Gh Quotes

The world's continual breathing is what we hear and call silence.

And I want to be held down. I don't know what to do with the horrifying freedom that can destroy me.

It is not worth an intelligent man's time to be in the majority. By definition, there are already enough people to do that.

Angel is right,"said Dr. G-H quickly. "This is my clumsy way of demonstrating." "Demonstrating what?" I was barely able to keep a snarl out of my voice. "How to get yourself beat up in one easy step?

What I want is to live of that initial and primordial something that was what made some things reach the point of aspiring to be human.

Reality prior to my language exists as an unthinkable thought. . . . life precedes love, bodily matter precedes the body, and one day in its turn language shall have preceded possession of silence.

Claire was going to hate me. Our son was sucked into the pits of hell while I was watching General Hospital. God damn you Brenda and Sonny for making me lose focus.

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.

The mystery of human destiny is that we are fated, but that we have the freedom to fulfill or not fulfill our fate: realization of our fated destiny depends on us. While inhuman beings like the cockroach realize the entire cycle without going astray because they make no choices.

Holding someone's hand was always my idea of joy. Often before falling asleep - in that small struggle not to lose consciousness and enter the greater world - often, before having the courage to go toward the greatness of sleep, I pretend that someone is holding my hand and I go, go toward the enormous absence of form that is sleep. And when even then I can't find the courage, then I dream.

I spend roughly $80 per year watching bananas go brown.

Real mathematics must be justified as art if it can be justified at all.

Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.

I' is merely one of the world's instantaneous spasms.

I'm so frightened that I shall be able to accept the notion that I have lost myself only if I imagine that someone is holding my hand.

In the world there exists no aesthetic plane, not even the aesthetic plane of goodness.

I don't want beauty, I want identity.

Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. “Immortality” may be a silly word, but probably a mathematician has the best chance of whatever it may mean.

Would it be simplistic to think the moral problem with regards to others consists in behaving as one ought to, and the moral problem with regards to oneself is managing to feel what one ought to?

Reality is the raw material, language is the way I go in search of it - and the way I do not find it. But it is from searching and not finding that what I did not know was born, and which I instantly recognise. Language is my human effort. My destiny is to search and my destiny is to return empty-handed. But - I return with the unsayable. The unsayable can only be given to me through the failure of my language. Only when the construction fails, can I obtain what I could not achieve.

Perder-se é um achar-se perigoso.

The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.

If a man has any genuine talent he should be ready to make almost any sacrifice in order to cultivate it to the full.

I, who called love my hope for love.

Oh, don't pull your hand away from me, I've promised myself that maybe by the end of this impossible narrative I shall understand, oh maybe it will be on Hell's road that I shall be able to find what we need—but don't pull your hand away, even though I now know that the finding has to come on the road of what we are, if I can succeed in not sinking completely into what we are.

Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune; and there are probably more people really interested in mathematics than in music. Appearances suggest the contrary, but there are easy explanations. Music can be used to stimulate mass emotion, while mathematics cannot; and musical incapacity is recognized (no doubt rightly) as mildly discreditable, whereas most people are so frightened of the name of mathematics that they are ready, quite unaffectedly, to exaggerate their own mathematical stupidity

Never again shall I understand anything I say. Since how could I speak without the word lying for me? How could I speak except timidly like this: life just is for me. Life just is for me, and I don't understand what I'm saying. And so I adore it.

Depersonalization like the deposing of useless individuality— the loss of everything that can be lost, while still being. To take away from yourself little by little, with an effort so attentive that no pain is felt, to take away from yourself like one who gets free of her own skim, her own characteristics. Everything that characterizes me is just the way I am most easily viewed by others and end up being superficially recognizable to myself.

Exposition, criticism, appreciation, is work for second-rate minds. I

Real life is so secret that not even I, who am dying of it, have been given the password, I am dying without knowing of what.

It (proof by contradiction) is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.

A note exists between two notes of music, between two facts exists a fact, between two grains of sand no matter how close together there exists an interval of space, a sense that exists between senses — in the interstices of primordial matter is the line of mystery and fire that is the breathing of the world, and the continual breathing of the world is what we hear and call silence.

A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.

Estou tão assustada que só poderei aceitar que me perdi se imaginar que alguém me está dando a mão.

Dá-me a tua mão desconhecida, que a vida está me doendo, e não sei como falar – a realidade é delicada demais, só a realidade é delicada, minha irrealidade e minha imaginação são mais pesadas.

No mathematician should ever allow him to forget that mathematics, more than any other art or science, is a young man's game. … Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty. There have been men who have done great work later; … [but] I do not know of a single instance of a major mathematical advance initiated by a man past fifty. … A mathematician may still be competent enough at sixty, but it is useless to expect him to have original ideas.

Life was taking its vengeance on me, and that vengeance consisted merely in coming back, nothing more. Every case of madness involves something coming back. People who are possessed are not possessed by something that just comes but instead by something that comes back. Sometimes life comes back. If in me everything crumbled before that power, it is not because that power was itself necessarily an overwhelming one: it in fact had only to come, since it had already become too full-flowing a force to be controlled or contained - when it appeared it overran everything. And then, like after a flood, there floated a wardrobe, a person, a loose window, three suitcases. And that seemed like Hell to me, that destruction of layers and layers of human archaeology.

—————— I’m searching, I’m searching. I’m trying to understand. Trying to give what I’ve lived to somebody else and I don’t know to whom, but I don’t want to keep what I lived. I don’t know what to do with what I lived, I’m afraid of that profound disorder. I don’t trust what happened to me. Did something happen to me that I, because I didn’t know how to live it, lived as something else? That’s what I’d like to call disorganization, and I’d have the confidence to venture on, because I would know where to return afterward: to the previous organization. I’d rather call it disorganization because I don’t want to confirm myself in what I lived — in the confirmation of me I would lose the world as I had it, and I know I don’t have the fortitude for another.

I am the cockroach, I am my leg, I am my hair, I am the section of brightest light on the wall plaster—I am every Hellish piece of myself—life is so pervasive in me that if they divide me in pieces like a lizard, the pieces will keep on shaking and writhing. I am the silence etched on a wall, and the most ancient butterfly flutters in and looks at me: just the same as always. From birth to death is what I call human in myself, and I shall never actually die. But this is not eternity, it is condemnation. How opulent this silence is. It is the accumulation of centuries. It is the silence of the cockroach looking. The world looks at itself in me. Everything looks at everything, everything experiences the other; in this desert things know things.

The seriousness of a theorem, of course, does not lie in its consequences, which are merely the evidence for its seriousness.

At the same time, Dad was working on a book arguing the case for phonetic spelling. He called it 'A Ghoti out of Water.' "Ghoti," he liked to point out, could be pronounced like "fish." The "gh" had the "f" sound in "enough," the "o" had the short "i" sound in "women," and "ti" had the "sh" sound in "nation.

...things are very delicate. People tread upon them with too many human feet, with too many sentiments. Only the delicacy of innocence or only the delicacy of the initiate senses its almost nonexistent taste. Before, I needed seasoning for everything, and in that way I skipped over the thing and tasted the taste of the seasoning.

I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our "creations," are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards, and I shall use the language which is natural to a man who holds it.

One rather curious conclusion emerges, that pure mathematics is on the whole distinctly more useful than applied. ... For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.

I'm blinder than before. I did see, I really did. I was terrified by the raw truth of a world whose greatest horror is that it is so alive that for me to admit that I am as alive as it is - and my most hideous discovery is that I am as alive as it is - I shall have to raise my consciousness of life outside to so high a point that it would amount to a crime against my personal life.

I was advised to read Jordan's 'Cours d'analyse'; and I shall never forget the astonishment with which I read that remarkable work, the first inspiration for so many mathematicians of my generation, and learnt for the first time as I read it what mathematics really meant.

Ignorance of the law of irreducibility was no excuse. I could no longer excuse myself with the claim that I didn't know the law -- for knowledge of self and of the world is the law that, even though unattainable, cannot be broken, and no one can excuse himself by saying that he doesn't know it. . . . The renewed originality of the sin is this: I have to carry out my unknowing, I shall be sinning originally against life.

her short stories read like perfect songs.

And my fear, too, was different now: not the fear of someone who is still about to go in but the so much greater fear of someone who has gone in.

Sou: o que vi.

The truly transformative power of language occurs when these descriptive root terms are used to form words that convey abstract concepts. A three-letter root compound used to name the spine (Q-W-M) is adapted to describe “flexibility.” The root term for a heated pot boiling over (Gh-Dh-B) constructs a word meaning “hot-headed.” A root term describing the process of carefully separating grains (D-R-S) evolves to express “analyzing” or “interpreting.” From physical sources emerge words for the intangible, like the Qur’an’s parable of the healthy tree with roots anchored in the ground while branches stretch toward the heavens.

The beauty of a mathematical theorem depends a great deal on its seriousness, as even in poetry the beauty of a line may depend to some extent on the significance of the ideas which it contains.

Poetry is more valuable than cricket, but Bradman would be a fool if he sacrificed his cricket in order to write second-rate minor poetry (and I suppose that it is unlikely that he could do better).

And giving myself over with the confidence of belonging to the unknown. For I can pray only to what I do not know. And I can love only the unknown evidence of things and can add myself only to what I do not know. Only that is a real giving of oneself.

At the same time, Dad was working on a book arguing the case for phonetic spelling. He called it A Ghoti out of Water. “Ghoti,” he liked to point out, could be pronounced like “fish.” The “gh” had the “f” sound in “enough,” the “o” had the short “i” sound in “women,” and “ti” had the “sh” sound in “nation.” Dad

Ambition is a noble passion which may legitimately take many forms; there was something noble in the ambitions of Attila or Napoleon; but the noblest ambition is that of leaving behind something of permanent value.

Regarding mathematics, there are now few studies more generally recognized, for good reasons or bad, as profitable and praiseworthy. This may be true; indeed it is probable, since the sensational triumphs of Einstein, that stellar astronomy and atomic physics are the only sciences which stand higher in popular estimation.

I do not think that G. H. Hardy was talking nonsense when he insisted that the mathematician was discovering rather than creating... The world for me is a necessary system, and in the degree to which the thinker can surrender his thought to that system and follow it, he is in a sense participating in that which is timeless or eternal.

Se eu me confirmar e me considerar verdadeira, estarei perdida porque não saberei onde engastar meu novo modo de ser - se eu for adiante nas minhas visões fragmentárias, o mundo inteiro terá que se transformar para eu caber nele.

Oh, but to reach silence, what a huge effort of voice. My voice is the way I go seek reality; reality prior to my language exists as an unthinkable thought, but I was and am fatefully impelled to have to know what thought thinks. Reality precedes the voice that seeks it, but like the earth precedes the tree, but like the world precedes the man, but like the sea precedes the view of the sea, life precedes love, bodily matter precedes the body, and one day in its turn language shall have preceded possession of silence. - Clarice Lispector, The Passion According to G.H.

I had reached nothingness, and the nothingness was live and moist.

My old life was necessary to me because it was precisely its error that made me take up imagining a hope that, without the life that I led, I wouldn't have known.

I shall never be able to understand it, but there must be someone who can. And I shall have to create that someone who can inside myself.

O erro é um dos meus modos fatais de trabalho.

Se não tivesse sido eu, eu não saberia, e tendo sido eu, eu soube.

Toda compreensão súbita é finalmente a revelação de uma aguda incompreensão. Todo momento de achar é um perder-se a si próprio.

Night is our latent state. And it is so moist that plants are born. In houses the lights go out in order to hear the crickets more clearly, and so the grasshoppers can walk atop the leaves almost without touching them, the leaves, the leaves, the leaves—in the night the soft anxiety is transmitted through the hollow of the air, the void is a means of transport.

I've always liked putting things in their places. I think it's my only true calling. By ordering things I create and understand at the same time ... Ordering is finding the best form.

The Greeks were the first mathematicians who are still ‘real’ to us to-day. Oriental mathematics may be an interesting curiosity, but Greek mathematics is the real thing. The Greeks first spoke a language which modern mathematicians can understand: as Littlewood said to me once, they are not clever schoolboys or ‘scholarship candidates’, but ‘Fellows of another college’. So Greek mathematics is ‘permanent’, more permanent even than Greek literature. Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. ‘Immortality’ may be a silly word, but probably a mathematician has the best chance of whatever it may mean.

Ah, meu amor, não tenhas medo da carência: ela é o nosso destino maior. O amor é tão mais fatal do que eu havia pensado, o amor é tão inerente quanto a própria carência, e nós somos garantidos por uma necessidade que se renovará continuamente. O amor já está, está sempre. Falta apenas o golpe da graça - que se chama paixão.

I still say to myself when I am depressed, and find myself forced to listen to pompous and tiresome people, 'Well, I have done one thing you could never have done, and that is to have collaborated with both Littlewood and Ramanujan on something like equal terms.

But I’m afraid to begin composing in order to be understood by the imaginary someone, I’m afraid to start to “make” a meaning, with the same tame madness that till yesterday was my healthy way of fitting into a system. Will I need the courage to use an unprotected heart and keep talking to the nothing and the no one? as a child thinks about the nothing. And run the risk of being crushed by chance.

Billy and Aaron found their way beyond the rain. Anyone who reads this story: Have faith in the resilience of the human spirit.........if I can tell you just one thing, please believe in yourself. God made you to be beautiful. You can find your way to the other side too. Just never give up! GH

Plenty of mathematicians, Hardy knew, could follow a step-by-step discursus unflaggingly—yet counted for nothing beside Ramanujan. Years later, he would contrive an informal scale of natural mathematical ability on which he assigned himself a 25 and Littlewood a 30. To David Hilbert, the most eminent mathematician of the day, he assigned an 80. To Ramanujan he gave 100.

A MAN who sets out to justify his existence and his activities has to distinguish two different questions. The first is whether the work which he does is worth doing; and the second is why he does it, whatever its value may be.

The geometer offers to the physicist a whole set of maps from which to choose. One map, perhaps, will fit the facts better than others, and then the geometry which provides that particular map will be the geometry most important for applied mathematics.

I shall need to courage to do what I'm about to do: speak. And risk the enormous surprise I shall feel at the poverty of the spoken thing. As soon as it's out of my mouth, I'll have to add: that's not it, that's not it! But I cannot be afraid of being ridiculous, I always preferred less to more also out of fear of the ridiculous: because there's also the shattering of modesty. I'm putting off having to speak to myself. Out of fear? And because I don't have a word to say. I don't have a word to say. So why don't I shut up? But if I do not force out the word muteness will swallow me forever in waves.

Twentieth-century British mathematician G.H. Hardy also believed that the human function is to "discover or observe" mathematics rather than to invent it. In other words, the abstract landscape of mathematics was there, waiting for mathematical explorers to reveal it.

Let the Angels lead me to you, if only they fly in this land.

Life just is for me, and I don’t understand what I’m saying. And so I adore it. ——————

I'm going to start my exercise in courage, living isn't courage, knowing that you're living, that's courage.

The best mathematics is serious as well as beautiful—‘important’ if you like, but the word is very ambiguous, and ‘serious’ expresses what I mean much better

Ouve, por eu ter mergulhado no abismo é que estou começando a amar o abismo de que sou feita.

Depersonalization as the great objectification of oneself. The greatest exteriorization one can reach. Whoever gets to oneself through depersonalization shall recognize the other in any disguise: the first step in relation to the other is finding inside oneself the man of all men. Every woman is the woman of all women, every man is the man of all men, and each of them could appear wherever man is judged. But only in immanence, because only a few reach the point of, in us, recognizing themselves. And then, by the simple presence of their existence, revealing ours.

Nostalgia is not for the God who is missing to us, it is a nostalgia for ourselves, for we do not sustain ourselves; we miss our impossible grandeur - my unreachable nowness is my paradise lost.

Never, until then, had life happened to me by day. Never in sunlight. Only in my nights did the world slowly revolve. Only that, whatever happened in the dark of night itself, would also happen at the same time in my own entrails, and my dark wasn't differentiated from the dark outside, and in the morning, when I opened my eyes, the world was still a surface: the secret life of the night soon reduced in my mouth to the taste of a nightmare that disappears. but now life was happening by day. undeniable and to be seen. unless I averted my eyes. and I could still avert my eyes.

Real life is so secret that not even I, who am dying of it, have been given the password, I am dying without knowing of what. And the secret is such that only if the mission is finally carried out do I, all of a sudden, see that I was born entrusted with it - all of life is a secret mission.

Immortality is often ridiculous or cruel: few of us would have chosen to be Og or Ananias or Gallio. Even in mathematics, history sometimes plays strange tricks; Rolle figures in the textbooks of elementary calculus as if he had been a mathematician like Newton; Farey is immortal because he failed to understand a theorem which Haros had proved perfectly fourteen years before; the names of five worthy Norwegians still stand in Abel’s Life, just for one act of conscientious imbecility, dutifully performed at the expense of their country’s greatest man. But on the whole the history of science is fair, and this is particularly true in mathematics. No other subject has such clear-cut or unanimously accepted standards, and the men who are remembered are almost always the men who merit it. Mathematical fame, if you have the cash to pay for it, is one of the soundest and steadiest of investments.

Eu tenho à medida que designo – e este é o esplendor de se ter uma linguagem . Mas eu tenho muito mais à medida que não consigo designar. A realidade é a matéria-prima, a linguagem é o modo como vou buscá-la – e como não acho. Mas é do buscar e não achar que nasce o que eu não conhecia, e que instantaneamente reconheço. A linguagem é o meu esforço humano. Por destino tenho que ir buscar e por destino volto com as mãos vazias. Mas – volto com o indizível . O indizível só me poderá ser dado através do fracasso de minha linguagem. Só quando falha a construção, é que obtenho o que ela não conseguiu.

I'm stalling. I know that everything I say is just to put it off-to put off the moment when I'll have to start talking, knowing that there is nothing more for me to say. I'm putting off my silence. Have I been putting off silence for my whole life? but now, in my disparagement of the word, perhaps I'll finally be able to start talking." (14)

I was now so much greater that I could no longer see myself. As great as a far-off landscape. I was far off. But perceptible in my furthest mountains and in my remotest rivers: the simultaneous present no longer scared me, and in the furthest extremity of me I could finally smile without even smiling. At last I was stretching beyond my sensibility.

666 HOW TO JOIN ILLUMINATI SECRET SOCIETY FOR MONEY. Get Rich Quick Join 666 now. JOIN THE BROTHERHOOD Money, Power, Fame and Love. For those who are interested in making money, every good thing comes with money, comes with extra effort. All u need do is a “Spiritual work” and every wicked power delaying your progress wants clear and good things will come to you like, money, favour from people, open doors, business breakthrough, good job. Note: It’s not a child’s play, it’s for those who are desperate and ready to make a change in their life. We are seeki¬ng that speci¬al wisdo¬m and knowl¬edge that would set us free from the bonda¬ge to dull and drear¬y every¬day life, while stren¬gthen¬ing us in body, mind and spiri¬t, and bring-ing us the mater¬ial rewar¬ds of wealt¬h, love, and succe¬ss. The Karis¬hika Broth¬erhoo¬d is a true broth¬erhoo¬d of secre¬t knowl¬edge and power¬. Me¬mber s¬hip into our frate¬rnity is free and norma¬lly throu¬gh a thoro¬ugh scree¬ning. We are here to liber¬ate those who need wealt¬h, riche¬s, power¬, prosp¬erity¬, prote¬ction and succe¬ss in all ramif¬icati¬on. Broth¬erhoo¬d offer¬s all initi¬ate membe¬rs growt¬h, wealt¬h, fame, power¬, prosp¬erity and succe¬ss in all areas of heart desir¬es. We don’t deman¬d human sacri¬fice, the use of any human parts or early perso¬nal death as a preco¬nditi¬on for you to becom¬e our membe¬r. W¬ant to join occul¬t in Switzerland how can I join secre¬t socie¬ty or cult to make money¬ how can join occul¬t for riche¬s I want to be rich but I don’t know how etc. how do I do money ritua¬l ho¬w do I join good occul¬t that will not affec¬t me and my famil¬y forev¬er w¬e are now here for you. K¬indly conta¬ct us on +41767918253 or email: info786@pm.me Contact Person Agent Adam Address: Kronenstrasse 25 9230 Flawil Switzerland

I am going to create what happened to me. Only because living isn't tellable. Living isn't livable. I shall have to create upon life. And without lying. Yes to creation, no to lying. Creation isn't imagination, it's running the huge risk of coming face to face with reality.

My life had been as continuous as death. Life is so continuous that we divide it into stages and call one of them death. I had always been in life, it mattered little that it was not I properly speaking, not that thing that I customarily call "I." I had always been in life.

When authors write best, or at least, when they write most fluently, an influence seems to waken in them which becomes their master, which will have its own way, putting out of view all behests but its own, dictating certain words, and insisting on their being used, whether vehement or measured in their nature; new moulding characters, giving unthought-of turns to incidents, rejecting carefully elaborated old ideas, and suddenly creating and adopting new ones. Is it not so? And should we try to counteract this influence? Can we indeed counteract it? from a letter to G.H. Lewes, 12 January 1848

I had better say something here about this question of age, since it is particularly important for mathematicians. No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game. To take a simple illustration at a comparatively humble level, the average age of election to the Royal Society is lowest in mathematics. We can naturally find much more striking illustrations. We may consider, for example, the career of a man who was certainly one of the world's three greatest mathematicians. Newton gave up mathematics at fifty, and had lost his enthusiasm long before; he had recognized no doubt by the time he was forty that his greatest creative days were over. His greatest idea of all, fluxions and the law of gravitation, came to him about 1666 , when he was twentyfour—'in those days I was in the prime of my age for invention, and minded mathematics and philosophy more than at any time since'. He made big discoveries until he was nearly forty (the 'elliptic orbit' at thirty-seven), but after that he did little but polish and perfect. Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty. There have been men who have done great work a good deal later; Gauss's great memoir on differential geometry was published when he was fifty (though he had had the fundamental ideas ten years before). I do not know an instance of a major mathematical advance initiated by a man past fifty. If a man of mature age loses interest in and abandons mathematics, the loss is not likely to be very serious either for mathematics or for himself.

My world today is raw, it is a world of great vital difficulty. Because, more than a star, today I want the thick and black root of the stars, I want the source that always seems dirty, and is dirty, and that is always incomprehensible. It is with pain that I bid farewell even to the beauty of a child - I want the adult who is more primitive and ugly and drier and more difficult, and who became a child-seed that cannot be broken between the teeth.

I lost something that was essential to me, and that no longer is. I no longer need it, as if I’d lost a third leg that up till then made it impossible for me to walk but that turned me into a stable tripod. I lost that third leg. And I went back to being a person I never was. I went back to having something I never had: just two legs. I know I can only walk with two legs. But I feel the useless absence of that third leg and it scares me, it was the leg that made me something findable by myself, and without even having to look for myself.

The mathematician is in much more direct contact with reality. This may seem a paradox, since it is the physicist who deals with the subject-matter usually described as 'real' ... A chair may be a collection of whirling electrons, or an idea in the mind of God : each of these accounts of it may have its merits, but neither conforms at all closely to the suggestions of common sense. ... neither physicists nor philosophers have ever given any convincing account of what 'physical reality' is, or of how the physicist passes, from the confused mass of fact or sensation with which he starts, to the construction of the objects which he calls 'real'. A mathematician, on the other hand, is working with his own mathematical reality. ... mathematical objects are so much more what they seem. ... 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way.

Published mathematical papers often have irritating assertions of the type: “It now follows that…,” or: “It is now obvious that…,” when it doesn't follow, and isn't obvious at all, unless you put in the six hours the author did to supply the missing steps and checking them. There is a story about the English mathematician G.H. Hardy, whom we shall meet later. In the middle of delivering a lecture, Hardy arrived at a point in his argument where he said, “It is now obvious that….” Here he stopped, fell silent, and stood motionless with furrowed brow for a few seconds. Then he walked out of the lecture hall. Twenty minutes later he returned, smiling, and began, “Yes, it is obvious that….” If he