Author Edwin A. Abbott’s ironies are so gentle, and seem to arise so organically from his world, it’s as if he were slyly saying, “Oh, does that conjure up some conceit or frailty of your own society? Well, now, what a coincidence...” He exhibits almost Machiavellian clearsightedness about how societies are constituted, neither lacking compassion for the oppressed nor becoming overwrought about them.
Thus, in Abbot’s world, Isosceles Triangles of the lowest orders occasionally beget better endowed Equilateral Triangles. Not surprisingly, this occasions rejoicing among them—but satisfaction, too, among the aristocracy; for here is a safe, slow means of upward mobility that neither weakens their control nor diminishes their privileges—“most useful barrier against revolution.”
All of this is good satire, good science fiction, good geometry. But Abbott takes it further yet: In the final moments of the Flatland calendar’s 20th century—corresponding, in other words, to its Millennium—our hero, our modest, two-dimensional Square, is granted a revelation:
“Straightaway I became conscious of a Presence in the room, and a chilling breath thrilled through my very being,” he writes. A priestly Circle appears in his home, unbidden, shimmering with an otherworldly glow, growing larger or smaller seemingly at will. This, he learns, is no ordinary Circle, but rather part of a great Sphere sectioned through by Flatland.
Through him—Him?—our Square learns of a third dimension beyond his own. Lifted into space, he’s treated to the rich panorama of a universe he’d previously experienced only as lines and edges. His mind stretched beyond its Flatland limits, he begins to preach the Gospel of Three Dimensions, of an elusive, higher world Flatlanders never see... Thus, abruptly, satire and science fiction expand into what today we call “higher consciousness.”
The author, an English theologian whose hobby was mathematics, created in Flatland a world. Within its narrow confines, he succeeded in commenting upon his own world, and ours.
Their Eyes Were Watching God
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By Zora Neale Hurston
First published in 1937
“He look like some ole skullhead in de grave yard.”
That’s how Janie Crawford, a young black woman growing up in western Florida, describes the older man her nanny wants to make her marry and for whom she cares nothing. So when big Jody Starks comes along, full of button-popping ambition and big plans for moving down to Eatonville, where black people are building a town of their own, she joins him. She aches to love and to experience wider vistas, and Jody supplies one if not the other: “He did not represent sun-up and pollen and blooming trees, but he spoke for far horizon.”
In Eatonville, Jody becomes mayor and starts a general store that becomes a focal point of the town’s life. And there, on the store’s front porch, much of the broad middle section of Zora Neale Hurston’s novel, Their Eyes Were Watching God, takes place. There the local men congregate of an evening to trade stories, insults and gossip. There they toy with poor Matt Bonner and the mule he fairly starves.
“Dat mule uh yourn, Matt. You better go see ‘bout him. He’s bad off.”
“Where ‘bouts? Did he wade in de lake and uh alligator ketch him?”
“Worser’n dat. De womenfolks got yo’ mule. When ah come round de lake ‘bout noontime mah wife and some others has ‘im flat on de ground usin’ his side fuh uh wash board.”
Some, conceivably, may feel embarrassed to see this rural dialect reduced to print, with no attempt to metamorphose it into English that is correct, grammatical—and white. But this novel, so full of dignity and strength, needs no excuses made on its behalf. Its characters are well drawn and distinct; some are foolish, some possess intelligence and grace. They may speak like Amos ‘n’ Andy, but the comparison ends there. Like William Faulkner, Hurston—a figure in that flowering of black culture and consciousness between the world wars known as the Harlem Renaissance— evokes the rich cultural roots of outwardly simple rural life.
Janie is not quite a “strong” woman, as we use the word today. She still largely depends on men and does what they expect her to do, whether running the store as Jody demands, or working in the fields as Tea Cake, her first real love, wishes. Still, she never merely settles, at least not permanently. And never for the sadly constrained life her Nanny, a former slave, has schooled her to accept. Instead, she clings to the vision that life can be fuller and richer. She finds such a life with the playful Tea Cake.
Outwardly, she conforms, but always consciously, knowingly. When Jody dies, she dons black and bears herself as a widow should. But then, too soon for the town’s tastes, she meets Tea Cake and takes to dressing in bright colors.
“Ah ain’t grievin’ so why do Ah hafta mourn?” she says to Phoeby, a friend who warns her of wagging tongues. “Tea Cake love me in blue, so Ah wears it. Jody ain’t never in his life picked out no color for me. De world picked out black and white for mournin’, Joe didn’t. So Ah wasn’t wearing it for him. Ah was wearin’ it for de rest of y’all.”
Hurston speaks in two voices: in the dialect of her characters, and in a narrator’s voice clear, and correct, and lush with poetry. When Jody upbraids Janie for a flash of independence—it’s his job to think for women, children, chickens and cows—she withdraws: “The bed was no longer a daisy-field for her and Joe to play in. It was a place where she went and laid down when she was sleepy and tired. She wasn’t petal-open anymore with him.”
Later, a hurricane stirs in the Everglades: “The monster began to roll in his bed. Began to roll and complain like a peevish world on a grumble.”
That “monster” ultimately tramples across the sunny, green field of Janie’s new life, and once again she must start over, returning to Eatonville. “Ah’m back home agin and Ah’m satisfied tuh be heah,” she tells Phoeby. “Ah done been tuh de horizon.”
A Mathematician’s Apology
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By G. H. Hardy
First published in 1940
Pity the poor mathematician stuck working out some problem in chemistry, ballistics, or another dreary subject, says Godfrey H. Hardy, author of this eloquent defense of pure mathematics; for the sheer utility of the work strands him from all that is mathematically loveliest. “’Imaginary’ universes are so much more beautiful than this stupidly constructed ‘real’ one,” he writes, “and most of the finest products of an applied mathematician’s fancy must be rejected, as soon as they have been created, for the brutal but sufficient reason that they do not fit the facts.”
For Hardy, the leading English mathematician of the early part of this century, known both for his own achievements and for introducing Indian mathematical prodigy Srinivasa Ramanujan to Cambridge University and the West, mathematics bore no relation to the tedium of long division we remember from grammar school; nor the angled ladders of high school trigonometry; nor the definite integrals memorized for college calculus.
No, for him mathematics was an art form, one demanding the same resources of creativity required of a composer or poet. (One review of A Mathematician’s Apology when it first appeared called it one of the best accounts of what being a creative artist was all about. The review was by novelist Graham Greene, who ought to know.) “There is no permanent place in the world for ugly mathematics,” writes Hardy, in a typically aestheticrooted assertion. “A mathematical proof should resemble a simple and clearcut constellation, not a scattered cluster in the Milky Way.”
In rhetorical tradition, an “apology” like Hardy’s is a formal defense of an idea, institution, or work; in this case, even more than most, the word means nothing like what it does in everyday speech. Hardy’s spirited argument reveals neither defensiveness nor diffidence. If anything, it verges on arrogance, full of cheerful mathematical chauvinism—as when Hardy says that the mathematician “Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not.”
The whole long essay is like that, shot
through with the serene assurance that pure mathematics is a noble endeavor. He abjures every superficially appealing and familiar defense of mathematics—that it is useful, that it does human good, and the like—and throws the weight of his argument on the very claim perhaps most difficult to sustain before a lay readership, that mathematics is beautiful.
What makes for beauty in a mathematical proof? Hardy offers simple examples from Euclid and Pythagoras that, he feels, exhibit a “high degree of unexpectedness, combined with inevitability and economy.” All three qualities, it need hardly be added, apply as well to other aesthetic realms, including writing. And all three qualities apply to A Mathematician’s Apology. Hardy’s readiness to dismiss the utility of mathematics is surely unexpected. And the book’s whole argument, though well-developed, is succinct, proceeding as if in response to natural law. For one with such a head for number and symbol, Hardy certainly has an ear for words—giving us the opportunity to taste through language what few of us can appreciate in mathematics itself.
In his foreword to the 1967 edition, English novelist C. P. Snow, a long-time younger friend of Hardy, asks us to see the Apology “as a book of desperate sadness,” the product of a mathematician long past his creative prime. “It is a melancholy experience for a professional mathematician to find himself writing about mathematics,” writes Hardy. But as an old man now, he comes close to saying, washed up as a mathematician, writing about mathematics is all that’s left for him.
But may Snow have been fooled by Hardy’s rhetorical flourishes? Or, more likely, by knowing Hardy, the man, too well in the years before his death? The sadness in the Apology is there all right, but more as the resignation accompanying a rich, full life inevitably nearing its end, than the bitter reflection of a life poorly spent. The overall feeling is one of satisfaction, not despair; of fullness, not the Void.
My Life
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By Isadora Duncan
First published in 1927
“Just as there are days when my life seems to have been a Golden Legend studded with precious jewels, a flowery field with multitudes of blossoms, a radiant morn with love and happiness crowning every hour ... so there are other days when, trying to recollect my life, I am filled only with a great disgust and a feeling of utter emptiness.”
That’s how much of this autobiography of Isadora Duncan reads—with grandiloquence and passion. Judged on narrow standards of literary merit alone, it might be dismissed easily. Yet somehow, for this author, we must make allowances... Imagine a lone figure on the stage, barefoot, garbed only in a simple white Greek tunic, her body flowing to the swelling sound of Wagner’s “The Ride of the Valkyries.” This is the great Isadora. Her native language is dance, not English prose. But many a reader will suffer willingly 300 pages of it for the chance to come under her spell.
It’s a bargain, one suspects, many struck with her during her lifetime— when friends, lovers, fellow artists and impresarios endured her excesses and idiosyncrasies, her narcissism and her single-mindedness, for the chance to touch the soul of one of the most distinctive personalities of the 20th century.
Born in 1878, Duncan was raised in San Francisco but spent most of her dance career in Europe. She regarded herself as a genius; she was certainly one-of-a-kind. She rejected classical ballet, which she viewed as unnatural and silly. She dismissed jazz and the dance it inspired, popular at the time she wrote, as barbaric. She favored instead a free and natural dance movement inspired by classical Greek forms.
Her life was as distinctive as her art. She was the honored guest of millionaires. She rebuffed the sexual advances of sculptor Auguste Rodin— only later to sweep aside her Puritan past and take numerous lovers, few of whom she describes in My Life as anything less than geniuses. On landing in Greece, she kissed its sacred earth. She and her brother Raymond, as romantic as she, set about building a Greek temple near Athens—a project they had to give up when they realized there was no fresh water for miles around.
Later, at the height of her fame, having impressed her genius on the world, she lost much of what gave meaning to her life on a country road outside Paris. There, her two children, born to different men, neither of whom she married, died in an apparent—it’s not quite clear from the text— automobile accident. Now she was no longer the great Isadora; she was a mother in agony.
Duncan was sure her approach to dance would, as she told one famous dancer, “revolutionize our entire epoch.” She had discovered it, she wrote, “by the Pacific Ocean, by the waving pine-forests of Sierra Nevada. I have seen the ideal figure of youthful America dancing over the top of the Rockies... I have discovered the dance that is worthy of the poem of Walt Whitman... “
My Life is one flourish like that after another, appeals to Art, Beauty and Truth fairly tumbling from the page, its author seemingly unmindful of how a steady diet of impassioned outpourings might affect her readers. Structurally, the book is a weakly linked chain of travels, performances, love affairs, money problems and devoted audiences held in her thrall.
It would be easy to write off all this as the ravings of a wild-eyed visionary, and to sneer at sensibilities that, at least on paper, lack all subtlety. On the other hand, we get a chance to see up close just the kind of intense and uninhibited personal vision—the supreme egotism, the sense of her own mission—from which many artists draw. Even as she scandalized the folks back home with her personal life, Duncan was the toast of Europe—a fact she delights in recounting through lengthy quotes from flattering reviews.
We can glimpse in this flawed account some of what Isadora might have been like as a dancer. We can sense the raw, uninhibited energy, the romantic flourishes, the willingness to shock. We can feel her life force.
The Structure of Scientific Revolutions
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By Thomas S. Kuhn
First appeared in 1962
By close study of rocks or trees, atoms or stars, scientists draw conclusions about how the world is made. From firm, neutral facts, they construct theories, then test them against new data. Gradually, over years and centuries, theory is bent to accommodate new evidence. And imperceptibly, like bricks fitted into a wall, science’s picture of the universe takes a new form.
This, roughly, is the conventional view of how science progresses. But it bears almost no relation to how science really does progress. Or so said Thomas Kuhn in The Structure of Scientific Revolutions; on its appearance almost 40 years ago, he virtually overnight changed how many saw science working.
Kuhn could scarcely have imagined the reception his long essay would receive. Issued by a university publisher as part of a series—the other two dozen remain largely obscure—the book, now in its third edition, has sold upwards of 600,000 copies. Each year, hundreds of scholarly treatises cite it. One publisher came out with a collection of essays devoted exclusively to issues it raised. And while, narrowly speaking, the book deals only with the physical sciences, its ideas are, as one critic has put it, “so seductive” that scholars in economics, political science and sociology have applied them to their own fields.
What really happens in science, said Thomas Kuhn, is that some longprevailing view of nature undergoes, abruptly and sometimes distressingly, a “paradigm shift”—a revolution in form not so different from a political one. Einstein, said Kuhn, changed the way we see the world: Relativity theory changed the kinds of experiments scientists perform, the instruments they use, the form of the questions they ask, even the types of problems considered important. Einstein ushered in a revolution. So did Newton, Lavoisier, Dalton.
While the old paradigm yet prevails, “normal science” is the rule. This he defines as “research firmly based upon one or more past scientific achievements ... that some particular scientific community acknowledges for a time as supplying the foundation for its further practice.”
The practitioners of normal science don’t run around, willy-nilly, gathering stray bits of data. Rather, all they
do is based on some prior scientific pattern— for example, the phlogiston theory of combustion, or Newtonian dynamics, or relativity theory. They seek particular kinds of facts, to fit particular gaps of knowledge, employing particular kinds of scientific apparatus. Moreover, as Kuhn writes, “other problems (than those the paradigm deems relevant) are rejected as metaphysical, as the concern of another discipline, or sometimes as just too problematic to be worth the time. A paradigm can even insulate the community from ... socially important problems.”
Now this paradigm, whatever it is, is logically self-contained. It explains past experimental results. It leads to the successful solution of new problems. But from time to time, discrepancies arise. In the end, most of these “anomalous” results are successfully explained away or otherwise dismissed. But some are shoved into a back corner of the scientific enterprise, there to nag the minds of a few.
With enough such anomalous results, the particular science may be thrown into crisis. Theories contend. And the competing theories are not just mildly differing interpretations but rather radically different views of nature. They are—to use another of the half-dozen words that Kuhn’s essay gave a special flavor—“incommensurable.” Finally, out of this clash of theories is built a new scientific order, a new paradigm that all but die-hard champions of the old paradigm accept.
Kuhn cites a classic psychological experiment from the 1940s: An experimenter briefly flashes upon a screen images of playing cards and asks the subject to say what he sees. Every so often, a black four of hearts, say, is substituted for the normal red one. At first, the subject is oblivious to the switch, continuing to see the kinds of cards for which all his previous experience prepared him. The conventional deck is his paradigm, the altered cards the anomaly he subconsciously dismisses. Ultimately, as exposure time to the cards increases, he may sense something wrong, perhaps experiencing great distress: He cannot discard his paradigm lightly.
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