“Of course. How eccentric-looking. What were these upside-down triangles again?”
All at once there came a horrible metallic banging and rattling from down at the street entry, accompanied, from beneath the window, by some tonedeaf beer-society in vulgar song. She stared at Kit, lips compressed, head nodding emphatically. “So—it’s all been a trick. Hasn’t it, yes. A squalid trick.”
“What?”
“Arranging for your little beer-mates to show up just as I was about to find the screamingly obvious fallacy in this . . . ‘proof’ of yours—”
“It’s only Humfried and some pals, trying to get a Hausknochen in the lock. If you want to hide someplace, I’d suggest that closet, there.”
“They . . . live here?”
“Not here, but none of them more than a couple-three blocks’ distance. Or do you Riemann folks say ‘metric interval’?”
“But why should your friend use his key—”
“Um actually, as it turns out, every Hausknochen fits pretty much every lock around here.”
“Therefore—”
“Social life is unpredictable.”
Shaking her head, eyes on the floor, “Auf wiedersehen, Herr Professor Traverse.” By mistake the door she chose to exit by was not the back door, though it looked—and from its swing, weighed—about the same, indeed seemed to be located in the same part of Kit’s rooms, as the back door, and yet, strangely, was not the back door. How could this be? Actually, it was not even a door to begin with, but something designed to allow the human brain to interpret it as a door, because it served a similar function.
On the other side of it, she found herself out on the corner of Prinzenstraße and Weenderstraße, known to mathematicians here as the origin of the city of Göttingen’s coördinate system. “Return to zero,” she muttered to herself. “Begin again.” She didn’t find this sort of excursion especially out of the ordinary—it had happened before, and once she had learned that no harm was likely to come of it, she had been able to shrug and get on with her day. It was no more upsetting than waking from a lucid dream.
Back in quotidian space, Kit, having observed Yashmeen apparently walk through a solid wall, had scarcely time to register puzzlement before up the stairs and into the room came thumping Humfried and his creature, Gottlob. They were indeed seldom noted apart, being driven by a common fascination with the details of others’ lives, no matter how trivial. “All right, where is she?”
“Where’s who, and speaking of where, Gottlob, where’s ’at twenty marks you owe me?”
“Ach, der Pistolenheld!” screamed Gottlob, attempting to hide behind Humfried, who as usual was looking for food.
“No, no, Gottlob, control yourself, he will not shoot at you, here, see, this interesting sausage—” Eating half of it immediately and offering the rest to Gottlob, who shook his head vigorously no.
Humfried had been obsessed for a while now with a connection he thought he saw between automorphic functions and the Anharmonic Pencil or, as he preferred, das Nichtharmonischestrahlenbündel, though he had decided to write all his papers in Latin, which no one had done since Euler.
Gottlob, on the other hand, had come to Göttingen from Berlin to study with Felix Klein, on the strength of Klein’s magisterial Mathematical Theory of the Top (1897), approached by way of functions of a complex variable, and also to get away from the sinister influence of the late Leopold Kronecker, keepers of whose flame regarded the complex domain with suspicion if not outright abhorrence—only to find at Göttingen a dwarf variety of the same monumental quarrel between Kronecker and Cantor then raging in the capital, not to mention the world. Fundamentalist Kroneckerites had been known to descend on Göttingen in periodic raids, from which not all of them returned.
“Ach, der Kronecker!” cried Gottlob, “he needed only to step out into the street, and mad dogs ran away or, knowing what was good for them, at once regained their sanity. Only five feet tall, but he enjoyed the abnormal strength of the possessed. Each time he appeared, one could count on weeks of panic.”
“But . . . folks say he was very sociable and outgoing,” said Kit.
“Perhaps, for an insane zealot who believed ‘the positive integers were created by God, and all else is the work of man.’ Of course, it is a religious war. Kronecker did not believe in pi, or the square root of minus one—”
“He did not even believe in the square root of plus two,” said Humfried.
“Against this, Cantor with his Kontinuum, professing an equally strong belief in just those regions, infinitely divisible, which lie between the whole numbers so demanding of all Kronecker’s devotion.”
“And that’s what has kept driving Cantor back into the Nervenklinik,” added Humfried, “and he was only worrying about line-segments. But out here in the four-dimensional space-and-time of Dr. Minkowski, inside the tiniest ‘interval,’ as small as you care to make it, within each tiny hypervolume of Kontinuum—there likewise must be always hidden an infinite number of other points—and if we define a ‘world’ as a very large and finite set of points, then there must be worlds. Universes!”
In fact, a mystical Cantorian cult of the very, indeed vanishingly, negligible, ever seeking escape into a boundless epsilonic world, was rumored to be meeting weekly at Der Finsterzwerg, a beer-hall just outside the old ramparts of the town, near the train station. “A sort of Geographical Society for the unlimited exploration of regions neighboring the Zero. . . .”
As Kit had rapidly discovered, this sort of eccentricity abounded at Göttingen. Discussion ran far into the night, insomnia was the rule, though if one did wish to sleep for some reason, there was always chloral hydrate, which had its own circle of devotees. He saw Yashmeen now and then, usually across the smoke-clouded depths of some disreputable Kneipe by the river, but seldom to talk to. One evening he happened to be walking along the promenade on top of the old fortifications, and near the statue of Gauss passing to Weber a remark forever among the pages of silence, noticed her gazing out over the red-tile roofs of the town, and the lights just coming on.
“How’s ’at old Zeta function?”
“Something amuses you, Kit?”
“Every time I see one them Zetas, it makes me think of a snake up on its tail being charmed by a snake-charmer, ever notice that?”
“These are the reflections that occupy your time?”
“Let me put it a different way. Whenever I see one, it reminds me of you. The ‘charmer’ part anyway.”
“Aaah! Even more trivial. Do none of you ever think beyond these walls? There is a crisis out there.” She scowled into the stained orange glow of the just-vanished sun, the smoke rising from hundreds of chimneys. “And Göttingen is no more exempt than it was in Riemann’s day, in the war with Prussia. The political crisis in Europe maps into the crisis in mathematics. Weierstrass functions, Cantor’s continuum, Russell’s equally inexhaustible capacity for mischief—once, among nations, as in chess, suicide was illegal. Once, among mathematicians, ‘the infinite’ was all but a conjuror’s convenience. The connections lie there, Kit—hidden and poisonous. Those of us who must creep among them do so at our peril.”
“Come on,” Kit said, “let a trivial fellow buy you a beer.”
THAT WINTER, IN St. Petersburg, troops at the Winter Palace fired on thousands of unarmed strikers who had marched there in respectfulnesss and innocence. Hundreds were killed and wounded. In Moscow the Grand Duke Sergei was assassinated. More strikes and fighting followed, along with peasant and military insurrections, on into the summer. The Navy mutinied at Kronstadt and Sebastopol. There was street-fighting in Moscow. The Black Hundreds carried out pogroms against Jews. The Japanese won the war in the East, obliterating the entire Baltic Fleet, which had just sailed halfway around the world to try to lift the siege of Port Arthur. A general strike in the autumn cut the country off for weeks from the rest of the world and, as people came slowly to realize, stopped history. In December the Army beat down
another major uprising. In the East there was fighting all up and down the railroad lines, banditry, eventually a Muslim rebellion in Inner Asia. If God had not forgotten Russia, He had turned His attention elsewhere.
For the rest of Europe, the year that followed was to be remembered as the year of Russians everywhere, fleeing into mass exile, as the Revolution went collapsing at their heels—the Peter and Paul Fortress and sooner or later death if they stayed. Who would have thought the Tsar had so many enemies?
Kit had begun to notice Russians in the Weenderstraße. Yashmeen was convinced they were in town to spy on her. They were trying to blend in, but certain telltale nuances—fur hats, huge unkempt beards, a tendency in the street to drop and begin dancing the kazatsky to music only they could hear—kept giving them away.
“Say, Yash, what’s with all those Russians?”
“I’m trying not to take it personally. My parents were Russian. When we lived on the frontier, my family and I one day were taken in a raid and sold as slaves. Some time later, Major Halfcourt found me in a bazaar in Waziristan and became my second father.”
Not feeling as surprised as he might’ve been, “And he’s still out there someplace?”
“Whatever he’s been up to, it is of enough political weight that someone thinks they can use me somehow.”
“Are you in touch?”
“We have our own means, which neither distance nor time can affect.”
“Telepathy or something.”
She frowned. “Perhaps you think I am a girl with Æther between her ears, easily influenced by the beliefs of the T.W.I.T.”
“Dang, Yash, you sure read my mind there,” with what he hoped was enough of a twinkle that she wouldn’t take offense, for her unannounced ferocities, however playful, continued to cause him some dismay.
She was fooling with her as-ever transcendentally interesting hair, always a sign of trouble down the line. “Even with the Revolution, news comes back. Thousands of miles, multitudes of tongues, unreliable witnesses, deliberate misinformation and all, it finds its way back to the T.W.I.T. people at Chunxton Crescent, and what comes out of their shop can surprisingly often be trusted—even the War Office admits it’s better ‘gen’ on the whole than their own.”
“Anything I can do, just fire away.”
She gave him a look. “To the world here, I enjoy a reputation as ‘my own person’ . . . yet I am also, ever . . . his. My other family have gone on to destinies I cannot imagine. Only in dreams do I catch glimpses of them, moments so fugitive, so slight, that afterward there is the sensible ache here, in my breast, of cruel incompletion. My true memories do not begin until the moment he first saw me in the market—I was a soul impaled, exactly upon the cusp between girl and young woman, a cusp I could literally feel as it penetrated me, as if to bisect me—I do hope that is not a blush, Kit.”
Well, sort of, but more from perplexity than desire. Today she wore an ancient coin, pierced and simply suspended from a fine silver chain around that ever-fascinating neck. . . . “It’s an Afghani dirhan, from the early days of the Ghaznivid Empire. He gave it to me, for luck.” Over its nine or ten centuries of circulation, thieves had nipped and shaved silver from around the outer border, but the inner circle survived, crowded with ancient writing. It was the outward emblem of a hidden history of assault and persistence, the true history of its region and perhaps of this young woman, through this life and who knew how many previous. “Thank you for the offer, Kit. If anything arises, I shall certainly seek your advice. I am ever so grateful,” with eyes a-dance in the luxury of believing little beyond the assumption that he would allow her to get away with this, yet expect no favors in return. He ate it all up like a fairground ice-cream cone, even if he had to pretend indifference. You sure never got this in New Haven. They didn’t know how to flirt like this even in New York. This is the world, Kit reflected, and a couple nights later, around three A.M., as an extra smack of the bamboo stick, She is the world.
Meanwhile Yashmeen, a fine one to scold the trivial, had taken up with a wealthy coffee scion named Günther von Quassel. On their first date, Günther, a devotee of the less than universally respected Ludwig Boltzmann, had tried to explain to her the Riemann problem by means of statistical mechanics.
“Here. Tell me please, as n grows infinitely large, what the nth prime is?”
Sighing, though not with desire, “Its value—as any Gymnasium child at all acquainted with the Prime Number Theorem knows—approaches n log n.”
“So. Looking at the entropy of a system—”
“Some sort of . . . steam-engine word, isn’t it? Am I a boiler engineer, Günni?”
“Except for the usual constants,” writing as he spoke, “one may express the entropy as . . . the summation, of p(Ek), times log p(Ek). All in order so far?”
“Of course, but this is only statistics. When do we get to the mathematics?”
“Ach, die Zetamanie . . . your Prime Number Theorem is not statistical?”
But she was looking at what he’d scribbled down, the two something-log-somethings. “This Ek . . . ?”
“The energy of a given system, you use the k to index if there is more than one, and there usually is.”
“And is there insanity in your family, Günther?”
“You do not find it odd that the Nth prime for very large N may be expressed as one measure of the chaos in a physical system?”
None of which kept Yashmeen from pursuing the attachment.
“AS A CRIME,” Humfried pointed out, “often of the gravest sort, committed in a detective story, may often be only a pretext for the posing and solution of some narrative puzzle, so romance in this town is often pursued as little beyond a pretext for running in and out of doors, not to mention up and down stairs, while talking nonstop and, on auspicious days, screaming.”
Yashmeen one day overheard Günther confessing to his intimate Heinrich, “There is only one girl in this town I have ever wanted to kiss.” It was doctoral-candidate talk, of course, though Yashmeen in her Riemannian obsession appeared to be unaware of the Göttingen tradition that required successful Ph.D.’s in mathematics to kiss the statue of the little goose-girl in the fountain of the Rathaus square, getting soaked and with luck delirious in the process.
Yashmeen grew exercised. “Who is this person?” she demanded of Heinrich, who assumed she was teasing.
“All I know is, he says that she waits every day near the Rathaus.”
“For whom? Not for Günther?”
Heinrich shrugged. “Geese were mentioned?”
“Real geese, or University students?” as she went storming out into the Platz, where she began to loiter menacingly. For days. Günther happened by, or did not happen by, but never in the company of any imaginable rival. Naturally she failed to pay much attention to the fountain nearby, or the little statue. One day she did hear him singing—
Her idea of banter
Likely isn’t Cantor,
Nor is she apt to murmur low
Axioms of Zermelo,
She’s been kissed by geniuses,
Amateur Frobeniuses,
One by one in swank array,
Bright as any Poincaré,
And . . . though she
May not care for Cauchy,
Any more than Riemann,
We’ll just have to dream on . . .
Let
it occur in spots in
Whittaker and Watson—
Unforeseen converging,
Miracles emerging,
Epsilonic dances,
Small but finite chances,
For love . . .
Concerned for her mental stability, everybody felt obliged to put in their two pfennigs, including Kit. “Yash, you want to forget this customer, he’s not for you. I mean, what if he is tall, muscular, even in some strange German way some’d think presentable—”
“You forgot brilliant, amusing, romantic—”
“But you are being used
by your racial memory here,” declared Humfried indignantly, “you are out looking for some Hun.”
“Are you saying I want to be overrun and conquered, Humfried?”
“Did I say that?”
“Well . . . suppose I do, is that, one, any business of either of you, two, anything I feel that I must apologize for, two point one—”
“Yash, you are flat correct,” Kit nodded, “we’re all just night-riders here miles up a posted trail, making pests of ourselves. Ought to be shot, well, shot at, anyway.”
“Günther may be all you say and worse, but until you experience emotions the way we women do, you will find in your relations with us much struggle and little success.”
“I could manage some sniffling maybe, would that help?”
She was already halfway out the door, scowling over her shoulder in reproof, when who should come bounding athletically up the stairway but the very Adonis under discussion, yes Günther von Quassel himself, brandishing a Hausknochen in menacing fashion, approaching, as the stairs brought him to their upper limit, a comparable level of brute rage. “Now Günni,” she greeted him, “you mustn’t murder Kit, must you?”
“What here is he doing?”
“I live here, you oversize bratwurst.”
“Oh. Ja. This is true.” He considered. “But Fräulein Yashmeen . . . she does not live here.”
“Say, Günther, that’s really interesting.”
Günther gazed at him, for what any but the erotically smitten would have considered far too long. Yashmeen, meanwhile, playful as Kit seldom saw her, kept snatching away Günther’s dueling-society cap and pretending to throw it down the stairs. Each time he would respond to the prank only after several seconds had gone by, though with as much alacrity as if it had just happened. In fact, according to Humfried, a disciple of Professor Minkowski, it ought to be obvious to all that Günther inhabited his own idiomatic “frame of reference,” in which time-discrepancies like this one were highly important, if not essential, features. “He is not ‘here,’” Humfried explained, “not completely. He is slightly . . . somewhere else. Enough so, to present some inconvenience to any who value his company.”
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