at the finest level by assuming that the basic constituents of the universe are not particles but tiny pieces of string. Theorists made progress with the theory but, despite a huge effort, they could not make a single prediction that experimenters could check. Sceptics like me began to believe that the theory would prove to be no more than mathematical science fiction.
I found it striking, however, that many of the leading theoretical physicists were not discouraged by the glaring absence of direct experimental support. Time and again, they stressed the theory’s potential and also the marvellous breadth and depth of its connections to mathematics, many of which were revelatory even to world-class mathematicians. This richness helped to shift collaborations between theoretical physicists and mathematicians into an even higher gear, and generated a welter of mind-blowing results, especially for mathematicians. It was clearer than ever not only that mathematics is indispensable to physics, but also that physics is indispensable to mathematics.
This intertwining of mathematics and physics seemed to exemplify the view expressed in the 1930s by the physicist Paul Dirac, sometimes described as ‘the theorist’s theorist’.^{7} He believed that fundamental physics advances through theories of increasing mathematical beauty.^{8} This trend convinced him—as ‘a matter of faith rather than of logic’—that physicists should always seek out examples of beautiful mathematics.^{9} It was easy to see why this credo had a special appeal for string experts: their theory had abundant mathematical beauty, so according to Dirac’s way of thinking, held commensurately huge promise.
The ascendancy of string theory did much to give modern fundamental physics a strong mathematical hue. Michael Atiyah, a brilliant mathematician who had switched his focus to theoretical physics, later wrote provocatively of the ‘mathematical takeover of physics’.^{10} Some physicists, however, were dismayed to see many of their most talented colleagues working on recondite mathematical theories that in many cases were impossible to test. In 2014, the American experimenter Burton Richter bluntly summarised his anxieties about this trend: ‘It seems that theory may soon be based not on real experiments done in the real world, but on imaginary experiments, done inside the heads of theorists.’^{11} The consequences could be disastrous, he feared: ‘Theoreticians would have to draw their inspiration not from new observations but from mathematics. In my view, that would be the end of research into fundamental physics as we now know it.’
Disenchantment with the state of modern theoretical physics has even become a public talking point. Over the past decade or so, several influential commentators have taken aim at string theory, describing it as ‘fairy tale physics’ and ‘not even wrong’, while a generation of theoretical physicists stand accused of being ‘lost in math’.^{12} It is now common to hear some critics in the media, especially in the blogosphere, complain that modern physics should get back on the straight and narrow path of real science.
This view is misguided and unnecessarily pessimistic. In this book, I shall argue that today’s theoretical physicists are indeed taking a path that is entirely reasonable and extremely promising. For one thing, their approach draws logically and creatively on centuries of achievements all the way back to Isaac Newton. By setting out mathematical laws that describe motion and gravity, he did more than anyone else to construct the first mathematically based and experimentally verifiable framework for describing the real world. As he made clear, the long-term aim is to understand more and more about the universe in terms of fewer and fewer concepts.^{13} Leading theorists today pursue this agenda by standing squarely on the two granitic foundation stones of the twentieth century: Einstein’s basic theory of relativity, a modification of Newton’s view of space and time, and quantum mechanics, which describes the behaviour of matter on the smallest scale. No experiment has ever disproved either of the two theories, so they form an excellent basis for research.
As Einstein often pointed out, quantum mechanics and the basic theory of relativity are devilishly difficult to meld. Physicists were eventually able to combine them into a theory that made impressively successful predictions, in one case agreeing with the corresponding experimental measurement to eleven decimal places.^{14} Nature seemed to be telling us loud and clear that it wanted both theories to be respected. Today’s theoretical physicists are building on that success, insisting that every new theory that aspires to be universal must be consistent with both basic relativity and quantum mechanics. This insistence led to consequences that nobody had foreseen: not only to new developments in physics—including string theory—but also to a host of links with state-of-the-art mathematics. It had never been clearer that physics and mathematics are braided: new concepts in fundamental physics shed light on new concepts in mathematics, and vice versa. It is for this reason that many leading physicists believe that they can learn not only from experiments but also from the mathematics that emerges when relativity and quantum mechanics are combined.
The astonishing effectiveness of mathematics in physics has enthralled me since I was a schoolboy. I remember being surprised that the abstract techniques we learned in our mathematics lessons were perfectly suited to solving the problems we were tackling in physics classes. Most remarkable for me was that some of the mathematical equations that linked unknown quantities x and y also applied to observations that describe the real world, with x and y standing for quantities that experimenters could measure. It amazed me that a few simple principles, underpinned by mathematics we had only recently learned, could be used to predict accurately everything from the paths of golf balls to the trajectories of planets.
As far as I recall, none of my schoolteachers commented on the way abstract mathematics lends itself to physics so exquisitely, one might even say miraculously.^{15} At university, I was even more impressed that theories that incorporated basic mathematics could describe so much about the real world—from the shapes of magnetic fields near current-carrying wires to the motion of particles inside atoms. It seemed something like a fact of scientific life that mathematics is utterly indispensable to physics. Only much later did I glimpse the other side of the story: that physics is indispensable to mathematics.
One of my main aims in this book is to highlight how mathematics, as well as proving useful to physicists, has supplied invaluable clues about how the universe ticks. I begin with Newton’s epoch-making use of mathematics to set out and apply the law of gravity, which he repeatedly tested against observations and careful measurements. Next, I explain how the mathematical laws of electricity and magnetism were discovered in the nineteenth century, using a mathematical framework that had huge implications for our understanding of nature.
I then move on to discuss two groundbreaking discoveries—first, basic relativity, and then quantum mechanics, the most revolutionary theory in physics for centuries. When Einstein used relativity to improve our understanding of gravity, he was forced to use mathematics that was new to him, and the success of this approach changed his view about the utility of advanced mathematics to physicists. Likewise, when physicists used quantum mechanics to understand matter, they were forced to use tranches of unfamiliar mathematics that changed their perspective on, for example, the behaviour of every one of nature’s smallest particles.
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}
Genre:
- "Will pure mathematics prevail over dogged experiments in investigating the physical world? Is the fulfillment of the Pythagorean dream of a mathematical universe finally at hand? Mr. Farmelo, deeply impressed with the accomplishments of Einstein, Dirac and string theorists, is optimistic."—Wall Street Journal
- "Farmelo shows that theoretical physics and pure mathematics thrive best together."—Scientific American
- "These are brilliant successes of the mathematical approach, and Farmelo leads us through them adeptly, with a mixture of contemporary accounts and scientific insight."—Nature
- "Despite the difficulties of writing about such specialised technical material, Farmelo has succeeded in writing a book for the general reader that gives insights into the motivation behind a theory developed by many of today's leading thinkers. His book provides as clear an account of the subject as I can imagine for a non-specialist reader."—Times Higher Education (UK)
- "Mathematics here becomes a brilliant laser beam illuminating the very frontiers of science!"—Booklist STARRED
- "A riveting account of one of the greatest stories of our time. Graham Farmelo has delved deep into this fascinating subject, combining original scholarship and lively interviews with leading contemporary theorists at the forefront of the field. The result is a masterful book, which gives us, for the first time, a behind-the-scenes look at how physicists and mathematicians, driven by their pursuit of ultimate Truth, have been drawn into common territory by mysterious intellectual forces seemingly beyond their control."—Nima Arkani-Hamed, professor, Institute of Advanced Study, Princeton
- "A thought-provoking look at a fierce, ongoing controversy over the future of theoretical physics."—Kirkus
- "Crisply-written, entertaining and extraordinarily well-informed. The most popular and elegant theories of what makes the cosmos tick are becoming harder---even impossible---to test directly but Graham Farmelo argues in this tour de force that they're still taking us deep into the mathematical heart of reality."—Roger Highfield, author, journalist, and Director of External Affairs at the Science Museum, London
- "This fascinating, splendidly readable, extensively researched, and remarkably up-to-date book takes readers from the days of Newton to the forefront of modern theoretical physics and shows how current research has reshaped the fields of physics and mathematics to the enrichment of both."—Jeremy Gray, emeritus professor, Open University
- "Farmelo expertly narrates the history of the dynamic dance between mathematics and theoretical physics, from Newton to Einstein to string theory and beyond. This book will be a must-read for anyone interested in either subject's history or present for many years ahead."—Jacob Bourjaily, associate professor of physics, Niels Bohr International Academy, Copenhagen University
- "A superbly written, riveting book. In elegant prose, and using virtually no equations, Farmelo describes the ongoing quest of great thinkers to understand the bedrock nature of reality, from the microworld to the cosmos."—Martin Rees, astronomer royal, emeritus professor of cosmology and astrophysics, University of Cambridge
- "I am overcome with admiration for its range and profundity. An amazing achievement."—Michael Frayn
- On Sale
- May 28, 2019
- Page Count
- 336 pages
- Publisher
- Basic Books
- ISBN-13
- 9780465056651
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