by E. T. Bell
Of course no Latin was taught in the school that Boole was permitted to attend. Making a pathetically mistaken diagnosis of the abilities which enabled the propertied class to govern those beneath them in the scale of wealth, Boole decided that he must learn Latin and Greek if he was ever to get his feet out of the mire. This was Boole’s mistake. Latin and Greek had nothing to do with the cause of his difficulties. He did teach himself Latin with his poor struggling father’s sympathetic encouragement. Although the poverty-stricken tradesman knew that he himself should never escape he did what he could to open the door for his son. He knew no Latin. The struggling boy appealed to another tradesman, a small bookseller and friend of his father. This good man could only give the boy a start in the elementary grammar. Thereafter Boole had to go it alone. Anyone who has watched even a good teacher trying to get a normal child of eight through Caesar will realize what the untutored Boole was up against. By the age of twelve he had mastered enough Latin to translate an ode of Horace into English verse. His father, hopefully proud but understanding nothing of the technical merits of the translation, had it printed in the local paper. This precipitated a scholarly row, partly flattering to Boole, partly humiliating.
A classical master denied that a boy of twelve could have produced such a translation. Little boys of twelve often know more about some things than their forgetful elders give them credit for. On the technical side grave defects showed up. Boole was humiliated and resolved to supply the deficiencies of his self-instruction. He had also taught himself Greek. Determined now to do a good job or none he spent the next two years slaving over Latin and Greek, again without help. The effect of all this drudgery is plainly apparent in the dignity and marked Latinity of much of Boole’s prose.
Boole got his early mathematical instruction from his father, who had gone considerably beyond his own meager schooling by private study. The father had also tried to interest his son in another hobby, that of making optical instruments, but Boole, bent on his own ambition, stuck to it that the classics were the key to dominant living. After finishing his common schooling he took a commercial course. This time his diagnosis was better, but it did not help him greatly. By the age of sixteen he saw that he must contribute at once to the support of his wretched parents. School teaching offered the most immediate opportunity of earning steady wages—in Boole’s day “ushers,” as assistant teachers were called, were not paid salaries but wages. There is more than a monetary difference between the two. It may have been about this time that the immortal Squeers, in Dickens’ Nicholas Nickleby, was making his great but unappreciated contribution to modern pedagogy at Dotheboys Hall with his brilliant anticipation of the “project” method. Young Boole may even have been one of Squeers’ ushers; he taught at two schools.
Boole spent four more or less happy years teaching in these elementary schools. The chilly nights, at least, long after the pupils were safely and mercifully asleep, were his own. He still was on the wrong track. A third diagnosis of his social unworthiness was similar to his second but a considerable advance over both his first and second. Lacking anything in the way of capital—practically every penny the young man earned went to the support of his parents and the barest necessities of his own meager existence—Boole now cast an appraising eye over the gentlemanly professions. The Army at that time was out of his reach as he could not afford to purchase a commission. The Bar made obvious financial and educational demands which he had no prospect of satisfying. Teaching, of the grade in which he was then engaged, was not even a reputable trade, let alone a profession. What remained? Only the Church. Boole resolved to become a clergyman.
In spite of all that has been said for and against God, it must be admitted even by his severest critics that he has a sense of humor. Seeing the ridiculousness of George Boole’s ever becoming a clergyman, he skilfully turned the young man’s eager ambition into less preposterous channels. An unforeseen affliction of greater poverty than any they had yet enjoyed compelled Boole’s parents to urge their son to forego all thoughts of ecclesiastical eminence. But his four years of private preparation (and rigid privation) for the career he had planned were not wholly wasted; he had acquired a mastery of French, German, and Italian, all destined to be of indispensable service to him on his true road.
* * *
At last he found himself. His father’s early instruction now bore fruit. In his twentieth year Boole opened up a civilized school of his own. To prepare his pupils properly he had to teach them some mathematics as it should be taught. His interest was aroused. Soon the ordinary and execrable textbooks of the day awoke his wonder, then his contempt. Was this stuff mathematics? Incredible. What did the great masters of mathematics say? Like Abel and Galois, Boole went directly to great headquarters for his marching orders. It must be remembered that he had had no mathematical training beyond the rudiments. To get some idea of his mental capacity we can imagine the lonely student of twenty mastering, by his own unaided efforts, the Mécanique céleste of Laplace, one of the toughest masterpieces ever written for a conscientious student to assimilate, for the mathematical reasoning in it is full of gaps and enigmatical declarations that “it is easy to see,” and then we must think of him making a thorough, understanding study of the excessively abstract Mécanique analytique of Lagrange, in which there is not a single diagram to illuminate the analysis from beginning to end. Yet Boole, self-taught, found his way and saw what he was doing. He even got his first contribution to mathematics out of his unguided efforts. This was a paper on the calculus of variations.
Another gain that Boole got out of all this lonely study deserves a separate paragraph to itself. He discovered invariants. The significance of this great discovery which Cayley and Sylvester were to develop in grand fashion has been sufficiently explained; here we repeat that without the mathematical theory of invariance (which grew out of the early algebraic work) the theory of relativity would have been impossible. Thus at the very threshold of his scientific career Boole noticed something lying at his feet which Lagrange himself might easily have seen, picked it up, and found that he had a gem of the first water. That Boole saw what others had overlooked was due no doubt to his strong feeling for the symmetry and beauty of algebraic relations—when of course they happen to be both symmetrical and beautiful; they are not always. Others might have thought his find merely pretty. Boole recognized that it belonged to a higher order.
Opportunities for mathematical publication in Boole’s day were inadequate unless an author happened to be a member of some learned society with a journal or transactions of its own. Luckily for Boole, The Cambridge Mathematical Journal, under the able editorship of the Scotch mathematician, D. F. Gregory, was founded in 1837. Boole submitted some of his work. Its originality and style impressed Gregory favorably, and a cordial mathematical correspondence began a friendship which lasted out Boole’s life.
It would take us too far afield to discuss here the great contribution which the British school was making at the time to the understanding of algebra as algebra, that is, as the abstract development of the consequences of a set of postulates without necessarily any interpretation or application to “numbers” or anything else, but it may be mentioned that the modern conception of algebra began with the British “reformers,” Peacock, Herschel, De Morgan, Babbage, Gregory, and Boole. What was a somewhat heretical novelty when Peacock published his Treatise on Algebra in 1830 is today a commonplace in any competently written schoolbook. Once and for all Peacock broke away from the superstition that the x, y, z, . . . in such relations as x + y = y + x, xy = yx, x(y + z) = xy + xz, and so on, as we find them in elementary algebra, necessarily “represent numbers”; they do not, and that is one of the most important things about algebra and the source of its power in applications. The x, y, z, . . . are merely arbitrary marks, combined according to certain operations, one of which is symbolized as +, another by × (or simply as xy instead of x × y), in accordance with postulates laid down at the beginnin
g, like the specimens x + y = y + x, etc., above.
Without this realization that algebra is of itself nothing more than an abstract system, algebra might still have been stuck fast in the arithmetical mud of the eighteenth century, unable to move forward to its modern and extremely useful variants under the direction of Hamilton. We need only note here that this renovation of algebra gave Boole his first opportunity to do fine work appreciated by his contemporaries. Striking out on his own initiative he separated the symbols of mathematical operations from the things upon which they operate and proceeded to investigate these operations on their own account. How did they combine? Were they too subject to some sort of symbolic algebra? He found that they were. His work in this direction is extremely interesting, but it is overshadowed by the contribution which is peculiarly his own, the creation of a simple, workable system of symbolic or mathematical logic.
* * *
To introduce Boole’s splendid invention properly we must digress slightly and recall a famous row of the first half of the nineteenth century, which raised a devil of a din in its own day but which is now almost forgotten except by historians of pathological philosophy. We mentioned Hamilton a moment ago. There were two Hamiltons of public fame at this time, one the Irish mathematician Sir William Rowan Hamilton (1805-1865), the other the Scotch philosopher Sir William Hamilton (1788-1856). Mathematicians usually refer to the philosopher as the other Hamilton. After a somewhat unsuccessful career as a Scotch barrister and candidate for official university positions the eloquent philosopher finally became Professor of Logic and Metaphysics in the University of Edinburgh. The mathematical Hamilton, as we have seen, was one of the outstanding original mathematicians of the nineteenth century. This is perhaps unfortunate for the other Hamilton, as the latter had no earthly use for mathematics, and hasty readers sometimes confuse the two famous Sir Williams. This causes the other one to turn and shiver in his grave.
Now, if there is anything more obtuse mathematically than a thickheaded Scotch metaphysician it is probably a mathematically thicker-headed German metaphysician. To surpass the ludicrous absurdity of some of the things the Scotch Hamilton said about mathematics we have to turn to what Hegel said about astronomy or Lotze about non-Euclidean geometry. Any depraved reader who wishes to fuddle himself can easily run down all he needs. It was the metaphysician Hamilton’s misfortune to have been too dense or too lazy to get more than the most trivial smattering of elementary mathematics at school, but “omniscience was his foible,” and when he began lecturing and writing on philosophy, he felt constrained to tell the world exactly how worthless mathematics is.
Hamilton’s attack on mathematics is probably the most famous of all the many savage assaults mathematics has survived, undented. Less than ten years ago lengthy extracts from Hamilton’s diatribe were vigorously applauded when a pedagogical enthusiast retailed them at a largely attended meeting of our own National Educational Association. Instead of applauding, the auditors might have got more out of the exhibition if they had paused to swallow some of Hamilton’s philosophy as a sort of compulsory sauce for the proper enjoyment of his mathematical herring. To be fair to him we shall pass on a few of his hottest shots and let the reader make what use of them he pleases.
“Mathematics [Hamilton always used “mathematics” as a plural, not a singular, as customary today] freeze and parch the mind”; “an excessive study of mathematics absolutely incapacitates the mind for those intellectual energies which philosophy and life require”; “mathematics can not conduce to logical habits at all”; “in mathematics dullness is thus elevated into talent, and talent degraded into incapacity”; “mathematics may distort, but can never rectify, the mind.”
This is only a handful of the birdshot; we have not room for the cannon balls. The whole attack is most impressive—for a man who knew far less mathematics than any intelligent child of ten knows. One last shot deserves special mention, as it introduces the figure of mathematical importance in the whole wordy war, De Morgan (18061871), one of the most expert controversialists who ever lived, a mathematician of vigorous independence, a great logician who prepared the way for Boole, the remorselessly good-humored enemy of all cranks, charlatans, and humbugs, and finally father of the famous novelist (Alice for Short, etc.). Hamilton remarks, “This [a perfectly nonsensical reason that need not be repeated] is why Mr. De Morgan among other mathematicians so often argues right. Still, had Mr. De Morgan been less of a Mathematician, he might have been more of a Philosopher; and be it remembered, that mathematics and dram-drinking tell especially, in the long run.” Although the esoteric punctuation is obscure the meaning is clear enough. But it was not De Morgan who was given to tippling.
De Morgan, having gained some fame from his pioneering studies in logic, allowed himself in an absent-minded moment to be trapped into a controversy with Hamilton over the latter’s famous principle of “the quantification of the predicate.” There is no need to explain what this mystery is (or was); it is as dead as a coffin nail. De Morgan had made a real contribution to the syllogism; Hamilton thought he detected De Morgan’s diamond in his own blue mud; the irate Scottish lawyer-philosopher publicly accused De Morgan of plagiarism—an insanely unphilosophical thing to do—and the fight was on. On De Morgan’s side, at least, the row was a hilarious frolic. De Morgan never lost his temper; Hamilton had never learned to keep his.
* * *
If this were merely one of the innumerable squabbles over priority which disfigure scientific history it would not be worth a passing mention. Its historical importance is that Boole by now (1848) was a firm friend and warm admirer of De Morgan. Boole was still teaching school, but he knew many of the leading British mathematicians personally or by correspondence. He now came to the aid of his friend—not that the witty De Morgan needed any mortal’s aid, but because he knew that De Morgan was right and Hamilton wrong. So, in 1848, Boole published a slim volume, The Mathematical Analysis of Logic, his first public contribution to the vast subject which his work inaugurated and in which he was to win enduring fame for the boldness and perspicacity of his vision. The pamphlet—it was hardly more than that—excited De Morgan’s warm admiration. Here was the master, and De Morgan hastened to recognize him. The booklet was only the promise of greater things to come six years later, but Boole had definitely broken new, stubborn ground.
In the meantime, reluctantly turning down his mathematical friends’ advice that he proceed to Cambridge and take the orthodox mathematical training there, Boole went on with the drudgery of elementary teaching, without a complaint, because his parents were now wholly dependent upon his support. At last he got an opportunity where his conspicuous abilities as an investigator and a lecturer could have some play. He was appointed Professor of Mathematics at the recently opened Queen’s College at what was then called the city of Cork, Ireland. This was in 1849.
Needless to say, the brilliant man who had known only poverty and hard work all his life made excellent use of his comparative freedom from financial worry and everlasting grind. His duties would now be considered onerous; Boole found them light by contrast with the dreary round of elementary teaching to which he had been accustomed. He produced much notable miscellaneous mathematical work, but his main effort went on licking his masterpeice into shape. In 1854 he published it: An Investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probabilities. Boole was thirty nine when this appeared. It is somewhat unusual for a mathematician as old as that to produce work of such profound originality, but the phenomenon is accounted for when we remember the long, devious path Boole was compelled to follow before he could set his face fairly toward his goal. (Compare the careers of Boole and Weierstrass.)
A few extracts will give some idea of Boole’s style and the scope of his work.
“The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to t
hem in the language of a Calculus, and upon this foundation to establish the science of Logic and construct its method; to make that method itself the basis of a general method for the application of the mathematical doctrine of probabilities; and, finally, to collect from the various elements of truth brought to view in the course of these inquiries some probable intimations concerning the nature and constitution of the human mind. . . .”
“Shall we then err in regarding that as the true science of Logic which, laying down certain elementary laws, confirmed by the very testimony of the mind, permits us thence to deduce, by uniform processes, the entire chain of its secondary consequences, and furnishes, for its practical applications, methods of perfect generality? . . .”
“There exist, indeed, certain general principles founded in the very nature of language, by which the use of symbols, which are but the elements of scientific language, is determined. To a certain extent these elements are arbitrary. Their interpretation is purely conventional: we are permitted to employ them in whatever sense we please. But this permission is limited by two indispensable conditions,—first, that from the sense once conventionally established we never, in the same process of reasoning, depart; secondly, that the laws by which the process is conducted be founded exclusively upon the above fixed sense or meaning of the symbols employed. In accordance with these principles, any agreement which may be established between the laws of the symbols of Logic and those of Algebra can but issue in an agreement of processes. The two provinces of interpretation remain apart and independent, each subject to its own laws and conditions.
