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Archive for the ‘Mathematics’ Category

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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The Great Mathematical Problems by Ian StewartThe Great Mathematical Problems: Marvels and Mysteries of Mathematics, Ian Stewart (Profile Books 2013)

If you asked me what the Riemann hypothesis was, I would be able to say that it was a very important mathematical problem that had something to do with prime numbers and zeros on a line. And that would be it. There’s an entire chapter in this book about the hypothesis, but with me as a reader Stewart might as well have been “teaching Urdu to a marmoset” (as Laurence Olivier said about trying to teach Marilyn Monroe to act). Stewart is a great popularizer of advanced mathematics, but he faces a big problem: if you understand the profundity, you’ll understand the popularization of it. If you don’t, you won’t.

So I don’t understand what the Riemann hypothesis is about. Or the Poincaré conjecture. Or the Hodge conjecture. Or the Birch–Swinnerton-Dyer conjecture (note long hyphen: it’s named after two mathematicians, not three). So more than half of the chapters in this book lost me almost immediately. The other chapters lost me later, because it’s easy to understand the questions behind the Goldbach conjecture, the four-colour problem and the three-body problem. Is every even integer greater than 2 equal to the sum of two primes? Can we colour all flat two-dimensional maps with four colours or fewer? Can we write an equation to predict the gravitational interaction of three celestial bodies?

The questions are easy to understand, but the answers are very difficult. In fact, only the four-colour problem has been solved. The mathematicians who solved it famously used computers to do so, and their solution can’t be held in or followed by a single human mind. That was something new and it raised interesting questions about the nature of mathematical proof. Stewart discusses them here and supports the idea that computer proofs are legitimate. He ends the book with a list of newer, less famous but perhaps, in some cases, even more important problems. Again, some of them are easy to understand, some aren’t.

So you can get a good sense of the size, scope and complexity of mathematics from this book. And the difficulty. I found a lot of it incomprehensible, but if Ian Stewart can’t explain it to me, no-one else could. And there’s fun amidst the befuddlement:

According to a time-honoured joke, you can tell how advanced a physical theory is by the number of interacting bodies it can’t handle. Newton’s law of gravity runs into problems with three bodies. General relativity has difficulty dealing with two bodies. Quantum theory is over-extended for one body and quantum field theory runs into trouble with no bodies – the vacuum. (ch. 8, “Orbital Chaos: Three-Body Problem”, pg. 136)

There are also ideas to explore for yourself, like Langton’s ant, because maths is like a mountain range. Even if you can’t get to the peaks, you can enjoy climbing some of the way. There are gentle slopes before the sheer, ice-sheened cliffs. Ian Stewart doesn’t get to the cliffs, but there’s some tough climbing here and I quickly fell off. A lot of amateurs will do much better and this book is worth trying anyway. Being baffled teaches you something both about a subject and about yourself.

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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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The Strange Adventures of Mr Andrew Hawthorn & Other Stories by John BuchanThe Strange Adventures of Mr Andrew Hawthorn & Other Stories, John Buchan (Penguin Books 2009)

“How the devil could one associate horror with mathematics?” A Lovecraft fan will answer: easily. But that question was asked by John Buchan in a story first published in 1911. Buchan is most famous for the character Richard Hannay, hero of The Thirty-Nine Steps (1915), but just as there is much more to Doyle than his detective, so there is much more to Buchan than his battler.

As you’ll see in this collection. Like Doyle, Buchan ranged from horror to humour, from realism to romance, from outdoors adventure to indoors introspection. He could write vivid descriptions of everything from dinner with the Devil to a storm at sea. Doyle was obviously an influence on him; so were Kipling and Stevenson. He doesn’t always match their quality, but that’s hardly surprising: writing formed only part of his very full and active life. According to the chronology here, he trained as a barrister, became President of the Oxford Union, worked as secretary to the High Commissioner of South Africa and served in the Intelligence Corps during the First World War, then became successively a director of Reuters, a Conservative member of parliament, President of the Scottish Historical Society, Lord High Commissioner to the General Assembly of the Church of Scotland, Governor-General of Canada and Chancellor of Edinburgh University.

During all that time, he was also hunting, fishing and tramping the wilderness of Scotland, South Africa and Canada. And he was reading in several languages on many subjects: there are quotes here from Suetonius, Shakespeare, the Bible, Burke, A.E. Housman, Verlaine, Pascal and Poincaré. The last two supply the seed for “Space” (1911), his proto-Lovecraftian story of mathematics and menace:

All Hollond’s tastes were on the borderlands of sciences, where mathematics fades into metaphysics and physics merges in the abstrusest kind of mathematics. Well, it seems he had been working for years at the ultimate problem of matter, and especially of that rarefied matter we call aether or space. I forget what his view was – atoms or molecules or electric waves. […] He claimed to have discovered — by ordinary inductive experiment — that the constituents of aether possessed certain functions, and moved in certain figures obedient to certain mathematical laws. Space, I gathered, was perpetually ‘forming fours’ in some fancy way. (“Space” in The Moon Endureth: Tales and Fancies – in this online version of the story, the opening quote is by Tertullian)

Like one of Lovecraft’s protagonists, Holland is doomed by his discovery. So is the antiquarian Dubellay in “The Wind in the Portico” (1928). He is visited by the narrator, who is “busy on a critical edition of Theocritus” and wants to see a rare codex owned by Dubellay:

I had made a portrait in my mind of a fastidious old scholar, with eye-glasses on a black cord, and a finical Weltkind-ish manner. Instead I found a man still in early middle age, a heavy fellow dressed in the roughest of country tweeds. […] His face was hard to describe. It was high-coloured, but the colour was not healthy; it was friendly, but it was also wary; above all, it was unquiet. He gave me the impression of a man whose nerves were all wrong, and who was perpetually on his guard. (“The Wind in the Portico” in The Runagates Club)

He’s right to be: having excavated an “old temple” in the woods, he’s foolishly renewed worship of a “British god of the hills” called Vaunus. What happens to him seemed startlingly Lovecraftian when I first read the story, but when I read it again the Lovecraftian charge was muted. It’s hard to be startled twice and a story with powerful images can be disappointing when you return to it.

Buchan uses a similar theme in another story, “The Grove of Ashtaroth”, but in that case the story holds its power when I read it again. It has a different ending too: the doom is averted and the deity is ambivalent. Baleful or beautiful? Grotesque or glorious? It depends partly on one’s race and the story is about atavism and the way ancestry can overthrow environment. Or rather: can re-emerge in the right environment. Like Doyle, Buchan accepted some shocking and long-exploded ideas about the influence of genetics on brains, bodies and behaviour. They’re shocking to modern sensibilities, at least, but they might prove less exploded than some suspect.

Buchan himself may be evidence for them, because he’s another example of the disproportionate Scottish influence on English-speaking culture and literature. He died in Montreal but he was born in Perth near the east coast of Scotland. This background means that some of the strangeness in this collection is a matter of perspective. If you’re not Scottish, it will be strange. If you are, it won’t be. Take “Streams of Water in the South” (1899) and the apparent tramp who suddenly appears and helps a shepherd get his flock across a deep and dangerous flood. The shepherd asks the narrator of the story if he knows who the tramp is:

I owned ignorance.

“Tut,” said he, “ye ken nocht. But Yeddie had aye a queer crakin’ for waters. He never gangs on the road. Wi’ him it’s juist up yae glen and doon anither and aye keepin’ by the burn-side. He kens every water i’ the warld, every bit sheuch and burnie frae Gallowa’ to Berwick. And then he kens the way o’ spates the best I ever seen, and I’ve heard tell o’ him fordin’ waters when nae ither thing could leeve i’ them. He can weyse and wark his road sae cunnin’ly on the stanes that the roughest flood, if it’s no juist fair ower his heid, canna upset him. Mony a sheep has he saved to me, and it’s mony a guid drove wad never hae won to Gledsmuir market but for Yeddie.” (“Streams of Water in the South”)

The mixture of formal literary English and broad Scots heightens the richness and earthiness of the Scots. But perhaps “earthiness” is the wrong word. Language is like water: fickle, fissile, rushing over the landscape of history and culture. So Scots runs through southern English like the streams after which, via the Bible, the story is named.

The tramp Yeddie is named after them too: his real name is Adam Logan but “maist folk ca’ him ‘Streams of Water’”. He both loves water and gains power from it. As he carries fifteen sheep, one by one, across the dangerous flood, he stands “straighter and stronger”, his eye flashes and his voice rings with command. He reminds me of Kipling’s jungle boy Mowgli, who’s at ease with natural forces in a way most people don’t understand and are disturbed by.

The power of this story is Kiplingesque too: it will stay with you, partly for its strangeness, partly for its sadness. Unlike his beloved streams, Logan can’t defy time and where he was once familiar, he will one day be forgotten.

Politics and the May-Fly” (1896) also involves water and also uses Scots. It’s memorable in a different way: not sad, but sardonic. It’s psychological too, involving a battle of wits between a Tory farmer and his radical ploughman. High-born Buchan, the future Governor-General of Canada, could understand and sympathize with all stations of men. But there are things common to all men: “Politics” is a Machiavellian tale in miniature and not something that Lovecraft could have written.

Lovecraft didn’t like fishing or the great outdoors, after all, and he couldn’t explain their appeal as Buchan can. Nor could he have written “Basilissa” (1914), a story that involves both life-long love and rib-cracking wrestling. You’d have to look to Robert E. Howard for a story like that. And this, from a story with a Lovecraftian title, is like Clark Ashton Smith:

Sometimes at night, in the great Brazen Palace, warders heard the Emperor walking in the dark corridors, alone, and yet not alone; for once, when a servant entered with a lamp, he saw his master with a face as of another world, and something beside him which had no face or shape, but which he knew to be that hoary Evil which is older than the stars. (“The Watcher by the Threshold”, 1900)

So Buchan could write like all of the Weird Big Three. I think he must have influenced them too. The Thirty-Nine Steps is a classic, but it doesn’t reveal Buchan’s full range, erudition and intelligence. This collection does. I don’t think all the stories are good, but at his best he isn’t so far behind Kipling, Doyle and H.G. Wells. With a less strenuous public life, perhaps he would have matched them. But if he’d had less appetite for work, he might have had less appetite for landscapes and ideas too. There are lots of them here, from Scottish hills to Canadian forests, from mathematical pandemonium to the “Breathing of God”.

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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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Plates from the GreatShots from the Front: The British Soldier 1914-18, Richard Holmes (HarperPress 2008; paperback 2010)

Math for the MistressA Mathematician’s Apology, G.H. Hardy (1940)

Sinister SinemaScalarama: A Celebration of Subterranean Cinema at Its Sleazy, Slimy and Sinister Best, ed. Norman Foreman, B.A. (TransVisceral Books 2015)

Rick PickingsLost, Stolen or Shredded: Stories of Missing Works of Art and Literature, Rick Gekoski (Profile Books 2013/2014)

Slug is a DrugCollins Complete Guide to British Coastal Wildlife, Paul Sterry and Andrew Cleave (HarperCollins 2012) (posted @ Overlord of the Über-Feral)


Or Read a Review at Random: RaRaR

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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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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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