Seeing Further (38 page)

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.

T
HE 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.

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 hemisphere 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.

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 1849 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.

The link to digital computation is natural, but Boole’s influence runs deeper. He was one of the first to realise that algebra need not be about numbers alone: it can be about any mathematical concepts or structures that can be manipulated symbolically according to a fixed system of rules. Boole was one of the earliest thinkers in a long tradition that includes the tragic figure of Évariste Galois, killed in a duel shortly before his twenty-first birthday. Today’s abstract algebra, with its key concepts of groups, rings, fields and vector spaces, represents the fruits of their early labours.

These ideas, if I were to explain them in any detail, would seem abstract and impractical – formal games played with symbols, to no clear purpose.
They look like that because they operate on a structural level and focus on deep generalities. But behind the scenes, the abstract algebra that Boole pioneered has taken over most areas of mathematics, because it organises concepts and provokes new ideas. The resulting mathematics can be found, embodied in computer chips, inside most of today’s electronic gadgets: CDs, DVDs, digital TVs, mobile phones, iPods, Nintendo Wiis, BlackBerries, SatNav, digital cameras …