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Posts Tagged ‘mathematics’

the-universe-in-100-key-discoveries-by-giles-sparrowThe Universe in 100 Key Discoveries, Giles Sparrow (Quercus 2012)

Possibly the best book I’ve ever read on astronomy: text and images complement each other perfectly. Even the solidness of the book was right. It’s a heavy book about heavy ideas, from the beginning of the universe to its possible endings, with everything astronomical in between.

And everything is astronomical, if it’s looked at right. The elements vital for life were cooked in stars before being blasted out by supernovae. We are star-stuff that has the unique privilege – so far as we know – of being able to understand stars.

Or trying to. This book was first published in 2012, so it’s inevitably out of date, but many of the mysteries it describes are still there. And when mysteries are solved, they sometimes create new ones. Even the behaviour and composition of a celestial body as close as the Moon is still impossible for us to explain. But sometimes it’s easier at a distance: the interior of the earth can harder to study than galaxies millions of light years away, as I pointed out in “Heart of the Mother”.

In every case, however, understanding depends on mathematics. Astronomers have been building models of the heavens with shapes and numbers for millennia, but the models had to wait for two things to really become powerful: first, the invention of the telescope; second, the development of modern chemistry and physics. Whether or not there is life out there, celestial light is full of messages about the composition and movement of the stars and other bodies that generate it.

But visible light is only a small part of the electromagnetic spectrum and modern astronomy probes the universe at wavelengths far above and below it. The more data astronomers can gather, the more they can test the mathematical models they’ve built of the heavens. The best models make the most detailed predictions, inviting their own destruction by ugly facts. But when predictions fail, it sometimes means that the observations are faulty, not the models. Cosmological models predicted much more matter in the universe than we can see. Is the gap accounted for by so-called “dark matter”, which “simply doesn’t interact with light or other electromagnetic radiations at all”? (ch. 98, “Dark Matter”, pg. 396)

Dark matter is a strange concept; so is dark energy. Astronomy may get stranger still, but the cover of this book is a reminder that human beings inhabit two kinds of universe. One is the universe out there: matter and radiation, moons, planets, stars, galaxies, supernovae. The other is the universe in here, behind the eyes, between the ears and above the tongue. The cover of this book offers a vivid contrast between the swirling complexity and colour of a star-field and the sans-serif font of the title and author’s name. But the contrast is ironic too. The stars look complex and the font looks simple, but language is actually far more complex and difficult to understand than stars.

Consciousness may be far more complex still. In the end, is the value of science that it expands consciousness, offering new physical and mental sensations of discovery and understanding? The powerful and beautiful images and ideas in this book could only have been generated by science, because the universe is more inventive than we are. But without consciousness, the universe might as well not exist. Without language, we’d never be able to try and understand it. Then again, the universe seems to have invented language and consciousness too.

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restless-creatures-by-matt-wilkinsonRestless Creatures: The Story of Life in Ten Movements, Matt Wilkinson (Icon 2016)

A fascinating book about a fascinating thing: the movement of plants and animals. It’s also a very familiar thing, but it’s far more complex than we often realize. Human beings have been watching horses gallop for thousands of years, but until the nineteenth century no-one was sure what was happening:

The man usually credited for ushering in the modern study of locomotion is the brilliant photographer Eadweard Muybridge. […] His locomotory calling came in 1872, when railroad tycoon and former California governor Leland Stanford invited him to his stock farm in Palo Alto, supposedly to settle a $25,000 bet that a horse periodically becomes airborne when galloping. (ch. 1, “Just Put One Foot in Front of Another”, pg. 16)

To answer the question, Muybridge used a series of still cameras triggered by trip-wires. And yes, galloping horses do become airborne: “not when the legs were at full stretch, as many had supposed, but when the forelimbs and hindlimbs were at their closest approach.” However, Matt Wilkinson calls another man “the true founding father of the science of locomotion”: the French scientist Étienne-Jules Marey, who had been investigating movement using a stylograph. In fact, it was Marey who first proved that galloping horses become airborne (ch. 1, pg. 19). Muybridge’s photographs were dramatic confirmation and the two men began to collaborate.

Marey also pioneered electromyography, or the recording of the electrical impulses generated by moving muscles. Like the rest of modern science, biokinesiology, as the study of animal movement might be called, depends on instruments that supplement or enhance our fallible senses. It also depends on mathematics: there is a lot of physics in this book. You can’t understand walking, flying or swimming without it. Walking is the most mundane, but also in some ways the most interesting, at least in its human form. Bipedalism isn’t an everyday word, but it’s an everyday sight.

What does it involve? How did it evolve? And how important was it in making us human? Wilkinson looks at all these questions and you’ll suddenly start seeing your legs and feet in a different way. What wonders of bioengineering they are! And what a lot of things happen in the simple process of “just putting one foot in front of another”. Scientists still don’t understand these things properly: for example, they can’t say whether or not sport shoes are dangerous, “lulling us into a false sense of security, causing us to pass dreadful shocks up our legs and spine without our being aware of them” (ch. 1, pg. 29).

But there’s much more here than horse and human locomotion: Wilkinson discusses everything from eels, whales, pterodactyls, bats and cheetahs to amoebas, annelid worms, fruit-flies, zombified ants and the “gliding seed of the Javan cucumber Alsomitra macrocarpa”. He also discusses the nervous systems, genes and evolution behind all those different kinds of movement. This book is both fascinating and fun, but I have one criticism: its prose doesn’t always move as lightly and gracefully as some of its subjects do. Wilkinson mentions both Stephen Jay Gould and Richard Dawkins. I wish he’d written more like the latter and less like the former. If he had, a good book would have become even better.

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The Invention of Science by David WoottonThe Invention of Science: A New History of the Scientific Revolution, David Wootton (Allen Lane 2015)

I picked up this book expecting to start reading, then get bored, start skimming for interesting bits, and sooner or later give up. I didn’t. I read steadily from beginning to end, feeling educated, enlightened and even enthralled. This is intellectual history at nearly its best, as David Wootton sets out to prove what is, for some, a controversial thesis: that “Modern science was invented between 1572, when Tycho Brahe saw a new star, and 1704, when Newton published his Opticks” (introduction, pg. 1).

He does this in a clever and compelling way: by looking at the language used in science across Europe. If there was indeed a scientific revolution and science was indeed a new phenomenon, we should expect to see this reflected in language. Were old words given new meanings? Did new words and phrases appear for previously inexpressible concepts? They were and they did. “Scientist” itself is a new word, replacing earlier and less suitable words like “naturalist”, “physiologist”, “physician” and “virtuoso”. The word “science” is an example of an old word given a new meaning. In Latin, scientia meant “knowledge” or “field of learning”, from the verb scire, “to know”.

But it didn’t mean a systematic collective attempt to investigate and understand natural phenomena using experiments, hypotheses and sense-enhancing, evidence-gathering instruments. Science in that sense was something new, Wootton claims. He assembles a formidable array of texts and references to back his thesis, which is part of why this book is so enjoyable to read. As Wootton points out, the “Scientific Revolution has become almost invisible simply because it has been so astonishingly successful.” Quotations like this, from the English writer Joseph Glanvill, make it visible again:

And I doubt not but posterity will find many things, that are now but Rumors, verified into practical Realities. It may be some Ages hence, a voyage to the Southern unknown Tracts, yea possibly the Moon, will not be more strange then one to America. To them, that come after us, it may be as ordinary to buy a pair of wings to fly into remotest Regions; as now a pair of Boots to ride a Journey. And to conferr at the distance of the Indies by Sympathetick conveyances, may be as usual to future times, as to us in a litterary correspondence. (The Vanity of Dogmatizing, 1661)

Glanvill’s prescience is remarkable and he’s clearly writing in an age of pre-science or proto-science. He wasn’t just a powerful thinker, but a powerful writer too. So was Galileo and Wootton, who has written a biography of the great Italian, conveys his genius very clearly in The Invention of Science. You can feel some of the exhilaration of the intellectual adventure Galileo and other early scientists embarked on. They were like buccaneers sailing out from Aristotle’s Mediterranean into the huge Atlantic, with a new world before them.

Wootton also emphasizes the importance of Galileo’s original speciality:

The Scientific Revolution was, first and foremost, a revolt by the mathematicians against the authority of the philosophers. The philosophers controlled the university curriculum (as a university teacher, Galileo never taught anything but Ptolemaic astronomy), but the mathematicians had the patronage of princes and merchants, of soldiers and sailors. They won that patronage because they offered new applications of mathematics to the world. (Part 2, “Seeing is Believing”, ch. 5, “The Mathematization of the World”, pg. 209)

But there’s something unexpected in this part of the book: he describes “double-entry bookkeeping” as part of that mathematical revolt: “the process of abstraction it teaches is an essential precondition for the new science” (pg. 164).

He also has very interesting things to say about the influence of legal tradition on the development of science:

Just as facts moved out of the courtroom and into the laboratory, so evidence made the same move at around the same time; and, as part of the same process of constructing a new type of knowledge, morality moved from theology into the sciences. When it comes to evidence, the new science was not inventing new concepts, but re-cycling existing ones. (Part 3, “Making Knowledge”, ch. 11, “Evidence and Judgment”, pg. 412)

Science was something new, but it wasn’t an ideology ex nihilo. That isn’t possible for mere mortals and Wootton is very good at explaining what was adapted, what was overturned and what was lost. Chapter 13 is, appropriately enough, devoted to “The Disenchantment of the World”; the next chapter describes how “Knowledge is Power”. That’s in Part 3, “Birth of the Modern”, and Wootton wants this to be a modern book, rather than a post-modern one. He believes in objective reality and that science makes genuine discoveries about that reality.

But he fails to take account of some modern scientific discoveries. The Invention of Science is a work of history, sociology, philology, and philosophy. It doesn’t discuss human biology or the possibility that one of the essential preconditions of science was genetic. Modern science arose in a particular place, north-western Europe, at a particular time. Why? The Invention of Science doesn’t, in the deepest sense, address that question. It doesn’t talk about intelligence and psychology or the genetics that underlie them. It’s a work of history, not of bio-history or historical genetics.

In 2016, that isn’t a great failing. History of science hasn’t yet been revolutionized by science. But I would like to see the thesis of this book re-visited in the light of books like Gregory Clark’s A Farewell to Alms (2007), which argues that the Industrial Revolution in England had to be preceded by a eugenic revolution in which the intelligent and prudent outbred the stupid and feckless. The Invention of Science makes it clear that Galileo was both a genius and an intellectual adventurer. But why were there so many others like him in north-western Europe?

I hope that historians of science will soon be addressing that question using genetics and evolutionary theory. David Wootton can’t be criticized for not doing so here, because bio-history is very new and still controversial. And he may believe, like many of the post-modernists whom he criticizes, in the psychic unity of mankind. The Invention of Science has other and less excusable flaws, however. One of them is obvious even before you open its pages. Like Dame Edna Everage’s bridesmaid Madge Allsop, it is dressed in beige. The hardback I read does not have an inviting front cover and Wootton could surely have found something equally relevant, but more interesting and colourful.

After opening the book, you may find another flaw. Wootton’s prose is not painful, but it isn’t as graceful or pleasant to read as it could have been. This is both a pity and a puzzle, because he is very well-read in more languages than one: “We take facts so much for granted that it comes as a shock to learn that they are a modern invention. There is no word in classical Greek or Latin for a fact, and no way of translating the sentences above from the OED [Oxford English Dictionary] into those languages.” (Part 3, “Facts”, pg. 254)

He certainly knows what good prose looks like, because he quotes a lot of it. But his own lacks the kind of vigour and wit you can see in the words of, say, Walter Charleton:

[I]t hath been affirmed by many of the Ancients, and questioned by very few of the Moderns, that a Drum bottomed with a Woolfs skin, and headed with a Sheeps, will yeeld scarce any sound at all; nay more, that a Wolfs skin will in short time prey upon and consume a Sheeps skin, if they be layed neer together. And against this we need no other Defense than a downright appeal to Experience, whether both those Traditions deserve not to be listed among Popular Errors; and as well the Promoters, as Authors of them to be exiled the society of Philosophers: these as Traitors to truth by the plotting of manifest falsehoods; those as Ideots, for beleiving and admiring such fopperies, as smell of nothing but the Fable; and lye open to the contradiction of an easy and cheap Experiment. (Physiologia Epicuro-Gassendo-Charltoniana, 1654)

The Invention of Science is also too long: its message often rambles home rather than rams. If Wootton suffers from cacoethes scribendi, an insatiable itch to write, then I feel an itch to edit what he wrote. It’s good to pick up a solid book on a solid subject; it would be even better if everything in the book deserved to be there.

But if the book weren’t so good in some ways, I wouldn’t be complaining that it was less than good in others. In fact, I wouldn’t have finished it at all and I wouldn’t be heartily recommending it to anyone interested in science, history or linguistics. But I did and I am. The Invention of Science is an important book and an enjoyable read. I learned a lot from it and look forward to reading it again.

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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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Chaotic Fishponds and Mirror Universes by Richard ElwesChaotic Fishponds and Mirror Universes: the maths that governs our world, Richard Elwes (Quercus 2013)

Most popular introductions to maths cover well-trodden ground: the prime numbers, the square root of 2, the Fibonacci sequence, Möbius strips, the Platonic polyhedra, and so on. Chaotic Fishponds and Mirror Universes covers some of those, but it lives up to the promise of its title and also talks about less familiar things: Voronoi tilings, Delaunay triangulation, neural networks, the simplex algorithm, discrete cosine functions, Pappus’s theorem, kinematic equations and the most effective ways to test blood samples for syphilis. Or coins for counterfeits.

Syphilis and counterfeits are both covered by the mathematics of group-testing, after all, but then maths covers everything. As Richard Elwes puts it: maths governs our world. He is good at explaining how and at demonstrating how it has, does and will shape the world. Some of the fields he discusses are very complex, so he can’t explain them properly in a popular introduction, but I couldn’t cope with a full explanation. It doesn’t matter: you don’t have to be able to climb Everest to be awed and enriched by the knowledge of its existence. Chaotic Fishponds and Mirror Universes is about what you might call hyperdimensional Himalayas: the mountains of maths and the men who climb them. The mountains rise for ever and contain everything that is, was or ever could be. Matter and energy are susceptible to mathematical modelling and may, in the final analysis, be maths, but maths is about much more. Richard Elwes is a mather placing a stethoscope to the heart not just of the world but of all possible worlds.

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Physics in Minutes by Giles SparrowPhysics in Minutes: 200 key concepts explained in an instant, Giles Sparrow (Quercus 2014)

In Borges’ story “The Book of Sand” (1975), the narrator acquires a heavy little book that has an infinite number of pages. When he opens it, he can never find the same page twice. The discrepancy between its finite size and its infinite contents begins to prey on his mind. He decides the book is a monstrous thing and wants to get rid of it: “I considered fire, but I feared that the burning of an infinite book might be similarly infinite, and suffocate the planet in smoke.”

It’s a good story, but the central idea doesn’t work, unless you assume magic is at work. A book with an infinite number of pages would be infinitely heavy. In fact, it would instantly become a black hole and start swallowing the universe.

So I assume, anyway. I’m interested in physics but I don’t know much about it. This book is aimed at people like me. It reminded me of Borges’ Book of Sand, partly because it’s small but heavy, partly because of the density of its ideas and the weight of history behind those ideas. Each page of explanation could easily become a hundred or a thousand: physics is daunting in its scope and complexity. Some of the greatest minds in history have put centuries of effort into understanding the behaviour of matter and energy.

That’s how we got astonishing things like electronics, X-rays and the atom bomb. Physics is an intellectual over-achiever, the super-star of the sciences, the most spectacular, powerful and difficult of all. But it’s the most difficult science because it’s also the simplest. Stars and steam-engines are much less complex than societies or brains, which is why you can’t get away with talking nonsense in physics. And although mathematics governs everything, it’s the simpler things – pendulums, light-rays, atoms, stars – that we can mathematize first.

Or some of us can, at least: the highly intelligent and obsessive men, like Galileo and Isaac Newton, who began modern physics by finding ways to extract abstract mathematics from concrete realities. If they’d tried to find maths in psychology or culture, they would have failed, because those things are too complex. They had to look at much simpler things like falling objects, planetary motion and light-rays. Galileo and Newton laid the foundations and later physicists have built on them, so that physics now towers into the scientific skies, the envy and awe of those working with more complex and intractable aspects of existence.

Giles Sparrow takes his readers on a tour of the tower. I suppose you could say he’s operating an express elevator, stopping briefly on the floors and offering a brief explanation of what it contains: elastic and inelastic collisions on one floor, fluid mechanics on another, mass spectrometry, electromagnetic induction and quantum electrodynamics on more. Then the doors snap shut and the elevator shoots up another floor. But one thing is found everywhere: mathematics. Sparrow quotes a lot of equations and uses a lot of numbers. If you want to understand physics, you have to know the maths. If you don’t, there’s no way to disguise your ignorance.

The maths is beyond me, so until brain-modification arrives I won’t be able to understand physics properly. Until then, this book is a good way of glimpsing the glories of the science. It’s also the closest you’ll get to handling Borges’ Book of Sand in real life.

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Discovering the UniverseDiscovering the Universe: The Story of Astronomy, Paul Murdin (Andre Deutsch 2014)

First published in 2011 as Mapping the Universe, this is a well-written, well-illustrated history of astronomy that begins in the Stone Age and ends with the Hubble Space Telescope and Large Hadron Collider. The photographs will stimulate your eyes as the text stimulates your mind. The universe is a big place and big things happen there, like gamma-ray bursts (GRBs):

Until 1997, astronomers didn’t know whether GRBs originated in some sort of explosions on the edge of our solar system, around our Galaxy, or far away. Two examples proved that the explosions occur the edge of the observable Universe. For their duration of a few seconds, the bursts had been over a million times brighter than their parent galaxy, the biggest bangs since the Big Bang. (ch. 17, “Exploding Stars”, pg. 87)

Ptolemy, Galileo and Newton would all be astonished by the technology that allows modern astronomers to study phenomena like gamma-ray bursts, but one thing has remained constant: the importance of mathematics and measurement in studying the sky. The story of astronomy is not just about seeing further and clearer, but also of measuring better and mathematizing more powerfully. Ptolemy’s geocentric universe entailed the arbitrary complexity of epicycles on epicycles, to explain how the planets sometimes seemed to move backwards against the stars. Then Copernicus resurrected the ancient Greek hypothesis of a heliocentric universe.

Back cover

Back cover


Planetary retrogression became easier to explain. Other hypotheses, like the steady state universe and Kepler’s planetary Platonic solids, haven’t proved successful, but data don’t explain themselves and astronomers have to be adventurous in mind, if not usually in body. This book contains the big names, the big sights and the big mysteries that are still awaiting explanation. More big names, sights and mysteries are on their way.

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Philosophy 100 Essential Thinkers by Philip StokesPhilosophy: 100 Essential Thinkers: The Ideas That Have Shaped Our World, Philip Stokes (Arcturus Publishing 2012)

Caricatures are compelling because they simplify and exaggerate. A good artist can create one in a few strokes. In fact, a good artist has to caricature if he can use only a few strokes. The image won’t be recognizable otherwise.

This also applies to philosophical ideas. If you have to describe them in relatively few words, you’ll inevitably caricature, making them distinct but losing detail and complexity. So this book is a series of caricatures. With only 382 pages of standard print, what else could it be? In each case, Philip Stokes uses a few strokes to portray “100 Essential Thinkers” from Thales of Miletus, born c. 620 B.C., to William Quine (1908-2000), with all the big names in between: Plato, Aristotle, Descartes, Pascal, Hume, Kant, Leibniz, Schopenhauer, Nietzsche, Russell, Wittgenstein and so on. The philosophical portraits are recognizable but not detailed. But that’s why they’re fun, like a caricature.

It’s also fun to move so quickly through time. There are nearly three millennia of Western philosophy here, but the schools and the civilizations stream by, from the Pre-Socratics and Atomists to the Scholastics and Rationalists; from pagan Greece and Rome to Christianity and communism. Bertrand Russell’s History of Western Philosophy, which inevitably comes to mind when you look at an over-view like this, moves much more slowly, but it’s a longer and more detailed book.

It’s also funnier and less inclusive. This book discusses men who are more usually seen as scientists or mathematicians, like Galileo and Gödel. But in a sense any historic figure could be included in an over-view of philosophy, because everyone has one. You can’t escape it. Rejecting philosophy is a philosophy too. Science and mathematics have philosophical foundations, but in some ways they’re much easier subjects. They’re much more straightforward, like scratching your right elbow with your left hand.

Philosophy can seem like trying to scratch your right elbow with your right hand. The fundamentals of existence are difficult to describe, let alone understand, and investigating language using language can tie the mind in knots. That’s why there’s a lot of room for charlatans and nonsense in philosophy. It’s easier to pretend profundity than to be profound. It’s also easy to mistake profundity for pseudery.

And, unlike great scientists or mathematicians, great philosophers should be read in the original. Reading Nietzsche in English is like looking at a sun-blasted jungle through tinted glass or listening to Wagner wearing earplugs. Or so I imagine: I can’t read him in German. But some philosophers suffer less by translation than others, because some philosophical ideas are universal. Logic, for example. But how important is logic? Is it really universal? And is mathematics just logic or is it something more?

You can ask, but you may get more answers than you can handle. Philosophy is a fascinating, infuriating subject that gets everywhere and questions everything. You can’t escape it and this book is a good place to learn why.

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Mapping the World by Beau RiffenburghMapping the World: The Story of Cartography, Beau Riffenburgh (Carlton Books 2011, 2014)

A good map is like a swan on a river. Beneath the elegance there is a lot of effort. This book is about that effort: all the millennia of research and refinement that have gone into perfecting maps. Not that any map can be perfect. As Beau Riffenburgh explains here, there are always choices to be made: what do you put in, what do you leave out? And how do you represent spherical geometry on flat paper?

The Flemish cartographer Gerardus Mercator came up with one famous answer to that question:

Mercartor’s major achievement came in 1569 with a new projection that represented a breakthrough in nautical cartography. Since known as the Mercator projection, it is cylindrical-like, with the meridians as equally spaced parallel lines and the lines of latitude as parallel, horizontal lines, which are spaced further apart as their distance from the equator increases. This projection is uniquely suited to navigation because a line of constant true bearing allows a navigator to plot a straight-line course. However, this projection grossly distorts geographical regions in high latitudes – thus Greenland is shown larger than South America, although it is actually less than one-eighth of the size. (“Cosmographies and the Development of Projection”, pg. 51)

So the map looks wrong, but leads right. So does the famous map of the London Underground, which ignores true distances and bearings: the designer Harry Beck made it look like an “electrical circuit, with straight lines and the inclusion of only one feature above ground – the Thames” (“Mapping for the Masses”, pg. 143). Maps are about abstraction: they condense and confine what people find interesting or important about the real world.

So minds mould maps and in writing about maps, Riffenburgh is also writing about culture and politics. About art too, because maps can be very beautiful things, sometimes deliberately, sometimes incidentally. Above all, however, he’s writing about mathematics. What was implicit from the beginning – the importance of maths in mapping – became more and more explicit, as he describes in the chapter “Men, Measurements and Mechanisms” (pp. 70-3). The men are drawn from the world’s most evil and energetic group: white Europeans. Galileo, Newton and Huygens are three of them: as they contributed to maths and science, they contributed to cartography.

Another man is the Yorkshire watchmaker John Harrison (1693-1776), the hero of Dava Sobel’s Longitude (1995). He was a remarkable personality and looks it in the portrait here: proud, determined and self-possessed. He needed all those qualities to get his due. He invented a chronometer that kept accurate time on long voyages and enabled navigators to determine longitude, but British officialdom “made him wait years for all of his prize-money” (pg. 73).

Elsewhere the names are obscurer and the stories sometimes sadder:

In the history of cartography, few individuals stand out for their work in so many geographical regions and aspects of science as James Rennell. Born in Devon in 1742, Rennell went to sea at the age of 14, learned maritime surveying and then, at the end of the Seven Years’ War, received a commission in the Bengal Army as an engineer. … Equipped with quadrant, compass and chain, Rennell began a thorough and scientific survey of [Bengal’s] major river systems, roads, plains, jungles, mangrove forests and mountains. (“James Rennell: Mapping India, Africa and Ocean Currents”, pg. 86)

However, he “never fully recovered from a severe wound received in an ambush” and retired to London to produce his “masterpiece – A Map of Hindoostan, or the Mogul Empire” (1782/1788). But en route to England, he had an “extended stay in Southern Africa” and developed an interest in ocean currents. So he became a pioneering hydrographer too: his posthumous An Investigation of Currents of the Atlantic Ocean (1832) “is often considered to form the historical basis of the study of currents” (pg. 89).

Later in the century, the German August Petermann worked for the Royal Geographical Society and was appointed “Physical Geographer Royal” by Queen Victoria. His assistant John Bartholomew said “no one has done more than he to advance modern cartography”, but Petermann committed suicide in 1878 after returning to Germany (“Maps reach a wider audience”, pg. 132).

Nietzsche would not have approved. But I think he would have applauded this:

Perhaps the most remarkable nautical drawings of all, considering the conditions under which they were produced, were those of William Bligh, captain of the British ship HMAV [His Majesty’s Armed Vessel] Bounty in 1789. Following the infamous mutiny, Bligh and 18 loyal seamen were set adrift in the ship’s launch. During the next 47 days, Bligh navigated approximately 3,600 nautical miles (6,660 km) to Timor, with only one stop. Throughout the journey, which is considered one of the most remarkable accomplishments in the history of open-boat travel, Bligh kept a detailed log and made sketches of his course. (“Mapping Australia and the Pacific”, pg. 77)

His chart is reproduced here. Using anecdotes like that with serious analysis and intellectual history, Riffenburgh tells the story of cartography from Mesopotamia and before to the moon and beyond. The story of maps is the story of man: even pre-literate societies like the ancient Polynesians have used maps to record the sea and its currents. In Europe, maps have reflected every advance in technology, like printing and photography. But as they’ve responded to technology, they’ve altered the way we see and interact with reality. When you look at a map, there’s a whole world of exploration, endeavour and ingenuity just beyond its margins. Mapping the World is about that world: the margins of mapness without which the maps themselves would not exist. It’s a book to stimulate the mind and delight the eye.

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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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