Vance regarded the men on the board, which were now in the position they had occupied at the time of Pardee’s resignation.*
“Out of curiosity,” he said quietly, “I played the game through to the checkmate the other night.—I say, Mr. Arnesson, would you mind doin’ the same. I could bear to hear your comment on it.”
Arnesson studied the position closely for a few minutes. Then he turned his head slowly and lifted his eyes to Vance. A sardonic grin overspread his face.
“I grasp the point. Gad! What a situation! Five moves for Black to win through. And an almost unheard-of finale in chess. Can’t recall a similar instance. The last move would be Bishop to Knight-7, mating. In other words, Pardee was beaten by the black bishop! Incredible!”*
Professor Dillard put down his book. “What’s this?” he exclaimed, joining us at the chess table. “Pardee was defeated by the bishop?” He gave Vance a shrewd, admiring look. “You evidently had good reason, sir, for investigating that chess game. Pray overlook an old man’s temper.” He stood gazing down at the board with a sad, puzzled expression.
Markham was frowning with deep perplexity. “You say it’s unusual for a bishop alone to mate?” he asked Arnesson.
“Never happens—almost unique situation. And that it should happen to Pardee of all people! Incomprehensible!” He gave a short ironic laugh. “Inclines one to believe in a nemesis. You know, the bishop has been Pardee’s bête noir for twenty years—it’s ruined his life. Poor beggar! The black bishop is the symbol of his sorrow. Fate, by Gad! It’s the one chessman that defeated the Pardee gambit. Bishop-to-Knight-5 always broke up his calculations—disqualified his pet theory—made a hissing and a mocking of his life’s work. And now, with a chance to break even with the great Rubinstein, the bishop crops up again and drives him back into obscurity.”
A few minutes later we took our departure and walked to West End Avenue, where we hailed a taxicab.
“It’s no wonder, Vance,” commented Markham as we rode downtown, “that Pardee went white the other afternoon when you mentioned the black bishop’s being at large at midnight. He probably thought you were deliberately insulting him—throwing his life’s failure in his face.”
“Perhaps… ” Vance gazed dreamily out into the gathering shadows. “Dashed queer about the bishop being his incubus all these years. Such recurring discouragements affect the strongest minds sometimes, create a desire for revenge on the world, with the cause of one’s failure exalted to an astraean symbol.”
“It’s difficult to picture Pardee in a vindictive role,” objected Markham. Then, after a moment: “What was your point about the discrepancy in time between Pardee’s and Rubinstein’s playing? Suppose Rubinstein did take forty-five minutes or so to work out his combination. The game wasn’t over until after one. I don’t see that your visit to Arnesson put us ahead in any way.”
“That’s because you’re unacquainted with the habits of chess players. In a clock game of that kind no player sits at the table all the time his opponent is figuring out moves. He walks about, stretches his muscles, takes the air, ogles the ladies, imbibes ice water, and even indulges in food. At the Manhattan Square Masters Tournament last year there were four tables, and it was a common sight to see as many as three empty chairs at one time. Pardee’s a nervous type. He wouldn’t sit through Rubinstein’s protracted mental speculations.”
Vance lighted a cigarette slowly. “Markham, Arnesson’s analysis of that game reveals the fact that Pardee had three-quarters of an hour to himself around midnight.”
Footnotes
*For the benefit of the expert chess player who may be academically interested I append the exact position of the game when Pardee resigned:—WHITE: King at QKtsq; Rook at QB8; Pawns at QR2 and Q2. BLACK: King at Q5; Knight at QKt5; Bishop at QR6; Pawns at QKt7 and QB7.
*The final five unplayed moves for Black to mate, as I later obtained them from Vance, were—45. RxP; KtxR. 46. KxKt; P—Kt8 (Queen). 47. KxQ; K—Q6. 48. K—Rsq; K—B7. 49. P—Q3; B—Kt7 mate.
CHAPTER TWENTY-ONE
Mathematics and Murder
(Saturday, April 16; 8.30 p.m.)
LITTLE WAS SAID about the case during dinner, but when we had settled ourselves in a secluded corner of the club lounge room, Markham again broached the subject.
“I can’t see,” he said, “that finding a loophole in Pardee’s alibi helps us very much. It merely complicates an already intolerable situation.”
“Yes,” sighed Vance. “A sad and depressin’ world. Each step appears to tangle us a little more. And the amazin’ part of it is, the truth is staring us in the face, only we can’t see it.”
“There’s no evidence pointing to anyone. There’s not even a suspect against whose possible culpability reason doesn’t revolt.”
“I wouldn’t say that, don’t y’ know. It’s a mathematician’s crime; and the landscape has been fairly cluttered with mathematicians.”
Throughout the entire investigation no one had been indicated by name as the possible murderer. Yet each of us realized in his own heart that one of the persons with whom we had talked was guilty; and so hideous was this knowledge that we instinctively shrank from admitting it. From the first we had cloaked our true thoughts and fears with generalities.
“A mathematician’s crime?” repeated Markham. “The case strikes me as a series of senseless acts committed by a maniac running amuck.”
Vance shook his head. “Our criminal is supersane, Markham. And his acts are not senseless: they’re hideously logical and precise. True, they have been conceived with a grim and terrible humor, with a tremendously cynical attitude; but within themselves they are exact and rational.”
Markham regarded Vance thoughtfully.
“How can you reconcile these Mother Goose crimes with the mathematical mind?” he asked. “In what way can they be regarded as logical? To me they’re nightmares, unrelated to sanity.”
Vance settled himself deeper in his chair, and smoked for several minutes. Then he began an analysis of the case, which not only clarified the seeming madness of the crimes themselves but brought all the events and the characters into a uniform focus. The accuracy of this analysis was brought home to us with tragic and overwhelming force before many days had passed.*
“In order to understand these crimes,” he began, “we must consider the stock-in-trade of the mathematician, for all his speculations and computations tend to emphasize the relative insignificance of this planet and the unimportance of human life.—Regard, first, the mere scope of the mathematician’s field. On the one hand, he attempts to measure infinite space in terms of parsecs and light-years and, on the other, to measure the electron, which is so infinitely small that he has to invent the Rutherford unit—a millionth of a millimicron. His vision is one of transcendental perspectives, in which this earth and its people sink almost to the vanishing point. Some of the stars—such as Arcturus, Canopus, and Betelgeuse—which he regards merely as minute and insignificant units, are many times more massive than our entire solar system. Shapleigh’s estimate of the diameter of the Milky Way is 300,000 light-years; yet we must place 10,000 Milky Ways together to get the diameter of the universe—which gives us a cubical content a thousand milliard times greater than the scope of astronomical observation. Or, to put it relatively in terms of mass:—the sun’s weight is 324,000 times greater than the weight of the earth; and the weight of the universe is postulated as that of a trillion*—a milliard times a milliard—suns… Is it any wonder that workers in such stupendous magnitudes should sometimes lose all sense of earthly proportions?”
Vance made an insignificant gesture.
“But these are element’ry figures—the everyday facts of journeyman calculators. The higher mathematician goes vastly further. He deals in abstruse and apparently contradict’ry speculations which the average mind can not even grasp. He lives in a realm where time, as we know it, is without meaning save as a fiction of the brain, and becomes a fourth
coordinate of three-dimensional space; where distance also is meaningless except for neighboring points, since there are an infinite number of shortest routes between any two given points; where the language of cause and effect becomes merely a convenient shorthand for explanat’ry purposes; where straight lines are nonexistent and insusceptible of definition; where mass grows infinitely great when it reaches the velocity of light; where space itself is characterized by curvatures; where there are lower and higher orders of infinities; where the law of gravitation is abolished as an acting force and replaced by a characteristic of space—a conception that says, in effect, that the apple does not fall because it is attracted by the earth, but because it follows a geodesic, or world-line…
“In this realm of the modern mathematician curves exist without tangents. Neither Newton nor Leibnitz nor Bernoulli even dreamed of a continuous curve without a tangent—that is, a continuous function without a differential co-efficient. Indeed, no one is able to picture such a contradiction,—it lies beyond the power of imagination. And yet it is a commonplace of modern mathematics to work with curves that have no tangents.—Moreover, pi— that old friend of our schooldays, which we regarded as immutable—is no longer a constant; and the ratio between diameter and circumference now varies according to whether one is measuring a circle at rest or a rotating circle… Do I bore you?”
“Unquestionably,” retorted Markham. “But pray continue, provided your observations have an earthly direction.”
Vance sighed and shook his head hopelessly but at once became serious again.
“The concepts of modern mathematics project the individual out of the world of reality into a pure fiction of thought, and lead to what Einstein calls the most degenerate form of imagination—pathological individualism. Silberstein, for instance, argues the possibility of five- and six-dimensional space, and speculates on one’s ability to see an event before it happens. The conclusions contingent on the conception of Flammarion’s Lumen—a fictive person who travels faster than the velocity of light and is therefore able to experience time extending in a reverse direction—are in themselves enough to distort any natural and sane point of view.* But there is another conceptual homunculus even weirder than Lumen from the standpoint of rational thinking. This hypothetical creature can traverse all worlds at once with infinite velocity, so that he is able to behold all human history at a glance. From Alpha Centauri he can see the earth as it was four years ago; from the Milky Way he can see it as it was four thousand years ago; and he can also choose a point in space where he can witness the ice age and the present day simultaneously!…”
Vance settled himself more deeply in his chair.
“Toying with the simple idea of infinity is enough to unhinge the average man’s mind. But what of the well-known proposition of modern physics that we cannot take a straight and ever-advancing path into space without returning to our point of departure? This proposition holds, in brief, that we may go straight to Sirius and a million times further without changing direction, but we can never leave the universe: we at last return to our starting point from the opposite direction! Would you say, Markham, that this idea is conducive to what we quaintly call normal thinking? But however paradoxical and incomprehensible it may seem, it is almost rudiment’ry when compared with other theorems advanced by mathematical physics. Consider, for example, what is called the problem of the twins. One of two twins starts to Arcturus at birth—that is, with accelerated motion in a gravitational field—and, on returning, discovers that he is much younger than his brother. If, on the other hand, we assume that the motion of the twins is Galilean and that they are therefore traveling with uniform motion relative to each other, then each twin will find that his brother is younger than himself!…
“These are not paradoxes of logic, Markham; they’re only paradoxes of feeling. Mathematics accounts for them logically and scientifically.* The point I’m trying to make is that things which seem inconsistent and even absurd to the lay mind are commonplaces to the mathematical intelligence. A mathematico-physicist like Einstein announces that the diameter of space—of space, mind you—is 100,000,000 light-years, or 700 trillion miles; and considers the calculation abecedarian. When we ask what is beyond this diameter, the answer is: ‘There is no beyond: these limitations include everything.’ To wit, infinity is finite! Or, as the scientist would say, space is unbounded but finite.—Let your mind meditate on this idea for half an hour, Markham, and you’ll have a sensation that you’re going mad.”
He paused to light a cigarette.
“Space and matter—that’s the mathematician’s speculative territ’ry. Eddington conceives matter as a characteristic of space—a bump in nothingness; whereas Weil conceives space as a characteristic of matter—to him empty space is meaningless. Thus Kant’s noumenon and phenomenon become interchangeable; and even philosophy loses all significance. But when we come to the mathematical conceptions of finite space, all rational laws are abrogated. De Sitter’s conception of the shape of space is globular, or spherical. Einstein’s space is cylindrical; and matter approaches zero at the periphery, or ‘border condition.’ Weyl’s space, based on Mach’s mechanics, is saddle-shaped… Now, what becomes of nature, of the world we live in, of human existence, when we weigh them against such conceptions? Eddington suggests the conclusion that there are no natural laws—namely, that nature is not amenable to the law of sufficient reason. Alas, poor Schopenhauer!* And Bertrand Russell sums up the inevitable results of modern physics by suggesting that matter is to be interpreted merely as a group of occurrences, and that matter itself need not be existent!…Do you see what it all leads to? If the world is noncausative and nonexistent, what is a mere human life?—or the life of a nation?—or, for that matter, existence itself?…”
Vance looked up, and Markham nodded dubiously.
“So far I follow you, of course,” he said. “But your point seems vague—not to say esoteric.”
“Is it surprising,” asked Vance, “that a man dealing in such colossal, incommensurable concepts, wherein the individuals of human society are infinitesimal, might in time lose all sense of relative values on earth and come to have an enormous contempt for human life? The comparatively insignificant affairs of this world would then become mere petty intrusions on the macrocosmos of his mental consciousness. Inevitably such a man’s attitude would become cynical. In his heart he would scoff at all human values and sneer at the littleness of the visual things about him. Perhaps there would be a sadistic element in his attitude, for cynicism is a form of sadism—”
“But deliberate, planned murder!” objected Markham.
“Consider the psychological aspects of the case. With the normal person, who takes his recreations daily, a balance is maintained between the conscious and the unconscious activities: the emotions, being constantly dispersed, are not allowed to accumulate. But with the abnormal person, who spends his entire time in intense mental concentration and who rigorously suppresses all his emotions, the loosening of the subconscious is apt to result in a violent manifestation. This long inhibition and protracted mental application, without recreation or outlet of any kind, causes an explosion which often assumes the form of deeds of unspeakable horror. No human being, however intellectual, can escape the results. The mathematician who repudiates nature’s laws is nevertheless amenable to those laws. Indeed, his rapt absorption in hyperphysical problems merely increases the pressure of his denied emotions. And outraged nature, in order to maintain her balance, produces the most grotesque fulminations—reactions which, in their terrible humor and perverted gaiety, are the exact reverse of the grim seriousness of abstruse mathematical theories. The fact that Sir William Crookes and Sir Oliver Lodge—both great mathematical physicists—became confirmed spiritists constitutes a similar psychological phenomenon.”
Vance took several deep inhalations on his cigarette.
“Markham, there’s no escaping the fact: these fantastic and seemingly incredible murders we
re planned by a mathematician as forced outlets to a life of tense abstract speculation and emotional repression. They fulfill all the indicated requirements: they are neat and precise, beautifully worked out, with every minute factor fitting snugly in place. No loose ends, no remainders, apparently no motive. And aside from their highly imaginative precision, all their indications point unmistakably to an abstrusely conceptive intelligence on the loose—a devotee of pure science having his fling.”
“But why their grisly humor?” asked Markham. “How do you reconcile the Mother Goose phase of them with your theory?”
“The existence of inhibited impulses,” explained Vance, “always produces a state favorable to humor. Dugas designates humor as a détente—a release from tension; and Bain, following Spencer, calls humor a relief from restraint. The most fertile field for a manifestation of humor lies in accumulated potential energy—what Freud calls Besetzungsenergie —which in time demands a free discharge. In these Mother Goose crimes we have the mathematician reacting to the most fantastic of frivolous acts in order to balance his superserious logical speculations. It’s as if he were saying cynically: ‘Behold! This is the world that you take so seriously because you know nothing of the infinitely larger abstract world. Life on earth is a child’s game—hardly important enough to make a joke about’… And such an attitude would be wholly consistent with psychology; for after any great prolonged mental strain one’s reactions will take the form of reversals—that is to say, the most serious and dignified will seek an outlet in the most childish games. Here, incidentally, you have the explanation for the practical joker with his sadistic instincts…
The Bishop Murder Case Page 22
