Gelf Magazine - Looking over the overlooked


April 9, 2009

Known Unknowns

John Derbyshire, more than most authors, contains multitudes; he is a novelist, conservative commentator, and author of a pair of singularly graceful popular mathematics books.

Konstantin Kakaes

Popular books about physics and math can be a recipe for frustration. There's a lot of basic ground that needs to be covered to explain any given modern topic—like what Newton did and why Aristotle was wrong, or how the Greeks invented geometry and the Indians the number zero. Not only is the same ground paced over, but the metaphors used to detail that ground are often identical. It's a hard thing to write a popularization of math that is original, and even harder to write one that is both correct to someone who knows about the subject, and understandable to someone who doesn't (which is the intended audience, after all). Sacrifices are necessary; as John Derbyshire writes in Unknown Quantity, his history of algebra published in 2006, "I hope only to show what algebraic ideas are like."

He succeeds in doing so, both in this book and in another, earlier book, Prime Obsession, an accounting of the Riemann hypothesis, which similarly conveys what analytic number theory is like.

John Derbyshire. Courtesy
"The extremes of the future of human cognition are represented by the Singularitarians on one side and Mike Judge's Idiocracy on the other. I'm inclined to the latter view."

John Derbyshire. Courtesy

Derbyshire (and a consensus of mathematicians) describe the Riemann hypothesis as the most important unsolved problem in mathematics. An exact statement of the hypothesis is beyond the scope of this interview. Let us say simply that it is a speculation on exactly how quickly the primes (numbers, like 3, 5, 7, and 29, that are divisible by only themselves and 1) thin out as integers get bigger. It takes a book to explain properly. Derbyshire studied mathematics as an undergraduate, but since then has turned to writing both novels and criticism. He is a columnist at the National Review and has written for a broad range of publications. He is a prolific literary critic, and as a political commentator he is both verbose and acerbic. But he seems never to be gratuitously courting controversy, even when expressing opinions far outside today's mainstream, and never without nuance.

The following interview was conducted by email; the questions have been re-ordered, but otherwise edited only minimally. (You can hear Derbyshire speak, along with noted enigmatologist Paul Hoffman and math museum founder Glen Whitney, at the debut of Gelf Magazine's Geeking Out reading series on April 15th at the Jan Larsen Art Studios in Brooklyn, New York.)

Gelf Magazine: What made you turn from studying mathematics to the mix of professions you now practice? Was the decision not to become a "practicing mathematician" one you consciously made?

John Derbyshire: There wasn't really any decision to be made. I simply wasn't good enough at math to be a working mathematician. My three years at university made that plain. I therefore decided to try other lines of work. My attitude to math is that of a lover whose love is unrequited. I love math, but it doesn't really care for me. Its not caring, of course, only sharpens my passion.

Gelf Magazine: In writing about mathematics for a lay audience, inevitably there comes a point past which it is impossible to be both lucid and accurate. What I wonder is how much this point moves over time. It's possible to give a pretty clear, simple explanation of say, proving there are an infinite number of primes, today; presumably in Euclid's time this was not accessible to whatever the equivalent of an "educated layman" was. Do you think that some time hence the average smart reader will be able to grasp the essence of theorems in Lie Algebra? Or do you think the cumulative nature of math will make this impossible? If so, what does this imply for the role that math can play in our society?

John Derbyshire: You are asking me to predict the future of human cognition. That is an immense topic, containing a range of current opinions spread out between two extremes. One of the extremes is represented by Ray Kurzweil and the "Singularitarians" arguing that sometime around the middle of this century—Kurzweil has actually said 2045 at latest—information science, neuroscience, and biotechnology will meet in a perfect storm and we shall be sharing the planet with, or shall have become, a higher kind of intelligence. At the other extreme you have Mike Judge's 2007 movie Idiocracy, in which the human race just gets dumber and dumber. I'm inclined to the latter view myself. Intelligence doesn't seem to be adaptive in an advanced society, and the indicators are all pointing towards a dysgenic future. You never know, though. If Kurzweil is right, the toddlers of 2050 will be proving theorems in motivic cohomology before they're out of diapers. If Judge is right, the ability to solve quadratic equations will soon be the height of human mathematical attainment. Or perhaps H.G. Wells's The Time Machine will come true at last: a clever but brutish race looking after a dumb but beautiful one, and using them for food. I think you're wrong about the educated layman of Euclid's time, though. I'd guess that the proportion of citizens who could grasp his proof for the infinity of primes was the same than as now, or not much smaller. Math is easier to grasp now because we have much better tools—Indian/Arabic numerals, place-holder zero, literal symbolism. I try to make the point in Unknown Quantity that literal symbolism in particular was a huge step forward, perhaps the greatest in the history of math, though modern numerals are a strong competitor.

Gelf Magazine: Aside from the difficulties of explaining it for a popular audience, how does the growing complication of mathematics affect its practice? It's become pretty much impossible for a precocious teenager to make a meaningful contribution, a la Galois and any number of others. (I think!) Given that mathematics is thought of as a "young man's game," does the fact that it's getting tough to learn the preliminaries in time to still have an agile mind affect the practice of mathematical research?

"The tremendous burst of mathematical creativity that followed the spread of literal symbolism in the late 1500s and the invention of calculus a century later may just have run out of steam. I don't see much low-hanging fruit."
John Derbyshire: How could it not do so? But you know, no field of intellectual or cultural endeavor progresses steadily through history. Stage drama in the West pretty much stopped after the Roman dramatists of the second century B.C. Even they were working slavishly from Greek models. Some classicists would tell you ancient stage drama really stopped with Sophocles and Euripides, who both died in 406 B.C. Things got going again in 16th-century England, but that's a pause of two thousand years. Or ask a Chinese person to recite a poem. It is most unlikely you will hear anything from later than the Sung Dynasty (12th century). Yes, I know, there were miracle plays and mumming shows in the Middle Ages, and Chinese poems written since the Sung, but nobody cares. The tremendous burst of mathematical creativity that followed the spread of literal symbolism in the late 1500s and the invention of calculus a century later may just have run out of steam. I don't see much low-hanging fruit. Taking on these big old problems in math is mainly a case of breaking rocks. I'd even put Wiles's proof of FLT under that head, without wishing to subtract anything from Wiles' achievement. I suppose one might reasonably hope that we'll turn some kind of corner, open up some vast new virgin territory in math, and leave the rock-breaking to those clever machines. I don't see that happening; but then, who in 1659 saw calculus happening?

Gelf Magazine: A related, but different question—what do you think of proofs that are so complicated that no one person can understand them? I'm thinking of both things like the classification of finite simple groups, which were done by people, if a lot of them, and the four-color theorem, done by computer.

John Derbyshire: Well, it takes a lot of the aesthetic pleasure out of math, for sure. Anyone with any feeling for the subject recalls the aesthetic thrill of grasping an elegant result—I recall it in number theory with Fermat's Little Theorem and in Classical mechanics with the derivation of Euler's equations. Contrariwise there are those annoying theorems that just don't seem to have really neat proofs, like Morley's Triangle in geometry and the Jordan Curve Theorem in analysis.
Any reduction in aesthetic pleasure is to be regretted, of course. Once again, let's hope that machines will take over the rock-breaking, while we open up some new vista in which gifted 19-year-olds can dazzle us with wonderful insights.

Gelf Magazine: Why a history of algebra rather than analysis, topology, geometry? What was it that drew you to this lens as a history of mathematics?

John Derbyshire: It was the challenge. Algebra was my weakest topic at university, and I wanted to have another go at it. The best way to learn something is to write a book about it.

Gelf Magazine: In the algebra book, you mention the difficulties of typesetting equations in Microsoft Word. That seems like a nightmare—I can't imagine writing a book with any measurable amount of math in anything but LaTeX. Did you really use Word?

John Derbyshire: Sure. I'd never written any math before, was on a tight schedule, and didn't want to give up time to learning anything comprehensive. MS Word had an add-on called MS Equation, which I found served perfectly well. I'm not alone, either. In my exchanges with Sir Michael Berry, he told me that he uses MS Equation too.

Gelf Magazine: A major figure towards the end of your algebra book is Alexander Grothendieck, who somewhat famously renounced mathematics and became a recluse in the Pyrenees. One of the (many) things I admired about your book is that you manage the deft trick of recognizing this as an archetype, without bowing to cliché. Nonetheless, the archetype of the holy fool, as it were, does hold a certain fascination. Have you thought of further pursuing Grothendieck?

John Derbyshire: No, mainly because, as I confess in the book, I don't understand his math. Also because, while holy fools are interesting from a distance—I mean sociologically interesting—the very small amount of time I have actually spent in close contact with this human type has convinced me not to seek more.

Gelf Magazine: What's your impression of Grigori Perelman, who recently proved the Poincaré conjecture, another long-standing mathematical problem? Had you thought at any point of writing on the Poincaré conjecture? Any regrets about tackling the Riemann hypothesis instead? To what extent do the arrival of solutions within the span of a decade or so to both Fermat's last theorem and the Poincaré conjecture herald a fruitful epoch for mathematics?

John Derbyshire: I can't claim any real impression, never having met the fellow, but on the published reports, and mathematicians' gossip, he seems like a very odd bird. On the Poincaré conjecture, I actually have a PowerPoint presentation for lay audiences, that I use as a filler when I have to lecture on cruises and such. Yes, I would have liked to have written a book about it, but others got there first.
Are we, as Keith Devlin predicted 10 years ago, in a new Golden Age for math? My knowledge of the field is neither wide enough nor deep enough to say. There are more centers of mathematical excellence—more Alexandrias, Göttingens, and Trinities—than ever before, especially with China and India coming online, so it's something that can be reasonably hoped for. I have the dark suspicion, though, that mathematical creativity may be the first kind of creativity to be "cracked" by the biotechnologists, so that clever machines may be solving problems in the future. The question is, shall we be able to understand the proofs? Back in Singularity territory here.

Gelf Magazine: Along those lines, has anything since the publication of the book caused you to revise your odds on the truth of the Riemann hypothesis to anything more than Andrew Odlyzko's "either it is true, or else it isn't"?

John Derbyshire: I'm embarrassed to admit I've paid very little attention to developments over the past six years, though the fact of six years having passed without any headlines suggests that the odds of its being true are somewhat better. (Since it seems to me a disproof would be easier to arrive at than a proof. One single counterexample—zero not on the critical line—would suffice!)

Gelf Magazine: At roughly the same time as your book on the Riemann hypothesis was published, another one by Karl Sabbagh was, as well. Have people confused the two books? Has this been bothersome to you?

John Derbyshire: I'm a bit ashamed to say I never read Karl's book. I did read Dan Rockmore's Stalking the Riemann Hypothesis, which I thought very good, and Marcus du Sautoy's The Music of the Primes, good on the history but, I thought, trying too hard to make the math reader-friendly, and not really succeeding there. I can recall no instance of anyone confusing my book with any of the others. We all took quite different approaches. That's a good thing: The determined reader who strikes out with one approach can try another.

Gelf Magazine: Aside from your mathematics writing, you have written a fair amount of political commentary, largely for right-of-center outfits. You say on your website that you have little interest in politics—why then the political commentary?

John Derbyshire: Who doesn't like sounding off on topics of the day? It seems wonderful to me that you can actually get paid for doing so. With a family to support, I naturally leapt on the opportunity. It's true, though, that I see it all—and write about it all—from a large and general point of view. I have no taste for the wonky stuff—5,000-word position papers on health saving plans or counterinsurgency warfare.
And yes, I'm a conservative. I very much dislike being bossed around. I believe strongly in individual liberty. I think modern governments do far too many things that are none of their business; and even when they do things that are their business, they usually do them badly. I detest dealing with paperwork and bureaucrats. Where social policy is based on the human sciences, I want the science to be right, or at least not flagrantly, defiantly wrong. I believe in nations, each with its own character, each fiercely jealous in defense of its sovereignty—not wishing other nations any harm, but always preferring its interests to theirs. I agree with Vladimir Nabokov that no portrait of any national leader should exceed the size of a postage stamp.

Gelf Magazine: Unfortunately, I wasn't able to get a hold of your novels in time for this interview. One of them, I notice, is self-published. What went into your decision to publish it yourself, rather than publish it not at all or with a publisher?

John Derbyshire: Fire from the Sun was too long for any publisher to take on. Having put the effort into writing it, though, I wanted to see it in print, and self-publishing is surprisingly cheap. So that was what I did. It wasn't a great success, as I have no marketing skills. The publisher didn't do a good job. At last I just put the whole thing on my website with a PayPal button so readers who like it can pay for the pleasure. Many do: I get $200 or $300 a year from that button. Hey, I'll take it.

Gelf Magazine: Speaking of publishers, I remember reading a comment somewhere (I think it was by Feynman, but I'm uncertain) that each instance of an equation in a book cuts the sales by half. I thought both math books were exceptionally lucid, but they definitely would be challenging to a reader who had no mathematics background. Was there pressure from the publisher towards a more superficial treatment?

"The political left has no intellectual content. It's just a money and status racket."
John Derbyshire: That was Stephen Hawking in A Brief History of Time, as I remember, though he may have borrowed it from Feynman. I took the approach, when writing my math books, that there is no avoiding the math, and I just had to make it as digestible as possible. The other approach, the one taken by du Sautoy, is to dress it all up in metaphors about landscapes and musical notes— but I didn't think I could make that work; and, as I've said, I don't think du Sautoy pulled it off, either, though all credit to him for trying. No, my publisher took the manuscript pretty much as delivered.

Gelf Magazine: I imagine that you set out to write the Reimann book before the announcement of the Clay prizes. [The Clay prizes are million-dollar prizes announced in 2000 for seven unsolved problems in math, one of them being the Reimann hypothesis.] I'm thankful you avoided a "win a million dollars" sort of narrative—nonetheless, the prizes, at least somewhat, brought things into the public eye. What kind of influence do you think they have had on the popularization of math? On math itself?

John Derbyshire: No, the Clay prizes had already been announced—see page 354 of Prime Obsession. They don't strike me as that big of a deal, and I doubt they have had much influence.
There have always been cash prizes in math—recall the one for Fermat's Last Theorem. But mathematicians of the first rank will do what they do regardless of prizes and such. It's the fascination of the hunt—and for many, the lust for glory amongst their peers. Cash prizes are peripheral and decorative.
I even wonder if the setting up of cash prizes has a negative effect, wasting the time of productive mathematicians. These prizes mainly attract cranks, and mathematicians have to deal with these people, if only at the minimal level practiced by Edmund Landau. He had postcards printed up with the words: "Dear Sir, Thank you for your proof of Fermat's Last Theorem. The first error is at line number _______." Any time a crank proof came in, Landau would hand it off to one of his graduate students with one of the postcards to be filled in and returned.

Gelf Magazine: If you'll permit me the observation, the style of conservatism you seem to believe seems very British, a classical Tory sensibility. How do you find it living in the United States, where the political spectrum spans an axis somewhat skew to the British one? In particular, there seems to be a real anti-intellectual strain in American conservatism, while the conservativism you practice is a nuanced one that relies on being smart. To put this perhaps more succinctly—what's it like being a latter-day Edmund Burke in the land of Sarah Palin? (If you disagree with the premise here that the anti-intellectual strain is a powerful one on the American right today, by all means I'd be interested to hear that case made.)

John Derbyshire: Yes, I'm essentially a Tory anarchist, though with some modifications. My parents and grandparents were coal-mining people, living on weekly paychecks, the men going down the pit to hack at coal, and the women going into domestic service if they couldn't find a husband. (My mother's first paid employment was in domestic service.) Early 20th-century British socialism did a great deal to improve their lives. I remember that, and do not reflexively scoff at all "schemes for political improvement." There was a point of diminishing returns sometime in the 1960s, though, and the modern welfare state is way overdue for a major overhaul.
I don't agree with much of the rest of what you say. Any large political movement is an alliance of people across the IQ spectrum. Eighteenth-century Toryism embraced not only Edmund Burke but also Squire Western (the thick-headed reactionary country squire in Tom Jones, who was generally drunk when not riding to hounds or debauching milkmaids in haystacks).
I have actually been a constituency worker in a local Tory Party chapter, and I can tell you, there's plenty of anti-intellectualism down there. A good thing, too, as intellectuals in politics do at least as much harm as good, and need to be balanced out with unintellectual, practical, worldly types. Robert Conquest notes somewhere that while the Tsar went to the Opera, Disraeli went to the races; then he asks—rhetorically, of course—which country was better governed?
I like Sarah Palin, and favored her over Joe Biden. She's run a town, and a state. What has Biden ever run, other than errands for Ted Kennedy? And if it's anti-intellectualism you want, check out some of the leftist websites, especially the comment threads. These people can't complete a sentence without a cuss word. Even with the aid of cuss words, in fact, they have trouble maintaining coherence at sentence length.
The political left has no intellectual content. It's just a money and status racket: jobs for kiss-up blowhards like Biden and race-guilt shake-down artists like the Obamas, the glow of moral superiority for self-regarding Stuff White People Like preeners, so-o-o-o pleased with themselves for being smarter, better, and more sensitive than those crude trailer-park conservative yahoos.
The left is a money pump and a status pump. It has no ideas. If you try to argue with leftists, they lose their temper and start with the cuss words. Tell me of one single original idea the political Left has come up with since Marxism faded away. [Sound of crickets chirping.]

Konstantin Kakaes

Konstantin Kakaes is a fellow of the International Reporting Project, a non-profit journalism organization based in Washington, DC.

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Article by Konstantin Kakaes

Konstantin Kakaes is a fellow of the International Reporting Project, a non-profit journalism organization based in Washington, DC.

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