Category Archives: pure mathematics

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:


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


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


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.


Nalin Pithwa

Wisdom of Bill Thurston, Fields Medallist, Topologist

Thinking is seeing. Bill Thurston

1+2+3+… = – 1/12 (Euler)

B.S. in Mathematics: IIT Bombay program:

Note that the admission is through IITJEE Advanced only.

Nalin Pithwa.

John Conway, Simons Foundation, Science Lives, Mathematics, Mathematicians

via John Conway, Simons Foundation, Science Lives, Mathematics, Mathematicians

best explanation of epsilon delta definition

Refer any edition of (i) Calculus and Analytic Geometry by Thomas and Finney (ii) recent editions which go by the title “Thomas’ Calculus”. If you need, you will have to go through the previous stuff (given in the text) on “preliminaries” and/or functions also. For Sets, Functions and Relations, I have also presented a long series of articles on this blog.


Another little portrait of Hermann Weyl

(I simply loved the following introduction of Hermann Weyl; I am sharing it verbatim; as described by Peter Pesic in the book Mind and Nature by Hermann Weyl, Princeton University Press; available in Amazon India)

“It’s a crying shame that Weyl is leaving Zurich. He is a great master”. Thus, Albert Einstein described Hermann Weyl (1885-1955) who remains a legendary figure, “one of the greatest mathematicians of the first half of the twentieth century…No other mathematician could claim to have initiated more of the theories that are now being explored,” as (Sir) Michael Atiyah (had) put it. Weyl deserves far wider renown not only for his importance in mathematics and physics but also because of his deep philosophical concerns and thoughtful writing. To that end, this anthology gathers together some of Weyl’s most important general writings, especially those that have become unavailable, have not previously been translated into English, or were unpublished. Together, they form a portrait of a complex and fascinating man, poetic and insightful, whose “vision has stood the test of time>”

This vision has deeply affected contemporary physics, though Weyl always considered himself a mathematician, not a physicist. The present volume emphasizes his treatment of philosophy and physics, but another complete anthology could be made of Weyl’s general writings oriented more directly toward mathematics. Here, I have chosen those writings that most accessibly show how Weyl synthesized philosophy, physics and mathematics.

Weyl’s philosophical reflections began in early youth. He recollects vividly the worn copy of a book about Immaneul Kant’s Critique of Pure Reason he found in the family attic and read avidly at age fifteen. “Kant’s teaching on the “ideality of space and time” immediately took powerful hold of me, with a jolt I was awakened from my ‘dogmatic slumber,’ and the mind of the body found the world being questioned in radical fashion.” At the same time, he was drinking deep of great mathematical works. As “a country lad of eighteen,” he fell under the spell of David Hilbert, whom he memorably described as a Pied Piper “seducing so many rats to follow him into the deep river of mathematics”; the following summer found Weyl poring over Hilbert’s Report on the Theory of Algebraic Numbers during the “happiest months of my life.” As these stories reveal, Weyl was a very serious man; Princeton students called him “holy Hermann” among themselves, mocking a kind of earnestness probably more common in Hilbert’s Gottingen. There, under brilliant teachers, who also included Felix Klein and Hermann Minkowski, Weyl completed his mathematical apprenticeship. Forty years later, at the Princeton Bicentennial in 1946, Weyl gave a personal overview of this period and of the first discoveries that led him to find a place of distinction at Hilbert’s side. This address. never before published, may be a good place to begin if you want to encounter the man and hear directly what struck him most. Do not worry if you find the mathematical references unfamiliar; Weyl’s tone and angle of vision express his feelings about the mathematics (and mathematicians) he cared for in unique and evocative ways; he describes “Koebe the rustic and Brouwer the mystic” and the “peculiar gesture of his hands” Koebe used to define Riemann surfaces, for which Weyl sought “a more diginfied definition.”

In this address, Weyl also vividly recollects how Einstein’s theory of general relativity affected him after the physical and spiritual desolation he experienced during the Great War. “In 1916, I had been discharged from the German army and returned to my job in Switzerland. My mathematical mind was as blank as any veteran’s and I did not know what to do. I began to study algebraic surfaces, but before I had gotten far, Einstein’s memoir came into my hands and set me afire. Both Weyl and Einstein had lived in Zurich and taught at its ETH (Eidgenoschule Technishce Hochschule) during the very period Einstein was struggling to find his generalized theory, for which he needed mathematical help. This was golden period for both men, who valued the freer spirit they found in Switzerland, compared to German, Einstein adopted Swiss citizenship, completed his formal education in his new country, and then worked at its patent office. Among the first to realize  the full import of Einstein’s work, especially its new, more general, phase, Weyl gave lectures on it at the ETH in 1917, published in his eloquent book Space Time and Matter (1918). Not only the first (and perhaps the greatest) extended account of Einstein’s general relativity, Weyl’s book was immensely influential because of its profound sense of perspective, great expository clarity, and indications of directions to carry Einstein’s work further. Einstein himself praised the book as a “symphonic masterpiece.”

As the first edition of Space Time Matter went to press, Weyl reconstructed Einstein’s ideas from his own mathematical perspective and came upon a new and intriguing possibility, which Einstein immediately called ” a first class stroke of genius”. Weyl describes this new idea in his essay “Electricity and Gravitation” (1921), much later recollecting some interesting personal details in his Princeton address. There, Weyl recalls explaining to a student, Willy Scherer, “that vectors when carried around by parallel displacement may return to their starting point in changed direction. And, he asked also with changed length? Of course, I gave him the orthodox answer {no} at that moment, but in my bosom gnawed the doubt.” To be sure. Weyl wrote this remembrance thirty years later, which thus may or may not be a perfectly faithful record of the events; nevertheless, it represents Weyl’s own self-understanding of the course of his thinking, even if long after the fact. Though Weyl does not mention it, this conversation was surrounded by a complex web of relationships: Weyl’s wife Helene was deeply involved with Willy’s brother Paul, while Weyl himself was the lover of Erwin Schrodinger’s wife, Amy. These personal details are significant here because Weyl himself was sensitive to the erotic aspects of scientific creativity in others, as we will see in his commentary on Schrodinger, suggesting that Weyl’s own life and were similarly intertwined.

In Space Time Matter, Weyl used the implications of parallel transport of vectors to illuminate the inner structure of the theory Einstein had originally phrased in purely metric terms, meaning the measurement of distances between points, on the model of the Pythagorean theorem. Weyl questioned the implicit assumption that behind this metrical structure is a fixed, given distance scale, or “gauge.” What if the direction as well as the length of meter sticks (and also the standard second given by clocks) were to vary at different places in space-time, just as railway gauge varies from country to country? Perhaps, Weyl’s concept began with this kind of homely observation about the “gauge relativity” in the technology of rail travel, well-known to travelers in those days, who often had to change trains at frontiers between nations having different, incompatible railway gauges. By considering this new kind of relativity, Weyl stepped even beyond the general coordinate transformations Einstein allowed in his general theory so as to incorporate what Weyl called relativity of magnitude.

In what Weyl called an “affinely connected space,” a vector could be displaced parallel to itself, at least to an infintely nearby point. As he realized after talking to Willy Scherrer, in such a space a vector transported around a closed curve might return to its starting point with changed direction as well as length (which he called “non-integrability,” as measured mathematically by the “affine connection.”) Peculiar as this changed length might seem, Weyl was struck by the mathematical generality of this possibility, which he explored in what he called his “purely infinitesimal geometry,” which emphasized infinitesimal displacement as the foundation in terms of which any finite-displacement needed to be understood. As he emphasized the centrality of the infinite in mathematics, Weyl also placed the infinitesimal before the finite.

Weyl also realized that his generalized theory gave him what seemed a natural way to incorporate electromagnetism into the structure of space-time, a goal that had eluded Einstein, whose theory treated electromagnetism along with matter as mass-energy sources that caused space-time curvature but remained separate from space-time itself. Here Weyl used the literal “gauging” of distances as the basis of a mathematical analogy; his reinterpretation of these equations led naturally to a gauge field he could then apply to electromagnetism, from which Maxwell’s equations now emerged as intrinsic to the structure of space-time. Though Einstein at first hailed this ‘stroke of genius’, soon he found what he considered a devastating objection: because of the non-integrability of Weyl’s gauge field, atoms would not produce the constant, universal spectral lines we actually observe: atoms of hydrogen on Earth give the same spectrum as hydrogen observed telescopically in distant stars. Weyl’s 1918 paper announcing his new theory appeared with an unusual postscript by Einstein, detailing his objection, along with Weyl’s reply that the actual behaviour of atoms in turbulent fields, not to speak of measuring rods and clocks, was not yet fully understood. Weyl noted that his theory used light signals as a fundamental standard, rather than relying on supposedly rigid measuring rods and idealized clocks, whose atomic structures was in some complex state of accommodation to ambient fields.

In fact, the atomic scale was the arena in which quantum theory was then emerging. Here began the curious migration of Weyl’s idea from literatly regauging length and time to describing some other realm beyond space-time. Theordor Kaluza (1922) and Oskar Klein (1926) proposed a generalization of general relativity using a fifth dimension to accommodate electromagnetism. In their theory, Weyl’s gauge factor turns into a phase factor, just as the relative phase of traveling waves depends on the varying dispersive properties of the medium they traverse. If so, Weyl’s gauge would no longer be immediately observable (as Einstein’s objection asserted) because the gauge affects only the phase, not the observable frequency of atomic spectra.

At first, Weyl speculated that his 1918 theory gave support to the radical possibility that “matter” is only a form of curved, empty space (a view John Wheeler championed forty years later). Here Weyl doubltess remembered the radical opinions of Michael Faraday and James Clerk Maxwell, who went so far as to consider so-called matter to be a nexus of immaterial lines of force.

Weyl then weighed these mathematical speculations against the complexities of physical experience. Though he still believed in his fundamental insight that gauge invariance was crucial, by 1922 Weyl realized that it needed to be re-considered in light of the emergent quantum theory. Already in 1922, Schrodinger pointed out that Weyl;s idea could lead to a new way to understand quantization. In 1927, Fritz London argued that gravitational scale factor implied by Weyl’s 1918 theory, which Einstein had argued was unphysical, actually makes sense as the complex phase factor essential to quantum theory.

As Schrodinger struggled to formulate his wave equation, at many points he relied on Weyl for mathematical help. In their liberated circles, Weyl remained a valued friend and colleague even while being Anny Schrodinger’s lover. From that intimate vantage point, Weyl observed that Erwin “did his great work during a late erotic outburst in his life,” an intense love affair simultaneously with Schrodinger’s struggle to find a quantum wave equation. But, then as Weyl inscribed his 1933 Christmas gift to Anny and Erwin (a set of erotic illustrations to Shakespeare’s Venus and Adonis), “the sea has bounds but deep desire has none.”

Weyl’s insight into the nature of quantum theory comes forward in a pair of letters he and Einstein wrote in 1922, here reprinted and translated for the first time, responding to a journalist’s question about the significance of the new physics. Einstein dismisses the question: for him in 1922, relativity theory changes nothing fundamental in out view of the world, and that is that. Weyl takes the question more seriously, finding a radically new insight not so much in relativity theory as in the emergent quantum theory, which Weyl already understood as asserting that “the entire physics of matter is statistical in nature,” showing how clearly he understood this decisive point several years before the formulation of the new quantum theory in 1923-26 by Max Born, Werner Heisenberg, Pascual Jordal, P.A.M. Dirac, and Schrodinger. In his final lines, Weyl also alludes to his view of matter as agent, in which he ascribed to matter an innate activity that may have helped him understand and accept the spontaneity and indeterminacy emerging in quantum theory. This view led Weyl to reconsider the significance of the concept of a field. As he wrote to Wolfgang Pauli in 1919, “field physics, I feel, really plays only the role of ‘world geometry’, in matter there resides still something different, (and) real, that cannot be grasped causally, but that perhaps should be thought of in the image of ‘independent decisions,’ and that we account for in physics by statistics.

In the years around 1920, Weyl continued to work out the consequences of this new approach. His conviction about the centrality of consciousness as intuition and activity deeply influenced his view of matter. As the ego drives the whole world known to consciousness, he argued that matter is analogous to the ego, the effects of which, despite the ego itself being non-spatial, originate via the body at a given point of the world continuum. Whatever the nature of this agents, which excites the field, might be — perhaps life and will — in physics we only look at the field effects caused by it. This took him in a direction very different from the vision of matter reduced to pure geometry he had entertained in 1918. Writing to Felix Klein in 1920, Weyl noted that “field physics no longer seems to me to be the key to reality; but, the field, the ether, is to me only a totally powerless transmitter by itself of the action, but matter rests beyond the field and is the reality that causes its states.” Weyl described his new view in 1923 using an even more striking image: “Reality does not move into space as into a right-angled tenement house along which all its changing play of forces glide past without leaving any trace; but, rather as the snail, matter itself builds and shapes this house of its own.” For Weyl now, fields were “totally powerless transmitters” that are not really existent or effectual in their own right, but only a way of talking about states of matter that are the locus of fundamental reality. Though Weyl still retained fields to communicate interactions, his emphasis that the reality of “matter rests beyond the field” may have influenced Richard Feynman and Wheeler two decades later in their own attempt to remove “fields” as independent beings. Weyl also raised the question of whether matter has some significant topological structure on the subatomic scale, as if topology were a kind of activation that brings static geometry to life, analogous to the activation ego infuses into its world. Such topological aspects of matter only emerged as an important frontier of contemporary investigation fifty years later. Looking back from 1955 at his original 1918 paper, Weyl noted that he “had no doubt” that the correct context of his vision of gauge theory was “not, as I believed in 1918, in the intertwining of electromagnetism and gravity” but in the “Schrodinger-Dirac potential \Psi of the electron-positron field…The strongest argument for my theory seems to be this that gauge-invariance corresponds to the conservation of electric charge in the same way that co-ordinate invariance corresponds to the conservation of energy and momentum,” the insight that Emmy Noether’s famous theorem put at the foundations of quantum field theory. Nor did Weyl himself stop working on his idea; in 1929 he published an important reformulating his idea in the language of what today are called gauge fields; these considerations also led him to consider fundamental physical symmetries long before the discovery of the violation of parity in the 1950’s. The “Weyl two-component neutrino field” remains a standard description of neutrinos, all of which are “left-handed” (spin always opposed to direction of motion), as all anti-neutrinos are “right-handed.” In 1954, (a year before Weyl’s death, but apparently not known to him), C. N. Yang, R. Mills, and others took the next steps in developing gauge fields, which ultimately became the crucial element in the modern “standard model” of particle physics that triumphed in the 1970’s, unifying strong, weak, and electromagnetic interactions in ways that realized Weyl’s distant hopes quite beyond his initial expectations.

In the years that Weyl continued to try to find a way to make his idea work, he and Einstein under went a curious exchange of positions. Originally, Einstein thought Weyl was not paying enough attention to physical measuring rods and clocks because Weyl used immaterial light beams to measure space-time. As Weyl recalled in a letter of 1952, “I thought to be able to answer his concrete objections, but in the end he said: “Well, Weyl let us stop this. For what I actually have against your theory is: ‘It is impossible to do physics like this (that is, in such a speculative fashion, without a guiding intuitive physical principle)!”‘ Today, we have probably changed our viewpoints in this respect. E. believes that in this domain the chasm between ideas and experience is so large, that only mathematical speculation (whose consequences, of course, have to be developed and confronted with facts) gives promise of success, while my confidence in pure speculation has diminished and a closer with quantum physical experience seems necessary, especially as in my view it is not sufficient to blend gravitation and electricity to one unity, but that the wave fields of the electron (and whatever there may still be of nonreducible elementary particles) must be included.

Ironically, Weyl the mathematician finally sided with the complex realities of physics, whereas Einstein the physicisit sought refuge in unified field theories that were essentially mathematical. Here is much food for thought about the philosophic reflections each must have undertaken in his respective soul-searching and that remain important now, faced with the possibilities and problems of string theory, loop quantum gravity, and other theoretical directions for which sufficient expeimental evidence may long remain unavailable.

Both here and through out his life, Weyl used philosophical reflection to guide his theoretical work, preferring “to approach a question through a deep analysis of the concepts it involves rather than by blind computations,” as Jean Dieudonne put it. Though others of his friends, such as Einstein and Schrodinger, shared his broad humanisitc education and philosophical bent, Weyl tended to go even further in this direction. As a young student in Gottingen, Weyl had studied with Edmund Husserl (who had been a mathematician before turning to philosophy), with whom Helene Weyl had also studied.

Weyl’s continuing interest in phenomenological philosophy marks many of his works, such as his 1927 essay on “Time Relations in the Cosmos, Proper Time, Lived Time, and Metaphysical Time,” here reprinted and translated for the first time. The essay’s title indicates its scope, beginning with his interpretation of the four-dimensional space-time Hermann Minkowski introduced in 1908, which Weyl then connects with human time consciousness (also a deep interest of Husserl’s). Weyl treats a world point not merely as a mathematical abstraction but as situating a “point-eye,” a living symbol of consciousness peering along its world line. Counterintuitively, that point-eye associates the objective with the relative, the subjective with the absolute.

Weyl used this striking image to carry forward a mathematical insight that had emerged earlier in the considerations about the nature of the continuum. During the early 1920’s, Weyl was deeply drawn to L.E.J. Brouwer’s advocacy of intuition as the critical touchstone of modern mathematics. Thus, Brouwer rejected Cantor’s transfinite numbers as not intuitable, despite Hilbert’s claim that “no one will drive us from paradise which Cantor created for us.” Hilbert argued that mathematics should be considered purely formal, a great game in which terms like “points” or “lines” could be replaced with arbitrary words like ‘beer mugs” or ‘tables” or with pure symbols, so long as the axiomatic relationships between the respective terms do not change. Was this, then, the “deep river of mathematics” into which Weyl thought this Pied Piper had lured him and so many other clever young rats?

By the mid-1920’s, Weyl was no longer an advocate of Brouwer’s views (though still reaffirming his own 1913 work on the continuum). In his magisterial Philosophy of Mathematics and Natural sciences (written in 1927 but extensively revised in 1949), Weyl noted that “mathematics with Brouwer gains its highest intuitive clarify…It cannot be denied, however, that in advancing to higher and more general theories the inapplicability of the simple laws of logic eventually results in an almost unbearable awkwardness. And the mathematician watches with pain the larger part of his towering edifice which he believed to be built of concrete blocks dissolve into a mist before his eyes.

Even so, Weyl remained convinced that we should not consider a continuum (such as the real numbers between 0 and 1) as an actually completed and infinite set but only as capable of endless subdivision. This understanding of the “potential infinite” recalls Aristotle’s critique of the ‘actual infinite. In his 1927 essay on “Time Relations,” Weyl applied this view to time as a continuum. Because an infinitely small point could be generated from a finite interval only through actually completing an infinite process of shrinkage, Weyl applies the same argument to the presumption that the present instant is a “point in time.” He concludes that “there is no pointlike Now and also no exact earlier and later.” Weyl’s arguments about the continuum have the further implication that the past is never completely determined, any more than a finite, continuous interval is ever exhaustively filled; both are potentially infinite because always further divisible. If so, the past is not fixed or unchangeable and continues to change, a luxuriant, ever-proliferating tangle of “world tubes,” as Weyl called them, “open into the future and again and again a fragment of it is lived through.” This intriguing idea is psychologically plausible: A person’s past seems to keep changing and ramifying as life unfolds, the past today seems different than it didi yesterday. As a character in Fauklner put it, “the past is never dead. It’s not even past.”

In Weyl’s view, a field is intrinsically continuous, endlessly subdividable, and hence, an abyss in which we never come to an ultimate point where a decision can be made. To be or not to be? Conversely pointlike, discrete matter is a locus of decisive spontaneity because it is not predictable through continuous field laws, only observable statistically. As Weyl wrote to Pauli in 1919, “I am firmly convinced that statistics is in principle something independent from causality, the ‘law’; because it is in general absurd to imagine a continuum as something like a finished being.” Because of this independence, Weyl continued in 1920:-

the future will act on and upon the present and it will determine the present more and more precisely; the past is not finished. Thus, the fixed pressure of natural causality disappears and there remains, irrespective of the validity of the natural laws, a space for autonomous and causally absolutely independent decisions. lI consider the elementary quanta of matter to be the place of these decisions.

“Lived time,” in Weyl’s interpretation, keeps evoked the past into furhter life, even as it calls the future into being. Weyl’s deep thoughts may still reap the further exploration, they have not received so far.

Weyl also contributed notably to the application of general relativity to cosmology. He found new solutions to Einstein’s equation and already in 1923 calculated a value for the radius of the universe of roughly one billion light years, six years before Edwin Hubble’s systematic measurements provided what became regarded as conclusive evidence that our galaxy is one among many. Weyl also reached a seminal insight, derived from both his matheamtical and his philosophical considerations, that the topology of the universe is “the first question in all speculations on the world as a whole.” This prescient insight was taken up only in the 1970’s and remains today at the forefront of cosmology, still unsolved and as important as Weyl thought. He also noted that relativistic cosmology indeed “left the door open for possibilities of every kind.” The mysteries of dark energy and dark matter remind us of how much still lies beyond that door. Then too, we still face the questions Weyl raised regarding the strange recurrence throughout cosmology of the “larger numbers” like 10^{20} and 10^{40} (seemingly as ratios between cosmic and atomic scales), later rediscovered by Dirac.

Other of Weyl’s ideas long ago entered and transformed the mainstream of physics, characteristically bridging the mathematical and physical through the philosophical. He considered his greatest mathematical work the classification of the semisimple groups of continuous symmetries (Lie groups), which he later surveyed in The Classical Groups: Their Invariants and Representations (1938). In the introduction to this first book, he wrote in English, Weyl noted that “the gods have imposed upon my writing the yoke of a foreign tongue that was not sung at my cradle.” But even in his adopted tongue he does not hesitate to criticize hte “too thorough technicalization of mathematical research” In America that has led to a ” mode of writing which must give the reader the impression of being shut up in a brightly illuminated cell where every detail sticks out with the same dazzling clarity, but without relief. I prefer the open landscape under a clear sky with its depth of perspective, where the wealth of sharply defined nearby details gradually fades away toward the horizon.” Such writing exemplifies Weyl’s uniquely eloquent style.

Soon after quantum theory had first been formulated, Weyl used his deep mathematical perspective to shape The Theory of Groups and Quantum Mechanics (1928). It is hard to overstate the importance of his marriage of the mathematical theory of symmetry to quantum theory, which has proved ever more fruitful, with no end in sight. At first, as eminent and hard-headed physicist as John Slater resisted the “group-pest” as if it were a plague of abstractness. But Weyl, along with Eugene Wigner, prevalied because the use of group theory gave access to the symmetries essential for formulating all kinds of physical theories, from crystal lattices to multiplets of fundamental particles. It was this depth and generality that moved Julian Schwinger to ‘read and to re-read that book, each time progressing a little farther, but I cannot say that I ever —- not even to this day — fully mastered it.”  Thus, Schwinger considered Weyl “one of my gods,” note merely an outstanding teacher, because “the ways of gods are mysterious, inscrutable, and beyond the comprehension of ordinary mortals.” This from someone regarded as god-like by many physicists because of his own inscrutable powers. Weyl’s insights about the fundamental mathematical symmetries led Schwinget and others decades later to formulate the TCP theorem, which expresses the fundamental indenticality between particles and anti-particles under the combined symmetries of time reversal (T), charge conjugation (reversal of the sign of charge, C) and parity (mirror) reversal (P), In one of his most powerful interventions in physics, Weyl used such symmetry principles to argue that Dirac’s newly proposed (and as yet unobserved) “holes” (anti-electrons) could not be (as Dirac had suggested) protons, which are almost two thousand times heavier than electrons. Weyl showed mathematically that anti-electrons had to have the same mass as electrons, though having opposite charge; this was later confirmed by cosmic ray observations. Weyl’s purely mathematical argument struck Dirac, who drew from this experience his often cited principle that “it is more important to have beauty in one’s equations than to have them fit experiment,” a principle that continues to be an important touchstone for many physicists. Even though Weyl’s mathematics moved Dirac to this radical declaration, Weyl’s own turn away from mathematical speculation about physics raises the question whether in the end to prefer beautiful mathematics to the troubling complexities of experience.

Whether a “god” or no, Weyl seemed to feel that the philosophical enterprise cannot remain on the godlike plane but really requires the occasions of human conversation. The two largest works in this anthology contain the rich harvest of Weyl’s long standing interest in expressing his ideas to a broader audience, both began as lecture series, thus doubly public, both spoken and written. To use the apt phrase of his son Michael, The Open World (1932) contains “Hermann’s dialogues with God” because here the mathematician confronts his ultimate concerns. These do not fall into the traditional religious traditions but are much closer in spirit to Spinoza’s rational analysis of what he called “God or nature,” so important for Einstein as well. As Spinoza considered the concept of infinity fundamental to the nature of God, Weyl defines “God as the completed infinite.” In Weyl’s conception, God is not merely a mathematician but is mathematics itself because “mathematics is the science of the infinite,” engaged in the paradoxical enterprise of seeking “the symbolic comprehension of the infinite with human, that is finite, means.” In the end, Weyl concludes that this God ‘cannot and will not be comprehended” by the human mind, even though “mind is freedom within the limitations of existence, it is open toward the infinite.” Nevertheless, “neither can God penetrate into man by revelation, nor man penetrate to Him by mystical perception. The completed infinite we can only represent in symbols.” In Weyl’s praise of openness, this freedom of the human mind begins to seem even higher than the completed infinity essential to the meaning of God. Does not his argument imply that God, as actual infinite, can never be actually complete, just an infinite time will never have passed, however long one waits? And if God’s actuality will never come to pass, in what sense could or does or will God exist at all? Perhaps, God, like the continuum or the field, is an infinite abyss that needs completion by the decisive seed of matter, of human choice.

Weyl inscribes this paradox and its possibilities in his praise of the symbol, which includes the mathematical no less than the literary, artistic, poetic, thus bridging the presumed chasm between the “two cultures.” At every turn in his writing, we encounter a man of rich and broad culture, at home in many domains of human thought and feeling, sensitive to its symbols and capable of expressing himself beautifully. He moves so naturally from quoting the ancients and moderns to talking about space-time diagrams, thus showing us something of his innate turn of mind, his peculiar genius. His quotations and reflections are not mere illustrations but show the very process by which his thought lived and moved. His philosophical turn of mind helped him reach his own finest scientific and mathematical ideas. His self-deprecating disclaimer that he thus “wasted his time” might be read as irony directed to those who misunderstood him, the hardheaded who had no feeling for those exalted ideas and thought his philosophizing idle or merely decorative. Weyl gained perspective, insight, and altitude by thinking back along the never unfolding past and studying its great thinkers, whom he used to help him soar, like a bird feeling the air under its wings.

In contrast, Weyl’s lectures on Mind and Nature, published only two years later (1934), have a less exalted tone. The difference shows his sensitivity to the changing times. Though invited to return to Gottingen in 1918, he preferred to remain in Zurich, finally in 1930, he accepted the call to succeed Hilbert, but almost immediately regretted it. The Germany he returned to had become dangerous for him, his Jewish wife, and his children. Unlike some who were unable to confront those ugly realities, Weyl was capable of political clear-sightedness, by 1933 he was seeking to escape Germany. His depression and uncertainty in the face of those huge decisions shows another side of his humanity, as Richard Courant put it, “Weyl is actually in spite of his enormously broad talents an inwardly insecure person, for whom nothing is more difficult than to make a decision which will have consequences for his life, and who mentally is not capable of dealing with the weight of such decisions, but needs a strong support somewhere. That anxiety and inner insecurity gives Weyl’s reflections their existential force. As he himself struggled along his own world line through endlessly ramifying doubts, he came to value the spontaneity and decisiveness he saw in the material world.

Weyl’s American lectures marked the start of a new life, beginning with a visiting professorship at Princeton (1928-1929), where he revised his book on group theory and quantum mechanics in the course of introducing his insights to this new audience.  Where in 1930 Weyl’s Open World began with God, in 1933 his lectures on Mind and Nature start with human subjectivity and sense perception. Here, symbols help us confront a world that “does not exist in itself, but is merely encountered by us as an object in the correlative variance of subject and object.” For Weyl, mathematical and poetic symbols may disclose a path through the labyrinth of “mirror land,” a world that may seem ever more distorted, unreal on many fronts. Though Weyl discerns “an abyss which no realistic conception of the world can span” between the physical processes of the brain and the perceiving subject, he finds deep meaning in “the enigmatic two fold nature of the ego, namely that I am both: on the one hand a real individual which performs real psychical acts, the dark, striving human being that is cast out into the world and its individual fate, on the other hand light which beholds itself, intuitive vision, in whose consciousness that is pregnant with images and endows with meaning, the world opens up. Only in this ‘meeting’ of consciousness and being both exist, the world and I.”

Weyl treats relativity and quantum theory as the latest and most suggestive symbolic constructions we make to meet the world. The dynamic character of symbolism endures, even if the particular symbols change, “their truth refers to a connected system that can be confronted with experience only as a whole.” Like Einstein, Weyl emphasized that physical concepts as symbols “are constructions within a free realm of possibilities,” freely created by the human mind. “Indeed, space and time are nothing in themselves, but only certain order of the reality existing and happening in them.” As he noted in 1947, “it has now become clear that physics needs no such ultimate objective entities as space, time, matter, or ‘events’, or the like, for its constructions symbols without meaning handled according to certain rules are enough.” In Mind and Nature, Weyl notes that in nature itself, as (quantum) physics constructs it theoretically, the dualism of object and subject, of law and freedom, is already most distinctly predesigned.” As Niels Bohr put it, this dualism rests on “the old truth that we are both spectators and actors in the great drama of existence.”

After Weyl left Germany definitively for Princeton in 1933, he continued to reflect on these matters. In the remaining selections, one notes him retelling some of the same stories, quoting the same passages from great thinkers of the past, repeating an idea he had already said elsewhere. These repetitions posed a difficult problem, for the latest essays contain some interesting new points along with the old, Because of this, I decided to include these later essays, for Weyl’s repetitions also show him reconsidering. Reiterating a point in a new or larger context may open further dimensions. Then too, we as readers are given another chance to think about Weyl’s points and also see where he held to his earlier ideas and where he may have changed. For he was capable of changing his mind, more so that Einstein, whose native stubbornness may well have contributed to his unyielding resistance to quantum theory. As noted above, Weyl was far more able to entertain and even embrace quantum views, despite their strangeness, precisely because of his philosophical openness.

Weyl’s close reading of the past and his philosophical bent inspired his continued openness. In his hitherto unpublished essay “Man and the Foundations of Science” (written about 1949), Weyl describes an ocean traveler who distrusts the bottomless sea and therefore clings to the view of the disappearing coast as long as there is in sight no other coast toward which he moves. I shall now try to describe the journey on which the old coast has long since vanished below the horizon. There is no use in staring in that direction any longer.” He struggles to find a way to speak about “a new coast [that] seems dimly discernible, to which I can point by dim words only and may be it is merely a bank of fog that deceives me.” Here symbols might be all we have, for “it becomes evident that now the words ‘in reality’ must be put between quotation marks, we have a symbolic construction, but nothing which we could seriously pretend to be the true real world.” Yet even legerdemain with symbols cannot hide the critical problem of the continuum. “The sin committed by the set theoretic mathematician is his treatment of the field of possibilities open into infinity as if it were a completed whole all members of which are present and can be overlooked with one glance. For those whose eyes have been opened to the problem of infinity, the majority of his statements carry no meaning. If the true aim of the mathematician is to master the infinite by the finite means, he has attained it by fraud only — a gigantic fraud which, one must admit, works as beautifully as paper money.” By his reaffirmation of his critique of the actual infinite, we infer that Weyl continued to hold his radical views about “lived time,” especially that “we stand at that intersection of bondage and freedom which is the essence of man himself.”

Indeed, Weyl notes that he had put forward this relation between being and time years before Martin Heidegger’s famous book on that subject appeared. Weyl’s account of Heidegger is especially interesting because of the intersection between their concerns, no less than their deep divergences. Yet Weyl seemingly could not bring himself to give a full account of Heidegger or of his own reactions, partly based on philosophical antipathy, partly (one infers) from his profound distaste for Heidegger’s involvement with the Nazi regime. Though he does not speak of it, Weyl may well also have known of the way Heidegger abandoned their teacher, Husserl, in those dark days. Most of all, Weyl conveys his annoyance that Heidegger had botched important ideas that were important to Weyl himself and, in the process, that Heidegger lost sight of the future of science. “Taking up a crucial term, they both use, Weyl asserts that ‘no other ground is left for science to build on than this dark but very solid rock which I once more call the concrete Dasein of man in his world.” Weyl grounds this Dasein, man’s being-in-the-world, in ordinary language, which is “neither tarnished poetry nor a blurred substitute for mathematical symbolism; on the contrary, neither the one nor the other would and could exist without the nourishing stem of the language of everyday life, with all its complexity, obscurity, crudenss, and ambiguity.” By thus connecting mathematical and poetic symbolism as both growing from the soil of ordinary human language, Weyl implicitly rejects Heidegger’s turn away from modern mathematical science.

In his late essay “The Unity of Knowledge” (1954), Weyl reviews the ground and concludes that “the shield of Being is broken beyond repair,” but does not take this disunity in a tragic sense because “on the side of Knowing there may be unity. Indeed, mind in the fullness of its experience has unity. Who says ‘I’ points to it. Here he reaffirms his old conviction that human consciousness is not simply the product of other, more mechanical forces, but is itself the luminous centre constituting that reality through its “complex symbolic creations which this lumen built up in the history of mankind.” Even though “myth, religion, and alas! also philosophy” fall prey to “man’s infinite capacity for self-deception,” Weyl implicitly holds our greater hope for the symbolic creations of mathematics and science, though he admits that he is still struggling to find clarity.

The final essay in this anthology, “Insight and Reflection” (1955) is Weyl’s rich Spatless, the intense, sweet wine made from grapes long on the vine. This philosophical memoir discloses his inner world of reflection in ways his other, earlier essays did not reveal quite so directly, perhaps aware of the skepticism and irony that may have met them earlier on. WE are reminded of his “point-eye,” disclosing his thoughts and feelings while creeping up his own world line. Nearing its end, Weyl seems freer to say what he feels, perhaps no longer caring who might mock. He gives his fullest avowal yet of what Husserl meant to him, but does not hold back his own reservations; Husserl finally does not help with Weyl’s own deep question about “the relation between the one pure I of immanent consciousness and the particular lost human being which I find myself to be in a world full of people like me (for example, during the afternoon rush hour on Fifth Avenue in New York).” Weyl is intrigues by Fichte’s mystic strain, but in the end Fichte’s program (analyzing everything in terms of I and not=I) strikes him as “preposterous.” Weyl calls Meister Eckhart “the deepest of the Occidental mystics…a man of high responsibility and incomparably higher nobility than Fichte.” Eckhart’s soaring theological flight beyond God toward godhead stirred Weyl alongwith Eckhart’s fervent simplicity of tone. Throughout his account, Weyl interweaves his mathematical work, his periods of soberness after the soaring flights of philosophical imagination, though he presents them as different sides of what seems to his point-eye a unified experience. Near the end, he remembers with particular happiness his book Symmetry (1952) which so vividly unites the poetic, the artistic, the mathematical, and the philosophical, a book no reader of Weyl should miss. In quoting T. S. Eliot that “the world becomes stranger, the pattern more complicated,” we are aware of Weyl’s faithful openness to their strangeness, as well as the ever more complex and beautiful symmetries he discerned in it.

Weyl’s book on symmetry shows the fundamental continuity of themes throughout his life and work. Thinking back on the theory of relativity, Weyl describes it not (as many of his contemporaries had) as disturbing or revolutionary but really as “another aspect of symmetry” because “it is the inherent symmetry of the four-dimensional continuum of space and time that relativity deals with.” Yet as beautifully as he evokes and illustrates the world of symmetry, Weyl still emphasizes the fundamental difference between perfect symmetry and life, with its spontaneity and unpredictability. “If nature were all lawfulness then every phenomenon would share the full symmetry of the universal laws of nature as formulated by the theory of relativity. The mere fact that this is not so proves that contingency is an essential feature of the world.” Characteristically, Weyl recalls the scene in Thomas Mann’s Magic Mountain in which his hero, Hans Castorp, nearly perishes when he falls asleep with exhaustion and leaning against a barn dreams his deep dream of death and love. An hour before when Hans sets out on his unwarranted expedition on skis he enjoys the play of the flakes “and among these myriads of enchanting little stars,” so he philosophizes, “in their hidden splendour, too small for man’s naked eye to see, there was not one like unto another, an endless inventiveness governed the development and unthinkable differentiation of one and the same basic scheme, the equilateral, equiangular hexagon. Yet each in itself — this was the uncanny, the antiorganic, the life-denying character of them all — each of them was absolutely symmetrical, icily regular in form. They were too regular, as substance adapted to life never was to this degree — the living principle shuddered at this perfect precision, found it deathly, the very marrow of death === Hans Castrop felt he understood how the reason why the builders of antiquity purposely and secretly introduced minute variation from absolute symmetry in their columnar structures.”

Weyl’s own life and work no less sensitively traced out this interplay between symmetry and life, field and matter, mathematics and physics, reflection and action.

So rich and manifold are Weyl’s writings that I have tried to include everything I could while avoiding excessive repetitiveness.

Not long after making his epochal contributions to quantum theory, Dirac was invited to visit universities across the United States. When he arrived in Madison, Wisconsin, in 1929, a reporter from the local paper interviewed him and learned from Dirac’s laconic replies that his favourite thing in America was potatoes, his favourite sport Chinese chess. Then the reporter wanted to ask him something more:

“They tell me that you and Einstein are the only two real sure-enough high brows and the only ones who can understand each other. I won’t ask you if this is straight stuff for I know you are too modest to admit it. But I want to know this — Do you ever run across a fellow that even you can’t understand?

“Yes”, replied Dirac.

“This will make a great reading for the boys down at the office,” says I (reporter). “Do you mind releasing to me who he is?”

“Hermann Weyl,” says he (Dirac).

The interview came to a sudden end just then, for the doctor pulled out his watch and I dodged and jumped for the door. But he let loose a smile as we parted and I knew that all the time he had been talking to me he was solving some problem that no one else could touch.

But if that fellow Professor Weyl ever lectures in this town again I sure am going to take a try at understanding him. A fellow ought to test his intelligence once in a while.

So should we —- and here is Professor Weyl himself, in his own words.

Reference: Amazon India link: Mind and Nature (Hermann Weyl) — Peter Pesic.


















A little portrait of Hermann Weyl

“A Proteus who transforms himself ceaselessly in order to elude the grip of his adversary, not becoming himself again until after the final victory.” Thus, Hermann Weyl (1885-1955) appeared to his eminent younger colleagues Claude Chevalley and Andre Weil. Surprising words to describe a mathematician, but apt for the amazing variety of shapes and forms in which Weyl’s extraordinary abilities revealed themselves, for “among all the mathematicians who began their working life in the twentieth century, Hermann Weyl was the one who made major contributions in the greatest number of different fields. He alone could stand comparison with the last great universal mathematicians of the nineteenth century, David Hilbert and Henri Poincare,” in the view of Freeman Dyson. “He was indeed not only a great mathematician but a great mathematical writer,” wrote another colleague.


Nalin Pithwa

Set Theory, Relations, Functions Preliminaries: II


Concept of Order:

Let us say that we create a “table” of two columns in which the first column is the name of the father, and the second column is name of the child. So, it can have entries like (Yogesh, Meera), (Yogesh, Gopal), (Kishor, Nalin), (Kishor, Yogesh), (Kishor, Darshna) etc. It is quite obvious that “first” is the “father”, then “second” is the child. We see that there is a “natural concept of order” in human “relations”. There is one more, slightly crazy, example of “importance of order” in real-life. It is presented below (and some times also appears in basic computer science text as rise and shine algorithm) —-

Rise and Shine algorithm: 

When we get up from sleep in the morning, we brush our teeth, finish our morning ablutions; next, we remove our pyjamas and shirt and then (secondly) enter the shower; there is a natural order here; first we cannot enter the shower, and secondly we do not remove the pyjamas and shirt after entering the shower. 🙂

Ordered Pair: Definition and explanation:

A pair (a,b) of numbers, such that the order, in which the numbers appear is important, is called an ordered pair. In general, ordered pairs (a,b) and (b,a) are different. In ordered pair (a,b), ‘a’ is called first component and ‘b’ is called second component.

Two ordered pairs (a,b) and (c,d) are equal, if and only if a=c and b=d. Also, (a,b)=(b,a) if and only if a=b.

Example 1: Find x and y when (x+3,2)=(4,y-3).

Solution 1: Equating the first components and then equating the second components, we have:

x+3=4 and 2=y-3

x=1 and y=5

Cartesian products of two sets:

Let A and B be two non-empty sets then the cartesian product of A and B is denoted by A x B (read it as “A cross B”),and is defined as the set of all ordered pairs (a,b) such that a \in A, b \in B.

Thus, A \times B = \{ (a,b): a \in A, b \in B\}

e.g., if A = \{ 1,2\} and B = \{ a,b,c\}, tnen A \times B = \{ (1,a),(1,b),(1,c),(2,a),(2,b),(2,c)\}.

If A = \phi or B=\phi, we define A \times B = \phi.

Number of elements of a cartesian product:

By the following basic counting principle: If a task A can be done in m ways, and a task B can be done in n ways, then the tasks A (first) and task B (later) can be done in mn ways.

So, the cardinality of A x B is given by: n(A \times B)= n(A) \times n(B).

So, in general if a cartesian product of p finite sets, viz, A_{1}, A_{2}, A_{3}, \ldots, A_{p} is given by n(A_{1} \times A_{2} \times A_{3} \ldots A_{p}) = n(A_{1}) \times n(A_{2}) \times \ldots \times n(A_{p})

Definitions of relations, arrow diagrams (or pictorial representation), domain, co-domain, and range of a relation:

Consider the following statements:

i) Sunil is a friend of Anil.

ii) 8 is greater than 4.

iii) 5 is a square root of 25.

Here, we can say that Sunil is related to Anil by the relation ‘is a friend of’; 8 and 4 are related by the relation ‘is greater than’; similarly, in the third statement, the relation is ‘is a square root of’.

The word relation implies an association of two objects according to some property which they possess. Now, let us some mathematical aspects of relation;


A and B are two non-empty sets then any subset of A \times B is called relation from A to B, and is denoted by capital letters P, Q and R. If R is a relation and (x,y) \in R then it is denoted by xRy.

y is called image of x under R and x is called pre-image of y under R.

Let A=\{ 1,2,3,4,5\} and B=\{ 1,4,5\}.

Let R be a relation such that (x,y) \in R implies x < y. We list the elements of R.

Solution: Here A = \{ 1,2,3,4,5\} and B=\{ 1,4,5\} so that R = \{ (1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)\} Note this is the relation R from A to B, that is, it is a subset of A x B.

Check: Is a relation R^{'} from B to A defined by x<y, with x \in B and y \in A — is this relation R^{'} *same* as R from A to B? Ans: Let us list all the elements of R^{‘} explicitly: R^{'} = \{ (1,2),(1,3),(1,4),(1,5),(4,5)\}. Well, we can surely compare the two sets R and R^{'} — the elements “look” different certainly. Even if they “look” same in terms of numbers, the two sets R and R^{'} are fundamentally different because they have different domains and co-domains.

Definition : Domain of a relation R: The set of all the first components of the ordered pairs in a relation R is called the domain of relation R. That is, if R \subseteq A \times B, then domain (R) is \{ a: (a,b) \in R\}.

Definition: Range: The set of all second components of all ordered pairs in a relation R is called the range of the relation. That is, if R \subseteq A \times B, then range (R) = \{ b: (a,b) \in R\}.

Definition: Codomain: If R is a relation from A to B, then set B is called co-domain of the relation R. Note: Range is a subset of co-domain.

Type of Relations:

One-one relation: A relation R from a set A to B is said to be one-one if every element of A has at most one image in B and distinct elements in A have distinct images in B. For example, let A = \{ 1,2,3,4\}, and let B=\{ 2,3,4,5,6,7\} and let R_{1}= \{ (1,3),(2,4),(3,5)\} Then R_{1} is a one-one relation. Here, domain of R_{1}= \{ 1,2,3\} and range of R_{1} is \{ 3,4,5\}.

Many-one relation: A relation R from A to B is called a many-one relation if two or more than two elements in the domain A are associated with a single (unique) element in co-domain B. For example, let R_{2}=\{ (1,4),(3,7),(4,4)\}. Then, R_{2} is many-one relation from A to B. (please draw arrow diagram). Note also that domain of R_{1}=\{ 1,3,4\} and range of R_{1}=\{ 4,7\}.

Into Relation: A relation R from A to B is said to be into relation if there exists at least one element in B, which has no pre-image in A. Let A=\{ -2,-1,0,1,2,3\} and B=\{ 0,1,2,3,4\}. Consider the relation R_{1}=\{ (-2,4),(-1,1),(0,0),(1,1),(2,4) \}. So, clearly range is \{ 0,1,4\} and range \subseteq B. Thus, R_{3} is a relation from A INTO B.

Onto Relation: A relation R from A to B is said to be ONTO relation if every element of B is the image of some element of A. For example: let set A= \{ -3,-2,-1,1,3,4\} and set B= \{ 1,4,9\}. Let R_{4}=\{ (-3,9),(-2,4), (-1,1), (1,1),(3,9)\}. So, clearly range of R_{4}= \{ 1,4,9\}. Range of R_{4} is co-domain of B. Thus, R_{4} is a relation from A ONTO B.

Binary Relation on a set A:

Let A be a non-empty set then every subset of A \times A is a binary relation on set A.

Illustrative Examples:

E.g.1: Let A = \{ 1,2,3\} and let A \times A = \{ (1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)\}. Now, if we have a set R = \{ (1,2),(2,2),(3,1),(3,2)\} then we observe that R \subseteq A \times A, and hence, R is a binary relation on A.

E.g.2: Let N be the set of natural numbers and R = \{ (a,b) : a, b \in N and 2a+b=10\}. Since R \subseteq N \times N, R is a binary relation on N. Clearly, R = \{ (1,8),(2,6),(3,4),(4,2)\}. Also, for the sake of completeness, we state here the following: Domain of R is \{ 1,2,3,4\} and Range of R is \{ 2,4,6,8\}, codomain of R is N.

Note: (i) Since the null set is considered to be a subset of any set X, so also here, \phi \subset A \times A, and hence, \phi is a relation on any set A, and is called the empty or void relation on A. (ii) Since A \times A \subset A \times A, we say that A \subset A is a relation on A called the universal relation on A. 

Note: Let the cardinality of a (finite) set A be n(A)=p and that of another set B be n(B)=q, then the cardinality of the cartesian product n(A \times B)=pq. So, the number of possible subsets of A \times B is 2^{pq} which includes the empty set.

Types of relations:

Let A be a non-empty set. Then, a relation R on A is said to be: (i) Reflexive: if (a,a) \in R for all a \in A, that is, aRa for all a \in A. (ii) Symmetric: If (a,b) \in R \Longrightarrow (b,a) \in R for all a,b \in R (iii) Transitive: If (a,b) \in R, and (b,c) \in R, then so also (a,c) \in R.

Equivalence Relation: 

A (binary) relation on a set A is said to be an equivalence relation if it is reflexive, symmetric and transitive. An equivalence appears in many many areas of math. An equivalence measures “equality up to a property”. For example, in number theory, a congruence modulo is an equivalence relation; in Euclidean geometry, congruence and similarity are equivalence relations.

Also, we mention (without proof) that an equivalence relation on a set partitions the set in to mutually disjoint exhaustive subsets. 

Illustrative examples continued:

E.g. Let R be an equivalence relation on \mathbb{Q} defined by R = \{ (a,b): a, b \in \mathbb{Q}, (a-b) \in \mathbb{Z}\}. Prove that R is an equivalence relation.

Proof: Given that R = \{ (a,b) : a, b \in \mathbb{Q}, (a-b) \in \mathbb{Z}\}. (i) Let a \in \mathbb{Q} then a-a=0 \in \mathbb{Z}, hence, (a,a) \in R, so relation R is reflexive. (ii) Now, note that (a,b) \in R \Longrightarrow (a-b) \in \mathbb{Z}, that is, (a-b) is an integer \Longrightarrow -(b-a) \in \mathbb{Z} \Longrightarrow (b-a) \in \mathbb{Z} \Longrightarrow (b,a) \in R. That is, we have proved (a,b) \in R \Longrightarrow (b,a) \in R and so relation R is symmetric also. (iii) Now, let (a,b) \in R, and (b,c) \in R, which in turn implies that (a-b) \in \mathbb{Z} and (b-c) \in \mathbb{Z} so it \Longrightarrow (a-b)+(b-c)=a-c \in \mathbb{Z} (as integers are closed under addition) which in turn \Longrightarrow (a,c) \in R. Thus, (a,b) \in R and (b,c) \in R implies (a,c) \in R also, Hence, given relation R is transitive also. Hence, R is also an equivalence relation on \mathbb{Q}.

Illustrative examples continued:

E.g.: If (x+1,y-2) = (3,4), find the values of x and y.

Solution: By definition of an ordered pair, corresponding components are equal. Hence, we get the following two equations: x+1=3 and y-2=4 so the solution is x=2,y=6.

E.g.: If A = (1,2), list the set A \times A.

Solution: A \times A = \{ (1,1),(1,2),(2,1),(2,2)\}

E.g.: If A = \{1,3,5 \} and B=\{ 2,3\}, find A \times B, and B \times A, check if cartesian product is a commutative operation, that is, check if A \times B = B \times A.

Solution: A \times B = \{ (1,2),(1,3),(3,2),(3,3),(5,2),(5,3)\} whereas B \times A = \{ (2,1),(2,3),(2,5),(3,1),(3,3),(3,5)\} so since A \times B \neq B \times A so cartesian product is not a commutative set operation.

E.g.: If two sets A and B are such that their cartesian product is A \times B = \{ (3,2),(3,4),(5,2),(5,4)\}, find the sets A and B.

Solution: Using the definition of cartesian product of two sets, we know that set A contains as elements all the first components and set B contains as elements all the second components. So, we get A = \{ 3,5\} and B = \{ 2,4\}.

E.g.: A and B are two sets given in such a way that A \times B contains 6 elements. If three elements of A \times B are (1,3),(2,5),(3,3), find its remaining elements.

Solution: We can first observe that 6 = 3 \times 2 = 2 \times 3 so that A can contain 2 or 3 elements; B can contain 3 or 2 elements. Using definition of cartesian product of two sets, we get that A= \{ 1,2,3\} and \{ 3,5\} and so we have found the sets A and B completely.

E.g.: Express the set \{ (x,y) : x^{2}+y^{2}=25, x, y \in \mathbb{W}\} as a set of ordered pairs.

Solution: We have x^{2}+y^{2}=25 and so

x=0, y=5 \Longrightarrow x^{2}+y^{2}=0+25=25

x=3, y=4 \Longrightarrow x^{2}+y^{2}=9+16=25

x=4, y=3 \Longrightarrow x^{2}+y^{2}=16+9=25

x=5, y=0 \Longrightarrow x^{2}+y^{2}=25+0=25

Hence, the given set is \{ (0,5),(3,4),(4,3),(5,0)\}

E.g.: Let A = \{ 1,2,3\} and B = \{ 2,4,6\}. Show that R = \{ (1,2),(1,4),(3,2),(3,4)\} is a relation from A to B. Find the domain, co-domain and range.

Solution: Here, A \times B = \{ (1,2),(1,4),(1,6),(2,2),(2,4),(2,6),(3,2),(3,4),(3,6)\}. Clearly, R \subseteq A \times B. So R is a relation from A to B. The domain of R is the set of first components of R (which belong to set A, by definition of cartesian product and ordered pair)  and the codomain is set B. So, Domain (R) = \{ 1,3\} and co-domain of R is set B itself; and Range of R is \{ 2,4\}.

E.g.: Let A = \{ 1,2,3,4,5\} and B = \{ 1,4,5\}. Let R be a relation from A to B such that (x,y) \in R if x<y. List all the elements of R. Find the domain, codomain and range of R. (as homework quiz, draw its arrow diagram);

Solution: Let A = \{ 1,2,3,4,5\} and B = \{ 1,4,5\}. So, we get R as (1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5). domain(R) = \{ 1,2,3,4\}, codomain(R) = B, and range(R) = \{ 4,5\}.

E.g. Let A = \{ 1,2,3,4,5,6\}. Define a binary relation on A such that R = \{ (x,y) : y=x+1\}. Find the domain, codomain and range of R.

Solution: By definition, R \subseteq A \times A. Here, we get R = \{ (1,2),(2,3),(3,4),(4,5),(5,6)\}. So we get domain (R) = \{ 1,2,3,4,5\}, codomain(R) =A, range(R) = \{ 2,3,4,5,6\}

Tutorial problems:

  1. If (x-1,y+4)=(1,2), find the values of x and y.
  2. If (x + \frac{1}{3}, \frac{y}{2}-1)=(\frac{1}{2} , \frac{3}{2} )
  3. If A=\{ a,b,c\} and B = \{ x,y\}. Find out the following: A \times A, B \times B, A \times B and B \times A.
  4. If P = \{ 1,2,3\} and Q = \{ 4\}, find the sets P \times P, Q \times Q, P \times Q, and Q \times P.
  5. Let A=\{ 1,2,3,4\} and \{ 4,5,6\} and C = \{ 5,6\}. Find A \times (B \bigcap C), A \times (B \bigcup C), (A \times B) \bigcap (A \times C), A \times (B \bigcup C), and (A \times B) \bigcup (A \times C).
  6. Express \{ (x,y) : x^{2}+y^{2}=100 , x, y \in \mathbf{W}\} as a set of ordered pairs.
  7. Write the domain and range of the following relations: (i) \{ (a,b): a \in \mathbf{N}, a < 6, b=4\} (ii) \{ (a,b): a,b \in \mathbf{N}, a+b=12\} (iii) \{ (2,4),(2,5),(2,6),(2,7)\}
  8. Let A=\{ 6,8\} and B=\{ 1,3,5\}. Let R = \{ (a,b): a \in A, b \in B, a+b \hspace{0.1in} is \hspace{0.1in} an \hspace{0.1in} even \hspace{0.1in} number\}. Show that R is an empty relation from A to B.
  9. Write the following relations in the Roster form and hence, find the domain and range: (i) R_{1}= \{ (a,a^{2}) : a \hspace{0.1in} is \hspace{0.1in} prime \hspace{0.1in} less \hspace{0.1in} than \hspace{0.1in} 15\} (ii) R_{2} = \{ (a, \frac{1}{a}) : 0 < a \leq 5, a \in N\}
  10. Write the following relations as sets of ordered pairs: (i) \{ (x,y) : y=3x, x \in \{1,2,3 \}, y \in \{ 3,6,9,12\}\} (ii) \{ (x,y) : y>x+1, x=1,2, y=2,4,6\} (iii) \{ (x,y) : x+y =3, x, y \in \{ 0,1,2,3\}\}

More later,

Nalin Pithwa









A leaf out of Paul Erdos’ biography: My Brain is Open: by Bruce Schechter


My Brain is Open: The Mathematical Journeys of Paul Erdos by Bruce Schechter, a TouchStone Book, Published by Simon and Schuster, New York.

Amazon India link:

Chapter One: Traveling.

The call might come at midnight or an hour before dawn — mathematicians are oddly unable to handle the arithmetic of time zones. Typically, a thickly accented voice on the other end of the line would abruptly begin:

“I am calling from Berlin. I want to speak to Erdos.”

“He’s not here, yet.”

“Where is he?”

“I don’t know.”

“Why don’t you know!” Click!

Neither are mathematicians always observant of social graces.

For more than sixty years mathematicians around the world have been roused from their abstract dreams by such calls, the first of the many disruptions that constituted a visit from Paul Erdos. The frequency of the calls would increase over the next several days and would culminate with a summons to the airport, where Erdos himself would appear, a short, frail man in a shapeless old suit, clutching two small suitcases that contained all of his worldly possessions. Stepping off the plane he would announce to the welcoming group of mathematicians, “My brain is open!”

Paul Erdos’ brain, when open, was one of the wonders of the world, an Ali Baba’s cave, glittering with mathematical treasures, gems of the most intricate cut and surpassing beauty. Unlike Ali Baba’s cave, which was hidden behind a huge stone in a remote desert, Erdos and his brain were in perpetual motion. He moved between mathematical meetings, universities, and corporate think tanks, logging hundreds of thousands of miles. “Another roof, another proof,” as he liked to say. “Want to meet Erdos?” mathematicians would ask. “Just stay here and wait. He’ll show up.” Along the way, in borrowed offices, guest bedrooms, and airplane cabins, Erdos wrote in excess of 1600 papers, books and articles, more than any other mathematician who ever lived. Among them are some of the great classics of the twentieth century, papers that opened up entire new fields and became the obsession and inspiration of generations of mathematicians.

The meaning of life, Erdos often said, was to prove and conjecture. Proof and conjecture are the tools with which mathematicians explore the Platonic universe of pure form, a universe that to many of them is as real as the universe in which they must reluctantly make their homes and livings, and far more beautiful. “If numbers aren’t beautiful, I don’t know what is,” Erdos frequently remarked. And although, like all mathematicians, he was forced to make his home in the temporal world, he rejected worldly encumbrances. He had no place on earth called home, nothing resembling a conventional year-round, nine-to-five job, and no family in the usual sense of the word. He arranged his life with only one purpose, to spend areas many hours a day as possible engaged in the essential, life-affirming business of proof and conjecture.

For Erdos, the mathematics that consumed most of his waking hours was not a solitary pursuit but a social activity, a movable feast. One of the greatest mathematical discoveries of the twentieth century was the simple equation that two heads are better than one. Ever since Archimedes traced his circles in sand, mathematicians, for the most part, have laboured alone — that is, until some forgotten soul realized that mathematics could be done anywhere. Only paper and pencil were needed, and those were not strictly essential. A table-cloth would do in a pinch, or the mathematician could carry his equations in his head, like a chessmaster playing blindfolded. Strong coffee, and in Erdos’ case even more powerful stimulants, helped too. Mathematicians began to frequent the coffeehouses of Budapest, Prague, and Paris, which led to the quip often attributed to Erdos:”A mathematician is a machine for turning coffee into theorems.” Increasingly, mathematical papers became the work of two, three, or more collaborators. That radical transformation of how mathematics is created is the result of many factors, not the least of which was the infectious example set by Erdos.

Erdos had more collaborators than most people have acquaintances. He wrote papers with more than 450 collaborators —- the exact number is still not known, since Erdos participated in the creation of new mathematics until the last day of his life, and his collaborators are expected to continue writing and publishing for years. The briefest encounter could lead to a publication — for scores of young mathematicians a publication that could become the cornerstone of their life’s work. He would work with anyone who could keep up with him, the famous or the unknown. Having been a child prodigy himself, he was particularly interested in meeting and helping to develop the talents of young mathematicians. Many of the world’s leading mathematicians owe their careers to an early meeting with Erdos.

Krishna Alladi, who is now a mathematician at the University of Florida, Gainesville, is one of the many young mathematicians whom Erdos helped. In 1974, when Alladi was an undergraduate in Madras, India, he began an independent investigation of a certain number theoretic function. His teachers could not help Alladi with his problem, nor could his father, who was a theoretical physicist and head of Madras Institute of Mathematics. Alladi’s father told some of his knowledgeable friends about his son’s difficulty, and they suggested that he write to Erdos.

Because Erdos was constantly on the move, Alladii sent a letter to the Hungarian Academy of Sciences. In an astonishingly short time, Alladi heard from Erdos, who said he would soon be lecturing in Calcutta. Could Alladi come there to meet him? Unfortunately, Alladi had examinations and could not attend, so he sent his father in his place to present the results of his research. After his father’s talk, Alladi recounts, “Erdos walked up to him and told him in very polite terms that he was not interested in the father but in the son.” Determined to meet with the promising young mathematician, Erdos, who was bound for Australia, rerouted his trip to stop briefly in Madras, which lies about 860 miles south of Calcutta.

Alladi was astonished that a great mathematician should change his plans to visit a student. He was nervous when he met Erdos at the airport, but that soon passed. “He talked to me as if he had known me since childhood,” Alladi recalls. The first thing Erdos asked was, “Do you know my poem about Madras?” And then he recited:

This the city of Madras

The home of the curry and the dhal,

Where Iyers speak only to Iyengars

And Iyengars speak only to God.

The Iyers and Iyengars are two Brahmin sects. The Iyers worship Shiva the Destroyer but will also worship in the temples of the Iyengars, who worship only Lord Vishnu, the Protector. Erdos explained that this was his variation on the poem about Boston and the pecking order among the Lowells, the Cabots, and God. Having put Alladi at ease, Erdos launched into a discussion of mathematics. Erdos was so impressed with Alladi, who was applying to graduate schools in the United States, that he wrote a letter on his behalf. Within a month, Alladi received the Chancellor’s Fellowship at the University of California, Los Angeles.

A celebrated magazine article about Erdos was called, “The Man Who Loved Only Numbers.” While it is true that Erdos loved numbers, he loved much more. He loved to talk about history, politics, and almost any other subject. He loved to take long walks and to climb towers, no matter how dismal the prospective view, he loved to play ping-pong, chess, and Go, he loved to perform silly tricks to amuse children and to make sly jokes and thumb his nose at authority. But, most of all, Erdos loved those who loved numbers, mathematicians. He showed that love by opening his pocket as well as his mind. Having no permanent job, Erdos also had little money, but whatever he had was at the service of others. If he heard of a graduate student who needed money to continue his studies, he would sent a cheque. Whenever he lectured in Madras, he would send his fee to the needy widow of the great Indian mathematician Srinivasa Ramanujan; he had never met Ramanujan or his wife, but the beauty of Ramanujan’s equations had inspired Erdos as a young mathematician. In 1984, he won the prestigious Wolf prize, which came with a cash reward of $at 50000, easily the most money Erdos had ever received at one time. He gave $30000 to endow a postdoctoral fellowship in the name of his parents at the Technion in Haifa, Israel, and used the remainder to help relatives, graduate students, and colleagues:”I kept only $720,” Erdos recalled.

In the years before the internet, there was Paul Erdos. He carried a shopping bag crammed with latest papers, and his brain was stuffed with the latest gossip as well as an amazing database of the world of mathematics. He knew everybody: what they were interested in; what they had conjectured, proved, or were in the midst of proving; their phone numbers; the names and ages of their wives, children, pets; and, much more. He could tell off the top of his head on which page in which obscure Russian journal a theorem similar to the one you were working on was proved in 1922. When he met a mathematician in Warsaw, say, he would immediately take up the conversation where they had left it two years earlier. During the iciest years of the Cold War Erdos’s fame allowed him freely to cross the Iron Curtain, so that he became vital link between the East and the West.

In 1938, with Europe on the brink of war, Erdos fled to the United States and embarked on his mathematical journeys. This book is the story of those adventures. Because they took Erdos everywhere mathematics is done, this is also the story of the world of mathematics, a world virtually unknown to outsiders.. Today perhaps the only mathematician most people can name is Theodore Kacznyski. The names of Karl Friedrich Gauss, Bernhard Riemann, Georg Cantor and Leonhard Euler, who are to Mathematics what Shakespeare is to literature and Mozart to music, are virtually unknown outside of the worlds of math and science.

For all frequent flier miles Erdos collected, his true voyages were journeys of the mind. Erdos carefully constructed his life to allow himself as much time as possible for those inward journeys, so a true biography of Erdos should spend almost as much time in the Platonic realm of mathematics as in the real world. For a layman this may seem to be a forbidding prospect. Fortunately, many of the ideas that fascinated Erdos can be easily grasped by anyone with a modest recollection of high school mathematics. The proofs and conjectures that made Erdos famous are, of course, far more difficult to follow, but that should not be of much concern to the reader. As Ralph Boas wrote, “Only professional mathematicians learn anything from proofs. Other people learn from explanations.” Just as it is not necessary to understand how Glenn Gould fingers a difficult passage to be dazzled by his performance of thee “Goldberg Variations,” one does not have to understand the details of Erdos’s elegant proofs to appreciate the beauty of mathematics. And, it is the nature of Erdos’s work that while his proofs are difficult, the questions he asks can be quite easy to understand. Erdos often offered money for the solution to problems he proposed. Some of those problems are enough for readers of this book to understand — and, perhaps, even solve. Those who decide to try should be warned that, as Erdos has pointed out, when the number of hours it takes to solve one of his problems is taken into account, the cash prizes rarely exceed minimum wage. The true prize is to share in the joy that Erdos knew so well, joy in understanding a page of the eternal book of mathematics.

— shared by Nalin Pithwa (to motivate his students and readers.)