Mathematical fiction. What can that mean? A novel or play with fictional characters who are mathematicians? Mathematics that is fictional? A novel based on the life of a real mathematician? We have recently seen examples of the first — David Auburn's play, Proof, which won both the Pulitzer Prize and the Tony Award, and Uncle Petros and Goldbach's Conjecture, by Apostolos K. Doxiadis. Of the second type I don't recall seeing any examples and it's probably a good thing since mathematics that is fictional doesn't sound like very good mathematics. Novels that purport to be based on the lives of real mathematicians are more common. The French Mathematician by Tom Petsinis (a novel about Galois), and D'Alembert's Principle/A Novel in Three Panels, by Andrew Crumey, come to mind.
With The Parrot's Theorem (Le Théorème du Perroquet) we have something else, a novel about someone who is not a mathematician but who becomes deeply involved with mathematics as he tries to solve a mystery. The cast consists of an elderly Parisian bookseller, Pierre Ruche, whose home is shared with a woman, Perrette Liard, her son Max (who is deaf), her twin children, Lea and Jonathan, a parrot, Sidney, and, though dead at the time the events of the novel take place, an ever-present, rather mysterious figure, M. Grosrouvre (who refers to M. Ruche as πR — to be pronounced as if in French, of course). The parrot has been rescued by Max at the famous flea market at Clignancourt, where it has been badly abused by some men of dubious character who play a threatening role later in the narrative. Curiously enough, the parrot seems to know some mathematics. The connection with the elusive figure, M. Grosrouvre, is not immediately clear. All we know about him is that he and M. Ruche were friends in their youth in Paris. At the Sorbonne, Ruche studied philosophy and wrote on being, whereas Grosrouvre studied mathematics, writing a thesis on zero. The two became known as "Being and Nothingness", preceding Sartre's L'ètre et le néant by years. Grosrouvre has somehow ended up in Manaus, Brazil, an old and wealthy man obsessed with some great unsolved problems in mathematics. There he has amassed a magnificent mathematical library with many rare and expensive volumes. The idea of such a collection deep in the Amazon jungle stimulates our imagination, as does Manaus itself, which was the financial center of the great rubber boom of the late nineteenth century and, until the boom collapsed and the center of rubber production moved to southeast Asia, was a thriving city that built itself a magnificent opera house as well as other accouterments of a great city of the world. Even today people go many miles up the Amazon just to visit its very grand but little used opera house.
M. Grosrouvre has apparently died in a disastrous fire in his vast house in Brazil, but just before this event he has sent his library to M. Ruche, who wonders what has really happened to his friend and delves into the library looking for clues. (The first book he pulls from a packing crate when the books arrive in Paris is the first edition of Euler's Introductio in Analysin Infinitorum! M. Grosrouvre clearly had excellent taste.) This search takes up much of the book — the exploration of the history of mathematics in search of parallels with what M. Ruche knows about his almost-forgotten friend of many years ago, in the hope of finding clues to explain how M. Grosrvouvre came to his rather bizarre end.
The author is a French professor of the history of science at the Université de Paris, so the history sections tend to be detailed and, alas, somewhat didactic. The structure of the novel has M. Ruche going through his newly acquired library systematically, but also searching for additional historical details at the Bibliothèque Nationale, the Institut du Monde Arabe, the Palais de la Déscouverte, and other familiar Paris settings. It is in these sections that it becomes not entirely clear what the audience is for the book. Mathematicians with even a passing interest in the history of their subject will find much of this explication very familiar stuff. All the old stories — indeed myths and legends — seem to be there. Readers with no mathematical background, on the other hand, may find these passages heavy going. Historians will no doubt squirm uneasily as they read stories about Pythagoras and Archimedes that I believe are long since discredited. They make wonderful stories but not very good history. But, this is a novel, so how can one complain about its telling stories?
Here's where I begin to feel uncomfortable about "mathematical fiction". How much of this is to be taken as fact, how much as fiction? A mathematician will also feel uneasy — and this may be no fault of the author but instead of the translator — because often mathematical statements are made without the precision they would have in a mathematical text. For example, "number" is used where "integer" would be more precise. A list of factors of 220 on page 70 omits the factor 20. In a list of polyhedra, a cube has six square sides but a dodecahedron has twelve facets. Sometimes we see "polyhedra", other times "polyhedrons". We read on page 106 that "there are wide triangles which have one obtuse angle, and narrow triangles which have one acute angle." What? On page 109, we read that "prime factors are the smallest prime numbers that divide into the whole number." And so on. Other comments are provocative. In the discussion of Galois's difficulty in getting anyone to read his work, Poisson is dismissed as someone "who was responsible for a nice little theory of probability." M. Ruche, in his lecture on Hypatia, claims that "she was not the only woman to be burned at the stake; religious authority often felt that a woman was better dead than powerful."
Now on a more positive note: this book is just full of mathematics. And some of the descriptions and comments are indeed charming. As M. Ruche is wandering around the market to buy the ingredients of the evening's osso bucco, he's trying to come up with an answer to Lea's question: "Why did maths begin in Greece and not somewhere else?" Does everyone know the story of Bhaskara and his daughter, Lilavati, and that he named for her his great book on his life's work, a list of mathematical problems? In the section on noneuclidean geometry we read about Menelaus' explanation of the difference between flat and curved planes: "A triangle spread onto the skin of an orange would be 'bigger' than the same triangle placed onto the leaf of an orange tree." Again, in discussing noneuclidean geometry after a ski trip, Lea proposes the following postulate for euclidean geometry: "For any given foot, there is only one possible ski and to that ski only one parallel ski." Lea is described as "thin as a rake — Euclid would have said she had 'length but no breadth'." On the usefulness of mathematics, we see the following exchange, reported by M. Ruche: "That reminds me of something I once heard Grosrouvre say to one of his mistresses. It was in a cafe near the Sorbonne where we always met. Grosrouvre arrived very late and she asked, 'What kept you, dearest?' Grosrouvre replied, 'I was finishing a maths problem.' The girl shrugged and said, 'I don't get it — how can you spend so much time on that stuff? What use is maths anyway?' Elgar [Grosrouvre] looked her straight in the eye and said, 'And love, darling, what use is that?' He never saw her again."
All in all, this is an entertaining book, deftly written but on occasion frustrating and occasionally very slow as it plows through the details of a mathematical discourse. How many novels have a six-page name index at the back (called a cast of characters, of course)? And all but a few are mathematicians.
Gerald L. Alexanderson (firstname.lastname@example.org) is Valeriote Professor of Science at Santa Clara University. He has served as Secretary and President of the MAA.