Book Reviews

Book Review

The Unfinished Game: Pascal, Fermat, and the Seventeenth-CenturyLetter That Made the World Modern by Keith Devlin (Basic Books, 2008).

I've been reading some books recently with a new sort of question in mind: would I ever give this book to a student I was teaching?

In the case of The Unfinished Game, Keith Devlin's little riff on the 17th century exchange between Blaise Pascal and Pierre de Fermat that gave rise to the development of probability theory, and its role in the history of risk management, the answer is definitely not.

I did find value in the book for myself. I read it because I've become increasingly interested as an educator in the history of math, especially at key moments, and in relation to that interest, the book did give me a useful sense of the historical trajectory of the idea of probability, and Fermat's and Pascal's roles in it. Also, it includesthe full text (translated, of course) of a key letter from Pascal to Fermat dated August 24, 1654, and this primary source document, with its human in addition to mathematical content, was for me the most exciting thing in the book.

However, I have several deep objections, that matter to me all the more as a person working to make math more readily accessed and appreciated. First, the book succumbs to the trade publisher's temptation to sensationalize a story that is plenty important enough already. This lends the whole book a "look what a big deal this is!!!!" quality. Since probability is at the core of every facet of modern life (as Devlin rightly notes), it is really not necessary to talk, for example, about its role in predicting terrorist attacks. This is especially true since the technical details are omitted, so Devlin's description of the use of probability theory in such high-stakes situations is in awed terms that mystify the actual math at work and so deny the reader the right to think about it for herself.

This is especially frustrating from the point of view of a math educator. The goal of math education is to have people see math as something that their reasoning applies to; that they have the right to engage with and expect to understand. Huddling in cowed submission in the face of a claim to scientific authority that hides the content of the ideas being presented is a posture most non-mathematical adults are quite used to, and it is the exact opposite of the posture they need if they are to actually understand. Devlin invites them right into this powerless role with his "I'm-not-gonna-tell-you-any-details-but-look-how-important-this-is!!!" discussions, especially the one about terrorism.

But actually, there is a second problem that is even greater. The book's framing device is the 1654 correspondence between Pascal and Fermat, and throughout the book, Devlin seems intent on judging them against each other. He repeats several times that Fermat was the greater mathematician, and that Pascal appreciated this. There is a fair amount of fawning over both of them and their prowess, which is exceedingly common in writings on math history, but the insistence on raising up Fermat by putting Pascal down strikes me as immature whether the book is seen as a work of literature or history.

More than immature, though, it is pernicious and damaging to mathematics and our society's relationship to it. Devlin is not personally responsible for this. He is only acting out a perversity that is already present in how math is understood, at least in the United States and England. We as a society insist on ranking mathematicians - "Archimedes was the greatest mathematician of antiquity;" "Euler was the greatest in his day;" "Gauss was the greatest ever"; and so on. This ranking starts at the top but extends all the way downward, to schoolchildren. I have been a teacher for close to a decade now and I have hardly met the student who has not gotten from somewhere a strong idea of where she stands - "I'm good at math" or "I suck at math" as the case may be. And good or bad, this idea only hurts her. For those whose mathematical self-opinion is negative, it is obvious that it is an obstacle; but for those who have the sense that they are greater, it is an obstacle as well. For if you think you're good at it, what happens when you hit a problem that really challenges you - that makes you struggle? Most kids with the idea that they're "good at math" do their best to avoid such problems altogether, because they automatically cause a crisis: if I'm struggling, doesn't that mean I wasn't so good after all?

The problem is compounded by the grounds on which Devlin declares Fermat the greater mathematician. Just as he harps on this relative evaluation of the two, he also harps on the fact that in the letter that frames the book, Pascal is visibly struggling to understand what Fermat is saying, and it is on this basis that Fermat is judged superior. Devlin gives us to understand that Fermat probably solved the problem quickly and had no confusions about it, but humored Pascal's slow process of making sense of his elegant solution. Now, the primary source Devlin provides us does not strongly suggest this interpretation of the story, but more importantly, Devlin is implying to his readership that struggling with an idea makes you lesser. In order to be truly great, we are being told, you have to solve the problem quickly, without struggle. A student fed on such ideas is doomed as a creator of original mathematics because no great idea was ever arrived at without struggle. There is a venerable history of hiding this fact, but the fact remains. (I have read elsewhere that the celebrated Simon de Laplace used to write "it is easy to see" about conclusions that he would wrestle with for an hour.)

The most disappointing aspect of this is that the story Devlin tells is clearly an opportunity for the opposite lesson. I was drawn to pick up the book partly because of this. It has a sub-subtitle: "A Tale of How Mathematics Is Really Done," and it shows us (in the formof the Fermat-Pascal correspondence) a snapshot of a mathematical idea in the process of being created. It is the natural state of such an idea to be not entirely worked out, not yet cleaned up, still being struggled with. Of course Pascal was struggling with Fermat's solution. I feel certain that Fermat struggled with it too, even if by the time of the correspondence he had sorted things out for himself.

The point is that this could have been a story that revealed the creation of original, groundbreaking, in fact world-changing mathematics to be a human activity that the reader can relate to. How empowering, how uplifting, for anyone who has ever struggled with a mathematical idea (and all of us have) to watch one of the inventors of that idea go through the same process? That inspiring possibility is what this book could have been. Instead, it is one more voice telling all of us (since we have all struggled) that we are fundamentally different and lesser beings than the Math Gods like Fermat who originated everything we obediently consume.

2010 Ben Blum-Smith, 2010

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