I’ve seen math described as a universal language, a prerequisite to philosophy, and the common sense equivalent of Iron Man’s suit. Most people, however, wouldn’t dream of trying to crack open a calculus textbook in their free time. This is understandable for a field so fraught with technical lingo high up and obvious concepts down low.
Nevertheless, math has spawned some more popular, commonplace, entry-level literature, which at times even allows for some of the sweeping, beautiful visions from upper-level math. Like that of any other field from art history to English literature, mathematics’ reading material is divided between the technical and the popular. It is true that math is underrepresented in the latter; I said to a friend the other day, “I’m learning how much good mathematical literature is out there.” He replied, “That’s like saying that there’s some Argon in the atmosphere” (0.93%, to be precise). I thought, subsequently, that compiling a list for people’s convenience would be a fitting service.
1. Division by Zero, by Ted Chiang (Short Story, 1991)
“Division” uses math to explore how people relate to each other. It contains some darker content, but is excellent. The expertly crafted piece prefaces each mini-chapter with a story from the history of mathematics and culminates in a masterful ending. Re-readings uncover new layers of understanding as parallels, contrasts, and the crux of the piece become ever clearer. It demands zero technical knowledge.
2. Mathematics and the Imagination by Kasner and Newman (Nonfiction, 1940)
A well-known math professor at my college once said that this was the first math book that really hooked him. It’s not hard to experience why: This glittering gem of engaging writing contains several chapters, each one painting scenes of breathtaking upper-level math subjects like rubber-sheet geometry, transcendental numbers, and famous math paradoxes. While the book indeed requires a lot of imagination, virtually no math background need be brought to it. The book lays out concepts and ideas in vivid, simple terms, dazzling the mind with what math has the power to reveal, model, and conclude. Kasner and Newman leave space for breathing room, interspersing the hard-hitting logic and abstractions with anecdotes and literary allusions. Measure for Measure is quoted at one place, while the chapter on paradoxes is entitled “Paradox Lost and Paradox Regained.”
The authors spend one chapter exploring rubber-sheet geometry (also known by the staler and commoner name “topology”). This field comprises the study of properties of shapes that do not change even when the shapes are distorted, properties called “invariants.” The authors recount the history of the field, discourse on some of the typical features of it, and recount some of its applications – a famous tube map of London, for instance, is representative of reality in almost no respect at all except topologically. No matter how much England’s distances are distorted and paths straightened from curves, keeping the order of points along the routes is invariant. Since this is also all the typical tube traveler needs to know, the format of the map has remained through decades unchallenged.
Readers may find themselves hung up on a particular part: The chapter on the infinite stalled me for months, until I charged through it and feasted on what remained for a week and a half. Also, the very final chapter delves into some particularly bad meta-mathematics – discussing what mathematics ultimately is and whatnot – getting, as I understand things, nearly everything wrong and undermining the entire rest of the book. I would almost recommend skipping it wholesale.
Everything apart from that, nonetheless, could scarcely be recommended more highly. If any other intriguing incentive were needed, this is also the book that introduced the world to the word “googol.”
3. How to Lie with Statistics by Darrell Huff (Nonfiction, 1954)
This short, well-crafted primer of about 115 pages is an informative, concise, and clear guide about what statistics mean. Playfully framed by the title and by such illustrations as the one shown here (some are superfluous jollity, but many are great for demonstrating points graphically), this work explains what most statistical jargon means, what goes into the creation of these number sausages, and how to identify fallacies epidemic in advertising, reporting, and common discourse. It is not only fascinating, but also earns the merit of being eminently practical.
4. Mathematical Creation by Henri Poincare (essay, 1910)
Poincare reflects for about fifteen pages on the nature of creation and invention. I’ve drawn on this piece in an another article and consider his essay highly insightful, although not necessarily written in the most engaging style. I’d recommend it for anyone who has heard and is curious about the idea that your brain works on problems subconsciously, even as you sleep. Whether this is the origin or a part of this tradition, Poincare explains the idea and corroborates with his own personal experience.
5. How Not to Be Wrong by Jordan Ellenberg (Nonfiction, 2014)
This is a truly fascinating, highly entertaining, and generally informative book. While the author lacks the sort of rigorous program the title might suggest, he does come equipped with beginner-friendly explanations of higher math concepts, various applications of upper level math to real life, and scores of anecdotes and examples from different fields to illustrate his ideas. His project is illustrated in this picture from his book’s beginning:
Math, he explains, can be subdivided one way into simple and complicated truths, and another way into shallow and profound truths. His book discusses simple and profound truths, introducing the reader to high-level concepts and ideas while at the same time outlining their applications. He staffs his book with anecdotes about information transmission, the lottery, Renaissance painting, baseball statistics, and smoking studies. Through this all, Ellenberg fascinates, entertains, and offers his readers tools for thinking more clearly in everyday life.
This book, however, is kept from being the unparalleled material that Mathematics and the Imagination is, and the chance of this book attaining lasting, classic status is doubtful. This is not because of its timely references or down to earth stories – these could function as artfully as Dante’s provincial cast of characters or Shakespeare’s peculiarities, which make their works both typical of their age and for all ages – but because of plain bad thinking. All throughout this work the author winks as blind spots and alleys gape, holes in arguments yawn. The effect is profound given the book’s lofty goal of teaching how not to be wrong – especially given the unfulfilled directive of acknowledging and/or rejecting assumptions. An early discussion in the book critiques the Laffer curve – in short, it considers the possibility that tax revenue may be increased by lowering tax rates – and only several pages in does the author even suggest that any other criterion might be appropriate for determining tax rates. This is tactless, but not as truly shoddy as the “syllogism” he chalks up to those who endorse the graph’s use (emphasis his):
“It could be the case that lowering taxes will increase government revenue; I want it to be the case that lowering taxes will increase government revenue. Therefore, it is the case that lowering taxes will increase government revenue.”
This tact is either extremely snarky or (as from context it seems to me) crudely wrong. This is laughable, but the author presents it straightforwardly. Other examples offer similar problems: While discussing Pascal’s wager about Christianity, the author omits many relevant factors and commits blatant common sense errors; he writes about baseball and mentions umpiring with the undefended (I would add, indefensible) claim that video replay feels “foreign to the spirit of the sport”; most egregiously, he revolves around the possibilities and the perspectives on “unusual” punishment without once defining “unusual.” (It is obvious from context that he interprets “unusual” to mean “strictly any more than 50%,” but this definition is never stated or defended against any of a number of alternate ones. His entire case – preachy, by the way – stands or falls on its legitimacy.) Other errors seep through – the author unapologetically interchanges the words “libertarian” and “conservative” early on – but this pattern of failing precisely where he purports to excel draws attention and is without excuse.
In spite of this, this book is highly recommended. It will introduce you to a number of fascinating ideas and give you a wealth of entertaining stories, wherein math may or may not play an explicit role. Even the errors, though they can only tarnish the book’s excellent name (and it is a good one) as time grants them more attention, add to the book a certain luster as a contemporary book of our times. You find yourself thrust into a schoolroom and the battlefield at once, learning critical thought while needing it in the same breath.
6. Flatland by Edwin A. Abbott (Romance Novella, 1884)
This is a curious classic. In case you’re unfamiliar with the quirky Flatland, do allow A. Square to give you a tour. The narrator, an inhabitant of the two-dimensional reality, introduces the reader to his world before his own story, unlooked for and mind-breaking, begins to unfold. The work was written as a social commentary in geometric garb, but in such a creative way that it can be read clearly from the angles of society or math. Depending on your perspective, it might appear as a geometry world where shapes have their own weird conventions, or as a social commentary through the utterly abstract, allegorical, and inoffensive vehicle of Euclidean geometry. It is not written in the most colorful or encouraging style, but is a rewarding and creative work nonetheless.
7. The Works of Tom Lehrer (Musical, Twentieth Century)
This is downright cheating, but I couldn’t help but add the musical works of Tom Lehrer. Best known, perhaps, for his song “Elements” (which, as he has said, is simply the atomic elements listed off to a Gilbert and Sullivan tune), Lehrer also composed and sang a number of songs about mathematics, the subject in which he majored at Harvard. His songs are humorous and informed but by no means trying. Zero technical knowledge is requested or imparted. The specifically mathematical melodies are “Lobachevsky,” “That’s Mathematics,” and “New Math.”
This post constitutes my first catalog of good literature about math (in some respect or other; here represented are basic guides of upper-level fields, philosophical discourses concerning math, handbooks to applying techniques in life, and fiction that integrally uses math as a vehicle); I expect to extend it at least once more someday in the future.