A Mathematician Reads the Newspaper, John Allen Paulos
Ah, unrequited love. I love maths, but maths doesn’t love me. Still, it likes me enough for me to learn a lot from books like this. And I, like most people, do need to learn a lot about maths, because not knowing about it can lead you to make all sorts of mistakes and fall into all kinds of misunderstandings.
So we need more people like the mathematician John Allen Paulos, who knows a lot about maths and can express what he knows simply and entertainingly. This book is one of those that divide your life into BR and AR – Before Reading and After Reading – because it changes the way you look at the world. Take politics and important questions like the way we vote and the way power blocs work. Paulos examines all sorts of paradoxes and contradictions in both and you should come out of that section understanding the imperfections and dangers of democracy a lot better, as well as knowing that it’s possible to create a set of four dice, A, B, C, and D, in which A beats B, B beats C, C beats D, and D beats A.
Impossible? No, it’s very simple – once you know how. Or take the much vexed question of discrimination. Women are 50% of the population and blacks are (depending where you live) 5% and you should therefore expect them to be 50% and 5%, respectively, of MPs or bishops or disc-jockeys or senior managers in confectionery factories, shouldn’t you? And if they aren’t, that’s clear proof of discrimination, isn’t it?
Paulos’s answers are, respectively, no, not necessarily, and no, not necessarily. What is true of a general population is not necessarily true of its extremes:
As an illustration, assume that two population groups vary along some dimension – height, for example. Although it is not essential to the argument, make the further assumption that the two groups’ heights vary in a normal or bell-shaped manner. Then even if the average height of one group is only slightly greater than the average height of the other, people from the taller group will constitute a large majority among the very tall (the right tail of the curve). Likewise, people from the shorter group will constitute a large majority among the very short (the left tail of the curve). This is true even though the bulk of the people from both groups are of roughly average stature. Thus if group A has a mean height of 5’8” and group B has a mean height of 5’7”, then (depending on the exact variability of the heights) perhaps 90 percent or more of the those over 6’2” will be from group A. In general, any differences between two groups will always be greatly accentuated at the extremes.Discrimination undoubtedly exists, but where it exists and how much of an effect it has are not questions that can always be answered in simple ways. Paulos even describes how taking measures against discrimination can make its supposed effects worse.
Look before you leap, in other ways, and look with mathematically trained eyes. It will help you in all sorts of ways, from not being taken in by fallacious political arguments to not being ripped off. Suppose, Paulos asks, a pile of potatoes is left out in the sun. It’s 99% water and weighs 100 pounds. A day later, it’s 98% water. How much does it weigh now?
If you can’t work out the answer then you might be on your way to losing a lot of money if someone who does know it looks after your money or investments. Paulos explains the answer – which, surprisingly (or not), is 50 pounds – very clearly and simply, the way he explains the answers of all the other little puzzles he drops into the text as he discusses gossip, celebrity, cooking, bargains, infectious disease, and a myriad of other subjects that maths can either illuminate or obfuscate, depending on how well you understand it and the logic that underlies it.
The Penguin Dictionary of Curious and Interesting Mathematics, David Wells
Many people don’t see the beauty and excitement of maths, often because they weren’t introduced to it in the right way as children. This book can introduce adults and children alike to it in the right way. It starts with -1 and i (the square root of -1) and goes all the way through to Graham’s Number, which is so big that you could drive yourself mad trying to grasp just a fraction of it. En route, it introduces topics and ideas suitable for everyone from absolute beginners to the most advanced mathematicians. That is one of the beauties of maths: someone once described as it like an ocean in which a child can paddle and an elephant can swim. Wells discusses odd numbers, even numbers, rational numbers, irrational numbers, transcendental numbers, primes, Mersenne primes, factorials, logarithms, magic squares, Pascal’s triangle, the Rhind Papyrus (1650 BC), and much, much more, seasoning it all with a sprinkling of folklore and numerology and lots of ideas for recreational maths and musing. The Fibonacci numbers get a little of the attention they deserve (a book ten – or a hundred – times longer could only give them a little of the attention they deserve) and there’s also the solution to the problem of the largest number you can represent using only three digits and no other symbols. If you know what it is or not, read this book.
An Adventure in Multidimensional Space: The Art and Geometry of Polygons, Polyhedra, and Polytopes, Koji Miyazaki
Two, three, four – or rather, two, three, ∞. Polygons are closed shapes in two dimensions (for example, the square), polyhedra closed shapes in three dimensions (the cube), and polytopes closed shapes in four or more (the hypercube). You could spend a lifetime exploring any one of these geometries, but unless you take psychedelic drugs or brain-modification becomes much more advanced, you’ll be able to see only two of them: the geometries of polygons and polyhedra. Polytopes are beyond imagining but you can glimpse the shadows of their wonder and beauty here – literally, because we can represent polytopes by the shadows they cast in 3-space or by the shadows of their shadows in 2-space.
Elsewhere Miyazaki doesn’t have to convey wonder and beauty by shadows: not only is this book full of beautiful shapes, it’s beautifully designed too and the way it alternates black-and-white pages with color actually increases the power of both. It isn’t restricted to pure mathematics either: Miyazaki also looks at the modern and ancient art and architecture inspired by or reflecting the influence of geometry, and at geometry in nature: the dodecahedral pollen of Gypsophilum elegans, for example, and the tetrahedral seeds of the Water Chestnut, which the Japanese spies and assassins called the ninja used as natural caltrops. A regular tetrahedron always lies on a flat surface with a vertex facing directly upward, and when a pursued ninja scattered the sharply pointed tetrahedral seeds of the Water Chestnut, they were regular enough to injure “the soles of feet of his pursuers”.
The slightly odd English there is another example of what I like about this book, because it proves the parochialism of language and the universality of mathematics. Miyazaki’s mathematics, as far as I can tell, is flawless, like that of many other Japanese mathematicians, but his self-translated English occasionally isn’t. Japanese mathematics was highly developed before Japan fell under strong Western influence, and would continue to develop if the West disappeared tomorrow. Language is something we have to absorb intuitively from the particular culture we’re born into, but mathematics is learnt and isn’t tied to any particular culture, which is why it’s accessible in the same way to minds everywhere in the world. Miyazaki’s pictures and prose are an extended proof of that, and the book is actually more valuable because it was written by a Japanese speaker. I think it’s probably more attractively designed for the same reason: the skill with which the pictures have been selected and laid out reflects something characteristically Japanese. Elegance and simplicity perhaps sum it up, and elegance and simplicity are central to mathematics and some of the greatest art.
Platonic and Archimedean Solids: The Geometry of Space, Daud Sutton
An excellent little introduction to the five Platonic and thirteen Archimedean polyhedra, plus their stellations and duals. Each page of lucid text is complemented by a page of elegant figures, and though it doesn’t pretend to be an exhaustive treatise — at 57 pages it couldn’t possibly be — it would be a perfect present for a budding mathematician or for someone of any age interested in design. That could be true of other introductions to polyhedra, but what makes this one out-of-the-ordinary is that Daud Sutton lives and works as a faithful Muslim in Cairo. For example, the title page includes the elegant Arabic cartouche Bismillah Ar-Rahman Ar-Rahiim — “In the name of Allah the Compassionate, the Merciful”. There’s a remainder here of the beauties of Muslim architecture and of the former contribution of Arabs to mathematics, but there’s also a fascinating contrast between the almost plant-like complexity of the Arabic script, governed by the contingencies of human culture and psychology, and the crystalline simplicity of the polyhedra, governed by universal mathematical necessity. Yet the polyhedra’s simplicity vanishes on closer examination, for each is a world in itself, and mathematics actually underlies Arabic and its script too.
The Man Who Knew Infinity: A Life of the Genius Ramanujan, Robert Kanigel
Reading the life of Ramanujan (pronounced something like Raa-MAA-nuh-jun) is likely to put those of an old-fashioned literary bent in mind of Gray’s “Elegy in a Country Churchyard”, lines fifty-three to fifty-six:
Full many a gem of purest ray serene,Or to put it in other, less poetic words: some people never realize a minute fraction of their very great potential. Many more geniuses have been born than have ever been heard of, and Ramanujan was nearly one of those born but never heard of, because he was born into a poor family in southern India in 1887. Without luck and a lot of hard work by his friends, he might have never taken his rightful place in history besides the likes of Gauss and Euler as one of the most intellectually gifted human beings who have ever lived.
The dark unfathom’d caves of ocean bear;
Full many a flower is born to blush unseen,
And waste its sweetness on the desert air.
And if you’re wondering what he was gifted in, then you’ve obviously never heard of Gauss or Euler, which is a pity. Gauss and Euler were mathematicians and mathematics is probably the greatest of all human intellectual achievements, perhaps, paradoxically, because it is also the simplest and most direct of all subjects. That is why maths is so accessible to anyone with the right kind of mind. It doesn’t depend on language or race or culture but on intellect, and that is why Ramanujan, despite his background, was able to climb to its peak.
Though even at its peak there were mists of prejudice and culture, which was why it took some time before the men who shared the peak with him – even those further from the summit than he was – were able to recognize him as one of themselves: a supremely gifted mountaineer of the mind. Ramanujan wrote three letters to mathematicians at Cambridge University and was ignored twice. The third letter, however, reached a mathematician called G.H. Hardy, who glimpsed something in it that his colleagues had missed, gave it more time and thought, and realized the truth: that the gods of mathematicians had chosen a new favorite in a country thousands of miles from the wealthy centres of intellectual life in Europe and America.
Because Hardy was powerful and had a great deal of influence, he was able to have this new favorite of the gods brought to England. By doing so, he very probably killed him: Ramanujan died before he was forty, in 1920, and his death almost certainly had a great deal to do with the cold and poor diet he endured in England during the First World War. Robert Kanigel weaves that story into the wider tapestry of Ramanujan’s life and the still wider tapestry of British and Indian and Anglo-Indian history and produces not just one of the best scientific biographies I have ever read, but one of the best biographies of any kind. You don’t need to know anything about mathematics beyond the fact that it exists to appreciate the romance and tragedy of Ramanujan’s life, or its greatness, and one of the book’s central messages – that genius can so easily go unnoticed or unappreciated – has been a theme of literature too.
As my quotation from Gray proves. Ramanujan was lucky, though as a Brahmin he was less lucky than he might have been. If you don’t understand that, it’s another reason to read this book, because it will teach you a lot not just about a genius, and genius itself, but about Indian and British culture and history too.
The Joy of π, David Blatner
A delightful little book about a delightful big number: the ratio of the circumference of a circle to its diameter, aka π. The Bible says it’s three and though we knew far better by the nineteenth century, we still had fewer than a thousand digits. We had 707, in fact, and it wasn’t until 1945 that we discovered that some of them, calculated with enormous labor and dedication by the English mathematician William Shanks, were wrong.
1945 was the year someone set to work calculating π with the aid of a desk calculator, and was the start of the electronic race to find π with greater and greater accuracy. Fifty years later, in 1995, the Japanese mathematician Yasumasa Kanada had calculated 6 billion digits – that’s 6,000,000,000 – only for the Russian-American brothers David and Gregory Chudnovsky to hit back the following year with 8 billion. Kananda took the lead again in 1997 with 51·5 billion digits (and holds the record as of May 2005 with 1·2 trillion digits).
The story of π is a story of competition too, you see, and Blatner devotes a chapter to the Chudnovskys and their attempts to build ever more powerful computers to win and then win back the π-digit record. For almost all practical purposes, the competition is useless, and this quotation from the nineteenth-century Canadian astronomer Simon Newcomb tells you why:
Ten decimals [of π] are sufficient to give the circumference of the earth to the fraction of an inch, and thirty decimals would give the circumference of the whole visible universe to a quantity imperceptible to the most powerful microscope.But the quest for more and more digits does test computers and their software and programmers to their limits and mathematically speaking the digits are interesting because they can be tested for what is called normality. That is, are the digits of π effectively random, like those one would expect from rolling a perfect ten-sided die (or n-sided die in base n)? So far it seems that they are, and that is one of the paradoxes of π. A circle is the complete opposite of a random shape, and the ratio of its circumference to its diameter has a completely fixed value. Yet the digits of that ratio seem to be completely unpredictable.
But the quest for more and more digits is valuable for two other reasons – symbolic ones. The English mountaineer George Malory said that he wanted to climb Everest because “it was there.” If π-nauts try to find the digits of π because they are there, they are only there because we have in fact found ways of “predicting” them. Mathematicians have discovered many finite formulae for an infinite sequence of digits, and perhaps the most astonishing so far is this:
The Σ part means the sum of k from one to infinity. In other words, you can find the kth hexadecimal digit of π simply by calculating the formula for k, without having to calculate all the digits before it. It can’t actually be converted to a decimal digit unless you know all the digits before it, but that doesn’t destroy the interest of the formula and perhaps a decimal equivalent will be discovered one day.
The second symbolic value of the quest for ever more digits of π is that the quest is being carried out by men. The story of π is a male story, or rather, the story of the human relationship with π is a male story. Mathematics is beyond sex and personality, but for various biological reasons mathematics, as practised and applied by human beings, is overwhelmingly dominated by men. The ethnicity of Kanada and (I presume) the Chudnovsky brothers is symbolically important too: East Asians like the Japanese have a higher-than-average IQ and Ashkenazi Jews have a much higher-than-average IQ – Ashkenazim are hugely over-represented among mathematicians, just as they are hugely over-represented among grandmasters of chess.
Blatner, who I presume is himself Jewish, doesn’t comment on race and biology, but it’s one of the most interesting aspects of mathematical contingency: the way the necessary truths of mathematics are discovered by and influence human beings. Much less interesting, for me, are other aspects of mathematical contingency: the appearance of π in popular culture, for example. Blatner looks at these too in passing, and includes a list of π mnemonics in various languages. My favorite is this one in Spanish, in which the number of letters in each word stands for a digit of π:
Sol y Luna y Mundo proclaman al Eterno Autor del Cosmo.With no accents and digraphs and every letter standing for exactly one sound, it’s about as close as language gets to the clarity and concision of mathematics. This book is an excellent popular insight into that clarity and concision, and more beside.(Sun and Moon and Earth acclaim the Eternal Creator of the Cosmos.)
Paradoxes in Probability Theory and Mathematical Statistics, Gábor J. Székely
A fascinating book in a number of ways. First the obvious way: probability contains some of the strangest and most counter-intuitive mathematics easily open to amateurs and dabblers. I can actually understand a lot of this book, but it still stretches and even re-shapes my mind and my understanding of the world more far deeply than almost all art has ever done. There are very odd things to be found even in something as simple as the patterns of heads-and-tails in coin-tossing. For example, although HH and HT are equally likely to occur first when you start tossing a fair coin, “more tosses are necessary, on average, for HH than for HT to turn up”. That just doesn’t make sense at first glance. It’s a paradox, in other words, and if you can understand it you’ve taken a step even the most intelligent human beings were once completely unable to take.
Much less subtle, but probably much more important in life, is this:
Consider two random events with probabilities of 99% and 99.99%, respectively. One could say that the two probabilities are nearly the same, both events are almost sure to occur. Nevertheless the difference may become significant in certain cases. Consider, for instance, independent events which may occur on any day of the year with probability p = 99%; then the probability that it will occur every day of year is less than P = 3%, while if p = 99.99%, then P = 97%. (ch. 1, “Classical paradoxes of probability theory”, pp. 54-5)Then there’s the question of why buses always seem to run “more frequently in the opposite direction”. The mathematics gets much trickier here, but that’s an example of how mathematical analysis, unlike so much of what passes for analysis in the modern humanities, extracts deep meaning from apparently simple things because it’s actually there to be extracted. Mathematics is both the most fundamental and the purest of all subjects, and is something that can unite minds across barriers of language, culture, and politics.
This book is actually a good example of that, because it was first published not only in a communist country but in what is, to almost all Europeans, a very strange European language: Hungarian. Hungarian isn’t related to the Indo-European family spoken almost everywhere else. If this book had been written in French or German or Spanish, its original title would look more or less familiar to an English-speaker. But its original title in Hungarian — Paradoxonok a véletlen matematikában — looks very odd. Even without being told you could guess from some of the English that the book is a translation, but that adds to its charm and helps prove the universality of mathematics. A Hungarian can speak mathematics to anyone without an accent, and vice versa — though I suspect that the mathematics here occasionally stutters because of typos. The English certainly does, but then the book was printed under communism, with all the inefficiency and carelessness that entailed. Communism is gone now, Hungarian and maths both continue, but maths will outlast Hungarian too, just as it will outlast all other languages spoken today. Vivat regina.
e: The Story of a Number, Eli Maor
The test of lucid writing isn’t that it is easy to understand but that it is as easy to understand as it can be. The writing in this book is not always easy to understand, but it’s still some of the most lucid I’ve ever come across. Less laudably, it was strangely repetitive too. This appears on page 124:
This makes the spiral a close relative of the circle, for which the angle of intersection is 90°. Indeed, the circle is a logarithmic spiral whose rate of growth is 0…And this on page 134:
This property [of intersecting any straight line through the pole at the same angle] endows the [logarithmic] spiral with perfect symmetry of the circle – indeed the circle is a logarithmic spiral for which the angle of intersection is 90° and the rate of growth is 0.That aside, I can recommend this book highly as a history and survey of the most overlooked of the three great mathematical constants. The most recently recognized too, but then there’s an obvious reason for all that. π and φ have simple definitions: the ratio of a circle’s circumference to its diameter and the ratio x/y such that (x+y)/x = x/y. e, the base of natural logarithms, doesn’t have such a simple definition: it’s the limit of the equation (1+1/n)n as n = ∞, and begins 2·7182182... That misleading double “182” is an artefact of its representation in base 10: e is not only irrational, like φ, which means its digits never begin repeating, but transcendental too, like π. But if e became familiar to mathematicians thousands of years later than π, it got a symbol of its own at nearly the same time. As David Blatner describes in The Joy of π, the symbol π was popularized, but not invented, by the great Swiss mathematician Leonhard Euler (pronounced “Oiler”), but Euler seems to have both invented and popularized e. Maor lays to rest an old story:
Why did he choose the letter e? There is no general consensus. According to one view, Euler chose it because it is the first letter of the word exponential. More likely, the choice came to him naturally as the first “unused” letter of the alphabet, since the letters a, b, c, and d frequently appear elsewhere in mathematics. It seems unlikely that Euler chose the letter because it was the initial of his own name, as has occasionally been suggested: he was an extremely modest man and often delayed publication of his own work so that a colleague or student of his could get the credit. (ch. 13, ‘eix: “The Most Famous of All Formulas”’, pg. 156)But Euler certainly deserved to have a mathematical constant named in his honor, if for no other reason – and there are certainly lots of other reasons – than his discovery of the relationship explored in this chapter: eix = -1, which “appeals equally to the mystic, the scientist, the philosopher, the mathematician”. Rather like this book as a whole, and though some of it was well beyond me, it’s a model of pop math, from the mathematically rigorous – its examination of the catenary, or the shape made by a hanging chain, for example – to the culturally quirky. I’ve often read before that Jakob Bernoulli, one of a Swiss family that was the mathematical equivalent of the Bachs, asked for a logarithmic spiral to be carved on his tombstone with the words Eadem mutata resurgo: “Even though changed I rise again”. But I read for the first time here that the engraver got it wrong out of ignorance or laziness and used an Archimedean spiral instead. Not only that, I got to see the tombstone itself. That’s dedicated research, and though dedicated research doesn’t guarantee a good book, it’s part of what makes this book so good.
What Shape Is A Snowflake? Magical Numbers in Nature, Ian Stewart
The book of a TV series that never was: lots of pretty pictures, lots of simplistic anecdotes, no hard information, no intellectual challenges. The ideas examined – fractals, complexity theory, chaos, animal gaits and skin patterns, and the relation of mathematics to reality – are fascinating, but their treatment is superficial and there are no footnotes to guide readers quickly to more detailed sources of information. Stewart seems to have boiled down books like Does God Play Dice?, Fearful Symmetry, and Life’s Other Secret, thrown away the residue, and used the condensation on the windows instead. What Shape Is A Snowflake? would be good as an introduction to these ideas for an intelligent teenager or an adult with an arts degree, but if you don’t fall into one of those categories you’d be much better off with one of the serious books named above.
The Golden Ratio: The Story of Phi, the Extraordinary Number of Nature, Art and Beauty, Mario Livio
A good short popular guide to perhaps the most fascinating, and certainly the most irrational, of all numbers: the golden ratio or phi, which is approximately equal to 1·6180339... Prominent in mathematics since at least the ancient Greeks and Euclid, phi is found in many places in nature too, from pineapples and sunflowers to the flight of hawks, and Livio catalogs its appearances in both realms, with particular attention to rabbit-breeding and the Fibonacci sequence, before going on to debunk mistaken claims of its appearances in art, music, and poetry. Dalí certainly used it, but da Vinci, Debussy, and Virgil almost certainly didn’t, and neither, almost certainly, did the builders of the Parthenon and pyramids. Finally, he examines what has famously been called (by the physicist Eugene Wiegner) the unreasonable effectiveness of mathematics: why is this human invention so good at describing the behavior of the Universe? Livio quotes one of the best short answers I’ve yet seen to the question:
Human logic was forced on us by the physical world and is therefore consistent with it. Mathematics derives from logic. That is why mathematics is consistent with the physical world. (ch. 9, “Is God a mathematician?”, pg. 252)Any book that can quote Jef Raskin, “the creator of the Macintosh computer”, with Johannes Kepler, William Blake, Lewis Carroll, and Christopher Marlowe, has to be recommended, and recreational mathematicians should find lots of ideas for further investigation, from fractal strings to the fascinating number patterns governed by Benford’s law. It isn’t just human beings who look after number one: as a leading figure, 1 turns up much more often in data from the real world, and in mathematical constructs like the Fibonacci sequence, than intuition would lead you to expect. If you’d like to learn more about that and about many other aspects of mathematics, hunt down a copy of this book.
© 2005-6 Simon Whitechapel