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Learn the language of reality

LOVE it or loathe it, there is no escape: mathematics is out there. We live in a world that is in some sense mathematical – although in precisely what sense is still hotly debated by mathematicians, physicists, philosophers and others. We are, as a result, innately mathematical beings. Crossing the road, catching a ball, stacking the dishwasher: in our everyday lives we are constantly, unconsciously, manipulating numbers, assessing shapes, calculating position, geometry and motion. We are doing maths. But most of us see the conscious pursuit of the subject as something best left to the experts. The abstractions, conjectures and proofs of formal mathematics belong to a higher, rarefied plane few of us can access. This latest issue of New Scientist: The Collection aims to bridge the divide between the…

THE ORIGIN OF MATHEMATICS

TO THE Iranian mathematician Maryam Mirzakhani, the first woman to win the Fields medal, mathematics often felt like “being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks”. “With some luck,” she added, “you might find a way out.” Mirzakhani, who died in July 2017 at the age of 40, ventured deeper into the mathematical jungle than most. Nonetheless, most of us have spent enough time on its periphery to have a sense of what the terrain looks like. Increasingly, it seems as if humans are the only animals with the cognitive ability to hack their way through the undergrowth. But where does this ability come from? Why did we develop it? And what is it for?…

THE ORIGIN OF MATHEMATICS

THE PILLARS OF MATHEMATICS

For most of us, maths means numbers, and that’s not wrong. The ability to understand and manipulate numbers in the abstract (think addition, subtraction, multiplication and division) is the foundation on which a formidable edifice has been built (see main story). Broadly speaking, this edifice consists of three pillars: geometry, analysis and algebra. Geometry is probably the most familiar to us. It begins with a sense of space, codified into principles that describe how static things in space relate to each other, like a triangle’s sides. When you have to consider things that move and change with time, you come to analysis, a field that includes calculus, whether it’s integral or differential calculus, or its many variations. Algebra is what allows us to process knowledge in terms of numbers, symbols…

ANIMAL INSTINCTS

The debate over whether our sense of exact numbers is innate has often turned to animals for support. If our distant cousins can be shown to share certain mathematical abilities, then that implies our own must predate the development of culture. Certainly, some individual animals have been shown to display remarkable talent. Alex, an African grey parrot trained by Irene Pepperberg, could correctly identify sets of between two and six objects 80 per cent of the time. Ai, a chimpanzee trained by Japanese primatologist Tetsuro Matsuzawa, could do much the same. But too much emphasis is placed on research involving animals, says Rafael Núñez at the University of California, San Diego, at the expense of data from human cultures that have sophisticated languages and yet don’t show exact numerosity (see…

ANIMAL INSTINCTS

WHY DO PEOPLE HATE MATHS?

“It is familiar to anyone writing about (or teaching) mathematics: no one very much likes the subject,” writes mathematician David Berlinski in his book One, Two, Three. This distaste, even fear, of mathematics is common – most of us know the feeling. Berlinski says this can be attributed to its use of arcane symbols. Symbols are strange, plus using them in the forms of theorems and proofs demands great attention, and the pay-off is never obvious (have you ever asked “how is learning algebra going to help me in real life?”). “In mathematics, something must be invested before anything is gained, and what is gained is never quite so palpable as what has been invested,” writes Berlinski. Maths anxiety, a tendency to panic when asked to perform mathematical tasks, is…

It doesn’t add up

” Gödel revealed the awkward possibility that arithmetic sometimes could not supply any answers at all” IF YOU were forced to learn long division at school, you might have had cause to curse whoever invented arithmetic. A wearisome whirl of divisors and dividends, of bringing the next digit down and multiplying by the number you first thought of, it almost always went wrong somewhere. And all the while you were plagued by that subversive thought – provided you were at school when such things existed – that any sensible person would just use a calculator. Well, here’s an even more subversive thought: are the rules of arithmetic, the basic logical premises underlying things like long division, unsound? Implausible, you might think. After all, human error aside, our number system delivers…

It doesn’t add up

Bound not to work

In the 1920s, David Hilbert laid down a grand challenge to his fellow mathematicians: to produce a framework for studying arithmetic, meaning the natural numbers together with addition, subtraction, multiplication and division, with Giuseppe Peano’s axioms as its backbone. Such a framework, Hilbert said, should be consistent, so it should never produce a contradiction such as 2 + 2 = 3. And it should be complete, meaning that every true statement about numbers should be provable within the framework. Kurt Gödel’s first incompleteness theorem, published in 1931, killed that aspiration dead by encoding in arithmetical terms the statement “this statement is unprovable”. If the statement could be proved using arithmetical rules, then the statement itself is untrue, so the underlying framework is inconsistent. If it could not be proved, the…

The colour of numbers

When Jeff Paris and Leo Harrington got their glimpse of arithmetical incompleteness in 1977, they were considering a variant on a classic mathematical result called Ramsey’s theorem. Suppose we have some scheme for assigning one of two colours, either red or blue, to every possible set of four natural numbers. So {1, 5, 8, 101} might be red for example, and {101, 187, 188, 189} might be blue. It is quite possible, then, that any given number will occur in some red sets and some blue sets. What Ramsey’s theorem says is that, despite this, we can always find an infinite collection of numbers that is monochromatic – coloured entirely red or blue. There’s nothing magic about sets of four numbers or two colours: change those to any figures you…

A ladder of infinities

How big is infinity? A silly question, you might say, as infinity is infinitely big. Perhaps, but as the 19th-century German mathematician Georg Cantor proved to his contemporaries’ dismay, the infinite comes in different sizes. Take the natural numbers: 0, 1, 2, 3, 4, 5… You can go on counting these till kingdom come, so there’s no doubting that the set of natural numbers is infinite. But this “countable” infinity occupies only the lowest rung of an infinite ladder. Ironically, larger infinities arise when you break down the natural numbers into subsets: the numbers 1 to 1,000,000, for example, or the odd numbers, the prime numbers, or pairs of numbers such as four and 1296. How many such subsets are there altogether? An infinite number, of course. Cantor was able…

No good with numbers

“Dyscalculics fail to see the connection between a set of objects and the numerical symbol that represents it” JILL, 19, from Michigan, wanted to go to university to read political science. There was just one problem: she kept failing the mathematics requirement. “I was an exceptional student in all other subjects, so my consistent failure at math made me feel very stupid,” she says. In fact, she stopped going to her college mathematics class after a while because, she says, “I couldn’t take the daily reminder of what an idiot I was.” Jill got herself screened for learning disabilities. She found that while her IQ was above average, her numerical ability was equivalent to that of an 11-year-old. The diagnosis came partly as a relief, because it explained a lot…

No good with numbers

Amazonian hunter-gatherers called the Mundurucú only have words for numbers up to 5. Does this affect the way they think about mathematical problems? Experts who think that the human concept of exact number is innate would predict not. However, Stan Dehaene of the Collège de France in Paris is among a growing number who believe that exact number is learned and therefore affected by our culture. He decided to test this idea with the Mundurucú. Working with his colleague in the field, Pierre Pica, and others, Dehaene has found that the Mundurucú can add and subtract with numbers under 5, and do approximate magnitude comparisons as successfully as a control group. But last year the team discovered a big cultural difference. They asked volunteers to look at a horizontal line…

Seduced by numbers

“Can intellectual pursuit be as compelling as bodily urges?” MATHEMATICIANS are famous for the lengths they go to when solving problems. To crack Fermat’s Last Theorem, Andrew Wiles worked in isolation for more than six years. And Thomas Hales produced a body of work consisting of 250 pages of notes and 3 gigaytes of computer programs to solve Kepler’s Conjecture, a problem open since 1611 regarding the most efficient way to stack cannonballs. What is it that motivates mathematicians to go to these extremes? It seems there is something compelling, almost seductive, about their subject. Could there be some sort of drive, similar to the sex drive? In other words, something that we could call a “maths drive” that urges us to find new mathematical explanations and truths? As strange…

Seduced by numbers

Pairing the primes

IT WAS the British mathematician G.H. Hardy who popularised the idea that youthful brains do the best maths. “I do not know of a major mathematical advance initiated by a man past fifty”, he wrote in A Mathematician’s Apology, a lament for the decline of his own creativity that he published in 1940 at the age of 62. “If a man of mature age loses interest in and abandons mathematics, the loss is not likely to be very serious either for mathematics or for himself.” If blooming youth is the rule, Yitang Zhang is a definite exception. For the best part of a decade after completing his PhD, he wasn’t even working as a mathematician, instead doing odd accounting jobs around Kentucky. At one point he did a stint working…

Pairing the primes

Goldbach’s conjecture This is the prediction that every even number above 4 can be written as a sum of two odd prime numbers – for example, 10 = 3 + 7, and 78 = 31 + 47. Proposed by Christian Goldbach in 1742, it remains unproven. Infinite Germains A Germain prime, named after Sophie Germain, is one that gives another prime if you double it and add 1. For example, 29 is prime, and (29 x 2) + 1 = 59 is also prime, so 29 is a Germain prime. Mathematicians expect that there are infinitely many Germain primes, but no one can prove it. The Riemann hypothesis In 1859, Bernhard Riemann put forward an idea about where the Riemann zeta function takes the value zero. Proving this conjecture would…

Wonders of numberland

i The imaginary number THE rules of mathematics say that two positive numbers multiply to give a positive, and two negative numbers also multiply to give a positive. So what number could you multiply by itself to give -1? This is not a trick question – it’s just that the answer is imaginary. The square roots of negative numbers were first called “imaginary” by René Descartes in 1637. But it wasn’t until the 18th century that they came to be represented as multiples of i, the square root of -1. Imaginary numbers don’t fit on the regular number line, so they are put on a second, independent line, with the two intersecting at zero. The lines can be treated as axes, making imaginary numbers handy for representing things that change…

Wonders of numberland

Euler’s number

PUT a pound in the bank. If the yearly interest were 100 per cent, then a year later you would have £2. That’s simple enough. But what if instead of calculating the interest at the end of the year, the bank worked it out more regularly? It turns out this question leads us to one of the most subtle numbers in mathematics. Say your bank paid interest twice a year but halved the rate to 50 per cent. That would take your £1 to £1.50 after 6 months, and at the end of the year you would get another 50 per cent, making £2.25 – a nice gain. If you got interest monthly but scaled down the rate accordingly, you would end up with £2.61. Do the same thing daily,…

The golden ratio

YOU have probably heard of the Fibonacci sequence, that list of numbers where the next digit is given by adding the previous two. It goes 1, 1, 2, 3, 5, 8, 13 and so on. But here’s something strange: work out the ratio of each number and its predecessor, and you start edging towards a specific number. Its first few digits are 1.618. This mysterious beast is the golden ratio, and it crops up a lot. Try drawing a diagonal line connecting two vertices of a regular pentagon. Divide the length of that line by the length of the pentagon’s sides and there it is. Something similar is possible with an equilateral triangle. It turns out to be a quirk of maths. Imagine you have a number, A, and a…

The golden ratio

Graham’s number

MOST numbers have never touched a human mind. There are an infinite number of numbers, after all, so it stands to reason that we have only bothered with the small ones. But in the 1970s, Ronald Graham, a mathematician now at the University of Califonia, San Diego, was working on a puzzle that proved to have a truly gargantuan answer. He was trying to solve a problem to do with cubes in higher dimensions, and when he finally got there, the answer involved a number so large we can’t write down its digits – there isn’t enough space in the universe. Yet there is a way to grasp at Graham’s number. A more concise way of writing 3×3×3 is exponentiation: 33 means “multiply three threes together”, giving 27. We can…

2 74,207,281 -1

MULTIPLY 2 by itself just over 74 million times, then subtract 1. This is the largest known prime number, with more than 22 million digits. It is also a Mersenne prime, one equal to a power of 2, minus 1. Other numbers in the Mersenne club include 3 and 31, but finding larger ones is no easy task. We have only discovered 49 of them. We rely on large primes like these to ensure that all sorts of online transactions are encrypted, so that only the intended recipient can unscramble them. The idea is that the receiver multiplies two big primes to create a new number called the public key. Anyone with this key can encrypt messages, but to turn them from gobbledegook to something meaningful requires knowledge of the…

2 74,207,281 -1

Ever wondered how a website knows you’ve typed your credit card number in wrongly before it’s sent it to your bank for verification? It’s down to the Luhn algorithm, brainchild of the German-born IBM engineer Hans Peter Luhn in 1954. A pioneer of mechanical data storage, Luhn was also a prolific inventor who held more than 80 patents, including one for ornamented, knitted stockings. But the algorithm is his enduring achievement. The digits of most major credit card numbers are chosen so that, if you apply the Luhn algorithm to them, the result will be divisible by 10. Get any single digit wrong, and the number that comes out the other end won’t end in a zero – and in the blink of an eye the computer says no. Richard…

The Laplace limit

IN 1609 the great astronomer Johannes Kepler published a book called Astronomia Nova. This “new astronomy” delivered a bombshell: planets revolve around the sun in ellipses, not perfect circles. But the equation at the heart of the revelations, Kepler’s equation, had astronomer’s heads spinning faster than the planets they were studying. The formula describes the relationship between the coordinates of an object in orbit and the time elapsed from an arbitrary starting point. Actually solving it to find that location is fiendishly tricky. It took 150 years to find a mathematical way to solve the formula. The laborious process involved long strings of mathematical expressions known as series expansions. But the French polymath Pierre-Simon Laplace showed that this method would not work if the orbit was too elliptical. You can…

The Laplace limit

The Lyapunov exponent

AND the weather on Tuesday will be exponential errors, followed by a loss in predictability. Because of Lyapunov exponents, it is impossible to accurately forecast the weather more than a few days ahead. Instead of predictability, there is chaos. In the late 19th century, the Russian mathematician Aleksandr Lyapunov invented these numbers to describe how sensitive a system is to its starting point. Imagine, for example, throwing a ball across a field. Provided you know the angle and speed at launch, you can calculate where the ball will land to a good degree of accuracy without worrying about small effects like air resitance. If your measurements of the angle are a bit off, that doesn’t matter either. This situation would have a Lyapunov exponent of 0, or perhaps a negative…

From zero to hero

I USED to have seven goats. I bartered three for corn; I gave one to each of my three daughters as dowry; one was stolen. How many goats do I have now? This is not a trick question. Oddly, though, for much of human history we have not had the mathematical wherewithal to supply an answer. There is evidence of counting that stretches back five millennia in Egypt, Mesopotamia and Persia. Yet even by the most generous definition, a mathematical conception of nothing – a zero – has existed for less than half that time. Even then, the civilisations that discovered it missed its point entirely. In Europe, indifference, myopia and fear stunted its development for centuries. What is it about zero that stopped it becoming a hero? This is…

From zero to hero

Nothing in common

THE mathematicians’ version of nothing is the empty set. This is a collection that doesn’t actually contain anything, such as my own collection of vintage Rolls-Royces. The empty set may seem a bit feeble, but appearances deceive; it provides a vital building block for the whole of mathematics. It all started in the late 1800s. While most mathematicians were busy adding a nice piece of furniture, a new room, even an entire storey to the growing mathematical edifice, a group of worrywarts started to fret about the cellar. Innovations like non-Euclidean geometry and Fourier analysis were all very well – but were the underpinnings sound? To prove they were, a basic idea needed sorting out that no one really understood. Numbers. Sure, everyone knew how to do sums. Using numbers…

Nothing in common

WHEN David Hilbert left the podium at the Sorbonne in Paris, France, on 8 August 1900, few of the assembled delegates seemed overly impressed. According to one contemporary report, the discussion following his address to the second International Congress of Mathematicians was “rather desultory”. Passions seem to have been more inflamed by a subsequent debate on whether Esperanto should be adopted as mathematics’ working language. Yet Hilbert’s address set the mathematical agenda for the 20th century. It crystallised into a list of 23 crucial unanswered questions, including how to pack spheres to make best use of the available space, and whether the Riemann hypothesis, which concerns how the prime numbers are distributed, is true. Today many of these problems have been resolved, sphere-packing among them. Others, such as the Riemann…

HOW TO THINK ABOUT INFINITY

I AN STEWART has an easy, if not particularly helpful, way of envisaging infinity. “I generally think of it as: (a) very big, but (b) bigger than that,” says the mathematician from the University of Warwick in the UK. “When something is infinite, there is always some spare room around to put things in.” Infinity is one of those things with a preprogrammed boggle factor. Mathematically, it started off as a way of expressing the fact that some things, like counting, have no obvious end. Count to 146 and there’s 147; count to a trillion and say hello to a trillion and one. There are two ways of dealing with this, says Stewart. “You can sum it up boldly as ‘there are infinitely many numbers’. But if you want to…

HOW TO THINK ABOUT INFINITY

The infinity illusion

INFINITY is a concept that defies imagination. We have a hard-enough time trying to wrap our minds around things that are merely extremely big: our solar system, our galaxy, the observable universe. But those scales are nothing compared with the infinite. Just thinking about it can make you queasy. But we cannot avoid it. Mathematics as we know it is riddled with infinities. The number line stretches to eternity and beyond, and is infinitely divisible: countless more numbers lurk between any two others. The number of digits in a constant like π is limitless. Whether geometry, trigonometry or calculus, the mathematical manipulations we use to make sense of the world are built on the idea that some things never end. Trouble is, once unleashed, these infinities are wild, unruly beasts.…

The infinity illusion

Careless pork costs lives... ...and other medical myths

TYPE the word “cancer” into the website search engine of the Daily Mail, a UK tabloid newspaper, and a wealth of information is just a mouse click away. Some of the reports are calming, most alarming – and all come with figures to back them up. Women who use talcum powder are 40 per cent more likely to develop ovarian cancer, says research. Cancer survival rates in the UK are among the worst in Europe, according to a study. The incidence of bowel cancer among the under-30s has soared by 120 per cent in 10 years, astonishing figures show. The figures might make us worry for our health, but somehow we feel the better for their existence. Numbers help us make sense of the world: if you can put a…

Careless pork costs lives... ...and other medical myths

” One-third of academic papers reporting medical benefits and harms used a mixture of statistical measures” “Bowel cancer soars by 120% among the under 30s” Daily Mail, 31 March 2009 137 people under 30 were diagnosed with bowel cancer in 2006 in England & Wales up from 63 cases in 1997 “Throat cancer among white men up 400% in 30 years” Percentage of white male population in the US diagnosed annually with adenocarcinoma of the oesophagus Journal of National Cancer Institute, vol 100, p 1184…

“Over 80 per cent of women say that this shampoo leaves their hair healthier and shinier.” Such claims are common in advertising for all manner of consumer products. What they might not tell you is that only five women tested the shampoo. And of the four who certified its miraculous effect, one or two probably ended up with nicer hair purely by chance, or simply imagined the results. Similar caveats apply to the effectiveness of medical treatments. Curing six out of 10 patients is promising. Curing 300 out of 500 is the same success rate, but far more convincing. “The sample size in a test is absolutely crucial in deciding whether any apparent improvement could have happened by chance alone,” says Spiegelhalter. The standard procedure for such trials is the…

HOW TO PLAY THE GAME

“One way to win at chicken is to throw the steering wheel out of the window” IN THE film A Beautiful Mind, mathematician John Nash and his Princeton grad student buddies are sitting in a smoky bar when a group of women walk in. As the men tease each other about their chances, Nash is struck with inspiration. Is there a logical, mathematical way of working out the best strategy for each man getting a date? Next thing you know he’s shambling out of the bar, and spends the night furiously scribbling equations. In a ham-fisted, Hollywood sort of way, this imagined episode does hint at how game theory, the branch of maths Nash helped to make famous, can apply to our everyday lives. In fact, we use it all…

HOW TO PLAY THE GAME

HOW NUMBERS GROW

THE greatest shortcoming of the human race is our inability to understand the exponential function.” These are the words of the late Albert Bartlett, a physicist at the University of Colorado, Boulder, whose lectures on the subject became a YouTube hit. Take saving for retirement. “Start early” is the mantra, but it is easy to overlook just how much difference a few years can make. It all comes down to exponential growth – an often abused term that refers to anything that grows in proportion to its current value. It dictates that a forward-thinking 18-year-old can retire as a millionaire at 65 by investing around £250 a month with an annual return of 7 per cent. That figure might sound high by today’s standards, but it’s a rough average of…

HOW NUMBERS GROW

HOW TO THINK ABOUT PROBABILITY THE MONTY HALL PROBLEM

PROBABILITY is one of those things we all get wrong… deeply wrong. The good news is we’re not the only ones, says John Haigh, formerly of the University of Sussex in Brighton, UK. “Many pure mathematicians claim that probability has many unreasonable answers.” Take the classic problem of a class of 25 schoolchildren. How likely is it that two of them share the same birthday? The common-sense answer is that it is not implausible, but quite unlikely. Wrong: it’s actually just under 57 per cent. Or the celebrated Monty Hall problem, named after the former host of US television game show Let’s Make a Deal. You’re playing a game in which there are three doors, one hiding a car, two of them goats (see diagram, left). You choose one door;…

HOW TO THINK ABOUT PROBABILITY THE MONTY HALL PROBLEM

PROBABILITY WARS

WE ARE in a bar, and agree to toss a coin for the next round. Heads, I pay; tails, the drinks are on you. What are your chances of a free pint? Most people – sober ones, at least – would agree: evens. Then I flip the coin and catch it, but hide in it the palm of my hand. What’s your probability of free beer now? Broadly speaking, there are two answers: (1) it is still 50 per cent, until you have reason to think otherwise; (2) assigning a probability to an event that has already happened is nonsense. Which answer you incline towards reveals where you stand in a 250-year-old, sometimes strangely vicious debate on the nature of probability and statistics. It is the spat between frequentist and…

PROBABILITY WARS

THE BOOKIE ALAN GLYNN

How do you calculate odds? You analyse the teams or competitors with the stats you have available. For a Premier League football game, for example, you look at which team is better in the long run, which one has played better recently, where the game is being played, which players are available and any other factors such as how important this particular game is to each team. Having weighed up all these things, you come up with a probability that each team might win. How much is down to maths and how much to human judgement? We use algorithms and mathematical models, but very few bookmakers would generate prices and be confident of them using a computer program alone. You need human intervention to make sure everything has been taken…

THE BOOKIE ALAN GLYNN

THE GAMBLER VANESSA SELBST

So, that old question: how much is poker about luck? Luck is a huge factor. Your job as a poker player is to identify the situations in which you have a very good chance of winning and risk as much money as possible. The skilful players will give themselves a better chance of winning, but no matter how good you are, there is so much luck involved in any specific hand. Of course, given the law of large numbers, the luck eventually runs out. So the more tournaments you play, the less luck is involved in the game. How important is it to understand probability theory? There is a lot of simple maths you need to know and memorise. For instance, what are the odds of making a flush [five…

THE GAMBLER VANESSA SELBST

Justice you can count on

SHAMBLING sleuth Columbo always gets his man. Take the society photographer in a 1974 episode of the cult US television series who has killed his wife and disguised it as a bungled kidnapping. It is the perfect crime – until the hangdog detective hits on a cunning ruse to expose it. He induces the murderer to grab from a shelf of 12 cameras the exact one used to snap the victim before she was killed. “You just incriminated yourself, sir,” says a watching police officer. If only it were that simple. Killer or not, anyone would have a 1 in 12 chance of picking the same camera at random. That kind of evidence would never stand up in court. Or would it? In fact, such probabilistic pitfalls are not limited…

Justice you can count on

Suppose you have a piece of evidence, E, from a crime scene – a bloodstain, or perhaps a clothing thread – that matches to a suspect. How should it affect your perception or hypothesis, H, of the suspect’s innocence? Bayes’s theorem tells you how to work out the probability of H given E. It is: (the probability of H) multiplied by (the probability of E given H) divided by (the probability of E). Or in standard mathematical notation: P(H I E) = P(H) × P(E I H)/ P(E) Say you are a juror at an assault trial, and so far you are 60 per cent convinced the defendant is innocent: P(H) = 0.6. Then you’re told that the blood of the defendant and blood found at the crime scene are…

FIVE FALLACIES TO FORGO

1. PROSECUTOR’S FALLACY “The prosecutor’s fallacy is such an easy mistake to make,” says Ian Evett of Principal Forensic Services, a UK forensics company. It confuses two subtly different probabilities that Bayes’s formula distinguishes: P(H I E), the probability that someone is innocent if they are a match to a piece of evidence, and P(E I H), the probability that someone is a match to a piece of evidence if they are innocent (see “Bayes on trial”, previous page). The first probability is what we would like to know; the second is what forensics usually tells us. Unfortunately, even professionals sometimes mix them up. In the 1991 rape trial of Andrew Deen in Manchester, UK, for example, an expert witness agreed on the basis of a DNA sample that “the…

FIVE FALLACIES TO FORGO

THINK OF A NUMBER

“OUR BRAINS HANDICAP US WHEN WE NEED TO THINK IN RANDOM AND UNPREDICTABLE WAYS” MADS HAAHR is in no doubt. “Generating randomness is not a task that should be left to humans,” he says. You might expect him to say that. A computer scientist at Trinity College Dublin, he is the creator of a popular online random number generator, hosted at random.org. But he has a point. Human brains are wired to spot and generate patterns. That is useful when it’s all about seeing predators on the savannah before they see you, but it handicaps us when we need to think in random and unpredictable ways (see “The mathematician”, right). That’s a problem, because true randomness is a useful thing to have. Random numbers are used in cryptography, computing, design…

THINK OF A NUMBER

THE MATHEMATICIAN DAVID HAND

In your recent book The Improbability Principle , you state that extremely unlikely events are commonplace. How so? At first glance, it sounds like a contradiction: if something is highly improbable, how can it possibly be commonplace? But as you dig deeper you see it is not a contradiction, and that you should expect what appear to be extremely improbable events to occur quite often. The principle itself is really an interweaving of five fundamental laws. Could you give an example of one of those laws? Take the law of truly large numbers. The most obvious example of this is a lottery. In a 49-ball game you have a 1 in 14 million chance of winning if you buy just one ticket. But of course if you…

THE MATHEMATICIAN DAVID HAND

Definitely not maybe

“ Probability and randomness are large sledgehammers to crack some small eggs” PROBABILITY theory is a quaint little piece of mathematics. It is about sets of non-negative numbers that are attached to actual and possible physical events, that sum to 1 and that obey certain rules. It has numerous practical applications. So does flat-Earth theory: for instance, it’s an excellent approximation when laying out your garden. Science abandoned the misconception that Earth extends over an infinite plane, or has edges, millennia ago. Probability insinuated itself into physics relatively recently, yet the idea that the world actually follows probabilistic rules is even more misleading than saying Earth is flat. Terms such as “likely”, “probable”, “typical” and “random”, and statements assigning probabilities to physical events are incapable of saying anything about what…

Definitely not maybe

THE HARDEST PROBLEM

“DEAR Fellow Researchers, I am pleased to announce a proof that P is not equal to NP, which is attached in 10pt and 12pt fonts.” So began an email sent in August 2010 to a group of leading computer scientists by Vinay Deolalikar, a mathematician at Hewlett-Packard Labs in Palo Alto, California. It was an incendiary claim. Deolalikar was saying he had cracked the biggest problem in computer science, a question concerning the complexity and fundamental limits of computation. Answer the question “Is P equal to NP?” with a yes, and you could be looking at a world transformed, where planes and trains run on time, accurate electronic translations are routine, the molecular mysteries of life are revealed at the click of a mouse – and secure online shopping becomes…

THE HARDEST PROBLEM

The world maker

YOU might not have heard of the algorithm that runs the world. Few people have, though it can determine much that goes on in our day-to-day lives: the food we have to eat, our schedule at work, when the train will come to take us there. Somewhere, in some server basement right now, it is probably working on some aspect of your life tomorrow, next week, in a year’s time. Perhaps ignorance of the algorithm’s workings is bliss. The door to Plato’s Academy in ancient Athens is said to have borne the legend “let no one ignorant of geometry enter”. That was easy enough to say back then, when geometry was firmly grounded in the three dimensions of space our brains were built to cope with. But the algorithm operates…

The world maker

2000 YEARS OF ALGORITHMS

c. 300 BC THE EUCLIDEAN ALGORITHM From Euclid’s mathematical primer Elements, this is the grandaddy of all algorithms, showing how, given two numbers, you can find the largest number that divides into both. It has still not been bettered. 820 THE QUADRATIC ALGORITHM The word algorithm is derived from the name of the Persian mathematician Al-Khwarizmi. Experienced practitioners today perform his algorithm for solving quadratic equations (those containing an x2 term) in their heads. For everyone else, modern algebra provides the formula familiar from school. 1936 THE UNIVERSAL TURING MACHINE The British mathematician Alan Turing equated algorithms with mechanical processes – and found one to mimic all the others, the theoretical template for the programmable computer. 1946 THE MONTE CARLO METHOD When your problem is just too hard to solve…

Electoral dysfunction

IN AN ideal world, elections should be two things: free and fair. Every adult, with a few sensible exceptions, should be able to vote for a candidate of their choice, and each single vote should be worth the same. Ensuring a free vote is a matter for the law. Making elections fair is more a matter for mathematicians. They have been studying voting systems for hundreds of years, looking for sources of bias that distort the value of individual votes, and ways to avoid them. Along the way, they have turned up many paradoxes and surprises. What they have not done is come up with the answer. With good reason: it probably doesn’t exist. The many democratic electoral systems in use around the world attempt to strike a balance between…

Electoral dysfunction

BORDERLINE CASE

In first-past-the-post or “plurality” systems, borders matter. To ensure that each vote has roughly the same weight, each constituency should have roughly the same number of voters. Threading boundaries between and through centres of population on the pretext of ensuring fairness is also a great way to cheat for your own benefit – a practice known as gerrymandering, after a 19th-century governor of Massachusetts, Elbridge Gerry, who created an electoral division whose shape reminded a local newspaper editor of a salamander. Suppose a city controlled by the Liberal Republican (LR) party has a voting population of 900,000 divided into three constituencies. Polls show that at the next election LR is heading for defeat – 400,000 people intend to vote for it but the 500,000 others will opt for the Democratic…

BORDERLINE CASE

PROPORTIONAL PARADOX

Although elections to the US House of Representatives use a first-past-the-post voting system, the constitution requires that seats be “apportioned among the several states according to their respective numbers” – that is, divvied up proportionally. In 1880, the chief clerk of the US Census Bureau, Charles Seaton, discovered that Alabama would get eight seats in a 299-seat House, but only seven in a 300-seat House. This “Alabama paradox” was caused by an algorithm known as the largest remainder method, which was used to round the number of seats a state would receive under strict proportionality to a whole number. Suppose for simplicity’s sake that a nation of 39 million voters has a parliament with four seats – giving a quota of 9.75 million voters per seat. The seats must, however,…

PROPORTIONAL PARADOX

POWER IN THE BALANCE

One criticism of proportional voting systems is that they make it less likely that one party wins a majority of the seats available, thus increasing the power of smaller parties as “king-makers” who can swing the balance between rival parties as they see fit. The same can happen in a plurality system if the electoral arithmetic delivers a hung parliament, in which no party has an overall majority – as might happen in the UK after its election next week. Where does the power reside in such situations? One way to quantify that question is the Banzhaf power index. First, list all combinations of parties that could form a majority coalition, and in all of those coalitions count how many times a party is a “swing” partner that could destroy…

POWER IN THE BALANCE

” Most mathematicians would have thought, ‘I’m not going to look at it.’ We were stupid enough to try” LUNCH with a colleague from work should be a time to unwind – the most taxing task being to decide what to eat, drink and choose for dessert. For Rick Mabry and Paul Deiermann it has never been that simple. They can’t think about sharing a pizza, for example, without falling headlong into the mathematics of how to slice it up. “We went to lunch together at least once a week,” says Mabry, recalling the early 1990s when they were both at Louisiana State University, Shreveport. “One of us would bring a notebook, and we’d draw pictures while our food was getting cold.” The problem that bothered them was this. Suppose…

What’s luck got to do with it?

IN 2004, Londoner Ashley Revell sold his house, all his possessions and cashed in his life savings. It raised £76,840. He flew to Las Vegas, headed to the roulette table and put it all on red. The wheel was spun. The crowd held its breath as the ball slowed, bounced four or five times, and finally settled on number seven. Red seven. Revell’s bet was a straight gamble: double or nothing. But when Edward Thorp, a mathematics student at the Massachussetts Institute of Technology, went to the same casino some 40 years previously, he knew pretty well where the ball was going to land. He walked away with a profit, took it to the racecourse, the basketball court and the stock market, and became a multimillionaire. He wasn’t on a…

What’s luck got to do with it?

OSLO, May 2008. King Harald of Norway presents mathematicians John Thompson and Jacques Tits with the Abel prize, one of the highest accolades in mathematics. There is a pleasing symmetry at the heart of the award. The winners are being honoured for ground-breaking work that led to the completion of a project started by Niels Abel, the 19th-century Norwegian mathematician after whom the prize is named. Appropriately enough, that project concerns mathematicians’ attempts to answer the question: what is symmetry? Most people’s response is to point to the left-right reflectional symmetry of the human face. Or a flower, or a snowflake. But a snowflake has additional symmetries to that of a human face: as well as looking at its two halves, you can also turn a snowflake 60 degrees to…

WE PHYSICISTS have a habit of depicting our discipline as “beautiful” or “elegant”, where an outsider might be forgiven for seeing no more than an endless morass of equations. In an ideal world, those equations would be unnecessary; the ultimate goal of physics – and science generally – is to describe the world as simply as possible. In 1915 Albert Einstein caught the attention of the mathematical world with the presentation of his general theory of relativity. But that same year, the excitement surrounding relativity spawned another seminal piece of work. Even among physicists, though, it is not nearly as famous as it should be. Perhaps that is down to the complexity of its mathematics, but perhaps the author’s sex and sadly short life played their parts too. Yet there…

Universe by numbers

WHAT is the meaning of life, the universe and everything? In the sci-fi spoof The Hitchhiker’s Guide to the Galaxy, the answer was found to be 42; the hardest part turned out to be finding the real question. Indeed, although our inquisitive ancestors undoubtedly asked such big questions, their search for a “theory of everything” evolved as their knowledge grew. As the ancient Greeks replaced myth-based explanations with mechanistic models of the solar system, their emphasis shifted from asking “why” to asking “how”. Since then, the scope of our questioning has dwindled in some areas and mushroomed in others. Some questions were abandoned as naive or misguided, such as explaining the sizes of planetary orbits from first principles, which was popular during the Renaissance. The same may happen to currently…

Universe by numbers

“OH, I am fortune’s fool,” says Romeo. Rest easy, lover boy; we all are. Or are we? Romeo, having killed Tybalt and realising he must leave Verona or risk death, was expressing a view common in Shakespeare’s time: that we are all marionettes, with some higher cause pulling the strings. Chance – let alone our own decision-making – plays little part in the unravelling of cosmic designs. Even processes that inherently involved chance were pre-determined. Long before dice were used for gaming, they were used for divination. Ancient thinkers thought the gods determined the outcome of a die roll; the apparent randomness resulted from our ignorance of divine intentions. Oddly, modern science at first did little to change that view. Isaac Newton devised laws of motion and gravitation that connected…

THE WEATHER MAN KEN MYLNE

How do you forecast the weather? We set up a model to represent the current state of the atmosphere based on many observations. From that, the model projects forward in time and calculates how the atmosphere may evolve. The outcome of the forecast is very sensitive to small errors in the initial state, so we run what we call an ensemble forecast. Instead of just running the model once, we make a series of small changes to the initial state and re-run the model a large number of times to get a set of forecasts. On some days the model runs may be similar, which gives us a high level of confidence in the forecast; on other days, the model runs can differ radically so we have to be more…

THE WEATHER MAN KEN MYLNE

SNATCH a toy from the tiniest of infants, and the reaction is likely to disappoint you. Most seem to conclude that the object has simply ceased to exist. This rapidly changes. Within the first year or so, playing peekaboo also becomes fun. As babies, we soon grasp that stuff persists unchanged even when we are not looking at it. Granted, at that age we know nothing of quantum theory. In the standard telling, this most well-tested of physical theories – fount of the computers, lasers and cellphones that our adult souls delight in – informs us that reality’s basic building blocks take on a very different, nebulous form when no one is looking. Electrons, quarks or entire atoms can easily be in two different places at once, or have many…

From a human perspective, physics has a problem with time. We have no difficulty defining a special moment called “now” that is distinct from the past and the future, but our theories cannot capture the essence of the moment. The laws of nature deal only with what happens between certain time intervals. David Mermin of Cornell University claims to have solved this problem using a principle similar to the one he and others have applied to quantum theory (see main story). We should simply abandon the notion that an objectively determinable space-time exists. Instead of forming a series of slices or layers that from some viewpoint correspond to a “now” or “then”, Mermin’s space-time is a mesh of intersecting filaments relating to the experiences of different people. “Why promote space-time…

Spaghetti functions

“There are probably only three basic topological shapes in pasta: cylinders, spheres, and ribbons” ALPHABETTI spaghetti: now there was a name to conjure with when I was a kid. Succulent little pieces of pasta, each shaped into a letter of the alphabet, served up in a can with lashings of tomato sauce. Delicious, nutritious – and best of all they made playing with your food undeniably educational. A few decades on, in an upscale Italian restaurant near the London offices of New Scientist, I decide against sharing this reminiscence of family mealtimes with my lunching partner. George Legendre doesn’t look quite the type. For one thing, he is French, and possibly indisposed to look kindly on British culinary foibles. For another, he is an architect, designer and connoisseur of all…

Spaghetti functions

The Italian designer Giorgetto Giugiaro has a string of supercars to his name, conceived for the likes of Ferrari, Maserati and Lamborghini. In 1999 he was voted “car designer of the century” by an international jury of motoring journalists. Less well known are his activities as a designer of pasta. In 1983, the Neapolitan manufacturer Voiello commissioned him to design a new shape compatible with the traditional manufacturing method of extrusion, in which the pasta dough is forced through a slit in a bronze die. In the event, his “Marille” design, consisting of two parallel tubes with a flap protruding from their join, rather landed him in hot water. While pleasing on the eye, its intricacy meant that different parts of the pasta cooked at vastly different rates. In 1987,…

Alice’s secrets in Wonderland

WHAT would Lewis Carroll’s Alice’s Adventures in Wonderland be without the Cheshire Cat, the trial, the Duchess’s baby or the Mad Hatter’s tea party? Look at the original story that the author told Alice Liddell and her two sisters one day during a boat trip near Oxford, though, and you will find that these famous characters and scenes are missing from the text. As I embarked a few years back on my DPhil investigating Victorian literature, I wanted to know what inspired these later additions. The critical literature focused mainly on Freudian interpretations of the book as a wild descent into the dark world of the subconscious. There was no detailed analysis of the added scenes, but from the mass of literary papers, one stood out: in 1984 Helena Pycior…

Alice’s secrets in Wonderland

IMAGINARY MATHEMATICS

The real numbers, which include fractions and irrational numbers like π that can nevertheless be represented as a point on a number line, are only one of many number systems. Complex numbers, for example, consist of two terms – a real component and an “imaginary” component formed of some multiple of the square root of -1, now represented by the symbol i. They are written in the form a + bi. The Victorian mathematician William Rowan Hamilton took this one step further, adding two more terms to make quaternions, which take the form a + bi + cj + dk and have their own strange rules of arithmetic.…

God, what a problem

“George Boolos was wont to pace the corridors solving problems with invisible chalk on an invisible blackboard” HERE’S a puzzle guaranteed to set your grey cells humming. “Three gods A, B, and C are called, in some order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language in which the words for ‘yes’ and ‘no’ are ‘da’ and ‘ja’, in some order. You do not know which word means which.” Welcome to the “Hardest Logic Puzzle…

God, what a problem

EXPLOSIVE LOGIC

Solving George Boolos’s “Hardest Logic Puzzle Ever” with “head-exploding” questions that have no true or false answer (see main story) puts the spotlight on the “law of non-contradiction”. This axiom of classical logic states that no proposition may be both true and false. Graham Priest of the City University of New York is one logician who thinks it is at best a half-truth. He has spent the past three decades developing “paraconsistent” logical systems that admit the existence of dialetheia, or true contradictions. The initial motivation was to get around the liar paradox – the 2500-year-old unsolved conundrum of what truth is in the statement “this statement is false”. “If you’re using a paraconsistent logic, you can tolerate that sort of contradiction without it causing havoc elsewhere,” says Priest. Some…

The real answer to life, the universe and everything

MAXIM MAKUKOV has an idea. It’s unorthodox; you might call it “out there”. Makukov understands that. He knew he’d have his critics the moment he began to develop it. But it’s there in the numbers, he says. And numbers don’t lie. A cosmologist and astrobiologist at the Fesenkov Astrophysical Institute in Almaty, Kazakhstan, Makukov says the numbers reveal that all terrestrial life came from outer space. Not only that, it was planted on Earth by intelligent aliens. Billions of years ago, the planet was barren and lifeless. But then, at some distant and unknowable moment, it was seeded with what Makukov calls an “intelligent-like signal” – a signal that is too orderly and intricate to have occurred randomly. This signal, he says, is in our genetic code. Highly preserved across…

The real answer to life, the universe and everything

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