Chapter One: Thinking by Numbers
This grand book, the universe...cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics.
Galileo
The World by Numbers
Of all the many abilities that have raised us from cave-dwellers using stone tools to creators of great cities and modern science, one of the most important is the ability to use numbers. It is also among the least understood. Traditional studies of the role of numbers have been concerned largely with their contribution to the development of science. However, numbers have affected almost all the aspects of our life which are most characteristically human. We have used numbers from the beginning of the historical record, and perhaps from way back in the prehistory of our species. This book is an attempt to explain why we all think about the world in terms of numbers.
Nowadays, we use numbers routinely. We use them to count things, to tell the time, as statistical data, to gamble, to buy and to sell. Even barter needs numbers: I'll give you six knives if you give me two pigs. We use them to rank competitors, in addresses (house numbers and postal codes), to grade examination candidates. Blood pressures, temperatures, and IQs are given numerical values. Cars and their engines, TV stations and telephone lines, all have their number labels. Goods in shops have bar codes. People have social security numbers, bank account numbers, and passport numbers. Your height, weight, and age are all denoted by some numerical multiple of a standard unit.
The importance of numbers lies not just in their obvious utility, but also in the way they have shaped how we think about the world. It is the language in which we formulate scientific theories. Numbers, as Einstein said, are the 'symbolic counterpart of the universe'; they are crucial to the measurements we take as the fundamental evidence for our theories. 'When you cannot measure...your knowledge is of a meagre and unsatisfactory kind,' to quote the great nineteenth-century scientist Lord Kelvin, who, not coincidentally, invented the scale for measuring absolute temperature.
To get some idea of just how fundamental numbers are, try to imagine our world without them. There would be no money, no counting, no income tax. Without numbers, trade would be restricted to face-to-face barter. I could see the knives you wished to trade, and you could see my pigs, but trade at a distance would be extraordinarily difficult: how could I convey the number of pigs I would be willing to trade for your knives, and the number of knives I would be willing to accept for my pigs?
Without numbers, there would be none of our familiar sports, such as football, baseball, and tennis, since they use numbers to define how many players may be in a team, and to keep the score. Many sports obsessively keep numerical records; thus athletics is a good example. Qualification for the Olympic Games depends on exceeding a numerical standard. This tradition goes back to the original Olympian Games, whose very period -- the Olympiad -- is a definite number of years, four. Numerical records were kept of some events. Phayllus of Croton was credited with a leap of 55 feet (16.8 metres), which you might think raises the question of just how good the Greeks were at measurement. Some modern competitions, such as the heptathlon, are defined by the number of events in which the athletes take part, seven in this case.
Without numbers, we could not frame basic theories of physical nature, such as Kepler's laws of planetary motion, Newton's laws of motion, or Einstein's E = mc2. Chemists would be at a distinct disadvantage without the numerically ordered periodic table of the elements. The study of human nature has also depended on numbers to quantify mental attributes such as intelligence, reading age, or degree of introversion.
The basic laws of perception are another good example of the use of numbers in the study of human nature. To see a bright light get brighter you need a bigger absolute increase in energy than to see a dim light get brighter; to hear a loud noise get louder you need a bigger absolute increase in energy than for a quiet noise. The scientist, with or without numbers, would ask whether there is some general law that will predict how big an increase is needed to make a noticeable difference to brightness or to loudness. It turns out that there is, and it was discovered in 1834 by Ernst Weber. The law states that a barely noticeable change in brightness depends on a proportional increase in energy, not an absolute increase. What is the value of the increase that yields a difference which is just detectable? Experimentation tells us that an increase of roughly 1% in energy gives us a noticeable difference in brightness and about 10% is needed for a noticeable difference in loudness. Weber's law could not be formulated, perhaps not even thought about, without numbers.
Clearly, numbers are extraordinarily useful, but there is still the question of how we came to describe and represent our world in terms of numbers. Is there something about the world that would oblige us to invent numbers if we didn't have them already? Many useful but difficult inventions, such as the alphabet, double entry bookkeeping, or the printed circuit,6 were invented just once and were diffused around the world. One possibility, then, is that there was one ancient Einstein who invented numbers. Once he had made the breakthrough, it became clear how valuable the idea was and so it was eagerly adopted by neighbours, and then neighbours of neighbours, and so on. This scenario implies that cultures distant from the inventor would get numbers later than those nearby, or perhaps not at all. This is what has happened with the alphabet, for example. Indeed, some Amazonian tribes still haven't learned to read or write.
On the other hand, many inventions seem to have been less difficult, like plant domestication, pottery, and fire, and arose independently in many areas. Is the idea of numbers a relatively easy invention that many societies could have developed by themselves? But even easy inventions depend on local circumstances. The invention of settled agriculture with crops and domesticated animals has depended on having plants and animals that are suitable and easy to domesticate. Although trading creates a need for numbering (as I discuss in Chapter 2), it is not clear what would make the invention of numbers harder or easier.
A third possibility is that the idea of numbers was not invented at all. Rather, they are something about us, an intrinsic part of human nature, like the ability to see colours, or schadenfreude.
Today using numbers for trading, ordering, and labelling seems very easy, convenient, and natural. In a way this is very surprising, since numbers are, in the words of Adam Smith, 'among the most abstract ideas which the human mind is capable of forming'. Numbers are not properties of objects. You cannot touch, see, or feel them. They are not like the properties of an orange. If an object is an orange it will have a characteristic colour, texture, size, shape, smell, and taste. You can check each of these properties to see whether a candidate object is an orange or, for example, a ball or a lemon. But a collection of five things doesn't possess characteristic colour, shape, or taste. What all such collections have in common is their fiveness, and this is abstract. To understand numbers -- to understand, for example, the difference between five and four -- is to understand something very abstract indeed.
To use numbers, therefore, we should require extensive training, as for other abstract concepts. Think how long it takes to learn and apply the principles of ch