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Have you ever wondered what humans did before numbers existed? How they organized their lives, traded goods, or kept track of their treasures? What would your life be like without them?
Numbers began as simple representations of everyday things, but mathematics rapidly took on a life of its own, occupying a parallel virtual world. In Are Numbers Real?, Brian Clegg explores the way that math has become more and more detached from reality, and yet despite this is driving the development of modern physics. From devising a new counting system based on goats, through the weird and wonderful mathematics of imaginary numbers and infinity, to the debate over whether mathematics has too much influence on the direction of science, this fascinating and accessible book opens the reader’s eyes to the hidden reality of the strange yet familiar entities that are numbers.
|Publisher:||St. Martin's Press|
|Product dimensions:||5.70(w) x 8.20(h) x 1.10(d)|
About the Author
BRIAN CLEGG is the author of Ten Billion Tomorrows, Final Frontier, Extra Sensory, Gravity, How to Build a Time Machine, Armageddon Science, Before the Big Bang, Upgrade Me, and The God Effect among others. He holds a physics degree from Cambridge and has written regular columns, features, and reviews for numerous magazines. He lives in Wiltshire, England, with his wife and two children.
Read an Excerpt
Are Numbers Real?
The Uncanny Relationship of Mathematics and the Physical World
By Brian Clegg
St. Martin's PressCopyright © 2016 by Brian Clegg
All rights reserved.
Our journey in this book will explore a question that is fundamentally important to scientists — and for that matter the rest of us. Yet it's a question that most people, including scientists, rarely give a moment's thought to. Are numbers, and is the wider concept of mathematics, real?
At first glance, this seems a crazy question to devote thirty seconds to, let alone a whole book. Of course numbers are real. You only have to take a look at my bank statement. It contains a whole load of numbers, most of which seem to be negative as cash flows out of the account. And as for mathematics, we all had plenty of homework when we were at school, and that seemed real enough at the time. But here I'm using a different definition of "real." It is essential to gain a better understanding of science to discover whether numbers and mathematics form real entities, whether they have a factual existence in the universe. Would numbers exist without people to think about them, or are they just valuable human inventions, the imaginary inhabitants of a useful fantasy world?
We know that it is perfectly possible to devise mathematics that does not have any underlying link with reality. Mathematicians do this all the time. Math, in the end, provides nothing more or less than a set of rules that are used to get from a starting point to an outcome. We can define those rules in such a way that they happen to match what we observe in the real world, or we can make them as bizarrely and wonderfully different from reality as we like. And some mathematicians delight in taking such fantasy journeys into alternative universes.
To take a simple example, the real world has three spatial dimensions (unless string theory, the attempt in physics to combine gravity and the other forces of nature that requires 9- or 10-dimensional space, has it right — see here) — but a mathematician is just as comfortable working with 1, 2, 4, 79, or 5,000 dimensions. Mathematicians delight in the existence of a mathematical construct called the Monster group, which is a group of ways you could rotate things if space had 196,883 dimensions. When working with the Monster, to quote Dorothy in The Wizard of Oz, "Toto, I've a feeling we're not in Kansas anymore."
For that matter, when mathematicians work on something as everyday as the shape of knots, they make their own definition of what a knot is that bears no resemblance to the things we use to tie up shoelaces. For reasons of practical convenience, the mathematicians set a rule that both ends of the string they are knotting must be joined together, making a continuous loop. We know real-world knots aren't like that — even mathematicians (admittedly not the most worldly people) know this — but they don't care, because that's the rule that they chose to use.
Similarly we could devise a mathematical system in which 2 + 2 = 5. It doesn't work with real-world objects, but there is no reason why it can't with a number system if we define it to work that way. Although not so extreme, there is a commonly used mathematical system where we can define 2 + 2 to be 0 or 1. It's called clock arithmetic. Instead of numbers adding constantly, they progress like the numbers on a clock, resetting to 0 at a specific value. Admittedly these do have a parallel in the world. We use clock arithmetic, as the name suggests, on analog clocks. On a twelve-hour clock, for instance, 9 + 6 = 3. Such arithmetic provides a better representation of anything cyclical than traditional counting. What this illustrates is both the arbitrariness of mathematics and how we have to be careful about definitions. The number 9 on a clock is not the same thing as the number 9 when we are counting goats, they just have some things in common, and use the same symbol.
To turn it around and consider things from the real-world viewpoint, it is possible to go through life without ever encountering much in the way of mathematics. For most of human existence, the vast majority of human beings have managed to do so. Some very basic arithmetic seems to be preprogrammed. Both dogs and babies react with surprise when, for instance, one item is put into a bowl, then another, but when they then look in the bowl, there is only one object, because the second was palmed. "1 + 1 = 2" seems a pretty low-level mammal programming, and is without doubt useful in calculating the odds when faced with more than one enemy to fight. Most of the rest of mathematics is a late add-on to our capabilities, but one that has proved extremely useful.
Without mathematics, hardly any of the science and technology that is essential for today's civilization would be produced. Math threads through our lives, from everyday functions like transactions in a store, to understanding the significance of the distribution of a disease or the outcome of an election. Because it is important that we have a feel for a discipline that is so useful in understanding the underlying structures and principles of the world around us and predicting its behavior, it's a shame that so many of us find getting into mathematics remarkably difficult, or even painful — something to be avoided if at all possible. A 2012 British article for World Math Day commented:
We know too that many adults simply don't like maths and don't see the point of it. Many have no qualms about saying so. Being "no good at maths" carries little stigma. That tends not to be the case in other parts of the world. Negative attitudes to maths set in early in the UK — some would say between the ages of seven and nine, when many children's interest and attainment dip, in most cases never to return. They switch off and decide maths is something to be borne until the moment they can give it up — for ever. ... The process is then cyclical, with parents (and in some cases — dare I say? — teachers) passing on their own lack of enthusiasm and confidence to the next generation.
The article suggests that this problem of having a negative attitude to mathematics is a particularly British one, but I suspect that it is one that is reflected not only in the United States, but also across many parts of the world. And this opinion is nothing new. St. Augustine of Hippo wrote back in AD 415, "The danger ... exists that the mathematicians have made a covenant with the devil to darken the spirit and confine man in the bonds of hell." He clearly did not have much fun in his geometry lessons. (The quote is a touch misleading. Augustine was usually more supportive of mathematics — the word translated as "mathematicians" referred to astrologers — but it still reflects the feeling that many seem to have.)
And yet mathematics can be both enjoyable, when presented the right way, and immensely powerful. The fun comes from mathematical puzzles and diversions. The delight can be particularly strong with mathematics that entertains by making your head spin — like the idea that there is more than one size of infinity (see here).
We might not need much math to go through our basic everyday lives, and the vast majority of us get by with a touch of arithmetic and little else. But when scientists and engineers try to understand how things work and to construct products based on that understanding, mathematics has proved an invaluable tool to gain insights. Without it, it would be very difficult to understand much about the natural world, or to predict how it is going to behave. Without it, we would not have the computer this was written on or much of the other technology that supports our modern lives.
Initially mathematics was intimately tied to natural behavior. Numbers, for instance, corresponded to tangible objects. But with time it has become separated from reality. There is still applied mathematics with a tie to the real world, but pure mathematics soared in the Renaissance as mathematicians realized that they were playing an immense game where they could set their own rules, play along, and see what happened. Sometimes some of the ideas and worlds they generated would have practical uses, sometimes they wouldn't. The distinction was arbitrary as far as they were concerned (and often remains so). The great game is paradoxical in that it is both totally open and surprisingly restrictive — what mathematics covers, what rules you set, are absolutely up to you. But once those rules have been established, the game says that you must stick to them. In math you can never cheat.
When we contemplate the nature of numbers and reality, the arbitrariness that lies beneath the surface of mathematics can lead to problems for the more rigidly minded. In the British Court of Appeal, the second highest court in the UK, in 2015, three judges were set the task of deciding exactly what the number "1" meant. And their decision certainly didn't equate to something that matched the understanding most of us (or even most mathematicians) would have.
The legal case was a wrangle between two pharmaceutical companies over a chemical solution used to reduce infection in wound dressings, and bizarrely the case led to a change in the legal definition of the number "1" in the UK. The problem was that one company, ConvaTec, had a patent covering a silver- based solution of "between 1 per cent and 25 per cent of the total volume of treatment." The rival pharma company, Smith & Nephew, had therefore devised a competing product containing a 0.77 percent solution, which they believed kept them safely outside the remit of their competitor's patent.
This dispute had already been taken to trial in 2013. A lower court agreed that the "1" at the lower end of the ConvaTec range did not simply represent the numerical value "1," corresponding to a single object. Instead, they adopted an approach not uncommon among chemists, but unusual mathematically, of defining the value as the split between the ranges of two significant figures. This "significant figure" aspect meant that "1" was defined by the lower court as anything between 0.95 and 1.5 — giving a conveniently asymmetric definition that left the Smith & Nephew product legal. Unhappy with this approach, the Court of Appeal judges went for a more familiar arithmetic approach of rounding to the nearest whole number, meaning that "1" now became anything between 0.5 and 1.4999. The result was to put Smith & Nephew into a difficult position. But it also demonstrates the arbitrary nature of mathematical decisions.
There is no "right" way to define something that is 1 unless you stick to exactly and only 1 — and the result is that, as far as the lawyers are concerned, the range "between 1 and 25" would have to include 0.5. One of the panel, Lord Justice Christopher Clarke, made the unhelpful statement, "A linguist may regard the word 'one' as meaning 'one — no more and no less.' To those skilled in the art it may, however, in context, imply a range of values extending beyond the integer." It's not clear exactly what dark art he had in mind.
Over time, mathematicians (as opposed to lawyers) have been distinctly creative in their handling of math. The way they operate is a bit like the way that some companies allow their employees to play around with different ideas and technologies in the hope that a new product will emerge. Often nothing relevant to the commercial world will be produced. But every now and then, something wonderful and genuinely new will be brought into being. Similarly, when mathematicians played around with an idea like the square root of a negative number (see chapter 8), called an imaginary number, they were initially simply enjoying a new direction to take their mathematical game. But, as it happened, because of the rules they decided to apply to this magical class of numbers, it became a hugely useful tool for physics and engineering.
No scientist or engineer ever said prior to the introduction of imaginary numbers, "What we want is the square root of a negative number. They would really help us with this problem we've got." Similarly, no one in mathematics thought, "How can we solve this particular problem that the physicists have?" before dreaming up imaginary numbers. The mathematicians just played with the implications of their new concept and the set of rules attached to it. The applications emerged later.
Generally speaking, up to the nineteenth century, the mathematics that was needed for all of science was within the grasp of pretty well anyone who hasn't got serious problems with numbers. In my experience, as long as you can get on top of the workings of fractions (something a scary number of people never achieve), you can manage everything up to basic calculus, which sounds worse than it is. But in the nineteenth century, it is arguable that two things happened in mathematics that drove a wedge between the general public and science.
The first of these was the use of increasingly complex mathematical techniques that take a considerable amount of post-school study to get a handle on. Pick up a modern physics paper at random, for instance, and the chances are it will use at least one approach that never made it into high school math. It is no surprise that when Einstein developed his general theory of relativity he had to get help with the mathematics because he found it too difficult alone. The science he was fine with, but the mathematics had moved beyond his experience.
The second development that has made science less approachable was putting the cart before the horse. Mathematics had always been the servant of scientists, but in the twentieth century it increasingly was put in charge. Attempts to unify the forces of nature, for instance, became driven from the mathematics of symmetry, along the way becoming very difficult to explain in laypeople's terms. Another example was matrix mechanics, where a form of mathematics then largely unfamiliar to scientists, let alone the rest of us, was used to explain the behavior of quantum particles in a way that made it impossible to visualize what was happening. All that remained were the numbers and the rules to manipulate them.
There is nothing wrong with these developments per se, but they bring some unfortunate baggage along with them. If your science can't be easily described to the person in the street, then it becomes harder to justify spending taxpayer dollars on it. Physicists often point to the U.S. government funding decision in the 1990s between spending on the International Space Station (ISS) and the Superconducting Super Collider (SSC). The SSC was a massive particle accelerator, already by then under construction in Texas. Nobel Prize–winning physicist Steven Weinberg has pointed out that the SSC, larger than the Large Hadron Collider (LHC) at CERN, the European nuclear research organization, could have produced results a good ten years earlier than its rival, that would have added to our fundamental understanding of the universe. The ISS, by contrast, which won the funds, leading to the cancellation of the SSC, has given us a better understanding of space and space travel, which may be valuable in the future, but has made little contribution to scientific research.
What's more, the ISS has now cost the United States ten times the SSC budget in the process of delivering very little. But the difference is that the ISS was doing something that all the politicians could clearly visualize and that could be easily explained to the people controlling the funds. Hence its prioritization and the cancellation of the SSC. No one could explain to the politicians really what was going to be achieved by the collider, because it was too complicated to do so. The result was that the science funding went to the project with almost zero scientific benefit, rather than the one that would have retained the U.S. position as a world leader in physics.
Instead that glory went to CERN and the LHC. And yet in a way it was a hollow victory, because that same math-based difficulty was looming in terms of what the LHC was trying to achieve. We hear about "re-creating the conditions near the Big Bang" or "searching for the Higgs boson," but when it comes down to explaining to the public what these handy labels actually mean, it is very difficult. The Higgs boson will typically be described as "the particle that gives all the others their mass," which has elements that verge on the correct, but is also wrong in several different ways. However, it is hard even to describe why this wording is wrong without resorting to the language of mathematics — one that will immediately lose the audience.
What I don't want to do is sound miserable and negative about mathematics. As you will discover as we take our journey into the reality of numbers, it is a topic that is as fascinating as it is powerful. And it has been central to the development of modern society and technology. But that should not stop us asking some fundamental questions. Does modern science give too much emphasis to mathematics? Some scientists, such as physicist Max Tegmark, go as far as to suggest that the universe is mathematical. That numbers aren't just real, but are what make everything happen. Could it be that scientists like Tegmark sometimes confuse math and reality? Is mathematics really at the heart of the universe, or is it just a great tool for helping us understand what's going on? These are the questions we will discover answers to as we explore how math has come to rule the scientific world.
Excerpted from Are Numbers Real? by Brian Clegg. Copyright © 2016 by Brian Clegg. Excerpted by permission of St. Martin's Press.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
Table of Contents
1 Counting Sheep 1
2 Counting Goats 13
3 All Is Number 27
4 Elegant Perfection 45
5 Counting Sand 55
6 The Emergence of Nothing 63
7 He Who Is Ignorant 79
8 All in the Imagination 95
9 The Amazing Mechanical Mathematical Universe 101
10 The Mystery of "Maybe" 117
11 Maxwell's Mathematical Hammer 145
12 Infinity and Beyond 161
13 Twentieth-century Mathematical Mysteries 195
14 Symmetry Games 219
15 Cargo Cult Science? 243