Recommended Reading

Since I intend to be writing posts about mathematics for a lay audience, it seems a good idea to provide a list of recommended books for the layperson to read. I haven’t read too many math books apart from textbooks, but there are several I can recommend and a few I haven’t read but want to. I figure I’ll keep this list linked on the front page and update it periodically when I think of something or read something new. And while we’re on the subject of recommendations, I might as well list other books I highly recommend as well, fiction and nonfiction!

Nonfiction:

Math books (for a lay audience)

Flatland and relatives: The original Flatland, by Edwin A. Abbot, is a (very short!) classic, a fantasy and social commentary about life in two-dimensions, how a two-dimensional being might imagine or interact with a three-dimensional one, and how, by extension, we can imagine four or more dimensions. It has inspired two direct sequels that I know of:

Sphereland, by Dionys Burger, extends the story of Flatland by exploring the notion of curved space, and also the idea of an expanding universe. The best of the three in my opinion, both for the excellent and plausible way it explains the discoveries of the main characters, and also for the real drama that develops.

Flatterland, by Ian Stewart, is another sequel to Flatland, although it takes more after Lewis Carroll in style. This one romps through just about every subject that can be even remotely considered “geometrical” (non-Euclidean, discrete, projective geometry), and then goes off into topology and beyond, and even into relativity. There are lots of bad math jokes, which is a plus!

Also by Ian Stewart (who really is an excellent writer; there are several of his books I’m anxious to read) is a great little book called Letters to a Young Mathematician, which traces a mathematician’s journey from grade school on up to becoming a full professor, doing research, teaching classes, going to conferences, and getting tenure; giving all sorts of great advice and encouragement along the way. For someone like me who is partway along that very journey, it’s a very inspiring read.

Expository math books:

Everything and More: A Compact History of Infinity, by David Foster Wallace. This gives a nice history of the ideas relating to infinity from the Greeks on up, including various opposing views, from people who thought the idea could never be made rigorous or simply couldn’t exist (or who thought mucking about with the infinite was treading on God’s toes). It builds up nicely to Cantor’s theory of sets and the idea of cardinals, including the fact that the real numbers are a larger infinity than the natural numbers, but it doesn’t go much beyond that, which is a shame.

The Equation That Couldn’t Be Solved, by Mario Livio. This one looks at the development of group theory from a historical perspective, telling the tragic stories of Évariste Galois and Neils Henrik Abel, who independently proved that no general formula could be found to determine the roots of a quintic polynomial. This one is rather heavy on the history, which isn’t bad, but it doesn’t go much at all into the math and I had a hard time finishing it.

Also by Mario Livio, The Golden Ratio. This one I enjoyed–it covers the history as well as the mathematics behind the golden ratio, including the Fibonacci series, its occurrences in nature and geometry, and its use in art (actually, the book makes a very convincing case that the use of the golden ratio in ancient art is WAY overhyped, and that it’s easy to find the ratio where it was never intended).

More math-intensive books:

The Colossal Book of Mathematics, by Martin Gardner. For several decades, Martin Gardner wrote a column for Scientific American called “Mathematical Games”, in which he explained interesting or obscure math concepts, major fields of mathematics, tricky puzzles and paradoxes, and whatever else caught his interest–most of what he covered would be termed recreational mathematics. This book is a compilation of many of them, organized by subject and including further discussion and reader responses from after the original article was published. Martin Gardner is (in my opinion and many others’) the greatest popularizer of mathematics who ever lived, and his articles have both accessibility for an interested layperson and mathematical depth and sophistication enough to attract a more experienced mathematician.

Gödel, Escher, Bach, by Douglas Hofstadter. This is a mind-expanding, beautiful, witty, difficult, utterly wonderful book. Its main focus is a discussion of Gödel’s Incompleteness Theorem and, more broadly, the idea of a “strange loop”, the types of paradoxes arising from self-reference and emergent meaning. More generally, the book attempts to answer and raise questions about consciousness, inherent meaning, and intelligence, especially as they relate to mathematical systems and computers. Along the way it discusses everything from music and art (hence Escher and Bach in the title), to DNA, to Zen. The thing that really strikes me about this book, however, is that it’s playful–most apparent in the wonderful dialogues between Achilles and the Tortoise (and their friends) that occur between each chapter, but also even in the “serious” parts. You can just tell Hofstadter is having fun with these ideas, and it’s catching.

I should note that as much as I love this book, it took me three attempts to get all the way through it–some of the logical concepts he delves into are challenging enough that for the level I was at three or four years ago, it got exhausting too quickly. Nevertheless, there is nothing in here that can’t be understood by everyone willing to put in the time and mental effort required.

I should mention this one, even though it really is intended as a textbook rather than a popularization: The Knot Book, by Colin Adams. Best introduction to knot theory I’ve ever read (not that I’ve read that many), this book covers an enormous range of topics in knot theory in a fairly intuitive way. Plus, from what I hear, Adams is a hilarious speaker. (He also wrote a great paper on Brunnian clothing that I intend to write a post about at some point.)

A couple others:

The Man Who Loved Only Numbers, by Paul Hoffman: This one I haven’t read, but I intend to do so very soon. It’s an often-recommended biography of one of the most prolific (and most eccentric) mathematicians of our time, Paul Erdös. Erdös had very few worldly possessions, and would travel around the country to show up on colleagues’ doorsteps with a suitcase, saying, “My brain is open.” He would stay long enough to collaborate on a few papers and then leave, with the result that he was one of the only mathematicians to publish more papers than Euler. Because he collaborated with so many others, mathematicians now refer to their “Erdös number”, a measure of their distance from Erdös collaboration-wise. Erdös himself is given Erdös number 0, anyone who directly collaborated with him has Erdös number 1, anyone who works with any of them has Erdös number 2, and so on. (One of my professors has Erdös number 1, and lots of stories to tell.)

Men of Mathematics, by E. T. Bell: Sexist language aside (it was a product of its time: 1937), this book is a great retelling of some of the most interesting stories of mathematical history. I’m still working my way through it, since it’s pretty long and requires small doses.

How Mathematicians Think, by William Byers: I’m not sure I should really list this, since I’m still on the first chapter (I read many books at once, and got fed up with this one for the moment). So far, he doesn’t seem to be able to make clear just what his thesis is, and from what I do understand of it, half the time I find myself wondering how he can possibly support it, and the other half I am nodding my head and saying, “Yes! This is what mathematics is all about, that people don’t understand!” I really should read the rest before I make a judgement.

Drifting away from mathematics and toward science in general:

Surely You’re Joking, Mr. Feynman! and What Do YOU Care What Other People Think? by Richard Feynman: Everybody ought to read these memoirs by the great physicist and all-around funny guy Feynman. If you haven’t heard of him there’s really nothing I can say except GO READ HIM NOW! I haven’t read his more in-depth scientific writings, though I’d like to.

The Blind Watchmaker, by Richard Dawkins: I actually didn’t think much of this book. It’s meant to explain the theory of evolution and present the evidence and arguments for the theory, but it didn’t convince me and I’m already convinced. I felt it made too many generalizations and relied too heavily on analogies that could be faulty. Nevertheless, I feel I should include it since a great many people HAVE found it effective, and anyway I intend to give it another chance.

Drifting way off into philosophy and religion:

The Mind’s I, edited by Douglas Hofstadter and Daniel Dennett: This one is just fun (the best philosophy often is). It’s a collection of essays, articles, and stories by various authors exploring the ideas of consciousness, intelligence, mind, and soul (the editors are very decidedly against dualism).

The God Delusion, by Richard Dawkins: I really think this one needs to be read, at least by atheists and agnostics (who are mostly its intended audience). I dislike the title, but I understand what Dawkins is trying to do, and the points he makes about raising consciousness are excellent. And if you do read it, wave the shiny cover around proudly.

Well, that about wraps it up for nonfiction for now (I think I’ve covered everything).

Fiction:

My taste in fiction is mostly fantasy, particularly light or comic fantasy, with some exceptions. The books I will read over and over again are usually ones where the writing itself is a work of art–I go for wit, grand imagery, a little silliness, and especially wordplay, and when I sit down to read I want to know I’m reading a story. I can’t guarantee that everyone will like these books and authors, but if you like any one of them, or if you share my taste in literature, you’ll probably enjoy them all. Here are some recommended books and authors, in no particular order:

Fool on the Hill, by Matt Ruff: A wonderful conglomeration of mythology, silliness, talking animals, action, adventure, and romance, set on a college campus. You wouldn’t expect such a stew of different characters and genres to come together as a cohesive whole, but it works beautifully. It’s all about the art of storytelling, actually, and plus, there are dragons.

American Gods, by Neil Gaiman: This is a dark, poignant travel story about Shadow, an ex-convict who gets entangled with all the gods people have brought with them to America (not a good land for gods, as they put it) over the centuries. Beautiful, beautiful storytelling, as always with Neil Gaiman, and the symbolism and mythical imagery is great.

Jonathan Strange & Mr. Norrell, by Susanna Clarke: Again with the great storytelling and fantastic imagery. This book is loooonnngg, written in the style of the 1800’s, and absolutely worth it. Set in an alternate England which was ruled for several centuries by the Raven King, who was raised in Faerie, it tells the story of the two magicians who brought magic back to England after several hundred years’ absence.

Favorite authors (and recommended books):

Jasper Fforde: Jasper’s seven books are all metafiction, playing with words, stories, classic literature, nursery rhymes, and more. The Thursday Next series, beginning with The Eyre Affair, combines satire, fantasy, metafiction, action, and comedy, with Thursday Next as a member of Jurisfiction, the police force inside books. The Nursery Crime series, beginning with The Big Over Easy, ties into the Thursday Next series although each series regards the other as fiction. The Nursery Crimes are mystery stories involving nursery rhyme characters (the first deals with the apparent suicide–or murder–of Humpty Dumpty), but they mysteries become so convoluted and over the top that by the end they’ve become utter silliness.

Terry Pratchett: The Discworld series started off as a parody of overly self-serious epic fantasy, but the later novels have become somewhat more serious (underneath the comedy) satire in their own right. There are somewhere around 33 novels, and they can mostly be read in any order (with the exception of some several-book mini series), and in fact I recommend skipping some of the earlier books until later. My absolute favorite is Small Gods, with Reaper Man a close second, but they’re all worth reading and rereading.

Douglas Adams: I shouldn’t even have to mention the author of The Hitchhiker’s Guide to the Galaxy. Everyone should read it, and the four sequels, and the Dirk Gently books, and The Salmon of Doubt. They are extremely silly, it’s true, but nearly every paragraph has at least one quotable statement about Life, the Universe, and Everything.

P. G. Wodehouse: The only non-fantasy author on the list, although the England his books describe has never really existed anyway. I say Wodehouse is the funniest author ever to have written in the English language, and his words really are works of art. I particularly recommend several of his Blandings stories, including Leave it to Psmith, Summer Lightning, and Heavy Weather. He is most known, of course, for the Jeeves stories, all of which I recommend. You really can’t go wrong with Wodehouse.

That’s all I can think of for the moment. If you have anything to recommend, by all means please comment!