Posts Tagged ‘maths’

Infinitesimal by Alexander AmirInfinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, Amir Alexander (Oneworld 2014)

Infinitesimal is an entertaining read on a fascinating topic: the pioneers of a new form of mathematics and those who opposed them. Amir Alexander claims that “the ultimate victory of the infinitely small helped open the way to a new and dynamic science, to religious toleration, and to political freedoms unknown in human history” (Introduction, pg. 14).

It’s an extraordinary claim and I don’t think he manages to provide extraordinary proof for it. In fact, he probably gets cause-and-effect reversed. Is it likelier that new mathematics opened minds, dynamized science and transformed politics or that open minds created new forms of mathematics, science and politics? I’d suggest that support for the new mathematics was a symptom, not a cause, of a new psychology. But Alexander makes a good case for his thesis and there is no doubt that the world was changed by the willingness of mathematicians to use infinitesimals. Calculus was one result, after all. The book begins in Italy and ends in England, because the pioneers lost in Italy:

For nearly two centuries, Italy had been home to perhaps the liveliest mathematical community in Europe. … But when the Jesuits triumphed over the advocates of the infinitely small, this brilliant tradition died a quick death. With Angeli silenced, and Viviani and Ricci keeping their mathematical views to themselves, there was no mathematician left in Italy to carry on the torch. The Jesuits, now in charge, insisted on adhering close to the methods of antiquity, so that the leadership in mathematical innovation now shifted decisively, moving beyond the Alps, to Germany, England, France and Switzerland. (ch. 5, “The Battle of the Mathematicians”, pg. 178)

Why were the Jesuits involved in an esoteric mathematical dispute? You might say that de minimis curat Loyola – Ignatius Loyola (1491-1556), founder of the Jesuits, cared about anything, no matter how small, that might undermine the authority of the Church. In the view of his successors, the doctrine of indivisibles did precisely that: “in its simplest form, the doctrine states that every line is composed of a string of points, or ‘indivisibles’, which are the line’s building blocks, and which cannot themselves be divided” (Introduction, pg. 9).

Indivisibles must be infinitesimally small, or they wouldn’t be indivisible, but then how does an infinitesimal point differ from nothing at all? And if it isn’t nothing, why can’t it be divided? These paradoxes were familiar to the ancient Greeks, which is why they rejected infinitesimals and laid the foundations of mathematics on what seemed to them to be solider ground. In the fourth century before Christ, Euclid used axioms and rigorous logic to create a mathematical temple for the ages. He proved things about infinity, like the inexhaustibility of the primes, but he didn’t use infinitesimals. When Archimedes broke with Greek tradition and used infinitesimals to make new discoveries, “he went back and proved every one of them by conventional geometrical means, avoiding any use of the infinitely small” (Introduction, pg. 11).

So even Archimedes regarded them as dubious. Aristotle rejected them altogether and Aristotle became the most important pre-Christian influence on Thomas Aquinas and Catholic philosophy. Accordingly, when mathematicians began to look at infinitesimals again, the strictest Catholics opposed the new development. Revolutionaries like Galileo were opposed by reactionaries like Urban VIII.

But the story is complicated: Urban had been friendly to Galileo until “the publication of Galileo’s Dialogue on the Copernican system and some unfavourable political developments” (pg. 301). So I don’t think the mathematics was driving events in the way that Alexander suggests. Copernicus didn’t use them and the implications of his heliocentrism were much more obvious to many more people than the implications of infinitesimals could ever have been. That’s why Copernicus was frightened of publishing his ideas and why Galileo faced the Inquisition for his astronomy, not his mathematics.

But Amir’s thesis makes an even more interesting story: the tiniest possible things had the largest possible consequences, creating a new world of science, politics and art. In Italy, two of the chief antagonists were Galileo and Urban; in England, two were the mathematician John Wallis (1616-1703) and the philosopher Thomas Hobbes (1588-1679). Alexander discusses Wallis and Hobbes in Part II of the book, “Leviathan and the Infinitesimal”. Hobbes thought that de minimis curat rex – “the king cares about tiny things”. Unless authority was absolute and the foundations of knowledge certain, life would be “nasty, brutish and short”.

However, there was a big problem with his reasoning: he thought he’d achieved certainty when he hadn’t. Hobbes repeatedly claimed to have solved the ancient problem of the “quadrature of the circle” – that is, creating a square equal in size to a given circle using only a compass and an unmarked ruler. Wallis demolished his claims, made Hobbes look foolish, and strengthened the case for religious toleration and political freedom. But I don’t think this new liberalism depended on new mathematics. Instead, both were products of a new psychology. Genetics will shed more light on the Jesuits and their opponents than polemics and geometry textbooks from the period. Alexander’s theory is fun but flawed.


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The Secret Footballer's Guide to the Modern GameThe Secret Footballer’s Guide to the Modern Game: Tips and Tactics from the Ultimate Insider, The Secret Footballer (Guardian Books 2014)

Who is the Secret Footballer? I don’t know. But he’s definitely a Guardianista. You can tell this by two things: 1) he’s passionately committed to the fight against “homophobia, sexism, racism and everything in between”; 2) he uses “in terms of” a lot. Interviewing another concealed component of the crypto-community, The Secret Physio, he asks this:

TSF: So would players need to train differently from one another in terms of the weights they lift and the core work they do? (ch. 1, “Getting Started”, pg. 14)

“Core” is also Guardianese and maybe he’s really interviewing himself, because the Secret Physio uses “in terms of” too. I didn’t spot the incendiary slam-dunk of a mixed metaphor anywhere, but he does claim that Wayne Rooney is “one of quite literally only a handful of players” who matter a lot to Manchester United’s profits (ch. 4, “It’s Football, But Not As We Know It”, pg. 116). So case proven: he’s a Guardianista.

But he’s also worth reading and this is his most interesting book. He talks about world football and the game in general, not just his life in the Premier League, and he seems to know his stuff. I don’t. To me football is like music: I appreciate it without understanding it. I know what players, teams and matches I like, but I don’t have a clue about tactics or formations.

The Secret Footballer combines appreciation with understanding, so it’s gratifying that he praises three of my favourite players: Glen Hoddle, Matt Le Tissier and Dennis Bergkamp. He says that Hoddle proved that “an entire football nation did not know what to do with skill and finesse” (Epilogue, pg. 218) and lists Le Tissier and Bergkamp among the scorers of “The goals that influenced me most”. This is Le Tissier’s:

…his finest goal, in my opinion, came against Newcastle in 1993. It is so skilful that it deserves to grace most lists. The three touches he takes to get the ball under control while beating a defender at the same time are by no means easy and all have to be perfect. I later read that the slightly scuffed finish had taken the gloss off it for Le Tissier himself, but, for me, it serves as a lesson in composure for every kid who wants to be a striker. (ch. 1, pp. 52-3)

This is Bergkamp’s, against Newcastle in 2002:

Almost every other player I have seen would try to control the horrible bouncing ball that comes into him. But Bergkamp, with his back to goal, flicks it to one side of the defender and runs the other, using his strength to outmuscle the defender and find the calmest of finishes. For a long time, some people debated whether or not Dennis had actually intended to do what he did here. Like so many others, those people don’t truly understand football. (Ibid., pg. 54)

But what does it mean to “truly understand football”? Ultimately, it means using mathematics. There’s maths everywhere in football and everywhere in this book, from the topspin on a free kick (ch. 1, pg. 41) to 4-2-3-1, “the most in-vogue formation in modern football” (ch. 6, “Formations”, pg. 158). A good footballer has to be both an athlete and an expert in reading and responding to patterns. The movement of players on the field sets constantly shifting problems in combinatorics, for example. There’s no entry for “Mathematics” in the index, but then there’s no entry for “English language” either. This book is written in English and is talking about maths, implicitly but intensively.

That’s as true in the section about diet as it is in the section about using spin in free-kicks. One is physiology, the other is physics, but they both involve the interaction of entity that is the essence of mathematics. The spin of the ball affects its interaction with the air. Chemicals in the body affect its interaction with play: its strength, stamina, flexibility and so on. That’s why diet is so important. But chemicals are important in other ways. To physiology and physics you can add physiognomy, as a recent scientific paper shows:

The structure of a soccer player’s face can predict his performance on the field – including his likelihood of scoring goals, making assists and committing fouls – according to a study led by a researcher at the University of Colorado Boulder.

The scientists studied the facial-width-to-height ratio (FHWR) of about 1,000 players from 32 countries who competed in the 2010 World Cup. The results, published in the journal Adaptive Human Behavior and Physiology, showed that midfielders, who play both offense and defense, and forwards, who lead the offense, with higher FWHRs were more likely to commit fouls. Forwards with higher FWHRs also were more likely to score goals or make assists. (Facial structure predicts goals, fouls among World Cup soccer players, ScienceDaily, 12/xi/2014)

Facial structure is influenced by testosterone, which also influences competitiveness and aggression. And testosterone itself is influenced by genetics. Football was invented and is still dominated by men. That won’t change until the human race changes. And it will be men who invent the means for the human race to change.

Or rather: the human races, because there are a lot of them. The big ones – Europeans, Africans and Asians – are all represented in this book and the Secret Footballer writes a lot about genetic differences, even though he doesn’t know it. And would be horrified by the claim that it matters. As a Guardianista, he knows we’re all the same under the skin and that environment is responsible for the way blacks contribute little to science and mathematics. Blacks contribute a lot to football, but not as managers and not as certain types of player: goalkeeper, for example.

Why not? The Secret Footballer would say it’s racism and lack of opportunity. I would say it’s lack of intelligence. But lack of intelligence is due to racism and lack of opportunity too, isn’t it? No, I’d say it’s due to genetics. Why is the performance of the brain less influenced by genes than the performance of the muscles? It isn’t. Sadly for Guardianistas, hateful stereotypes like this are based on a hateful genetic reality:

Speedboat, no driver: Refers to a player who has blistering pace but no clue where he is supposed to be running or when. Controversially, this phrase is typically used for young black players. There are lots of managers who do not trust black players with the disciplined side of the game and just tell them to run instead – I even had a manager who did not want to play black centre-halves because he was convinced that they had tunnel vision and didn’t read the game well. I can’t disprove it one way or another, though it sounds ridiculous to me. However, I’m here to tell you that lots of managers feel this way and I’ve lost count of managers, coaches, academy coaches and players who describe young black players using this term. It’s even been said to me on the pitch by an opposition player when we brought on a young black player in the second half. (“Appendix: The Guide to Modern Football Language”, pg. 228)

Genetics at work, in my opinion: the environment of Africa selected for athletic ability but not high intelligence. Football is not just a beautiful game. It’s a bountiful one too, because it offers so many patterns to analyse: patterns of play, of history, of culture, race, human behaviour and biology in general. The Secret Footballer discusses all of them, sometimes without realizing it. He’s interesting, opinionated and obsessed with the game. I’m not and never have been, but this book woke memories of the days when I cared much more about twenty-two men chasing an inflated sphere around a rectangular field.

Perhaps I should care more now, because the game has never stopped evolving and improving, as the Secret Footballer will show you. There are some exciting names in his list of the “ten best players of the last twenty years”: Lionel Messi, Zinedine Zidane, Cristiano Ronaldo, Xavi Hernández, Ronaldinho, Paul Scholes, Paolo Maldini, Thierry Henry, Ryan Giggs, Andrés Iniesta (ch. 6, pg. 186). He also offers his “ten best players of the future playing now” (ch. 7, “Coaching”, pg. 206) and lists the “best young players you probably haven’t heard of… yet” (ch. 3, “Fashion in Football”, pg. 104) And where does he stand on one of the great questions of our time? Here:

Cristiano Ronaldo once said that God put him on this planet to play football. We’ll just have to ask Lionel Messi if he remembers doing that. (ch. 8, “Whatever Happens, Never, Ever Give Up”, pg. 215)

There’s also Nike vs Adidas, Mark Viduka singing Monty Python in Middlesbrough and an explanation of why England are so bad. And for once a good popular book isn’t spoilt by a bad literary omission, because there’s a detailed index. I don’t like the Guardian, but it occasionally comes up with good things and this guide is one of them.

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Numericon by Marianne Freiberger and Rachel ThomasNumericon, Marianne Freiberger and Rachel Thomas (Quercus Editions 2014)

An easy-to-read introduction to some profound and difficult ideas. You can start climbing the foothills of Mount Maths here and train for the icy cliffs and avalanche-scoured slopes above. And perhaps the mountain rises for ever. There may be important mathematical questions that are impossible to answer. But some answers we already have, like the irrationality of √2 and the infinitude of the primes, are astonishing to contemplate. How are finite creatures able to discover facts about infinity? Logic is a lever that can move more than the universe. Euclid described 2,300 years ago how it could be used to prove that primes never ended: for every integer, there is a prime.

Numericon explains his proof, then goes on to describe how the German Georg Cantor (1845-1918) used logic to prove that there is more than one kind of infinity. In fact, there’s an infinite hierarchy of infinities, climbing endlessly into the metaphysical empyrean. That seems like an insane idea and Cantor did end his life in a sanatorium. He’s an extreme example of something Freiberger and Thomas note in their introduction: that mathematicians are often eccentric. It’s a strange subject that doesn’t come naturally to human beings. We can’t escape it, because our brains and bodies are governed by mathematics, but we don’t need to apply it explicitly and consciously to get through life. Those who lift the surface of reality and gaze upon its brilliant mathematical core can find that the light troubles and even subverts their brains.

And some avert their gaze. The German Carl Friedrich Gauss (1777-1855) may have been the greatest mathematician who ever lived, but he declined to publish his revolutionary ideas about Euclid’s fifth axiom: that parallel lines never meet. New geometries arise if the axiom is eliminated, but Gauss didn’t claim the credit for discovering them. As this book describes, it was the younger mathematicians János Bolyai (1802-60) and the Russian Nikolai Lobachevsky (1792-1865). And Bolyai’s mathematician father had tried to warn his son off.

Mathematicians need a sense of adventure. They need recognition and support too, but sometimes even the great ones find both hard to win. Sometimes their greatness is part of the problem, because the work they produce is too new and powerful to be properly appreciated. The Norwegian Niels Henrik Abel (1802-29) died at 26 “from tuberculosis and in abject poverty” (pg. 183). The Frenchman Évariste Galois (1811-32) died even younger, at 21, fatally wounded in a duel. Perhaps it was an affair de cœur, perhaps a plot by his enemies: Galois was revolutionary in both his mathematics and his politics. The Indian Srinivasa Ramanujan (1887-1920) is another great who died young after making discoveries of permanent value.

This book discusses all three and many more, but mathematicians aren’t truly important. Mathematics is necessary; mathematicians are contingent. If advanced civilizations exist on other planets or in parallel universes, their art and literature might be entirely different from our own or might not exist at all. But their mathematics would be recognizable, whatever the nature of the brains that had discovered it. Advanced civilization is impossible to imagine without mathematics and although we can ignore mathematics if we choose, we’re missing something central to existence if we do. As Darwin said: it seems to give one an extra sense. It’s a mind’s eye that can see into infinity – and into infinitesimality.

As though to reflect the importance of mind in maths, this book has relatively few and simple illustrations. Maths can yield gorgeous complexity for real eyes, but the mind’s eye is more important. So this book is eye-candy for the mind. Eye-brandy too: maths can both dazzle you and make you drunk. It’s appropriate that Numericon echoes Necronomicon, because new worlds wait within the pages of both.

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Classic Horror Stories by H.P. LovecraftH.P. Lovecraft: The Classic Horror Stories, edited by Roger Luckhurst (Oxford University Press 2013)

Lovecraft has come a long way. From the margins to the mountebanks, you might say, because he’s getting serious interest from American and British academics nowadays. In France, he got it a long time ago:

In the late 1960s, the French academic Maurice Lévy wrote a thesis on Lovecraft as a serious fantasiste, continuing the French love of all things tinged with Poe. In turn, the radical philosophers Gilles Deleuze and Félix Guattari used Lovecraft as a touchstone for notions of unstable being and becoming-other in their revolutionary manifesto, A Thousand Plateaus (1980). (“Introduction”, pg. xiii)

I didn’t realize it was as bad as that. Then again, I already knew that the Trotskyist gasbag China Miéville had been influenced by Lovecraft and had intensively interrogated issues around Lovecraft’s racism and xenophobia. Roger Luckhurst interrogates them too. After all, they’re a glaring flaw in an important and highly influential writer. How could HPL have been so egregiously wrong and in such an offensive way?

Well, perhaps he wasn’t wrong and perhaps he wouldn’t have written so imaginatively and powerfully without his crime-think. The psychologist Hans Eysenck suggested that psychoticism — which is distinct from psychosis – was essential to genius. But was HPL a genius? In his way, I think he was. It wasn’t a purely literary way and perhaps HPL is bigger than literature. He wasn’t a genius like Dickens or Kipling, because you don’t read Lovecraft for literary skill, psychological subtlety and clever characterization. No, you read him for sweep and scale, grandeur and grotesqueness, darkness and density. You should also read him for humour:

In February the McGregor boys from Meadow Hill were out shooting woodchucks, and not far from the Gardner place bagged a very peculiar specimen. The proportions of its body seemed slightly altered in a queer way impossible to describe, while its face had taken on an expression which no one ever saw in a woodchuck before. (“The Colour out of Space”, 1927)

Like J.G. Ballard, Lovecraft is often misread as lacking humour. In fact, like Ballard, he’s often very funny. This book is a joke he would have appreciated: there’s something blackly humorous about his posthumous elevation to hard covers and high-quality paper under the auspices of the Oxford University Press. His work is now getting more care than his body did: as Luckhurst notes in the introduction, HPL died of stomach cancer at 47 as “an unknown and unsuccessful pulp writer” (pg. xii). Is he better in a pulp paperback, with battered covers, yellowing paper and no notes? Yes, I think he is, but he’s best of all when he’s both paperback and hardback. I don’t like literary studies in their modern form, but Roger Luckhurst doesn’t slather HPL in jargon or suffocate the stories with notes.

So the notes aren’t intrusive, but they are instructive – for example, about HPL’s modesty and self-doubt. Did he really think “At the Mountains of Madness” (1936) “displayed evidence of a ‘lack of general ability’ and a mind corrupted by ‘too much reading of pulp fiction’” (“Explanatory Notes”, pg. 470)? Then he was a giant who mistook himself for a pygmy. But that’s better than the reverse. Most of his greatness is collected here, from “The Call of Cthulhu” to “The Shadow Out of Time”, though I would have dropped “The Horror at Red Hook” and included “The Music of Erich Zann”. I would also like to drop China Miéville and include J.G. Ballard, but unfortunately HPL didn’t influence Ballard. I wish he had. Mutual influence would have been even better.

Nietzsche did influence Lovecraft and Lovecraft’s work can be read as, in part, an attempt to confront the death of God. Spirit departs the world; science invades. Where are wonder and horror to be found now? In “The Call of Cthulhu” or “At the Mountains of Madness”, stories that draw on astronomy, geology and biology to awe us with space, time and organic possibility. And Lovecraft, unlike Nietzsche or Ballard, recognized the importance of mathematics. That’s most evident here in “The Dreams in the Witch-House” (1933), which mixes trans-Euclidean geometry with ancient superstition. But maths isn’t the only influence on this story: so is M.P. Shiel’s novel The House of Sounds (1896). I didn’t know about that and I’m glad to have learnt it. That’s good scholarship, introducing readers to older authors and deeper influences. It still doesn’t feel right to read Lovecraft on clean white paper in a heavy book, but it’s good that he’s come up in the world. Let him bask in the sun before the Übermensch arrives.

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Front cover of A Mathematician's Apology by G.H. HardyA Mathematician’s Apology, G.H. Hardy (1940)

The World Wide Web is also the Random Reading Reticulation – the biggest library that ever existed. Obscure texts and ancient manuscripts are now a mouse-click away. A Mathematician’s Apology is neither obscure nor ancient, but it wasn’t easy to get hold of before it became available online. I’ve wanted to read it for a long time. And now I have.

Alas, I was disappointed. G.H. Hardy (1877-1947) was a very good mathematician, but he’s not a very good writer about mathematics. And in fact, he didn’t want to be a writer at all, good, bad or indifferent:

It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds. (Op. cit., Section 1)

But is philosophy of mathematics work for second-rate minds? At the highest level, I don’t think it is. The relation of mathematics to reality, and vice versa, is a profoundly interesting and important topic, but Hardy doesn’t have anything new or illuminating to say about it:

It may be that modern physics fits best into some framework of idealistic philosophy — I do not believe it, but there are eminent physicists who say so. Pure mathematics, on the other hand, seems to me a rock on which all idealism founders: 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is, because mathematical reality is built that way. (Section 24)

He experienced and explored that mathematical reality, but he can’t communicate the excitement or importance of doing so very well. I wasn’t surprised by his confession that, as a boy: “I thought of mathematics in terms of examinations and scholarships: I wanted to beat other boys, and this seemed to be the way in which I could do so most decisively” (sec. 29). He says it wasn’t until he had begun his degree at Cambridge that he “learnt for the first time … what mathematics really meant” (ibid.).

In this, he was very different from someone he helped make much more famous than he now is: an unknown and struggling Indian mathematician called Srinivasa Ramanujan, who sparked Hardy’s interest by sending him theorems of startling originality and depth before the First World War. Hardy brought Ramanujan to England, but barely mentions his protégé here. All the same, his respect and even perhaps his affection are still apparent:

I still say to myself when I am depressed, and find myself forced to listen to pompous and tiresome people, ‘Well, I have done one thing you could never have done, and that is to have collaborated with both [J.E.] Littlewood and Ramanujan on something like equal terms.’ (sec. 29)

Very few people could have done that: a mere handful of the many millions who lived at the time. So it would be wrong to expect that Hardy could both ascend to the highest peaks of mathematics and write well about what he experienced there. He couldn’t and A Mathematician’s Apology supplies the proof. That’s a shame, but the text is short and still worth reading. Hardy had no false modesty, but he had no delusions of grandeur either:

I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world. I have helped to train other mathematicians, but mathematicians of the same kind as myself, and their work has been, so far at any rate as I have helped them to it, as useless as my own. Judged by all practical standards, the value of my mathematical life is nil; and outside mathematics it is trivial anyhow. I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating. And that I have created is undeniable: the question is about its value.

It was valuable: that is why Hardy is still remembered and celebrated, sixty-seven years after his death. He is also still famous as an atheist, but you could say that he spent his life in the service of Our Lady – Mathematica Magistra Mundi, Mathematics Mistress of the World.

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Book in BlackBlack Sabbath: Symptom of the Universe, Mick Wall (Orion Books 2013)

Critical Math – A Mathematician Reads the Newspaper, John Allen Paulos (Penguin 1996)

Rude BoysRuthless: The Global Rise of the Yardies, Geoff Small (Warner 1995)

K-9 KonundrumDog, Peter Sotos (TransVisceral Books 2014)

Ghosts in the CathedralThe Neutrino Hunters: The Chase for the Ghost Particle and the Secrets of the Universe, Ray Jayawardhana (Oneworld 2013) (posted @ Overlord of the Über-Feral)

Or Read a Review at Random: RaRaR

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A Mathematician Reads the Newspaper by John Allen PaulosA Mathematician Reads the Newspaper, John Allen Paulos (Penguin 1996)

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 writers 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. You’ll also know 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 horrors of discrimination in terms of issues around race and gender. Women are about 50% of the British population and non-whites are about 10% and you should therefore expect them to be 50% and 10%, respectively, of MPs or judges 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 always 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, who it’s being exercised against 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 a conman 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 host of other subjects that maths can either illuminate or obfuscate, depending on how well you understand it and the logic that underlies it.

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Brit GritGranite and Grit: A Walker’s Guide to the Geology of British Mountains, Ronald Turnbull (Francis Lincoln 2011)

Singh Summing SimpsonsThe Simpsons and Their Mathematical Secrets, Simon Singh (Bloomsbury 2013)

Go with the QuoStatus Quo: Still Doin’ It – The Official Updated Edition, compiled by Bob Young, edited by Francis Rossi and Rick Parfitt (Omnibus Press 2013)

Breeding BunniesThe Golden Ratio: The Story of Phi, the Extraordinary Number of Nature, Art and Beauty, Mario Livio (Headline Review 2003) (posted @ Overlord of the Über-Feral)

Brit Bot BookReader’s Digest Field Guide to the Wild Flowers of Britain, J.R. Press et al, illustrated Leonora Box et al (1981) (@ O.o.t.Ü.-F.)

Or Read a Review at Random: RaRaR

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Front cover of The Simpsons and Their Mathematical Secrets by Simon SinghThe Simpsons and Their Mathematical Secrets, Simon Singh (Bloomsbury 2013)

I don’t like The Simpsons and I don’t think Simon Singh is a very good writer. But there is some interesting maths in this book. As the Emperor Vespasian said when criticized for taxing urinals in Rome: Pecunia non olet – “money doesn’t smell”. And simple sources can yield riches in other ways. There’s a good example of that in chapter 9 of this book, “To Infinity and Beyond”, where Singh looks at the mathematics of pancake-sorting. It was first discussed in 1975 by the geometer Jacob E. Goodman of the City College of New York. Suppose there’s a pile of pancakes of different sizes. You can insert a spatula at any point in the pile and flip the block of pancakes above it. Goodman posed this question about sorting the pancakes into order of size:

If there are n pancakes, what is the maximum number of flips (as a function of n) that I will ever have to use to rearrange them? (ch. 9, pg. 110)

It sounds simple, but isn’t. As the pile gets higher, the problem gets harder. The answer is 20 flips for 18 pancakes and 22 flips for 19. And 20 pancakes? Surprisingly, mathematicians don’t know: “nobody has been able to sidestep the brute computational approach by finding a clever equation that predicts pancake numbers”. The best mathematicians can do is find the upper limit: pancake(n) < (5n + 5)/3 flips.

This limit was proved in a paper “co-authored by William H. Gates and Christos H. Papadimitriou” in 1979 (pg. 112). The first co-author is better known now as Bill Gates of Microsoft. The Simpsons enter the story because David S. Cohen, a writer for the series, extended the problem in a mathematical paper published in 1995: the pancakes don’t just come in different sizes, they’re burnt on one side and have to be flipped both in order of size and with the burnt side down. Now the number of flips is “between 3n/2 and 2n – 2” (pg. 113). The source of the problem may seem trivial, but the maths of the solution isn’t. Pancake-flipping has important parallels with “rearranging data” in computer science.

Cohen has degrees in both computer science and physics, but his expertise isn’t unique: “the writing team of The Simpsons have equally remarkable backgrounds in mathematical subjects” (ch. 0 (sic), “The Truth about the Simpsons”, pg. 3). They have degrees and doctorates in tough subjects from colleges like Harvard, Berkeley and Princeton. And they’ve been engaged, according to Cohen, in a “decades-long conspiracy to secretly educate cartoon viewers” (back cover). They haven’t had much success with that, but they’ve succeeded in other ways: TV is no good at education, but very good at propaganda and manipulation. That’s one reason I dislike The Simpsons, which is obviously inspired by cultural Marxism, despite its occasional un-PC jokes. Another reason is that I think the characters and colours are ugly and dispiriting. Or is that cultural Marxism again? But I have to admit that the series is cleverly done. To appeal to so many people for so long takes skill, but explicit maths has been low in the mix.

It had to be, because The Simpsons wouldn’t have been successful otherwise. It has a lot of stupid fans and stupid people aren’t interested in Fermat’s Last Theorem, strategies for rock-scissors-paper or equations for pancake-numbers. That’s why you need to freeze the frame to find a lot of the explicit maths in The Simpsons. Or you did before Singh wrote this book and froze the frames for you. The implicit maths in The Simpsons is everywhere, but that’s because maths is everything, including an ugly cartoon and its science-fiction offshot. Singh discusses Futurama too and the “taxi-cab numbers” inspired by the Indian mathematician Srinivasa Ramanujan (1887-1920). I’ve never seen Futurama and I wish I could say the same of The Simpsons. I certainly hope I never see it again. But it’s an important programme and this is an interesting book.

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