by Bill Bryson
What, then, is our place in the universe as currently understood? As far as we can tell, our planetary system, galaxy and galactic environs are unexceptional out as far as our most powerful instruments can penetrate, over twelve billion light years. But our biological situation remains unresolved. The universe might be teeming with life, or it may turn out that life is very rare – intelligent life more so. It is even conceivable that we are alone in the vastness of space. If so, history will have turned a curious full circle. Before Copernicus, people believed that humans and their planet occupied pole position in the universe. It may yet be that we are privileged after all, in being the only place in the universe with intelligent life.
Is that as far as we can take the Copernican principle, to the edge of the observable universe? As I have commented, each new advance in astronomy has unveiled a universe even larger and more majestic than previously realised, but with instruments like the Hubble Space Telescope we are approaching a fundamental limit due to the finite speed of light. When we see a galaxy 12 billion light years away, we see it as it was 12 billion years ago. Light can have travelled at most 13.7 billion light years since the big bang, so if that explosive event represented the true origin of the universe, then there is an ultimate horizon beyond which we cannot see. That does not mean the universe comes to an end there, any more than a horizon at sea signals the edge of the world. But it does mean we cannot directly observe what lies beyond. An uncritical application of the Copernican principle would suggest that if by some magic we could be transported across the horizon we would find a region of the universe that looked much the same as our region, with stars, galaxies and galactic clusters uniformly distributed on the largest scale of size. But inevitably this raises the question of how far we can extrapolate. Does this pattern continue to infinity, or is there some variation?
The attempt to construct proper mathematical models of the universe based on the best understanding of gravitation began shortly after Einstein published his general theory of relativity in 1915. For many decades the default assumption was that the Copernican principle applied all the way to infinity (it is called the cosmological principle when applied to gravitational models of the universe). But in the 1970s this conventional wisdom was challenged. The basis for the challenge was the development of a theory of the big bang based on the application of quantum mechanics to the very early stages of the universe. Quantum mechanics is normally reserved for microscopic systems like atoms and molecules, but the theory predicts that, at a sufficiently early time, it would affect the evolution of the universe too. That time is about a hundred trillion-trillion-trillionths of a second after the big bang. According to some variants of the theory, there would not be a single big bang, but a countless number of them scattered randomly throughout space and time. Each quantum event would nucleate a universe with a big bang, ‘like bubbles in an uncorked bottle of champagne’, to use the words of the physicist Leonard Susskind. The space between the bubbles would expand so rapidly that, even though the bubbles themselves expand, they would rarely intersect. Our own universe would be just one of those bubbles. The entire collection is known as the multiverse. In the most popular multiverse theory, the size of the bubbles is stupendous – about 1010,000,000,000 km across. So once again, the scale of the universe has leapt dramatically, but by a far larger factor than the jump from pre-Copernican cosmology to the time of Hubble. Now we confront the same Copernican principle on a mega-scale: do we live in a typical bubble? Will the other bubbles be similar to ours?
The evidence from theory suggests no. Physicists are convinced that many features of the laws of physics, such as the masses of subatomic particles, the nature and number of forces, and the density of dark energy (the mysterious stuff that seems to be making the expansion of our universe accelerate) are ‘frozen accidents’ locked in when the universe cooled from the searing heat of the big bang. If the experiment were done again, so to speak, the masses and forces would come out differently; there might even be a different number of spatial dimensions. Einstein once famously expressed his distaste for quantum mechanics by declaring that ‘God does not play dice with the universe’. In the multiverse theory He plays dice with universes (I am tempted to say He plays at randomly blowing bubbles). Taking a God’s-eye-view, the multiverse is a patchwork quilt, featuring bubble universes of all hues and textures, distributed across a fantastic range of possibilities. What we had taken to be universal immutable laws of physics turn out to be more like ‘local bylaws, valid only in our cosmic patch’, to use Martin Rees’ evocative description.
A key feature of the multiverse’s cosmic smorgasbord is that only a tiny fraction of bubble universes will possess the right laws of physics to permit life and observers to arise. Many prerequisites needed for life, such as abundant carbon, stable stars and a universe neither too hot or chaotic, but cool and inhomogeneous enough to permit galaxies to form, depend very sensitively on the precise values of the parameters that characterise the laws and the initial conditions of the quantum universe-nucleation process. The ‘Goldilocks enigma’ – why our universe’s laws and initial conditions are, amazingly, just right for life – has been a source of puzzlement for a long time. The multiverse theory could explain what otherwise looks suspiciously like a cosmic fix, in terms of an observer selection effect. It is no surprise that we find ourselves living in one of those very rare universes that have bio-friendly laws; we obviously could not inhabit a bio-hostile one.
With the multiverse theory – which, it has to be cautioned, remains extremely speculative and hard to test – the Copernican principle decisively fails. Although we are most likely living in a typical bio-friendly bubble universe, the overall number of life-permitting bubbles is an infinitesimal fraction of the whole multiverse. Earth’s address within our bubble universe might well be typical, but our cosmic coordinates in the broader multiverse place us at a very exceptional location indeed.
15 IAN STEWART
BEHIND THE SCENES:THE HIDDEN MATHEMATICS THAT RULES OUR WORLD
Ian Stewart FRS is Professor of Mathematics at the University of Warwick and a leading populariser of mathematics. He has published more than 80 books including From Here to Infinity, Nature’s Numbers and The Collapse of Chaos. Recent popular science books include Why Beauty is Truth, Letters to a Young Mathematician, The Magical Maze and the series The Science of Discworld (with Terry Pratchett and Jack Cohen). His most recent book is Professor Stewart’s Cabinet of Mathematical Curiosities. He has also written the science fiction novels Wheelers and Heaven (with Jack Cohen). Among his many other popular writings, he wrote the monthly ‘Mathematical Recreations’ column of Scientific American for ten years.
THE STUFF SCIENCE ALLOWS US TO MAKE IS VISIBLE. BUT THE WAYS APPLIED INTELLIGENCE ALLOWS IT TO DO WHAT IT DOES REMAIN HIDDEN FROM MOST OF US – WHEN IT INVOLVES MATHEMATICS. SOMETIMES, AS IAN STEWART REVEALS, IT IS EVEN HIDDEN FROM THE PEOPLE WHO BUILD THE THINGS WHICH EMBODY THE MATHS.
HOW IMPORTANT IS MATHEMATICS IN TODAY’S WORLD?
The role of most sciences is relatively obvious, but mathematics is far less visible than engineering or biology. However, this lack of visibility does not imply that mathematics has no useful applications. On the contrary, mathematics underpins much of today’s technology, and is vital to virtually all areas of human activity. To explore how this has happened, and explain why it has gone unnoticed, I’m going to look at two of the great historical figures in British mathematics – both Fellows of the Royal Society – and trace some of the practical consequences of their work. Along the way, we’ll see why mathematics is so important, and why hardly anyone outside the subject seems to be aware of that.
My story begins with a strange event, which took place on 4 January 2004, on Mars. A Martian wandering around near Gusev Crater on that particular day would have undergone a life-changing experience. First, a streak of fire high in the sky would have heralded the arrival of an alien artefact, descending rapidly beneath a hemi
sphere of fabric. Then, as the artefact neared the ground, the fabric would have torn away, allowing it to fall the final hundred metres. And bounce. In fact, it bounced twenty-seven times before finally coming to rest. It would certainly have been a sight to remember.
The bouncy visitor was Mars Exploration Rover A, otherwise known as Spirit. After a journey of 487 million kilometres it entered the Martian atmosphere at a speed of 19,000 kilometres per hour. It was still travelling at a healthy 50 kilometres per hour a few seconds before impact when its airbags inflated and it made its touchdown. Spirit and its companion Opportunity have now spent more than four years exploring the surface of Mars, nearly twenty times as long as originally planned, leading to a wealth of new scientific information about Earth’s sister planet. They may not have finished yet.
Much of the credit for this stunning success must go to NASA’s engineers and managers, but other disciplines were also essential – among them, mathematics. The spacecraft’s trajectories were calculated using Newton’s laws of motion and gravity; Einstein’s later refinements were not needed. Isaac Newton was elected a Fellow of the Royal Society in 1672, twelve years after the Society was founded. His role in the development of space travel is not hard to identify, even though he died 240 years before the first Moon landing. Less obvious is the influence of a Fellow from the Victorian era, George Boole, whose pioneering ideas in logic and algebra proved fundamental to computer science. His influence can be detected in the error-correcting codes that made it possible for the Rovers (and most other space missions) to send images and scientific data back to Earth. Mathematics, both ancient and modern, is deeply embedded in today’s science, and makes vital contributions on a daily basis to many aspects of human society.
The importance of mathematics in the space programme should be evident even to a casual observer. Yet when the Rovers landed, and the American mathematician Philip Davis pointed out that the mission ‘would have been impossible without a tremendous underlay of mathematics’ – so tremendous, in fact, that ‘it would defy the most knowledgeable historian of mathematics to discover and describe all the mathematics that was involved’ – he found it necessary to add that ‘The public is hardly aware of this.’
This remark was an understatement. In 2007 two Danes with postgraduate mathematics degrees, Uffe Jankvist and Björn Toldbod, decided to uncover the hidden mathematics in the Mars Rover programme. They visited NASA’s Jet Propulsion Laboratory at Pasadena, which ran the mission, and discovered that it is not only the general public that lacks awareness of the mathematics used in the Rover mission. Many of the scientists most intimately involved were also unaware of the mathematics being used. Some denied that there was any.
‘We don’t do any of that,’ said one. ‘We don’t really use any abstract algebra, group theory, and that sort.’
‘Except in the channel coding,’ one of the Danish mathematicians pointed out.
‘They use abstract algebra and group theory in that?’
‘The Reed–Solomon codes are based on Galois fields.’
‘That’s news to me. I didn’t know that.’
This story is fairly typical. Few people are aware of the mathematics that makes their world work. Indeed, few are aware that mathematics is involved in their world at all. But – as the history of the Royal Society exemplifies – mathematics has long been central to science, and science has long been a major driving force for social change.
What causes this lack of awareness of the importance of mathematics in the modern world? One of the main reasons, as the NASA story shows, is that you don’t have to know any mathematics, or even be aware of its existence, to use the technology that it enables. This is entirely sensible – you don’t need to understand computer programming to buy CDs over the Internet, and you don’t need a degree in engineering to drive a car. However, most computer users are aware that someone had to write the software, and most drivers realise that someone had to design and build the car. With mathematics, it seems to be different.
Why? The story of the Mars Rovers is instructive. JPL scientists did not realise how deeply mathematics was involved in the Rover mission because the mathematical techniques were built into dedicated computer chips and programs. The resulting hardware and software carried out the necessary calculations without human intervention. Moreover, most of the chips and software were designed and manufactured by external subcontractors.
In actual fact, the Rover mission rested on a huge variety of mathematical techniques. These included dynamical systems and numerical analysis to calculate and control the spacecraft’s trajectory on its way to Mars, signal processing methods to compress data and eliminate transmission errors caused by electrical interference, even the design and deployment of the airbags. These techniques did not come into being overnight, and they were not, initially, developed with the space programme in mind. The work of Newton makes this very clear.
Newton’s father was a Lincolnshire farmer, who died three months before his son was born. The boy did not impress some of his schoolteachers, who reported that he was idle and inattentive, but he did impress his headmaster, who persuaded Isaac’s mother to send him to university. At Cambridge he studied law, but he also read books on physics, philosophy and mathematics. In 1665 the university was closed because of plague, and he returned to Lincolnshire. There, in a few years, he made huge advances in several areas of mathematics and physics, which led to his election as a Fellow of Trinity College.
Newton is famous for many things – his laws of motion, calculus (also discovered by Gottfried Wilhelm Leibniz), the beginnings of numerical analysis. All of this work leaves fingerprints on the Mars Rover mission, but the most significant is the law of gravitation. Every body in the universe, Newton declared, attracts every other body with a force that is proportional to their masses and inversely proportional to the square of the distance between them. When coupled to his laws of motion, the law of gravitation provided accurate descriptions of the motion of the assorted planets and moons of the solar system, and much more. It explained the curious way in which the Moon wobbled on its axis, and the paths of comets. It made the future of the solar system predictable, millions of years ahead.
Newton’s motivation was ‘natural philosophy’, the scientific study of Nature. If he had practical objectives in mind, they were related to things like navigation, and were secondary to understanding what he called ‘the system of the world’, which was the subtitle to his epic Principia Mathematica (Mathematical Principles of Natural Philosophy). At that time, the idea that humans might travel to the Moon was considered absurd, when anyone considered it at all. Yet such is the power of mathematics that when spacecraft began to leave the Earth in the 1960s, the tools needed to calculate their orbits and plan their re-entry trajectories through the atmosphere were those developed by Newton and his successors. In particular, since the law of gravitation applies to every particle of matter in the universe, it must apply to spacecraft.
NATURAL PHILOSOPHY HAS BORNE FRUIT AS TECHNOLOGY
Once pointed out, it’s no great surprise that esoteric mathematics can be used in esoteric applications like Martian space probes, even if no one notices … But what does that have to do with the everyday life of the ordinary citizen? Next time you listen to a CD while driving along the motorway in your car, and hit a bump, you may care to ask yourself why the CD player skips tracks only if it’s a really big bump – big enough to risk damaging your wheel. After all, a CD player is an extremely delicate device, with a tiny laser that hovers a few millionths of a metre away from a plastic disc covered in tiny dots.
The answer goes back to George Boole and the other nineteenth-century mathematicians who founded modern abstract algebra. Boole also hailed from Lincolnshire, being born in Lincoln in 1815; his father was a cobbler who was also interested in making scientific instruments, and his mother was a lady’s maid. He did not take a university degree, but his talent for mathematics attracted attention, and in 184
9 he became Professor of
Mathematics at Queen’s College, Cork. His most significant work was his 1854 book An Investigation of the Laws of Thought. In it, he reformulated logic in terms of algebra – but a very strange kind of algebra. Most of the familiar algebraic rules, such as x+y = y+x, are valid in Boole’s logical realm, but there are some surprises, such as 1+1 = 0. Here 1 means ‘true’, 0 means ‘false’, and x+y means what computer scientists now call ‘exclusive or’: either x is true, or y is true, but not both. The first formula says that this statement does not depend on the order in which the two statements x and y are considered. The second says that if x and y are both true, then x+y is false – because the definition of + includes the requirement ‘not both’. More elaborate algebraic laws, such as (x+y)z = xz+yz, are also true in Boole’s system; now the product xy means ‘x and y’. So Boole’s algebraic rules follow from sensible logical ones.
It is a striking and surprising discovery. Logic, previously thought of as being more basic than mathematics, can actually be reduced to mathematics. And the reduction is so natural that the algebra of logic is almost the same as traditional algebra. The new rules do make a difference, but you soon get used to that. Boole knew he was on to something important, but it took a while for most mathematicians to appreciate it. ‘Boolean algebra’ really took off when digital computers started to appear. Computers are basically logic engines, and Boole is widely recognised as a founder of theoretical computer science.
