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.


Elephants on Acid and Other Bizarre Experiments by Alex Boese

There is a connection between vaudeville and science, and it is more profound than people credit. Alex Boese’s collection of bizarre scientific anecdotes illuminates this connection, claims far too much for it, and loses the thread of it entirely.

This probably doesn’t matter – by Boese’s own estimation, Elephants on Acid is a book you dip into in the bathroom. There’s even an entire chapter, ‘Toilet Reading’, dedicated to this very idea.

But Boese, quietly meticulous, is a champion of the idea of science. So, at the risk of taking a mallet to a sugar-coated almond, let’s take him seriously here.

Boese is the curator of a splendid on-line museum of hoaxes – museumofhoaxes.com. To move from deliberate fakery to science gone awry, deliberately or not, is, Boese argues, but a small step.

Hoaxers and experimenters are both manipulators of reality. But only experimenters wrap themselves in the authority of science. ‘This sense of gravity is what lends bizarre experiments their particularly surreal quality.’ More charitably, he might have added: only scientists run a serious and career-busting risk of hoaxing themselves.

Boese’s accounts of unlikely experiments include sensible and legitimate studies into risible subjects (how could studies into human ticklishness not sound silly?) Elsewhere, accounts of doubtful ‘discoveries’ reveal how badly credulousness and ambition will misdirect the enquiring mind.

Wandering among Boese’s carnival of curiosities we learn, for example, the precise weight of a human soul and acquire a method for springing crystalline insects out of rocks.

Less convincing are his stories of research misinterpreted by gullible or hostile media. A sharper editor would have spotted when Boese’s eye for a good tale was leading him astray.

In 1943 the behaviourist Burrhus Skinner invented a comfortable, labour-saving crib for his baby daughter – only to be pilloried for imprisoning her in an experimental ‘box’. This is a tale of irony and injustice, deftly told. But it is not ‘bizarre science’.

It’s devilishly difficult to get good at something unless you can find the fun in it. The more intellectually serious a work is, the more likely it is to have playful, even mischievous aspects. Science is no exception.

The more entertaining, and less troubling, of Boese’s tales involve ingenious, self-aware acts of scientific folly. We learn a truly magnificent (and wrong) formula for working out the moment at which cocktail parties become too loud.

A study that involves erotically propositioning young men on a wobbly bridge must surely have fallen out of the bottom of an Atom Egoyan movie. And pet owners should heed a slapstick 2006 study entitled ‘Do Dogs Seek Help in an Emergency?’ (‘Pinned beneath the shelves, each owner let go of his or her dog’s leash and began imploring the animal to get help from the person in the lobby.’)

Yet, for all its hilarity, Elephants on Acid proves to be an oddly disturbing experience when read cover-to-cover.

The decision to put all the truly gut-wrenching vivisection stories in the first chapter was foolhardy. Robert White’s 1962 attempt to isolate a monkey’s brain by removing, piece by piece, the face and skull, absolutely belongs in this book – but it is delivered so early that it’s one hell of a hurdle to clear in the first five minutes of reading.

Other horrors lurk in wait for those who persevere (Ewen Cameron’s brainwashing experiments of the 1950s are particularly horrendous). Boese’s off-the-cuff observation that the Cold War had its surgical and psychological aspects is not staggeringly original but it does mollify our easy outrage at such past ‘mistakes’.

Quite rightly so, for most of what we primly label ‘maverick science’ is no such thing; it is simply science that served a long-since-vanished purpose.

Most disturbing of all, however are those celebrated and familiar behavioural experiments that, while harming no one, reveal human gullibility, spite, vanity and witlessness.

Philip Zimbardo’s prison-psychology experiment at Stanford University had to be terminated, so keenly did his volunteers brutalise each other. Testing the limits of obedience (clue: there aren’t any), Stanley Milgram invited volunteers to inflict what they thought were potentially lethal electric shocks to people. Few demurred. Ironically, these kinds of experiments share methods with many stage magic routines.

The connection between vaudeville and science is profound, all right – and not particularly funny. Boese is right to invite us to dip in and out of his book. His facetious mask cannot hide for long the underlying seriousness of such striking material.


Glimpses of the Wonderful

Glimpses of the Wonderful by Anne Thwaite, reviewed for New Scientist, 2 November 2002

WHY do we study the natural world? Today, we might answer: to uncover life’s underlying principles. In the mid-19th century, those underlying principles were thought to be already established: life was a Creation of God’s.

Ann Thwaite, a literary biographer best known for her lives of Edmund Gosse and A. A. Milne, forays into the history of science with this life of Edmund Gosse’s father, the naturalist Philip Henry Gosse.

Thwaite shows that Gosse’s believed that “the gratification of scientific curiosity is worse than useless if we ignore God”. After all, what is science for, if not veneration? What Gosse never could do was abandon his belief in the revealed Word and take up the un-anthropomorphic “search for underlying principles” which would become the defining feature of modern science.

A self-styled Puritan who famously called Christmas pudding “the devil’s sweetmeat”, Gosse was also the finest naturalist of his age. He enjoyed a lifelong, friendly correspondence with Charles Darwin, and popularised the science of his day with rigour and intelligence. For most of us, though, Gosse is best known through his son’s memoir Father and Son – a poignant account of Edmund’s father’s “strange severities and eccentric prohibitions”, to which Thwaite provides a robust response.

Dogmatic belief shields us from the inevitability of death. Gosse, born into a millennial age, believed that Christ would return before he died. He spent the last hours of his life in a state of heart-rending and terrible dejection. Who can say they do not share Gosse’s terrible fear of death? His unenviable distinction was to hold fast to conventional comforts in a revolutionary age.

Thwaite weaves together Gosse’s professional studies and personal convictions, not into some dead synthesis, but into a story of a man caught in the toils of the scientific establishment as it re-geared itself for the modern age. Omphalos, Gosse’s great – and greatly lampooned – attempt to marry creationist dogma with the evolutionary record, is the work by which he is best known. It is a measure of Thwaite’s intellectual grasp that we understand how well considered that book really is, and at the same time how unworthy of him. Better that we remember Gosse as a friend remembered him: a figure to embody all the contradictions of his day, “plunging into a pool in full sacerdotal black, after a sea anemone”.