In Pursuit of the Unknown (27 page)

It is a mistake to think about a mathematical model as if it were the reality. In the physical sciences, where the model often fits reality very well, this may be a convenient way of thinking that causes little harm. But in the social sciences, models are often little better than caricatures. The choice of title for
The Bell Curve
hints at this tendency to conflate model with reality. The idea that IQ is some sort of precise measure of human ability, merely because it has a mathematical pedigree, makes the same error. It is not sensible to base sweeping and highly contentious social policy on simplistic, flawed mathematical models. The real point about
The Bell Curve
, one that it makes extensively but inadvertently, is that cleverness, intelligence, and wisdom are not the same.

Probability theory is widely used in medical trials of new drugs and treatments to test the statistical significance of data. The tests are often, but not always, based on the assumption that the underlying distribution is normal. A typical example is the detection of cancer clusters. A cluster, for some disease, is a group within which the disease occurs more frequently than expected in the overall population. The cluster may be geographical, or it may refer more metaphorically to people with a particular lifestyle, or a specific period of time. For example, retired professional wrestlers, or boys born between 1960 and 1970.

Apparent clusters may be due entirely to chance. Random numbers are seldom spread out in a roughly uniform way; instead, they often cluster together. In random simulations of the UK National Lottery, where six numbers between 1 and 49 are randomly drawn, more than half appear to
show some kind of regular pattern such as two numbers being consecutive or three numbers separated by the same amount, for example 5, 9, 13. Contrary to common intuition, random is clumped. When an apparent cluster is found, the medical authorities try to assess whether it is due to chance or whether there might be some possible causal connection. At one time, most children of Israeli fighter pilots were boys. It would be easy to think of possible explanations – pilots are very virile and virile men sire more boys (not true, by the way), pilots are exposed to more radiation than normal, they experience higher
g
-forces – but this phenomenon was short-lived, just a random cluster. In later data it disappeared. In any population of people, it is always likely that there will be more children of one sex or the other; exact equality is very improbable. To assess the significance of the cluster, you should keep observing and see whether it persists.

However, this procrastination can't be continued indefinitely, especially if the cluster involves a serious disease. AIDS was first detected as a cluster of pneumonia cases in American homosexual men in the 1980s, for instance. Asbestos fibres as a cause of a form of lung cancer, mesothelioma, first showed up as a cluster among former asbestos workers. So statistical methods are used to assess how probable such a cluster would be if it arose for random reasons. Fisher's methods of significance testing, and related methods, are widely used for that purpose.

Probability theory is also fundamental to our understanding of risk. This word has a specific, technical meaning. It refers to the potential for some action to lead to an undesirable outcome. For example, flying in an aircraft could result in being involved in a crash, smoking cigarettes could lead to lung cancer, building a nuclear power station could lead to the release of radiation in an accident or a terrorist attack, building a dam for hydroelectric power could cause deaths if the dam collapses. ‘Action' here can refer to not doing something: failing to vaccinate a child might lead to its death from a disease, for example. In this case there is also a risk associated with vaccinating the child, such as an allergic reaction. Over the whole population this risk is smaller, but for specific groups it can be larger.

Many different concepts of risk are employed in different contexts. The usual mathematical definition is that the risk associated with some action or inaction is the probability of an adverse result, multiplied by the loss that would then be incurred. By this definition a one in ten chance of killing ten people has the same level of risk as a one in a million chance of killing a million people. The mathematical definition is rational in the sense that there is a specific rationale behind it, but that doesn't mean that it is necessarily sensible. We've already seen that ‘probability' refers to the
long run, but for rare events the long run is very long indeed. Humans, and their societies, can adapt to repeated small numbers of deaths, but a country that suddenly lost a million people at once would be in serious trouble, because all public services and industry would simultaneously come under a severe strain. It would be little comfort to be told that over the next 10 million years, the total deaths in the two cases would be comparable. So new methods are being developed to quantify risk in such cases.

Statistical methods, derived from questions about gambling, have a huge variety of uses. They provide tools for analysing social, medical, and scientific data. Like all tools, what happens depends on how they are used. Anyone using statistical methods needs to be aware of the assumptions behind those methods, and their implications. Blindly feeding numbers into a computer and taking the results as gospel, without understanding the limitations of the method being used, is a recipe for disaster. The legitimate use of statistics, however, has improved our world out of all recognition. And it all began with Quetelet's bell curve.

The acceleration of a small segment of a violin string is proportional to the average displacement of neighbouring segments.

It predicts that the string will move in waves, and it generalises naturally to other physical systems in which waves occur.

Big advances in our understanding of water waves, sound waves, light waves, elastic vibrations. . . Seismologists use modified versions of it to deduce the structure of the interior of the Earth from how it vibrates. Oil companies use similar methods to find oil. In
Chapter 11
we will see how it predicted the existence of electromagnetic waves, leading to radio, television, radar, and modern communications.

W
e live in a world of waves. Our ears detect waves of compression in the air: we call this ‘hearing'. Our eyes detect waves of electromagnetic radiation: we call this ‘seeing'. When an earthquake hits a town or a city, the destruction is caused by waves in the solid body of the Earth. When a ship bobs up and down on the ocean, it is reacting to waves in the water. Surfers use ocean waves for recreation; radio, television, and large parts of the mobile telephone network use waves of electromagnetic radiation, similar to those that we see by, but of differing wavelengths. Microwave ovens . . . well, the name gives it away, doesn't it?

With so many practical instances of waves impinging on daily life, even centuries ago, the mathematicians who decided to follow up Newton's epic discovery that nature has laws could hardly fail to start thinking about waves. What got them started, though, came from the arts: specifically, music. How does a violin string create sound? What does it
do?

There was a reason for starting with violins, the kind of reason that appeals to mathematicians, though not to governments or businessmen considering investing in mathematicians and expecting a quick payback. A violin string can sensibly be modelled as an infinitely thin line, and its motion – which is clearly the cause of the sound that the instrument makes – can be assumed to take place in a plane. This makes the problem ‘low-dimensional', which means you have a chance of solving it. Once you have understood this simple example of waves, there's a good chance that the understanding can be transferred, often in small stages, to more realistic and more practical instances of waves.

The alternative, to plough headlong into highly complex problems, may appear attractive to politicians and captains of industry, but it usually gets bogged down in complexities. Mathematics thrives on simplicities, and if necessary mathematicians will invent them artificially to provide an entry route into more complex problems. They deprecatingly refer to such models as ‘toys', but these are toys with a serious purpose. Toy models of waves led to today's world of electronics and high-speed global communications, wide-bodied passenger jets and artificial satellites, radio, television, tsunami warning systems. . . but we'd never have
achieved any of those if a few mathematicians hadn't started to puzzle out how a violin works, using a model that wasn't realistic, even for a violin.

The Pythagoreans believed that the world was based on numbers, by which they meant whole numbers or ratios between whole numbers. Some of their beliefs tended towards the mystical, investing specific numbers with human attributes: 2 was male, 3 female, 5 symbolised marriage, and so on. The number 10 was very important to the Pythagoreans because it was 1 + 2 + 3 + 4 and they believed there were four elements: earth, air, fire, water. This kind of speculation strikes the modern mind as slightly crazy – well, my mind, at least – but it was reasonable in an age when humans were only just starting to investigate the world around them, seeking crucial patterns. It just took a while to work out which patterns were significant and which were dross.

One of the great triumphs of the Pythagorean world view came from music. Various stories circulate: according to one, Pythagoras was passing a blacksmith's shop and he noticed that hammers of different sizes made noises of different pitch, and that hammers related by simple numbers – one twice the size of the other, for instance – made noises that harmonised. Charming though this tale is, anyone who actually tries it out with real hammers will discover that a blacksmith's operations are not especially musical, and hammers are too complicated a shape to vibrate in harmony. But there's a grain of truth: on the whole, small objects make higher-pitched noises than large ones.

The stories are on stronger ground when they refer to a series of experiments that the Pythagoreans performed using a stretched string, a rudimentary musical instrument known as a canon. We know about these experiments because Ptolemy reported them in his
Harmonics
around 150
AD.
By moving a support to various positions along the string, the Pythagoreans found that when two strings of equal tension had lengths in a simple ratio, such as 2 : 1 or 3 : 2, they produced unusually harmonious notes. More complex ratios were discordant and unpleasant to the ear. Later scientists pushed these ideas much further, probably a bit too far: what seems pleasant to us depends on the physics of the ear, which is more complicated than that of a single string, and it also has a cultural dimension because the ears of growing children are trained by being exposed to the sounds that are common in their society. I predict that today's children will be unusually sensitive to differences in mobile phone ringtones. However, there is a solid scientific story behind these
complexities, and a lot of it confirms and explains the early Pythagorean discoveries with their single-stringed experimental instrument.

Musicians describe pairs of notes in terms of the interval between them, a measure of how many steps separate them in some musical scale. The most fundamental interval is the octave, eight white notes on a piano. Notes an octave apart sound remarkably similar, except that one note is higher than the other, and they are extremely harmonious. So much so, in fact, that harmonies based on the octave can seem a bit bland. On a violin, the way to play the note one octave above an open string is to press the middle of that string against the fingerboard. A string half as long plays a note one octave higher. So the octave is associated with a simple numerical ratio of 2 : 1.

Other harmonious intervals are also associated with simple numerical ratios. The most important for Western music are the fourth, a ratio of 4 : 3, and the fifth, a ratio of 3 : 2. The names make sense if you consider a musical scale of whole notes C D E F G A B C. With C as base, the note corresponding to a fourth is F, the fifth is G, and the octave C. If we number the notes consecutively with the base as 1, these are respectively the 4th, 5th, and 8th notes along the scale. The geometry is especially clear on an instrument like a guitar, which has segments of wire, ‘frets', inserted at the relevant positions. The fret for the fourth is one-quarter of the way along the string, that for a fifth is one-third of the way along, and the octave is halfway along. You can check this with a tape measure.