I’ve got a mathematical problem. Birth of a Theorem is by one of the great geniuses of today, a cosmopolitan, liberal-minded man who helps his wife look after their children, likes big-hearted folk songs, welcomes diversity and wears the same jewellery as I do. But as a contribution to the genre of popular maths, the book stinks.

To give the problem extra calculus, my favourite maths writer is a sour-faced white supremacist with a mouth the shape of staple, who thinks women in America should be deprived of the vote and apparently calls himself ‘Derb’. An honest reviewer should obey his prejudices, so I’ve tried to find a way to cover up my dislike of Cédric Villani’s book, just as I tried to find a way I could slag off John Derbyshire’s excellent Prime Obsession (about the Riemann Hypothesis) when it came out. It’s not possible. There’s hardly a chapter in Birth of a Theorem that I could enjoy. But I’ve had a breakthrough. I have realised that I’ve had been looking at Villani’s Theorem in the wrong light.

There are three ways to write a popular maths book. The first is to ignore mathematics. G.H Hardy’s 1940 classic, A Mathematician’s Apology, is the best example of this approach. Another tactic is to get down your old university textbooks, take a deep breath and train the reader up through the mathematical basics of your chosen subject, using a combination of technical exercises and metaphors. (This is John Derbyshire’s style: his Unknown Quantity is one of the best histories of algebra there is.)

The third, and most common, method is to toy with the subject using a combination of anecdotes and interviews. This is to mathematics what a Smarties McFlurry is to haute cuisine. Often such books blow up two-thirds of the way through, when the author tries to bring the subject up to date in a explosion of impenetrable jargon that leaves the reader feeling panic-stricken and utterly stupid. But when done well, such as by Simon Singh or Alex Bellos, this approach is revelatory.

Cédric Villani has come up with a fourth method. It is entirely new: his book is impenetrable from page four. It is an account of how Professor Villani won the Fields medal, the greatest prize in mathematics, for a theorem about… to be honest, I have no idea what his theorem (or most of his book) is about, because the professor never explains it.

The story begins with what sounds like a duff episode of Star Trek:

‘So what’s up? Your message was pretty vague.’
‘My old demon’s back again — regularity for the inhomogeneous Boltzmann.’
‘Conditional regularity? You mean, modulo minimal regularity bounds?’
‘No, unconditional.’
‘Completely? Not even in a perturbative framework? You really think it’s possible?’

‘Never,’ says the blurb, ‘has mathematics seemed so magical or so exciting.’ Several of the chapters contain undiluted segments of very advanced integral equations. There are 12 solid pages of them at one point. Emails between mathematicians, with text that looks like the burbling of machines, are dumped in wholesale — a total of 35 pages of those:

Date: Sun, 22 Mar 2009 12:04:36 +800
From: Clement Mouhot <clement.mouhot@ceremade.dauphine.fr>
To: Cedric Villani <Cedric.VILLANI@umpa.ens-lyon.fr>
Subject: Re: finishing touches

Okay … here’s the plan:
a. we replace $F[h^{n+1}]\circ\Omega^n_{t,\tau}\circ S^0_{\tau,t}$ by $F[h^{n+1}]\circ S^0_{\tau,t}$, the remainder has the right time decay thanks to the estimates on $\Omega^n – Id$ …

On the few occasions when Villani does get close to discussing what the meanings of these terms are, he throws up his hands in genial exasperation and settles for an exclamation mark.

‘Landau Damping!’
‘Tartar’s T4 configuration!’
‘Ah, the Boltzmann! … Some people say I understand the mathematical world of this equation better than anyone alive.’

But he doesn’t want to let us understand it too.

I have yet to speak to a single person, and that includes my ex-landord (who won the International Mathematics Olympiad three times with a top score) who knows what Villani is talking about. The book has been a bestseller in France.

Birth of a Theorem is the popular maths equivalent of a novel that refuses to let the reader enter the story, except the bad writer usually does this by explaining everything — every thought the characters have, every connection in the plot, every clever authorial perception — so that readers aren’t allowed to have a single idea of their own. In mathematics, careful explanation has the opposite effect: it’s what allows you to begin thinking about the subject, however trivially, yourself. The first duty of a popular maths book author is to give the reader a tiny peek inside the subject by explaining the basics carefully.

Villani can be a very good writer when not goobledegooking. He gives a thrilling (and unforgiving) sense of the tension and jealous ambition for prizes that motivate him. At one point his co-researcher and former PhD student telephones him with a startling new idea. He wants to ‘give up on regularisation’? Villani thinks to himself. ‘He wants to forget about making up for the loss of regularity encoded in the time interval?’

Cedric, you’ve got to start paying attention, the young guys are brilliant. If you don’t watch out, you’re going to be left behind!
Okay, there’s nothing you can do about it, the next generation always ends up winning … but … already?

Villani is 36 years old.

It was a telephone conversation with a younger man that gave me my breakthrough with Villani’s Theorem, too. I rang Joe, the publicist for the book, to complain; and then Stuart, the publisher, to continue complaining; after that I sent an email to Cédric, the author (also, alas, a younger man), who apparently never listens to complaints.

It was Joe who put me right. ‘I want to like this,’ I explained. ‘I believe in supporting publications by admirable people who share my political opinions and taste in jewellery, especially when they’re mathematicians making an effort to bring their subject to a wider public. But this! Has he made it at all clear to you what the Boltzmann equation, which is at the heart of his theorem, actually is? Have you any idea what Landau Damping is about, even though he mentions it on every page?’

‘Not the foggiest,’ said Joe. ‘I look at all the mathematics in this book as illustrations.’

‘But then nine-tenths of the book is illustrations!’

‘Yes,’ said brave Joe.

And that is the way to think of Birth of a Theorem. It is not a book about mathematics; it is an art project. It is the brief story of an astonishing adventure, told with many pictures. It could also be read as a 15-minute play script about a man’s hunt for a medal, quirkily interleaved between 200 pages of instructions to the set-designer. The one thing it has almost nothing to do with is mathematics. Looked at in this light, Birth of a Theorem is a remarkable book and I advise everyone to buy it.

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