Leonard Mlodinow
Euclid’s Window: The Story of Geometry from Parallel Lines to Hyperspace
(Touchstone, 2001)

This is a book that arrived as a birthday present, along with another book by Mlodniow, from Aunt Chris. I am not entirely why she sent these two particular books: over Christmas, we’d been discussing the problem of math education, and she said she had a couple of books about the problem of math education, but I am not sure that they were actually these ones. But I’m always interested in math, though I don’t know very much about it, due in large part to having grotesquely incompetent math teachers for almost all of my high school education. My geometry teacher was the only competent one; he should have taught calculus senior year, but he’d been elbowed aside for a rookie teacher who was fresh out of college and probably had never learned calculus to begin with. But aside from geometry, all the math I learned was self-taught; this worked out reasonably well, and I probably could have taught myself calculus, but by senior year it didn’t seem to matter any more.

And now I am mathless. That said: I like the idea of math, and I generally like histories of mathematics, though I don’t actively seek them out as often as I should. This book, a history of geometry from the Greeks to the present, reads easily, maybe too easily. But it’s hard for me to like this book. The problems start with the paratext: the book is set in Times New Roman, which is dispiriting: math, of course, is more difficult to typeset than ordinary text, but there are plenty of tools for the job, tools which give much better looking results. These weren’t used here. Early in the book (p. 26), the square root of 2 comes up; this is displayed not with the 2 under the radical sign as it should be:

[latex size=”-2″]\sqrt{2}[/latex]

but rather as the Unicode radical sign, then the 2:

√2

(Looking up the history of the radical sign in Wikipedia does point me to Antonio J. Duran, George Ifrah, and Alberto Manguel’s The Life of Numbers, which looks like the sort of math I’m interested in – one of the books I’ve always wanted is a history of mathematical typography.) But this looseness with typography bothers me mostly because of the lack of attention to detail: it seems cheap, and it makes me wonder about a math writer who doesn’t insist on this kind of precision, in the same way that misspellings make one wonder about the quality of a writer. At one point, we learn that the distance in flat space is the sum of the squares of the differences in x, y, and z: I’m pretty sure that value should be squared, but I could be missing something. Doubt has crept in.

Mlodinow’s style also grates. He’s clearly aiming for the popular audience, but that’s a hard thing to hit. There are a lot of jokes, which aren’t worth mentioning. The writing suffers from plenty of choppy declarative statements: “It sure seemed radical at the time.” (p. 217). Historical characters are given thoughts and dramatic struggles; frequently, we hear about their family life, which seems to serve as a counterpoint to Mlodinow’s own two sons, Nicolai and Alexei, who turn up again and again. They serve both as the protagonists of thought experiments and as the source of minor anecdotes; the former undercut the latter. If Alexei and Nicolai have been floating around in dimensionless space, why does it matter if they dye their hair blue to see what their teacher will say? At a certain point, they move the narrative forward by appearing in their father’s dream; at this point the reader wants to toss the book across the room. The author himself makes appearances toward the end of the book, but never to much obvious purpose; perhaps the author is a serious physicist, but it’s hard to know this from the way he appears in his text.

I critique the book’s style because it’s difficult for me to assess how accurate the book’s presentation of material might be. The book is structured as five great revolutions, as personified by five revolutionaries – Euclid, Decartes, Gauss, Einstein, and Edward Witten. Certainly the book seems to be in a rush to leave pure mathematics behind for physics. This is where the narrative loses interest for me; for whatever reason, I find myself more interested in pure mathematics than the glorious world of string theory. Not unrelated: at the same time, the examples in the book become less comprehensible. It’s hard for me to know how well this sort of focus works: I picked up the book wanting to know more about non-Euclidean geometry, never having properly had an introduction to the subject, but Lobachevsky, Bolyai, and Riemann don’t figure as major players. Klein bottles never appear, which made me sad; nor does topology, except as two paragraphs of background for string theory. Instead, we get personal narratives: a great deal of time is spent on Gauss’s relationship with his father and his teachers. And, of course, the sense that everything is leading to the present, the trap of any historical narrative. Certainly math does build on the past; but I wonder how useful this reductionist view of everything building to the present moment is.

Above all else, this is a book that needed an editor and didn’t have enough of one; it’s also unclear why there weren’t more illustrations, as the ones that do exist are generally helpful. A lengthy review by Robert Langlands tears the book apart from the opening line: “This is a shallow book on deep matters, about which the author knows next to nothing.” But this is an extremely useful review, if perhaps overly stern. “One should not ask about the scientific or mathematical achievements of Pythagoras but of the Pythagoreans, whose relation to him is not immediately evident,” says Langlands; but it’s hard to follow this advice when your project is to make Pythagoras come alive as a person. Langlands’s review is a useful corrective. As mentioned before, I do have a copy of another of Mlodinow’s book, The Drunkard’s Walk; I’ll probably make my way through it, but not without some trepidity.