Alex’s Adventures in Numberland (69 page)

Regression to the mean is not a complicated idea. All it says is that if the outcome of an event is determined at least in part by random factors, then an extreme event will probably be followed by one that is less extreme. Yet despite its simplicity, regression is not appreciated by most people. I would say, in fact, that regression is one of the least grasped but most useful mathematical concepts you need for a rational understanding of the world. A surprisingly large number of simple misconceptions about science and statistics boil down to a failure to take regression to the mean into account.

Take the example of speed cameras. If several accidents happen on the same stretch of road, this could be because there is one cause – for example, a gang of teenage pranksters have tied a wire across the road. Arrest the teenagers and the accidents will stop. Or there could be many random contributing factors – a mixture of adverse weather conditions, the shape of the road, the victory of the local football team or the decision of a local resident to walk his dog. Accidents are equivalent to an extreme event. And after an extreme event, the likelihood is of less extreme events occurring: the random factors will combine in such a way as to result in fewer accidents. Often speed cameras are installed at spots where there have been one or more serious accidents. Their purpose is to make drivers go more slowly so as to reduce the number of crashes. Yes, the number of accidents tends to be reduced after speed cameras have been introduced, but this might have very little to do with the speed camera. Because of regression to the mean, whether or not one is installed, after a run of accidents it is already likely that there will be fewer accidents at that spot. (This is not an argument against speed cameras, since they may indeed be effective. Rather it is an argument about the argument for speed cameras, which often displays a misuse of statistics.)

My favourite example of regression to the mean is the ‘curse of
Sports Illustrated
’, a bizarre phenomenon by which sportsmen suffer a marked drop in form immediately after appearing on the cover of America’s top sports magazine. The curse is as old as the first issue. In August 1954 basball player Eddie Mathews was on the cover after he had led his team, the Milwaukee Braves, to a nine-game winning streak. Yet as soon as the issue was on the news-stands, the team lost. A week later Mathews picked up an injury that forced him to miss seven games. The curse struck most famously in 1957, when the magazine splashed on the headline ‘Why Oklahoma is Unbeatable’ after the Oklahoma football team had not lost in 47 games. Yet sure enough, on the Saturday after publication, Oklahoma lost 7–0 to Notre Dame.

One explanation for the curse of
Sports Illustrated
is the psychological pressure of being on the cover. The athlete or team becomes more prominent in the public eye, held up as the one to beat. It might be true in some cases that the pressure of being a favourite is detrimental to performance. Yet most of the time the curse of
Sports Illustrated
is simply an illustration of regression to the mean. For someone to have earned their place on the cover of the magazine, they will usually be on top form. They might have had an exceptional season, or just won a championship or broken a record. Sporting performance is due to talent, but it is also reliant on many random factors, such as whether your opponents have the flu, whether you get a puncture, or whether the sun is in your eyes. A best-ever result is comparable to an extreme event, and regression to the mean says that after an extreme event the likelihood is of one less extreme.

Of course, there are exceptions. Some athletes are so much better than the competition that random factors have little sway on their performances. They can be unlucky and still win. Yet we tend to underestimate the contribution of randomness to sporting success. In the 1980s statisticians started to analyse scoring patterns of basketball players. They were stunned to find that it was completely random whether or not a particular player made or missed a shot. Of course, some players were better than others. Consider player A, who scores 50 percent of his shots, on average; in other words, he has an equal chance of scoring or missing. Researchers discovered that the sequence of baskets and misses made by player A appeared to be totally random. In other words, instead of shooting he might as well have flipped a coin.

Consider player B, who has a 60 percent chance of scoring and a 40 percent chance of missing. Again, the sequence of baskets was random, as if the player was flipping a coin biased 60–40 instead of actually throwing the ball. When a player makes a run of baskets pundits will eulogize him for playing well, and when he makes a run of misses he will be criticized for having an off day. Yet making or missing a basket in one shot has no effect on whether he will make or miss it on the next shot. Each shot is as random as the flip of a coin. Player B can be genuinely praised for having a 60–40 score ratio on average over many games, but praising him for any sequence of five baskets in a row is no different from praising the talent of a coin flipper who gets five consecutive heads. In both cases, they had a lucky streak. It is also possible – if not entirely probable – that player A, who is not as good overall at making baskets as player B, might have a longer run of successful shots in a match. This does not mean he is a better player. It is randomness giving A a lucky streak and B an unlucky one.

More recently, Simon Kuper and Stefan Szymanski looked at the 400 games the England football team has played since 1980. They write, in
Why England Lose
: ‘England’s win sequence…is indistinguishable from a random series of coin tosses. There is no predictive value in the outcome of England’s last game, or indeed in any combination of England’s games. Whatever happened in one match appears to have no bearing on what will happen in the next one. The only thing you can predict is that over the medium to long term, England will win about half its games outright.’

The ups and downs of sporting performance are often explained by randomness. After a very big up you might get a call from
Sports Illustrated
. And you are almost guaranteed that your performance will slump.

A few ye
ars ago Daina Taimina was reclining on the sofa at home in Ithaca, New York, where she teaches at Cornell University. A family member asked her what she was doing.

‘I’m crocheting the hyperbolic plane,’ she replied, referring to a concept that has mystified and fascinated mathematicians for almost two centuries.

‘Have you ever seen a mathematician do
crochet
?’ came the dismissive response.

The rebuff, however, made Daina even more determined to use handicraft in the course of scientific advancement. Which is just what she did, inventing what is known as ‘hyperbolic crochet’, a method of looping yarn that produces objects as intricate and beautiful as anything produced by the WI, and that has also contributed to an understanding of geometry in a way that mathematicians once never thought possible.

I’ll come shortly to a detailed definition of
hyperbolic
and the insights gleaned by Daina’s crochet models, but for the moment all you need to know is that hyperbolic geometry is an utterly counterintuitive type of geometry that emerged in the early nineteenth century in which the set of rules that Euclid laid out so carefully in
The Elements
are taken as being false. ‘Non-Euclidean’ geometry was a watershed for mathematics in that it described a theory of physical space that totally contradicted our experience of the world, and therefore was hard to imagine, but nevertheless contained no mathematical contradictions, and so was as mathematically valid as the Euclidean system that came before.

Later that century an intellectual breakthrough of similar significance was made by Georg Cantor, who turned our intuitive understanding of the infinite on its head by proving that infinity comes in different sizes. Non-Euclidean geometry and Cantor’s set theory were gateways into two strange and wonderful worlds, and I’ll visit them both in the following pages. Arguably, together they marked the beginning of modern mathematics.

Hyperbolic crochet.

The Elements
, to recap from much earlier, is easily the most influential maths textbook of all time, having set out the basics of Greek geometry. It also established the
axiomatic method
, by which Euclid began with clear definitions of the terms to be used and the rules to be followed, and then built up his body of theorems from them. The rules, or
axioms
, of a system are the statements that are accepted without proof, so mathematicians always try to make them as simple and self-evident as possible.

Euclid proved all 465 theorems of
The Elements
with only five axioms, which are more commonly known as his five
postulates
:

When we get to number 5, something does not feel right. The postulates start briskly enough. The first four are easy to state, easy to understand and easy to accept. Yet who invited the fifth to the party? It is long-winded, complicated and not especially, if at all, self-evident. And it is not even as clearly fundamental: the first time
The Elements
requires it is for Proposition 29.

Despite their love of Euclid’s deductive method, mathematicians loathed his fifth postulate; not only did it go against their sense of aesthetics, they felt that it assumed too much to be an axiom. In fact, for 2000 years many great minds attempted to change the status of the fifth postulate by trying to deduce it from the other postulates so that it could be reclassified as a theorem instead of remaining as a postulate or axiom. But none succeeded. Perhaps the greatest evidence of Euclid’s own genius was that he understood that the fifth postulate had to be accepted without proof.

Mathematicians had more success with restating the postulate in different terms. For example, the Englishman John Wallis in the seventeenth century realized that all
The Elements
could be proved by keeping the first four postulates as they were but by replacing the fifth postulate with the following alternative:
given any triangle, the triangle can be blown up or shrunk to any size so that the lengths of the sides stay in the same proportion to each other and the angles between the sides remain unaltered
. While it was quite an insight to realize that the fifth postulate could be rephrased as a statement about triangles rather than a statement about lines, it did not resolve mathematicians’ concerns: Wallis’s alternative postulate was perhaps more intuitive than the fifth postulate, though perhaps only marginally so, but it still wasn’t as simple or self-evident as the first four. Other equivalents for the fifth postulate were also discovered; Euclid’s theorems still held true if the fifth postulate was substituted by the statement that the sum of angles in a triangle is 180 degrees, that Pythagoras’s Theorem is true, or that for all circles the ratio of the circumference to diameter is pi. Extraordinary as it might sound, each of these statements is mathematically interchangeable. The equivalent that most conveniently expressed the essence of the fifth postulate, however, concerned the behaviour of parallel lines. From the eighteenth century mathematicians studying Euclid began to prefer using this version, which is known as the parallel postulate:

It can be shown that the parallel postulate refers to the geometry of two distinct types of surface, hinging on the phrase ‘at most one line’ – which is mathspeak for ‘either one line or no lines’. In the first case, illustrated by the diagram, for any line L and point P, there is
only one
line parallel to L (which is marked L’) that goes through P. This version of the parallel postulate applies to the most obvious type of surface, a flat surface, such as a sheet of paper on your desk.

The parallel postulate.

Now let’s consider the second version of the postulate, in which for any line L and point P not on that line there are
no
lines through P and parallel to L. At first it is hard to think on what type of surface this might be the case. Where on Earth…? On Earth is exactly where! Imagine, for example, that our line L is the equator, and imagine that point P is the North Pole. The only straight lines through the North Pole are the lines of longitude, such as the Greenwich Meridian, and all lines of longitude cross the equator. So, there are no straight lines through the North Pole that are parallel to the equator.

The parallel postulate provides us with a geometry for two types of surface: flat surfaces and spherical surfaces.
The Elements
was concerned with flat surfaces, so for 2000 years this was the main focus of mathematical enquiry. Spherical surfaces like the Earth were of less interest to theoreticians than they were to navigators and astronomers. It was only at the beginning of the nineteenth century that mathematicians found a wider theory that encompassed flat and spherical surfaces – and this happened only after they encountered a
third
kind of surface, the hyperbolic one.

One of the most determined aspirants in the quest to prove the parallel postulate from the first four postulates, and therefore show that it is not a postulate at all but a theorem, was Janós Bolyai, an engineering undergraduate from Transylvania. His mathematician father Farkas knew the scale of the challenge from his own failed attempts and implored his son to stop: ‘For God’s sake, I beseech you, give it up. Fear it no less than sensual passions because it too may take all your time and deprive you of your health, peace of mind and happiness in life.’ But Janós stubbornly ignored his father’s advice, and that was not his only rebellion: Janós dared to consider that the postulate might be false.
The Elements
was to mathematics what the Bible was to Christianity, a book of unchallengeable, sacred truths. While there was debate about whether the fifth postulate was an axiom or a theorem, no one had the temerity to suggest that it might actually not be true. As it turned out, doing so was the key to a new world.

The parallel postulate states that for any given line and a point not on that line there is
at most one
parallel line through that point. Janós’s audacity was to suggest that for any given line and a point not on the line,
more than one
parallel line passes through that point. Even though it was not at all clear how to visualize a surface for which this statement was true, Janós realized that eometry created by this statement, together with the first four postulates, was still mathematically consistent. It was a revolutionary discovery, and he recognized its momentousness. In 1823 he wrote to his father announcing that ‘Out of nothing I have created a new universe’.

Janós was probably helped by the fact that he was working in isolation from any major mathematical institution, and so was less indoctrinated by traditional views. Even after making his discovery, he opted not to become a mathematician. On graduating, he joined the Austro-Hungarian army, where he was reportedly the best swordsman and dancer among his colleagues. He was also an outstanding musician, and it is said that he once challenged thirteen officers to duels on the condition that upon victory he could play the loser a piece on his violin.

Unbeknownst to Janós, another mathematician in an outpost even more distant from the hubs of European academia than Transylvania was making similar advances independently, but he had his work rejected by the mathematical establishment. In 1826 Nikolai Ivanovich Lobachevsky, a professor at Kazan University in Russia, submitted a paper that disputed the truth of the parallel postulate to the internationally renowned St Petersburg Academy of Sciences. It was turned down, so Lobachevsky then decided to submit it for publication in the local newspaper
Kazan Messenger
, and consequently no one took any notice.

The greatest irony about the toppling of Euclid’s fifth postulate from the plinth of inviolable truth, however, is that several decades beforehand someone at the very heart of the mathematical establishment had indeed made the same discovery as Janós Bolyai and Nikolai Lobachevsky, yet this man had withheld his results from his peers. Quite why Carl Friedrich Gauss, the greatest mathematician of his day, decided to keep his work on the parallel postulate secret is not understood, although the received view is that he wanted to avoid getting embroiled in a feud about the primacy of Euclid with faculty members.

It was only on reading about Janós’s results, which were published in 1831 as an appendix in a book by his father Farkas, that Gauss revealed to anyone that he had also considered the falsity of the parallel postulate. Gauss wrote a letter to Farkas, an old university classmate, in which he described Janós as a ‘genius of the first order’, yet added that he was unable to praise his breakthrough: ‘For to praise it would be to praise myself. The entire content of the essay…coincide[s] with my own discoveries, some of which date back 30 to 35 years…I had intended to write all this down later so that at least it would not perish with me. It is therefore a pleasant surprise for me that I am spared this trouble, and I am especially glad that it is just the son of my old friend who takes precedence to me in this matter.’ Janós was distressed when he learned that Gauss had got there first. And when, years later, Janós learned that Lobachevsky had also preceded him, he became haunted by the ludricrous notion that Lobachevsky was a fictional character invented by Gauss as a cunning ruse in order to deprive him of the credit for his work.