What's the Use? Read online

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  Originally published in 2021 by Profile Books in the United Kingdom

  First US Edition: August 2021

  Published by Basic Books, an imprint of Perseus Books, LLC, a subsidiary of Hachette Book Group, Inc. The Basic Books name and logo is a trademark of the Hachette Book Group.

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  Typeset in Sabon by MacGuru Ltd

  Library of Congress Control Number: 2021930966

  ISBNs: 978-1-5416-9948-9 (hardcover); 978-1-5416-9949-6 (ebook)

  E3-20210701-JV-NF-ORI

  Contents

  Cover

  Title Page

  Copyright

  1 Unreasonable Effectiveness

  2 How Politicians Pick Their Voters

  3 Let the Pigeon Drive the Bus

  4 The Kidneys of Königsberg

  5 Stay Safe in Cyberspace

  6 The Number Plane

  7 Papa, Can You Multiply Triplets?

  8 Boing!

  9 Trust Me, I’m a Transform

  10 Smile, Please!

  11 Are We Nearly There Yet?

  12 De-Ising the Arctic

  13 Call the Topologist

  14 The Fox and the Hedgehog

  Discover More

  Notes

  Picture Credits

  About the Author

  Also by Ian Stewart

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  1

  Unreasonable Effectiveness

  The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure even though perhaps also to our bafflement, to wide branches of learning.

  Eugene Wigner, The Unreasonable Effectiveness of Mathematics in the Natural Sciences

  What is mathematics for?

  What is it doing for us, in our daily lives?

  Not so long ago, there were easy answers to these questions. The typical citizen used basic arithmetic all the time, if only to check the bill when shopping. Carpenters needed to know elementary geometry. Surveyors and navigators needed trigonometry as well. Engineering required expertise in calculus.

  Today, things are different. The supermarket checkout totals the bill, sorts out the special meal deal, adds the sales tax. We listen to the beeps as the laser scans the barcodes, and as long as the beeps match the goods, we assume the electronic gizmos know what they’re doing. Many professions still rely on extensive mathematical knowledge, but even there, we’ve outsourced most of the mathematics to electronic devices with built-in algorithms.

  My subject is conspicuous by its absence. The elephant isn’t even in the room.

  It would be easy to conclude that mathematics has become outdated and obsolete, but that view is mistaken. Without mathematics, today’s world would fall apart. As evidence, I’m going to show you applications to politics, the law, kidney transplants, supermarket delivery schedules, Internet security, movie special effects, and making springs. We’ll see how mathematics plays an essential role in medical scanners, digital photography, fibre broadband, and satellite navigation. How it helps us predict the effects of climate change; how it can protect us against terrorists and Internet hackers.

  Remarkably, many of these applications rely on mathematics that originated for totally different reasons, often just the sheer fascination of following your nose. While researching this book I was repeatedly surprised when I came across uses of my subject that I’d never dreamed existed. Often they exploited topics that I wouldn’t have expected to have practical applications, like space-filling curves, quaternions, and topology.

  Mathematics is a boundless, hugely creative system of ideas and methods. It lies just beneath the surface of the transformative technologies that are making the twenty-first century totally different from any previous era – video games, international air travel, satellite communications, computers, the Internet, mobile phones.1 Scratch an iPhone, and you’ll see the bright glint of mathematics.

  Please don’t take that literally.

  *

  There’s a tendency to assume that computers, with their almost miraculous abilities, are making mathematicians, indeed mathematics itself, obsolete. But computers no more displace mathematicians than the microscope displaced biologists. Computers change the way we go about doing mathematics, but mostly they relieve us of the tedious bits. They give us time to think, they help us search for patterns, and they add a powerful new weapon to help advance the subject more rapidly and more effectively.

  In fact, a major reason why mathematics is becoming ever more essential is the ubiquity of cheap, powerful computers. Their rise has opened up new opportunities to apply mathematics to real-world issues. Methods that were hitherto impractical, because they needed too many calculations, have now become routine. The greatest mathematicians of the pencil-and-paper era would have flung up their hands in despair at any method requiring a billion calculations. Today, we routinely use such methods, because we have technology that can do the sums in a split second.

  Mathematicians have long been at the forefront of the computer revolution – along with countless other professions, I hasten to add. Think of George Boole, who pioneered the symbolic logic that forms the basis of current computer architecture. Think of Alan Turing, and his universal Turing machine, a mathematical system that can compute anything that’s computable. Think of Muhammad al-Khwarizmi, whose algebra text of AD 820 emphasised the role of systematic computational procedures, now named after him: algorithms.

  Most of the algorithms that give computers their impressive abilities are firmly based on mathematics. Many of the techniques concerned have been taken ‘off the shelf’ from the existing store of mathematical ideas, such as Google’s PageRank algorithm, which quantifies how important a website is and founded a multibillion dollar industry. Even the snazziest deep learning algorithm in artificial intelligence uses tried and tested mathematical concepts such as matrices and weighted graphs. A task as prosaic as searching a document for a particular string of letters involves, in one common method at least, a mathematical gadget called a finite-state automaton.

  The involvement of mathematics in these exciting developments tends to get lost. So next time the media propel some miraculous new ability of computers to centre stage, bear in mind that hiding in the wings there will be a lot of mathematics, and a lot of engineering, physics, chemistry, and psychology as well, and that without the support of this hidden cast of helpers, the digital superstar would be unab
le to strut its stuff in the spotlight.

  *

  The importance of mathematics in today’s world is easily underestimated because nearly all of it goes on behind the scenes. Walk along a city street and you’re overwhelmed by signs proclaiming the daily importance of banks, greengrocers, supermarkets, fashion outlets, car repairs, lawyers, fast food, antiques, charities, and a thousand other activities and professions. You don’t find a brass plaque announcing the presence of a consulting mathematician. Supermarkets don’t sell you mathematics in a can.

  Dig a little deeper, however, and the importance of mathematics quickly becomes apparent. The mathematical equations of aerodynamics are vital to aircraft design. Navigation depends on trigonometry. The way we use it today is different from how Christopher Columbus used it, because we embody the mathematics in electronic devices instead of pen, ink, and navigation tables, but the underlying principles are much the same. The development of new medicines relies on statistics to make sure the drugs are safe and effective. Satellite communications depend on a deep understanding of orbital dynamics. Weather forecasting requires the solution of equations for how the atmosphere moves, how much moisture it contains, how warm or cold it is, and how all of those features interact. There are thousands of other examples. We don’t notice they involve mathematics, because we don’t need to know that to benefit from the results.

  What makes mathematics so useful, in such a broad variety of human activities?

  It’s not a new question. In 1959 the physicist Eugene Wigner gave a prestigious lecture at New York University,2 with the title ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’. He focused on science, but the same case could have been made for the unreasonable effectiveness of mathematics in agriculture, medicine, politics, sport… you name it. Wigner himself hoped that this effectiveness would extend to ‘wide branches of learning’. It certainly did.

  The key word in his title stands out because it’s a surprise: unreasonable. Most uses of mathematics are entirely reasonable, once you find out which methods are involved in solving an important problem or inventing a useful gadget. It’s entirely reasonable, for instance, that engineers use the equations of aerodynamics to help them design aircraft. That’s what aerodynamics was created for in the first place. Much of the mathematics used in weather forecasting arose with that purpose in mind. Statistics emerged from the discovery of large-scale patterns in data about human behaviour. The amount of mathematics required to design spectacles with varifocal lenses is huge, but most of it was developed with optics in mind.

  The ability of mathematics to solve important problems becomes unreasonable, in Wigner’s sense, when no such connection exists between the original motivation for developing the mathematics, and the eventual application. Wigner began his lecture with a story, which I’ll paraphrase and embellish slightly.

  Two former school classmates met up. One, a statistician working on population trends, showed the other one of his research papers, which began with a standard formula in statistics, the normal distribution or ‘bell curve’.3 He explained various symbols – this one is the population size, that one is a sample average – and how the formula can be used to infer the size of the population without having to count everyone. His classmate suspected his friend was joking, but he wasn’t entirely sure, so he asked about other symbols. Eventually he came to one that looked like this: π.

  ‘What’s that? It looks familiar.’

  ‘Yes, it’s pi – the ratio of the circumference of the circle to its diameter.’

  ‘Now I know you’re pulling my leg,’ said the friend. ‘What on earth can a circle have to do with population sizes?’

  The first point about this story is that the friend’s scepticism was entirely sensible. Common sense tells us that two such disparate concepts can’t possibly be related. One is about geometry, the other about people, for heaven’s sake. The second point is that despite common sense, there’s a connection. The bell curve has a formula, which happens to involve the number π. It’s not just a convenient approximation; the exact number really is good old familiar π. But the reason it appears in the context of the bell curve is far from intuitive, even to mathematicians, and you need advanced calculus to see how it arises, let alone why.

  Let me tell you another story about π. Some years ago we had the downstairs bathroom renovated. Spencer, an amazingly versatile craftsman who came to fit the tiles, discovered that I wrote popular mathematics books. ‘I’ve got a maths problem for you,’ he said. ‘I’ve got to tile a circular floor, and I need to know its area to work out how many tiles I’ll need. There was some formula they taught us…’

  ‘Pi r squared,’ I replied.

  ‘That’s the one!’ So I reminded him how to use it. He went away happy, with the answer to his tiling problem, a signed copy of one of my books, and the discovery that the mathematics he’d done at school was, contrary to his long-held belief, useful in his present occupation.

  The difference between the two stories is clear. In the second story, π turns up because it was introduced to solve exactly that kind of problem in the first place. It’s a simple, direct story about the effectiveness of mathematics. In the first story, π also turns up and solves the problem, but its presence is a surprise. It’s a story of unreasonable effectiveness: an application of a mathematical idea to an area totally divorced from that idea’s origins.

  *

  In What’s the Use? I’m not going to say much about reasonable uses of my subject. They’re worthy, they’re interesting, they’re as much a part of the mathematical landscape as everything else, they’re equally important – but they don’t make us sit up and say ‘Wow!’ They can also mislead the Powers That Be into imagining that the only way to advance mathematics is to decide on the problems and then get the mathematicians to invent ways to solve them. There’s nothing wrong with goal-oriented research of this kind, but it’s fighting with one arm tied behind your back. History repeatedly shows the value of the second arm, the amazing scope of human imagination. What gives mathematics its power is the combination of these two ways of thinking. Each complements the other.

  For instance, in 1736, the great mathematician Leonhard Euler turned his mind to a curious little puzzle about people taking walks across bridges. He knew it was interesting, because it seemed to require a new kind of geometry, one that abandoned the usual ideas of lengths and angles. But he couldn’t possibly have anticipated that in the twenty-first century the subject that his solution kick-started would help more patients get life-saving kidney transplants. For a start, kidney transplants would have seemed pure fantasy at that time, but even if they hadn’t, any connection with the puzzle would have seemed ridiculous.

  And who would ever have imagined that the discovery of space-filling curves – curves that pass through every point of a solid square – could help Meals on Wheels to plan its delivery routes? Certainly not the mathematicians who studied such questions in the 1890s, who were interested in how to define esoteric concepts like ‘continuity’ and ‘dimension’, and initially found themselves explaining why cherished mathematical beliefs can be wrong. Many of their colleagues denounced the entire enterprise as wrong-headed and negative. Eventually everyone realised that it’s no good living in a fool’s paradise, assuming that everything will work perfectly when in fact it won’t.

  It’s not just the mathematics of the past that gets used in this way. The kidney transplant methods rely on numerous modern extensions of Euler’s original insight, among them powerful algorithms for combinatorial optimisation – making the best choice from a huge range of possibilities. The myriad mathematical techniques employed by movie animators include many that go back a decade or less. An example is ‘shape space’, an infinite-dimensional space of curves that are considered to be the same if they differ only by a change of coordinates. They’re used to make animation sequences appear smoother and more natural. Persistent homology, another very rece
nt development, arose because pure mathematicians wanted to compute complicated topological invariants that count multidimensional holes in geometric shapes. Their method also turned out to be an effective way to ensure that sensor networks provide full coverage when protecting buildings or military bases against terrorists or other criminals. Abstract concepts from algebraic geometry – ‘supersingular isogeny graphs’ – can make Internet communications secure against quantum computers. These are so new that they currently exist only in rudimentary form, but they will trash today’s cryptosystems if they can fulfil their potential.

  Mathematics doesn’t just spring such surprises on rare occasions. It makes a positive habit of it. In fact, as far as many mathematicians are concerned, these surprises are the most interesting uses of the subject, and the main justification for considering it to be a subject, rather than just a rag-bag of assorted tricks, one for each kind of problem.

  Wigner went on to say that ‘the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and… there is no rational explanation for it.’ It is, of course, true that mathematics grew out of problems in science in the first place, but Wigner wasn’t puzzled by the subject’s effectiveness in areas it was designed for. What baffled him was its effectiveness in apparently unrelated ones. Calculus grew from Isaac Newton’s research on the motion of the planets, so it’s not greatly surprising that it helps us to understand how planets move. However, it is surprising when calculus lets us make statistical estimates of human populations, as in Wigner’s little story, explains changes in the numbers of fish caught in the Adriatic Sea during the First World War,4 governs option pricing in the financial sector, helps engineers to design passenger jets, and is vital for telecommunications. Because calculus wasn’t invented for any such purpose.

  Wigner was right. The way mathematics repeatedly turns up uninvited in the physical sciences, and in most other areas of human activity, is a mystery. One proposed solution is that the universe is ‘made of’ mathematics, and humans are just digging out this basic ingredient. I’m not going to argue the toss here, but if this explanation is correct it replaces one mystery by an even deeper one. Why is the universe made of mathematics?