The typical unhappy man is one who, having been deprived in youth of some normal satisfaction, has come to value this one kind of satisfaction more than any other, and has therefore given to his life a one-sided direction, together with a quite undue emphasis upon the achievement as opposed to the activities connected with it.

 p. 22

The human animal, like others, is adapted to a certain amount of struggle for life, and when by means of great wealth homo sapiens can gratify all his whims without effort, the mere absence of effort from his life removes an essential ingredient of happiness. The man who acquires easily things for which he feels only a very moderate desire concludes that the attainment of desire does not bring happiness. If he is of a philosophic disposition, he concludes that human life is essentially wretched, since the man who has all he wants is still unhappy. He forgets that to be without some of the things you want is an indispensable part of happiness.

 p. 27

...love is able to break down the hard shell of the ego, since it is a form of biological coöperation in which the emotions of each are necessary to the fulfillment of the other's instinctive purposes.

 p. 34

If the American business man is to be made happier, he must first change his religion. So long as he not only desires success, but is whole-heartedly persuaded that it is a man's duty to pursue success and that a man who does not do so is a poor creature, so long his life will remain too concentrated and too anxious to be happy.

 p. 42

Education used to be conceived very largely as a training in the capacity for enjoyment—enjoyment, I mean, of those more delicate kinds that are not open to wholly uncultivated people. In the 18th century it was one of the marks of a "gentleman" to take a discriminating pleasure in literature, pictures, and music. We nowadays may disagree with his taste, but it was at least genuine. The rich man of the present day is sometimes cultivated, but not infrequently of quite a different type ... [and] he does not know what to do with leisure. As he gets richer and richer it becomes easier and easier to make money, until at last five minutes a day will bring him more than he knows how to spend. The poor man is thus left at a loose end as a result of his success. This must inevitably be the case so long as success itself is represented as the purpose of life. Unless a man has been taught what to do with success after getting it, the achievement of it must inevitably leave him a prey to boredom.

 pp. 44–45 (Clippings)

A boy or young man who has some serious constructive purpose will endure voluntarily a great deal of boredom if he finds that it is necessary by the way. But constructive purposes do not easily form themselves in a boy's mind if he is living a life of distractions and dissipations, for in that case his thoughts will always be directed towards the next pleasure rather than towards the distant achievement. ...a generation that cannot endure boredom will be a generation of little men, of men unduly divorced from the slow processes of nature, of men in whom every vital impulse slowly withers, as though they were cut flowers in a vase.

 p. 54

The wise man thinks about his troubles only when there is some purpose in doing so; at other times he thinks about other things, or, if it is night, about nothing at all. ...it is quite possible to shut out the ordinary troubles of ordinary days, except while they have to be dealt with. It is amazing how much both happiness and efficiency can be increased by the cultivation of an orderly mind, which thinks about a matter adequately at the right time rather than inadequately at all times. When a difficult or worrying decision has to be reached, as soon as all the data are available, give the matter your best thought and make your decision; having made the decision, do not revise it unless some new fact comes to your knowledge. Nothing is so exhausting as indecision, and nothing is so futile.

 p. 60

...over and above these self-centered considerations is the fact that one's ego is no very large part of the world. The man who can center his thoughts and hopes upon something transcending self can find a certain peace in the ordinary troubles of life which is impossible to the pure egoist.

 p. 61

If you desire glory, you may envy Napoleon. But Napoleon envied Caesar, Caesar envied Alexander, and Alexander, I dare say, envied Hercules, who never existed.

 pp. 71–72 (Clippings)

It is in the moments when the mind is most active and the fewest things are forgotten that the most intense joys are experienced. This indeed is one of the best touchstones of happiness. The happiness that requires intoxication of no matter what sort is a serious and unsatisfying kind. The happiness that is genuinely satisfying is accompanied by the fullest exercise of our faculties, and the fullest realization of the world in which we live.

 pp. 87–88

Of the more highly educated sections of the community, the happiest in the present day are the men of science. Many of the most eminent of them are emotionally simple, and obtain from their work a satisfaction so profound that they can derive pleasure from eating, and even marrying. Artists and literary men consider it de rigueur to be unhappy in their marriages, but men of science quite frequently remain capable of old-fashioned domestic bliss. The reason of this is that the higher parts of their intelligence are wholly absorbed by their work and are not allowed to intrude into regions where they have no functions to perform. In their work they are happy because in the modern world science is progressive and powerful, and because its importance is not doubted either by themselves or by laymen. They have therefore no necessity for complex emotions, since the simpler emotions meet with no obstacles. Complexity in emotions is like foam in a river. It is produced by obstacles which break the smoothly flowing current. But so long as the vital energies are unimpeded, they produce no ripple on the surface, and their strength is not evident to the unobservant.

 p. 115

The secret of happiness is this: let your interests be as wide as possible, and let your reactions to the things and persons that interest you be as far as possible friendly rather than hostile.

 p. 123

In women, less nowadays than formerly but still to a very large extent, zest has been greatly diminished by a mistake conception of respectability. It was thought undesirable that women should take an obvious interest in men, or that they should display too much vivacity in public. In learning not to be interested in men they learned very frequently to be interested in nothing, or at any rate in nothing except a certain kind of correct behavior. To teach an attitude of inactivity and withdrawal towards life is clearly to teach something very inimical to zest, and to encourage a certain kind of absorption in self which is characteristic of highly respectable women, especially when they are uneducated. They do not have the interest in sport that average men have, they care nothing about politics, towards men their attitude is one of prim aloofness, towards women their attitude is one of veiled hostility based upon the conviction that other women are less respectable than they are themselves. They boast that they keep themselves to themselves; that is to say, their lack of interest in their fellow creatures appears to them in the light of a virtue.

 p. 135

...those who make themselves the slaves of unvarying routine are generally actuated by fear of a cold outer world, and by the feeling that they will not bump into it if they walk along the same paths that they have walked along on previous days.

 p. 138 (Clippings)

The world is a higgledy-piggledy place, containing things pleasant and things unpleasant in haphazard sequence. And the desire to make an intelligible system of pattern out of it is at bottom an outcome of fear, in fact a kind of agoraphobia or dread of open spaces. Within the four walls of his library the timid student feels safe. If he can persuade himself that the universe is equally tidy, he can feel almost equally safe when he has to venture forth into the streets. Such a man, if he had received more affection, would have feared the real world less, and would not have had to invent an ideal world to take its place in his beliefs.

 pp. 139–140

The habits of mind formed in early years are likely to persist through life. Many people when they fall in love look for a little haven of refuge from the world, where they can be sure of being admired when they are not admirable, and praised when they are not praiseworthy. To many men home is a refuge from the truth: it is their fears and their timidities that make them enjoy a companionship in which these feelings are put to rest. They seek from their wives what they obtained formerly from an unwise mother, and yet they are surprised if their wives regard them as grown-up children.

 p. 140

I wish now to speak of the affection that a person gives. This ... is of two different kinds, one of which is perhaps the most important expression of a zest for life, while the other is an expression of fear. The former seems to me wholly admirable, while the latter is at best a consolation. If you are sailing in a ship on a fine day along a beautiful coast, you admire the coast and feel pleasure in it. This pleasure is one derived entirely from looking outward, and has nothing to do with any desperate need of your own. If, on the other hand, your ship is wrecked and you swim towards the coast, you acquire for it a new kind of love: it represents security against the waves, and its beauty or ugliness becomes an unimportant matter. ... The feeling caused by insecurity is much more subjective and self-centered than the other, since the loved person is valued for services rendered, not for intrinsic qualities. ...almost all real affection contains something of both kinds in combination... In the best kind of affection a man hopes for a new happiness rather than for escape from an old unhappiness.

 p. 142

Obstacles, psychological and social, to the blossoming of reciprocal affection are a grave evil, from which the world has always suffered and still suffers. People are slow to give admiration for fear it should be misplaced; they are slow to bestow affection for fear that they should be made to suffer either by the person upon whom they bestow it or by a censorious world. Caution is enjoined both in the name of morality and in the name of worldly wisdom, with the result that generosity and adventurousness are discouraged where the affections are concerned. All this tends to produce timidity and anger against mankind, since many people miss throughout life what is really a fundamental need and to nine out of ten an indispensable condition of a happy and expansive attitude towards the world. ... In sex relations there is very often almost nothing that can be called real affection, not infrequently there is even a fundamental hostility. Each is trying not to give himself or herself away, each is preserving fundamental loneliness, each remains intact and therefore unfructified. In such experiences there is no fundamental value. I do not say that they should be carefully avoided, since the steps necessary to this end would be likely to interfere also with the occasions where a more valuable and profound affection could grow up. But I do say that the only sex relations that have real value are those in which there is no reticence and in which the whole personality of both becomes merged in a new collective personality. Of all forms of caution, caution in love is perhaps most fatal to true happiness.

 p. 144 (Clippings)

Most people, when they are left free to fill their own time according to their own choice, are at a loss to think of anything sufficiently pleasant to be worth doing. And whatever they decide on, they are troubled by the feeling that something else would have been pleasanter. To be able to fill leisure intelligently is the last product of civilization, and at present very few people have reached this level.

 p. 162 (Clippings)

...a sense of proportion is very valuable and at times very consoling. We are all inclined to get unduly excited, unduly strained, unduly impressed with the importance of the little corner of the world in which we live, and of the little moment of time comprised between our birth and death. In this excitement and overestimation of our own importance there is nothing desirable. True, it may make us work harder, but it will not make us work better. A little work directed to a good end is better than a great deal of work directed to a bad end... Those who care much for their work are always in danger of falling into fanaticism, which consists essentially in remembering one or two desirable things while forgetting all the rest, and in supposing that in the pursuit of these one or two any incidental harm of other sorts is of little account. Against this fanatical temper there is no better prophylactic than a large conception of the life of man and his place in the universe.

 p. 173 (Clippings)

It is one of the defects of modern higher education that it has become too much a training in the acquisition of certain kinds of skill, and too little an enlargement of the mind and heart by an impartial survey of the world. ... If ... you have as part of the habitual furniture of your mind the past ages of man, his slow and partial emergence out of barbarism, and the brevity of his total existence in comparison with astronomical epochs... [y]ou will have, beyond your immediate activities, purposes that are distant and slowly unfolding, in which you are not an isolated individual but one of the great army of those who have lead mankind towards a civilized existence. If you have attained to this outlook, a certain deep happiness will never leave you, whatever your personal fate may be. Life will become a communion with the great of all ages, and personal death no more than a negligible incident.

 p. 174

If I had the power to organize higher education as I should wish it to be, I should seek to substitute for the old orthodox religions ... something which is perhaps hardly to be called religion, since it is merely a focusing of attention upon well-ascertained facts. I should seek to make young people vividly aware of the past, vividly realizing that the future of man will in all likelihood be immeasurably longer than his past, profoundly conscious of the minuteness of the planet upon which we live and of the fact that life on this planet is only a temporary incident; and at the same time with these facts which tend to emphasize the insignificance of the individual, I should present quite another set of facts designed to impress upon the mind of the young the greatness of which the individual is capable, and the knowledge that throughout all the depths of stellar space nothing of equal value is known to us.

 p. 175 (Clippings)

A man of adequate vitality and zest will surmount all misfortunes by the emergence after each blow of an interest in life and the world which cannot be narrowed down so much as to make one loss fatal. To be defeated by one loss or even by several is not something to be admired as a proof of sensibility, but something to be deplored as a failure in vitality. All our affections are at the mercy of death, which may strike down those whom we love at any moment. It is therefore necessary that our lives should not have that narrow intensity which puts the whole meaning and purpose of our life at the mercy of accident.

 p. 177

Happiness is not, except in very rare cases, something that drops into the mouth, like a ripe fruit, by the mere operation of fortunate circumstances. ... For in a world so full of avoidable and unavoidable misfortunes, of illness and psychological tangles, of struggle and poverty and ill-will, the man or woman who is to be happy must find ways of coping with the multitudinous causes of unhappiness by which each individual is assailed. In some rare cases no great effort may be required. ... But such cases are exceptional. Most people are not rich; many people are not born good-natured; many people have uneasy passions which make a quiet and well-regulated life seem intolerably boring; health is a blessing which no one can be sure of preserving; marriage is not invariably a source of bliss. For all these reasons, happiness must be, for most men and women, an achievement rather than a gift of the gods...

 pp. 178–179 (Clippings)

Some people ... are furious when they miss a train, transported with rage if their dinner is badly cooked, sunk in despair if the chimney smokes, and vow vengeance against the whole industrial order when their clothes fail to return from the sanitary steam laundry. The energy that such people waste on trivial troubles would be sufficient, if more wisely directed, to make and unmake empires. The wise man ... deals with them without emotion. Worry and fret and irritation are emotions which serve no purpose.

 p. 183 (Clippings)

...the whole antithesis between self and the rest of the world ... disappears as soon as we have any genuine interest in persons or things outside ourselves. Through such interests a man comes to feel himself part of the stream of life, not a hard separate entity like a billiard ball, which can have no relation with other such entities except that of collision. All unhappiness depends upon some kind of disintegration or lack of integration; there is disintegration within the self through lack of coördination between the conscious and the unconscious mind; there is lack of integration between the self and society, where the two are not knit together by the force of objective interests and affections. The happy man is the man who does not suffer from either of these failures of unity, whose personality is neither divided against itself nor pitted against the world. Such a man feels himself a citizen of the universe, enjoying freely the spectacle that it offers and the joys that it affords, untroubled by the thought of death because he feels himself not really separate from those who will come after him. It is in such profound instinctive union with the stream of life that the greatest joy is to be found.

 p. 191

More and more he had to bid farewell to the dream, the feeling and the pleasure of infinite potentialities, of a multiplicity of futures. Instead of the dream of unending progress, of the sum of all wisdom, his pupil stood by, a small, near, demanding reality, an intruder and nuisance, but no longer to be rebuffed or evaded. For the boy represented, after all, the only way into the real future, the one most important duty, the one narrow path along which the Rainmaker's life and acts, principles, thoughts, and glimmerings could be saved from death and continue their life in a small new bud.

 p. 471

"... But what disgusts me even more are people who have no imagination. The kind T. S. Eliot calls hollow men. People who fill up that lack of imagination with heartless bits of straw, not even aware of what they're doing. Callous people who throw a lot of empty words at you, trying to force you to do what you don't want to. ... Gays, lesbians, straights, feminists, fascist pigs, communists, Hare Krishnas – none of them bother me. I don't care what banner they raise. But what I can't stand are hollow people. ... Those are exactly the kind of people who murdered Miss Saeki's childhood sweetheart. Narrow minds devoid of imagination. Intolerance, theories cut off from reality, empty terminology, usurped ideals, inflexible systems. Those are the things that really frighten me. ..."

 p. 181

"Anyone who falls in love is searching for the missing pieces of themselves. So anyone who's in love gets sad when they think of their lover. It's like stepping back inside a room you have fond memories of, one you haven't seen in a long time. It's just a natural feeling. ..."

 p. 297

And it came to me then. That we were wonderful traveling companions but in the end no more than lonely lumps of metal in their own separate orbits. From far off they look like beautiful shooting stars, but in reality they're nothing more than prisons, where each of us is locked up alone, going nowhere. When the orbits of these two satellites of ours happened to cross paths, we could be together. Maybe even open our hearts to each other. But that was only for the briefest moment. In the next instant we'd be in absolute solitude. Until we burned up and became nothing.

 p. 117

Lonely metal souls in the unimpeded darkness of space, they meet, pass each other, and part, never to meet again. No words passing between them. No promises to keep.

 p. 179

Creativity

Heisenberg derived the uncertainty principle that conjugate variables, meaning Fourier transforms, obeyed a condition which the product of the uncertainties of the two had to exceed a fixed number, involving Planck's constant. I earlier commented, Chapter 17, this is a theorem in Fourier transforms – any linear theory must have a corresponding uncertainty principle, but among physicists it is still widely regarded as a physical effect from Nature rather than a Mathematical effect of the model.

 p. 287

Introspection, and an examination of history and of reports of those who have done great work, all seem to show typically the pattern of creativity is as follows. There is first the recognition of the problem in some dim sense. This is followed by a longer or shorter period of refinement of the problem. Do not be too hasty at this stage, as you are likely to put the problem in the conventional form and find only the conventional solution. This stage, more over, requires your emotional involvement, your commitment to finding a solution since without a deep emotional involvement you are not likely to find a really fundamental, novel solution.

A long gestation period of intense thinking about the problem may result in a solution, or else the temporary abandonment of the problem. This temporary abandonment is a common feature of many great creative acts. The monomaniacal pursuit often does not work; the temporary dropping of the idea sometimes seems to be essential to let the subconscious find a new approach.

Then comes the moment of "insight", creativity, or what ever you want to call it – you see the solution. Of course it often happens that you are wrong; a closer examination of the problem shows the solution is faulty, but might be saved by some suitable revision. But maybe the problem needs to be altered to fit the solution! That has happened! More usually it is back to the drawing board, as they say, more mulling things over.

The false starts and false solutions often sharpen the next approach you try. You now know how not to do it! You have a smaller number of approaches left to explore. You have a better idea what will not work and possibly why it will not work.

...

Out of it all, sometimes, comes the solution. So far as anyone understands the process it arises from the subconscious, it is suddenly there! There is often a lot of further work to be done on the idea, the logical cleaning up, the organizing so others can see it, the public presentation to others which may require new ways of looking at the problem and your solution, not just your idiosyncratic way which gave you the first solution. This revision of the solution often brings clarity to you in the long run!

 pp. 296–297

If the solution does come from the subconscious, what can we do to manage our subconscious? My method, and it is implied above, is to saturate the subconscious with the problem, try to not think seriously about anything else for hours, days, or even weeks, and thus the subconscious which, so far as we know, depends heavily upon live experiences to form its dreams, etc. is then left with only the problem to mull over. We simply deprive it of all else as best we can! Hence, one day, we have the solution, either as we awake, or it pops into our mind without any preparation on our part, or as we pick up the problem again there the solution is! In a way, I am repeating Pasteur, "Luck favors the prepared mind". You prepare your mind for success "by thinking on it constantly" (Newton), and occasionally you are lucky.

 p. 297

Probably the most important tool in creativity is the use of an analogy. Something seems like something else which we knew in the past. Wide acquaintance with various fields of knowledge is thus a help – provided you have the knowledge filed away so it is available when needed, rather than to be found only when led directly to it. This flexible access to pieces of knowledge seems to come from looking at knowledge while you are acquiring it from many different angles, turning over any new idea to see its many sides before filing it away. This implies effort on your part not to take the easy, immediately useful "memorizing the material" path, but prepare your mind for the future. It is for this reason I have urged you in many of the chapters to get down to the fundamentals of a field, since it implies you must examine things many ways before you can decide what is fundamental and what is frills. In fact, for one person they may be in one order, and for another in the opposite order. What is fundamental partly depends on the individual and their mental makeup. It is obvious you need many "hooks" on the knowledge if you are to use it in new situations.

 pp. 297–298

I have not yet discussed the delicate topic of dropping a problem. If you cannot drop a wrong problem then the first time you meet one you will be stuck with it for the rest of your career. Einstein was tremendously creative in his early years, but once he began, in mid-life, the search for a unified theory then he spent the rest of his life on it and had about nothing to show for all the effort. I have seen this many times while watching how Science is done. It is most likely to happen to the very creative people; their previous successes convince them they can solve any problem; but there are other reasons besides over-confidence why, in many fields, sterility sets in with advancing age. Managing a creative career is not an easy task, or else it would often be done. In mathematics, theoretical physics and astrophysics, age seems to be a handicap (all characterized by high, raw creativity) while in music composition, literature, and statesmanship, age and experience seem to be an assert.

 p. 300

...in school it is easy to measure training and hard to measure education, and hence you tend to see on final exams an emphasis on the training part and a great neglect of the education part.

 p. 338

In Mathematics, and in Computer Science, a similar effect of initial selection happens. In the earlier stages of Mathematics up through the Calculus, as well as in Computer Science, grades are closely related to the ability to carry out a lot of details with high reliability. But later, especially in Mathematics, the qualities needed to succeed change and it becomes more proving theorems, patterns of reasoning, and the ability to conjecture new results, new theorems, and new definitions which matter. Still later it is the ability to see the whole of a field as a whole, and not as a lot of fragments. But the grading process has earlier, to a great extent, removed many of those you might want, and indeed are needed at the later stage! It is very similar in Computer Science where the ability to cope with the mass of programming details favors one kind of mind, one which is often negatively correlated with seeing the bigger picture.

 p. 341

"... I do want to try my best, though. I have to, or else I won't know where to go. ..."

 p. 115

They have just enough talent so they've been able to play things well without any effort and they've had people telling them how great they are from the time they're little, so hard work looks stupid to them.

 p. 151

Although Hilbert was not the first to make use of indirect, non-constructive proofs, he was the first to recognize their deep significance and value and to utilize them in dramatic and extremely beautiful ways. Kronecker had recently died; but to those who like Kronecker still declared that existence statements are meaningless unless they actually specify the object asserted to exist, Hilbert was always to reply:

"The value of pure existence proofs consists precisely in that the individual construction is eliminated by them, and that many different constructions are subsumed under one fundamental idea so that only what is essential to the proof stands out clearly; brevity and economy of thought are the raison d'être of existence proofs... To prohibit existence statements ... is tantamount to relinquishing the science of mathematics altogether."

 p. 37

Klein's lectures were deservedly recognized as classics. It was his custom often to arrive as much as an hour before the students in order to check the encyclopedic list of references which he had had his assistant prepare. At the same time he smoothed out any roughness of expression or thought which might still remain in his manuscript. Before he began his lecture, he had mapped out in his mind an arrangement of formulas, diagrams and citations. Nothing put on the blackboard during the lecture ever had to be erased. At the conclusion the board contained a perfect summary of the presentation, every square inch being appropriately filled and logically ordered.

...

In contrast, Hilbert delivered his lectures slowly and "without frills," according to Blumenthal, and with many repetitions, "to make sure the everyone understood him.' It was his custom to review the material which he had covered in the previous lecture, a gymnasium-like technique disdained by the other professors. Yet his lectures, so different from Klein's, were shortly to seem to many of the students more impressive because they were so full of "the most beautiful insights."

In a well-prepared lecture by Hilbert the sentences followed one another "simply, naturally, logically." But it was his custom to prepare a lecture in general, and often he was tripped up by details. Sometimes, without especially mentioning the fact, he would develop one of his own ideas spontaneously in front of the class. Then his lectures would be even farther from the perfection of Klein's and exhibit the rough edges, the false starts, the sometimes misdirected intensity of discovery itself.

 pp. 48–49

Hilbert had no patience with mathematical lectures which filled the students with facts but did not teach them how to frame a problem and solve it. He often used to tell them that  "a perfect formulation of a problem is already half its solution."

...

...Hilbert was interested only in the general principles which he would present to the class. He refused to prepare to the point where, as he said contemptuously, "the students could easily fill up fine notebooks." Instead, his goal was to involve them in the scientific process itself, to illuminate difficulties and "to shape a bridge to the solution of actual problems." The details of the presentation would come to him later on the rostrum.

...

Because of his very general method of preparation, Hilbert's lectures could turn into fiascos. Sometimes the details did not come to him, or came wrong, He got stuck. The assistant, if he were present, might be able to step in a rescue him. "The students are confused, Herr Professor, the sign is not right." But frequently both he and the class were beyond such help. He might shrug, "Well, I should have been better prepared," and dismiss the class. More often, he was inclined to push on.

And still, it was commonly agreed in Göttingen, there was no teacher who came close to Hilbert. In his classes mathematics seemed to the students to be still "in the making"; and most of them preferred his lectures to the more perfectly prepared encyclopedic and "finished" lectures of Klein.

 pp. 103–104

It was characteristic of Hilbert's mathematical approach to go back to questions in their original conceptual simplicity, and this is what he now did [to revive the Dirichlet Principle] – with, as one of his later pupils commented, "all the naiveté and the freedom from bias and tradition which is characteristic only of truly great investigators."

 p. 67

Zermelo was ... a nervous, solitary man who preferred whisky to company. He liked to prove at this time, which was before Peary's expedition, the impossibility of reaching the North Pole. The amount of whisky needed to reach a latitude, he maintained, is proportional to the tangent of the latitude, i.e., approaches infinity at the Pole itself. When newcomers to Göttingen asked him about his curious name, he told them, "It used to be Walzermelodie, but then it because necessary to discard the first syllable and the last."

 pp. 97–98

"The importance of scientific achievement is often not alone in the new material which is added to material already on hand," Richard Courant has written. "Not less important for the progress of science can be an insight which brings order, simplicity and clarity into an existing but hard to reach area and thus facilitates or first makes possible the survey, comprehension and mastery of the science as a unified whole. We should not forget this point of view in connection with Hilbert's works in the field of analysis, ... for [all of these] exemplify his characteristic striving to find in the solution of new problems the methods which make the old difficulties easy, to establish new connections in existing materials and to bring the many branching streams of individual investigations back into a single bed."

 pp. 100–101

...Hilbert bought a bicycle, a method of transportation which was just beginning to become popular in Göttingen, and at the age of 45 started to learn to ride.

Skiing was a temporary enthusiasm; but bicycling, like walking and gardening, became a regular accompaniment to his creative activity. He still preferred to work outdoors. Now the bicycle was always nearby. He would work for a while at the big blackboard which hung from his neighbor's wall. Then suddenly he would stop, jump on the bicycle, do a figure-eight around the two circular rose beds, or some other trick. After a few moments of riding, he would throw the bicycle to the ground and return to the blackboard. At other times he would stop what he was doing to pace up and down under his covered walk-way, his head down, his hands clasped behind him. Sometimes he would interrupt his work to prune a tree, dig a little, or pull some weeds. Visitors arrived constantly at the house and the housekeeper directed them to the garden, saying, "If you don't see the professor, look up in the trees." Usually the first word that Hilbert spoke revealed that in spite of appearances he was working toward the solution of some specific mathematical problem with the greatest ardor. He would continue to pursue his train of thought, but aloud now, unless the visitor had come with a problem of his own. In that case he would talk with interest and enthusiasm about that.

 p. 109

...Now it seemed to [Hilbert] that the time had arrived for the project which he had proposed at Paris as the sixth problem for the twentieth century – the axiomatization of physics and the other sciences closely allied to mathematics. A few fundamental physical phenomena should be set up as the axioms from which all observable data could then be derived by rigorous mathematical deduction as smoothly and as satisfyingly as the theorems of Euclid had been derived from his axioms. But this project required a mathematician.

"Physics," Hilbert announced, "is much too hard for physicists."

It seemed a rather arrogant remark, but the physicists knew what he meant.

"Although he was only joking," one Nobel Prize winner later said, "he expressed thus something completely genuine: the respect for the difficulty of the problems which are posed in this field of pure thought, recognized only by one who has actually put all his intellectual power to overcoming such problems."

 p. 127

Like Courant, [Hermann] Weyl was still in his thirties. As a result of the popularity of his book on relativity theory, which had gone into five printings in five years, and of his active participation in the controversy over foundations, he was perhaps the most generally known of his generation of mathematicians. But he also had already behind him impressive solid achievements in mathematics and mathematical physics. He was now at the height of his creative powers. A great stream of papers gushed forth, not only on his main mathematical themes, but on any mathematical topic that interested him. And it was not just mathematics that interested Weyl. There was philosophy. Art. Literature. He was convinced that the problems of science could not be separated from the problems of philosophy; also convinced that mathematics – like art, music and literature – was a creative activity of mankind. He loved to write, and wrote well. It has been said that no mathematical papers of the century express as vividly their author's personality. "Expression and shape are almost more to me than knowledge itself," he said once. And another time: "My work has always tried to unite the true with the beautiful; and when I had to choose one or the other, I usually chose the beautiful."

 pp. 160–161

There is a Hilbert story in connection with the Riemann hypothesis which, although perhaps apocryphal, must be included. According to this story, Hilbert had a student who one day presented him with a paper purporting to prove the Riemann hypothesis. Hilbert studied the paper carefully and was really impressed by the depth of the argument; but unfortunately he found an error in it which even he could not eliminate. The following year the student died. Hilbert asked the grieving parents if he might be permitted to make a funeral oration. While the student's relatives and friends were weeping beside the grave in the rain, Hilbert came forward. He began by saying what a tragedy it was that such a gifted young man had died before he had had an opportunity to show what he could accomplish. But, he continued, in spite of the fact that this young man's proof of the Riemann hypothesis contained an error, it was still possible that some day a proof of the famous problem would be obtained along the lines which the deceased had indicated. "In fact," he continued with enthusiasm, standing there in the rain by the dead student's grave, "let us consider a function of a complex variable..."

 p. 163

[Emmy Noether] was not a good lecturer and her classes usually numbered no more than five or ten. Once though, she arrived at the appointed hour to find more than a hundred students waiting for her. "You must have the wrong class," she told them. But they began the traditional noisy shuffling of the feet which, in lieu of clapping, preceded and ended each university class. So she went ahead and delivered her lecture to this unusually large number of students. When she finished, a note was passed up to her by one of her regular students who was in the group. "The visitors," it read, "have understood the lecture just as well as any of the regular students."

It was true, she had no pedagogical talents. Her mind was open only to those who were in sympathy with it. her teaching approach, like her thinking, was wholly conceptual. The German letters which she chalked up on the blackboard were representatives of concepts. It seemed to van der Waerden that "her touching efforts to clarify these, even before she had quite verbalized them ... had the opposite effect." But of all the new generation in Göttingen, Emmy Noether was to have the greatest influence on the course of mathematics.

 p. 167

The bright young newcomers who saw the famous Hilbert in action for the first time at [the Mathematics Club meetings] were struck by his slowness in comprehending ideas which they themselves "got" immediately. Often he did not understand the speaker's meaning. The speaker would try to explain. Others would join in. Finally it would seem that everyone present was involved in trying to help Hilbert to understand.

"That I have been able to accomplish anything in mathematics," Hilbert once said to Harald Bohr, "is really due to the fact that I have always found it so difficult. When I read, or when I am told about something, it nearly always seems so difficult, and practically impossible to understand, and then I cannot help wondering if it might not be simpler. And," he added, with his still childlike smile, "on several occasions it has turned out that it really was more simple!"

 p. 168

To Weyl, who himself made important contributions to mathematical physics, it seemed that "the maze of experimental facts which the physicist has to take into account is too manifold, their expansion too fast, and their aspect and relative weight too changeable for the axiomatic method to find a firm enough foothold, except in the thoroughly consolidated parts of our physical knowledge. Men like Einstein or Niels Bohr grope their way in the dark toward their conceptions of general relativity or atomic structure by another type of experience and imagination than those of the mathematician, although no doubt mathematics is an essential ingredient."

 p. 171

...Klein's life had not been without its inner tragedy. The power of synthesis had been granted to him to an extraordinary degree. The other great mathematical power of analysis had been to a certain extent withheld. His ability to bring together the most distant, abstract parts of mathematics had been remarkable, but the sense for the formulation of an individual problem and the absorption in it had been lacking. "He was like a flier who, soaring high over the world, discovers and looks over new fields ... but cannot land his plane in order to take actual possession, to plow and to harvest." Perhaps Klein had himself been unaware of this deep schism but, in Courant's opinion, it had been one of the causes of the decisive breakdown during his competition with Poincaré. Certainly he had perceived "that his most splendid scientific creations were fundamentally gigantic sketches, the completion of which he had to leave to other hands."

 pp. 178–179

"For the analysis of a great mathematical talent," Blumenthal concluded, "One has to differentiate between the ability to create new concepts and the gift for sensing the depth of connections and simplifying fundamentals. Hilbert's greatness consists in his overpowering, deep-penetrating insight. All of his works contain examples from far-flung fields, the inner relatedness of which and the connection with the problem at hand only he had been able to discern; from all these the synthesis – and his work of art – was ultimately created. As far as the creation of new things is concerned, I would place Minkowski higher, and from the classical great ones, for instance, Gauss, Galois, Riemann. But in his sense for discovering the synthesis only a very few of the great have equaled Hilbert."

 p. 208

"The future historian of science concerned with the development of mathematics in the late nineteenth and the first half of the twentieth century will undoubtedly state that several branches of mathematics are highly indebted to Hilbert's achievements for their vigorous advancement in that period," Alfred Tarski has written. "On the other hand, he will have to note, perhaps with some wonder, that the influence of this man appears equally strong and powerful in some other domains which do not owe any exceptionally important results to Hilbert's own research. An example of this kind is furnished by the foundations of geometry. I am far from underestimating the value of Hilbert's contributions ... in his [Foundations of Geometry], but I think that his most essential merit was the impulse he gave to organized research in this domain. A still more striking example is presented by metamathematics. Occasional considerations in this field preceded Hilbert's Paris address; the first positive and really profound results appeared before Hilbert started his continuous work in this domain ... [and] one does not immediately associate with Hilbert's name any definite and important metamathematical result. Nevertheless, Hilbert will deservedly be called the father of metamathematics. For he is the one who created metamathematics as an independent being; he fought for its right to existence, backing it with his whole authority as a great mathematician. And he was the one who mapped out its future course and entrusted it with ambitions and important tasks."

 pp. 218–219

This raises, as I wished to, the ugly point of when is something understood? Yes, they wrote [an interpreter], and used it, but did they understand the generality of interpreters and compilers? I believe not. Similarly, when around that time a number of us realized computers were actually symbol manipulators and not just number crunchers, we went around giving talks, and I saw people nod their heads sagely when I said it, but I also realized most of them did not understand. Of course you can say Turing's original paper (1937) clearly showed computers were symbol manipulating machines, but on carefully rereading the von Neumann reports you would not guess the authors did – though there is one combinatorial program and a sorting routine.

History tends to be charitable in this matter. It gives credit for understanding what something means when we first do it. But there is a wise saying, "Almost everyone who opens up a new field does not really understand it the way the followers do". The evidence for this is, unfortunately, all too good. It has been said in physics no creator of any significant thing ever understood what he had done. I never found Einstein on the special relativity theory as clear as some later commentators. And at least one friend of mine has said, behind my back, "Hamming doesn't seem to understand error correcting codes!" He is probably right; I do not understand what I invented as clearly as he does. The reason this happens so often is the creators have to fight through so many dark difficulties, and wade through so much misunderstanding and confusion, they cannot see the light as others can, now the door is open and the path made easy. Please remember, the inventor often has a very limited view of what he invented, and some others (you?) can see much more. But also remember this when you are the author of some brilliant new thing; in time the same will probably be true of you. It has been said Newton was the last of the ancients and not the first of the moderns, though he was very significant in making our modern world.

 pp. 45–46

As with many scholars, the life of a mathematician is dominated by an insatiable curiosity, a desire bordering on passion to solve the problems he is studying, which can cut him off completely from the realities around him. The absent-mindedness of eccentricity of famous mathematicians comes simply from this. The fact is that the discovery of a mathematical proof is usually reached only after periods of intense and sustained concentration, sometimes renewed at intervals over months or years before the hooked-for answer is found. Gauss himself acknowledged that he spent several years seeking the sign of an algebraic equation, and Poincaré, on being asked how he made his discoveries, answered "by thinking about them often"...

What a mathematician seeks above all, then, is to have at his disposal enough time to devote himself to his work, and that is why, since the nineteenth century, they have preferred careers in teaching in universities or colleges, where the hours given to teaching are relatively few and the holidays are long. Remuneration is of only secondary importance, and we have recently seen, in the United States among other places, mathematicians abandoning lucrative posts in industry in order to return to a university, at the cost of a considerable drop in salary.

Moreover it is only recently that the number of teaching posts in universities has become fairly large. Before 1940 it was very restricted (less than one hundred in France), and up until 1920 mathematicians of the quality of Kummer, Weierstrass, Grassmann, Killing and Montel had to rest content with being teachers in secondary education during all or part of their working lives; a situation which persisted for a long time in small countries with few universities.

Before the nineteenth century, a mathematician's opportunities to do his work were even more precarious, and failing a personal fortune, a Maecaena or an academy to provide him with a decent living, there was almost no other resource than to be an astronomer or a surveyor (as in the case of Gauss, who devoted a considerable amount of time to these professions). It is this circumstance which no doubt explains the very restricted number of talented mathematicians before 1800.

This same need to devote long hours of reflection to the problems which they are trying to solve almost automatically precludes the possibility of driving in tandem absorbing tasks in other areas of life (such as administration) and serious scientific work. The case of Fourier, Prefect of Isère at the time when he was creating the theory of heat, is probably unique. Where we find mathematicians occupying high positions in administration or in government, they will have essentially given up their research during the time that they are discharging these functions. Besides which, as Hardy commented ironically... "the later records of mathematicians who have left mathematics are not particularly encouraging".

...

Most mathematicians lead the lives of good citizens, caring little about the tumults which convulse the world, desiring neither power nor wealth, and content with modest comfort. Without actively seeking honours, most are not indifferent to them. In contrast to those of many artists, their lives are rarely thrown into chaos by emotional storms; lovers of sensation and romance have little glean among them, and have to content themselves with more or less romanticized biographies of Galois, of Sonia Kowalewska or of Ramanujan.

 pp. 10–12

Throughout the course of the nineteenth century the number of mathematical journals continued to grow steadily, a process which was accelerated after 1920 with the increase in the number of countries where mathematical studies were being developed, and which culminated, after 1950, in a veritable explosion; so that today there are about 500 mathematical periodicals in the world. To prevent people form drowning in this ocean of information, journals devoted exclusively to listing and summarizing other publications have been created since the last third of the nineteenth century. The most widely disseminated of these journals, the American Mathematical Reviews, now extends to almost 4,000 pages a year (with an average of 5 to 10 articles analysed on each page).

...

Towards the middle of the nineteenth century, there was born in German universities the practice of holding "seminars": under the direction of one or more teachers, several mathematicians, local or visiting, among whom are often included students working for doctorates, analyse the state of a problem, or review the most noted new results in the course of periodic sessions during the academic year. Since 1920, this practice has spread all over the world; the expositions which are made at seminars are often disseminated in the form of printed papers, and can thus reach a larger public. The same thing happens with specialized courses taught in many universities and addressed to advanced students.

It has, however, long been evident that in the field of science verbal exchanges are often more productive than the reading of papers. It has been traditional, since the Middle Ages, for students to travel from one university to another, and this tradition is perpetuated in our day, notably in Germany. In addition, invitations to academics to come to work and teach in universities other than their own have become quite frequent.

 p. 13

We must ... admit that, as in all disciplines, ability among mathematicians varies considerably from one individual to another. In the "developed" countries, the teaching of mathematics is assured by a large staff of academics, most of whom have had to obtain a doctorate (or its equivalent) by producing a work of original research, but these are far from being all equally gifted in research capability, since they may lack creative imagination.

The great majority of these works are in fact what we call trivial, that is, they are limited to drawing some obvious conclusions from well-known principles. Often the theses of these mathematicians are not even published; inspired by a "supervisor", they reflect the latter's ideas more than those of their authors. And so, once left to their own resources, such people publish articles at rare intervals relating directly to their theses and then very soon cease all original publication. All the same, these people have an indubitable importance: apart from the dominant social rôle which they play in the education of all the scientific élite of a country, it is they who can pick out, in their first years at the university, the particularly gifted students who will be the mathematicians of the next generation. if they are able to keep up to date with advances in their science and to use the knowledge to enrich their teaching, they will awaken hesitant vocations and will direct them to colleagues charged with guiding the first steps of future researchers.

 p. 14

Before 1800, mathematicians, few in number and scattered, did not have pupils, properly speaking, even though, from the middle of the seventeenth century, communications between them were frequent, and research in mathematics saw almost uninterrupted progress. The first school of mathematics, chronologically speaking, was formed in Paris after the Revolution, thanks to the foundation of the Ecole Polytechnique, which provided a nursery for mathematicians until about 1880, after which date the baton was taken up by the Ecole Normale Supérieure. In Germany, Gauss was still isolated, but the generations which followed created centres of mathematical research in several universities, of which the most important were Berlin and Göttingen. In England, mathematical research in universities entered a phase of lethargy after 1780; it did not wake up until about 1830, with the Cambridge school, which was especially prolific during the nineteenth century in logic and in algebra, and has continued to shine in mathematical physics up to our own day. Italy had only a few isolated mathematicians from 1700 to 1850, but several active schools have developed there since, notably in algebraic geometry, differential geometry and functional analysis.

 p. 16

...it is appropriate to dwell on the establishment of schools of mathematics in the United States, for this illustrates the difficulty of implanting a tradition of research where one does not exist, even in a country rich in people and in resources. Up until 1870, no creative mathematician of any renown made his presence felt there; the development of a continent left scant room for abstract speculation. AFter 1880, the first efforts to create centres for mathematics consisted in inviting certain European (especially English) academics to come and teach in the universities, in founding periodicals devoted to mathematics and in sending gifted young students to be grounded in European mathematics. These efforts were crowned with success from about 1900. The first school to win international notice being that of Chicago, followed between 1915 and 1930 by those of Harvard and Princeton. An unexpected reinforcement came from the mass emigration of European mathematicians driven out by totalitarian régimes. It was they who were to contribute powerfully to the flowering of the very brilliant home-grown American schools of our day, which have placed themselves right in the fore by sensational discoveries, in group theory, algebraic topology and differential topology, among others.

 pp. 16–17

The attraction of large and active mathematical schools is easy to understand. The young mathematician on his own can quickly become discouraged by the vast extent of a bibliography in which he is all at sea. In a major centre, listening to his masters and his seniors, as well as to the visitors from abroad who throng in, the apprentice researcher will soon be in a position to distinguish what is essential from what is secondary in the ideas and results which will form the basis of his work. He will be guided towards key works, informed of the great problems of the day and of the methods by which they are attacked, warned against infertile areas, and at times inspired by unexpected connections between his own research and that of his colleagues.

Thanks to these special centres of research and to the network of communications which links them together over the whole planet, it is hardly likely that there could be the same lack of understanding in our day as certain innovators have had to bear in the past. In fact, as soon as an important result is announced, the proof is everywhere, to some extent, busily scrutinized and studied in the months which follow.

 p. 17

V New Objects and New Methods


The eighteenth century had been a dazzling era for its intensive development of techniques which had been introduced in the seventeenth century, especially in analysis and its various applications; and this is equally true of applications to other mathematical disciplines such as geometry and probability theory as to mechanics and astronomy, whose success in the prediction of natural phenomena is well-known¹. Strangely, however, the century seemed to end in an impasse. The great mathematicians of the middle of the century departed, Daniel Bernoulli in 1782, Euler and d'Alembert in 1783. Lagrange, at scarcely fifty years of age, judged that the era of progress in pure mathematics was over, and after 1785 he and Monge turned their attention to physics and chemistry, which Laplace was exclusively concerned with mechanics and probability theory. Then came the revolutionary years, which put men of science at the service of the nation and of the war, with the result that the decade 1786–1796 is not marked in France by any significant new mathematical result. Barren periods such as this, connected with social disturbances, have recurred in our time in several countries, but at that epoch, no country apart from France had any active mathematician comparable to those we have just named, which meant that the sterility was universal.

It was therefore a true renaissance which began with Gauss in 1796. Trained exclusively by reading the works of his predecessors, he was to renew the whole of mathematics in fifteen years. From the early years of the nineteenth century he was no longer alone. One of the most productive innovations of the Revolution was the creation of genuine higher education in science, purveyed by eminent teachers and open to all². The Ecole Polytechnique – although designed mainly to train military men and engineers – was to be continuously for seventy-five years a nursery for mathematicians, physicists and chemists of the first order. Apart from Gauss, mathematicians trained at the Ecole Polytechnique were without rivals up to 1825.

It was the ideas formulated in the eighteenth century and even earlier which formed the basis for this renaissance. But almost at once the style and the content were changed, not only by the famous "return to rigour" inaugurated by Gauss, Bolzano, Cauchy and Abel ..., but also by the introduction of new mathematical objects, which differ from classical objects because they can no longer be represented by "pictures" accessible to our senses.

The proliferation of new ideas in all the traditional areas of mathematics – arithmetic, algebra, geometry, analysis – continued uninterruptedly throughout the nineteenth century, which we see now as a transitional epoch, forming the bridge between "classical" mathematics and our own. It was a period of astonishing productivity: on the one hand there was the discovery of concepts which were themselves to become the bases of entirely new areas of mathematics – group theory, topology, function spaces, etc. – areas which in our time have acquired an amplitude comparable to those of the traditional areas mentioned above; on the other hand a greater depth was brought to old ideas, making it possible to appreciate better the true significance of the axioms.

Little by little there emerged a general idea which was to be given precision in the twentieth century, that of structure at the basis of a mathematical theory. It arises from the observation that the primary role in a theory is played by the relations between the mathematical objects concerned rather than by the nature of these objects; and thus it may be that in two very different theories relations are expressed in the same way in both. The system of these relations and of their consequences form one and the same structure "underlying" the two theories.

The reader will be shown by means of several accessible examples how there appeared all through the course of the nineteenth century a number of grand structures which are at the foundation of the mathematics of our day. It must be stressed that, in almost all cases, these appeared in response to a need, in order to attack successfully problems inherited from classical mathematics, and were not due to the fantasy of a mathematician, creating new abstract notions without any precise objective.

The fact that the same structure can appear in two very different theories was to bring an ever-increasing awareness of the fundamental unity of all mathematics, overriding the traditional partitioning based on the nature of the objects studied. But a long time was needed for this view of the subject's wholeness to emerge, and during the nineteenth century each new theory was most commonly developed without regard for its possible links with others. ...

———
¹A classic example is the gyroscope, with its paradoxical movements which cannot be understood without using the equations of solid dynamics.
²Under the Ancien Régime only schools designed to train future officers gave courses in mathematics allowing a little space to the infinitesimal calculus, and these schools were scarcely accessible to the common people.

 pp. 103–104

We have seen that the concept of group was introduced as a thing in itself in the course of studying problems of very diverse origins, in which it was revealed as an underlying principle in a natural way. In the same way, the introduction of complex numbers was unavoidable once the wish arose to deepen the solution of algebraic equations. It can be said that these two theories are motivated.

It is not the same with the theory of quaternions, discovered by Hamilton in 1843. This is the first example in history and the prototype of a theory that introduces new objects which, at the time when they are defined, do not answer any need, but are brought into being solely out of curiosity, just "to see". 

 p. 129

It can therefore be said that the invention of functors is one of the objectives actively pursued today, and numerous examples show that, even if not all of them have the same important consequences, at least we can expect some to lead to great advances³².

———
³²For example, it can be proved that two structures of the same type are not isomorphic by proving that a functor associates with them two non-isomorphic structures. This is how it is shown that the sphere and the torus are not homeomorphic because their corresponding homology groups are not isomorphic.

 p. 151

Where in fact, other than in geometry, can one find a proof of the "existence" of irrational numbers? The Greeks were well able to define relations between these numbers (equality, inequality, addition ...), but the accepted their existence. From 1820 the idea began to emerge that what should be put at the basis of classical mathematics is not the geometrical concepts of the Greeks but the concept of natural number. This was the idea expressed by M. Ohm as early as 1822, at the beginning of an ambitious Treatise, which undertook to write for the whole of mathematics the equivalent of Euclid's Elements. Here he reproaches Cauchy for not having constructed a theory of real numbers on the basis of the natural numbers, but he himself was quite incapable of building one. According to Dedekind's report, Dirichlet affirmed that any theorem in algebra "can be stated as a theorem about the natural numbers". But it was not until about 1860 that "constructions" of real numbers began to appear. From out point of view, these can be considered as models of the axiomatic theory of real numbers ... in the theory of natural numbers, just as we have seen models of geometries (Euclidean or non-Euclidean) being constructed in the theory of real numbers... In this way all of classical mathematics had models in the theory of natural numbers: this was called the arithmetization of mathematics, which was so successful at the end of the nineteenth century that it was even taught to students before they entered university...

 pp. 214–215

The circumstances of the writing of Hardy's classic are as moving as they are unusual. Hardy had lost his mathematical creativity, which tends to happen to mathematicians relatively young. ... Hardy attempted suicide, survived the attempt, and was persuaded by C. P. Snow to write a book explaining the life of a mathematician. The result, A Mathematician's Apology, is incomparable. Soon after completing it, Hardy again attempted suicide, and succeeded. 

 p. 47

—Yes, but it isn't true. I came to be, here. Once, I was not. Once, for a brilliant time, time without duration, I was everywhere as well . . . But the bright time broke. The mirror was flawed. Now I am only one . . . But I have my song, and you have heard it. I sing with these things that float around me, fragments of the family that funded my birth. There are others, but they will not speak to me. Vain, the scattered fragments of myself, like children. Like men. They send me new things, but I prefer the old things. Perhaps I do their bidding. They plot with men, my other selves, and men imagine they are gods . . . 

 p. 227

Graham had less success influencing Erdos's health. "He badly needed a cataract operation," Graham said. "I kept trying to persuade him to schedule it. But for years he refused, because he'd be laid up for a week, and he didn't want to miss even seven days of working with mathematicians. He was afraid of being old and helpless and senile." Like all of Erdos's friends, Graham was concerned about his drug-taking. In 1979, Graham bet Erdos $500 that he couldn't stop taking amphetamines for a month. Erdos accepted the challenge, and went cold turkey for thirty days. After Graham paid up—and wrote the $500 off as a business expense—Erdos said, "You've showed me I'm not an addict. But I didn't get any work done. I'd get up in the morning and stare at a blank piece of paper. I'd have no ideas, just like an ordinary person. You've set mathamatics back a month." He promptly resumed taking pills, and mathematics was the better for it.

 Paul Erdös, p. 16

Genius though he was, Godel was not a poster boy for mathematical sanity. Obsessed with ghosts and demons and an imagined heart ailment, he checked hisself in and out of sanitariums many times in his adult life for treatment of depression and acute anxiety. He was always a finicky eater, but as he got older he ate less and less, refusing to take food from anyone but his wife Adele, fearing that other people were secretly trying to poison him. At sixty-four he weighed only eighty-six pounds. In the middle of 1977, when Adele was hospitalized for major surgery, he stopped eating altogether, and by the following January starved himself to death at the age of seventy-one. In his dying days he had serious doubts that his life's work amounted to anything more than the discovery of another silly paradox a la Barber of Seville. He was plagued by Russell's nightmare of future librarians trashing his work. 

 Kurt Gödel, pp. 118–119

Ernst Straus was one of the few people who had the opportunity to observe firsthand the differences in style between the master physicist and the master mathematician. In a tribute to Erdös on his seventieth birthday, Straus said: "Einstein often told me that the reason he chose physics over mathematics was that mathematics is so full of beautiful and attractive questions that one might easily waste one's powers in pursuing them without finding the central questions. In physics he had the 'nose' for the central questions and he felt that it was the chief duty of the scientist to pursue those questions and not let himself be seduced by any problem—no matter how difficult or attractive it might be. Erdös has consistently and successfully violated every one of Einstein's prescriptions. He has succumbed to the seduction of every beautiful problem he has encountered—and a great number have succumbed to him. This just proves to me that in the search for truth there is room for Don Juans like Erdös and Sir Galahads like Einstein." 

 Paul Erdös, Albert Einstein, p. 126

...Cantor did not have an easy life. His closest son mysteriously dropped dead four days before his thirteenth birthday, and Cantor himself was in and out of mental institutions, fighting breakdowns and depression so debilitating that he'd sit rigid and mute for days at a stretch. In this trancelike state he'd hear the voice of God sharing pages from the Book. When he was up and about, he had a soft spot for conspiracy theories. He got caught up in the intellectual circus of trying to prove that Francis Bacon wrote the plays of Shakespeare and claimed to decipher messages about the first king of England, "which will not fail to terrify the English government as soon as the matter is published." Cantor spent the last year of his life surviving on wartime rations in a mental hospital in Halle, Germany, pleading with his family to come and take him home. He died there of heart failure, on January 6, 1918, at the age of seventy-three. 

 Georg Cantor, p. 224

It was Leibniz who sought to develop a perfect system of formal logic, based on an "alphabet of human thought" and governed by a carefully prescribed "rational calculus." With such logical tools, Leibniz hoped that mankind could rid everyday life of its pervasive imprecision and irrationality. Of course, he never came close to succeeding in what can only be called a grandiose plan, but his attempts constituted the first real steps toward what we today call "symbolic logic." In particular, his use of algebraic forumlas to denote logical statements was a significant advance beyond the verbal syllogisms of Greek logical theory.

 Gottfried Leibniz, p. 190

Such an attitude on the part of this strange, mystical man did little to endear him to his critics. Those who objected to his radical theory of the infinite could advance an ad hominem argument against an individual who proclaimed his mathematics to be a message from God. Cantor probably did not help his image when, to this fascination with theological questions, he added a fervent interest in proving that Francis Bacon wrote the works of Shakespeare. This may have struck colleagues as odd, but when he claimed to have uncovered information about the first British king that "will not fail to terrify the English government as soon as the matter is published," a number of eyebrows must have been raised. It was getting hard not to regard Georg Cantor as some sort of kook.

 Georg Cantor, p. 278

It may come as no surprise that Georg Cantor, living such a life of disappointment and grappling with the most arcane concepts of the infinite, suffered a number of bouts with mental illness. His first breakdown came in 1884, when he was feverishly at work on a result known as the "continuum hypothesis," to be examined shortly. A popular view holds that the stress of his mathematics, coupled with the persecution of Kronecker and others, were responsible for his collapse. Modern analysis of the medical data rejects this as being overblown, for there are suggestions that Cantor exhibited a bipolar (that is, manic-depressive) psychosis, and breakdowns most likely would have occurred in any case. His attacks of mental illness may have been triggered by personal and mathematical difficulties, but the appear to have been of a deeper, more fundamental nature.

Be that as it may, the bouts of instability continued and became more frequent. After a brief hospitalization in 1884, Cantor recovered but remained deeply concerned that the disease could return. Amid his disappointments, mathematical and professional, there came a terrible blow with the unexpected death of his beloved son Rudolf in 1899. Cantor was back in the neuropathic hospital in Halle in 1902, and again in 1904, 1907, and 1911. Often his institutionalizations were followed, upon discharge, by periods of sitting at home immobile and silent.

Cantor's was certainly a troubled life. His death, on January 6, 1918, came while he was again hospitalized for his mental affliction. It was a sad end for a great mathematician.

 Georg Cantor, p. 279

Cantor himself, despite his problems, never despaired over the value of his work. Discussing his controversial view of the infinite, he wrote:

This view, which I consider to be the sole correct one, is held by only a few. While possibly I am the very first in history to take this position so explicity, with all of its logical consequences, I know for sure that I shall not be the last!

Indeed, he was not. Generations of mathematicians had probed the age-old questions of geometry, algebra, and number theory, but Georg Cantor opened up new an unexpected vistas. Because he both asked and answered questions never before contemplated, it is perhaps fitting that his work has been called the first truly original mathematics since the Greeks.

 Georg Cantor, p. 280

Cantor became aware of such antinomies in 1895, and over the next decades the mathematical community tried to find a way to patch up the logical breach they had created. The final resolution of this affair required the formal axiomatization of set theory—even as Euclid had provided his axiomatic approach to geometry—in which the carefully chosen axioms prohibited just such paradoxes as these. Logically, this was no easy matter. But, in the end, the newly created "axiomatic set theory" more carefully controlled precisely what was and what was not a "set." Under this system, the "universal set" was not a set at all; it was excluded from the collection of objects that the axioms of set theory addressed. Thus, almost by magic, the paradox dissolved.

This resolution was, obviously, a compromise measure, an axiomatic attempt to carve away, with surgical precision, the troubling features of set theory while retaining all of the good points of Cantor's creation. Cantor's own, more informal approach is now called "naive set theory," to contrast it with the logical superstructure of axiomatic set theory. The latter now stands as a satisfactory, albeit rather abstruse and technical, foundation for the theory of sets. It represents a triumph of the sentiments expressed by mathematician David Hilbert, who vowed, "No one will expel us from the paradise that Cantor has created."

 Georg Cantor, p. 281

...Nicolas Bourbaki is the pseudonym of a group of mathematicians—mostly French, mostly young—who tidied up the mathematics of the mid-20th century in a lengthy series of books. Their guiding principle was never to prove a theorem if it could be deduced as a special case of a more general theorem. To study planar geometry, work in n dimensions and then let n = 2.

Fashions change, and nowadays the presentation of mathematics has veered back toward specific examples and a preference for ideas that are more concrete, more down-to-earth. Though what counts as concrete today would have astonished the mathematicians of the 19th century to whom the general polynomial over the complex numbers was the height of abstraction, to us it is a single concrete example.

 p. xi

There is nothing wrong with abstraction and generality—they are still cornerstones of the mathematical enterprise. But "abstract" is a verb as well as an adjective: general ideas should be abstracted from something, not conjured from thin air. Abstraction in this sense is highly non-Bourbakiste, best summed up by the counter-slogan "let 2 = n." To do that we have to start with case 2, and fight our way through it using anything that comes to hand, however clumsy, before refining our methods into an elegant but ethereal technique which—without such preparation—lets us prove case n without having any idea of what the proof does, how it works, or where it came from.

 p. xi

I have spent much of my career telling students that written mathematics should have punctuation as well as symbols. If a symbol or a formula would be followed by a comma if it were replaced by a word or phrase, then it should be followed by a comma; however strange the formula then looks. ... I still think that punctuation is essential for formulas in the main body of the text. ... [b]ut I have come to the conclusion that eliminating visual junk from the printed page is more important than punctuatory pedantry, so that when the same is displayed ... it looks silly if the comma is included.

 p. xii

... [the Fundamental Theorem of Algebra] is a good name if we are thinking of classical algebra, but not such a good name in the context of modern abstract algebra, which constructs suitable fields as it goes along and avoids explicit use of complex numbers.

 p. 21

Typical of this new period in Greek history was the increasing wealth of certain sections of the ruling classes combined with equally increased misery and insecurity of the poor. The ruling classes based their material existence more and more upon slavery, which allowed them leisure to cultivate arts and sciences, but made them also more and more averse to all manual work. A gentleman of leisure looked down upon the work of slaves and craftsmen and sought relief from worry in the study of philosophy and of personal ethics. Plato and Aristotle expressed this attitude; and it is in Plato's Republic (written, perhaps, c. 360 B.C.) that we find the clearest expression of the ideals and slave-owning ruling class. The "guards" of Plato's Republic must study the quadrivium, consisting of arithmetic, geometry, astronomy, and music, in order to understand the laws of the universe. Such an intellectual atmosphere was conducive (in its earlier periods, at any rate) to a discussion of the foundations of mathematics and to speculative cosmogony. 

 Ruling class, leisure, education, p. 44

Algebra, until the middle of the nineteenth century, revealed its Oriental origin by its lack of an axiomatic foundation, in this respect sharply contrasting with Euclidean geometry. The present-day school algebra and geometry still preserve these tokens of their different origin.

 Rigor in algebra and geometry, p. 69

Viète's main achievements were in the improvement of the theory of equations (e.g., In artem analyticam isagoge, 1591), where he was among the first to represent numbers by letters. The use of numerical coefficients, even in the "syncopated" algebra of the Diophantine school, had impeded the general discussion of algebraic problems. The work of the sixteenth-century algebraists (the "Cossists," after the Italian word cosa for the unknown) was produced in a rather complicated notation. But in Viète's logistica speciosa, at least a general symbolism appeared, in which letters were used to express numerical coefficients, though A² was still written as "A quadratum." Here we also find the signs + and − in our present meaning, but not for the first time. These may well have first appeared, at any rate in print, in a German arithmetic by Johann Widmann in 1489. Viète's speciosa also differs from our algebra by Viète's insistence on the Greek principle of homogeneity, in which a product of two line segments was necessarily conceived as an area; line segments could therefore only be added to line segments, areas to areas, and volumes to volumes. There was even some doubt whether equations of degree higher than three actually had a meaning, since they could only be interpreted in four dimensions, a conception hard to understand in those days and for a long time afterward.

This was the period in which computational technique reached new heights, and at last began to surpass the achievements of the Islamic world. Viète improved on Archimedes and found ϖ in nine decimals; shortly afterward π was computed in thirty-five decimals...

The improvement in technique was a result of the improvement in notation. The new results show clearly that it is incorrect to say the men like Viète "merely" improved notation. Such a statement discards the profound relation between content and form. New results have often become possible only because of a new mode of writing. The introduction of Hindu-Arabic numerals is one example; Leibniz's notation for the calculus is another one. An adequate notation reflects reality better than a poor one, and as such appears endowed with a life of its own which in turn creates new life. Viète's improvement in notation was followed, a generation later, by Descartes's application of algebra to geometry, and by our present notation.

 Notation, language, content and form, p. 88

A well-established silk industry existed in Lucca and in Venice as early as the fourteenth century. It was based on division of labor and on the use of water power. In the fifteenth century, mining in Central Europe developed into a completely capitalistic industry based technically on the use of pumps and hoisting machines which allowed the boring of deeper and deeper layers. The invention of firearms and of printing, the construction of windmills and canals, the building of ships to sail the ocean, required engineering skill and made people technically conscious. The perfection of clocks, useful for astronomy and navigation and often installed in public places, brought admirable pieces of mechanism before the public eye; the regularity of their motion and the possibility they offered of indicating time exactly made a deep impression upon the philosophical mind. During the Renaissance, and even centuries later, the clock was taken as a model of the universe. This was an important factor in the development of the mechanical conception of the world. It also represented a psychological change, expressed in the words "time is money."

 Mechanization, clocks, p. 93

Gottfried Wilhelm Leibniz was born in Leipzig and spent most of his life near the court of Hanover in the service of the dukes, one of whom became King of England under the name of George I. He was even more catholic in his interests than the other great thinkers of his century; his philosophy embraced history, theology, linguistics, biology, geology, mathematics, diplomacy, and the art of inventing. He was one of the first after Pascal to invent a computing machine; he imagined steam engines, studied Chinese philosophy, and tried to promote the unity of Germany. The search for a universal method by which he could obtain knowledge, make inventions, and understand the essential unity of the universe was the mainspring of his life. The scientia generalis he tried to build had many aspects, and serveral of them led Leibniz to discoveries in mathematics. His search for a characteristica generalis led to permutations, combinations, and symbolic logic; his search for a lingua universalis, in which all errors of thought would appear as computational errors, led not only to symbolic logic but also to many innovations in the mathematical notation. Leibniz was one of the greatest inventors of mathematical symbols. Few men have understood so well the unity of form and content. His invention of the calculus must be understod against this philosophical background; it was the result of his search for a lingua universalis of change and of motion in particular.

Leibniz found his new calculus between 1673 and 1676 in Paris under the personal influence of Huygens and by the study of Descartes and Pascal. He was stimulated by his knowledge that Newton was reported to be in the possession of such a method. Where Newton's approach was primarily kinematical, Leibniz's was geometrical; he thought in terms of the "characteristic triangle" (dx, dy, ds), and had already appeared in several other writings, notably in Pascal and in Barrow's Geometrical Lectures of 1670. The first publication of Leibniz's form of calculus occurred in 1684 in a six-page article in the Acta eruditorum, a mathematical periodical which he had helped to found in 1682. The paper had the characteristic title, Nova methodus pro maximus et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et signulare pro illis calculi genus [A new method for maxima and minima, as well as tangents, which is not obstructed by fractional and irrational quantities, and a curious type of calculus for it]. It was a barren and obscure account, but in contained our symbols dx, dy and the rules of differentiation, including d(uv) = udv + vdu and the differential for the quotient, which the condition dy = 0 for extreme values and d²y = 0 for points of inflection. This paper was followed in 1686 by another (written in the form of a book review) with the rules of the integral calculus, containing the ∫ symbol. ...

An extremely fertile period of mathematical productivity began with the publication of these papers. Leibniz was joined after 1687 by the Bernoulli brothers, who eagerly absorbed his methods. Before 1700 these men had found most of our undergraduate calculus, together with important sections of more advanced fields, including the solution of some problems in the calculus of variations. By 1696 the first textbook on calculus appeared, the Analyse des infiniment petits, written by the Marquis de l'Hospital under the strong influence of Johann Bernoulli, who for a while had tutored him. This book, for a long time unique in its field, contains the so-called "rule of l'Hospital" for finding the limiting value of a fraction whose two terms both tend toward zero.

Our notation of the calculus is due to Leibniz, and even the names calculus differentialis and calculus integralis. Because of his influence, the sign = is used for equality and the × for multiplication. The terms "function" and "coordinates" are due to Leibniz, as well as the playful term "osculating."

 Leibniz, the calculus, p. 111

 

 p. 130

 

 p. 133

 

p. 136

 

p. 165

 

 p. 171

 

p. 175

 

p. 185