A kind of “symbol knitting”

Mathematics began when practical thinkers like Archimedes decided to ignore naysayers like Zeno (whose paradoxes were meant to bury mathematics, not to praise it) and deal with nonsenses like pi and the square root of 1. How do such monstrosities yield such sensible results? Because mathematics is magical. Deal with it.

Reviewing new books by Paul Lockhart and Ian Stewart for The Spectator 

D’Arcy Wentworth Thompson, the man who shaped biology and art

Biomorphic portrait of D'Arcy Thompson

Darren McFarlane, Scarus, Pomacanthus, 2012, oil on canvas. (University of Dundee Museum Services © the artist)

Even as geneticists like Ernst Mayr and Theodosius Dobzhansky were revealing the genetic mechanisms that constrain how living things evolve, Thompson was revealing the constraints and opportunities afforded to living things by physics and chemistry. Crudely put, genetics explains why dogs, say, look like other dogs. Thompson did something different: he glimpsed why dogs look the way they do.

For New Scientist, 1 February 2017

Let maths illuminate your life!

Thanks to the review I wrote of Thinking in Numbers, an excellent collection of essays about the psychology and culture of numbers, the RSA has invited me along to talk with the author Daniel Tammett on 27 Nov 2012 at 13:00. Follow this link for details of venue etc; you can also follow the event remotely through the following links

http://www.thersa.org/events/listen-live

http://www.thersa.org/events/watch-live

Maths into English

One to Nine by Andrew Hodges and The Tiger that Isn’t by Michael Blastland and Andrew Dilnot
reviewed for the Telegraph, 22 September 2007

Twenty-four years have passed since Andrew Hodges published his biography of the mathematician Alan Turing. Hodges, a long-term member of the Mathematical Physics Research Group at Oxford, has spent the years since exploring the “twistor geometry” developed by Roger Penrose, writing music and dabbling with self-promotion.

Follow the link to One to Nine’s web page, and you will soon be stumbling over the furniture of Hodges’s other lives: his music, his sexuality, his ambitions for his self?published novel – the usual spillage. He must be immune to bathos, or blind to it. But why should he care what other people think? He knows full well that, once put in the right order, these base metals will be transformed.

“Writing,” says Hodges, “is the business of turning multi?dimensional facts and ideas into a one?dimensional string of symbols.”

One to Nine – ostensibly a simple snapshot of the mathematical world – is a virtuoso stream of consciousness containing everything important there is to say about numbers (and Vaughan Williams, and climate change, and the Pet Shop Boys) in just over 300 pages. It contains multitudes. It is cogent, charming and deeply personal, all at once.

“Dense” does not begin to describe it. There is extraordinary concision at work. Hodges covers colour space and colour perception in two or three pages. The exponential constant e requires four pages. These examples come from the extreme shallow end of the mathematical pool: there are depths here not everyone will fathom. But this is the point: One to Nine makes the unfathomable enticing and gives the reader tremendous motivation to explore further.

This is a consciously old-fashioned conceit. One to Nine is modelled on Constance Reid’s 1956 classic, From Zero to Infinity. Like Reid’s, each of Hodges’s chapters explores the ideas associated with a given number. Mathematicians are quiet iconoclasts, so this is work that each generation must do for itself.

When Hodges considers his own contributions (in particular, to the mathematics underpinning physical reality), the skin tightens over the skull: “The scientific record of the past century suggests that this chapter will soon look like faded pages from Eddington,” he writes. (Towards the end of his life, Sir Arthur Eddington, who died in 1944, assayed a “theory of everything”. Experimental evidence ran counter to his work, which today generates only intermittent interest.)

But then, mathematics “does not have much to do with optimising personal profit or pleasure as commonly understood”.

The mordant register of his prose serves Hodges as well as it served Turing all those years ago. Like Turing: the Enigma, One to Nine proceeds, by subtle indirection, to express a man through his numbers.

If you think organisations, economies or nations would be more suited to mathematical description, think again. Michael Blastland and Andrew Dilnot’s The Tiger that Isn’t contains this description of the International Passenger Survey, the organisation responsible for producing many of our immigration figures:

The ferry heaves into its journey and, equipped with their passenger vignettes, the survey team members also set off, like Attenboroughs in the undergrowth, to track down their prey, and hope they all speak English. And so the tides of people swilling about the world?… are captured for the record if they travel by sea, when skulking by slot machines, half?way through a croissant, or off to the ladies’ loo.

Their point is this: in the real world, counting is back-breaking labour. Those who sieve the world for numbers – surveyors, clinicians, statisticians and the rest – are engaged in difficult work, and the authors think it nothing short of criminal the way the rest of us misinterpret, misuse or simply ignore their hard-won results. This is a very angry and very funny book.

The authors have worked together before, on the series More or Less – BBC Radio 4’s antidote to the sort of bad mathematics that mars personal decision-making, political debate, most press releases, and not a few items from the corporation’s own news schedule.

Confusion between correlation and cause, wild errors in the estimation of risk, the misuse of averages: Blastland and Dilnot round up and dispatch whole categories of woolly thinking.

They have a positive agenda. A handful of very obvious mathematical ideas – ideas they claim (with a certain insouciance) are entirely intuitive – are all we need to wield the numbers for ourselves; with them, we will be better informed, and will make more realistic decisions.

This is one of those maths books that claims to be self?help, and on the evidence presented here, we are in dire need of it. A late chapter contains the results of a general knowledge quiz given to senior civil servants in 2005.

The questions were simple enough. Among them: what share of UK income tax is paid by the top one per cent of earners? For the record, in 2005 it was 21 per cent. Our policy?makers didn’t have a clue.

“The deepest pitfall with numbers owes nothing to the numbers themselves and much to the slack way they are treated, with carelessness all the way to contempt.”

This jolly airport read will not change all that. But it should stir things up a bit.

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.