THE MATHEMATICS OF MAGIC
One
“Steak,” said Harold Shea.
“Porterhouse, sirloin—?” asked the waitress.
“Both, so long as they’re big and rare.”
“Harold,” said Gertrude Mugler, “whatever this is all about, please be careful of your diet. A large protein intake for a man who doesn’t do physical labor—”
“Physical labor!” barked Shea. “The last meal I had was twenty-four hours ago, and it was a little dish of oatmeal mush. Sour, too. Since then I’ve fought a duel with a couple of giants, done acrobatics on a magic broomstick, had a ride on a god’s enchanted brewery-horse—Well, anyway, I’ve been roasted and frozen and shaken and nearly scared to death, and by Thor’s hammer I want food!”
“Harold, are you—are you feeling well?”
“Fine, toots. Or I will be when I surround some grub.” He turned to the waitress again: “Steak!”
“Listen, Harold,” persisted Gertrude. “Don’t! You pop out of nowhere in that crazy costume; you talk wildly about things you couldn’t expect anyone to believe—”
“You don’t have to believe I popped out of nowhere, either,” said Shea.
“Then can’t you tell me what’s wrong?”
“Nothing’s wrong, and I’m not going to talk about it until I’ve consulted Dr. Chalmers.”
“Well,” said Gertrude, “if that’s your attitude—come on, Walter, let’s go to a movie.”
“But,” bleated Walter Bayard, “I want to listen—”
“Oh, be a gentleman for once in your life!”
“Oh, all right, Gert.” He leered back at Shea as he went. “Anyway you didn’t bring back any dreamgirls.”
Shea grinned after them. “There goes the guy who used to kid me about how Gert had gotten the psychological jump on me,” he said to Chalmers. “I hope she rides herd on him.”
Reed Chalmers smiled faintly. “You forget—uh—Walter’s infallible defense mechanism.”
“What’s that?”
“When the pressure becomes too great, he can simply go to sleep on her.”
Shea gave a suppressed snort. “You know not what you—ah, food!” He attacked his plate, working his mouth around a piece of steak big enough to choke a horse; with effort, like a snake engulfing a toad. An expression of pure bliss spread over his face as he chewed. Chalmers noted that his colleague ignored the fact that half the restaurant was staring at the tableaux of a long-faced young man in baggy Norse woolens.
“A—uh—somewhat less rapid rate of ingestion—” Chalmers began.
Shea shook a finger, gulped down his mouthful, and spoke: “Don’t worry about me.” Between mouthfuls he told his story.
Reed Chalmers’ mild eyes bugged as he watched and listened to his young friend. “Good gracious! That’s the third of those steaks, somewhat inadequately called small. You’ll—uh—render yourself ill.”
“This is the last one. Hey, waitress! May I please have an apple pie? Not just a segment; I want a whole pie.” He turned back to Chalmers. “So the spook said, ‘Go on back to where you came from,’ and here I am!”
Chalmers mused: “While I have known you, Harold, to commit venial sins of rhetorical exaggeration incompatible with true scientific accuracy, I have never known you to engage in deliberate fabrication. So I believe you. The general alteration in your appearance and bearing furnishes persuasive corroboration.”
“Have I changed?” asked Shea.
“You show the effects of physical hardship, as well as exposure to the sun and wind.”
“That all?”
Chalmers pondered: “You would like me to say, would you not, that your air of self-conscious brashness has been replaced by one of legitimate self-confidence?”
“Well, uh . . .”
Chalmers continued: “Those conscious of shortcomings are always eager to be informed of radical improvement. Actually such improvements, when they occur at all in an adult, take place slowly. No miraculous change is to be expected in a couple of weeks.” He twinkled at Shea’s discomfort, and added: “I will admit that you seem to show some alteration of personality, and I think in the right direction.”
Shea laughed. “At least I learned to appreciate the value of theory. If you’d been along we’d really have gotten somewhere in applying the screwy laws of the world of Scandinavian myth.”
“I—” Chalmers stopped.
“What?”
“Nothing.”
“Of course,” said Shea, “you’d never have stood the physical end of it.”
Chalmers sighed. “I suppose not.”
Shea went on: “It checked your theory of paraphysics all right. In that universe the laws of similarity and contagion held good—at least, the magic spells I figured out with their help worked.”
Chalmers brushed his gray mop out of his eyes. “Amazing! I asserted that the transfer of the physical body, to another space-time frame by symbolic logic—what did you call it? A syllogismobile!—was possible. But it is a shock to have so—uh—far-fetched a deduction confirmed by experimental proof.”
Shea said: “Sure, we’ve got something all right. But what are we going to do with it?”
Chalmers frowned. “It is rather obscure. Presents a whole new world-picture, unlike anything but some of the Oriental religions. An infinity of universes, moving along parallel but distinct space-time vectors. But, as you put it, what can be done with it? If I publish the results of your experiment they’ll simply say poor old Chalmers has . . . uh . . . a tile loose, and in any case an experimental psychologist has no business venturing into physics. Think of Oliver Lodge!” He shuddered. “The only satisfactory proof would be to send some of the doubters to another universe. Unfortunately, we could hardly count on their encountering Grua with a handful of enchanted snow. They would be unable to return, and the doubters left behind would be doubters still. You perceive the difficulty.”
“Huh-uh. Wonder how the fight came out? It might be worthwhile going back to see.”
“It would be inadvisable. The Ragnarök was only beginning when you left. You might return to find the giants had won and were in charge. If you wish adventure, there are plenty of other and less—” The voice trailed off.
“Other what?”
“Well, perhaps nothing of importance. I was about to say—systematic attainable universes. Since you left I have been engaged in the development of the structural theory of a multiple-universe cosmology, and—”
Shea interrupted. “Listen, Dr. Chalmers. We both know too much psychology to kid each other. Something’s eating you besides paraphysical mathematics.”
“Harold”—Chalmers gave a sigh—“I’ve always maintained that you’d make a better . . . uh . . . salesman or politician than psychologist. You’re weak on theory, but in off-hand, rule-of-thumb diagnosis of behavior patterns, you are incomparable.”
“Don’t evade, Doctor.”
“Very well. Were you perhaps thinking of making another journey soon?”
“Why, I just got back and haven’t had time to think. Say! You aren’t suggesting you’d like to go along, are you?”
Reed Chalmers rolled a fragment of bread into a precise gray pill. “As a matter of fact that’s what I was suggesting, Harold. Here I am, fifty-six years old, without family or intimate friends—except you young men of the Garaden Institute. I have made—or believe I have—the greatest cosmological discovery since Copernicus, yet its nature is such that it cannot be proved, and no one will credit it without the most exhaustive proof.” He shrugged slightly. “My work is done, but to a result that will afford me no appreciation in this world. May I not . . . uh . . . be permitted the foible of seeking a fuller life elsewhere?”
###
Back in Shea’s room and seated in the best armchair, Chalmers stretched his legs and meditatively sipped a highball. “I’m afraid your suggestion of Cuchulinn’s Ireland does not meet with my approval. An adventurous life, no doubt—but culturally a barbarism, with an elaborate system of taboos, violations of which are punished by the removal of heads.”
“But the girls—” protested Shea. “Those piano-legged Scowegian blondes—they all remind me of Gertrude—”
“For a person of my age amorous adventure has few attractions. And as my partner in this enterprise I must ask you to remember that while you have . . . uh . . . certain physical skills that would be useful anywhere, I am limited to fields where intellectual attainments would be of more value than in ancient Ireland. The only non-warriors who got anywhere in those days were minstrels—and I can neither compose lays nor play the harp.”
Shea grinned maliciously. “All right, you leave the girls to me, then. But I guess you’re right; we’ll have to drop Queen Maev and Ossian.” He peered around the bookshelves. “How about this?”
Chalmers examined the volume he handed down. “Spenser’s Faerie Queene. Mm-m-m—‘vision unrolled after vision to the sound of varying music,’ as Dr. Johnson said. Certainly a brilliant and interesting world, and one in which I personally might have some place. But I am afraid we should find it uncomfortable if we landed in the latter half of the story, where Queen Gloriana’s knights are having a harder and harder time, as though Spenser were growing discouraged, or the narrative for some reason were escaping from his hands, taking on a life of its own. I’m not sure we could exercise the degree of selectivity needed to get into the story at the right point. After all, in your last experience, you attempted Ireland and arrived in Scandinavian myth.”
“But,” protested Shea, “if you’re going adventuring you can’t avoid—” and then stopped, his mouth open.
“You were about to say ‘danger,’ were you not?” said Chalmers, with a smile. “I confess—”
Shea got to his feet. “Doctor . . . Doc—” he burst out. “Listen: why shouldn’t we jump right into that last part of the Faerie Queene and help Gloriana’s knights straighten things out? You said you had worked out some new angles. We ought to be better than anyone else in the place. Look what I was able to do in the Ragnarök with the little I know!”
“You are immodest, Harold,” replied Chalmers, but he was leaning forward. “Still, it is an . . . uh . . . attractive plan; to look in another world for the achievement denied in this. Suppose you fill my glass again while we consider details.”
“Well, the first detail I’d like to know something about is what new wrinkles in theory you have in mind.”
Chalmers settled himself and took on his lecture-room manner. “As I see it, our universes have a relation analogous to that of a pencil of parallel vectors,” said he. “The vectors themselves represent time, of course. That gives us a six-dimensional cosmos—three in space, one in time, and two which define the relationships of one universe of the cosmos to another.
“You know enough mathematics to be aware that the ‘fourth dimension,’ so called, is only a dimension in the sense of a measurable quality, like color or density. The same applies for interuniversal dimensions. I maintain—”
“Whoa!” said Shea. “Is there an infinite number of universes?”
“Ahem—I wish you would learn to avoid interruptions, Harold. I used to believe so. But now I consider the number finite, though very large.
“Let me continue. I maintain that what we call ‘magic’ is merely . . . the physics of some of these other universes. This physics is capable of operating along the interuniversal dimensions—”
“I see,” Shea interrupted again. “Just as light can operate through interplanetary space, but sound requires some such conducting medium as air or water.”
“The analogy is not perfect. Let me continue. You know how the theme of conjuring things up and making them disappear constantly recurs in fairy tales. These phenomena become plausible if we assume the enchanter is snatching things from another universe or banishing them to one.”
Shea said: “I see an objection. If the laws of magic don’t operate in the conducting medium of our universe, how’s it possible to learn about them? I mean, how did they get into fairy stories?”
“The question is somewhat obvious. You remember my remarking that dements suffered hallucinations because their personalities were split between this universe and another? The same applies to the composers of fairy stories, though to a lesser degree. Naturally, it would apply to any writer of fantasy, such as Dunsany or Hubbard. When he describes some strange world, he is offering a somewhat garbled version of a real one, having its own set of dimensions quite independent of ours.”
Shea sipped his highball in silence. Then he asked: “Why can’t we conjure things into and out of this universe?”
“We can. You successfully conjured yourself out of this one. But it is probable that certain of these parallel universes are easier of access than others. Ours—”
“Would be one of the hard kind?”
“Ahem. Don’t interrupt, please. Yes. Now as to the time dimension, I’m inclined to think we can travel among universes only at right angles to the pencil of space-time vectors, if you follow my use of a . . . a somewhat misleading analogy.
“However, it appears likely that our vectors are curved. A lapse of time, along the inner side of the curve would correspond to a greater lapse of time along the outer. You know the theme in certain fairy tales—the hero comes to fairyland, spends three days, and returns to find he has been gone three minutes or three years.
“The same feature would account for the possibility of landing in someone’s imagined idea of the future. This is clearly a case where a mind has been running along one of the outer curved vectors at a speed which has outstripped the passage of time along our own inner side of the curve. The result—Harold, are you following me?”
Shea’s highball glass had rolled onto the rug with a gentle plunk, and the suspicion of a snore came from his chair. Fatigue had caught up with him at last.
###
Next weekend, Harold Shea went up to Cleveland. He was approaching his second time-journey with some misgivings. Chalmers was an astute old bird—no doubt about that. A good theorist. But it was the pursuit of the theory rather than its result that interested the old boy. How would he work out as a companion in a life of arduous adventure—a man of fifty-six, who had always led a sedentary life, and for that matter, who always seemed to prefer discussion to experience.
Well, too late to pull out now, Shea told himself, as he entered the shop of the Montrose Costume Co. He asked to see medieval stuff. A clerk, who seemed to think that the word “medieval” had something to do with pirates, finally produced an assortment of doublets and hose, feathered hats, and floppy boots of thin yellow leather. Shea selected a costume that had once been worn by the leading man in De Koven’s Robin Hood. It had no pockets, but a tailor could be found to remedy that. For Chalmers, he bought a similar but plainer outfit, with a monkish robe and attached hood. Chalmers was to go as a palmer, or pilgrim, a character which both felt would give him some standing.
The costume company’s assortment of arms and armor proved not only phony but impractical. The chain mail was knitted woolens dipped in aluminum paint. The plate was sheets of tin-can thinness. The swords had neither edge, balance, nor temper. The antique shops had nothing better; their antique weapons were mostly Civil War cavalry sabers. Shea decided to use his own fencing épée. It had a rather stiff blade, and if he unscrewed the point d’arrêt, ground the end down to a sharp point, and contrived some kind of sheath, the weapon would do till he got something better.
The most serious question, as he explained to Chalmers on his return, was concern with the formulas of the magic they intended to use on their arrival. “How do you expect to read English in the land of Faerie when I couldn’t in Scandinavia?” he demanded.
“I’ve allowed for that,” Chalmers replied. “You forgot that mathematics is a . . . a universal language, independent of words.”
“All right. But will your mathematical symbols mean the same things?”
“Glance at this sheet, Harold. Knowing the principles of symbolic logic to begin with, I can look over this pictured equation with an apple at the left and a great many apples at the right, and thus realize it means that an apple belongs to the class of apples. From that I shall infer that the horseshoe-shaped symbol in the center means ‘is a member of the class of.’ ”
“You think that’ll work, honest? But, say, how do we know that you and I will land in the same part of the Faerie world?”
Chalmers shrugged. “For that matter, how do we know we shan’t land in Greek mythology? There are still laws of this method of transference to be worked out. We can only hold onto each other, read the formulas in chorus, and hope for the best.”
Shea grinned. “And if it doesn’t work, what the hell? Well, I guess we’re ready.” He inhaled deeply. “If P equals not-Q, Q implies not-P, which is equivalent to saying either P or Q but not both. But if not-P is not implied by not-Q—Come in, Mrs. Ladd.”
Shea’s landlady opened the door, and opened her mouth to say something. But the something failed to come forth. She stared agape at a pair of respectable psychologists, standing side by side in medieval costumes, with rucksacks on their shoulders. They were holding hands and, with their free hands, holding sheets of paper. Chalmers purpled with embarrassment.
Shea bowed easily. “We’re doing an experiment, Mrs. Ladd. We may be away for some time. If Mr. Bayard asks for us, let him in and tell him he can look at the papers in the top right-hand drawer. And you might mail this letter. Thanks.” He explained to Chalmers: “It’s to Gert; to tell her not to waste her money by setting Johnson after us.”
“But Mr. Shea—” said the landlady.
“Please, Mrs. Ladd. You can sit down and watch if you like. Let’s go, Doctor—a conclusion can be drawn concerning the relation between two classes even if the evidence refers only to a part of some third class to which both are related. Whatever is predicated affirmatively or negatively of a class may be predicated in like manner of everything asserted to be contained in that class—”
Mrs. Ladd watched, ample bosom heaving. Her eyes bulged from her head; she’d have material for backfence conversation for months to come.
Pfump! There was a movement of air, fluttering the papers on the table and whirling ashes from the ashtrays. Mrs. Ladd, pulling herself together, moved a trembling hand through the space where her strangely dressed lodgers had stood.
It met no resistance.