Most mathematical examples illustrate the truth of a statement; conversely, counterexamples demonstrate a statement's falsity if changing the conditions. Mathematicians have always prized counterexamples as intrinsically enjoyable objects of study as well as valuable tools for teaching, learning, and research. This third edition of the definitive book on counterexamples in probability and stochastic processes presents the author's revisions and corrections in addition to a substantial new appendix.
Suitable as a supplementary source for advanced undergraduates and graduate courses in the field of probability and stochastic processes, this volume features a wide variety of topics that are challenging in both content and detail. The text consists of four chapters and twenty-five sections. Each section begins with short introductory notes of basic definitions and main results. Counterexamples related to the main results follow, along with motivation for questions and counterstatements that range in difficulty. A familiarity with basic notions and results in probability and stochastic processes is assumed, and a chapter of supplementary remarks provides a wealth of information about original sources as well as references for further studies.
About the Author
Formerly a Senior Research Fellow at the Bulgarian Academy of Sciences and a Professor at the University of Sofia, Jordan Stoyanov is on the faculty of the School of Mathematics and Statistics, University of Newcastle, United Kingdom.
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W. E. B. Du Bois
Selections from His Writings
By Bob Blaisdell
Dover Publications, Inc.Copyright © 2014 Dover Publications, Inc.
All rights reserved.
STRIVINGS OF THE NEGRO PEOPLE (1897)
"Then it dawned upon me with a certain suddenness that I was different from the others; or like, mayhap, in heart and life and longing, but shut out from their world by a vast veil," writes Du Bois in this, his first article for the famous Boston-based magazine, The Atlantic Monthly. Though he evokes compelling particular details of his boyhood, his real subject is an overview of Black American history and the social humiliations that had left extraordinary people seeing life split in two: "[The American Negro] simply wishes to make it possible for a man to be both a Negro and an American without being cursed and spit upon by his fellows, without losing the opportunity of self-development." As in many of his essays he explains an overlooked problem while advocating a moral solution. Du Bois later revised this essay into the first chapter of his renowned The Souls of Black Folk: Essays and Sketches (1903).
Between me and the other world there is ever an unasked question: unasked by some through feelings of delicacy; by others through the difficulty of rightly framing it. All, nevertheless, flutter round it. They approach me in a half-hesitant sort of way, eye me curiously or compassionately, and then, instead of saying directly, How does it feel to be a problem? they say, I know an excellent colored man in my town; or, I fought at Mechanicsville; or, Do not these Southern outrages make your blood boil? At these I smile, or am interested, or reduce the boiling to a simmer, as the occasion may require. To the real question, How does it feel to be a problem? I answer seldom a word.
And yet, being a problem is a strange experience,—peculiar even for one who has never been anything else, save perhaps in babyhood and in Europe. It is in the early days of rollicking boyhood that the revelation first bursts upon one, all in a day, as it were. I remember well when the shadow swept across me. I was a little thing, away up in the hills of New England, where the dark Housatonic winds between Hoosac and Taghanic to the sea. In a wee wooden schoolhouse, something put it into the boys' and girls' heads to buy gorgeous visiting-cards—ten cents a package—and exchange. The exchange was merry, till one girl, a tall newcomer, refused my card,—refused it peremptorily, with a glance. Then it dawned upon me with a certain suddenness that I was different from the others; or like, mayhap, in heart and life and longing, but shut out from their world by a vast veil. I had thereafter no desire to tear down that veil, to creep through; I held all beyond it in common contempt, and lived above it in a region of blue sky and great wandering shadows. That sky was bluest when I could beat my mates at examination-time, or beat them at a foot-race, or even beat their stringy heads. Alas, with the years all this fine contempt began to fade; for the world I longed for, and all its dazzling opportunities, were theirs, not mine. But they should not keep these prizes, I said; some, all, I would wrest from them. Just how I would do it I could never decide: by reading law, by healing the sick, by telling the wonderful tales that swam in my head,—some way. With other black boys the strife was not so fiercely sunny: their youth shrunk into tasteless sycophancy, or into silent hatred of the pale world about them and mocking distrust of everything white; or wasted itself in a bitter cry, Why did God make me an outcast and a stranger in mine own house? The "shades of the prison-house" closed round about us all: walls strait and stubborn to the whitest, but relentlessly narrow, tall, and unscalable to sons of night who must plod darkly on in resignation, or beat unavailing palms against the stone, or steadily, half hopelessly watch the streak of blue above.
After the Egyptian and Indian, the Greek and Roman, the Teuton and Mongolian, the Negro is a sort of seventh son, born with a veil, and gifted with second-sight in this American world,—a world which yields him no self-consciousness, but only lets him see himself through the revelation of the other world. It is a peculiar sensation, this double-consciousness, this sense of always looking at one's self through the eyes of others, of measuring one's soul by the tape of a world that looks on in amused contempt and pity. One ever feels his two-ness,—an American, a Negro; two souls, two thoughts, two unreconciled strivings; two warring ideals in one dark body, whose dogged strength alone keeps it from being torn asunder.
The history of the American Negro is the history of this strife,—this longing to attain self-conscious manhood, to merge his double self into a better and truer self. In this merging he wishes neither of the older selves to be lost. He does not wish to Africanize America, for America has too much to teach the world and Africa; he does not wish to bleach his Negro blood in a flood of white Americanism, for he believes—foolishly, perhaps, but fervently—that Negro blood has yet a message for the world. He simply wishes to make it possible for a man to be both a Negro and an American without being cursed and spit upon by his fellows, without losing the opportunity of self-development.
This is the end of his striving: to be a co-worker in the kingdom of culture, to escape both death and isolation, and to husband and use his best powers. These powers, of body and of mind, have in the past been so wasted and dispersed as to lose all effectiveness, and to seem like absence of all power, like weakness. The double-aimed struggle of the black artisan, on the one hand to escape white contempt for a nation of mere hewers of wood and drawers of water, and on the other hand to plough and nail and dig for a poverty-stricken horde, could only result in making him a poor craftsman, for he had but half a heart in either cause. By the poverty and ignorance of his people the Negro lawyer or doctor was pushed toward quackery and demagogism, and by the criticism of the other world toward an elaborate preparation that overfitted him for his lowly tasks. The would-be black savant was confronted by the paradox that the knowledge his people needed was a twice-told tale to his white neighbors, while the knowledge which would teach the white world was Greek to his own flesh and blood. The innate love of harmony and beauty that set the ruder souls of his people a-dancing, a-singing, and a-laughing raised but confusion and doubt in the soul of the black artist; for the beauty revealed to him was the soul-beauty of a race which his larger audience despised, and he could not articulate the message of another people.
This waste of double aims, this seeking to satisfy two unreconciled ideals, has wrought sad havoc with the courage and faith and deeds of eight thousand thousand people, has sent them often wooing false gods and invoking false means of salvation, and has even at times seemed destined to make them ashamed of themselves. In the days of bondage they thought to see in one divine event the end of all doubt and disappointment; eighteenth-century Rousseauism never worshiped freedom with half the unquestioning faith that the American Negro did for two centuries. To him slavery was, indeed, the sum of all villainies, the cause of all sorrow, the root of all prejudice; emancipation was the key to a promised land of sweeter beauty than ever stretched before the eyes of wearied Israelites. In his songs and exhortations swelled one refrain, liberty; in his tears and curses the god he implored had freedom in his right hand. At last it came,—suddenly, fearfully, like a dream. With one wild carnival of blood and passion came the message in his own plaintive cadences:—
"Shout, O children!
Shout, you're free!
The Lord has bought your liberty!"
Years have passed away, ten, twenty, thirty. Thirty years of national life, thirty years of renewal and development, and yet the swarthy ghost of Banquo sits in its old place at the national feast. In vain does the nation cry to its vastest problem,—
"Take any shape but that, and my firm nerves
Shall never tremble!"
The freedman has not yet found in freedom his promised land. Whatever of lesser good may have come in these years of change, the shadow of a deep disappointment rests upon the Negro people,—a disappointment all the more bitter because the unattained ideal was unbounded save by the simple ignorance of a lowly folk.
The first decade was merely a prolongation of the vain search for freedom, the boom that seemed ever barely to elude their grasp,—like a tantalizing will-o'-the-wisp, maddening and misleading the headless host. The holocaust of war, the terrors of the Kuklux Klan, the lies of carpet-baggers, the disorganization of industry, and the contradictory advice of friends and foes left the bewildered serf with no new watchword beyond the old cry for freedom. As the decade closed, however, he began to grasp a new idea. The ideal of liberty demanded for its attainment powerful means, and these the Fifteenth Amendment gave him. The ballot, which before he had looked upon as a visible sign of freedom, he now regarded as the chief means of gaining and perfecting the liberty with which war had partially endowed him. And why not? Had not votes made war and emancipated millions? Had not votes enfranchised the freedmen? Was anything impossible to a power that had done all this? A million black men started with renewed zeal to vote themselves into the kingdom. The decade fled away,—a decade containing, to the freedman's mind, nothing but suppressed votes, stuffed ballot-boxes, and election outrages that nullified his vaunted right of suffrage. And yet that decade from 1875 to 1885 held another powerful movement, the rise of another ideal to guide the unguided, another pillar of fire by night after a clouded day. It was the ideal of "book-learning"; the curiosity, born of compulsory ignorance, to know and test the power of the cabalistic letters of the white man, the longing to know. Mission and night schools began in the smoke of battle, ran the gauntlet of reconstruction, and at last developed into permanent foundations. Here at last seemed to have been discovered the mountain path to Canaan; longer than the highway of emancipation and law, steep and rugged, but straight, leading to heights high enough to overlook life.
Up the new path the advance guard toiled, slowly, heavily, doggedly; only those who have watched and guided the faltering feet, the misty minds, the dull understandings of the dark pupils of these schools know how faithfully, how piteously, this people strove to learn. It was weary work. The cold statistician wrote down the inches of progress here and there, noted also where here and there a foot had slipped or some one had fallen. To the tired climbers, the horizon was ever dark, the mists were often cold, the Canaan was always dim and far away. If, however, the vistas disclosed as yet no goal, no resting-place, little but flattery and criticism, the journey at least gave leisure for reflection and self-examination; it changed the child of emancipation to the youth with dawning self-consciousness, self-realization, self-respect. In those sombre forests of his striving his own soul rose before him, and he saw himself,—darkly as through a veil; and yet he saw in himself some faint revelation of his power, of his mission. He began to have a dim feeling that, to attain his place in the world, he must be himself, and not another. For the first time he sought to analyze the burden he bore upon his back, that deadweight of social degradation partially masked behind a half-named Negro problem. He felt his poverty; without a cent, without a home, without land, tools, or savings, he had entered into competition with rich, landed, skilled neighbors. To be a poor man is hard, but to be a poor race in a land of dollars is the very bottom of hardships. He felt the weight of his ignorance,—not simply of letters, but of life, of business, of the humanities; the accumulated sloth and shirking and awkwardness of decades and centuries shackled his hands and feet. Nor was his burden all poverty and ignorance. The red stain of bastardy, which two centuries of systematic legal defilement of Negro women had stamped upon his race, meant not only the loss of ancient African chastity, but also the hereditary weight of a mass of filth from white whoremongers and adulterers, threatening almost the obliteration of the Negro home.
A people thus handicapped ought not to be asked to race with the world, but rather allowed to give all its time and thought to its own social problems. But alas! while sociologists gleefully count his bastards and his prostitutes, the very soul of the toiling, sweating black man is darkened by the shadow of a vast despair. Men call the shadow prejudice, and learnedly explain it as the natural defense of culture against barbarism, learning against ignorance, purity against crime, the "higher" against the "lower" races. To which the Negro cries Amen! and swears that to so much of this strange prejudice as is founded on just homage to civilization, culture, righteousness, and progress he humbly bows and meekly does obeisance. But before that nameless prejudice that leaps beyond all this he stands helpless, dismayed, and well-nigh speechless; before that personal disrespect and mockery, the ridicule and systematic humiliation, the distortion of fact and wanton license of fancy, the cynical ignoring of the better and boisterous welcoming of the worse, the all-pervading desire to inculcate disdain for everything black, from Toussaint to the devil,—before this there rises a sickening despair that would disarm and discourage any nation save that black host to whom "discouragement" is an unwritten word.
They still press on, they still nurse the dogged hope,—not a hope of nauseating patronage, not a hope of reception into charmed social circles of stock-jobbers, pork-packers, and earl-hunters, but the hope of a higher synthesis of civilization and humanity, a true progress, with which the chorus "Peace, good will to men,"
"May make one music as before,
Thus the second decade of the American Negro's freedom was a period of conflict, of inspiration and doubt, of faith and vain questionings, of Sturm und Drang. The ideals of physical freedom, of political power, of school training, as separate all-sufficient panaceas for social ills, became in the third decade dim and overcast. They were the vain dreams of credulous race childhood; not wrong, but incomplete and over-simple. The training of the schools we need to-day more than ever,—the training of deft hands, quick eyes and ears, and the broader, deeper, higher culture of gifted minds. The power of the ballot we need in sheer self-defense, and as a guarantee of good faith. We may misuse it, but we can scarce do worse in this respect than our whilom masters. Freedom, too, the long-sought, we still seek,—the freedom of life and limb, the freedom to work and think. Work, culture, and liberty,—all these we need, not singly, but together; for to-day these ideals among the Negro people are gradually coalescing, and finding a higher meaning in the unifying ideal of race,—the ideal of fostering the traits and talents of the Negro, not in opposition to, but in conformity with, the greater ideals of the American republic, in order that some day, on American soil, two world races may give each to each those characteristics which both so sadly lack. Already we come not altogether empty-handed: there is to-day no true American music but the sweet wild melodies of the Negro slave; the American fairy tales are Indian and African; we are the sole oasis of simple faith and reverence in a dusty desert of dollars and smartness. Will America be poorer if she replace her brutal, dyspeptic blundering with the light-hearted but determined Negro humility; or her coarse, cruel wit with loving, jovial good humor; or her Annie Rooney with "Steal Away"?
Merely a stern concrete test of the underlying principles of the great republic is the Negro problem, and the spiritual striving of the freedmen's sons is the travail of souls whose burden is almost beyond the measure of their strength, but who bear it in the name of an historic race, in the name of this land of their fathers' fathers, and in the name of human opportunity.
Excerpted from W. E. B. Du Bois by Bob Blaisdell. Copyright © 2014 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
Strivings of the Negro People (1897),
A Negro Schoolmaster in the New South (1899),
The Song of the Smoke (1899),
The Black North: A Social Study (1901),
The Sorrow Songs (1903),
The Talented Tenth (1903),
Address of the Niagara Movement to the Country (1906),
Religion in the South (1907),
The Value of Agitation (1907),
The Case (1907),
The Burden of Black Women (1907),
Evolution of the Race Problem (1909),
Jesus Christ in Georgia (1911),
The Upbuilding of Black Durham: The Success of the Negroes and Their Value to a Tolerant and Helpful Southern City (1912),
Socialism and the Negro Problem (1913),
Woman Suffrage (1915),
Booker T. Washington (1915),
The Shadow of Years (1918),
Returning Soldiers (1919),
Let Us Reason Together (1919),
The Souls of White Folk (1920),
The Damnation of Women (1920),
Again, Social Equality (1920),