Wednesday, October 28, 2009

Simplicio and Salviati


Who could dislike a book on math with a chapter inside it called "High School Geometry: Instrument of the Devil"?

This is a radical and subversive work in the best sense; it's brilliantly and comically irreverent toward that which is inane. What this particular author finds so inane is the standard K-12 math curriculum, and in a very concise and lucidly clear little treatise, he tells you why in surprisingly convincing terms. I say that as someone who liked my math education, who felt that calculus helped to show me the beauty of the universe--but Lockhart made me rethink the way I'd been compelled to spend much of my time up until that epiphany in calculus. He made me wonder if I might not have seen the same beauty or more beauty without so much time burned up in the acquisition of basically trivial skills.

More than anything else, A Mathematician's Lament is a wholesale attack on learning by rote. "By concentrating on WHAT, and leaving out WHY, mathematics is reduced to an empty shell," Lockhart writes. "Mathematics is THE ART OF EXPLANATION.... Math is not about following directions, it's about making new directions." Lockhart anticipates all the objections he's likely to encounter as he assails the math education of "accurate yet mindless manipulation of symbols" (where students are trained to perform mathematical operations they don't really understand and will not likely use ever again). And he dispenses with these objections in a Socratic dialogue between "Simplicio" and "Salviati," who, not so coincidentally, are also the fictional interlocutors in Galileo's 1632 Dialogue Concerning the Two Chief World Systems. In Galileo's dialogue, Simplicio clings to the (then) traditional Ptolemaic view of the heavens, and Salviati advocates for Copernicus. In a way, the revolution Lockhart proposes feels just that radical and that necessary. It's an argument that makes you feel very engaged with intellectual battles of our own moment in time. I hear the gears of intellectual history turning as I read Simplicio asking questions in defense of traditional education and Salviati answering them in a way that's often hilariously blunt:

SIMPLICIO: But don't we need people to learn those useful consequences of math? Don't we need accountants and carpenters and such?

SALVIATI: How many people actually use any of this 'practical math' they supposedly learn in school? Do you think carpenters are out there using trigonometry? How many adults remember how to divide fractions, or solve a quadratic equation? Obviously the current practical training program isn't working, and for good reason: it is excruciatingly boring, and nobody ever uses it anyway.


Lockhart certainly has the credibility to make any argument about math he wants. He's a published mathematician and he's lived his iconoclastic creed; he dropped out of college after a semester and, on the basis of his independent mathematical investigations, was admitted to graduate school at Columbia, from which he received a PhD in 1990.

I am a little puzzled by the extent of Lockhart's hostility to applied math. I do understand that pure math research can or perhaps must proceed with the sole object of deducing new and interesting principles from propositions and definitions. But Newton invented calculus as a very precise language for describing the laws that govern change in the real world--i.e., for describing physics--and I think physics is pretty damn beautiful and awe-inspiring and important. Even the much-maligned "quadratic formula" is beautiful to me, because it describes the parabolic motion of projectiles under the influence of gravity. Furthermore, what's so wrong with getting excited about the POWER that math gives to human beings to do interesting things, like walking on the moon or predicting how fast a drug will clear from the bloodstream. Finally, at the risk of exceeding my philosophical depth, I want to ask whether it's really possible to pose mathematical problems and to consider abstractions without ANY reference to reality. Lockhart mentions an abstract problem of finding the shortest path from a point to a line and on to another point; but isn't the whole notion of a "shortest path" inspired and given value by the experience of literally moving through space and caring when you get where you're going?

But all in all, the degree of emphasis placed on applications (and the pedagogical value of real-life examples) is probably not as important as Lockhart's bigger point. He wants us to witness truth and beauty, to enjoy creation and discovery, and to cease indulging those authorities who give us hoops to jump through simply because it makes their jobs easier. "I will not serve," as Stephen Dedalus says!

Check out the book that, according to NPR's math columnist, "is already a recognized landmark in the world of mathematics education" and you may get the same feeling Paul Lockhart gets from the "art" of pure mathematics—a feeling of penetration to what really matters.

Friday, October 23, 2009

Ulysses is a Modern Novel, Not a Postmodern Novel

Somewhere, in the last ten or twenty years or so, Dale Peck wrote that the development of the novel in English took a wrong turn somewhere in the middle of James Joyce's A Portrait of the Artist as a Young Man. His attack was really on post-modernism, and all literature that appears complex and self-referential and obscurantist. Apparently Peck blames James Joyce for this development in literature, and his attack has produced much bluster on the internet, most of it in defense of post-modernism. I have no wish to defend post-modernism, but I would like to defend Ulysses, which in my view deserves a throne among modern novels for reasons that have nothing whatsoever to do with the obscurantist literature that came after it.

Post-modernism is primarily concerned with skepticism, with what is false, with what we cannot know, with what makes us confused; the modernism that preceded it was above all else concerned with knowing reality in more depth than ever before, with depicting both the physical world and inner psychological experience with an unprecedented level of realism. A.D. Nuttall noted the profound engagement with reality in modern art when he observed, “Joyce, the disciple of Homer, listened to real Dubliners and C├ęzanne, the disciple of Poussin, stared at apples till they rotted on the cloth….” Joyce's naturalism is his real triumph. You see, hear, feel, smell, and taste Dublin on those pages. As Joyce's brother Stanislaus said, "Dublin lies stretched out before the reader, the minute living incidents start out of the pages. Anybody who reads can hear the people talk and feel himself among them." You see deep into the memory and the psychology of its three main characters.

In that sense I can't think of two more dissimilar writers than Thomas Pynchon and James Joyce (to whom Pynchon is sometimes compared). Both are hard to understand, but trying to understand Pynchon is like trying to put together a jigsaw puzzle without the benefit of most of the pieces; he once told his friend Jules Siegel that "I was so fucked up while I was writing it [Gravity's Rainbow], that now I go back over some of those sequences and I can’t figure out what I could have meant." Joyce's complexity, by contrast, rewards study and persistence, it illuminates new vistas of reality. W.B. Yeats at first dismissed Ulysses as "a mad book!" but on closer examination he recanted: "I have made a terrible mistake. It is a work perhaps of genius. I now perceive its coherence." Joyce's allusions are meaningful. The equivocations in his work honor the real fact of psychological ambivalence and the real nuance and complexity of reality itself. That is nothing so very new, in fact. Walter Pater noted this respect for the nuance of reality, which Francis Bacon called "subtilitas naturae," in the paintings of Leonardo Da Vinci. Post-modern equivocation and obscurantism rests on the thesis that reality is not knowable in the first place, that paradox and blindness is the order of the day. That's a million miles away from James Joyce, who considered himself, according to his biographer Richard Ellmann, to be in hot pursuit of a classical ideal of truth and beauty as articulated by Thomas Aquinas.

Don't be daunted by Ulysses. Don't be ashamed to skip a lot of it either. Joyce had acrobatic powers as a writer and he couldn't resist showing them off in Ulysses and Finnegans Wake. Some of what he did can be appreciated by dipping into it without doing a years-long archaeological dig to unearth everything that's there. The parts of Ulysses I love and remember best are lucid descriptions of Bloom's kitchen, and Molly's memories of singing by the piano, Stephen's apprehensions about the future mixed with his sensations of the sea and the sand as he walks along the beach, that sort of thing. While Yeats came to appreciate the book's coherence and genius, he also admitted later that he never finished it. Neither did George Bernard Shaw. So if you skip some healthy chunks of it, you'll be in good company.

Wednesday, October 21, 2009

Sylvia Plath, Maestro of Despair

Sylvia Plath's autobiographical portrait of Esther Greenwood, the 19 yr old protagonist of The Bell Jar (Plath's only published novel), must be one of the most intimate and realistic accounts of depression in all literature. Esther's a worthy heir to Hamlet in that respect. And like Hamlet, she's also acerbically funny, self-deprecating (to the point of self-destruction, unfortunately), and like the melancholy Dane, she's brilliant with words. Her brilliance is of course on loan from Sylvia Plath just as Hamlet's was on loan from Shakespeare. Plath is an absolute sorceress with a metaphor. I'm hard-pressed to think of anyone, any time, any place who does it better. In fact, she's a poet who can embarrass some of the giants of poetry by juxtaposition; try reading Plath's best poems side-by-side with some of her counterparts and you might be surprised by how much her rivals suffer the direct comparison in the brilliant light of her images. The feeling I get is like when I saw an exhibition of Matisse and Picasso side-by-side at MOMA; to my surprise, Matisse made Picasso's paintings look a little like subway graffiti, unsubtle and even cartoony. She's as good as Dante in depicting her descent into Hell in vivid terms:


I was the only girl on the beach in a skirt and high heels, and it occurred to me I must stand out. I had removed my patent leather shoes after a while, for they foundered badly in the sand. It pleased me to think they would be perched there on the silver log, pointing out to sea, like a sort of soul-compass, after I was dead.... The drench seemed to come off the sea floor itself, where blind white fish ferried themselves by their own light through the great polar cold. I saw sharks' teeth and whales' earbones littered about down there like gravestones.

The Bell Jar also contains some important social commentary on male chauvinism, particularly the sort that predominated in the 1950s, as well as a related paternalism in the field of psychiatry. Fundamentally, however, I think Esther's feeling of subjection to the whims of men conveys her larger despair about her place in the world as she gets more and more depressed and begins to see herself as more and more powerless and the world as more and more hostile.

The overarching metaphor of a "bell jar" as a crippling field of distortion surrounding someone's mind in a state of depression is an unforgettable and important contribution to literature in its general project of putting words to human experience. And ironically, given Plath's self-destructive, self-deprecating tendencies, she was an intrepid writer who never backed down from any challenge. After some significant early successes, she wrote a letter that said, "for the few little outward successes I may seem to have, there are acres of misgivings and self-doubt." And yet, even Shakespeare couldn't cow her; in her fabulous poem "Full Fathom Five" she goes toe to toe with him and she's utterly formidable right there in the face of the sea god.