Feeds:
Posts
Comments

Posts Tagged ‘mathematics’

Ficciones, Jorge Luis Borges

If you want a good reason to learn Spanish, here’s one: you’ll be able to read Borges in the original. Learning won’t be very difficult, but it would be worth it even if it were. Spanish is a clear and elegant language and Borges is a clear and elegant writer. He puts his stories together like mosaics, using words as chips of coloured stone to create the strangest of worlds and situations.

This collection, which combines El Jardín de Senderos que se Bifurcan (1941) (The Garden of Forking Paths) and Artificios (1944) (Artifices), has the very strange world known as “La Biblioteca de Babel” or “The Library of Babel”, an infinite library of hexagonal rooms whose books are a kind of drunkard’s walk through alphabetic possibility:

Uno, que mi padre vio en un hexágono del circuito quince noventa y cuatro, constaba de las letras MCV perversamente repetidas desde el renglón primero hasta el último.

One book, which my father once saw in a hexagon in circuit 15-94,consisted of the letters M C V perversely repeated from the first line to the last.

Borges was fascinated by concepts like randomness and infinity, which is why he drew on mathematics so often in his stories. “The Library of Babel” is an exploration of those ideas, but amid the abstraction and universality of mathematics there are haunting images like this:

Muerto, no faltarán manos piadosas que me tiren por la baranda; mi sepultura será el aire insondable; mi cuerpo se hundirá largamente y se corromperá y disolverá en el viento engenerado por la caída, que es infinita.

When I am dead, compassionate hands will throw me over the railing; my tomb will be the unfathomable air, my body will sink for ages, and will decay and dissolve in the wind engendered by my fall, which shall be infinite.

That’s both horrible and beautiful. The first words of the quote – “Muerto, no faltarán…” – are an example of how Spanish can be more precise than English. A literal translation would be: “Dead, there shall not lack caring hands to cast me over the railing…” But in English the referent of “dead” hangs in the air and doesn’t settle very readily on “me”. In Spanish, muerto is masculine singular and clearly refers to the speaker.

English has to paraphrase, just as it does with the title of Gautier’s «La Morte Amoureuse» (1836). One of the strange titles in the Library of Babel, Trueno peinado, translates well into English: Combed Thunder. Another title doesn’t: Calambre de Yeso, or Plaster Cramp. I think Sandstone Cramp or Onyx Cramp would work better in English: the translation fails by being too faithful.

But Borges survives translation better than most writers, because his prose is precise and his themes are universal. Or perhaps you could say fundamental. He’s playing with words and ideas, exploring the relationship between language and reality, between reality and imagination, between imagination and mathematics. “The Library of Babel” is an excellent example, which is why it’s perhaps his most famous story.

But there’s a melancholy and even a terror in the story too, which come across more clearly when you’re reading more slowly and with closer attention. That’s one reason it’s good to read in other languages: people whose mother tongue isn’t Spanish can find things in Borges that native speakers can’t.

But that applies to every language: in some ways the natives are trapped by their own familiarity and fluency. Borges was aware of questions like that and in “The Library of Babel” he suddenly throws a door open to an infinity of mirrors. If the relation between symbol and sense is arbitrary, then any combination of letters can have any meaning. That’s why the narrator of the story suddenly asks:

Tú, que me lees, ¿estás seguro de entender mi lenguaje?

You who read me — are you certain you understand my language?

In other stories, like “La Muerte y la Brújula”, or “Death and the Compass”, Borges’ games with symbols and coincidence can begin to seem like self-parody. This is the story of a series of murders committed to form the letters of the Tetragrammaton, or great and unspeakable name of God in Hebrew. I think the title in Spanish is better than the story, because brújula has an enticing echo of brujo, “wizard”, or bruja, “witch”. Borges was a profound writer, not a broad one, and he repeated himself, like a garden of forking paths or an echoing labyrinth. But my Spanish isn’t good enough to appreciate him fully or get the most out of his humour.

Whatever language you read him in, you’ll probably agree that he is among the greatest writers of the twentieth century. But one of his biggest services to literature may have been to encourage more people to try G.K. Chesterton, one of his own heroes and inspirations. He would certainly have been pleased to do so, because you don’t get ego with Borges. Instead, you get ideas, some of the strangest and most haunting ever set to cellulose. As I said in one of my own attempts at Borgesian weirdness:

Black Aikkos the God is eternally blind,
But he sees with the eyes of the infinite mind… (“The Dice of Aikkos”)

Homer, at the beginning of European literature, is said to have been blind. Borges certainly was, and if he proves to have been at the end of European literature, he is great enough to bear the comparison.

Read Full Post »

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.

Read Full Post »

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.

Read Full Post »

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.

Read Full Post »

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.

Read Full Post »

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.

Read Full Post »

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.

Read Full Post »

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.

Read Full Post »

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.

Read Full Post »

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.

Read Full Post »

Older Posts »