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Reviews
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Birth of a Theorum: A mathematical adventure

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Kester Brewin

Birth of a Theorem: A mathematical adventure

Cédric Villani

The Bodley Head, 272pp

Is mathematics created, or discovered? This is the question that haunts this diverting and surprisingly moving book by French mathematician Cédric Villani. Over the past few years there has been something of a rash of books attempting to rehabilitate mathematics. Alex's Adventures in Numberland, for example, sought - as most of these works do - to convince the reader that this odd world of mathematics is really a lovely and welcoming place. It might be that they have found it rather gnarly and unfriendly in the past, but they'd just met the wrong people… Villani is not interested in this at all. He is less of a contrite tourist guide than a Mad Hatter, pulling you down a rabbit hole but making no apology for the fact that the place he is going to show you is bizarre, ridiculous in places, and very very difficult to understand. Villani himself is all of these things, except the last. Flamboyant, a self-confessed dandy and bon-viveur, he is a delightful chaperone. As he leads the reader, it turns out that the world that he inhabits is rather wonderful. Yes the mathematics that he describes is unspeakably hard - 'You break down the solution in terms of the replicas of the torus… you change the variables in each piece… and finally you end up with convergence in 1/t. Slow, but it looks about right.' - but the community of people with which he works on these problems are generous, humorous and infectiously passionate about what they do, even if only a handful of people in the world understand - or care. There is thus an interesting parallel here. The work of the professional mathematician - incomprehensible to many, uninteresting to many more, yet done within a community who consider it a genuine vocation - is very similar to that of the theologian. Indeed, there are those who consider the similarities to run much deeper. At a conference I attended in Belfast last week, John Caputo, one of the leading voices in the philosophy of religion, spoke of how mathematics belongs to the psychoanalytic class of 'the Real', encompassing that which exists in a state that is always diminished by mediation into another form. This would put it firmly into the state of something that we discover, rather than create ourselves, something pre-existing that, like God, is gradually revealed in our tenacious engagement with it. It is this tenacity that so impresses. The Birth of a Theorem is essentially a Genesis narrative, the story of how Villani worked on, and came to prove a certain set of new results that arise from the Boltzmann equation - a highly complex means of describing the behaviour of gases that are not in thermodynamic equilibrium. It is no spoiler to tell you that, by the end of the book, he has received a Fields Medal - the equivalent of a Nobel Prize in mathematics - for this work, because the real thrill, as we know in parallel, is the journey to that point. For Villani, this means sitting with colleagues in his home academy, swapping potential lines of attack over emails with mathematicians across the globe and absorbing new proofs at conferences in rather glamorous locations. All of this frenetic activity towards the goal of understanding this mind-blowingly complex problem is told with great pace and vigour, but the meat that makes the sandwich worth savouring is the history that comes with it. Unashamedly, Villani positions himself on the shoulders of giants, and introduces us to each titan that supports his work as he goes along. More than that though, however much this is essentially a work about success - the newborn is delivered, and survives the forensic examination by his peers - the spice is given by the many moments of failure along the way. Months are spent following particular lines of thought, only for them to prove fruitless. Methods are dragged in from other theorems, only for them to be shown to be useless. It is in these stories of wrongdoing, of miscarriage, of heresy if you will, that make the book worth reading at all. No one wants to read the tale of the mountaineer who scaled the summit with ease. It is in the Paris Metro breaking down, the food on offer at a certain conference, watching his children sleeping while he works or life in unwavering service to a problem that no one else can see that the book sings. This is not to say that it is an easy read. The regular changes in typeface are annoying, and the gushing style can be overblown at times. (Terry Pratchett once quipped that 'five exclamation marks are the sure sign of an insane mind,' and this might, in fact, be the proof of this.) This aside, the book is worth reading, even if mathematics is not your 'thing.' Why? Because we are all creatures of language, and the question of creation vs discovery is one that should haunt each of us. Mathematics is just one way of describing the reality we experience. Poetry is another, but would anyone argue that poems pre-exist us, lie latent, waiting for the poet to unearth them? Perhaps, but questions remains as to the site at which this archaeology is done. Is it within the human unconscious, or the divine mind of an eternal creator? The theorem that Villani brings to birth thus ends up as both a creation and a discovery, for it is surely in the attempt to rein in and control this wild thing he has discovered that he creates the means by which the wildness within himself can be expressed. For that story, his account is well worth reading.