Just how much does the world follow laws?


How the Zebra Got its Stripes and Other Darwinian Just So Stories by Léo Grasset
The Serengeti Rules: The quest to discover how life works and why it matters by Sean B. Carroll
Lysenko’s Ghost: Epigenetics and Russia by Loren Graham
The Great Derangement: Climate change and the unthinkable by Amitav Ghosh
reviewed for New Scientist, 15 October 2016

JUST how much does the world follow laws? The human mind, it seems, may not be the ideal toolkit with which to craft an answer. To understand the world at all, we have to predict likely events and so we have a lot invested in spotting rules, even when they are not really there.

Such demands have also shaped more specialised parts of culture. The history of the sciences is one of constant struggle between the accumulation of observations and their abstraction into natural laws. The temptation (especially for physicists) is to assume these laws are real: a bedrock underpinning the messy, observable world. Life scientists, on the other hand, can afford no such assumption. Their field is constantly on the move, a plaything of time and historical contingency. If there is a lawfulness to living things, few plants and animals seem to be aware of it.

Consider, for example, the charming “just so” stories in French biologist and YouTuber Léo Grasset’s book of short essays, How the Zebra Got its Stripes. Now and again Grasset finds order and coherence in the natural world. His cost-benefit analysis of how animal communities make decisions, contrasting “autocracy” and “democracy”, is a fine example of lawfulness in action.

But Grasset is also sharply aware of those points where the cause-and-effect logic of scientific description cannot show the whole picture. There are, for instance, four really good ways of explaining how the zebra got its stripes, and those stripes arose probably for all those reasons, along with a couple of dozen others whose mechanisms are lost to evolutionary history.

And Grasset has even more fun describing the occasions when, frankly, nature goes nuts. Take the female hyena, for example, which has to give birth through a “pseudo-penis”. As a result, 15 per cent of mothers die after their first labour and 60 per cent of cubs die at birth. If this were a “just so” story, it would be a decidedly off-colour one.

The tussle between observation and abstraction in biology has a fascinating, fraught and sometimes violent history. In Europe at the birth of the 20th century, biology was still a descriptive science. Life presented, German molecular biologist Gunther Stent observed, “a near infinitude of particulars which have to be sorted out case by case”. Purely descriptive approaches had exhausted their usefulness and new, experimental approaches were developed: genetics, cytology, protozoology, hydrobiology, endocrinology, experimental embryology – even animal psychology. And with the elucidation of underlying biological process came the illusion of control.

In 1917, even as Vladimir Lenin was preparing to seize power in Russia, the botanist Nikolai Vavilov was lecturing to his class at the Saratov Agricultural Institute, outlining the task before them as “the planned and rational utilisation of the plant resources of the terrestrial globe”.

Predicting that the young science of genetics would give the next generation the ability “to sculpt organic forms at will”, Vavilov asserted that “biological synthesis is becoming as much a reality as chemical”.

The consequences of this kind of boosterism are laid bare in Lysenko’s Ghost by the veteran historian of Soviet science Loren Graham. He reminds us what happened when the tentatively defined scientific “laws” of plant physiology were wielded as policy instruments by a desperate and resource-strapped government.

Within the Soviet Union, dogmatic views on agrobiology led to disastrous agricultural reforms, and no amount of modern, politically motivated revisionism (the especial target of Graham’s book) can make those efforts seem more rational, or their aftermath less catastrophic.

In modern times, thankfully, a naive belief in nature’s lawfulness, reflected in lazy and increasingly outmoded expressions such as “the balance of nature”, is giving way to a more nuanced, self-aware, even tragic view of the living world. The Serengeti Rules, Sean B. Carroll’s otherwise triumphant account of how physiology and ecology turned out to share some of the same mathematics, does not shy away from the fact that the “rules” he talks about are really just arguments from analogy.

“If there is a lawfulness to living things, few plants and animals seem to be aware of it”
Some notable conservation triumphs have led from the discovery that “just as there are molecular rules that regulate the numbers of different kinds of molecules and cells in the body, there are ecological rules that regulate the numbers and kinds of animals and plants in a given place”.

For example, in Gorongosa National Park, Mozambique, in 2000, there were fewer than 1000 elephants, hippos, wildebeest, waterbuck, zebras, eland, buffalo, hartebeest and sable antelopes combined. Today, with the reintroduction of key predators, there are almost 40,000 animals, including 535 elephants and 436 hippos. And several of the populations are increasing by more than 20 per cent a year.

But Carroll is understandably flummoxed when it comes to explaining how those rules might apply to us. “How can we possibly hope that 7 billion people, in more than 190 countries, rich and poor, with so many different political and religious beliefs, might begin to act in ways for the long-term good of everyone?” he asks. How indeed: humans’ capacity for cultural transmission renders every Serengeti rule moot, along with the Serengeti itself – and a “law of nature” that does not include its dominant species is not really a law at all.

Of course, it is not just the sciences that have laws: the humanities and the arts do too. In The Great Derangement, a book that began as four lectures presented at the University of Chicago last year, the novelist Amitav Ghosh considers the laws of his own practice. The vast majority of novels, he explains, are realistic. In other words, the novel arose to reflect the kind of regularised life that gave you time to read novels – a regularity achieved through the availability of reliable, cheap energy: first, coal and steam, and later, oil.

No wonder, then, that “in the literary imagination climate change was somehow akin to extraterrestrials or interplanetary travel”. Ghosh is keenly aware of and impressively well informed about climate change: in 1978, he was nearly killed in an unprecedentedly ferocious tornado that ripped through northern Delhi, leaving 30 dead and 700 injured. Yet he has never been able to work this story into his “realist” fiction. His hands are tied: he is trapped in “the grid of literary forms and conventions that came to shape the narrative imagination in precisely that period when the accumulation of carbon in the atmosphere was rewriting the destiny of the Earth”.

The exciting and frightening thing about Ghosh’s argument is how he traces the novel’s narrow compass back to popular and influential scientific ideas – ideas that championed uniform and gradual processes over cataclysms and catastrophes.

One big complaint about science – that it kills wonder – is the same criticism Ghosh levels at the novel: that it bequeaths us “a world of few surprises, fewer adventures, and no miracles at all”. Lawfulness in biology is rather like realism in fiction: it is a convention so useful that we forget that it is a convention.

But, if anthropogenic climate change and the gathering sixth mass extinction event have taught us anything, it is that the world is wilder than the laws we are used to would predict. Indeed, if the world really were in a novel – or even in a book of popular science – no one would believe it.

Beware the indeterminate momentum of the throbbing whole


Graham Harman (2nd from right) and fellow speculative materialists in 2007


In 1942, the Argentine writer Jorge Luis Borges cooked up an entirely fictitious “Chinese” encyclopedia entry for animals. Among its nonsensical subheadings were “Embalmed ones”, “Stray dogs”, “Those that are included in this classification” and “Those that, at a distance, resemble flies”.

Explaining why these categories make no practical sense is a useful and enjoyable intellectual exercise – so much so that in in 1966 the French philosopher Michel Foucault wrote an entire book inspired by Borges’ notion. Les mots et les choses (The Order of Things) became one of the defining works of the French philosophical movement called structuralism.

How do we categorise the things we find in the world? In Immaterialism, his short and very sweet introduction to his own brand of philosophy, “object-oriented ontology”, the Cairo-based philosopher Graham Harman identifies two broad strategies. Sometimes we split things into their ingredients. (Since the enlightenment, this has been the favoured and extremely successful strategy of most sciences.) Sometimes, however, it’s better to work in the opposite direction, defining things by their relations with other things. (This is the favoured method of historians and critics and other thinkers in the humanities.)

Why should scientists care about this second way of thinking? Often they don’t have to. Scientists are specialists. Reductionism – finding out what things are made of – is enough for them.

Naturally, there is no hard and fast rule to be made here, and some disciplines – the life sciences especially – can’t always be reducing things to their components.

So there have been attempts to bring this other, “emergentist” way of thinking into the sciences. One of the most ingenious was the “new materialism” of the German entrepreneur (and Karl Marx’s sidekick) Friedrich Engels. One of Engels’s favourite targets was the Linnaean system of biological classification. Rooted in formal logic, this taxonomy divides all living things into species and orders. It offers us a huge snapshot of the living world. It is tremendously useful. It is true. But it has limits. It cannot record how one species may, over time, give rise to some other, quite different species. (Engels had great fun with the duckbilled platypus, asking where that fitted into any rigid scheme of things.) Similarly, there is no “essence” hiding behind a cloud of steam, a puddle of water, or a block of ice. There are only structures, succeeding each other in response to changes in the local conditions. The world is not a ready-made thing: it is a complex interplay of processes, all of which are ebbing and flowing, coming into being and passing away.

So far so good. Applied to science, however, Engels’ schema turn out to be hardly more than a superior species of hand-waving. Indeed, “dialectical materialism” (as it later became known) proved so unwieldy, it took very few years of application before it became a blunt weapon in the hands of Stalinist philosophers who used it to demotivate, discredit and disbar any scientific colleague whose politics they didn’t like.

Harman has learned the lessons of history well. Though he’s curious to know where his philosophy abuts scientific practice (and especially the study of evolution), he is prepared to accept that specialists know what they are doing: that rigor in a narrow field is a legitimate way of squeezing knowledge out of the world, and that a 126-page A-format paperback is probably not the place to reinvent the wheel.

What really agitates him, fills his pages, and drives him to some cracking one-liners (this is, heavens be praised, a *funny* book about philosophy) is the sheer lack of rigour to be found in his own sphere.

While pillorying scientists for treating objects as superficial compared with their tinest pieces, philosophers in the humanities have for more than a century been leaping off the opposite cliff, treating objects “as needlessly deep or spooky hypotheses”. By claiming that an object is nothing but its relations or actions they unknowingly repeat the argument of the ancient Megarians , “who claimed that no one is a house-builder unless they are currently building a house”. Harman is sick and tired of this intellectual fashion, by which “‘becoming’ is blessed as the trump card of innovators, while ‘being’ is cursed as a sad-sack regession to the archaic philosophies of olden times”.

Above all, Harman has had it with peers and colleagues who zoom out and away from every detailed question, until the very world they’re meant to be studying resembles “the indeterminate momentum of the throbbing whole” (and this is not a joke — this is the sincerely meant position statement of another philosopher, a friendly acquaintance of his, Jane Bennett).

So what’s Harman’s solution? Basically, he wants to be able to talk unapologetically about objects. He explores a single example: the history of the Dutch East India Company. Without toppling into the “great men” view of history – according to which a world of inanimate props is pushed about by a few arbitrarily privileged human agents – he is out to show that the EIC was an actual *thing*, a more-or-less stable phenomenon ripe for investigation, and not simply a rag-bag collection of “human practices”.

Does his philosophy describe the Dutch East India Company rigorously enough for his work to qualify as real knowledge? I think so. In fact I think he succeeds to a degree which will surprise, reassure and entertain the scientifically minded.

Be in no doubt: Harman is no turncoat. He does not want the humanities to be “more scientific”. He wants them to be less scientific, but no less rigorous, able to handle, with rigour and versatility, the vast and teeming world of things science cannot handle: “Hillary Clinton, the city of Odessa, Tolkein’s imaginary Rivendell… a severed limb, a mixed herd of zebras and wildebeest, the non-existent 2016 Chicago Summer Olympics, and the constellation of Scorpio”.

Graham Harman
Polity, £9.99

The tomorrow person


You Belong to the Universe: Buckminster Fuller and the future by Jonathon Keats
reviewed for New Scientist, 11 June 2016.


IN 1927 the suicidal manager of a building materials company, Richard Buckminster (“Bucky”) Fuller, stood by the shores of Lake Michigan and decided he might as well live. A stern voice inside him intimated that his life after all had a purpose, “which could be fulfilled only by sharing his mind with the world”.

And share it he did, tirelessly for over half a century, with houses hung from masts, cars with inflatable wings, a brilliant and never-bettered equal-area map of the world, and concepts for massive open-access distance learning, domed cities and a new kind of playful, collaborative politics. The tsunami that Fuller’s wing flap set in motion is even now rolling over us, improving our future through degree shows, galleries, museums and (now and again) in the real world.

Indeed, Fuller’s”comprehensive anticipatory design scientists” are ten-a-penny these days. Until last year, they were being churned out like sausages by the design interactions department at the Royal College of Art, London. Futurological events dominate the agendas of venues across New York, from the Institute for Public Knowledge to the International Center of Photography. “Science Galleries”, too, are popping up like mushrooms after a spring rain, from London to Bangalore.

In You Belong to the Universe, Jonathon Keats, himself a critic, artist and self-styled “experimental philosopher”, looks hard into the mirror to find what of his difficult and sometimes pantaloonish hero may still be traced in the lineaments of your oh-so-modern “design futurist”.

Be in no doubt: Fuller deserves his visionary reputation. He grasped in his bones, as few have since, the dynamism of the universe. At the age of 21, Keats writes, “Bucky determined that the universe had no objects. Geometry described forces.”

A child of the aviation era, he used materials sparingly, focusing entirely on their tensile properties and on the way they stood up to wind and weather. He called this approach “doing more with less”. His light and sturdy geodesic dome became an icon of US ingenuity. He built one wherever his country sought influence, from India to Turkey to Japan.

Chapter by chapter, Keats asks how the future has served Fuller’s ideas on city planning, transport, architecture, education. It’s a risky scheme, because it invites you to set Fuller’s visions up simply to knock them down again with the big stick of hindsight. But Keats is far too canny for that trap. He puts his subject into context, works hard to establish what would and would not be reasonable for him to know and imagine, and explains why the history of built and manufactured things turned out the way it has, sometimes fulfilling, but more often thwarting, Fuller’s vision.

This ought to be a profoundly wrong-headed book, judging one man’s ideas against the entire recent history of Spaceship Earth (another of Fuller’s provocations). But You Belong to the Universe says more about Fuller and his future in a few pages than some whole biographies, and renews one’s interest – if not faith – in all those graduate design shows.

The past is like Baltimore: there is no there there


Longing for the Bomb: Oak Ridge and atomic nostalgia, Lindsey A. Freeman (University of North Carolina Press)
Seeing Green: The use and abuse of American environmental images, Finis Dunaway (Chicago University Press)
for New Scientist  (4 April 2015),

THE past can’t be re-experienced. It leaves only traces and artefacts, which we constantly shuffle, sort, discard and recover, in an obsessive effort to recall where we have come from. This is as true of societies as it is of individuals.

Lindsey Freeman, an assistant professor of sociology at the State University of New York, Buffalo, is the grandchild of first-generation residents of Oak Ridge, Tennessee. Once a “secret city”, where uranium was enriched for the US’s Manhattan Project, Oak Ridge opened its gates to the world in 1949 as America’s first “Atomic City”: a post-war utopia of universal healthcare, zero unemployment and state-owned housing.

In Longing for the Bomb, Freeman describes how residents of Oak Ridge dreamed up an identity for themselves as a new breed of American pioneer. He visits Oak Ridge’s Y-12 National Security Complex (an “American Uranium Center of Excellence”) during its Secret City Festival, boards its Scenic Excursion Train and cannot decide if converting a uranium processing site into a wildlife reserve is good or bad.

It would have been easy to turn the Oak Ridge story into something sinister, but Freeman is too generous a writer for that. Oak Ridge owes its existence to the geopolitical business of mass destruction, but its people have created stories that keep them a proud and happy community. Local trumps global, every time.

This is good for the founders of communities, but a problem for those who want to wake up those communities to the need for change. As historian Finis Dunaway puts it in Seeing Green, his history of environmental imagery, “even as media images have made the environmental crisis visible to a mass public, they often have masked systemic causes and ignored structural inequalities”.

Reading this, I was reminded of a talk by author Andrew Blackwell, where he told us just how hard it is to take authentic pictures of some of the world’s most polluted places. Systemic problems do not photograph well. Some manipulation is unavoidable.

Dunaway knows this. Three months after the nuclear accident at Three Mile Island in 1979the worst radioactive spill in US history occurred near Church Rock, New Mexico, on lands held by the Navajo nation. It took a week for the event to be reported, once, on a single news channel.

The remoteness of the site and a lack of national interest in Native American affairs might explain the silence but, as Dunaway points out, the absence of an iconic and photogenic cooling tower can’t have helped.

The iconic environmental images Dunaway discusses are essentially advertisements, and adverts address individuals. They assume that radical social change will catch on like any other consumer good. For example, the film An Inconvenient Truth, chock full of eye-catching images, is the acme of the sincere advertiser’s art, and its maker, former US vice-president and environmental campaigner Al Gore, is a vocal proponent of carbon offset and other market initiatives.

Dunaway, though, argues that you cannot market radical social action. For him, the moral seems to be that sometimes, you just have to give the order – as Franklin Roosevelt did when he made Oak Ridge a city.

A comic novel about the death of God


F by Daniel Kehlmann reviewed for the Guardian

It cannot be an easy thing to write a comic novel about the death of God. Still, the German novelist Daniel Kehlmann may just have pulled it off. “F” is the protagonist of a book within a book, the debut novel of Arthur Friedland, a rather disorganised buffoon who never had any success as a writer until an encounter with a hypnotist gave his life its chilly purpose: “This is an order, and you’re going to follow it because you want to follow it, and you want to because I’m ordering you, and I’m ordering you because you want me to give the order. Starting today, you’re going to make an effort. No matter what it costs. Repeat!”

My Name Is No One is so exuberantly nihilistic, its readers are throwing themselves off TV transmission towers. As Kehlmann says: “The sentences are well constructed, the narrative has a powerful flow, the reader would be enjoying the text were it not for a persistent feeling of somehow being mocked.”

If Kehlmann played this intertextual game to the hilt – if F itself were as unforgiving as Arthur’s novel – then we would be looking at a less important book, as well as a less enjoyable one: some Johnny-come-lately contribution to the French nouvelle vague. The spirit of Alain Robbe-Grillet, the movement’s greatest exponent, illuminates the scene in which Arthur takes his granddaughter to an art museum to study a picture by her missing uncle: “She stepped even closer, and immediately everything dissolved. There were no more people any more, no more little flags, no anchor, no bent watch. There were just some tiny bright patches of colour above the main deck. The white of the naked canvas shone through in several places, and even the ship was a mere assemblage of lines and dots. Where had it all gone?”

Sign up for Bookmarks: discover new books in our weekly email
Read more
There are many such moments, they are all as beautifully judged as this one, and they are not the point. The point of F is not its humour (though Kehlmann, like Robbe-Grillet, can be very funny indeed), but its generosity. Arthur’s three sons, in their turn, make superhuman efforts to give their lives significance, and these efforts tangle and trip over each other to generate the comic business of the book. The eldest, Martin, a Rubik’s Cube expert, embraces the priesthood despite his lack of faith. Of Arthur’s two sons by his second marriage, Eric enters the glass-and-steel world of high finance to help control his fear of cramped spaces. His twin brother, Ivan, is a would-be painter turned art dealer, and author of Mediocrity As an Aesthetic Phenomenon.

“When I was young, vain, and lacking all experience,” he recalls, “I thought the art world was corrupt. Today I know that’s not true. The art world is full of lovable people, full of enthusiasts, full of longing and truth. It is art itself as a sacred principle that unfortunately doesn’t exist.”

Ivan, like all the others, lives in a nihilistic universe, but he is not himself nihilistic. It worries him that the world cannot live up to his expectations and those of the people he admires. These people include his lover Heinrich Eulenboeck, an artist with a true calling but only mediocre ability. What kind of world is it that plays such a trick on a person? “How do you live with that, why do you keep on going?”

The answer seems to be love. In a godless world, love counts for a great deal. And failing love, ordinary human decency goes a long way. Since Kurt Vonnegut died, there has really been no one to tell us this; the reminder is welcome.

F is again translated by Carol Brown Janeway, but it is a better book than Kehlmann’s last, Fame, whose narrative gymnastics caused characters to lose or swap their identities, and even to topple into their own or other people’s fictions. Fame was knowing, driven by its own absurdity. F is about the world’s absurdity, and this makes a huge difference morally. The world is big, and ultimately unknowable, and life is short and memory pitifully limited.

In the absence of God, Kehlmann’s protagonists hold themselves to account, and they give themselves hell. Sometimes, they give each other hell. “Something terrible has happened and the people seem to be wanting to cover it up. If you were to look a little longer, hunt a little better for clues, you’d be able to figure it out, or at least you think so. But if you step back, the details disappear and all that remains is a colourful street scene: bright, cheerful, full of life.”

It is very hard to express how funny this all is. But laughter matters most in the dark.

Life signs

Image: @LydNicholas (swiped from her twitter feed)

Image: @LydNicholas (swiped from her twitter feed)

Alexandra Daisy Ginsberg and the editors of Synthetic Aesthetics pulled no punches when they launched their new book at a “Friday late” at London’s Victoria and Albert Museum. A couple of audience members interrupted to bemoan the sheer abstractness of the enterprise. Why couldn’t the panel explain what synthetic biologists actually did? A rather unfair criticism of an event that scattered living biological materials across every floor of the museum. The task of explaining where beauty sits in the world of synthetic biology fell to Drew Endy, assistant professor of bioengineering at Stanford University, California. Endy explained how, when synthetic biology began, its self-styled “engineers” treated living things as wayward and overcomplicated machines, in need of radical simplification. Now, researchers are learning to appreciate and harness biological complexity. “Ford’s original Model T motor car was simple, in engineering terms, but it was hell to operate. A Tesla is complicated but a pleasure to drive.” Standards of beauty are fuzzy, personal and intuitive. They inspire real conversations. So I imagine talking about beauty in design is useful for a discipline that’s constantly struggling with its own hype, never mind other people’s panic.

Unknown Quantity: a Real and Imagined History of Algebra by John Derbyshire

Unknown Quantity: a Real and Imagined History of Algebra by John Derbyshire
reviewed for the Telegraph,  17 May 2007

In 1572, the civil engineer Rafael Bombelli published a book of algebra, which, he said, would enable a novice to master the subject. It became a classic of mathematical literature. Four centuries later, John Derbyshire has written another complete account. It is not, and does not try to be, a classic. Derbyshire’s task is harder than Bombelli’s. A lot has happened to algebra in the intervening years, and so our expectations of the author – and his expectations of his readers – cannot be quite as demanding. Nothing will be mastered by a casual reading of Unknown Quantity, but much will be glimpsed of this alien, counter-intuitive, yet extremely versatile technique.

Derbyshire is a virtuoso at simplifying mathematics; he is best known for Prime Obsession (2003), an account of the Riemann hypothesis that very nearly avoided mentioning calculus. But if Prime Obsession was written in the genre of mathematical micro-histories established by Simon Singh’s Fermat’s Last Theorem, Derbyshire’s new work is more ambitious, more rigorous and less cute.

It embraces a history as long as the written record and its stories stand or fall to the degree that they contribute to a picture of the discipline. Gone are Prime Obsession’s optional maths chapters; in Unknown Quantity, six “maths primers” preface key events in the narrative. The reader gains a sketchy understanding of an abstract territory, then reads about its discovery. This is ugly but effective, much like the book itself, whose overall tone is reminiscent of Melvyn Bragg’s Radio 4 programme In Our Time: rushed, likeable and impossibly ambitious.

A history of mathematicians as well as mathematics, Unknown Quantity, like all books of its kind, labours under the shadow of E T Bell, whose Men of Mathematics (1937) set a high bar for readability. How can one compete with a description of 19th-century expansions of Abel’s Theorem as “a Gothic cathedral smothered in Irish lace, Italian confetti and French pastry”?

If subsequent historians are not quite left to mopping-up operations, it often reads as though they are. In Unknown Quantity, you can almost feel the author’s frustration as he works counter to his writerly instinct (he is also a novelist), applying the latest thinking to his biography of the 19th-century algebraist Évariste Galois – and draining much colour from Bell’s original.

Derbyshire makes amends, however, with a few flourishes of his own. Also, he places himself in his own account – a cultured, sardonic, sometimes self-deprecating researcher. This is not a chatty book, thank goodness, but it does possess a winning personality.

Sometimes, personality is all there is. The history of algebra is one of stops and starts. Derbyshire declares that for 269 years (during the 13th, 14th and early 15th centuries) little happened. Algebra is the language of abstraction, an unnatural way of thinking: “The wonder, to borrow a trope from Dr Johnson, is not that it took us so long to learn how to do this stuff; the wonder is that we can do it at all.”

The reason for algebra’s complex notation is that, in Leibniz’s phrase, it “relieves the imagination”, allowing us to handle abstract concepts by manipulating symbols. The idea that it might be applicable to things other than numbers – such as sets, and propositions in logic – dawned with tantalising slowness. By far the greater part of Derbyshire’s book tells this tale: how mathematicians learned to let go of number, and trust the terrifying fecundity of their notation.

Then, as we enter the 20th century, and algebra’s union with geometry, something odd happens: the mathematics gets harder to do but easier to imagine. Maths, of the basic sort, is a lousy subject to learn. Advanced mathematics is rich enough to sustain metaphor, so it is in some ways simpler to grasp.

Derbyshire’s parting vision of contemporary algebra – conveyed through easy visual analogies, judged by its applicability to physics, realised in glib computer graphics – is almost a let-down. The epic is over. The branches of mathematics have so interpenetrated each other, it seems unlikely that algebra, as an independent discipline, will survive.

This is not a prospect Derbyshire savours, which lends his book a mordant note. This is more than an engaging history; it records an entire, perhaps endangered, way of thinking.