Once Upon a Prime, page 27
4. Derek Ball’s dissertation, Mathematics in George Eliot’s Novels, is at the time of writing available from the University of Leicester, in the UK. You can download it from https://leicester.figshare.com/articles/thesis/Mathematics_in_George_Eliot_s_Novels/10239446/1. If you are interested in the broader links between mathematics, science, and creativity in the nineteenth century, Professor Alice Jenkins of the University of Glasgow has written extensively on these topics. Her book, Space and the “March of Mind”: Literature and the Physical Sciences in Britain, 1815–1850 (Oxford University Press, 2007), is an academic exploration of the conversation between science and literature in nineteenth-century Britain.
5. Can I tell you a secret? I haven’t read Finnegans Wake; I’ve just dipped in and out. I felt much better about this when I read a brilliant article called “Finnegans Wake for Dummies,” by Sebastian D. G. Knowles, which I highly recommend if you can get hold of the fall 2008 issue of the James Joyce Quarterly. The first sentence is this: “I begin with a confession: in September 2003, after attending two decades of Joyce symposia, teaching over a dozen courses on Joyce, writing a book entirely devoted to Joyce’s work, and editing another, I had still not yet read Finnegans Wake.” In desperation, he signed up to teach a course on it, and that’s what finally did the trick.
6. In Chapter 3, I stated the fifth postulate of Euclid in a different way: that given a line and a point not on the line, there is exactly one line through that point, parallel to the given line. This version, which is known as Playfair’s axiom, after the Scottish mathematician John Playfair, who publicized it in the eighteenth century, is logically equivalent to the one Joyce is referencing, but much easier to work with. It’s the version that Hilbert used in his Foundations of Geometry that we mentioned in Chapter 3. Joyce’s version is what was in the original Greek text.
7: Travels in Fabulous Realms
1. The tallest person who has ever lived is Robert Wadlow (1918–1940). He had a pituitary gland disorder, which meant that his body produced too much growth hormone, and this condition continued throughout his life. By the time he was eight, he was taller than his father; when he died, aged twenty-two, he was 8 feet 11.1 inches (2.72 meters) tall and weighed 315 pounds (199 kilograms). He had a job with the International Shoe Company doing promotional work—the schtick was that if they could make the size 37 shoes that Wadlow wore, then they could make shoes for anyone. Wadlow required leg braces when walking and had little feeling in his legs and feet—ultimately this was what caused his death, because he didn’t feel that a minor abrasion from one of his leg braces had become infected until it was too late to stop the sepsis from spreading.
2. Some editions have “algebraists” rather than “geometers.”
3. The essay appeared in the book Possible Worlds and Other Essays, published by Chatto and Windus in 1927, but is easy to find online too. Haldane also has bad news about angels during a discussion of flight in which he explains that if you scale up something like a bird by a factor of four, the power required for it to fly increases by a factor of 128. He goes on to say that an angel “whose muscles developed no more power, weight for weight, than those of an eagle or a pigeon would require a breast projecting for about four feet to house the muscles engaged in working its wings, while to economize in weight, its legs would have to be reduced to mere stilts.”
4. There have been a couple of real scientific papers on the plausibility of tiny people like Lilliputians and Borrowers, which make for entertaining reading if you like that sort of thing (and who doesn’t?). I didn’t want to make the calculations of Lilliputians’ recommended calorie intake even more complicated, but a 2019 paper suggests that Lilliputians actually need 57 calories, not the 9.3 in my rough estimate, because of Quetelet’s observations about how mass changes with height. But that just makes things even worse in terms of Lilliput’s economy. Check out T. Kuroki, “Physiological Essay on Gulliver’s Travels: A Correction After Three Centuries,” in The Journal of Physiological Sciences 69 (2019): 421–24. Meanwhile, in “What Would the World Be Like to a Borrower?” (Journal of Interdisciplinary Science Topics 5, 2016), J. G. Panuelos and L. H. Green give more detail about several aspects of life for Borrowers, including discussions of their voices—likely too high-pitched and faint for us to hear them.
8: Taking an Idea for a Walk
1. Just as any picture of a three-dimensional cube cannot render every side as a square, any drawing of a hypercube necessarily distorts some of the lengths. One way to get around the challenge of representing a three-dimensional cube on a two-dimensional page is to draw what’s called the net of the cube. This is a diagram with six squares that can be cut out and folded up in three dimensions to make a cube. In the same way, we can make a three-dimensional net of eight cubes that could be folded up in four dimensions to make a hypercube. It’s this version of the hypercube that appears in Salvador Dalí’s famous 1954 painting Crucifixion (Corpus Hypercubus).
2. These strange new geometries seemed incomprehensibly alien to many. Even Ivan Karamazov, the most intellectual of the Karamazov brothers in Dostoyevsky’s 1880 novel, struggled with them. Ivan compares trying to understand the divine to trying to understand non-Euclidean geometry. There are geometers and philosophers, he says, who “dare to dream that two parallel lines, which according to Euclid can never meet on earth, may meet somewhere in infinity. I have come to the conclusion that, since I can’t understand even that, I can’t expect to understand about God. I acknowledge humbly that I have no faculty for settling such questions, I have a Euclidean, earthly mind, and how could I solve problems that are not of this world?”
3. Flatland wasn’t the first time that two-dimensional beings trying to grasp a three-dimensional world had been invoked as an analogy. The German physicist Hermann von Helmholtz had discussed how we would interpret the world if we were two-dimensional beings living on the surface of a sphere. The most important precursor to Flatland is the article “What Is the Fourth Dimension?” by Charles Howard Hinton, which was almost certainly read by Abbott. Hinton was a mathematician, teacher, and writer, a great popularizer of science. The article asks us to imagine “a being confined to a plane” and to suppose “some figure, such as a circle or rectangle, to be endowed with the power of perception.” Does that sound familiar?
4. Angles in triangles on the surface of a sphere really do add up to more than 180° We don’t have to worry about it in everyday life, because it turns out that the amount by which the sum exceeds 180° is proportional to how much of the sphere’s surface is contained in the triangle. But this knowledge was vital for one of the more impressive technical feats of the nineteenth century: the Great Trigonometrical Survey, a seventy-year undertaking with the aim of mapping the whole of India with high precision. The distances being measured were so vast that the curvature of the earth, and its effect on the angles in a triangle, had to be taken into account in the calculations.
5. The story is “The Unparalleled Adventure of One Hans Pfaall.”
6. “Cryptographic mathematicians,” the book says, are “by nature high-strung workaholics.” Are they, though? Anecdote is not data, but I met some of the mathematicians at Royal Holloway (where Brown’s other cryptographer heroine, Sophie Neveu, is supposed to have trained) when I gave an invited seminar there a few years ago, and they were a lovely bunch who gave every appearance, over tea and biscuits, of being relaxed and genial, and not at all addicted to workahol.
7. Okay fine, RSA stands for Rivest-Shamir-Adleman. It’s really not their fault that they get all the credit and fame. They earned it. Clifford Cocks doesn’t begrudge them it either. As he said, “You don’t get into this business for public recognition.” You can read more about RSA in Simon Singh’s excellent history of cryptography, The Code Book, which includes this Clifford Cocks quote in its chapter on public key cryptography.
9: The Real Life of Pi
1. Life of Pi is far from being the only work of literature to mention this fascinating number. For maximum erudition, check out this passage from the beginning of Umberto Eco’s Foucault’s Pendulum (a novel I’ve heard described as “the thinking person’s Da Vinci Code”): “That was when I saw the Pendulum.… I knew—but anyone could have sensed it in the magic of that serene breathing—that the period was governed by the square root of the length of the wire and by π, that number which, however irrational to sublunary minds, through a higher rationality binds the circumference and diameter of all possible circles. The time it took the sphere to swing from end to end was determined by an arcane conspiracy between the most timeless of measures: the singularity of the point of suspension, the duality of the plane’s dimensions, the triadic beginning of π, the secret quadratic nature of the root, and the unnumbered perfection of the circle itself.” You don’t get that in Dan Brown.
2. My quotations are from James E. Irby’s English translation of Jorge Luis Borges, Labyrinths, Penguin Classics edition (2000).
10: Moriarty Was a Mathematician
1. Fermat’s Last Theorem has been rolled out in this way on several occasions over the years. Pre-Wiles, your characters could gain fame and fortune by finding any proof. Post-Wiles, they have to find the elusive “short” proof. There was an episode of the British TV show Doctor Who in 2010 in which the Doctor tells the “real proof”—in other words, the short one—of Fermat’s Last Theorem to a group of geniuses as proof that they should trust his intelligence. By contrast, Jorge Luis Borges, with his customary erudition, carefully doesn’t claim that Unwin, the mathematician in “Ibn Hakkan al-Bokhari, Dead in His Labyrinth,” has proved the theorem. He has simply published a paper on “the theory supposed to have been written by Pierre Fermat in a page of Diophantus.” There’s another twist on it in the 1954 short story “The Devil and Simon Flagg,” by Arthur Porges, in which the mathematician Simon Flagg tricks the Devil into a wager. If the Devil can answer one question, he can have Flagg’s soul. If not, he must give Flagg wealth, health, and happiness all his days and leave him in peace for eternity. The question: “Is Fermat’s Last Theorem true?” The Devil fails to answer, and Flagg gets his reward.
2. Beth is good at math, as it happens—Tevis tells us that she is at the top of her class at the orphanage. That’s actually a crucial part of the story, because it means that she is the one chosen to go down to the basement after the Tuesday arithmetic class to clean the board erasers—something considered a privilege. It’s there that she first sees the janitor playing chess and persuades him to teach her. I’m pleased to report that Tevis does not, as the TV adaptation did, give Beth’s mother a backstory as a suicidal mathematician.
3. Hardy also made a major contribution to mathematics by taking the time to read a letter he received from a complete stranger one day, a clerk from India with no formal mathematical education, that was filled with crazy-looking formulae like 1 + 2 + 3 + 4 + … = There is a context in which this formula makes sense, and Hardy recognized that whoever had written this letter, and derived this mathematics more or less by pure intuition, had a rare talent. He managed to get funding to bring his correspondent, whose name was Srinivasa Ramanujan, over to England to work with him. Ramanujan proved to be one of the most brilliantly original mathematical thinkers of the twentieth century, and it is to Hardy’s credit that he recognized his gift and did all in his power to support him. Ramanujan’s story is told in the wonderful 2007 play A Disappearing Number, by Simon McBurney and his theater company, Complicité.
4. It shows even more in Doxiadis’s bestselling 2009 graphic novel Logicomix, co-written with the computer scientist Christos Papadimitriou, about the twentieth-century quest for the foundations of mathematical truth, which is also highly recommended. At the start of the twentieth century, there was a concerted attempt to put the whole of mathematics on the strictest possible logical foundation. The idea was to try to create a sort of mathematical language in which every possible mathematical statement could be expressed. Then you could agree on a list of initial assumptions, or axioms, and, proceeding in accordance with strictly defined rules of inference, either prove or disprove each statement. But the mathematician Kurt Gödel blew the whole thing out of the water in 1931 when he proved that any such mathematical system must be inadequate in that there would be true statements you could make that could not be proved within the system. This development was a profound disappointment to the logicians who had made it their life’s work to try to systematize the whole of mathematics. Perhaps that’s why the New York Times review of Logicomix was titled “Algorithm and Blues.”
5. For example, Jimi Hendrix, Kurt Cobain, Janis Joplin, Jim Morrison, Amy Winehouse, and Brian Jones died at twenty-seven. But Elvis Presley, John Lennon, David Bowie, and hundreds of others did not.
6. There is some ambiguity about how best to transliterate Софья ВасильевнаКовалевская into the English alphabet. Russian full names have three parts, the first name, the patronymic (based on the father’s first name), and the surname. So because Sofya’s father’s name was Vasily, and her husband’s surname was Kovalevsky, her full name was Sofya Vasilyevna Kovalevskaya. Both the patronymic and the surname have a male and a female form. Quite often, people are addressed by their first name and patronymic, and, to add to the fun, lots of first names have diminutive forms. Anyone who’s ever read a Russian novel knows the problem—you read ten increasingly confusing pages about this Sasha chap who’s suddenly appeared before you realize it was just Aleksandr Petrovich all along. Anyway, with Sofya Kovalevskaya I’ve gone for what current consensus seems to believe is the most accurate rendering. But you’ll also see Sofia, Sophia, Sophie, or even the diminutive Sonya, as well as Kovalevsky, Kovalevski, Kovalevskaia, and even Kovalevskaja. Alice Munro went for Sophia Kovalevsky.
A Mathematician’s Bookcase
We have come to the end of our mathematical guided tour of the house of literature. Mathematics is there in the foundation, in the rhythms of poetry and the structures of prose. It is there in the decoration of the house, the metaphors and allusions. And it is there in the characters who move through the house, who bring it to life.
I’ve gathered here a collection of some of the books on my shelves that we have discussed—with a few bonus recommendations thrown in for good measure. I hope that I have given you a new perspective on both mathematics and literature, and new ways to enjoy them both. May this be just the start of your journey. Happy reading!
1: One, Two, Buckle My Shoe
Tom Chivers (editor), Adventures in Form: A Compendium of Poetic Forms, Rules and Constraints (Penned in the Margins, 2012).
Jordan Ellenberg, Shape: The Hidden Geometry of Absolutely Everything (Penguin Press, 2021). He has also written a well-received novel, The Grasshopper King (Coffee House Press, 2003).
Michael Keith’s pilish poem “Near a Raven” is available on his website, cadaeic.net (that strange word “cadaeic” is not in any dictionary—but if you let a = 1, b = 2, and so on, you will see what’s going on). He has also written an entire pilish book, the only one I know of: Michael Keith, Not a Wake: A Dream Embodying (Pi)’s Digits Fully for 10000 Decimals (Vinculum Press, 2010).
Raymond Queneau’s Cent mille milliards de poèmes has been translated into English more than once. Stanley Chapman’s version uses the rhyme scheme abab, cdcd, efef, gg, and Queneau’s reaction was apparently “admiring stupefaction,” so that seems a good place to start. It appears in Oulipo Compendium, edited by Harry Mathews and Alastair Brotchie (Atlas Press, 2005).
Murasaki Shikibu, The Tale of Genji, translated by Royall Tyler (Penguin Classics, 2002).
For poetry with explicitly mathematical themes, check out these three collections:
Madhur Anand, A New Index for Predicting Catastrophes (McClelland and Stewart, 2015).
Sarah Glaz, Ode to Numbers (Antrim House, 2017).
Brian McCabe, Zero (Polygon, 2009).
2: The Geometry of Narrative
Eleanor Catton, The Luminaries (Little, Brown, 2013).
Georges Perec, Life: A User’s Manual, translated by David Bellos (Collins Harvill, 1987).
Hilbert Schenck, “The Geometry of Narrative,” Analog Science Fiction / Science Fact (Davis Publications, August 1983).
Catherine Shaw has written several Vanessa Duncan novels. The first is The Three-Body Problem (Allison and Busby, 2004).
Laurence Sterne, The Life and Opinions of Tristram Shandy, Gentleman (1759–1767).
Amor Towles, A Gentleman in Moscow (Viking, 2016).
3: A Workshop for Potential Literature
Christian Bök, Eunoia (Coach House Books, 2001).
Alastair Brotchie (editor), Oulipo Laboratory: Texts from the Bibliothèque Oulipiènne (Atlas Anti-Classics, 1995).
Italo Calvino, If on a Winter’s Night a Traveler, translated by William Weaver (Harcourt Brace Jovanovich, 1982).
Italo Calvino, Invisible Cities, translated by William Weaver (Harcourt Brace Jovanovich, 1978).
