by E. T. Bell
E. T. BELL
THEY SAY, WHAT THEY SAY, LET THEM SAY
(Motto of Marischal College, Aberdeen)
The science of Pure Mathematics, in its modern developments, may claim to be the most original creation of the human spirit.—A. N. WHITEHEAD (Science and the Modern World, 1925)
A mathematical truth is neither simple nor complicated in itself, it is.—ÉMILE LEMOINE
A mathematician who is not also something of a poet will never be a complete mathematician.—KARL WEIERSTRASS
I have heard myself accused of being an opponent, an enemy of mathematics, which no one can value more highly than I, for it accomplishes the very thing whose achievement has been denied me.—GOETHE
Mathematicians are like lovers. . . . Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another.—FONTENELLE
It is easier to square the circle than to get round a mathematician.—AUGUSTUS DE MORGAN
I regret that it has been necessary for me in this lecture to administer such a large dose of four-dimensional geometry. I do not apologise, because I am really not responsible for the fact that nature in its most fundamental aspect is four-dimensional. Things are what they are . . . .—A. N. WHITEHEAD (The Concept of Nature, 1920)
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Number rules the universe.—THE PYTHAGOREANS
Mathematics is the Queen of the Sciences, and Arithmetic the Queen of Mathematics.—C. F. GAUSS
Thus number may be said to rule the whole world of quantity, and the four rules of arithmetic may be regarded as the complete equipment of the mathematician.—JAMES CLERK MAXWELL
The different branches of Arithmetic—Ambition, Distraction, Uglification, and Derision.—THE MOCK TURTLE (Alice in Wonderland)
God made the integers, all the rest is the work of man.—LEOPOLD KRONECKER
[Arithmetic] is one of the oldest branches, perhaps the very oldest branch, of human knowledge; and yet some of its most abstruse secrets lie close to its tritest truths.—H. J. S. SMITH
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Plato’s writings do not convince any mathematician that their author was strongly addicted to geometry. . . . We know that he encouraged mathematics. . . . But if—which nobody believes—the [Let no man ignorant of geometry enter] of Tzetzes had been written over his gate, it would no more have indicated the geometry within than a warning not to forget to bring a packet of sandwiches would now give promise of a good dinner.—AUGUSTUS DE MORGAN
There is no royal road to geometry.—MENAECHMUS (to ALEXANDER THE GREAT)
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He studied and nearly mastered the six books of Euclid since he was a member of Congress.
He began a course of rigid mental discipline with the intent to improve his faculties, especially his powers of logic and language. Hence his fondness for Euclid, which he carried with him on the circuit till he could demonstrate with ease all the propositions in the six books; often studying far into the night, with a candle near his pillow, while his fellow-lawyers, half a dozen in a room, filled the air with interminable snoring.—ABRAHAM LINCOLN (Short Autobiography, 1860)
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Strange as it may sound, the power of mathematics rests on its evasion of all unnecessary thought and on its wonderful saving of mental operations.—ERNST MACH
A single curve, drawn in the manner of the curve of prices of cotton, describes all that the ear can possibly hear as the result of the most complicated musical performance. . . . That to my mind is a wonderful proof of the potency of mathematics.—LORD KELVIN
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The mathematician, carried along on his flood of symbols, dealing apparently with purely formal truths, may still reach results of endless importance for our description of the physical universe.—KARL PEARSON
Examples . . . which might be multiplied ad libitum, show how difficult it often is for an experimenter to interpret his results without the aid of mathematics.—LORD RAYLEIGH
But there is another reason for the high repute of mathematics: it is mathematics that offers the exact natural sciences a certain measure of security which, without mathematics, they could not attain.—ALBERT EINSTEIN
Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field. For this reason a book on the new physics, if not purely descriptive of experimental work, must be essentially mathematical.—P. A. M. DIRAC (Quantum Mechanics, 1930)
As I proceeded with the study of Faraday, I perceived that his method of conceiving the phenomena [of electromagnetism] was also a mathematical one, though not exhibited in the conventional form of mathematical symbols. I also found that these methods were capable of being expressed in the ordinary mathematical forms, and thus compared with those of the professed mathematicians.—JAMES CLERK MAXWELL (A Treatise on Electricity and Magnetism, 1873)
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Query 64 . . . . Whether mathematicians . . . have not their mysteries, and, what is more, their repugnances and contradictions?—BISHOP BERKELEY
To create a healthy philosophy you should renounce metaphysics but be a good mathematician.—BERTRAND RUSSELL (in a lecture, 1935)
Mathematics is the only good metaphysics.—LORD KELVIN
How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality?—ALBERT EINSTEIN (1920)
Every new body of discovery is mathematical in form, because there is no other guidance we can have.—C. G. DARWIN (1931)
The infinite! No other question has ever moved so profoundly the spirit of man.—DAVID HILBERT (1921)
The notion of infinity is our greatest friend; it is also the greatest enemy of our peace of mind. . . . Weierstrass taught us to believe that we had at last thoroughly tamed and domesticated this unruly element. Such however is not the case; it has broken loose again. Hilbert and Brouwer have set out to tame it once more. For how long? We wonder.—JAMES PIERPONT (Bulletin of the American Mathematical Society, 1928)
In my opinion a mathematician, in so far as he is a mathematician, need not preoccupy himself with philosophy—an opinion, moreover, which has been expressed by many philosophers.—HENRI LEBESGUE (1936)
God ever geometrizes.—PLATO
God ever arithmetizes.—C. G. J. JACOBI
The Great Architect of the Universe now begins to appear as a pure mathematician.—J. H. JEANS (The Mysterious Universe, 1930)
Mathematics is the most exact science, and its conclusions are capable of absolute proof. But this is so only because mathematics does not attempt to draw absolute conclusions. All mathematical truths are relative, conditional.—CHARLES PROTEUS STEINMETZ (1923)
It is a safe rule to apply that, when a mathematical or philosophical author writes with a misty profundity, he is talking nonsense.—A. N. WHITEHEAD (1911)
CHAPTER ONE
Introduction
THIS SECTION IS HEADED Introduction rather than Preface (which it really is) in the hope of decoying habitual preface-skippers into reading—for their own comfort—at least the following paragraphs down to the first row of stars before going on to meet some of the great mathematicians. I should like to emphasize first that this book is not intended, in any sense, to be a history of mathematics, or any section of such a history.
The lives of mathematicians presented here are addressed to the general reader and to others who may wish to see what sort of human beings the men were who created modern mathematics. Our object is to lead up to some of the dominating ideas governing vast tracts of mathematics as it exists today and to do this through the lives of the men responsible for those ideas.
Two criteria have been applied in selecting names for inclusion: the importance for modern mathematics of a man’s work; the human appeal of the man’s life and character. Some qualify under both heads, for example Pascal, Abel, and Galois; others, like Gauss and Cayley, chiefly
under the first, although both had interesting lives. When these criteria clash or overlap in the case of several claimants to remembrance for a particular advance, the second has been given precedence as we are primarily interested here in mathematicians as human beings.
Of recent years there has been a tremendous surge of general interest in science, particularly physical science, and its bearing on our rapidly changing philosophical outlook on the universe. Numerous excellent accounts of current advances in science, written in as untechnical language as possible, have served to lessen the gap between the professional scientist and those who must make their livings at something other than science. In many of these expositions, especially those concerned with relativity and the modern quantum theory, names occur with which the general reader cannot be expected to be familiar—Gauss, Cayley, Riemann, and Hermite, for instance. With a knowledge of who these men were, their part in preparing for the explosive growth of physical science since 1900, and an appreciation of their rich personalities, the magnificent achievements of science fall into a truer perspective and take on a new significance.
The great mathematicians have played a part in the evolution of scientific and philosophic thought comparable to that of the philosophers and scientists themselves. To portray the leading features of that part through the lives of master mathematicians, presented against a background of some of the dominant problems of their times, is the purpose of the following chapters. The emphasis is wholly on modern mathematics, that is, on those great and simple guiding ideas of mathematical thought that are still of vital importance in living, creative science and mathematics.
It must not be imagined that the sole function of mathematics—“the handmaiden of the sciences”—is to serve science. Mathematics has also been called “the Queen of the Sciences.” If occasionally the Queen has seemed to beg from the sciences she has been a very proud sort of beggar, neither asking nor accepting favors from any of her more affluent sister sciences. What she gets she pays for. Mathematics has a light and wisdom of its own, above any possible application to science, and it will richly reward any intelligent human being to catch a glimpse of what mathematics means to itself. This is not the old doctrine of art for art’s sake; it is art for humanity’s sake. After all, the whole purpose of science is not technology—God knows we have gadgets enough already; science also explores depths of a universe that will never, by any stretch of the imagination, be visited by human beings or affect our material existence. So we shall attend also to some of the things which the great mathematicians have considered worthy of loving understanding for their intrinsic beauty.
Plato is said to have inscribed “Let no man ignorant of geometry enter here” above the entrance to his Academy. No similar warning need be posted here, but a word of advice may save some over-conscientious reader unnecessary anguish. The gist of the story is in the lives and personalities of the creators of modern mathematics, not in the handful of formulas and diagrams scattered through the text. The basic ideas of modern mathematics, from which the whole vast and intricate complexity has been woven by thousands of workers, are simple, of boundless scope, and well within the understanding of any human being with normal intelligence. Lagrange (whom we shall meet later) believed that a mathematician has not thoroughly understood his own work till he has made it so clear that he can go out and explain it effectively to the first man he meets on the street.
This of course is an ideal and not always attainable. But it may be recalled that only a few years before Lagrange said this the Newtonian “law” of gravitation was an incomprehensible mystery to even highly educated persons. Yesterday the Newtonian “law” was a commonplace which every educated person accepted as simple and true; today Einstein’s relativistic theory of gravitation is where Newton’s “law” was in the early decades of the eighteenth century; tomorrow or the day after Einstein’s theory will seem as “natural” as Newton’s “law” seemed yesterday. With the help of time Lagrange’s ideal is not unattainable.
Another great French mathematician, conscious of his own difficulties no less than his readers’, counselled the conscientious not to linger too long over anything hard but to “Go on, and faith will come to you.” In brief, if occasionally a formula, a diagram, or a paragraph seems too technical, skip it. There is ample in what remains.
Students of mathematics are familiar with the phenomenon of “slow development,” or subconscious assimilation: the first time something new is studied the details seem too numerous and hopelessly confused, and no coherent impression of the whole is left on the mind. Then, on returning after a rest, it is found that everything has fallen into place with its proper emphasis—like the development of a photographic film. The majority of those who attack analytic geometry seriously for the first time experience something of the sort. The calculus on the other hand, with its aims clearly stated from the beginning, is usually grasped quickly. Even professional mathematicians often skim the work of others to gain a broad, comprehensive view of the whole before concentrating on the details of interest to them. Skipping is not a vice, as some of us were told by our puritan teachers, but a virtue of common sense.
As to the amount of mathematical knowledge necessary to understand everything that some will wisely skip, I believe it may be said honestly that a high school course in mathematics is sufficient. Matters far beyond such a course are frequently mentioned, but wherever they are, enough description has been given to enable anyone with high school mathematics to follow. For some of the most important ideas discussed in connection with their originators—groups, space of many dimensions, non-Euclidean geometry, and symbolic logic, for example —less than a high school course is ample for an understanding of the basic concepts. All that is needed is interest and an undistracted head. Assimilation of some of these invigorating ideas of modern mathematical thought will be found as refreshing as a drink of cold water on a hot day and as inspiring as any art.
To facilitate the reading, important definitions have been repeated where necessary, and frequent references to earlier chapters have been included from time to time.
The chapters need not be read consecutively. In fact, those with a speculative or philosophical turn of mind may prefer to read the last chapter first. With a few trivial displacements to fit the social background the chapters follow the chronological order.
It would be impossible to describe all the work of even the least prolific of the men considered, nor would it be profitable in an account for the general reader to attempt to do so. Moreover, much of the work of even the greater mathematicians of the past is now of only historical interest, having been included in more general points of view. Accordingly only some of the conspicuously new things each man did are described, and these have been selected for their originality and importance in modern thought.
Of the topics selected for description we may mention the following (among others) as likely to interest the general reader: the modern doctrine of the infinite (chapters 2, 29); the origin of mathematical probability (chapter 5); the concept and importance of a group (chapter 15); the meanings of invariance (chapter 21); non-Euclidean geometry (chapter 16 and part of 14); the origin of the mathematics of general relativity (last part of chapter 26); properties of the common whole numbers (chapter 4), and their modern generalization (chapter 25); the meaning and usefulness of so-called imaginary numbers—like (chapters 14, 19); symbolic reasoning (chapter 23). But anyone who wishes to get a glimpse of the power of the mathematical method, especially as applied to science, will be repaid by seeing what the calculus is about (chapters 2, 6).
Modern mathematics began with two great advances, analytic geometry and the calculus. The former took definite shape in 1637, the latter about 1666, although it did not become public property till a decade later. Though the idea behind it all is childishly simple, yet the method of analytic geometry is so powerful that very ordinary boys of seventeen can use it to prove results which would have baffled the greatest o
f the Greek geometers—Euclid, Archimedes, and Apollonius. The man, Descartes, who finally crystallized this great method had a particularly full and interesting life.
In saying that Descartes was responsible for the creation of analytic geometry we do not mean to imply that the new method sprang fullarmed from his mind alone. Many before him had made significant advances toward the new method, but it remained for Descartes to take the final step and actually to put out the method as a definitely workable engine of geometrical proof, discovery, and invention. But even Descartes must share the honor with Fermat.
Similar remarks apply to most of the other advances of modern mathematics. A new concept may be “in the air” for generations until some one man—occasionally two or three together—sees clearly the essential detail that his predecessors missed, and the new thing comes into being. Relativity, for example, is sometimes said to have been the great invention reserved by time for the genius of Minkowski. The fact is, however, that Minkowski did not create the theory of relativity and that Einstein did. It seems rather meaningless to say that So-and-so might have done this or that if circumstances had been other than they were. Any one of us no doubt could jump over the moon if we and the physical universe were different from what we and it are, but the truth is that we do not make the jump.
In other instances, however, the credit for some great advance is not always justly placed, and the man who first used a new method more powerfully than its inventor sometimes gets more than his due. This seems to be the case, for instance, in the highly important matter of the calculus. Archimedes had the fundamental notion of limiting sums from which the integral calculus springs, and he not only had the notion but showed that he could apply it. Archimedes also used the method of the differential calculus in one of his problems. As we approach Newton and Leibniz in the seventeenth century the history of the calculus becomes extremely involved. The new method was more than merely “in the air” before Newton and Leibniz brought it down to earth; Fermat actually had it. He also invented the method of Cartesian geometry independently of Descartes. In spite of indubitable facts such as these we shall follow tradition and ascribe to each great leader what a majority vote says he should have, even at the risk of giving him a little more than his just due. Priority after all gradually loses its irritating importance as we recede in time from the men to whom it was a hotly contested cause of verbal battles while they and their partisans lived.