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 any one. 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.*
* I am obviously unable to set down Vance’s exact words, despite the
completeness of my notes; but I sent him a proof of the following
passages with a request that he revise and edit them; so that, as
they now stand, they represent an accurate paraphrase of his
analysis of the psychological factors of the Bishop murders.
“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 was here using the English connotation of “trillion,” which
is the third power of a million, as opposed to the American and
French system of numeration which regards a trillion as a mere
million millions.
Vance made an insignificant gesture.
“But these are element’ry figures—the every-day 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 co-ordinate 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 non-existent 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 school-days,
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 4,000 years ago;
and he can also choose a point in space where he can witness the
ice-age and the present day simultaneously! . . .”
* Lumen was invented by the French astronomer to prove the
possibility of the reversal of time. With a speed of 250,000 miles
per second, he was conceived as soaring into space at the end of the
battle of Waterloo, and catching up all the light-rays that had left
the battlefield. He attained a gradually increasing lead, until at
the end of two days he was witnessing, not the end, but the
beginning of the battle; and in the meantime he had been viewing
events in reverse order. He had seen projectiles leaving the objects
they had penetrated and returning to the cannon; dead men coming to
life and arranging themselves in battle formation. Another
hypothetical adventure of Lumen was jumping to the moon, turning
about instantaneously, and seeing himself leaping from the moon to
the earth backwards.
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 travelling
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.”
* Vance requested me to mention here A. d’Abro’s recent scholarly
work, “The Evolution of Scientific Thought,” in which there is an
excellent discussion of the paradoxes associated with space-time.
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 non-causative and non-existent, what is a mere human
life?—or the life of a nation?—or, for that matter, existence
itself? . . .”
* Vance’s M. A. thesis, I recall, dealt with Schopenhauer’s “Ueber
die vierfache Wurzel des Satzes vom zureichenden Grunde.”
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 were planned by a mathematician as forced outlets
to a life of tense abstract speculation and emotional repression. They
fulfil 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. . . .
“Moreover, all sadists have an infantile complex. And the child is
totally amoral. A man, therefore, who experiences these infantile
psychological reversals is beyond good and evil. Many modern
mathematicians even hold that all convention, duty, morality, good,
and the like, could not exist except for the fiction of free will. To
them the science of ethics is a field haunted by conceptual ghosts;
and they even arrive at the disintegrating doubt as to whether truth
itself is not merely a figment of the imagination. . . . Add to these
considerations the sense of earthly distortion and the contempt for
human life which might easily result from the speculations of higher
mathematics, and you have a perfect set of conditions for the type of
crimes with which we are dealing.”
When Vance had finished speaking Markham sat silent for a long time.
Finally he moved restively.
“I can understand,” he said, “how these crimes might fit almost any of
the persons involved. But, on the basis of your argument, how do you
account for the notes to the press?”
“Humor must be imparted,” returned Vance. “‘A jest’s prosperity lies
in the ear of him who hears it.’ Also, the impulse toward
exhibitionism enters into the present case.”
“But the ‘Bishop’ alias?”
“Ah! That’s a most vital point. The _raison d’être_ of this terrible
orgy of humor lies in that cryptic signature.”
Markham turned slowly.
“Does the chess player and the astronomer fulfil the conditions of
your theory as well as the mathematical physicist?”
“Yes,” Vance replied. “Since the days of Philidor, Staunton and
Kieseritzki, when chess was something of a fine art, the game has
degenerated almost into an exact science; and during Capablanca’s
régime it has become largely a matter of abstract mathematical
speculation. Indeed, Maroczy, Doctor Lasker and Vidmar are all
well-known mathematicians. . . . And the astronomer, who actually
views the universe, may get an even more intense impression of the
unimportance of this earth than the speculative physicist. Imagination
runs riot through a telescope. The mere theory of existing life on
distant planets tends to reduce earthly life to second’ry
consideration. For hours after one has looked at Mars, for instance,
and dallied with the notion that its inhabitants outnumber and surpass
in intelligence our own population, one has difficulty in readjusting
oneself to the petty affairs of life here on earth. Even a reading of
Percival Lowell’s romantic book* temporarily takes away from the
imaginative person all consciousness of the significance of any single
planet’ry existence.”
* I do not know whether Vance was here referring to “Mars and Its
Canals” or “Mars as the Abode of Life.”
There was a long silence. Then Markham asked:
“Why should Pardee have taken Arnesson’s black bishop that night
instead of one from the club where it would not have been missed?”
“We don’t know enough of the motive to say. He may have taken it with
some deliberate purpose in view.—But what evidence have you of his
guilt? All the suspicions in the world would not permit you to take
any step against him. Even if we knew indubitably who the murderer
was, we’d be helpless. . . . I tell you, Markham, we’re facing a
shrewd brain—one that figures out every move, and calculates all the
possibilities. Our only hope is to create our own evidence by finding
a weakness in the murderer’s combination.”
“The first thing in the morning,” declared Markham grimly, “I’m going
to put Heath to work on Pardee’s alibi that night. There’ll be twenty
men checking it up by noon, questioning every spectator at that chess
game, and making a door-to-door canvass between the Manhattan Chess
Club and the Drukker house. If we can find some one who actually saw
Pardee in the vicinity of the Drukkers’ around midnight, then we’ll
have a very suspicious piece of circumstantial evidence against him.”
“Yes,” agreed Vance; “that would give us a definite starting-point.
Pardee would have considerable difficulty in explaining why he was six
blocks away from the club during his set-to with Rubinstein at the
exact hour that a black bishop was being left at Mrs. Drukker’s
door. . . . Yes, yes. By all means have Heath and his minions tackle
the problem. It may lead us forward.”
But the Sergeant was never called upon to check the alibi. Before nine
o’clock on the following morning Markham called at Vance’s house to
inform him that Pardee had committed suicide.