- Mathematics and Credulity, The Journal of Philosophy 22 (17) (1925), 449-458.
In a pathetic outburst in the Introduction to his fallacious "New Theory of Parallels," the creator of matter-of-fact Alice and the elusive Snark exclaims: "The charm (of pure mathematics) lies chiefly ... in the absolute certainty of its results: for that is what, beyond all mental treasures, the human intellect craves for. Let us only be sure of something! More light, more light! . . . 'And if our fate be death, give light and let us die!' This is the cry that, through all the ages, is going up from perplexed Humanity, and Science has little else to offer, that will really meet the demands of its votaries, than the conclusions of Pure Mathematics." The journeyman mathematician may not, in his sober interludes, feel certain of the coy snark concealed in twice two, but the ever- buoyant populariser of the "inhuman science" has no such qualms. He finds the object of his search everywhere, whether it is there or not. To him the reputed "Queen of the Sciences" appears to rule her unwieldy domain with a rod of steel and a head of cast iron, and to be withal generous. When the enthusiast's moth-eaten creed rots and leaves him naked, the gracious Queen presents him with another, more durable than the first, for half of it is cotton and unpalatable to maggots. ... Alfred Pringsheim, a professional mathematician, offers us the cliché: "It is true that mathematics, owing to the fact that its whole content is built up by means of purely logical deduction from a small number of universally comprehended principles, has not un- justly been designated as the science of the self-evident." If to be true is formally equivalent to being self-evident, then we have found the inestimable "something," and Pilate is effectively answered. Is the "axiom of infinity" to be received into Pringsheim's select society of "universally comprehended principles"? ... Although mathematics with its infinite fails to satisfy the hyperrational aspirations of a rational mind (granting that such a mind can be capable of such nonsense), it is otherwise with the pragmatic test of applicability to the world as mathematicians perceive it. Admitting that logically the position of the finitists is very strong, Borel justifies the use of the doubtful element as follows, and probably most students of mathematics will agree with him. "It suffices to have studied the elements of mathematical analysis to form an estimate of the degree of complication to which the pretence of excluding the infinite, as it is considered in these elements, would lead. "Granted; but this does not justify a mathematical philosopher in deifying a common utensil, or in mistaking, as some of them do occasionally, a hod for a god. Superstition has been kicked out of the front door by the physical and biological sciences. Why should mathematicians permit it to re-enter the House of Solomon by the cellar window? Charity, perhaps, or possibly just plain sloth. So far as mathematics is concerned in these brawling days of peace, when Fundamentalists and Modernists are at each other's throats, it may be well to remember that no cause, scientific or spiritual, was ever helped for long by fraudulent claims in its behalf. That some mathematicians have stooped to make of mathematics a specious pleader in a cause where its voice is at best irrelevant, is a poor compliment to their science.
- Mathematics and Speculation, The Scientific Monthly 32 (3) (1931), 193-209.
The rapid advancement of science is nowhere more strikingly apparent than in the numerous excellently contrived accounts of current science pre- pared for the scientific layman by professional scientists and by those who believe they understand what scientists are talking about in their ingenious theories. A reasonably critical mind, contemplating these brilliant expositions of fact or fascinating speculation on the apparent state, purpose and destiny of man and the universe, may become slightly confused by the subtly conflicting testimony of so many witnesses to the truth, but this is only a minor blemish on an otherwise encouraging picture of progress and universal enlightenment. The downright sceptic, looking for a light in his darkness, who closes more than one of these books or articles with the ejaculation, "God help the layman!" may be forgiven, for that help and no other is precisely what some authors abandon their readers with in their concluding chapters. And the reasonably critical mind will neither affirm nor deny, but continue to seek answers to such of its questions as seem to make sense. Science has at last become articulate, not to say garrulous. Mathematics is not classed by some with the sciences, but this is of no importance here. What does matter is the fact that mathematics can not descend to untechnical language so readily as the sciences. Non-Euclidean geometry, with all of its deep implications for metaphysics no less than for physics, began to pass into the common stock of knowledge only with the popularity of Einstein's general relativity long after it had been a commonplace t, the geometers. Modern algebra, eve now, is only beginning to influence the speculations of physical science, and the theory of algebraic numbers and ideals of no less philosophic interest than non Euclidean geometry, is still all but un known outside a narrow circle of arithmeticians.
- A Suggestion Regarding Foreign Languages in Mathematics, The American Mathematical Monthly 40 (5) (1933), 287.
While following a mathematical discussion in which English, French, and German were used by various speakers, I was struck by the remarkably small number of common, non-technical words sufficient to lubricate the technical terms and to present a mathematical argument intelligibly. The common words include conjunctions, prepositions, personal pronouns, a few nouns, and a handful of verbs, particularly the auxiliaries. Some of the common verbs, like "put," or "take," of course are used with uncommon meanings, but these idioms could be included in the technical vocabulary. For an argument in analysis, algebra, or the theory of numbers, it would seem that 300 common words are more than sufficient; for geometry the number may be higher. For a philosophical discussion dealing, say, with mathematical logic, much more would be necessary. To the 300 or so common words must be added the technical vocabulary of the subject discussed. It should be possible, by sampling pages of foreign mathematical books and periodicals to form a reasonable estimate of the number of common words necessary for a given branch of mathematics. If the guess of 300 is anywhere near the truth, it would be worth someone's time to compile such common vocabularies for the languages in which mathematics is alive. By concentrating on the minimum lists, supplemented by dictionaries if necessary, a working competence in two or three languages might readily be acquired by reading minutely two or three fairly long papers of which the mathematical subject matter was already familiar to the reader.
- The History of Blissard's Symbolic Method, with a Sketch of its Inventor's Life, The American Mathematical Monthly 45 (7) (1938), 414-421.
It sometimes happens in the history of mathematics that the credit for a particular method is commonly ascribed to another than its originator. In the interests of historical justice, such oversights should be corrected, where the facts are known. A conspicuous instance is the extremely useful symbolic method expounded and widely applied by Edouard Lucas and usually attributed to him. The kernel of this symbolic method is described below. This representative notation, to use the designation of its inventor, or symbolic method, as Lucas called it, or umbral calculus, was fully developed in 1861 by John Blissard, in a mathematical journal with a wide circulation among mathematicians, fifteen years before Lucas published his first papers on the subject. Although J W L Glaisher called attention to Blissard's indisputable priority forty years ago, it seems still to be not generally known to those who use or refer to the symbolic method. Thus, for example, the symbolic treatment of the Bernoulli numbers, first given by Blissard, is assigned to Lucas. Regarding this treatment, L E Dickson expressed the opinion of most students of the Bernoulli (and allied) numbers when he stated that, "Bernouillian numbers are most conveniently employed in the symbolic notation of Lucas. ... Mathematically, Blissard was 'a man of one book.' He did in fact project a book on his invention, to unite his papers and give further applications of his method; but the constant demands on his time as a clergyman in an English village of the mid-nineteenth century postponed the book indefinitely, and it was never even begun. His life takes us back to an era that now seems almost as remote as the seventeenth century.
- Buddha's Advice to Students and Teachers of Mathematics, The Mathematics Teacher 33 (1940), 252-261.
Reprinted in: The Mathematics Teacher 62 (5) (1969), 373-383.
Stripped of all false trappings of respectability, the chief objection to what I said about the teaching of elementary mathematics stands forth without a rag to cover its nudity in the following somewhat startling proposition: It is good for young people to be taught lies. Now we may or may not agree with that proposition as a sound principle of pedagogy. But whether we agree or disagree, it is a readily verifiable fact that some of the most influential organizations of teachers in the history of civilization have proceeded deliberately on that very proposition. The teachers themselves - all except the incorrigible dupes who would believe in the flatness of the earth if so ordered - have known that what they taught was not so. Yet they have "honestly" believed that they were doing the best possible thing for their pupils, "under the circum stances." The "honestly" is in quotes be cause a candid analysis might reveal the well known fact that those who dictate the finances have a great deal to do with our moral convictions. Among "the circum stances" usually cited as a justification for filling children with Mother Goose rhymes or similar nonsense instead of offering them some sound, useful mathematics and science, one of the tritest is that "the pupils are not sufficiently mature to accept any thing closer to the facts." Possibly; but I have yet to meet the adolescent who still believes that babies are found by their mammas in cabbage patches, while I know scores of teachers of all sorts, from the grades to the university, who believe that mathematics is an image of the Eternal Truth, and who teach the subject accordingly.
It is not necessary, of course, to go out of one's way to tempt the vigilante committee by inciting the young to think for themselves on social problems; the habit of independent, critical thinking can be inculcated by quite innocuous examples. The rest, including vigilante committees, will then be taken care of automatically. This may all sound visionary and Utopian ; but history is our witness. One lump - a small fraction of one percent of any generation - time after time has leavened the whole lot. Is education so impotent that it cannot turn out one free and independent mind for every ten thousand robots? Doubtless some have been wondering what the title of this article has to do with its content. Buddha's parting injunction to his followers is said to have been this: "Believe nothing on hearsay. Do not believe in traditions because they are old, or in anything on the mere authority of my self or any other teacher." Mathematics might be more valuable than it is if it were taught and learned in that spirit.
- Newton After Three Centuries, The American Mathematical Monthly 49 (9) (1942), 553-575.
Christmas Day, 1942 is the three-hundredth anniversary of the birth of Isaac Newton. The two-hundredth anniversary of Newton's death was suitably commemorated in 1927 in nearly every civilized country of the world. To estimate adequately the influence of this unique mind on our present civilization would require the labours of several men, and incidentally traverse much of the histories of the astronomical and physical sciences, of industry and engineering, and of philosophic thought during the last two centuries. Until such an estimate, worthy of the man, is undertaken, a short survey of some major items of Newton's work that are still vital in science and mathematics, with a glance at what has been abandoned, and a brief indication of present problems originating in his work, may be of passing interest. Few of even the greatest men of science have left the world much that retained its full life for more than two centuries after they had died. Newton's work, though modified in detail, continues to inspire men gifted with some of his genius to carry it on, and to extend its scope beyond anything he knew or could have imagined. This is his immortality. Newton was a man of three masterpieces: the calculus, the Opticks, the Principia. The few items of his work noted here may be conveniently classified under pure mathematics, optics, gravitation, metaphysics, all with the limitation stated above.
By 1942, western civilization had experienced three centuries, more or less, of the modern science which developed from the experimental-mathematical method of Galileo and Newton. Among other things this science has taught open minds is a decent humility in the presence of nature. The old assurances and arrogances are gone; the universe is not a book to be read in a cloister, nor is the solar system the simple parish it was in the middle ages. If little is known now, less was known then. Yet the majority, if the choice were theirs, would probably return to the centuries before Galileo and Newton were born. Not all are envious reactionaries; many sincerely and ignorantly believe that science has showered the world with a wealth of material comforts while robbing it of what they call spiritual values. If they ever think at all about their place and function in society, scientists may be inclined to overestimate their importance as shapers of public opinion and educators of the mass of mankind. The mass of mankind knows next to nothing of them or their work, and if it knew more, might think even less. What little it does know fosters a sullen distrust. Men who hold their hypotheses lightly or who, like Newton, glory that they frame none, are not popular. They never were. And while science goes its indifferent way, the world it would serve yearns for the futilities of a nostalgic humanism that knew better days three hundred years ago, and surrenders its intelligence to the unreason of credulous mysticisms. If the world is to abandon science and return to the past, somewhere on its way in the next three centuries Newton's estimate of his scientific work will be confirmed; and his Observations upon the Prophecies of Daniel and the Apocalypse of John, with The Chronology of Ancient Kingdoms amended, on which he lavished his intellectual powers in the latter half of his life, will outlive the Opticks and the Principia.
- Gauss and the Early Development of Algebraic Numbers, Part 1, National Mathematics Magazine 18 (5) (1944), 188-204.
Gauss and the Early Development of Algebraic Numbers, Part 2, National Mathematics Magazine 18 (6) (1944), 219-233.
An unexpected turn in twentieth-century mathematics was the abrupt change in the motivation and objectives of algebra. The change became evident by 1925 at the latest, and in about ten years made some of the algebra of the nineteenth and early twentieth centuries seem rococo and strangely antiquated to algebraists of the younger generation. The transition from individually developed theories, overloaded with masses of intricate theorems - often the seemingly fortuitous outcome of elaborate calculations carried through with consummate manipulative skill - to the deliberate search for unifying abstract principles was sudden. It did not occur without preparation, of course; but the passage from calculation to preoccupation with fundamental concepts was accomplished within a decade. Without the vast accumulations of special results, like the data in Darwin's notebooks which suggested the theory of evolution, the problems and methods of abstract algebra might never have emerged. However that may be, interest in the algoristic type of algebra declined rapidly after 1921. Theorems that had been obtained primarily by modes of calculation germane to the particular theory in which the theorems originated, were seen to be instances of underlying structures independent of the particular theory. To cite but one striking example, the Jordan-Holder theorem and its refinements became more intuitive when analyzed structurally than they had seemed in their traditional settings. Actually the change might have happened earlier than it did. Hilbert's basis theorem') of 1890, in both its statement and its proof, was plainly in an order of ideas different from that of the algebra of its epoch. Yet earlier, in 1877, Dedekind clearly formulated the strategy of abstract algebra, in a statement of the methodology which he had applied in developing his theory of ideals. Having noted that his initial attempt at a general theory of algebraic numbers could, conceivably, be successfully completed, but only (he believed) by overcoming serious obstacles in calculation, Dedekind continued as follows: "But even if a theory [does not encounter great difficulties when developed algoristically], it seems to me that such a theory, based on calculation, still does not offer the highest degree of perfection. As in the modern theory of functions, it is preferable to seek to extract proofs, not from calculation, but immediately from characteristic fundamental concepts, and to construct the theory in such a manner that, on the contrary, it shall be in a position to predict the results of calculation." That declaration of intention might well stand as the party manifesto of the modern abstract algebraists. Dedekind himself gave numerous and brilliant examples of his grand strategy. A typical simple specimen, which has passed unaltered into current usage, is his postulational definition of an ideal, and there are many others.
- What Mathematics Has Meant to Me, Mathematics Magazine 24 (3) (1951), 161.
My interest in mathematics began with two school prizes, one in Greek, the other for physical laboratory, both richly bound in full calf. The Greek prize was Clerk Maxwell's classic on electricity and magnetism, the other, Homer's Odyssey. My cousin got the prize for Greek, I got the other. He read mine, I tried, and failed, to read his. The integral signs were particularly baffling to one who had not gone beyond the binomial theorem for a positive integral exponent. The calculus was not a school subject at the time, so my mother paid for private lessons from a man - the late E M Langley - who was the best teacher I ever had. From him I learned what
dy/dxand y dx mean. The rest was comparatively easy, and I found myself in possession of a key that unlocks a hundred doors. Although I have never done anything in mathematical physics, I have been able to read some of the great classics which, without the calculus, would have been incomprehensible. This has been one thing that has made life interesting. How some philosophers of science and others have the audacity to write on relativity and the quantum theory without a reading knowledge of the calculus is the wonder of the ages. Another thing I got from mathematics has meant more to me than I can say. No man who has not a decently sceptical mind can claim to be civilized. Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions. Then, in the alleged proof, be alert for inexplicit assumptions. Euclid's notorious oversights drove this lesson home. Thanks to him, I am (I hope!) immune to all propaganda, including that of mathematics itself. Mathematical 'truth' is no 'truer' than any other, and Pilate's question is still meaningless. There are no absolutes, even in mathematics.