Category Archives: mathematicians

On Georg Cantor: Paradise Lost: E T Bell’s views

Mathematics, like all other subject, has now to take its turn under the microscope and reveal to the world any weaknesses there may be in its foundations. F. W. Westaway.

The controversial topic of Mengenlehre (theory of sets, or classes, particularly of infinite sets) created in 1874-1895 by Georg Cantor (1945-1918) may well be taken, out of its chronological order, as the conclusion of the whole story. This topic typifies for mathematics the general collapse of those principles which the prescient seers of the nineteenth century, foreseeing everything but the grand debacle, believed to be fundamentally sound in all things from physical science to democratic government.

If “collapse” is perhaps too strong to describe the transition the world is doing its best to enjoy, it is nevertheless true that the evolution of scientific ideas is now proceeding so vertiginously that evolution is barely distinguishable from revolution.

Without the errors of the past as a deep-seated focus of disturbance the present upheaval in physical science would perhaps not have happened; but to credit our predecessors with all the inspiration for what our own generation is doing, is to give them more than their due. This point is worth a moment’s consideration, as some may be tempted to say that the corresponding “revolution” in mathematical thinking, whose beginnings are now plainly apparent, is merely an echo of Zeno and other doubters of ancient Greece.

The difficulties of Pythagoras over the square root of 2 and the paradoxes of Zeno on continuity (or “infinite divisibility”) are — so far as we know — the origins of our present mathematical schism. Mathematicians today who pay any attention to the philosophy (or foundations) of their subject are split into at least two factions, apparently beyond present hope of reconciliation, over the validity of the reasoning used in mathematical analysis, and this disagreement can be traced back through the centuries to the Middle Ages and thence to ancient Greece. All sides have had their representatives in all ages of mathematical thought, whether that thought was disguised in provocative paradoxes as with Zeno, or in logical subtleties, as with some of the most exasperating logicians of the Middle Ages. The root of these differences is commonly accepted by mathematicians as being a matter of temperament: any attempt to convert an analyst like Weierstrass to the skepticism of a doubter like Kronecker is bound to be as futile as trying to convert a Christian fundamentalist to rabid atheism.

A few dated quotations from leaders in a dispute may serve as a stimulant — or, sedative according to taste — for our answer to the singular intellectual career of Georg Cantor, whose “positive theory of the infinite” precipitated in our generation the fiercest frog mouse battle (as Einstein once called it) in history over the validity of traditional mathematical reasoning.

In 1831 Gauss expressed his “horror of the actual infinite” as fpllows: “I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction.”

Thus, if x denotes a real number, the fraction 1/x diminishes as x increases, and we can find a value of x such that 1/x differs from zero y any preassigned amount (other than zero) which may be as small as we please, and as x continues to increase, the difference remains less than ths preassigned amount; the limit of 1/x, “as x tends to infinity,” is zero. The symbol of infinity \infty ; the assertion \frac{1}{\infty}=0 is nonsensical for two reasons:”division by infinity” is an operation which is undefined, and hence, has no meaning; the second reason was stated by Gauss. Similarly, the symbol \frac{1}{0} = \infty is meaningless.

Cantor agrees and disagrees with Gauss. Writing in 1886 on the problem of the actual (what Gauss called completed) infinite, Cantor says that “in spite of the essential difference between the concepts of the potential and the actual infinite, the former meaning a variable finite magnitude increasing beyond all finite limits (like x in 1/x above), while the latter is a fixed, constant magnitude lying beyond all finite magnitudes, it happens only too often that they are confused.”

Cantor goes on to state that misuse of the infinite in mathematics had justly inspired a horror of the infinite among careful mathematicians of his day, precisely as it did in Gauss. Nevertheless, he maintains that the resulting “uncritical rejection of the legitimate actual infinite is no lesser a violation of the nature of things (whatever that may be — it does not appear to have been revealed to mankind as a whole), which must be taken as they are”—- however, that may be. Cantor thus definitely aligns himself with the great theologians of the Middle Ages, of whom he was a deep student and an ardent admirer.

Absolute certainties and complete solutions of age-old problems always go down better if well salted before swallowing. Here is what Bertrand Russell had to say in 1901 about Cantor’s Promethean attack on the infinite.

“Zeno was concerned with three problems…These are the problem of the infinitesimal, the infinite, and continuity….From his day to our own, the finest intellects of each generation in turn attacked these problems, but achieved, broadly speaking, nothing. …Weierstrass, Dedekind, and Cantor …have completely solved them. Their solutions…are so clear as to leave no longer the slightest doubt of difficulty. This achievement is probably the greatest of which the age can boast…The problem of the infinitesimal was solved by Weierstrass, the solutions of the other two was begun by Dedekind and definitely accomplished by Cantor.”

The enthusiasm of this passage warms us even today, although we know that Russell in the second edition of his and Whitehead’s Principia Mathematica admitted that all was not well with the Dedekind “cut”, which is the spinal cord of analysis. Nor is it well today. More is done for or against a particular creed in science or mathematics in a decade than was accomplished in a century of antinquity, the Middle Ages, or the late renaissance. More good minds attack on outstanding scientific or mathematical problem today than ever before, and finality has become the private property of fundamentalists. Not one of the finalities in Russell’s remarks of 1901 has survived. A quarter of a century agon those who were unable to see the great light which the prophets assured them was blazing overhead like the noonday sun in a midnight sky were called merely stupid. Today for every competent expert on the side of the prophets there is an equally competent and opposite expert against them. If there is stupidtiy anywhere it is so evenly distributed that it has ceased to be a mark of distinction. We are entering a new era, one of doubt and decent humility.

On the doubtful side about the same time (1905) we find Poincare “I have spoken of …of our need to return continually to the first principles of our science, and of the advantages of this for the study of the human mind. This need has inspired two enterprises which have assumed a very prominent place in the most recent development of mathematics. The first is Cantorism…Cantor introduced into science a new way of considering the mathematical infinite…but it has come about that we have encountered certain paradoxes, certain apparent contradictions that would have delighted Zeno the Eleatic and the school of Megara. So each must seek the remedy. I, for my part, and I am not alone — think that the important thing is never to indtroduce entities not completely definable in a finite number of words. Whatever be the cure adopted, we may promise ourselves the joy of the physician called in to treat a beautiful pathologic case.”

A few years later Poincare’s interest in pathology for its own sake had abated somewhat. At the International Mathematical Congress of 1908 at Rome, the satiated physician delivered himself of this prognosis: “Later generations will regard Mengenlehre as a disease from which one has recovered.”

It was Cantor’s greatest merit to have discovered in spite of himself and against his own wishes in the matter that the “body mathematic” is profoundly diseased and that the sickness with which Zeno infected it has not yet been alleviated. His disturbing discovery is a curious echo of his own intellectual life. We shall first glance at his material existence, not of much interest in themselves, perhaps, but singularly illuminative in their later aspects of his theory.

Of pure Jewish descent on both sides, Georg Ferdinand Ludwig Philipp Cantor was the first child of the prosperous merchant Georg Waldemar Cantor and his artistic wife Maria Bohm. The father was born in Copenhagen, Denmark, but migrated as a young man to St. Petersburg, Russia, where the mathematician Georg Cantor was born on March 3, 1845. Pulmonary disease caused the father to move in 1856 to Frankfurt, Germany where he lived in comfortable retirement till his death in 1863. From this curious medley of nationalities, it is possible for several fatherlands to call Cantor as their son. Cantor himself favoured Germany, but it cannot be said that Germany favoured him very cordially.

Georg had a brother Constantin, who became a German army officer (very few Jews ever did), and a sister Sophie Nobiling. The brother was a fine pianist; the sister an accomplished designer. Georg’s pent-up artistic nature found its turbulent outlet in mathematics and philosophy, both classical and scholastic. The marked artistic temperaments of the children were inherited from their mother, whose grandfather was a musical conductor, one of whose brothers, living in Vienna, taught the celebrated violinist Joachim. A brother of Maria Cantor was a musician, and one of her nieces a painter. If it is true, as claimed by the psychological proponents of drab mediocrity, that normality and phlegmatic stability are equivalent, all this artistic brilliance in his family may have been the root of Cantor’s instability.

The family were Christians, the father having been converted to Protestantism; the mother was a born Roman Catholic. Like his arch enemy Kronecker, Cantor favoured the Protestant side and acquired a singular taste for the endless hairsplitting of medieval theology. Had he not become a mathematician it is quite possible that he would have left his mark on theology or philosophy. As an item of interest that may be noted in this connection, Cantor’s theory of the infinite was eagely pounced upon by the Jesuits, whose keen logical minds detected in the mathematical imagery beyong their theological comprehension indubitable proofs of the existence of God and the self-consistency of the Holy Trinity with its three-in-one, one-in-three, co-equal and co-eternal. Mathematics has strutted to some pretty queer tunes in the past 2500 years, but this takes the cake. It is only fair to say that Cantor, who had a sharp wit and a sharper tongue when he was angered, ridiculed the pretentious absurdity of such “proofs,” devout Christian and expert theologian though he himself was.

Cantor’s school career was like that of most highly gifted mathematicians — an early recognition (before the age of fifteen) of his greatest talent and an absorbing interest in mathemtical studies. His first instruction was under a private tutor, followed by a course in an elementary school in St. Petersburg. When the family moved to Germany, Cantor first attended private schools at Frankfurt and the Darmstadt nonclassical school, entering the Wiesbaden Gymnasium in 1860 at the age of fifteen.

Georg was determined to become a mathematician, but his practical father, recognizing the boy’s mathematical ability, obstinately tried to force him into engineering as a more promising bread-and-butter profession. On the occasion of Cantor’s confirmation in 1860, his father wrote to him expressing the high hopes he and all Georg’s numerous aunts, uncles, and cousins in Germany, Denmark, and Russia had placed on the gifted boy””They expect from you nothing less than that you become a Theodor Schaeffer and later, perhaps, if God so wills, a shining star in the engineering firmament. ” When will parents recognize the presumptuous stupidity of trying to make a cart horse out of a born racer?

The pious appeal to God which was intended to blackjack the sensitive, religious boy of fifteen into submission in 1860 would today (thank God!) rebound like a tennis ball from the harder heads of our own younger generation. But it hit Cantor pretty hard. In fact, it knocked him out cold. Loving his father devotedly and being of a deeply religious nature, young Cantor could not see that the old man was merely rationalizing his own absurd ambition. Thus began the warping of Georg Cantor’s acutely sensitive mind. Instead of rebelling, as a gifted boy today might do with some hope of success, Georg submitted till it became apparent even to the obstinate father that he was wrecking his son’s disposition. But in the process of trying to please his father against the promptings of his own instincts Georg Cantor sowed the seeds of the self-distrust which was to make him an easy victim for Kronecker’s vicious attack in later life and cause him to doubt the value of his work. Had Cantor been brought up as an independent human being he would never have acquired the timid deference to men of established reputation which made his life wretched.

The father gave in the mischief was already done. On Georg’s completion of his school course with distinction at the age of seventeen, he was permitted by “dear papa” to seek a university career in mathematics. ‘My dear papa!” Georg writes in his boyish gratitude:”You can realize for yourself how greatly your letter delighted me. The letter fixes my future…Now I am happy when I see that it will not displease you if I follow my feelings in the choice. I hope you will live to find joy in me, dear father; since my soul, my whole being, lives in my vocation; what a man desires to do, and that to which an inner compulsion drives him, that will he accomplish!” Papa no doubt deserves a vote of thanks, even if Georg’s gratitude is a shade too servile for a modern taste.

Cantor began his university studies at Zurich in 1862, but migrated to the University of Berlin the following year, on the death of his father. At Berlin he specialized in mathematics, philosophy, and physics. The first two divided his interests about equally; for physics he never had any sure feeling. In mathematics his instructors were Kummer, Weierstrass, and his future enemy Kronecker. Following the usual German custom, Cantor spent a short time at another university, and was in residence for one semester of 1866 at Gottingen.

With Kummer and Kronecker at Berlin the mathematical atmosphere was highly charged with arithmetic. Cantor made a profound study of the Disquisitiones Arithmeticae of Gauss and wrote his dissertation, accepted for the Ph.D. degree in 1867, on a difficult point which Gauss had left aside concerning the solution in integers x, y, z of the indeterminate equation:

ax^{2}+by^{2}+cz^{2}=0

where a, b, c are any given integers. This was a fine piece of work, but it is safe to say that no mathematician who read it anticipated that the conservative author of twenty two was to become one of the most radical originators in the history of mathematics. Talent no doubt is plain enough in his first attempt, but genius —- no. There is not a single hint of the great originator in this severely classical dissertation.

The like may be said for all of Cantor’s earliest work published before he was twenty nine. It was excellent, but might have been done by any brilliant man who had thoroughly absorbed, as Cantor had the doctrine of rigorous proof from Gauss and Weierstrass. Cantor’s first love was the Gaussian theory of numbers, to which he was attracted by the sharp, hard, clear perfection of the proofs. From this, under the influence of the Weierstrassians, he presently branched off into rigorous analysis, particularly in the theory of trigonometric series. (Fourier series).

The subtle difficulties of this theory (where question of convergence of infinite series are less easily approachable than in the theory of power series) seem to have inspired Cantor to go deeper for the foundations of analysis than any of his contemporaries had cared to look, and he was led to his grand attack on the mathematics and philosophy of the infinite itself, which is at the bottom of all questions concerning continuity, limits and convergence. Just before he was thirty, Cantor published his first revolutionary paper (in Crelle’s Journal) on the theory of infinite sets. This will be described presently. The unexpected and paradoxical result concerning the set of all algebraic numbers which Cantor established in this paper and the complete novelty of the methods employed immediately marked the young author as a creative mathematician of extraordinary originality. Whether all agreed that the new methods were sound or not is beside the point: it was universally admitted that a man had arrived with something fundamentally new in mathematics. He should have been given an influential position at once.

Cantor’s material career was that of any of the less eminent German professors of mathematics. He never achieved his ambition of a professorship at Berlin, possibly the highest German distinction during the period of Cantor’s greatest and most original productivity (1874-1884, age twenty nine to thirty nine). All his active professional career was spent at the University of Halle, a distinctly third-rate institution, where he was appointed Privatdozent (a lecturer who lives by what fees he can collect from his students) in 1869 at the age of twenty four. In 1872 he was made assistant professor and in 1879 — before the criticism of his work had begun to assume the complexion of a malicious personal attack on himself — he was appointed full professor. His earliest teaching experience was in a girls’ school in Berlin. For this curiously inappropriate task he had qualified himself by listening to dreary lectures on pedagogy by an uninspired mathematical mediocrity before securing his state license to teach children. More social waste.

Rightly or wrongly, Cantor blamed Kronecker for his failure to obtain the coveted position at Berlin. When two academic specialists disagree violently on purely scientific matters, they have a choice, if discretion seems the better part of valour, of laughing their hatreds of and not making a fuss about them, or of acting in any of the number of belligerent ways that other people resort to when confronted with situations of antagonism. One way is to go at the other in an efficient, underhand manner, which often enables one to gain his spiteful end under the guise of sincere friendship. Nothing of this sort here ! When Cantor and Kronecker fell out, they disagreed all over, threw reserve to the dogs, and did everything but slit the other’s throat. Perhaps all this is a more decent way of fighting — if men must fight — than the sanctimonious hypocrisy of the other. The object of any war is to destroy the enemy, and being sentimental or chivalrous about the unpleasant business is the mark of an incompetent of fighter. Kronecker was one of the most competent warriors in the history of scientific controversy; Cantor, one of the least competent. Kronecker won. But, as will appear later, Kronecker’s bitter animosity toward Cantor was not wholly personal but at least partly scientific and disinterested.

The year 1874 which saw the appearance of Cantor’s first revolutionary paper on the theory of sets was also that of his marriage, at the age of twenty nine, to Vally Guttmann. Two sons and four daughters were born out of this marriage. None of the children inherited their father’s mathematical ability.

On their honeymoon at Interlaken the young couple saw a lot of Dedekind, perhaps the one first rate mathematician of the time who made a serious and sympathetic attempt to understand Cantor’s subversive doctrine.

Himself somewhat of a persona non grata to the leading German overloads of mathematics in the last quarter of the nineteenth century, the profoundly original Dedekind was in a position to sympathize with the scientifically disreputable Cantor. It is sometimes imagined by outsiders that originality is always assured of a cordial welcome in science. The history of mathematics contradicts this happy fantasy; the way of the transgressor in a well established science is likely to be the hard as it is in any other field of human conservatism, even when the transgressor is admitted to have found something valuable by overstepping the narrow bounds of bigoted orthodoxy.

Both Dedekind and Cantor got what they might have expected had they paused to consider before striking out in new directions. Dedekind spent his entire working life in mediocre positions; the claim — now that Dedekind’s work is recognized as one of the most important contributions to mathematics that Germany has ever made — that Dedekind preferred to stay in obscure holes while men who were in no sense his intellectual superiors shone like tin plates in the glory of the public and academic esteem, strikes observers who are themselves “Aryans” but not Germans as highly diluted eyewash.

The ideal of German scholarship in the nineteenth century was the lofty one of a thoroughly coordinated “safety first,” and perhaps rightly it showed an extreme Gaussian caution toward radical originality — the new thing might conceivably be not quite right. After all an honestly edited encyclopaedia is in general a more reliable source of information about the soaring habits of skylarks than a poem, say Shelley’s, on the same topic.

In such an atmosphere of cloying alleged fact, Cantor’s theory of the infinite —- one of the most disturbingly original contributions in mathematics in the past 2500 years — felt about as much freedom as a skylark trying to soar up through an atmosphere of cold glue. Even if the theory was totally wrong — and there are some who believe it cannot be salvaged in any shape resembling the thing Cantor thought he had launched — it deserved something better than the brickbats which were hurled at it chiefly because it was new and unbaptized in the holy name of orthodox mathematics.

The pathbreaking paper of 1874 undertook to establish a totally unexpected and highly paradoxical property of the set of all algebraic numbers. Definition: If r satisfies an algebraic equation of degree n with rational integer (common whole number) coefficients, and if r satisfies no such equation of degree less than n, then r is an algebraic number of degree n.

This can be generalized. For it is easy to prove that any root of an equation of the type

c_{0}x^{n}+c_{1}x^{n-1}+\ldots+c_{n-1}x+c_{n}=0,

in which the c’s are any given algebraic numbers (as defined above ), is itself an algebraic number. For example, according to this theorem, all roots of

(1-3\sqrt{-1})x^{2}-(2+5\sqrt{17})x+\sqrt[3]{90}=0

are algebraic numbers, since the coefficients are. (The first coefficient satisfies x^{2}-2x+10=0, the second, x^{2}-4x-421=0, the third, x^{3}-90=0, of the respective degrees 2, 2, and 3.)

Imagine (if you can) the set of all algebraic numbers. Among these will be all the positive rational integers 1, 2, 3, …, since any one of them, say, n, satisfies an algebraic equation, x-n=0, in which the coefficients (1 and -n) are rational integers. But in addition to these the set of all algebraic equations will include all roots of all quadratic equations with rational integer coefficients, and all roots of all cubic equations with rational integer coefficients, and so on, indefinitely. Is it not intuitively evident that the set of all algebraic numbers will contain infinitely more members than its subset of the rational integers 1, 2, 3, …? It might indeed be so, but it happens to be false.

Cantor proved that the set of all rational integers 1, 2, 3, …contains precisely as many members as the “infinitely more inclusive” set of all algebraic numbers.

A proof of this paradoxical statement cannot be given here, but the kind of device — that of “one-to-one correspondence” — upon which the proof is based can be easily be made intelligible. This should induce in the philosophical mind an understanding of what a cardinal number is. Before describing this simple but somewhat elusive concept it will be helpful to glance at an expression of opinion on this and other definitions of Cantor’s theory which emphasizes a distinction between the attitudes of some mathematicians and many philosophers toward all questions regarding “number” or “magnitude.”

“A mathematician never defines magnitudes in themselves, as a philosopher would be tempted to do; he defines their equality, their sum and their product, and these definitions determine, or rather constitute, all the mathematical properties of magnitudes. In a yet more abstract and more formal manner he lays down symbols and at the same time prescribes the rules according to which they must be combined; these rules suffice to characterize these symbols and to give them a mathematical value. Briefly, he creates mathematical entities by means of arbitrary conventions, in the same way that the several chessmen are defined by the conventions which govern their moves and the relations between them. Not all schools of mathematical thought would subscribe to these opinions, but they suggest at least one “philosophy” responsible for the following definition of cardinal numbers.

Note that the initial stage in the definition is the description of “same cardinal number,” in the spirit of Couturat’s opening remarks; “cardinal number” then arises phoenix-like from the ashes of its “sameness.” It is all a matter of relations between concepts not explicitly defined.

Two sets are said to have the same cardinal number when all the things in the sets can be paired off one-to-one. After the pairing there are to be no unpaired things in either set.

Some examples will clarify this esoteric definition. It is one of those trivially obvious and fecund nothings which are so profound that they are overlooked for thousands of years. The sets \{x,y,z \} and \{ a, b, c\} have the same cardinal number (we shall not commit the blunder of saying, “Of course! Each contains three letters”) because we can pair off the things x, y, z in the first set with those a, b, c in the second as follows: x with a, y with b, z with c, and having done so, find that none remain unpaired in either set. Obviously there are other ways for effecting the pairing. Again, in a Christian community practising technical monogamy, if twenty married couples sit down together to dinner, the set of husbands will have the same cardinal number as the set of wives.

As another instance of this obvious sameness, we recall Galileo’s example of the set of all squares of positive integers and the set of all positive integers:

1^{2}, 2^{2}, 3^{2}, 4^{2}, \ldots , n^{2}, \ldots

1, 2, 3, 4, \ldots, n, \ldots

The “paradoxical” distinction between this and the preceding example is apparent. If all the wives retire to the drawing room, leaving their spouses to sip port and tell stories, there will be precisely twenty human beings sitting at the table, just half as many as there were before. But if all the squares desert the natural numbers, there are just as many left as there were before. Dislike it or not we may (we sashould not, if we are rational animals), the crude miracle stares us in the fact that a part of a set may have the same cardinal number as the entire set. If anyone dislikes the “pairing” definition of “same cardinal number,” he may be challenged to produce a comelier. Intuition (male, female or mathematical) has been greatly overrated. Intuition is the root of all superstitions.

Notice at this stage that a difficulty of the first magnitude has been glossed. What is a set, or a class ? “That,” in the words of Hamlet,, “is the question.” We shall return to it, but we shall not answer it. Whoever succeeds in answering that innocent question to the entire satisfaction of Cantor’s critics will quite likely dispose of the more serious objections against his ingenious theory on the infinite and at the same time establish mathematical analysis on a non-emotional basis. To see that the difficulty is not trivial, try to imagine the set of all positive rational integers 1, 2, 3, …, and ask yourself whether, with Cantor, you can hold this totality — which is a class — in your mind as a definite object of thought, as easily apprehended as the class x, y, z of three letters. Cantor requires us to do just this thing in order to reach the transfinite numbers which he created.

Proceeding now to the definition of “cardinal number,” we introduce a convenient technical term; two sets or classes whose members can be paired off one-to-one (as in the examples given previously) are said to be similar. How many things are there in the set (or class) x, y, z? Obviously three. But what is “three”? An answer is contained in the following definition: “The number of things in a given class is the class of all classes that are similar to the given class.”

This definition gains nothing from attempted explanation: it must be grasped as it is. It was proposed in 1879 by Gottlob Frege, and again (independently) by Bertrand Russell in 1901. One advantage which it has over other definition of “cardinal number of a class” is its applicability to both finite and infinite classes. Those who believe the definition too mystical for mathematics can avoid it by following Couturat’s advice and not attempting to define “cardinal number.” However, that way also leads to difficulties.

Cantor’s spectacular result that the class of all algebraic numbers is similar (in the technical sense defined above) to its subclass of all the positive rational integers was but the first of many wholly unexpected properties of infinite classes. Granting for the moment that his reasoning in reaching these properties is sound, or, if not unobjectionable in the form in which Cantor left it, that it can be made rigorous, we must admit its power.

Consider for example the “existence” of transcendental numbers. It is known that it cost Hermite a tremendous effort to prove the transcendence of a particular number of this kind. Even today there is no general method known whereby the transcendence of any number which we suspect is transcendental can be proved; each new type requires the invention of special and ingenious methods. It is suspected. for example, that the number (it is a constant, although it looks as if it might be a variable from its definition) which is defined as the limit of

1+\frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n} -\log{n}

as n tends to infinity is transcendental, but we cannot prove that it is. What is required is to show that this constant is not a root of any algebraic equation with rational integer coefficients.

All this suggests the question “How many transcendental number are there/” Are they more numerous than the integers, or the rationals, or the algebraic numbers as a whole, or are they less numerous? Since (by Cantor’s theorem) the integers, the rationals, and all algebraic numbers are equally numerous, the question amounts to this: can the transcendental numbers be counted off 1, 2, 3, …? Is the class of all transcendental numbers similar to the class of all positive rational integers? The answer is no; the transcendentals are infinitely more numerous than the integers.

Here we begin to get into the controversial aspects of the theory of sets. The conclusion just stated was like a challenge to a man of Kronecker’s temperament. Discussing Lindemann’s proof that $latex $\pi$ is transcendental, Kronecker had asked, “Of what use is your beautiful investigation regarding \pi? Why study such problems, since irrational (and hence, transcendental numbers) do not exist? ” We can imagine the effect of such a skepticism on Cantor’s proof that transcendentals are infinitely more numerous than the integers 1, 2, 3, …, which, according to Kronecker, are the noblest work of God and the only numbers that do exist.

Even a summary of Cantor’s proof is out of the question here, but something of the kind of reasoning he used can be seen from the following simple considerations. If a class is similar (in the above technical sense) to the class of all positive rational integers, the class is said to be denumerable. The things in a denumerable class can be counted off 1, 2, 3, …; the things in a non-denumerable class cannot be counted off 1, 2, 3, ….; there will be more things in a nondenumerable class than in a denumerable class. Do non-denumerable classes exist? Cantor proved that they do. In fact, the class of all points on any line-segment, no matter how small the segment is (provided it is more than a single point), is non-denumerable.

From this we see a hint of why the transcendentals are non-denumerable. We assume we know that any root of any algebraic equation is representable by a point on the plane of Cartesian geometry. All these roots constitute the set of all algebraic numbers, which Cantor proved to be denumerable. But if the points on a mere line segment are non-denumerable, it follows that all the points on the Cartesian plane are likewise non-denumerable. The algebraic numbers are spotted over the plane like stars against a black sky; the dense blackness is the firmament of the transcendentals.

The most remarkable thing about Cantor’s proof is that it provides no means whereby a single one of the transcendentals can be constructed. To Kronecker any such proof was sheer nonsense. Much milder instances of “existence proofs” roused his wrath. One of these in particular is of interest as it prophesied Brouwer’s objection to the full use of classical (Aristotelean logic) in reasoning about infinite sets.

A polynomial ax^{n}+bx^{n-1}+\ldots+l in which coefficients a, b, …, l are rational numbers is said to be irreducible if it cannot be factored into a product of two polynomials both of which have rational number coefficients. Now, it is a meaningful statement to most human beings to assert, as Aristotle would, that a given polynomial either is irreducible or is not irreducible.

Not so for Kronecker. Until some definite process, capable of being carried out in a finite number of nontentative steps, is provided whereby we can settle the reducibility of any given polynomial, we have no logical right, according to Kronecker, to use the concept of irreducibility in our mathematical proofs. To do otherwise, according to him, is to court inconsistencies in our conclusions and, at best, the use of “irreducibility” without the process described, can give us only a Scotch verdict of “not proven.” All such non-constructive reasoning is — according to Kronecker — illegitimate.

As Cantor’s reasoning in this theory of infinite classes is largely non-constructive, Kronecker regarded it as a dangerous type of mathematical insanity. Seeing mathematics headed for the madhouse under Cantor’s leadership, and being passionately devoted to what he considered the truth of mathematics, Kronecker attacked “the positive theory of infinity” and its hypersensitive author vigorously and viciously with every weapon that came to his hand, and the tragic outcome was that not the theory of sets went to the asylum, but Cantor. Kronecker’s attack broke the creator of the theory.

In the spring of 1884, in his fortieth year, Cantor experienced the first of those complete breakdowns which were to recur with varying intensity throughout the rest of his long life and drive him society to the shelter of a mental clinic. His explosive temper aggravated his difficulty. Profound fits of depression humbled himself in his own eyes and he came to doubt the soundness of his work. During one lucid interval he begged the authorities at Halle to transfer him from his professorship of mathematics to a chair of philosophy. Some of his best work on the positive theory of the infinite was done in intervals between one attack and the next. On recovering from a seizure he noticed that his mind became extraordinarily clear.

Kronecker perhaps has been blamed too severely for Cantor’s tragedy: his attack was but one of many contributing causes. Lack of recognition embittered the man who believed he had taken the first — and last — steps towards a rational theory of the infinite and he brooded himself into melancholia and irrationality. Kronecker however does appear to have been largely responsible for Cantor’s failure to obtain the position he craved in Berlin. It is usually considered not quite sporting for one scientist to deliver a savage attack on the work of a contemporary to his students. The disagreement can be handled objectively in scientific papers. Kronecker laid himself out in 1891 to criticize Cantor’s work to his students at Berlin, and it became obvious that there was no room for both under one roof. As Kronecker was already in possession, Cantor resigned himself to staying out in the cold.

However, he was not without some comfort. The sympathetic Mittag-Leffler not only published some of Cantor’s work in his journal (Acta Mathematica) but comforted Cantor in his fight against Kronecker. In one year alone Mittag-Leffler received no less than fifty two letters from the suffering Cantor. Of those who believed in Cantor’s theories, the genial Hermite was one of the most enthusiastic. His cordial acceptance of the new doctrine warmed Cantor’s modest heart: “The praises which Hermite pours out to me in this letter ….on the subject of the theory of sets are so high in my eyes, so unmerited, that I should not care to publish them lest I incur the reproach of being dazzled by them.”

With the opening of the new century Cantor’s work gradually came to be accepted as a fundamental contribution to all mathematics and particularly to the foundations of analysis. But unfortunately for the theory itself the paradoxes and antinomies which still infect it began to appear simultaneously. These may in the end be the greatest contribution which Cantor’s theory is destined to make to mathematics, for their unsuspected existence in the very rudiments of logical and mathematical reasoning about the infinite was the direct inspiration of the present critical movement in all deductive reasoning. Out of this we hope to derive a mathematics which is both richer and “truer” — freer from inconsistency — than the mathematics of the pre-Cantor era.

Cantor’s most striking results were obtained in the theory of non-denumerable sets, the simplest example of which is the set of all points on a line segment. Only one of the simplest of his conclusions can be stated here. Contrary to what intuition would predict, two unequal line segments contain the same number of points. Remembering that two sets contain the same number of things if, and only if, things in them can be paired off one-to-one, we easily see the reasonableness of Cantor’s conclusion.

An even more unexpected result can be proved. Any line-segment, no matter how small, contains as many points as an infinite straight line. Further, the segment contains as many points as there are in an entire plane, or in the whole 3-dimensional space, or in the whole of space of n dimensions (where n is any integer greater than zero) or, finally, in a space of denumerably infinite number of dimensions.

In this we have not yet attempted to define a class or a set. Possibly (as Russell held in 1912) it is not necessary to do so in order to have a clear conception of Cantor’s theory or for that theory to be consistent with itself — which is enough to demand of any mathematical theory. Nevertheless, present disputes seem to require that some clear, self-consistent definition be given. The following used to be thought satisfactory.

A set is characterized by three qualities: it contains all things to which a certain definite property (any redness, volume, or taste) belongs; no thing not having this property belongs to the set; each thing in the set is recognizable as the same thing and as different from all other things in the set — briefly, each thing in the set has a permanently recognizable individuality. The set itself is to be grasped as a whole. This definition may be too drastic for use.. Consider, for example, what happens to Cantor’s set of all transcendental numbers under the third demand.

At this point we may glance back over the whole history of mathematics — or as much of it as is revealed by the treatises of the master mathematicians in their purely technical works — and note two modes of expression which recur constantly in nearly all mathematical exposition. The reader perhaps has been irritated by the repetitious use of phrases such as “we can find a whole number greater than 2,” or “we can choose a number less than n and greater than n-2.” The choice of such phraseology is not merely stereotyped pedantry. There is a reason for its use, and careful writers mean exactly what they say when they assert that “we can find, etc.” They mean that “they can do what they say.”

In sharp distinction to this is the other phrase which is reiterated over and over again in mathematical writing: “there exists.” For example, some would say “there exists a whole number greater than 2,” or “there exists a number less than n and greater than n-2.” The use of such phraseology definitely commits its user to the creed which Kronecker held to be untenable, unless, of course, the “existence” is proved by a construction. The existence is not proved for the sets (as defined above) which appear in Cantor’s theory.

These two ways of speaking divide mathematicians into two types: the “we can” men believe (possibly subconsciously) that mathematics is a purely human invention; the “there exists” men believe that mathematics has an extra-human “existence” of its own, and that “we” merely come upon the “eternal truths” of mathematics in our journey through life, in much the same way that a man taking a walk in a city comes across a number of streets with whose planning he had nothing whatever to do.

Theologians are “exist” men; cautious skeptics for the most part “we” men. “There exist an infinity of even numbers, or of primes,” say the advocates of extra-human “existence”; “produce them,” says Kronecker and the “we” men.

That the distinction is not trivial can be seen from a famous instance of it in the New Testament. Christ asserted that the Father “exists”; Philip demanded “Show us the Father and it sufficeth us.” Cantor’s theory is almost wholly on the “existence” side. Is it possible that Cantor’s passion for theology determined his allegiance? If so, we shall have to explain why Kronecker, also a connoisseur of Christian theology, was the rabid “we” man that he was. As in all such questions ammunition for either side can be filched from any pocket.

A striking and important instance of the “existence” way of looking at the theory of sets is afforded by what is known as Zermelo’s postulate (stated in 1904). “For every set M whose elements are sets P (that is, M is a set of sets, or a class of classes), the sets P being non-empty and non-overlapping (no two contain things in common), there exists at least one set N which contains precisely one element from each of the sets P which constitute M.” Comparison of this with the previously stated definition of a set (or class) will show that the “we” men would not consider the postulate self-evident if the set M consisted, say, of an infinity of non-overlapping line segments. Yet the postulate seems reasonable enough. Attempts to prove it have failed. It is of considerable importance in all questions relating to continuity.

A word as how to this postulate came to be introduced into mathematics will suggest another of the unsolved problems of Cantor’s theory. A set of distinct, countable things, like all the bricks in a certain wall, can easily be ordered; we need only count them off 1, 2, 3, …, in any dozens of different ways that will suggest themselves. But how would we go about ordering all the points on a straight line? They cannot be counted off 1, 2, 3, ….The task appears hopeless when we consider that between any two points of the line “we can find,” or “there exists” another point of the line. If every time we counted two adjacent bricks another sprang in between them in the wall our counting would become slightly confused. Nevertheless, the points on a straight line do appear to have some sort of order: we can say whether one point is to the right or the left of another, and so on. Attempts to order the points of a line have not succeeded. Zermelo proposed his postulate as a means for making the attempt easier, but it itself is not universally accepted as a reasonable assumption or as one which it is safe to use.

Cantor’s theory contains a great deal more about the actual infinite and the “arithmetic” of transfinite (infinite) numbers than what has been indicated here. But, as the theory is still in the controversial stage, we may leave it with the statement of a last riddle. Does there “exist,” or can we “construct,” an infinite set which is not similar (technical sense of one-to-one matching) either to the set of all the positive rational integers or to the set of all points of a line? The answer is unknown.

Cantor died in a mental hospital in Halle on January 6, 1913, at the age of seventy three. Honours and recognition were his at the last, and even the old bitterness against Kronecker was forgotten. It was no doubt a satisfaction to Cantor to recall that he and Kronecker’s death in 1891. Could Cantor have lived till today he might have taken a just pride in the movement toward more rigorous thinking in all mathematics for which his own efforts to found analysis (and the infinite) on a sound basis were largely responsible.

Looking back over the long struggle to make the concepts of real number, continuity, limit, and infinity precise and consistently usable in mathematics, we see that Zeno and Eudoxus were not so far in time from Weierstrass, Dedekind and, Cantor as the twenty four or twenty five centuries which separate modern Germany from ancient Greece might seem to imply. There is no doubt that we have a clearer conception of the nature of the difficulties involved than our predecessors had, because we see the same unsolved problems cropping up in new guises and fields the ancients never dreamed of, but to say that we have disposed of those hoary old difficulties is a gross mis-statement of fact. Nevertheless, the net score records a greater gain than any which our predecessors could rightfully claim. We are going deeper than they ever imagined necessary, and we are discovering that some of the “laws” — for instance those of Aristotelian logic — which they accepted in their reasoning are better replaced by others — pure conventions — in our attempts to correlate our experiences. As has already been said, Cantor’s revolutionary work gave our present activity, its initial impulse. But, it was soon discovered — twenty one years before Cantor’s death — that his revolution was either too revolutionary or not revolutionary enough. The latter now appears to be the case.

The first shot in the counter revolution was fired in 1897 by the Italian mathematician Burali-Forti who produced a flagrant contradiction by reasoning of the type used by Cantor in his theory of infinite sets. This particular paradox was only the first of several, and as it would require lengthy explanations to make it intelligible, we shall state instead Russell’s of 1908.

Frege had given the “class of all classes similar to a given class” definition of the cardinal number of the given class. Frege had spent years trying to put the mathematics of numbers on a sound logical basis. His life work is Grundgesetze der Arithmetik (the Fundamental Laws of Arithmetic), of which the first volume was published in 1893, the second in 1903. In this work the concept of sets is used. There is also a considerable use of more or less sarcastic invective against previous writers on the foundations of arithmetic for their manifest blunders and manifold stupidities. The second volume closes with the following acknowledgement:

” A scientist can hardly encounter anything more undesirable than to have the foundation collapse just as the work is finished. I was put in this position by a letter from Bertrand Russell when the work was almost through the press.”

Russell had sent Frege his ingenious paradox of “the set of all sets which are not members of themselves.” Is this set a member of itself? Either answer can be puzzled out with a little thought to be wrong. Yet Frege had freely used “sets of all sets.”

Many ways were proposed for evading or eliminating the contradictions which began exploding like a barrage in and over the Frege-Dedekind-Cantor theory of the real numbers, continuity, and the infinite. Frege, Cantor, and Dedekind quit the field, beaten and disheartened. Russell proposed his “vicious circle principle” as a remedy; “whatever involves all of a collection must not be a one of the collection”; later he put forth his “axiom of reducibility,” which as it is now practically abandoned, need not be described. For a time these restoratives were brilliantly effective (except in the opinion of the German mathematicians, who never swallowed them). Gradually, as the critical examination of all mathematical reasoning gained headway, physic was thrown to the dogs and a concerted effort was begun to find out what really ailed the patient in his irrational and real number system before administering further nostrums.

The present effort to understand our difficulties originated in the work of David Hilbert of Gottingen in 1899 and in that of L.E.J. Brouwer of Amsterdam in 1912. Both of these men and their numerous followers have the common purpose of putting mathematical reasoning on a sound basis. although in several respects their methods and philosophies are violently opposed. It seems unlikely that both can be as wholly right as each appears to believe he is.

Hilbert returned to Greece for the beginning of his philosophy of mathematics. Resuming the Pythagorean program of a rigidly and fully stated set of postulates from which a mathematical argument must proceed by strict deductive reasoning, Hilbert made the program of the postulational development of mathematics more precise than it had been with the Greeks, and in 1899 issued the first edition of his classic on the foundations of geometry. One demand which Hilbert made, and which the Greeks do not seem to have thought of, was that the proposed postulates for geometry shall be proved to be self-consistent (free of internal, concealed contradictions). To produce such a proof for geometry it is shown that any contradiction in the geometry developed from the postulates would imply a contradiction in arithmetic. The problem is thus shoved back to proving the consistency of arithmetic, and there it remains today.

Thus we are back once more asking the sphinx to tell us what a number is. Both Dedekind and Frege fled to the infinite — Dedekind with his infinite classes defining irrationals, Frege with his class of all classes similar to a given class defining a cardinal number — to interpret the numbers that puzzled Pythagoreans. Hilbert, too, would seek the answer in the infinite which, he believes, is necessary for an understanding of the finite. He is quite emphatic in his belief that Cantorism will ultimately be redeemed from the purgatory in which it now tosses. “This (Cantor’s theory) seems to me the most admirable fruit of the mathematical mind and indeed one of the highest achievements of man’s intellectual processes.” But he admits that the paradoxes of Burali-Forti, Russell, and others are not resolved. However, his faith surmounts all doubts: “No one shall expel us from the paradise which Cantor has created for us.”

But at this moment of exaltation Brouwer appears with something that looks suspiciously like a flaming sword in his strong right hand. The chase is on: Dedekind, in the role of Adam, and Cantor disguised as Eve at his side, are already eyeing the gate apprehensively under the stern regard of the uncomprimising Dutchman. The postulational method for securing freedom from contradiction proposed by Hilbert will, says Brouwer, accomplish its end — produce no contradictions, “but nothing of mathematical value will be attained in this manner; a false theory which is not stopped by a contradiction is none the less false, just as a criminal policy unchecked by a reprimanding court is none the less criminal.”

The root of Brouwer’s objection to the criminal policy of his opponents is something new — at least in mathematics. He objects to an unrestricted use of Aristotelian, particularly in dealing with infinite sets , and he maintains that such logic is bound to produce contradictions when applied to sets which cannot be definitely constructed in Kronecker’s sense (a rule of procedure must be given whereby the things in the set can be produced). The law of excluded middle (a thing must have a certain property or must not have that property, as for in the example in the assertion that a number is prime or not prime) is legitimately usable only when applied to finite sets. Aristotle devised his logic as a body of working rules for finite sets, basing his method on human experience of finite sets , and there is no reason whatever for supposing that a logic which is adequate for the finite will continue to produce consistent (non contradictory) results when applied to the infinite. This seems reasonable enough when we recall that the very definition of an infinite set emphasizes that a part of an infinite set may contain precisely as many things as the whole set, a situation which never happens for a finite set when “part” means some, but not all (as it does in the definition of an infinite set).

Here we have what some consider the root of the trouble in Cantor’s theory of the actual infinite. For the definition of a set (as stated some time back), by which all things having a certain property are “united” to form a “set”(or “class”), is not suitable as a basis for the theory of sets, in that definition either is not constructive (in Kronecket’s sense) or assumes a constructability which no mortal can produce. Brouwer claims that the use of the law of excluded middle in such a situation is at best merely a heuristic guide to propositions which may be true, but which are not necessarily so, even when they have been deduced by a rigid application of Aristotlelian logic, and he says that numerous false theories (including Cantor’s) have been erected on this rotten foundation during the past half century.

Such a revolution in the rudiments of mathematical thinking does not go unchallenged. Brouwer’s radical move to the left is speeded by an outraged roar from the reactionary right. “What Weyl and Brouwer are doing (Brouwer is the leader, Weyl his companion in revolt) is mainly following in the steps of Kronecker,” according to Hilbert, the champion of the status quo. “They are trying to establish mathematics by jettisoning everything which does not suit them and setting up an embargo. The effect is to dismember our science and to run the risk of losing a large part of our most valuable possessions. Weyl and Brouwer condemn the general notion of irrational numbers, of functions — even of such functions as occur in the theory of numbers — Cantor’s transfinite numbers, etc., the theorem that an infinite set of positive integers has a least, and even the law of the excluded middle, as for example the assertion: Either there is only a finite number of primes or there are infinitely many. These are examples of (to them) forbidden theorems and modes of reasoning. I believe that impotent as Kronecker was to abolish irrational numbers (Weyl and Brouwer do permit us to retain the torso), no less impotent will their efforts prove today. No! Brouwer’s program is not a revolution, but merely the repetition of a futile coup de main with old methods, but which was then undertaken with greater verve, yet failed utterly. Today the State (“mathematics”) is thoroughly armed and strengthened through the labours of Frege, Dedekind, and Cantor. The efforts of Brouwer and Weyl are foredoomed to futility.

To which the other side replies by a shrug of shoulders and goes ahead with its great and fundamentally new task of reestablishing mathematics (particularly the foundations of analysis) on a firmer basis than any laid down by the men of the past 2500 years from Pythagoras to Weierstrass.

What will mathematics be like a generation hence when — we hope — these difficulties will have been cleared up? Only a prophet or the seventh son of a prophet sticks his head into the noose of prediction. But if there is any continuity at all in the evolution of mathematics — and the majority of dispassionate observers believe that there is — we shall find that the mathematics which is to come will be broader, firmer, and richer in content than that which we or our predecessors have known.

Already the controversies of the past third of a century have added new fields — including totally new logics — to the vast domain of mathematics, and the new is being rapidly consolidated and coordinated with the old. If we may rashly venture a prediction, what is it to come will be fresher, younger in every respect, and closer to human thought and human needs — freer of appeal for its justification to extra-human “existence”— than what is now being vigorously refashioned. The spirit of mathematics is eternal youth. As Cantor said,

Already the controversies of the past third of a century have added new fields — including totally new logics — to the vast domain of mathematics, and the new is being rapidly consolidated and coordinated with the old. If we may rashly venture a prediction, what is it to come will be fresher, younger in every respect, and closer to human thought and human needs — freer of appeal for its justification to extra-human “existence”— than what is now being vigorously refashioned. The spirit of mathematics is eternal youth. As Cantor said, “The essence of mathematics resides in its freedom”; the present “revolution” is but another assertion of this freedom.

Cheers,

Nalin Pithwa

Sir Andrew Wiles on being bright at math:

Yes, some people are brighter than others in math. But I really believe that most people can really reach a good level in math if they are prepared to handle pyschological issues arising out of situations when being stuck.

A mathematical person: the last universalist: Henri Poincare

Reference: E. T. Bell “Men of Mathematics”.

PS: Of course, some one might say that Prof John von Neumann was the last universalist. I leave you the dear reader to think…

Below I want to share the words of Prof. E. T. Bell in his classic “Men of Mathematics”. I am doing it for my own learning more rather than even sharing it with the general public. Perhaps, these are good times to be introverted 🙂

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Quote: (Henri Poincare): A scientist worthy of the name, above all a mathematician, experiences in his work, the same impression as an artist; his pleasure is as great and of the same nature.

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In The History of his Life and Times, the astrologer William Lilly (1602-1681) records an amusing — if incredible — account of the meeting between John Napier (1559-1617), of Merchiaton, the inventor of logarithms, and Henry Briggs (1561-1631) of Gresham College, London, who computed the first table of common logarithms. One John Marr, “an excellent mathematician and geometrician,” had gone ‘into Scotland before Mr Briggs, purposely to be there when these two so learned persons should meet.Mr Briggs appoints a certain day when to meet in Edinburgh, but failing thereof, the Lord Napier was doubtful he would not come. It happened one day as John Marr and Lord Napier were speaking of Mr Briggs, “Ah John (said Merchiaton), Mr Briggs will not now come.” At the very moment, one knocks at the gate; John Marr hastens down, and it proved Mr. Briggs to his great contentment. He brings Mr. Briggs up into my lord’s chamber, where almost one quarter of an hour was spent, each beholding the other with admiration, before one word was spoke.”

Recalling this legend, Sylvester tells how he himself went after Briggs’ world record for flabbergasted admiration when, in 1885, he called on the author of numerous astonishingly mature and marvelously original papers on a new branch of analysis which had been swamping the editors of mathematical journals since the early 1880’s.

“I quite entered into Briggs’ feelings at his interview with Napier,” Sylvester confesses, “when I recently paid a visit to Poincare (1854-1912) in his airy perch in the Rue Gay-Lussac…In the presence of that mighty reservoir of pent-up intellectual force my tongue at first refused its office, and it was not until I had taken some time (it may be two or three minutes) to peruse and absorb as it were the idea of his external youthful lineaments that I found myself in a condition to speak.”

Elsewhere Sylvester records his bewilderment when, after having toiled up the three flights of narrow stairs leading to Poincare’s “airy perch,” he paused, mopping his magnificent bald head, in astonishment at beholding a mere boy, “so blond, so young,” as the author of the deluge of papers which had heralded the advent of a successor to Cauchy.

A second anecdote may give some idea of the respect in which Poincare’s work is held by those in a position to appreciate its scope. Asked by some patriotic British brass hat in the rabidly nationalistic days of the World War — when it was obligatory on all academic patriots to exalt their esthetic allies and debase their boorish enemies — who was the greatest man France had produced in modern (recent) times, Bertrand Russell answered instantly, “Poincare.” “What ! That man! ” his uninformed interlocutor exclaimed, believing Russell meant Raymond Poincare, President of the French Republic. “Oh,” Russell explained when he understood the other’s dismay, “I was thinking of Raymond’s cousin, Henri Poincare.”

Poincare was the last man to take practically all mathematics, both pure and applied, as his province. It is generally believed that it would be impossible for any human being starting today to understand comprehensively, much less do creative work of high quality in more than two of the four main divisions of mathematics — arithmetic, algebra, geometry, analysis, to say nothing of astronomy and mathematical physics. However, even in the 1880’s. when Poincare’s great career opened, it was commonly thought that Gauss was the last of the mathematical universalists, so it may not prove impossible for some future Poincare once more to cover the entire field.

As mathematics evolves, it both expands and contracts, somewhat like one of Lemaitre’s models of the universe. At present, the phase is one of explosive expansion, and it is quite impossible for any man to familiarize himself with the entire inchoate mass of mathematics that has been dumped on the world since the year 1900. But already to certain important sectors a most welcome tendency toward contraction is plainly apparent. This is so, for example, in algebra, where the wholesale introduction of postulational methods is making the subject at once more abstract. more general, and less disconnected. Unexpected similarities — in some instances amounting to disguised identity— are being disclosed by the modern attack, and it is conceivable that the next generation of algebraists will not need to know much that is now considered valuable, as many of these particular, difficut things will have been subsumed under simpler general principles of wider scope. Something of this sort happened in classical mathematical physics when relativity put the complicated mathematics of the ether on the shelf.

Another example of this contraction in the midst of expansion is the rapidly growing use of the tensor calculus in preference to that of numerous special brands of vector analysis. Such generalizations and condensations are often hard for older men to grasp at first and frequently have a severe struggle to survive, but in the end it is usually realized that general methods are essentially simpler and easier to handle than miscellaneous collections of ingenious tricks devised for special problems. When mathematicians assert that such a thing as the tensor calculus is easy — at least in comparison with some of the algorithms that preceded it — they are not trying to appear superior or mysterious but are stating a valuable truth which any student can verify for himself. This quality of inclusive generality was a distinguishing trait of Poincare’s vast output.

If abstractness and generality have obvious advantages of the kind indicated, it is also true that they sometimes have serious drawbacks for those who must be interested in details. Of what immediate use is it to a working physicist to know that a particular differential equation occuring in his work is solvable, because some pure mathematician has proved that it is, when neither he nor the mathematician can perform the Herculean labour demanded by a numerical solution capable of application to specific problems?

To take an example from the field in which Poincare did some of his most original work, consider a homogeneous, incompressible fluid mass held together by the gravitation of its particles and rotating about an axis. Under what conditions will the motion be stable and what will be the possible shapes of such a stably rotating fluid? MacLaurin, Jacobi, and others proved that certain ellipsoids will be stable; Poincare, using more intuitive, “less arithmetical” methods than his predecessors, once thought he had determined the criteria for the stability of a pear-shaped body. But he had made a slip. His methods were not adapted to numerical computations and later workers, including G. H. Darwin, son of the famous Charles, undeterred by the horrific jungles of algebra and arithmetic that must be cleared out of the way before a definite conclusion be reached, undertook a decisive solution.

The man interested in the evolution of binary stars is more comfortable if the findings of the mathematicians are presented to him in a form to which he can apply a calculating machine. And since Kronecker’s fiat of “no construction, no existence”, some pure mathematicians themselves have been less enthusiastic than they were in Poincare’s day for existence theorems which are not constructive. Poincare’s scorn for this kind of detail that users of mathematics demand and must have before they can get on with their work was one of the most important contributory causes to his universality. Another was his extraordinarily comprehensive grasp of all the machinery of the theory of functions of a complex variable. In this he had no equal. And it may be noted that Poincare turned his universality to magnificent use in disclosing hitherto unsuspected connections between distant branches of mathematics, for example, between (continuous groups) and linear algebra.

One more characteristic of Poincare’s outlook must be recalled for completeness before we go on to his life: few mathematicians have led the breadth of philosophical vision that Poincare had, and none is his superior in the gift or clear exposition. Probably, he had always been deeply interested in the implications of science and mathematics, but it was only in 1904, when his greatness as a technical mathematician was established beyond all cavil, that he turned as a side-interest in what may be called the popular appeal of mathematics and let himself go in a sincere enthusiasm to share with the non-professionals the meaning and human importance of his subject. Here his liking for the general in preference to the particular aided him in telling intelligent outsiders what is of more than technical importance in mathematics without talking down to his audience. Twenty or thirty years ago workmen and shopgirls could be seen in the parks and cafes of Paris avidly reading one or other of Poincare’s popular masterpieces in its cheap print and shabby paper cover. The same works in a richer format could also be found — well thumbed and evidently read — on the tables of the professionally cultured. These books were translated into English, German, Spanish, Hungarian, Swedish and Japanese. Poincare spoke the universal languages of mathematics and science to all in accents which they recognized. His style, peculiarly his own, loses much by translation.

For the literary excellence of his popular writings Poincare was awarded the highest honour a French writer can get, membership in the literary section of the Institut. He has been somewhat spitefully said by envious novelists that Poincare achieved this distinction, unique for a man of science, because one of the functions of the literary Academy is the constant compilation of a dictionary of the French language, and the universal Poincare was obviously the man to help out the poets and grammarians in their struggle to tell the world what automorphic functions are. Impartial opinion, based on a study of Poincare’s writings, agrees that the mathematician deserved no less than he got.

Closely allied to his interest in the philosophy of mathematics was Poincare’s preoccupation with the psychology of mathematical creation. How do mathematicians make their discoveries? Poincare will tell us later his own observations on this mystery in one of the most interesting narratives of personal discovery that was ever written. The upshot seems to be that mathematical discoveries more or less make themselves after a long spell of hard labour on the part of the mathematician. As in literature — according to Dante Gabriel Rossetti — “a certain amount of fundamental brainwork” is necessary before a poem can mature, so in mathematics there is no discovery without preliminary drudgery, but this is by no means the whole story. All “explanations” of creativeness that fail to provide a recipe whereby a gifted human being can create are open to suspicion. Poincare’s excursion into practical psychology, like some others in the same direction, failed to bring back the Golden Fleece, but it did at least suggest that such a thing is not wholly mythical and may some day be found when human beings grow intelligent enough to understand their own bodies.

Poincare’s intellectual heredity on both sides was good. We shall notgo farther back than his paternal grandfather. During the Napoleonic campaign of 1814 this grandfather, at the early age of twenty, was attached to the military hospital at Saint-Quentin. On settling in 1817 at Rouen he married and had two sons: Leon Poincare, born in 1828, who became a first rate physician and a member of a medical faculty, and Antoine, who rose to the inspector-generalship of the department of roads and bridges. Leon’s son Henri, born on April 29, 1854, at Nancy, Lorraine, became the leading mathematician of the early 20th century; one of Antoine’s two sons, Raymond, went in for law and rose to the presidency of the French Republic during the World War; Antoine’s other son became director of secondary education. A great uncle who had followed Napoleon into Russia disappeared and was never heard of after the Moscow fiasco.

From this distinguished list it might be thought that Henri would have exhibited some administrative ability, but he did not, except in his early childhood when he freely invented political games for his sister and young friends to play. In these games he was always fair and scrupulously just, and just seeing that each of his playmates got his or her full share of officeholding. This perhaps is conclusive evidence that “child is father to the man” and that Poincare was constitutionally incapable of understanding the simplest principle of administration, which his cousin Raymond applied intuitively.

Poincare’s biography was written in great detail by his fellow countryman Gaston Darboux (1842-1917), one of the leading geometers of modern times, in 1913 (the year following Poincare’s death). Something may have escaped the present writer but it seems that Darboux, after having stated that Poincare’s mother “coming from a family in the Meuse district whose (“the mother’s”) parents lived in Arrancy, was a very good person, very active and very intelligent “, blandly omits to mention her maiden name. Can it be possible that the French took over the doctrine of the “the three big K’s” —- noted in connection with Dedekind — from their late instructors after the kultural drives of Germany and France in 1870 and 1914? However, it can be deduced from an anecdote told later by Darboux that the family name may have been Lannois. We learn that the mother devoted her entire attention to the education of her two young children, Henri and his younger sister (name not mentioned). The sister was to become wife of Emile Boutroux and the mother of a mathematician (who died young).

Due partly to his mother’s constant care, Poincare’s mental development as a child was extremely rapid, but also very badly at first because he thought more rapidly than he could get the words out. From infancy his motor coordination was poor. When he learned to write it was discovered that he was ambidexterous and that he could write or draw as badly as with the left hand as with the right hand. Poincare never outgrew this physical awkwardness. As an item of some interest in this connection it may be recalled that when Poincare was acknowledged as the foremost mathematician and leading popularizer of science of his time he submitted to the Binet tests and made such a disgraceful showing that, had he been judged as a child instead of the famous mathematician that he was, he would have been rated — by the tests — as an imbecile.

At the age of five, Henri suffered a setback from diptheria which left him for nine months with a paralyzed larynx. This misfortune made him for long delicate and timid, but it also turned him back on his own resources as he was forced to shun the rougher games of children of his own age.

His principal diversion was reading, where his unusual talents first showed up. A book once read — at incredible speed — became a permanent possession, and he could always state the page and line where a particular thing occured. He retained this powerful memory all his life. This rare faculty, which Poincare shared with Euler who had it in a lesser degree, might be called visual or spatial memory. In temporal memory — the ability to recall with uncanny precision a sequence of events long passed — he was also unusually strong. Yet he unblushingly describes his memory as “bad.” His poor eyesight perhaps contributed to a third peculiarity of his memory. The majority of mathematicians appear to remember theorems and formulas mostly by the eye; with Poincare it was mostly by the ear. Unable to see the board distinctly when he became a student of advanced mathematics, he sat back and listened, following and remembering perfectly without taking notes — an easy feat for him, but one incomprehensible to most mathematicians. Yet he must have had a vivid memory of the “inner eye” as well, for much of his work, like a good deal of Riemann’s, was of the kind that goes with facile space-intuition and acute visualization. His inability to use his fingers skillfully of course handicapped in laboratory exercises, which seems a pity, as some of his own work in mathematical physics might have been closer to reality had he mastered the art of experiment. Had Poincare been as strong in practical science as he was in theoretical he might have made a fourth with the incomparable three, Archimedes, Newton and Gauss.

Not many of the great mathematicians have been the asbsentminded dreamers that popular fancy likes to picture them. Poincare was one of the exceptions, and then only in comparative trifles, such as carrying off hotel linen in his baggage. But many persons who are anything but absentminded do the same, and some of the most alert mortals living have even been known to slip restaurant silver into their pockets and get away with it.

One phase of Poincare’s absentmindedness resembles something quite different. Thus (Darboux does not tell the story, but it should be told, as it illustrates a certain brusqueness of Poincare’s later years), when a distinguished mathematician had come all the way from Finland to Paris to confer with Poincare on scientific matters> Poincare did not leave his study to greet his caller when the maid notified him, but continued to pace back and forth — as was his custom when mathematicizing —- for three solid hours. All this time the diffident caller sat quietly in the adjoining room, barred from the master only by flimsy portieres. At last the drapes parted and Poincare’s buffalo head was thrust for an instant in to the room. “vous me derangen beaucoup” (you are disturbing me greatly) the head exploded, and disappeared. The caller departed without an interview, which was exactly what the “absentminded” professor wanted.

Poincare’s elementary school career was brilliant, although he did not at first show any marked interest in mathematics. His earliest passion was for natural history, and all his life he remained a great lover of animals. The first time he tried out a rifle he accidentally shot a bird at which he had not aimed. The mishap affected his so deeply that thereafter nothing (except compulsory military drill) could induce him to touch firearms. At the age of nine he showed the first promise of what was to be one of his major successes. The teacher of French composition declared that a short execise, original in both form and substance, which young Poincare had handed in, was a “little masterpiece,” and kept it as one of his treasures. But he also advised his pupil to be conventional — stupider — if he wished to make a good impression on the school examiners.

Being out of the more boisterous games of his schoolfellows, Poincare invented his own. He also became an indefatigable dancer. As all his lessons came to him as easily as breathing he spent most of his time on amusements and helping his mother about the house. Even at this early age of his career Poincare exhibited some of the more suspicious features of his mature “absentmindedness”; he frequently forgot his meals and almost never remembered whether or not he had breakfasted. Perhaps, he did not care to stuff himself as most boys do.

The passion for mathematics seized him at adolescence or shortly before (when he was about fifteen). From the first he exhibited a lifelong peculiarity : his mathematics was done in his head as paced restlessly about, and was committed to paper only when all had been thought through. Talking or other noise never disturbed his while he was working. In later life, he wrote his mathematical memoirs at one dash without looking back to see what he had written and limiting himself to but a very few erasures as he wrote. Cayley also composed in this way, and probably Euler, too. Some of Poincare’s work shows the marks of hasty composition, and he said himself that he never finished a paper without regretting either its form or its substance. More than one man who has written well has felt the same. Poincare’s flair for classical studies, in which he excelled at school, taught him the importance of both form and substance.

The Franco-Prusssian war broke over France in 1870 when Poincare was sixteen. Although he was too young and too frail for active service, Poincare nevertheless got his full share of the horrors, for Nancy, where he lived, was submerged by the full tide of the invasion, and the young boy accompanied his physician-father on his rounds of the ambulances. Later he went with his mother and sister, under terrible difficulties, to Arrancy to see what happened to his maternal grandparents, in whose spacious country garden the happiest days of his childhood had been spent during the long school vacations. Arrancy lay near the battle-field of Saint-Privat. To reach the town the three had to pass “in glacial cold” through burned and deserted villages. At last they reached their destination, only to find that the house had been thoroughly pillages, “not only of things of value but of things of no value,” and in addition had been defiled in the bestial manner made familiar to the French by the 1914 sequel to 1870. The grandparents had been left nothing; their evening meal on the day they viewed the great purging was supplied by a poor woman who had refined to abandon the ruins of her cottage and who insisted on sharing her meagre supper with them.

Poincare never forgot this, nor did he ever forget the long occupation of Nancy by the enemy. It was during the war that he mastered German. Unable to get any French news, and eager to learn what the Germans had to say of France and for themselves, Poincare taught himself the language. What he had seen and what he learned from the official accounts of the invaders themselves made him a flaming patriot for life but, like Hermite, he never confused the mathematics of his country’s enemies with their more practical activities. Cousin Raymond, on the other hand, could never say anything about les Allemands (the Germans) without an accompanying scream of hate. In the bookkeeping of hell which balances the hate of one patriot against that of another, Poincare may be checked off against Kummer, Hermite against Gauss, thus producing that perfect zero implied in the scriptural contract “an eye for an eye and a tooth for a tooth.”

Following the usual French custom Poincare took the examinations for his first degree (bachelor of letters, and of science) before specializing. These he passed in 1871 at the age of seventeen — after almost failing in mathematics ! He had arrived late and flustered at the examination and had fallen down on the extremely simple proof of the formula giving the sum of a convergent geometrical progression. But his fame had preceded him. “Any student other than Poincare would have been plucked,” the head examiner declared.

He next prepared for the entrance examination to the School of Forestry, where he astonished his companions by capturing the first prize in mathematics without having bothered to take any lecture notes. His classmates had previously tested him out, believing him to be a trifler, by delegating a fourth year student to quiz him on a mathematical difficulty which had seemed particularly tough. Without apparent thought, Poincare gave the solution immediately and walked off, leaving his crestfallen baiters asking “How does he do it?” Others were to ask the same question all through Poincare’s career. He never seemed to think when a mathematical difficulty was submitted to him by a colleague. “The reply came like an arrow.”

At the end of his year, he passed first into the Ecole Polytechnique. Several legends of his unique examination survive. One tells how a certain examiner forewarned that young Poincare was a mathematical genius, superseded the examination for three quarters of an hour in order to devise a “nice” question — a refined torture. But Poincare got the better of him and the inquisitor “congratulated the examinee warmly, telling him he had won the highest grade.” Poincare’s experiences with his tormentors would seem to indicate that the French mathematical examiners have learned something since they ruined Galois and came within an ace of doing the like by Hermite.

At the Polytechnique Poincare was distinguished for his brilliance in mathematics, his superb incompetence in all physical exercises, including gymnastics and military drill, and his utter inability to make drawings that resembled anything in heaven and earth. The last was more than a joke; his score of zero in the entrance examination in drawing had almost kept him out of the school. This had greatly embarassed his examiners: “…a zero is eliminatory. In everything else (“but drawing”) he is absolutely without an equal. “If he is admitted, it will be as first, but can he be admitted?” As Poincare was admitted the good examiners probably put a decimal point before the zero and placed a 1 after it.

In spite of his ineptitude for physical exercises Poincare was extremely popular with his classmates. At the end of the year they organized a public exhibition of his artistic masterpieces, carefully labelling them in Greek, “this is a horse,” and so on — not always accurately. But Poincare’s inability to draw also had its serious side when he came to geometry, and he lost his first place, passing out of the school second in rank.

On leaving the Polytechnique in 1875 at the age of twenty one Poincare entered the School of Mines with the intention of becoming an engineer. His technical studies, although faithfully carried out, left him some leisure to do mathematics, and he showed what was in him by attacking a general problem in differential equations. Three years later he presented a thesis, on the same subject, but concerning a more difficult and yet more general question, to the Faculty of Sciences at Paris for the degree of doctor of mathematical sciences. “At the first glance,” says Darboux, who had been asked to examine the work, “it was clear to me that the thesis was out of the ordinary and simply merited acceptance. Certainly, it contained results enough to supply material for several good theses. But, I must not be afraid to say, if an accurate idea of the way Poincare worked is wanted, many points called for corrections or explanations. Poincare was an intuitionist. Having once arrived at the summit, he never retraced his steps. He was satisfied to have crashed through the difficulties and left to others the pains of mapping the royal roads *(“There is no royal road to Geometry,” as Menaechinus is said to have told Alexander the Great when the latter wished to conquer geometry in a hurry. ) destined to lead more easily to the end. He willingly enough made the corrections and tidying up which seemed necessary to me. But he explained to me when I asked him to do it that he had many other ideas in his head; he was already occupied with some of the great problems whose solutions he was to give us.”

Thus, young Poincare, like young Gauss, was overwhelmed by the host of ideas which besieged his mind but, unlike Gauss, his motto was not “Few, but ripe.” It is an open question whether a creative scientist who hoards the fruit of his labour so long that some of them go stale does more for the advancement of science than the more impetuous man who scatters broadcast everything he gathers, green or ripe, to fall where it may to ripen or rot as wind and weather take it. Some believe one way, some another. As a decision is beyond the reach of objective criteria everyone is entitled to his own purely subjective opinion.

Poincare was not destined to become a mining engineer, but during his apprenticeship he showed that he had at least the courage of a real engineer. After a mine explosion and fire which had claimed sixteen victims he went down at once with the rescue crew. But the calling was uncongenial and he welcomed the opportunity to become a professional mathematician which his thesis and other early work opened up to him. His first academic appointment was at Caen on December 1, 1879 as Professor of Mathematical Analysis. Two years later he was promoted (at the age of 27 years) to the University of Paris where, in 1886, he was again promoted, taking charge of the course in mecnanics and experimental physics (the last seems rather strange, in view of Poincare’s exploits as a student in the laboratory). Except for trips to scientific congresses in Europe and a visit to the United States in 1904 as an invited lecturer at the St. Louis Exposition, Poincare spent the rest of his life in Paris as the ruler of French mathematics.

Poincare’s creative period opened with the thesis of 1878 and closed with his death in 1919 — when he was the apex of his powers. Into this comparatively brief span of thirty-four years he crowded a mass of work that is sheerly incredible when we consider the difficulty of most of it. His record is nearly five hundred papers on new mathematics, many of those extensive memoirs, and more than thirty books covering practically all branches of mathematical physics, theoretical physics, and theoretical astronomy as they existed in his day. This leaves out of account his classics on the philosophy of science and his popular essays. To give an adequate idea of this immense labour one would have to be a second Poincare, so we shall presently select two or three of his most celebrated works for brief description, apologizing here once for all for the necessary inadequacy.

Poincare’s first successes were in the theory of differential equations, to which he applied all the resources of the analysis of which he was absolute master. This early choice for a major effort already indicates his leaning towards the applications of mathematics, for differential equations have attracted swarms of workers since the time of Newton chiefly because they are of great importance in the exploration of the physical universe. “Pure” mathematicians sometimes like to imagine that all their activities are dictated by their own tastes and that the applications of science suggest nothing of interest to them. Nevertheless some of the purest of the pure drudge away their life over differential equations that first appeared in the translation of physical situations into mathematical symbolism, and it is precisely these practically suggested equations which are the heart of the theory. A particular equation suggested by science may be generalized by the mathematicians and then be turned back to the scientists (frequently without a solution in any form that they can use) to be applied to new physical problems, but first and last the motive is scientific. Fourier summed up this thesis in a famous passage which irritates one type of mathematician, but which Poincare endorsed and followed in much of his work.

“The profound study of nature,” Fourier declared, “is the most fecund source of mathematical discoveries. Not only does this study, by offering a definite goal to research, have the advantage of excluding vague questions and futile calculations, but it is also a sure means molding analysis itself and discovering those elements in it which it is essential to know and which science might always to conserve. These fundamental elements are those which recur in all natural phenomena.” To which some might retort: No doubt, but what about the arithmetic in the sense of Gauss? However, Poincare followed Fourier’s advice whether he believed in it or not — even his researches in the theory of numbers were more or less remotely inspired by others closer to the mathematics of physical science.

The investigations on differential equations led out in 1880, when Poincare was twenty six, to one of his most brilliant discoveries, a generalization of the elliptic functions (and of some others). The nature of a (uniform) periodic function of a single variable has frequently been described, but to bring out what Poincare did, we may repeat the essentials. The trigonometric function \sin{x} has the period 2\pi, namely, \sin{x+2\pi}=\sin{x}, that is, when the variable x is increased by 2\pi, the sine function of x returns to the initial value. For an elliptic function, say, E(x), there are two distinct periods, say p_{1} and p_{2} such that E(x+p_{1})=E(x) and E(x+p_{2})=E(x). Poincare found this periodicity is merely a special case of a more general property: the value of certain functions is restored when the variable is replaced by any one of a denumerable infinity of linear fractional transformationsof itself, and all these transformations form a group. A few symbols will clarify this statement.

Let x be replaced by \frac{ax+b}{cx+d}. Then, for a denumerable infinity of sets of values a, b, c, d, there are uniform functions of x, say F(x) is one of them, such that

F(\frac{ax+b}{cx+d})=F(x)

Further, if a_{1}, b_{1}, c_{1}, d_{1} and a_{2}, b_{2}, c_{2}, d_{2} are two sets of values of a, b, c, d, and if x be replaced first by \frac{a_{1}x+b_{1}}{c_{1}x+d_{1}} and in this x, be replaced by \frac{a_{2}x+b_{2}}{c_{2}x+d_{2}}, giving say, \frac{Ax+B}{Cx+D} then not only do we have

F(\frac{a_{1}x+b_{1}}{c_{1}x+d_{1}})=F(x), and F(\frac{a_{2}x+b_{2}}{c_{2}x+d_{2}}) = F(x),

but also F(\frac{Ax+B}{Cx+D}) = F(x)

Further the set of all substitutions

x \rightarrow \frac{ax+b}{cx+d}

which leave the value of F(x) unchanged as just explained form a group; the result of the successive performance of two substitutions in the set

x \rightarrow \frac{a_{1}x+b_{1}}{c_{1}x+d_{1}} and x \rightarrow \frac{a_{2}x+b_{2}}{c_{2}x+d_{2}}

is in the set; there is an “identity substitution” in the set, namely x \rightarrow x where a=1, b=0, c=0, d=1 and finally each substitution has a unique “inverse” — that is, for each substitution in the set there is a single other one which, if applied to the first, will produce the identity substitution. In summary, using this terminology, we see that F(x) is a function which is invariant under an infinite group of linear fractional transformations. Note that the infinity of substitutions is a denumerable infinity, as first stated: the substitutions can be counted off 1, 2, 3, …and are not as numerous as the points on a line. Poincare actually constructed such functions and developed their most important properties in a series of papers in the 1880’s. Such functions are called automorphic.

Only two remarks be made here to indicate what Poincare achieved by this wonderful creation. First, his theory includes that of the elliptic functions as a detail. Second, as the distinguished French mathematician Georges Humbert said, Poincare found two memorable propositions which “gave him the keys of the algebraic cosmos.”

Two automorphic functions invariant under the same group are connected by an algebraic equation.

Conversely, the coordinates of a point on any algebraic curve can be expressed in terms of automorphic functions, and hence by uniform functions of a single parameter (variable).

An algebraic curve is one whose equation is of the type P(x,y)=0, where P(x,y) is a polynomial in x and y. As a simple example, the equation of the circle whose centre is at the origin —- (0,0) — and whose radius is a is x^{2}+y^{2}=a^{2}. According to the second of Poincare’s “keys”, it must be possible to express x, y as automorphic functions of a single parameter, say t. It is, for if x=a\cos{t}, y=a\sin{t}, then squaring and adding, we get rid of t (since \sin^{2}{t}+\cos^{2}{t}=1) and find x^{2}+y^{2}=a^{2}. But, the trigonometric functions \cos{t} and $latex\sin{t}$ are special cases of elliptic functions, which in turn are special cases of automorphic functions.

The creation of this vast theory of automorphic functions was but one of many astounding things in analysis which Poincare did before he was thirty. Nor was all his time devoted to analysis; the theory of numbers, parts of algebra, and mathematical astronomy also shared his attention. In the first he recast the Gaussian theory of binary quadratic forms in a geometrical shape which appeals particularly to those who, like Poincare, prefer the intuitive approach. This of course was not all that he did in the higher arithmetic, but limitations of space forbid further details.

Work of this calibre did not pass unappreciated. At the unusually early age of thirty-two (in 1887) Poincare was elected to the Academy. His proposer said some pretty strong things, but most mathematicians will subscribe to their truth. “(Poincare’s) work is above ordinary praise and reminds us inevitably of what Jacobi wrote of Abel — that he had settled questions which before him were unimagined. It must indeed be recognized that we are witnessing a revolution in Mathematics comparable in every way to that which manifested itself, half a century ago, by the accession of elliptic functions.”

To leave Poincare’s work in pure mathematics here is like rising from a banquet table after having just sat down, but we must turn to another side of his universaility.

Since the time of Newton and his immediate successors astronomy has generously supplied mathematicians with more problems than they can solve. Until the late nineteenth century the weapons used by mathematicians in their attack on astronomy were practically all immediate improvements of those invented by Newton himself, Euler, Lagrange and Laplace. But all through the nineteenth century, particularly since Cauchy’s development of the theory of functions of a complex variable and the investigations of himself and others on the convergence of infinite series, a huge arsenal of untried weapons had been accumulating from the labours of pure mathematicians. To Poincare, to whom analysis came as naturally as thinking, this vast pile of unused mathematics seemed the most natural thing in the world to use in a new offensive on the outstanding problems of celestial mechanics and planetary evolution. He picked and chose what he liked out of the heap, improved it, invented new weapons of his own, and assaulted theoretical astronomy in a general fashion it had not been assaulted in for a century. He modernized the attack; indeed his campaign was so extremely modern to the majority of experts in celestial mechanics that even today, forty years or more after Poincare opened his offensive, few have mastered his weapons and some, unable to open his bow, insinuate that it is worthless in a practical attack. Nevertheless Poincare is not without forceful champions whose conquests would have been impossible to the men of the pre-Poincare era.

Poincare’s first (1889) great success in mathematical astronomy grew out of an unsuccessful attack on “the problem of n bodies.” For n=2, the problem was completely solved by Newton; the famous problem of “three bodies” will be noticed later, when n exceeds 3 some of the reductions applicable to the case n=3 can be carried over.

According to the Newtonian law of gravitation, two particles of masses, m and M at a distance D apart attract one another with a force proportional to \frac{m \times M}{D^{2}}. Imagine n material particles distributed in space; the masses, the initial motions, and the mutual distances of all the particles are assumed known at a given instant. If they attract one another according to the Newtonian law, what will be their positions and motions (velocities) after any stated lapse of time? For the purposes of mathematical astronomy, the stars in a cluster, or in a galaxy, on in a cluster of galaxies, may be thought of as material particles attracting one another according to the Newtonian law. The “problem of n bodies” thus amounts to — in one of its applications — to asking what will be the aspect of the heavens a year from now, or a billion years hence, it being assumed that we have sufficient astronomical data to describe the general configuration now. The problem of course is tremendously complicated by radiation — the masses of stars do not remain constant for millions of years, but a complete, calculable solution of the problem of n bodies in its Newtonian form would probably give results of an accuracy sufficient for all human purposes — the human race will likely be extinct long before radiation can introduce observable inaccuracies.

This was substantially the problem proposed for the prize offered by King Oscar II of Sweden in 1887. Poincare did not solve the problem, but in 1889 he was awarded the prize by a jury consisting of Weierstrass, Hermite, and Mittag-Leffler for his general discussion of the differential equations of dynamics and an attack on the problem of three bodies. The last is usually considered the most important case of the n body problem, as the Earth, the Moon and the Sun furnish an example of three body problem. In his report to Mittag-Leffler, Weierstrass wrote, ” You may tell your Sovereign that this work cannot indeed be considered as furnishing the complete solution of the question proposed, but that it is nevertheless of such importance that the publication will inaugurate a new era in the history of Celestial Mechanics. The end which his Majesty had in view in opening the competition may therefore be considered as having been attained.” Not to be outdone by the King of Sweden, the French Government followed up the prize by making Poincare a Knight of the Legion of Honour — a much less expensive acknowledgement of the young mathematician’s genius than the King’s 9500 crowns and gold medal.

As we have mentioned the problem of three bodies we may now report one item from its fairly recent history; since the time of Euler it has been considered one of the most difficult problems in the whole range of mathematics. Stated mathematically, the problem boils down to a system of nine differential equations(all linear, each of the second order). Lagrange succeeded in reducing this system to a simpler. As in the majority of physical problems, the solution is not be expected in finite terms; if a solution exists at all, it will be given by an infinite series. The solution will “exist” if these series satisfy the equations (formally) and moreover converge for certain values of the variables. The central difficulty is to prove the convergence. Up till 1905 various special solutions had been found, but the existence of anything that could be called general had not been proved.

In 1906 and 1908 a considerable advance came from a rather unexpected quarter — Finland, a country which sophisticated Europeans even today consider barely civilized, especially for its queer customs or paying its debts, and which few Americans thought advanced beyong the Stone Age till Paavo Nurmi ran the legs off the United States. Excepting only the rare case when all three bodies collide simultaneously, Karl Frithiof Sundman of Helsingfors, utlizing analytical methods due to the Italian Levi-Civita and the French Painleve, and making an ingenious transformation of his own, proved the existence of a solution in the sense described above. Sundman’s solution is not adapted to numerical computation, nor does it give much information regarding the actual motion, but that is not the point of interest here: a problem which had not been known to be solvable was proved to be so. Many had struggled desperately to prove this much; when the proof was forthcoming, some, humanly enough, hastened to point out that Sundman had done nothing much because he had not solved some problem other than the one he had. This kind of criticism is as common in mathematics as it is in literature and art, showing once more that mathematicians are as human as anybody.

Poincare’s most original work in astronomy was summed up in his great treatise Les methodes nouvelles de la mecanique celeste (New methods of celestial mechanics; three volumes 1892, 1893, 1899). This was followed by another three volume work in 1905-1910 of a more immediately practical nature, Lecons de mecanique celeste, and a little later by the publication of his course of lectures Sur les figures d’equilibre d’une masse fluids (on the figures of equilibrium of a fluid mass), and a historical critical book Sur les hypotheses cosmogoniques (On cosmological hypotheses).

Of the first of these works, Darboux (seconded by many others) declares that it did indeed start a new era in celestial mechanics and that it is comparable to the Mecanique celeste of Laplace and the earlier work of D’Alembert on the precession of the equinoxes. “Following the road in analytical mechanics opened up by Lagrange,” Darboux says, “…Jacobi had established a theory which appeared to be one of the most complete in dynamics. For fifty years, we lived on the theorems of the illustrious German mathematician, applying them and studying them from all angles, but without adding anything essential. It was Poincare who first shattered these rigid frames in which the theory seemed to be encased and contrived for it vistas and new windows on the external world. He introduced or used, in the study of dynamical problems, different notions: the first, which had been given before and which, moreover, is applicable not solely to mechanics, is that of variational equations, namely, linear differential equations that determine solutions of a problem infinitely near to a given solution; the second, that of integral invariants, which belong entirely to him and play a capital part in these researches. Further fundamental notions were added to these, notably those concerning so-called periodic solutions, for which the bodies whose motion is studied return after a certain time to their initial positions and original relative velocities.

The last started a whole department of mathematics, the investigation of periodic orbits: given a system of planets, or of stars, say, with a complete specification of the initial position and relative velocities of all members of the system at a stated epoch, it is required to determine under what conditions the system will return to its initial state at some later epoch, and hence, continue to repeat the cycle of motions indefinitely. For example, is the solar system of this recurrent type, or if not, would it be were it isolated and not subject to perturbations by external bodies ? Needless to say the general problem has not yet been solved completely.

Much of Poincare’s work in his astronomical researches was qualitative rather than quantitative, as befitted an intuitionist, and this characteristic led him, as it had Riemann, to the study of analysis situs. On this he published six famous memoirs which revolutionized the subject as it existed in his day. The work on analysis situs in its turn was freely applied to the mathematics of astronomy.

We have already alluded to Poincare’s work on the problemof rotating fluid bodies — of obvious importance in cosmogony, one brand of which assumes that the planets were once sufficiently like such bodies to be treated as if they actually were without patent absurdity. Whether they were or not is of no importance for the mathematics of the situation, which is of interest in itself. A few extracts from Poincare’s own summary will indicate more clearly than any paraphrase the nature of what he mathematicized about in this difficult subject.

“Let us imagine a (rotating) fluid body contracting by cooling, but slowly enough to remain homogeneous and for the rotation to be the same in all its parts.

“At first, very approximately a sphere, the figure of this mass will become an ellipsoid of revolution which will flatten more and more, then, at a certain moment, it will be transformed into an elllipsoid with three unequal axes. Later, the figure will cease to be an ellipsoid and will become pear-shaped until at last the mass, hollowing out more and more at its “waist,” will separate into two distinct and unequal bodies.

“The preceding hypothesis certainly can not be applied to the solar system. Some astronomers have thought that it might be true for certain double stars and that double stars of the type of Beta Lyrae might present transitional forms analogous to those we have spoken of.”

He then goes on to suggest an application to Saturn’s rings, and he claims to have proved that the rings can be stable only if their density exceeds 1/16 that of Saturn. It may be remarked that these questions were not considered as fully settled as late as 1935. In particular a more drastic mathematical attack on poor old Saturn seemed to show that he had not been completely vanquished by the great mathematicians, including Clerk Maxwell, who have been firing away at him off and on for the past seventy years.

Once more we must leave the banquest having barely tasted anything and pass on to Poincare’s voluminous work in mathematical physics. Here his luck was not so good. To have cashed in on his magnificent talents he should have been thirty years later or have lived twenty years longer. He had the misfortune to be in his prime just when physics had reached one of its recurrent periods of senility, and he was so thoroughly saturated with nineteenth century theories when physics began to recover its youth — after Planck, in 1900, and Einstein in 1905, had performed the difficult and delicate operation of endowing the decrepit roue with its first pair of new glands — that he had barely time to digest the miracle before his death in 1912. All his mature life Poincare seemed to absorb knowledge through his pores without a conscious effort. Like Cayley, he was not only a prolific creator but also a profoundly erudite scholar. His range was probably wider than ever Cayley’s, for Cayley never professed to be able to understand everything that was going on in applied mathematics. This unique erudition may have been a disadvantage when it came to a question of living science as opposed to classical.

Everything that boiled up in the melting pots of physics was grasped instantly as it appeared to Poincare and made the topic of several purely mathematical investigations. When wireless telegraphy was invented he seized the new thing and worked out its mathematics. While others were either ignoring Einstein’s early work on the (special) theory of relativity or passing it by as a mere curiosity, Poincare was already busy with its mathematics, he was the first scientific man of high standing to tell the world what had arrived and urge it to watch Einstein as probably the most significant phenomenon of the new era which he foresaw but could not himself usher in. It was the same with Planck’s early form of the quantum theory. Opinions differ, of course; but, at this distance it is beginning to look as if mathematical physics did for Poincare what Ceres did for Gauss; and although Poincare accomplished enough in mathematical physics to make half a dozen great reputations, it was not the trade to which he had been born and science would have got more out of him if he had stuck to pure mathematics — his astronomical work was nothing else. But science got enough, and a man of Poincare’s genius is entitled to his hobbies.

We pass on now to the last phase of Poincare’s universality for which we have space: his interest in the rationale of mathematical creation. In 1902 and 1904 the Swiss mathematical periodical L’Enseignement Mathematique undertook an enquiry into the working habits of mathematicians. Questionnaires were issued to a number of mathematicians,of whom over a hundred replied. The answers to the questions and an analysis of general trends were published in final form in 1912. Anyone wishing to look into the psychology of mathematics will find much of interest in this unique work and many confirmations of the views at which Poincare had arrived independently before he saw the results of the questionnaire. A few points of general interest may be noted before we quote from Poincare.

The early interest in mathematics of those who were to become great mathematicians has been frequently exemplified. To the question “At what period…and under what circumstances did mathematics seize you?” 93 replies to the first part were received; 35 said before the age of ten; 43 said eleven to fifteen; 11 said sixteen to eighteen; 3 said nineteen to twenty, and the lone laggard said twenty six.

Again, anyone with mathematical friends will have noticed that some of them like to work early in the morning (one distinguished mathematician began work at the inhuman hour of 5a.m. daily) while others do nothing till after dark. The replies on this point indicated a curious trend — possibly significant, although there are numerous exceptions: mathematicians of the northern races prefer to work in the night, while Latins favour the morning. Among night-workers prolonged concentration often brings on insomnia as they grow older and they change — reluctantly — to the morning. Felix Klein, who worked day and night as a young man, once indicated a possible way out of this difficulty. One of his American students complained that he could not sleep for thinking of his mathematics. “Can’t sleep, eh?” Klein snorted. “What choral for?” However, this remedy is not to be recommended indiscriminately; it probably had something to do with Klein’s own tragic breakdown.

Probably the most significant of the replies were those received on the topic of inspiration versus drudgery as the source of mathematical discoveries. The conclusion is that “mathematical discoveries small or great…are never born of spontaneous generation. They always presuppose a soil seeded with preliminary knowledge and will prepared by labour, both conscious and subconscious.”

Those who, like Thomas Alva Edison, have declared that genius is ninety nine percent perspiration, one percent inspiration are contradicted by those who reverse the figures. Both are right. One man remembers the drudgery while another forgets it all in the apparently discovery but both, when they analyze their impressions, admit that without drudgery and a flash of “inspiration” discoveries are not made. If drudgery alone sufficed, how is it that many gluttons for hard work who seem to know everything about some branch of science, while excellent critics and commentators, never themselves make even a small discovery! On the other hand, those who believe in “inspiration” as the sole factor in discovery or invention — scientific or literary — may find it instructive to look at an early draft of any of Shelley’s “completely spontaneous” poems (so far as these have been preserved and reproduced), or the successive versions of any of the greater levels that Balzacx inflicted on his maddened printer.

Poincare stated his views on mathematical discovery in an essay first published in 1908 and reproduced in Science et Methode. The genesis of mathematical discovery, he says, is a problem which should interest psychologists immensely, for it is the activity in which human mind seems to borrow least from the external world, and by understanding the process of mathematical thinking we may hope to reach what is most essential in the human mind.

How does it happen, Poincare asks, that there are persons who do not understand mathematics? “This should surprise us, or rather it would surprise us if we were not so accustomed to it.” If mathematics is based only on the rules of logic, such as all normal minds accept, and which only a lunatic would deny (according to Poincare), how is it that so many are mathematically impermeable? To which it may be answered that no exhaustive set of experiments substantatiating mathematical incompetence as the normal human mode has yet been published. “And, further,” he adds. “how is error possible in mathematics?” Ask Alexander Pope: “To err is human,” which is as unsatisfactory a solution as any other. The chemistry of the digestive system may have something to do with it, but Poincare prefers a more subtle explanation — one which could not be tested by feeding the “vile” body hashish and alcohol.

“The answer seems to me evident,” he declares. Logic has little to do with discovery or invention, and memory plays tricks. Memory however is not importtant as it might be. His own memory, he says without blush is bad:”Why then does not it desert me in a difficult piece of mathematical reasoning where most chess players (whose memories he assumes excellent) would be lost? Evidently because it is guided by the general course of reasoning. A mathematical proof is not a mere juxtaposition of syllogisms; it is syllogisms arranged in a certain order, and the order is more important than the elements themselves.” If he has the “intuition” of this order, memory is at a discount, for each syllogism will take its place automatically in the sequence.

Mathematical creation however does not consist merely in making new combinations of things already known, “anyone could do that, but the combinations thus made would be infinite in number and most of them entirely devoid of interest. To create consists precisely in avoiding useless combinations and in making those which are useful and which constitute only a small minority. Invention is discernment, selection.” But has not all this been said thousands of times before? What artist does not know that selection — an intangible — is one of the secrets of success? We are exactly where we were before the investigations began.

To conclude this part of Poincare’s observations it may be pointed out that much of what he says is based on an assumption which may indeed, be true but for which there is not a particle of scientific evidence. To put it bluntly he assumes that majority of human beings are mathematical imbeciles. Granting him this, we need not even accept his purely romantic theories. They belong to inspirational literature not to science. Passing to something less controversial, we shall now quote the famous passage in which Poincare describes how one of his own “greatest inspirations” came to him. It is meant to substantiate his theory of mathematical creation. Whether it does or not is left to the reader.

He first points out that technical terms need not be understood in order to follow his narrative. “What is of interest to the pyschologist is not the theorem but the circumstances.

“For fifteen days I struggled to prove that no functions analogous to those I have since called Fuchsian functions could exist. I was then very ignorant. Every day I sat down at the work table where I spent an hour or two, I tried a great number of combinations and arrived at no result.One evening, contrary to my custom, I took black coffee, I could not go to sleep, ideas swarmed up in clouds: I sensed them clashing until, to put it so, a pair would hook together to form a stable combination. By morning I had established the existence of a class of Fuchsian functions, those derived from the hypergeometric series. I had only to write up the results, which took me a few hours.

“Next I wished to represent these functions by the quotient of two series; this idea was perfectly conscious and thought out, analogy with elliptic functions guided me. I asked myself what must be the property of these series if they existed, and without difficulty I constructed the series which I called the thetafuchsian.

“I then left Caen, where I was living at the time, to participate in a geological trip sponsored by the School of Mines. The exigencies of travel made me forget my mathematical labours; reaching Coutances we took a bus for some excursion or another. The instant I put my foot on the step of the bus the idea came to me, apparently with nothing whatsoever in my previous thoughts having prepared me for it, that the transformations which I had used to define Fuchsian functions were identical with those of non-Euclidean geometry. I did not make the verifications; I should have not had the time, because once in the bus I resumed an interrupted conversation, but I felt an instant and complete certainty. On returning to Caen, I verifed the result at my leisure to satisfy my conscience.

“I then undertook the study of certain arithmetical questions without much apparent success and without suspecting that such matters should have the slightest connection with my previous studies. Disgusted at my lack of success, I went to spend a few days at the seaside and thought of something else. One day, while walking along the cliffs, the idea came to me, again with the same characterisitcs of brevity, suddenness, and immediate certainty, that the transformations of indefinite ternary quadratic forms were identical with those of non-Euclidean geometry.

“On returning to Caen, I reflected on this result and deduced its consequences; the examples of quadratic forms showed me that there were Fuchsian groups other than those corresponding to the hypergeometric series: I saw that I could apply to them the theory of thetafuchsian functions, and hence that there existed thetafuchsian functions other than those derived from the hypergeometric series, the only ones I had known up till then. Naturally, I set myself the task of constructing all these functions. I conducted a systematic siege and, one after another, carried all the outworks; there was however one which still held out and whose fall would bring about that of the whole position. But all my efforts were served only to make me better acquainted with the difficulty, which in itself was something. All this work was perfectly conscious.

“At this point, I left for Mont-Valerien, where I was to discharge my military service. I had therefore very different preoccupations. One day, while crossing the boulevard, the solution of the difficulty which had stopped me appeared to me all of a sudden. I did not seek to go into it immediately, and it was only after my service that I resumed the question. I had all the elements, and had only to assemble and order them. So I wrote out my definitive memoir at ohe stroke and with no difficulty.”

Many other examples of this sort of thing could be given from his own work, he says, and from that of other mathematicians as reported in L’Enseignement Mathematique. From his experiences, he believes that this semblance of “sudden illumination {is} a manifest sign of previous long subconscious work,” and he proceeds to elaborate his theory of the subconscious mind and its part in mathematical creation. Conscious work is necessary as a trigger to fire off the cumulated dynamite which the subconscious has been emitting — he does not put it so, but what he says amounts to the same. But what is gained in the way of rational explanation if, following Poincare, we foist off the “subconscious mind,” or the “subliminal self,” the very activities which it is our object to understand? Instead of endowing this mysterious agent with a hypothetical tact enabling it to discriminate between the “exceedingly numerous” possible combinations presented (how, Poincare does not say) for its inspection, and calmly saying that the “subconscious” rejects all but the “useful” combinations because it has a feeling for symmetry and beauty, sounds suspciously like solving the initial value problem by giving it a more impressive name. Perhaps this is exactly what Poincare intended, for he once defined mathematics as the art of giving the same name to different things; so here he may be rounding out the symmetry of his view by giving different names to the same thing. It seems strange that a man who could have been satisfied with such a “psychology” of mathematical invention was the complete skeptic in religious matters that Poincare was. AFter Poincare’s brilliant lapse into psychology skeptics may well despair of ever disbelieving anything.

During the first decade of the twentieth century Poincare’s fame increased rapidly and he came to be looked upon, especially in France, as an oracle on all things mathematical. His pronouncements on all manner of questions, from politics to ethics, were usually direct and brief, and were as accepted as final by the majority. As almost invariably happens after a great man’s extinction, Poincare’s dazzling reputation during his lifetime passed throught a period of partial eclipse in the decade following his death. But his intuition for what was likely to be of interest to a later generation is already justifying itself. To take but one instance of many, Poincare was a vigorous of opponent of the theory that all mathematics can be rewritten in terms of the most elementary notions of classical logic; something more than logic, he believed, makes mathematics what it is. Although he did not go so far as the current intuitionist school he seems to have believed, as that school does, that at least some mathematical notions precede logic, and if one is to be derived from the other it is logic which came out of mathematics, not the other way about. Whether this is to the ultimate creed remains to be seen, but at present it appears as if the theory which Poincare assailed with all the irony at his command is not the final one, whatever may be its merits.

Except for a distressing illness during his last four years Poincare’s busy life was tranquil and happy. Honors were showered upon him by all the leading learned societies of the world, and in 1906, at the age of 52, he achieved the highest distinction possible to a French scientist, the Presidency of the Academy of Sciences. None of all this inflated his ego, for Poincare was truly humble and unaffectedly simple. He knew of course that he was without a close rival in the days of his maturity, but he could also say without a trace of affection that he knew nothing compared to what it is to be known. He was happily married and had a son and three daughters in whom he took much pleasure, especially when they were children. His wife was a great-granddaughter of Etienne Geoffroy Saint-Hilaire, remembered as the antagonist of that pugnacious comparative anatomist Cuvier. One of Poincare’s passions was symphonic music.

At the International Mathematical Congress of 1908, held at Rome, Poincare was prevented by illness from reading his stimulating (if premature) address on The Future of Mathematical Physics. His trouble was hypertrophy of the prostate, which was relieved by Italian surgeons, and it was thought that he was permanently cured. On his return to Paris he resumed his work as energetically as ever. But in 1911 he began to have presentiments that he might not live long, and on December 9 wrote asking the editor of a mathematical journal iwhether he would accept an unpublished memoir — contrary to the usual custom —- on a problem which Poincare considered of the highest importance:”….at my age, I may not be able to solve it, and the results obtained, susceptible or putting researchers on a new and unexpected path, seem to me full of promise, in spite of the deceptions they have caused me, that I should resign myself to sacrificing them…” He had spent the better part of two fruitless years trying to overcome the difficulties.

A proof of the theorem which he conjectured would have enabled him to make a striking advance in the problem of three bodies; in particular it would have permitted him to give the existence of an infinity of periodic solutions in cases more general than those hitherto considered. The desired proof was given shortly after the publication of Poincare’s “unfinished symphony” by a young American mathematician, George David Birkhoff.

In the spring of 1912, Poincare fell ill again and underwent a second operation on July 9. The operation was successful, but on July 17 he died very suddenly from an embolism while dressing. He was in the fifty ninth year of his age and at the height of his powers —- “the living brain of the rational sciences,” in the words of Painleve.

Regards,

Nalin Pithwa

STEM for Australia

http://www.stemaustralia.edu.au/#:~:text=STEM%20Australia%20is%20a%20combination,Engineering%20and%20Mathematics%20in%20Australia.

Wisdom of George Polya

A great discovery solves a great problem. But there is a grain of discovery in the solution of any problem.

— George Polya, How to Solve it

Wisdom of Yajurveda

As are the crests on the crowns of peacocks,

As are the gems on the hoods of cobras,

So is mathematics at the helm of all Sciences.

— Yajurveda circa 400 BC

Wisdom of Bertrand Russell

Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of paintings or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. — Bertrand Russell (1902)

Euclid, the one and only one

Euclid alone

Has looked on Beauty bare Fortunate they

Who, though once only and then but far away,

Have heard her massive sandal set on stone.

— Edna St. Vincent Millay (1923)

Wisdom of Paul Erdos

If numbers are not beautiful, I don’t know what is. Paul Erdos

Wisdom of Bill Thurston, Fields Medallist, Topologist

Thinking is seeing. Bill Thurston