The wisdom of equations

From Neptune to the neutrino: stories of how “sitting and thinking” has come up with ground-breaking scientific predictions

The Magicians

Great minds and the central miracle of science

Marcus Chown Faber & Faber 2020 Pb, 294pp, £12.99, notes, ind, ISBN 9780571346394

The Universe Speaks in Numbers

How modern maths reveals nature’s deepest secrets Graham Farmelo Faber & Faber 2020 Pb, 320pp, £10.99, notes, illus, ind, ISBN 9780571321827

It’s conventional to say that for a theory to qualify as “real science” it has to come from practical observation and experiment, not just sitting and thinking, which is what mystics and pseudoscientists do. As striking as that view is, it’s not true. Some of the greatest discoveries in physics have been made by people sitting at a desk with pen and paper, rather than looking through a microscope or telescope. That’s one of the themes shared by these two books.

The other theme is that “the book of nature is written in the language of mathematics”, as Farmelo quotes Galileo as saying. Neither book contains any maths as such, but narrative accounts of important discoveries that happened to be prompted by mathematical musings. There’s a difference between the “sitting and thinking” you or I might do and that of people like James Clerk Maxwell, Albert Einstein and Paul Dirac, who all feature prominently in both books. Their genius lay in developing mathematical equations to describe known phenomena, and then spotting that those same equations predicted new, hitherto unsuspected phenomena – which, when searched for by experimentalists, turned out to exist.

Despite the similarity of subject matter, the two books are very different in style. Chown is an experienced pop-science author, who has also written science fiction, and The Magicians is as enjoyable an account of physics history as you’ll find anywhere, often reading more like a novel than non-fiction. Farmelo is a professional physicist, and while his writing is competent and readable, it’s much more pedestrian. Compare this from Farmelo – “the mathematics enabled creative theorists to make many precise and surprising predictions about the real world” – with this from Chown: “The central magic of science [is] its ability to predict the existence of things previously undreamt of which, when people went out and looked for them, turned out to actually exist in the real universe”. They’re essentially saying the same thing, but I find Chown’s version far more exciting and inspiring.

Chown’s book is divided into nine self-contained chapters, each dealing with a specific prediction and its experimental verification – often many decades later – with the two narratives woven together in non-linear style. The first chapter deals with the discovery of Neptune in 1846, following a prediction of its existence by the mathematician Urbain Le Verrier. As the astronomer Flammarion wrote: “Without leaving his study, without even looking at the sky, Le Verrier had found the unknown planet solely by mathematical calculation.” After that, there’s Maxwell’s prediction of radio waves and their subsequent discovery by Heinrich Hertz, followed by many of the key moments in 20th and 21st-century physics, from antimatter and black holes to the Higgs boson and gravitational waves. In some cases, the prediction was so bizarre that experimentalists were highly sceptical it would ever be verified. A case in point was the elusively tiny subatomic particle called the neutrino – “a fleeting ghost that barely haunted the world of physical reality”, as Chown puts it – which was considered such a dubious proposition that the search for it was dubbed Project Poltergeist.

The “magicians” of Chown’s title are the theoretical physicists who wrote down the equations which eventually proved so useful, but I can’t help thinking the real protagonists are the equations themselves. Chown quotes Heinrich Hertz: “One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers.” Paul Dirac, whose equation predicted the existence of antimatter, puts it even more bluntly: “My equation was smarter than I was”.

The first third of Farmelo’s book covers similar ground to Chown’s, but then it goes off into murkier and more controversial territory – the modern trend for mathematics-driven research, such as string theory, which is admired more for its formal elegance than its practical usefulness. Farmelo traces this idea back to Dirac too, who argued that “the research worker, in trying to express the fundamental laws of nature in mathematical form, should strive for mathematical beauty” and that “the rules which the mathematician finds interesting are the same as those which nature has chosen”.

In summary, Chown’s book is a brilliant read that should be on the bookshelf of anyone interested in the history of ideas. Farmelo’s book has more limited appeal, but if you’re a fan of abstract science like string theory you may want to check it out. Andrew May

★★★★★ ★★★