by Neil Turok
I recall being upset as a young scientist by the words of one of my heroes, the great U.S. physicist Richard Feynman, who recounted how he overcame his worries about working on the nuclear bomb: “[The Hungarian-American mathematician John] von Neumann gave me an interesting idea: that you don’t have to be responsible for the world that you’re in. So I have developed a very powerful sense of social irresponsibility as a result of von Neumann’s advice. It’s made me a very happy man ever since. But it was von Neumann who put the seed in that grew into my active irresponsibility!”4 At the time, Feynman’s cop-out seemed incompatible with what I knew of his persona. His humanity shone through in his writing, his teaching, and all his interactions. Only later, I realized he was in denial. Feynman loved his physics; he just couldn’t face thinking about the far more difficult questions of the uses to which it might be put.
THE DISCONNECTION BETWEEN SCIENCE and society is harmful, especially when you consider that science is, in general, open-minded, tolerant, and democratic. In its opposition to dogma and its willingness to live with uncertainty, science is in many ways a model for society. Many scientists are energized by the sense that their work is of wider interest and might contribute to progress. Back in the eighteenth century, the Scottish philosopher David Hume wrote these wise words: “It seems, then, as nature has pointed out a mixed kind of life as most suitable to the human race . . . Indulge your passion for science, says she, but let your science be human, and such as may have a direct reference to action and to society.”5 Equally, he argued, society in its aesthetic and moral concerns can benefit from science: “Accuracy is in every case advantageous to beauty, and just reasoning to gentle sentiment.”6
Hume had entered Edinburgh University as a lad of twelve — starting so young was not uncommon at the time — during the period known as the Scottish Enlightenment. His independence of mind is nicely illustrated in a letter he wrote at the end of his time at Edinburgh: “There is nothing to be learnt from a Professor, which is not to be met with in Books.”7 Nevertheless, it was at university that he discovered his passion for philosophy. He spent the eight years following his graduation writing his philosophical masterpiece, A Treatise of Human Nature, the first volume of which would later appear as An Enquiry Concerning Human Understanding.
Hume’s Enquiry reads, even today, as a breath of fresh air. His modesty, his originality, his accessible style are models of the art of gentle persuasion. His powers of reason worked wonders as he calmly overturned two millennia of doctrinaire thinking.
Hume’s revolutionary views, though far-reaching, were based on the simple suggestion that our existence, our feelings, and our experience are the foundation for all our ideas. Imagination is powerful, but it is no substitute for our natural impressions and instincts: “The most lively thought is still inferior to the dullest sensation,”8 and again, “It is impossible for us to think of any thing, which we have not antecedently felt, either by our external or internal senses.” Even mathematical abstractions like number or shape are, Hume argued, ultimately based upon our experience of interacting with natural phenomena.9
Hume believed our perceptions and feelings — our external and internal experiences — to be the foundation for our knowledge. It was a profoundly democratic idea, that knowledge is based on capacities which everyone shares. While recognizing the power of mathematics, Hume warned against reasoning too far removed from the real world: “If we reason a priori, anything may appear able to produce anything. The falling of a pebble may, for aught we know, extinguish the sun; or the wish of a man control the planets in their orbits. It is only experience, which teaches us the nature of cause and effect, and enables us to infer the existence of one object from that of another.”10 In his constant emphasis on experience, Hume helped to bring science back to earth, to connect it to our humanity, to who we are and what we can do.
Hume’s skepticism and frankness brought him into conflict with the Church. Hume’s Dialogues Concerning Natural Religion (echoing Galileo’s Dialogue Concerning the Two Chief World Systems) is framed as a debate between three protagonists in an ancient Greek setting, Dialogues addresses the validity of beliefs, such as the existence of a creator, the immortality of the soul, and the moral benefits of religion. It does so in a subtle and respectful way, encouraging open discourse without belittling the protagonists. Nevertheless, even as they recognized the book as a landmark, Hume’s friends persuaded him that it would be dangerous to publish. The book only appeared in print three years after his death, anonymously and with no publisher credited.
Hume took a unified approach to natural and moral philosophy — which he called “the science of man.” He expressed a balanced view of the advantages and limitations of both: “The chief obstacle, therefore, to our improvement in the moral or metaphysical sciences is the obscurity of the ideas, and the ambiguity of the terms. The principal difficulty in the mathematics is the length of inferences and compass of thought, requisite to the forming of any conclusion. And perhaps, our progress in natural philosophy is chiefly retarded by the want of proper experiments and phaenomena, which are often discovered by chance, and cannot always be found, when requisite, even by the most diligent and prudent enquiry.”11 In this last point, he had great foresight. In the nineteenth century, experiments and observations drove an age of discovery. Even in the twentieth century, Einstein was influenced by Hume and expressed his core beliefs in the same language.12
What Hume put forward continues to resonate today. Our ability to do science is rooted in our relationship with the universe, our nature as living beings. Our feelings and instincts are far more profound than our ideas. Our ideas allow us to imagine many things, but they can be unreliable, misguided, or misleading. It is the real world that keeps us honest.
Science is about discovering things: about the universe and about ourselves. We are looking for answers, for explanations that will open new doors. What is the meaning of life in the universe, or the purpose of our existence? Scientists typically refrain from any discussion of these notions, saying they are beyond science’s realm. But to me, such questions are profoundly important. Why do we decide to do the things we do? Are we, as some scientists would say, merely biological machines, driven by the need to replicate our selfish genes? If we can, as I believe, be much more than this, from where can we draw our wisdom?
Hume’s philosophy of knowledge was closely connected to his notions of ethics and of society. Our strength as scientists rests on our character and honesty as human beings, the same traits that make us good citizens. And all of these capabilities arise from our connection with the universe.
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AS A CHILD, I spent many hours watching ants, amazed at how these tiny creatures determinedly followed paths away from or back to their nests, and wondering how they coped with unexpected changes, like a stick across their path, being soaked in rain, or being blown off course. Like us, they must be constantly extracting the essential information they need from their surroundings, updating their mental models of the world, weighing their options, and taking decisions.
Our brains seem to work this way. We each have an internal model of the world, which we are constantly comparing against our perceptions. This internal model is a selective representation designed to capture reality’s most essential elements, the ones that are most important for us, and predict their behaviour. In receiving data from our senses, what we notice are the surprises — the discrepancies between our experiences and the predictions of our internal model, which force us to correct it. Science is the extension of this instinctive ability, allowing us to create explanatory knowledge at ever deeper and more far-reaching levels.
Mathematics is one of our most valuable tools, and perhaps the most valuable tool, in this reduction of nature down to its key elements. It is founded on mental abstractions like number, shape, and dimension, which are distillations of the properties of objects in the real world. It
complements our natural instincts and intuition in magical, unexpected ways. For example, when perspective and shadowing, which are entirely geometrical concepts, were first employed by artists in medieval Italy, paintings suddenly leaped from the flat two-dimensional world of medieval icons to the infinitely richer three-dimensional world of Renaissance art.
Leonardo da Vinci mastered these techniques, combining art and science in equal measure. Most famous for his paintings — some of the finest ever made — he also made a great number of drawings, of imaginary machines and inventions, of plants and animals, and of cadavers dissected illegally to reveal the inner workings of the human body.
Leonardo never published his writings, but he did keep personal notes that survived, although in complete disorder. Written in mirror-image cursive, from right to left, they open with this rejection of authority: “I am fully aware that the fact of my not being a man of letters may cause certain presumptuous persons to think that they may with reason blame me, alleging that I am a man without learning. Foolish folk! . . . they do not know that my subjects require for their exposition experience rather that the words of others.”13
He was not at all against theory, however — on the contrary, he states: “Let no man who is not a Mathematician read the elements of my work.”14 And elsewhere: “The Book of the science of Mechanics must precede the Book of useful inventions.”15 Like the ancient Greeks, he was strongly asserting the power of reason.
As an artist, Leonardo was understandably obsessed with light, perspective, and shadow. In his notebooks he explained how light is received, with the eye at the apex of a “pyramid” (or cone) of converging straight rays. Likewise, he discussed in detail how shadows are produced by the obstruction of light. Many of his mathematical ideas may be traced back to those of Alhazen (Ibn al-Haytham, 965–1040), one of the most famous Islamic scientists, who worked in both Egypt and Iraq at the end of the first millennium and whose Book of Optics (Kitab al-Manazir), written in 1021, was published in Italy in the fourteenth century.
Leonardo’s careful use of geometry and scientific employment of perspective and shadow, as well as his deep appreciation of anatomy, allowed him to create stunning works of art which not only captured the real world but playfully represented imaginary landscapes (as in the background of the Mona Lisa) or historical scenes (like The Last Supper). To see the effect of these advances, one has only to look at the way art was transformed. Before the Renaissance, paintings were little more than cartoon representations of the world; after it, realistic representations became normal.
Mathematics can take us far beyond our natural instinct for understanding the world. A mathematical model is a representation of reality, which we improve by an iterative process of trial and error, adaptation and refinement. Our models evolve, much as life does, and as they develop they change and are steadily improved. They are never final. As Einstein said, “As far as the laws of mathematics refer to reality, they are not certain: and as far as they are certain, they do not refer to reality.”16 Stated differently, being creatures of limited capability living in a very complex world, the best we can do is to focus on and understand nature’s underlying regularities.
From the motion of the planets, to the structure of atoms and molecules, to the expansion of the cosmos, many of the world’s most basic properties are accurately predictable from beautifully simple mathematical rules. Italian mathematician Galileo Galilei is reported to have said, “Mathematics is the language with which God wrote the universe.”17 It is an especially powerful language, a set of logical rules that allow no contradiction.
As an example, the circumference of a circle is its diameter times a number called π (pi). π is a peculiar number, first estimated by the Babylonians as about 3, shown by Greek scientist Archimedes (287–212 B.C.) to be between 3 1⁄7 and 3 10⁄71, then approximated by a Chinese mathematician, Zu Chongzhi (A.D. 429–500), as 355⁄113. But the point is, it doesn’t matter which circle you choose, π always comes out the same — 3.14159 . . . , with digits that go on and on and never repeat themselves. Well, all right, you say, π is a useful little rule. And handily enough, it turns up again in the volume of a sphere, any sphere of any size, anywhere in the universe — from a basketball to a planet. In physics, it turns up everywhere: in the formula for the period of a pendulum, or the force between two electric charges, or the power in a shockwave. And that’s only the beginning.
We do not understand why mathematics works to describe the world, but it does.18 One of its most remarkable features is that it transcends culture or history or religion. Whether you are Mexican or Nigerian, Catholic or Muslim, speak French or Arabic or Japanese, whether you lived two millennia ago or will live two millennia in the future, a circle is round and two plus two is four.
The reliable, seemingly timeless character of mathematical knowledge has allowed us to build our societies. We count, plan, and draw diagrams. From water and electrical supplies; to architecture, the internet, and road-building; to financial, insurance, and market projections; and even to electronic music, mathematics is the invisible plumbing of modern society. We normally take it for granted, and we don’t notice it until the pipes burst. However, mathematical models are only as good as their assumptions. When those assumptions are faulty or corrupted by wishful thinking or greed, as they were in the recent financial crisis, our whole world fails with them.
PHYSICISTS, ON THE OTHER hand, are interested in discovering the basic laws that govern the universe. Theoretical physics is the application of mathematics to the fundamental description of reality. It is the gold standard of mathematical science, and our most powerful internal model of the world.
Again we return to late sixteenth- and early seventeenth-century Renaissance Italy, where Galileo Galilei took the first steps towards founding the field of physics. He realized that mathematics, when used in conjunction with careful experiments and accurate measurements, could provide a powerful description of the real world. Mathematics allows us to form conceptions of the world far beyond our everyday experience, to delve deeply into our models of reality, and to search for contradictions in our descriptions, which often suggest new phenomena. But in the end, the only true test of the correctness or falsity of our ideas is, as Galileo first fully appreciated, experiment and observation.
So, through a combination of logical reasoning, observation, and painstaking experiment, Galileo developed physics as a new, universal discipline. His experiments with soot-blackened balls rolling on inclined planes, and his observations of the moons of Jupiter and the phases of Venus, provided the vital clues to ruling out the ancient Ptolemaic picture, in which the Earth lay at the centre of the universe, and establishing instead a Copernican universe with the sun at the centre of the solar system. That was the first step on the path to a Newtonian universe.
Galileo was a prodigious inventor: of a geometrical compass, a water clock, a new type of thermometer, telescopes, and microscopes, all the instruments that allowed him to accurately observe and measure the world. He risked his life in pursuit of his ideas. His notion of universal mathematical laws of motion, which could be uncovered by reason, was very threatening to religious authority. When his observations supported the Copernican, heliocentric picture of the solar system and directly contradicted the views of the Catholic Church, he was tried by the Inquisition and was forced to recant and then to live under house arrest for the rest of his life. He used the period of his imprisonment to write his final masterpiece, Two New Sciences, which laid the ground for Newton’s theory of mechanics. These achievements inspired Albert Einstein to call Galileo “the father of modern physics — indeed of modern science.”19
The combination of mathematical theory and real experience, pioneered by Galileo, drove the development of every modern technology, from electronics to construction engineering, from lasers to space travel. And it opened up the universe to our understanding, from far below the size of an ato
m right up to the entire visible cosmos. To be sure, there are still great gaps in our knowledge. But when we look at how rapidly and how far physics has come since Galileo, who can say what its future limits are?
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MY OWN ATTRACTION TO maths and physics began when I was about seven years old. Upon my father’s release from prison in 1966, he realized he was in serious danger of rearrest. So he escaped across the border with Botswana and made his way overland to Kenya. After a considerable delay, my mother, my two older brothers, and I were granted permission to join him, under the condition that we never return to South Africa. However, as a refugee, my father had no passport and could not obtain employment. Neighbouring Tanzania, under President Julius Kambarage Nyerere, was far more strongly committed to supporting the struggle against apartheid. So, after a brief stay in Nairobi, we were granted asylum in Tanzania and moved to Dar es Salaam, the country’s largest city.
I was sent to a government school, where I had a wonderful Scottish teacher named Margaret Carnie. She encouraged me to undertake many scientific activities, like making maps of the school, building electric motors, and playing around with equations. She was passionate about teaching, extraordinarily supportive and not at all prescriptive, and she gave me a lot of freedom. Most of all, she believed in me.