How Numbers Work Read online

  How Numbers Work

  Discover the strange and beautiful world of mathematics

  NEW SCIENTIST

  Contents

  Series introduction

  Contributors

  Guest contributors

  Introduction

  1 What is mathematics?

  2 Zero

  3 Infinity

  4 Prime numbers

  5 π, ϕ, e and i

  6 Probability, randomness and statistics

  7 The greatest problems of maths

  8 Everyday maths

  9 Numbers and reality

  Conclusion

  Forty-nine ideas

  Glossary

  Picture credits

  Series introduction

  New Scientist’s Instant Expert books shine light on the subjects that we all wish we knew more about: topics that challenge, engage enquiring minds and open up a deeper understanding of the world around us. Instant Expert books are definitive and accessible entry points for curious readers who want to know how things work and why. Look out for the other titles in the series:

  The End of Money

  How Your Brain Works

  The Quantum World

  Where the Universe Came From

  How Evolution Explains Everything about Life

  Why the Universe Exists

  Your Conscious Mind

  Machines That Think

  Scheduled for publication in 2018:

  Human Origins

  A Journey through the Universe

  This is Planet Earth

  Contributors

  Editor: Richard Webb, Chief Features Editor at New Scientist

  Instant Expert Series Editor: Alison George

  Instant Expert Editor: Jeremy Webb

  Guest contributors

  Richard Elwes wrote parts of the chapter on infinity and ‘The algorithm that runs the world’ in Chapter 8. He is a writer, teacher and researcher in mathematics, and a visiting fellow at the University of Leeds, UK. His latest book is Chaotic Fishponds and Mirror Universes (2013).

  Vicky Neale wrote about the twin primes conjecture in Chapter 5. She is the Whitehead Lecturer at the Mathematical Institute and Balliol College at the University of Oxford, UK, and author of Closing the Gap: The Quest to Understand Prime Numbers (2017).

  Regina Nuzzo wrote the section on frequentist and Bayesian statistics in Chapter 6. She is a writer, statistician and professor at Gallaudet University in Washington, DC.

  Ian Stewart wrote the sections on the empty set in Chapter 2 and the maths of elections in Chapter 8, as well as the conclusion on what makes maths special. He is Emeritus Professor at the University of Warwick, UK, and the author of numerous books on mathematics, the latest of which is Calculating the Cosmos (2017).

  Thanks also to the following writers and editors:

  Gilead Amit, Anil Ananthaswamy, Jacob Aron, Michael Brooks, Matthew Chalmers, Catherine de Lange, Marianne Freiberger, Amanda Gefter, Lisa Grossman, Erica Klarreich, Dana Mackenzie, Stephen Ornes, Timothy Revell, Bruce Schechter, Rachel Thomas, Helen Thomson.

  Introduction

  In 2014 the Iranian Maryam Mirzakhani became the first woman to win the highest honour of mathematics, the Fields Medal. To her, mathematics often felt like ‘being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks’. ‘With some luck’, she added, ‘you might find a way out.’

  Mirzakhani, who died in July 2017 at the age of 40, ventured deeper into the mathematical jungle than most. This New Scientist Instant Expert book is for those wandering on the periphery looking for a way in.

  Willingly or unwillingly, most of us have gleaned some idea of what the mathematical terrain looks like. There are symbols, equations and geometrical shapes. There are problems with right answers, truths that are seemingly universal, and proofs that are logically watertight. Above all, there are numbers.

  But how does it all hang together? What makes numbers and mathematics special – and some numbers and bits of mathematics more special than others? This is too broad a subject to hope to give a comprehensive overview but, by drawing on the thoughts of leading researchers and the very best of New Scientist, we hope to build up a picture.

  After a brief introduction to the nature and scope of mathematics itself, we start where it all started: with the fascinating properties of numbers. We look at zero and infinity, the prime numbers, and at inescapable oddball numbers such as the ‘transcendentals’ e and π and the imaginary unit i. Via a brief diversion through the problems of probability and statistics, we arrive at the cutting edge of modern mathematical methods and examples of how they are applied in some unexpected areas of our lives, before considering the deepest problem of all: how exactly does mathematics relate to reality?

  For many on the outside, the wonder of mathematics lies in the way it seems to be a universal language that helps us better understand the world. Many practitioners would agree, but they add that its beauty lies in how, from simple beginnings and using only the tools of purest abstract logic, you can create worlds that seem to transcend our own.

  Mirzakhani studied the geometry of a thing called moduli space, which can be envisaged as a universe in which every point is itself a universe. She described the number of ways a beam of light can travel a closed loop in a two-dimensional universe – an answer you cannot find by staying in your ‘home’ universe, but only by zooming out and navigating an entire multiverse.

  That’s further than most of us can aspire to go. But I hope that this book will provide you with your own satisfying journey of mathematical discovery – a way in, at the very least.

  Richard Webb, Editor

  1

  What is mathematics?

  What does mathematics consist of? Is it an invention or a discovery? Does it come naturally to us, or must we learn it? When it comes to the true character of mathematics, many questions remain unresolved…

  The pillars of mathematics

  For most of us, mathematics means numbers. The manipulation of numbers is certainly where humanity’s mathematical journey started. But we have built a formidable, far more extensive edifice on that foundation.

  Arithmetic is what we all know: addition, subtraction, division, multiplication and so on. The ability to understand and manipulate numbers in the abstract was the bit of mathematics that we began to develop first, in a formal way as much as six millennia ago. But watertight logical rules of arithmetical manipulation were devised only from the mid-nineteenth century onwards, with the development of set theory.

  You can read more about the development of set theory in Chapters 2 and 3 on zero and infinity, and about numbers themselves in Chapters 4 and 5, which deal with the prime numbers, the atoms of the number system, and other particularly intriguing numbers, π, ϕ, e and i.

  Probability theory, developed from the seventeenth century onwards, builds on the rules of arithmetic to create its own set of laws for dealing with the chance and uncertainty that is everywhere around us in the world. Originally applied to games of chance, it gained new significance in the twentieth century with the application of statistical methods to analyse large sets of data, and also with the development of quantum theory, which suggests that reality itself is ruled by chance.

  Probability and statistics are the subject of Chapter 6, and you will find more on the connection with quantum theory in Chapter 9 on the relationship between numbers and reality.

  Beyond the manipulation of numbers is ‘higher’ mathematics, with three main pillars:

  1 Geometry is probably the most familiar. It begins with a sense of space: formal geometry codifies the principles for describing how things in space can be related to each other, for example to form a triangle. But it’s a static description of things.

  2 Analysis is the second pillar of higher mathematics. It deals with things that move and change with time. It notably includes integral and differential calculus, together with many other sophisticated variations on the theme.

  3 Algebra allows us to represent and manipulate knowledge in terms of numbers, symbols and equations, and as such is the broadest pillar of formal higher mathematics. It encompasses esoteric subjects such as group theory (the study of groups, where groups are sets of elements which satisfy certain properties), graph theory (which studies how things are interconnected, such as the computers on the Internet or neurons in the brain) and topology (the mathematics of shapes that can be deformed continuously, without breaking and re-forming them).

  Each of these sprawling subjects would be worthy of a book in its own right, but you will gain a flavour of the insights they give and problems they present throughout this book, and particularly in Chapters 7 and 8, dealing with the great unsolved problems of mathematics and the application of mathematics to problems in the everyday world.

  Before all that, though, we turn our attention to one of the hardest philosophical questions of mathematics: where does it all come from?

  Mathematics: invention or discovery?

  Whenever we run to catch a ball or dart through heavy traffic, we do mathematics – entirely unconsciously. That makes sense. The natural world is a complex and unpredictable place. Habitats change, predators strike, food runs out. An organism’s survival depends on its ability to make sense of its surroundings, whether by counting down to nightfall, triangulating the quickest way out of danger, or assessing the spots most likely to have food. That means doing mathematics: manipulating numbers, assessing position
and movement using trigonometry and calculus, and weighing up probabilities.

  This points to a truth that is both profound and difficult to pin down: reality is in some sense mathematical. Karl Friston, a computational neuroscientist and physicist at University College London, observes that there is simplicity, parsimony and symmetry in mathematics. If you were treating it as a language, it would win hands down over all other ways of describing the world.

  One immediate consequence is that we are not the only organisms to have ‘mathematical’ abilities. From dolphins to slime moulds, organisms across the evolutionary tree seem to analyse the world mathematically, deciphering its patterns and regularities in order to survive. If the environment unfolds according to mathematical principles, Friston argues, then the anatomy of the brain must also recapitulate those mathematical principles.

  But human brains, with their seemingly unique ability for symbolic representation and abstract thinking, have taken that further. We have made mathematics a conscious activity that, to a greater or lesser extent, must be learned. The exact moment when culture transformed our instinctive senses into a recognizable, conscious mathematical ability is lost in the mists of time, but in the 1970s archaeologists investigating the Border Cave on the western scarp of the Lebombo Mountains in South Africa discovered a series of bones with notches, including the fibula of a baboon etched with 29 such marks. Dated to some 40,000 years ago, they seem to have been an aid to counting – the oldest evidence we have for an emerging conscious understanding for representing and manipulating numbers.

  Systems for counting and measuring reached new heights in the fourth millennium BCE, in the sophisticated Mesopotamian culture of the Tigris–Euphrates valleys. Here, in what today we call Iraq, the first consistent symbolic representations of numbers were used to keep track of days, months and years, to measure areas of land and amounts of grain, and perhaps even to record weights. As humans took to the seas and studied the skies, we began developing numerical methods for navigation and for tracking celestial objects.

  This conscious mathematics was a product of cultural necessity: an invention that helped to make sense of the world and do things such as trade and travel. With the help of mathematical tools, we have over the past 6,000 years built an immense pyramid of mathematical knowledge. Ancient Greek mathematicians such as Euclid formalized rules of geometry (see Figure 1.1) in around 300 BCE; Hindu and Arabic mathematicians a thousand or so years later began creating the number systems we are familiar with today and developing tools for the symbolic representation and manipulation of numerical quantities: algebra.

  FIGURE 1.1 Euclid’s Elements, seen here in its first printed edition from 1482, was a seminal primer of geometry.

  But even the great blooming of modern mathematics in the seventeenth-century Age of Enlightenment served only to further our understanding of things within our experience. The calculus of Isaac Newton and Gottfried Leibniz, for example, allowed us to calculate the trajectory of moving bodies on Earth and in the heavens. The coordinate system invented by René Descartes provided an algebraic representation of geometric shapes. Emerging theories of chance and probability helped us to deal with uncertainty and lack of information.

  But mathematics has since expanded into ever more abstract domains, and told us things we could not have hoped to understand by observation alone. As it has done so, it has assumed less and less the character of an invention, a product purely of human brains, and more and more that of a revealed truth, a discovery waiting to be made.

  When, for example, at the turn of the twentieth century, the mathematician David Hilbert extended the algebra of conventional 3D space to one with an infinite number of dimensions, it seemed a purely abstract development with little application to the real world. But a couple of decades later it turned out that the state of a quantum particle could best be described using such a ‘Hilbert space’. The underlying mathematics remains key to our attempts to make sense of quantum mechanics – a theory of which we have as yet no intuitive physical understanding.

  To many physicists today, the success of mathematics as a language to describe reality speaks to a prime role it has in the organization of the universe. Others would not go so far, arguing that we still just invent mathematics to satisfy our need to describe the world differently in different contexts.

  Consider the following sequence of events. The most famous of the geometrical axioms that Euclid laid down is that parallel lines never meet. But on the curved surface of the globe, for instance, parallel lines do meet – all lines of longitude meet at the North and South Poles. The exploration by German mathematician Bernhard Riemann and others of such non-Euclidean geometries led to the discovery – or invention – of a rich vein of mathematics that Einstein would use to formulate his general theory of relativity. The warping of space-time by massive bodies in general relativity is dictated by the rules of Riemann’s geometry, not Euclid’s.

  For Andy Clark, a cognitive philosopher at the University of Edinburgh, the universe is filled with all kinds of patterns and regularities and ways of behaving. So any creature that wants to build a mathematics is going to have to build it on top of regularities that are constraining the behaviour of the stuff they encounter. Follow this logic, and if mathematics is an organizing principle, it is one we impose on the world.

  Gödel’s incompleteness theorems, an ironically rather precise bit of mathematics developed by the Austrian mathematician Kurt Gödel in the 1930s, show that there will always be questions that mathematics will never have the tools to answer (see Chapter 3). That also suggests that it is too early for us to make any sweeping statements about mathematics being a universal truth. We’ll return to these thoughts at the end of the book, but in the meantime we are far from what mathematicians would regard as a proof one way or the other.

  Our mathematical brains

  We all have an innate ability to do a form of mathematics unconsciously, to navigate our way around the world and survive. But the origin of our ability to manipulate numbers is a more intriguing case. Is it learned, or does it tap into something inbuilt? Counting things, after all, has no obvious survival value.

  In 1997 the cognitive psychologist Stanislas Dehaene proposed that we are born with a conscious sense of number, in the same way that we are conscious of colours: evolution had endowed humans and other animals with ‘numerosity’, an ability to immediately perceive the number of objects in some pile of objects. Three red marbles would produce a sense of the number three just as they would produce a sense of the colour red.

  Evidence quickly started to accumulate in support of this ‘nativist’ view of numerical ability, with experiments showing, for example, that six-month-old infants could distinguish between arrays of different numbers of dots. Other studies suggested that humans come with a built-in mental number line – that we instinctively represent numbers spatially, with values increasing from left to right. Experiments appearing to demonstrate that some other animals, from chimps to chickens, can distinguish small numbers seemed to provide supplementary evidence.

  The development of numbers

  The development of numbers

  But, before long, some researchers grew uncomfortable with the conclusions of these studies. The subjects might, for example, be distinguishing arrays of dots based not on number but on other attributes such as their spatial distribution or area of coverage. Tali Leibovich of the University of Haifa in Israel points out that it makes sense that we would have evolved to assess these things: if you are hunting or being hunted, you need to act quickly, which would mean using all available cues.

  Soon, a different hypothesis emerged: instead of being born with an innate sense of number, we are born with a sense for quantities such as size and density that are correlated with the numbers of things, and our conscious mathematical ability builds on this. ‘It takes time and experience to develop and understand this correlation,’ says Leibovich.

  More refined tests in children tend to support this view. Children younger than about four years of age cannot understand that five oranges and five watermelons have something in common: the number five. To them, a bunch of watermelons simply represents more ‘stuff’ than the same number of oranges.