Pop math comes to NRO.


And now for something completely different. National Review’s own John Derbyshire is the author of a new book. The book is on math. Talk about a Renaissance man (which John really truly is).

The book is Unknown Quantity: A Real And Imaginary History of Algebra and Derb, as we know him around here, talks about it below.

Kathryn Jean Lopez: So thanks a ton for this book, John. I almost told someone at a cocktail party the other night that “Nine Zulu Queens Ruled China” thanks to you.

John Derbyshire: That, gentle reader, is a reference to a nonsense mnemonic I introduce in the book. It’s a good thing this is NRO, or I’d be getting e-mails accusing me of racism, homophobia, and anti-Chinese sentiment. Which, come to think of it, would make this a pretty normal day.

Lopez: When I first flipped through your book, my first thought was: Why the hieroglyphics throughout? Not to suggest we’re not well-rounded, but…do you really expect your NR-colleague types to read your book?

Derbyshire: I don’t see why not. Now, we all have a great deal to read–I am currently about 20 books behind–so you’d need to have some fairly keen interest in the subject matter. The actual math, though–the hieroglyphs–is not demanding. I have kept it all at the level of a high school or college-basics course, so it’s stuff that shouldn’t be seriously strange to anyone smart enough to read NRO. I just want to show how it all came up, who originated it, in what historical circumstances.

Lopez: What is the unknown quantity?

Derbyshire: The first step in algebra was taken thousands of years ago, when math turned from the purely declarative (two plus two equals four) to the interrogative (what plus two equals four?) That’s when the unknown quantity first showed up. The ancient Babylonians were so impressed with it, they used words from Sumerian–then a thoroughly dead language–to signify it, calling it by the semantically problematic, but indisputably Sumerian words “igum” or “igibum.” The medieval writers mainly called it “the thing” (shai in Arabic, res or cosa in Latin and Italian–”cossist” was a synonym for “mathematician” in medieval and early-modern Europe). Nowadays we just call the darn thing “x,” but it took us ages to get here. That’s part of the story I tell.

Lopez: And…not to be stuck at start (and stupid) here, but…what are you doing telling an “imaginary” history?

Derbyshire: [Sigh] That is my publisher’s people. I am hopeless at titles, subtitles, and cover art, things like that. I just do text; I leave the other stuff to them. I guess they think the word “imaginary” will add a romantic gloss, countering the common impression of math as an utterly unimaginative, dry-as-dust discipline. Since they did a pretty good job of marketing my previous book, I assume they know what they are doing. Every man to his own work, and marketing isn’t mine.

Lopez: You wind up in Sodom and Gomorrah in the early pages. This is the John Derbyshire I know! What do they have to do with the history of algebra?

Derbyshire: Ancient joke: “Dad, I think I have Sodom figured out, but what went on in Gomorrah?” Well, those unfortunate cities actually play no part in the history of algebra. It is only that the very beginnings of the subject lie in the early second millennium B.C, a time so remote that, as I say in my text: “Outside a small circle of specialists, the only widespread knowledge of [it] is the fragmentary and debatable account given in the Book of Genesis. … This was the world of Abraham and Isaac, Jacob and Joseph, Ur and Haran, Sodom and Gomorrah.” I’m just trying to key the world of that very remote time to things people might know.

Lopez: Please explain the “power and beauty” of algebra for the unacquainted.

Derbyshire: The second great milestone in the history of algebra (taking that turn from the declarative to the interrogative as the first) was the adoption, around A.D. 1600, of a systematic literal symbolism–the use of letters to stand for numbers, both for the quaesita (things sought) and the data (things given). “Universal arithmetic,” Sir Isaac Newton called it. It was one of the greatest advances in human intellectual history, lifting up our powers of thought to a new level of abstraction, “relieving the imagination” (Leibniz). And then, once this “universal arithmetic” had been fully internalized by everyone–which took about 200 years–a peculiar and wonderful thing happened. Those letter-symbols began to detach themselves from the world of numbers, revealing an entirely new world of mathematical objects never known before–groups, matrices, manifolds, operators. And astoundingly (it seems to me), these newly discovered mental objects turned out to be just the ticket for describing and understanding the natural world. That’s the power. The beauty is more difficult to describe in brief. It centers mainly on the notion of symmetry, and–for me at least–has a large geometrical component. My book is long on geometry, probably too long for some algebraists.

Lopez: What could possibly be “fascinating” about the theory of finite groups?

Derbyshire: I think the main fascination is, that it satisfies the human urges to classify and enumerate. At a slightly higher level, there is the wonder that these perfectly abstract mental objects should have so much inner structure, and so many connections to much older regions of math–number theory and geometry.

Lopez: “The Leap into the Fourth Dimension” sounds like it could be the chapter title of a sci-fi novel, not a math-history book.

Well, at least that is one chapter title I don’t have to apologize for, since it is a quote from a real professional algebraist, Bartel van der Waerden. And yes, the fourth dimension has great imaginative appeal, though it is widely misunderstood. Though for my money, the eleventh is more fun. (Kidding.)

What was the great debacle of 1847?

In very brief, a bunch of French mathematicians thought they were within an ace of cracking Fermat’s Last Theorem. Then a German mathematician, Eduard Kummer, sent a letter to the French Academy pointing out a fatal flaw in all their assumptions, bursting their bubble. It was a sweet moment for Kummer, whose father had died from typhus he contracted from the French armies retreating through Prussia in 1813. The Germans nursed great resentment at their humiliations under Napoleon, and Kummer’s letter was a harbinger of the growing power of German math, which was to dominate the later 19th and early 20th centuries. France had been the great power in math up to this point–a thing not much remembered nowadays. Cauchy’s entry in the Dictionary of Scientific Biography is 17 pages–same as Gauss’s.

Feminists are forever whining about there supposedly not being enough women in math and science. But you’ve got a few lady mathematicians in your book, dontcha?

I have two and a half. Well to the fore is Emmy Noether, my “Lady of the Rings” (a math joke you will get after reading the book). Emmy was a true genius, a great algebraist. Sophie Germain I mention only in passing, just because, although she was a formidable mathematician, her main work was not in pure algebra. The half is poor Hypatia, whose original contributions to algebra are unknown, but whom mathematicians honor as a martyr, and who features, au naturel, in a dramatic Victorian painting I could not resist including. My proud boast is, that this is the only book ever written about algebra to include a picture of a naked woman.

Lopez: What are math geeks going to like best about your book? And the rest of us?

Derbyshire: That picture of Hypatia in the buff, probably, if I know my math geeks. But I hope, also, that math geeks will appreciate this unified overview of algebra, relating in (I hope) a convincing way the labors of the ancient algebraists to the weird and wonderful developments of the 20th century. The rest of you will learn how human thought struggled out of the mere mud of numbers and geometrical figures, to soar up into a stratosphere of pure abstractions, and abstractions of abstractions, and abstractions of abstractions of abstractions… (repeat for a page or so).

Can you solve our current conservative malaise through math?

I doubt it. Though an instinctive Platonist in math–I write about new mathematical objects being “discovered,” not “invented”–I am not much of one in human affairs, which I think are best understood by empirical inquiries in history, anthropology, psychology, and biology. While not of direct value in this area, though, math does–as our high-school Latin teachers used to assure us (truthfully, I now see) about our instruction in that language–provide very valuable training in clear thought. It also instills a skeptical attitude towards pronouncements not supported by convincing arguments, and towards judgments based on the mere appearance of things. In math, and especially in algebra, very little is only what it appears to be. I doubt that any intellectual discipline uses, and cherishes, the word “counterintuitive” as much as math does.

Math as math, however, belongs in the math departments of our schools and universities. Somewhere in my book I mention the Neoplatonist philosopher Marinus, who sighed: “I wish everything were mathematics.” Given the dire state of the world in Marinus’s time, it is not surprising he felt that way. This is not, however, a wish I share.

<title>Unknown Quantity: A Real And Imaginary History of Algebra, by John Derbyshire</title>
<author>John Derbyshire</author>