Mathematical Calculation Quotes

It's not because I want to make out with her." Hold on." He grabbed a pencil and scrawled excitedly at the paper as if he'd just made a mathematical breakthrough and then looked back up at me. "I just did some calculations, and I've been able to determine that you're full of shit

Hold on." He grabbed a pencil and scrawled excitedly at the paper as if he'd just made a mathematical breakthrough and then looked back up at me. "I just did some calculations, and I've been able to determine that you're full of shit.

The game of life does not proceed like a mathematical calculation on the principle that two and two make four. Sometimes they make five, or minus four, and sometimes the blackboard topples over in the middle of the sum and the pedagogue is left with a black eye.

In fact a few simple mathematical calculations reveal that if reference librarians were paid at market rates for all the roles they play, they would have salaries well over $200,000.

How many times have I wondered if it is really possible to forge links with a mass of people when one has never had strong feelings for anyone, not even one's own parents: if it is possible to have a collectivity when one has not been deeply loved oneself by individual human creatures. Hasn't this had some effect on my life as a militant--has it not tended to make me sterile and reduce my quality as a revolutionary by making everything a matter of pure intellect, of pure mathematical calculation?

[The Old Astronomer to His Pupil] Reach me down my Tycho Brahe, I would know him when we meet, When I share my later science, sitting humbly at his feet; He may know the law of all things, yet be ignorant of how We are working to completion, working on from then to now. Pray remember that I leave you all my theory complete, Lacking only certain data for your adding, as is meet, And remember men will scorn it, 'tis original and true, And the obloquy of newness may fall bitterly on you. But, my pupil, as my pupil you have learned the worth of scorn, You have laughed with me at pity, we have joyed to be forlorn, What for us are all distractions of men's fellowship and smiles; What for us the Goddess Pleasure with her meretricious smiles. You may tell that German College that their honor comes too late, But they must not waste repentance on the grizzly savant's fate. Though my soul may set in darkness, it will rise in perfect light; I have loved the stars too fondly to be fearful of the night. What, my boy, you are not weeping? You should save your eyes for sight; You will need them, mine observer, yet for many another night. I leave none but you, my pupil, unto whom my plans are known. You 'have none but me,' you murmur, and I 'leave you quite alone'? Well then, kiss me, -- since my mother left her blessing on my brow, There has been a something wanting in my nature until now; I can dimly comprehend it, -- that I might have been more kind, Might have cherished you more wisely, as the one I leave behind. I 'have never failed in kindness'? No, we lived too high for strife,-- Calmest coldness was the error which has crept into our life; But your spirit is untainted, I can dedicate you still To the service of our science: you will further it? you will! There are certain calculations I should like to make with you, To be sure that your deductions will be logical and true; And remember, 'Patience, Patience,' is the watchword of a sage, Not to-day nor yet to-morrow can complete a perfect age. I have sown, like Tycho Brahe, that a greater man may reap; But if none should do my reaping, 'twill disturb me in my sleep So be careful and be faithful, though, like me, you leave no name; See, my boy, that nothing turn you to the mere pursuit of fame. I must say Good-bye, my pupil, for I cannot longer speak; Draw the curtain back for Venus, ere my vision grows too weak: It is strange the pearly planet should look red as fiery Mars,-- God will mercifully guide me on my way amongst the stars.

Naturally, we are inclined to be so mathematical and calculating that we look upon uncertainty as a bad thing...Certainty is the mark of the common-sense life. To be certain of God means that we are uncertain in all our ways, we do not know what a day may bring forth. This is generally said with a sigh of sadness; it should rather be an expression of breathless expectation.

His [Thomas Edison] method was inefficient in the extreme, for an immense ground had to be covered to get anything at all unless blind chance intervened and, at first, I was almost a sorry witness of his doings, knowing that just a little theory and calculation would have saved him 90 per cent of the labor. But he had a veritable contempt for book learning and mathematical knowledge, trusting himself entirely to his inventor's instinct and practical American sense. In view of this, the truly prodigious amount of his actual accomplishments is little short of a miracle.

The complexity of economics can be calculated mathematically. Write out the algebraic equation that is the human heart and multiply each unknown by the population of the world.

What music is to the heart, mathematics is to the mind.

And so to read is, in truth, to be in the constant act of creation. The old lady on the bus with her Orwell, the businessman on the Tube with Patricia Cornwell, the teenager roaring through Capote -- they are not engaged in idle pleasure. Their heads are on fire. Their hearts are flooding. With a book, you are the landscape, the sets, the snow, the hero, the kiss -- you are the mathematical calculation that plots the trajectory of the blazing, crashing zeppelin. You -- pale, punchable reader -- are terraforming whole worlds in your head, which will remain with you until the day you die. These books are as much a part of you as your guts and your bone. And when your guts fail and your bones break, Narnia, or Jamaica Inn, or Gormenghast will still be there; as pin-sharp and bright as the day you first imagined them -- hiding under the bedclothes, sitting on the bus. Exhausted, on a rainy day, weeping over the death of someone you never met, and who was nothing more than words until you transfused them with your time, and your love, and the imagination you constantly dismiss as "just being a bit of a bookworm.

But is it not already an insult to call chess anything so narrow as a game? Is it not also a science, an art, hovering between these categories like Muhammad's coffin between heaven and earth, a unique yoking of opposites, ancient and yet eternally new, mechanically constituted and yet an activity of the imagination alone, limited to a fixed geometric area but unlimited in its permutations, constantly evolving and yet sterile, a cogitation producing nothing, a mathematics calculating nothing, an art without an artwork, an architecture without substance and yet demonstrably more durable in its essence and actual form than all books and works, the only game that belongs to all peoples and all eras, while no one knows what god put it on earth to deaden boredom, sharpen the mind, and fortify the spirit?

The calculative exactness of practical life which the money economy has brought about corresponds to the ideal of natural science: to transform the world by mathematical formulas. Only money economy has filled the days of so many people with weighing, calculating, with numerical determinations, with a reduction of qualitative values to quantitative ones.

Don’t you think he may be pursuing an ideal that is hidden in a cloud of unknowing—like an astronomer looking for a star that only a mathematical calculation tells him exists?

And that is not all: even if man really were nothing but a piano-key, even if this were proved to him by natural science and mathematics, even then he would not become reasonable, but would purposely do something perverse out of simple ingratitude, simply to gain his point. And if he does not find means he will contrive destruction and chaos, will contrive sufferings of all sorts, only to gain his point! He will launch a curse upon the world, and as only man can curse (it is his privilege, the primary distinction between him and other animals), may be by his curse alone he will attain his object--that is, convince himself that he is a man and not a piano-key! If you say that all this, too, can be calculated and tabulated--chaos and darkness and curses, so that the mere possibility of calculating it all beforehand would stop it all, and reason would reassert itself, then man would purposely go mad in order to be rid of reason and gain his point!

Le Verrier—without leaving his study, without even looking at the sky—had found the unknown planet [Neptune] solely by mathematical calculation, and, as it were, touched it with the tip of his pen!

There is scarcely a subject that cannot be mathematically treated and the effects calculated or the results determined beforehand from the available theoretical and practical data.

Don't you think he may be pursuing an ideal that is hidden in a cloud of unknowing — like an astronomer looking for a star that only a mathematical calculation tells him exists?

...The pages and pages of complex, impenetrable calculations might have contained the secrets of the universe, copied out of God's notebook. In my imagination, I saw the creator of the universe sitting in some distant corner of the sky, weaving a pattern of delicate lace so fine that that even the faintest light would shine through it. The lace stretches out infinitely in every direction, billowing gently in the cosmic breeze. You want desperately to touch it, hold it up to the light, rub it against your cheek. And all we ask is to be able to re-create the pattern, weave it again with numbers, somehow, in our own language; to make the tiniest fragment our own, to bring it back to eart.

He knew by heart every last minute crack on its surface. He had made maps of the ceiling and gone exploring on them; rivers, islands, and continents. He had made guessing games of it and discovered hidden objects; faces, birds, and fishes. He made mathematical calculations of it and rediscovered his childhood; theorems, angles, and triangles. There was practically nothing else he could do but look at it. He hated the sight of it.

He walked straight out of college into the waiting arms of the Navy. They gave him an intelligence test. The first question on the math part had to do with boats on a river: Port Smith is 100 miles upstream of Port Jones. The river flows at 5 miles per hour. The boat goes through water at 10 miles per hour. How long does it take to go from Port Smith to Port Jones? How long to come back? Lawrence immediately saw that it was a trick question. You would have to be some kind of idiot to make the facile assumption that the current would add or subtract 5 miles per hour to or from the speed of the boat. Clearly, 5 miles per hour was nothing more than the average speed. The current would be faster in the middle of the river and slower at the banks. More complicated variations could be expected at bends in the river. Basically it was a question of hydrodynamics, which could be tackled using certain well-known systems of differential equations. Lawrence dove into the problem, rapidly (or so he thought) covering both sides of ten sheets of paper with calculations. Along the way, he realized that one of his assumptions, in combination with the simplified Navier Stokes equations, had led him into an exploration of a particularly interesting family of partial differential equations. Before he knew it, he had proved a new theorem. If that didn't prove his intelligence, what would? Then the time bell rang and the papers were collected. Lawrence managed to hang onto his scratch paper. He took it back to his dorm, typed it up, and mailed it to one of the more approachable math professors at Princeton, who promptly arranged for it to be published in a Parisian mathematics journal. Lawrence received two free, freshly printed copies of the journal a few months later, in San Diego, California, during mail call on board a large ship called the U.S.S. Nevada. The ship had a band, and the Navy had given Lawrence the job of playing the glockenspiel in it, because their testing procedures had proven that he was not intelligent enough to do anything else.

Kaufman calculated the risks of his situation: the mathematics of panic.

The first man to understand the extraordinary magical power of applying mathematical calculation to things in nature was an Italian called Galileo Galilei.

Something else gets under your skin, keeps you working days and nights at the sacrifice of your sleeping and eating and attention to your family and friends, something beyond the love of puzzle solving. And that other force is the anticipation of understanding something about the world that no one has ever understood before you. Einstein wrote that when he first realized that gravity was equivalent to acceleration -- an idea that would underlie his new theory of gravity -- it was the "happiest thought of my life." On projects of far smaller weight, I have experienced that pleasure of discovering something new. It is an exquisite sensation, a feeling of power, a rush of the blood, a sense of living forever. To be the first vessel to hold this new thing. All of the scientists I've known have at least one more quality in common: they do what they do because they love it, and because they cannot imagine doing anything else. In a sense, this is the real reason a scientist does science. Because the scientist must. Such a compulsion is both blessing and burden. A blessing because the creative life, in any endeavor, is a gift filled with beauty and not given to everyone, a burden because the call is unrelenting and can drown out the rest of life. This mixed blessing and burden must be why the astrophysicist Chandrasekhar continued working until his mid-80's, why a visitor to Einstein's apartment in Bern found the young physicist rocking his infant with one hand while doing mathematical calculations with the other. This mixed blessing and burden must have been the "sweet hell" that Walt Whitman referred to when he realized at a young age that he was destined to be a poet. "Never more," he wrote, "shall I escape.

He calculated the number of bricks in the wall, first in twos and then in tens and finally in sixteens. The numbers formed up and marched past his brain in terrified obedience. Division and multiplication were discovered. Algebra was invented and provided an interesting diversion for a minute or two. And then he felt the fog of numbers drift away, and looked up and saw the sparkling, distant mountains of calculus.

I know that the chances for success are slim. [...] But odds are mathematical formulas calculated to give people a reason not to try.

But that would mean it was originally a sideways number eight. That makes no sense at all. Unless..." She paused as understanding dawned. "You think it was the symbol for infinity?" "Yes, but not the usual one. A special variant. Do you see how one line doesn't fully connect in the middle? That's Euler's infinity symbol. Absolutus infinitus." "How is it different from the usual one?" "Back in the eighteenth century, there were certain mathematical calculations no one could perform because they involved series of infinite numbers. The problem with infinity, of course, is that you can't come up with a final answer when the numbers keep increasing forever. But a mathematician named Leonhard Euler found a way to treat infinity as if it were a finite number- and that allowed him to do things in mathematical analysis that had never been done before." Tom inclined his head toward the date stone. "My guess is whoever chiseled that symbol was a mathematician or scientist." "If it were my date stone," Cassandra said dryly, "I'd prefer the entwined hearts. At least I would understand what it means." "No, this is much better than hearts," Tom exclaimed, his expression more earnest than any she'd seen from him before. "Linking their names with Euler's infinity symbol means..." He paused, considering how best to explain it. "The two of them formed a complete unit... a togetherness... that contained infinity. Their marriage had a beginning and end, but every day of it was filled with forever. It's a beautiful concept." He paused before adding awkwardly, "Mathematically speaking.

He could not believe that any of them might actually hit somebody. If one did, what a nowhere way to go: killed by accident; slain not as an individual but by sheer statistical probability, by the calculated chance of searching fire, even as he himself might be at any moment. Mathematics! Mathematics! Algebra! Geometry! When 1st and 3d Squads came diving and tumbling back over the tiny crest, Bell was content to throw himself prone, press his cheek to the earth, shut his eyes, and lie there. God, oh, God! Why am I here? Why am I here? After a moment's thought, he decided he better change it to: why are we here. That way, no agency of retribution could exact payment from him for being selfish.

Memory cannot be understood, either, without a mathematical approach. The fundamental given is the ratio between the amount of time in the lived life and the amount of time from that life that is stored in memory. No one has ever tried to calculate this ratio, and in fact there exists no technique for doing so; yet without much risk of error I could assume that the memory retains no more than a millionth, a hundred-millionth, in short an utterly infinitesimal bit of the lived life. That fact too is part of the essence of man. If someone could retain in his memory everything he had experienced, if he could at any time call up any fragment of his past, he would be nothing like human beings: neither his loves nor his friendships nor his angers nor his capacity to forgive or avenge would resemble ours. We will never cease our critique of those persons who distort the past, rewrite it, falsify it, who exaggerate the importance of one event and fail to mention some other; such a critique is proper (it cannot fail to be), but it doesn't count for much unless a more basic critique precedes it: a critique of human memory as such. For after all, what can memory actually do, the poor thing? It is only capable of retaining a paltry little scrap of the past, and no one knows why just this scrap and not some other one, since in each of us the choice occurs mysteriously, outside our will or our interests. We won't understand a thing about human life if we persist in avoiding the most obvious fact: that a reality no longer is what it was when it was; it cannot be reconstructed.

He’s probably quite good at mathematical calculations, but he doesn’t seem to have ever handled accounts of money. Because we have always been poor,

I'm angry because you think everything happened by chance but there are billions of people on this planet and I found you so if you're saying I could just as well have found someone else then I can't bear your bloody mathematics!" Her fists had been clenched. He stood there looking at her for several minutes. Then he said that he loved her. It was the first time. They never stopped arguing and they never slept apart; he spent an entire working life calculating probabilities and she was the most improbable person he ever met. She turned him upside-down.

It is by a mathematical point only that we are wise, as the sailor or fugitive slave keeps the polestar in his eye; but that is sufficient guidance for all our life. We may not arrive at our port within a calculable period, but we would preserve the true course.

He grabbed a pencil and scrawled excitedly at the paper as if he'd just made a mathematical breakthrough and the looked back back at me. "I just did some calculations, and I've been able to determine that you're full of shit.

[All phenomena] are equally susceptible of being calculated, and all that is necessary, to reduce the whole of nature to laws similar to those which Newton discovered with the aid of the calculus, is to have a sufficient number of observations and a mathematics that is complex enough.

The aim of Mathematical Physics is not only to facilitate for the physicist the numerical calculation of certain constants or the integration of certain differential equations. It is besides, it is above all, to reveal to him the hidden harmony of things in making him see them in a new way.

Nothing comforted Sabine like long division. That was how she had passed time waiting for Phan and then Parsifal to come back from their tests. She figured the square root of the date while other people knit and read. Sabine blamed much of the world's unhappiness on the advent of calculators.

The Electoral College was a concession to slave owners, an affair of both mathematical and political calculation.

Deep Blue didn't win by being smarter than a human; it won by being millions of times faster than a human. Deep Blue had no intuition. An expert human player looks at a board position and immediately sees what areas of play are most likely to be fruitful or dangerous, whereas a computer has no innate sense of what is important and must explore many more options. Deep Blue also had no sense of the history of the game, and didn't know anything about its opponent. It played chess yet didn't understand chess, in the same way a calculator performs arithmetic bud doesn't understand mathematics.

The history of human knowledge has so uninterruptedly shown that to collateral, or incidental, or accidental events we are indebted for the most numerous and most valuable discoveries, that it has at length become necessary, in any prospective view of improvement, to make not only large, but the largest allowances for inventions that shall arise by chance, and quite out of the range of ordinary expectation. It is no longer philosophical to base, upon what has been, a vision of what is to be. Accident is admitted as a portion of the substructure. We make chance a matter of absolute calculation. We subject the unlooked for and unimagined, to the mathematical formulae of the schools.

Yeah, well. If you're staying here in hopes of making out with Alaska, I sure wish you wouldn't. If you unmoor her from the rock that is Jake, God have mercy on us all. That would be some drama, indeed. And as a rule, I like to avoid drama." "It's not because I want to make out with her." "Hold on." He grabbed a pencil and scrawled excitedly at the paper as if he'd just made a mathematical breakthrough and then looked back up at me. "I just did some calculations, and I've been able to determine that you're full of shit." And he was right.

I understood the infamous spiritual terror which this movement exerts, particularly on the bourgeoisie, which is neither morally nor mentally equal to such attacks; at a given sign it unleashes a veritable barrage of lies and slanders against whatever adversary seems most dangerous, until the nerves of the attacked persons break down … This is a tactic based on precise calculation of all human weaknesses, and its result will lead to success with almost mathematical certainty …

And so to read is, in truth, to be in the constant act of creation. The old lady on the bus with her Orwell, the businessman on the Tube with Patricia Cornwell, the teenager roaring through Capote -- they are not engaged in idle pleasure. Their heads are on fire. Their hearts are flooding. With a book, you are the landscape, the sets, the snow, the hero, the kiss -- you are the mathematical calculation the plots the trajectory of the blazing, crashing zeppelin. You -- pale, punchable reader -- are terraforming whole worlds in your head, which will remain with you until the day you die. These books are as much a part of you as your guts and your bone. And when your guts fail and your bones break, Narnia, or Jamaica Inn, or Gormenghast will still be there; as pin-sharp and bright as the day you first imagined them -- hiding under the bedclothes, sitting on the bus. Exhausted, on a rainy day, weeping over the death of someone you never met, and who was nothing more than words until you transformed them with your time, and your love, and the imagination you constantly dismiss as "just being a bit of a bookworm.

...and his analysis proved him to be the first of theoretical astronomers no less than the greatest of 'arithmeticians.

Only three constants are significant for star formation: the gravitational constant, the fine structure constant, and a constant that governs nuclear reaction rates.

There is a certain tyranny about perfection, a certain exhaustion about it even, something that denies the viewer a role in its creation and that asserts itself with all the dogmatism of an unambiguous statement. True beauty cannot be measured because it is fluctuating, it has only a few angles from which it may be seen, and then not in all lights and at all times. It flirts dangerously with ugliness, it takes risks with itself, it does not side comfortably with mathematical rules of proportion, it draws its appeal from precisely those areas that will also lend themselves to ugliness. Nothing can be beautiful that does not take a calculated risk with ugliness.

... those who seek the lost Lord will find traces of His being and beauty in all that men have made, from music and poetry and sculpture to the gingerbread men in the pâtisseries, from the final calculation of the pure mathematician to the first delighted chalk drawing of a small child.

The longer they stayed on, and the better they knew each other, the better she at least could see their mistake, and the more misguided their lives became—like a long proof in mathematics in which the first calculation is wrong, following which all other calculations move you further away from how things were when they made sense.

We should not conclude from this that everything depends on waves of irrational psychology. On the contrary, the state of long-term expectation is often steady, and, even when it is not, the other factors exert their compensating effects. We are merely reminding ourselves that human decisions affecting the future, whether personal or political or economic, cannot depend on strict mathematical expectation, since the basis for making such calculations does not exist; and that it is our innate urge to activity which makes the wheels go round, our rational selves choosing between the alternatives as best we are able, calculating where we can, but often falling back for our motive on whim or sentiment or chance.

The deep study of nature is the most fruitful source of mathematical discoveries. By offering to research a definite end, this study has the advantage of excluding vague questions and useless calculations; besides it is a sure means of forming analysis itself and of discovering the elements which it most concerns us to know, and which natural science ought always to conserve.

Moreover: then, you say, science itself will teach man (though this is really a luxury in my opinion) that in fact he has neither will nor caprice, and never did have any, and that he himself is nothing but a sort of piano key or a sprig in an organ;14 and that, furthermore, there also exist in the world the laws of nature; so that whatever he does is done not at all according to his own wanting, but of itself, according to the laws of nature. Consequently, these laws of nature need only be discovered, and then man will no longer be answerable for his actions, and his life will become extremely easy. Needless to say, all human actions will then be calculated according to these laws, mathematically, like a table of logarithms, up to 108,000, and entered into a calendar; or, better still, some well-meaning publications will appear, like the present-day encyclopedic dictionaries, in which everything will be so precisely calculated and designated that there will no longer be any actions or adventures in the world.

There was, I think, a feeling that the best science was that done in the simplest way. In experimental work, as in mathematics, there was 'style' and a result obtained with simple equipment was more elegant than one obtained with complicated apparatus, just as a mathematical proof derived neatly was better than one involving laborious calculations. Rutherford's first disintegration experiment, and Chadwick's discovery of the neutron had a 'style' that is different from that of experiments made with giant accelerators.

And that is not all: even if man really were nothing but a piano-key, even if this were proved to him by natural science and mathematics, even then he would not become reasonable, but would purposely do something perverse out of simple ingratitude, simply to gain his point. And if he does not find means he will contrive destruction and chaos, will contrive sufferings of all sorts, only to gain his point! He will launch a curse upon the world, and as only man can curse (it is his privilege, the primary distinction between him and other animals), may be by his curse alone he will attain his object--that is, convince himself that he is a man and not a piano-key! If you say that all this, too, can be calculated and tabulated--chaos and darkness and curses, so that the mere possibility of calculating it all beforehand would stop it all, and reason would reassert itself, then man would purposely go mad in order to be rid of reason and gain his point! I believe in it, I answer for it, for the whole work of man really seems to consist in nothing but proving to himself every minute that he is a man and not a piano-key! It may be at the cost of his skin, it may be by cannibalism! And this being so, can one help being tempted to rejoice that it has not yet come off, and that desire still depends on something we don't know?

understood the infamous spiritual terror which this movement exerts, particularly on the bourgeoisie, which is neither morally nor mentally equal to such attacks; at a given sign it unleashes a veritable barrage of lies and slanders against whatever adversary seems most dangerous, until the nerves of the attacked persons break down… This is a tactic based on precise calculation of all human weaknesses, and its result will lead to success with almost mathematical certainty… I achieved an equal understanding of the importance of physical terror toward the individual and the masses… For while in the ranks of their supporters the victory achieved seems a triumph of the justice of their own cause, the defeated adversary in most cases despairs of the success of any further resistance.49 No more precise analysis of Nazi tactics, as Hitler was eventually to develop them, was ever written.

The only gain of civilisation for mankind is the greater capacity for variety of sensations--and absolutely nothing more. And through the development of this many-sidedness man may come to finding enjoyment in bloodshed. In fact, this has already happened to him. Have you noticed that it is the most civilised gentlemen who have been the subtlest slaughterers, to whom the Attilas and Stenka Razins could not hold a candle, and if they are not so conspicuous as the Attilas and Stenka Razins it is simply because they are so often met with, are so ordinary and have become so familiar to us. In any case civilisation has made mankind if not more bloodthirsty, at least more vilely, more loathsomely bloodthirsty. In old days he saw justice in bloodshed and with his conscience at peace exterminated those he thought proper. Now we do think bloodshed abominable and yet we engage in this abomination, and with more energy than ever. Which is worse? Decide that for yourselves. They say that Cleopatra (excuse an instance from Roman history) was fond of sticking gold pins into her slave-girls' breasts and derived gratification from their screams and writhings. You will say that that was in the comparatively barbarous times; that these are barbarous times too, because also, comparatively speaking, pins are stuck in even now; that though man has now learned to see more clearly than in barbarous ages, he is still far from having learnt to act as reason and science would dictate. But yet you are fully convinced that he will be sure to learn when he gets rid of certain old bad habits, and when common sense and science have completely re-educated human nature and turned it in a normal direction. You are confident that then man will cease from INTENTIONAL error and will, so to say, be compelled not to want to set his will against his normal interests. That is not all; then, you say, science itself will teach man (though to my mind it's a superfluous luxury) that he never has really had any caprice or will of his own, and that he himself is something of the nature of a piano-key or the stop of an organ, and that there are, besides, things called the laws of nature; so that everything he does is not done by his willing it, but is done of itself, by the laws of nature. Consequently we have only to discover these laws of nature, and man will no longer have to answer for his actions and life will become exceedingly easy for him. All human actions will then, of course, be tabulated according to these laws, mathematically, like tables of logarithms up to 108,000, and entered in an index; or, better still, there would be published certain edifying works of the nature of encyclopaedic lexicons, in which everything will be so clearly calculated and explained that there will be no more incidents or adventures in the world.

As you all know, love is… complicated for me. It is something that adheres to no scientific or mathematic principles—it cannot be measured, weighed, or calculated. As such, it’s hard to credit that it even exists. I was certain we could find life in the stars before we could find scientific evidence that love was real.

She swore she could prove it mathematically, but the calculations required were so involved that they would have required a computer the size of the universe, running for a length of time that would have taken them past the projected heat-death of the universe, to work them out. It was pretty much the definition of moot.

When Dad wasn’t telling us about all the amazing things he had already done, he was telling us about the wondrous things he was going to do. Like build the Glass Castle. All of Dad’s engineering skills and mathematical genius were coming together in one special project: a great big house he was going to build for us in the desert. It would have a glass ceiling and thick glass walls and even a glass staircase. The Glass Castle would have solar cells on the top that would catch the sun’s rays and convert them into electricity for heating and cooling and running all the appliances. It would even have its own water-purification system. Dad had worked out the architecture and the floor plans and most of the mathematical calculations. He carried around the blueprints for the Glass Castle wherever we went, and sometimes he’d pull them out and let us work on the design for our rooms. All we had to do was find gold, Dad said, and we were on the verge of that. Once he finished the Prospector and we struck it rich, he’d start work on our Glass Castle.

Now I ask you: what can be expected of man since he is a being endowed with strange qualities? Shower upon him every earthly blessing, drown him in a sea of happiness, so that nothing but bubbles of bliss can be seen on the surface; give him economic prosperity, such that he should have nothing else to do but sleep, eat cakes and busy himself with the continuation of his species, and even then out of sheer ingratitude, sheer spite, man would play you some nasty trick. He would even risk his cakes and would deliberately desire the most fatal rubbish, the most uneconomical absurdity, simply to introduce into all this positive good sense his fatal fantastic element. It is just his fantastic dreams, his vulgar folly that he will desire to retain, simply in order to prove to himself—as though that were so necessary—that men still are men and not the keys of a piano, which the laws of nature threaten to control so completely that soon one will be able to desire nothing but by the calendar. And that is not all: even if man really were nothing but a piano-key, even if this were proved to him by natural science and mathematics, even then he would not become reasonable, but would purposely do something perverse out of simple ingratitude, simply to gain his point. And if he does not find means he will contrive destruction and chaos, will contrive sufferings of all sorts, only to gain his point! He will launch a curse upon the world, and as only man can curse (it is his privilege, the primary distinction between him and other animals), may be by his curse alone he will attain his object—that is, convince himself that he is a man and not a piano-key! If you say that all this, too, can be calculated and tabulated—chaos and darkness and curses, so that the mere possibility of calculating it all beforehand would stop it all, and reason would reassert itself, then man would purposely go mad in order to be rid of reason and gain his point! I believe in it, I answer for it, for the whole work of man really seems to consist in nothing but proving to himself every minute that he is a man and not a piano-key! It may be at the cost of his skin, it may be by cannibalism! And this being so, can one help being tempted to rejoice that it has not yet come off, and that desire still depends on something we don’t know?

To a scholar, mathematics is music.

I admire the elegance of his calculation method; it must be great to ride those fields on the horse of genuine mathematics while we have to do our hard work on foot

But real-world questions aren't like word problems. A real-world problem is something like "Has the recession and its aftermath been especially bad for women in the workforce, and if so, to what extent is this the result of Obama administration policies?" Your calculator doesn't have a button for this. Because in order to give a sensible answer, you need to know more than just numbers. What shape do the job-loss curves for men and women have in a typical recession? Was this recession notably different in that respect? What kind of jobs are disproportionately held by women, and what decisions has Obama made that affect that sector of the economy? It's only after you've started to formulate these questions that you take out the calculator. But at that point the real mental work is already finished. Dividing one number by another is mere computation; figuring out what you should divide by what is mathematics.

The youth may build or plant or sail, only let him not be hindered from doing that which he tells me he would like to do. It is by a mathematical point only that we are wise, as the sailor or the fugitive slave keeps the polestar in his eye; but that is sufficient guidance for all our life. We may not arrive at our port within a calculable period, but we would preserve the true course.

Look, this island is an attempt to re-create a natural environment from the past. To make an isolated world where extinct creatures roam freely. Correct?” “Yes.” “But from my point of view, such an undertaking is impossible. The mathematics are so self-evident that they don’t need to be calculated. It’s rather like my asking you whether, on a billion dollars in income, you had to pay tax. You wouldn’t need to pull out your calculator to check. You’d know tax was owed. And, similarly, I know overwhelmingly that one cannot successfully duplicate nature in this way, or hope to isolate it.

probably heard that math is the language of science, or the language of Nature is mathematics. Well, it’s true. The more we understand the universe, the more we discover its mathematical connections. Flowers have spirals that line up with a special sequence of numbers (called Fibonacci numbers) that you can understand and generate yourself. Seashells form in perfect mathematical curves (logarithmic spirals) that come from a chemical balance. Star clusters tug on

Music can be appreciated from several points of view: the listener, the performer, the composer. In mathematics there is nothing analogous to the listener; and even if there were, it would be the composer, rather than the performer, that would interest him. It is the creation of new mathematics, rather than its mundane practice, that is interesting. Mathematics is not about symbols and calculations. These are just tools of the tradequavers and crotchets and five-finger exercises. Mathematics is about ideas. In particular it is about the way that different ideas relate to each other. If certain information is known, what else must necessarily follow? The aim of mathematics is to understand such questions by stripping away the inessentials and penetrating to the core of the problem. It is not just a question of getting the right answer; more a matter of understanding why an answer is possible at all, and why it takes the form that it does. Good mathematics has an air of economy and an element of surprise. But, above all, it has significance.

One of the ideas of this book is to give the reader a possibility to develop problem-solving skills using both systems, to solve various nonlinear PDEs in both systems. To achieve equal results in both systems, it is not sufficient simply “to translate” one code to another code. There are numerous examples, where there exists some predefined function in one system and does not exist in another. Therefore, to get equal results in both systems, it is necessary to define new functions knowing the method or algorithm of calculation.

In learning any subject of a technical nature where mathematics plays a role, one is confronted with the task of understanding and storing away in the memory a huge body of facts and ideas, held together by certain relationships which can be “proved” or “shown” to exist between them. It is easy to confuse the proof itself with the relationship which it establishes. Clearly, the important thing to learn and to remember is the relationship, not the proof. In any particular circumstance we can either say “it can be shown that” such and such is true, or we can show it. In almost all cases, the particular proof that is used is concocted, first of all, in such form that it can be written quickly and easily on the chalkboard or on paper, and so that it will be as smooth-looking as possible. Consequently, the proof may look deceptively simple, when in fact, the author might have worked for hours trying different ways of calculating the same thing until he has found the neatest way, so as to be able to show that it can be shown in the shortest amount of time! The thing to be remembered, when seeing a proof, is not the proof itself, but rather that it can be shown that such and such is true. Of course, if the proof involves some mathematical procedures or “tricks” that one has not seen before, attention should be given not to the trick exactly, but to the mathematical idea involved.

What I am talking about may be difficult for you to understand, but it is really quite important. You see, technicians are not creators; and there are more and more technicians in the world, people who know what to do and how to do it, but who are not creators. In America there are calculating machines capable of solving in a few minutes mathematical problems which would take a man, working ten hours every day, a hundred years to solve. These extraordinary machines are being developed. But machines can never be creators—and human beings are becoming more and more like machines. Even when they rebel, their rebellion is within the limits of the machine and is therefore no rebellion at all.

Je me rends parfaitement compte du desagreable effet que produit sur la majorite de l'humanité, tout ce qui se rapporte, même au plus faible dègré, á des calculs ou raisonnements mathematiques. I am well aware of the disagreeable effect produced on the majority of humanity, by whatever relates, even at the slightest degree to calculations or mathematical reasonings.

Now we can see what makes mathematics unique. Only in mathematics is there no significant correction-only extension. Once the Greeks had developed the deductive method, they were correct in what they did, correct for all time. Euclid was incomplete and his work has been extended enormously, but it has not had to be corrected. His theorems are, every one of them, valid to this day. Ptolemy may have developed an erroneous picture of the planetary system, but the system of trigonometry he worked out to help him with his calculations remains correct forever. Each great mathematician adds to what came previously, but nothing needs to be uprooted. Consequently, when we read a book like A History of Mathematics, we get the picture of a mounting structure, ever taller and broader and more beautiful and magnificent and with a foundation, moreover, that is as untainted and as functional now as it was when Thales worked out the first geometrical theorems nearly 26 centuries ago. Nothing pertaining to humanity becomes us so well as mathematics. There, and only there, do we touch the human mind at its peak.

We know, however, that the mind is capable of understanding these matters in all their complexity and in all their simplicity. A ball flying through the air is responding to the force and direction with which it was thrown, the action of gravity, the friction of the air which it must expend its energy on overcoming, the turbulence of the air around its surface, and the rate and direction of the ball's spin. And yet, someone who might have difficulty consciously trying to work out what 3 x 4 x 5 comes to would have no trouble in doing differential calculus and a whole host of related calculations so astoundingly fast that they can actually catch a flying ball. People who call this "instinct" are merely giving the phenomenon a name, not explaining anything. I think that the closest that human beings come to expressing our understanding of these natural complexities is in music. It is the most abstract of the arts - it has no meaning or purpose other than to be itself. Every single aspect of a piece of music can be represented by numbers. From the organization of movements in a whole symphony, down through the patterns of pitch and rhythm that make up the melodies and harmonies, the dynamics that shape the performance, all the way down to the timbres of the notes themselves, their harmonics, the way they change over time, in short, all the elements of a noise that distinguish between the sound of one person piping on a piccolo and another one thumping a drum - all of these things can be expressed by patterns and hierarchies of numbers. And in my experience the more internal relationships there are between the patterns of numbers at different levels of the hierarchy, however complex and subtle those relationships may be, the more satisfying and, well, whole, the music will seem to be. In fact the more subtle and complex those relationships, and the further they are beyond the grasp of the conscious mind, the more the instinctive part of your mind - by which I mean that part of your mind that can do differential calculus so astoundingly fast that it will put your hand in the right place to catch a flying ball- the more that part of your brain revels in it. Music of any complexity (and even "Three Blind Mice" is complex in its way by the time someone has actually performed it on an instrument with its own individual timbre and articulation) passes beyond your conscious mind into the arms of your own private mathematical genius who dwells in your unconscious responding to all the inner complexities and relationships and proportions that we think we know nothing about. Some people object to such a view of music, saying that if you reduce music to mathematics, where does the emotion come into it? I would say that it's never been out of it.

No one had ever fought with him like she had. Their very first fight had been about the universe; he explained how it had been created and she refused to accept it. He raised his voice, she got angry, he couldn’t understand why, and she shouted, “I’m angry because you think everything happened by chance but there are billions of people on this planet and I found you so if you’re saying I could just as well have found someone else then I can’t bear your bloody mathematics!” Her fists had been clenched. He stood there looking at her for several minutes. Then he said that he loved her. It was the first time. They never stopped arguing and they never slept apart; he spent an entire working life calculating probabilities and she was the most improbable person he ever met. She turned him upside-down.

Further, the same Arguments which explode the Notion of Luck, may, on the other side, be useful in some Cases to establish a due comparison between Chance and Design: We may imagine Chance and Design to be, as it were, in Competition with each other, for the production of some sorts of Events, and many calculate what Probability there is, that those Events should be rather be owing to the one than to the other.

Pythagoras was born around 570 B.C. in the island of Samos in the Aegean Sea (off Asia Minor), and he emigrated sometime between 530 and 510 to Croton in the Dorian colony in southern Italy (then known as Magna Graecia). Pythagoras apparently left Samos to escape the stifling tyranny of Polycrates (died ca. 522 B.C.), who established Samian naval supremacy in the Aegean Sea. Perhaps following the advice of his presumed teacher, the mathematician Thales of Miletus, Pythagoras probably lived for some time (as long as twenty-two years, according to some accounts) in Egypt, where he would have learned mathematics, philosophy, and religious themes from the Egyptian priests. After Egypt was overwhelmed by Persian armies, Pythagoras may have been taken to Babylon, together with members of the Egyptian priesthood. There he would have encountered the Mesopotamian mathematical lore. Nevertheless, the Egyptian and Babylonian mathematics would prove insufficient for Pythagoras' inquisitive mind. To both of these peoples, mathematics provided practical tools in the form of "recipes" designed for specific calculations. Pythagoras, on the other hand, was one of the first to grasp numbers as abstract entities that exist in their own right.

And are we not guilty of offensive disparagement in calling chess a game? Is it not also a science and an art, hovering between those categories as Muhammad’s coffin hovered between heaven and earth, a unique link between pairs of opposites: ancient yet eternally new; mechanical in structure, yet made effective only by the imagination; limited to a geometrically fixed space, yet with unlimited combinations; constantly developing, yet sterile; thought that leads nowhere; mathematics calculating nothing; art without works of art; architecture without substance – but nonetheless shown to be more durable in its entity and existence than all books and works of art; the only game that belongs to all nations and all eras, although no one knows what god brought it down to earth to vanquish boredom, sharpen the senses and stretch the mind. Where does it begin and where does it end? Every child can learn its basic rules, every bungler can try his luck at it, yet within that immutable little square it is able to bring forth a particular species of masters who cannot be compared to anyone else, people with a gift solely designed for chess, geniuses in their specific field who unite vision, patience and technique in just the same proportions as do mathematicians, poets, musicians, but in different stratifications and combinations. In the old days of the enthusiasm for physiognomy, a physician like Gall might perhaps have dissected a chess champion’s brain to find out whether some particular twist or turn in the grey matter, a kind of chess muscle or chess bump, is more developed in such chess geniuses than in the skulls of other mortals. And how intrigued such a physiognomist would have been by the case of Czentovic, where that specific genius appeared in a setting of absolute intellectual lethargy, like a single vein of gold in a hundredweight of dull stone. In principle, I had always realized that such a unique, brilliant game must create its own matadors, but how difficult and indeed impossible it is to imagine the life of an intellectually active human being whose world is reduced entirely to the narrow one-way traffic between black and white, who seeks the triumphs of his life in the mere movement to and fro, forward and back of thirty-two chessmen, someone to whom a new opening, moving knight rather than pawn, is a great deed, and his little corner of immortality is tucked away in a book about chess – a human being, an intellectual human being who constantly bends the entire force of his mind on the ridiculous task of forcing a wooden king into the corner of a wooden board, and does it without going mad!

The faculty of re-solution is possibly much invigorated by mathematical study, and especially by that highest branch of it which, unjustly, and merely on account of its retrograde operations, has been called, as if par excellence, analysis. Yet to calculate is not in itself to analyse. A chess-player, for example, does the one without effort at the other. It follows that the game of chess, in its effects upon mental character, is greatly misunderstood. I am not now writing a treatise, but simply prefacing a somewhat peculiar narrative by observations very much at random; I will, therefore, take occasion to assert that the higher powers of the reflective intellect are more decidedly and more usefully tasked by the unostentatious game of draughts than by a the elaborate frivolity of chess. In this latter, where the pieces have different and bizarre motions, with various and variable values, what is only complex is mistaken (a not unusual error) for what is profound. The attention is here called powerfully into play. If it flag for an instant, an oversight is committed resulting in injury or defeat. The possible moves being not only manifold but involute, the chances of such oversights are multiplied; and in nine cases out of ten it is the more concentrative rather than the more acute player who conquers. In draughts, on the contrary, where the moves are unique and have but little variation, the probabilities of inadvertence are diminished, and the mere attention being left comparatively unemployed, what advantages are obtained by either party are obtained by superior acumen. To be less abstract, let us suppose a game of draughts where the pieces are reduced to four kings, and where, of course, no oversight is to be expected. It is obvious that here the victory can be decided (the players being at all equal) only by some recherché movement, the result of some strong exertion of the intellect. Deprived of ordinary resources, the analyst throws himself into the spirit of his opponent, identifies himself therewith, and not unfrequently sees thus, at a glance, the sole methods (sometime indeed absurdly simple ones) by which he may seduce into error or hurry into miscalculation.

Relativistic twins? When one looks at the paths that Newton and Einstein followed while pursuing their theories of gravity, one is struck by the many similarities: the unexplained data on orbits, the sudden insight about falling objects, the need for a new mathematics, the calculational difficulties, the retroactive agreements, the controversy, the problem-plagued expeditions, and the final triumph and acclaim.. Both men had worked in the same eccentric and lonely way, divorced from other scientists, armed with a great feeling of self-reliance while struggling with new concepts and difficult mathematics, and both produced earth-shaking results. One can't help but wonder if these two greatest of scientists, born 237 years apart, were "relativistically related", conceived as twins in some ethereal plane in a far-off galaxy and sent to earth to solve a matter of some gravity.

I understood the infamous spiritual terror which this movement exerts, particularly on the bourgeoisie, which is neither morally nor mentally equal to such attacks; at a given sign it unleashes a veritable barrage of lies and slanders against whatever adversary seems most dangerous, until the nerves of the attacked persons break down… This is a tactic based on precise calculation of all human weaknesses, and its result will lead to success with almost mathematical certainty…

Elliptic curve multiplication is a type of function that cryptographers call a “trap door” function: it is easy to do in one direction (multiplication) and impossible to do in the reverse direction (division). The owner of the private key can easily create the public key and then share it with the world knowing that no one can reverse the function and calculate the private key from the public key. This mathematical trick becomes the basis for unforgeable and secure digital signatures that prove ownership of bitcoin funds.

Now I ask you: what can be expected of man since he is a being endowed with strange qualities? Shower upon him every earthly blessing, drown him in a sea of happiness, so that nothing but bubbles of bliss can be seen on the surface; give him economic prosperity, such that he should have nothing else to do but sleep, eat cakes and busy himself with the continuation of his species, and even then out of sheer ingratitude, sheer spite, man would play you some nasty trick. He would even risk his cakes and would deliberately desire the most fatal rubbish, the most uneconomical absurdity, simply to introduce into all this positive good sense his fatal fantastic element. It is just his fantastic dreams, his vulgar folly that he will desire to retain, simply in order to prove to himself—as though that were so necessary—that men still are men and not the keys of a piano, which the laws of nature threaten to control so completely that soon one will be able to desire nothing but by the calendar. And that is not all: even if man really were nothing but a piano-key, even if this were proved to him by natural science and mathematics, even then he would not become reasonable, but would purposely do something perverse out of simple ingratitude, simply to gain his point. And if he does not find means he will contrive destruction and chaos, will contrive sufferings of all sorts, only to gain his point! He will launch a curse upon the world, and as only man can curse (it is his privilege, the primary distinction between him and other animals), may be by his curse alone he will attain his object—that is, convince himself that he is a man and not a piano-key! If you say that all this, too, can be calculated and tabulated—chaos and darkness and curses, so that the mere possibility of calculating it all beforehand would stop it all, and reason would reassert itself, then man would purposely go mad in order to be rid of reason and gain his point! I believe in it, I answer for it, for the whole work of man really seems to consist in nothing but proving to himself every minute that he is a man and not a piano-key!

firstly, what "really" attracted me to Indo-European, as well as to English, Polish, and Russian philology, wasn't the seductive variety of linguistic forms, or the infinitely picturesque accidents that fill the histories of words and dialects, but rather the fact that these obey lays that can be rigorously described, and that these laws, such as Grimm's Law in Germanic philology, or the principles of Slavic palatalization, which lie behind all those wonderful alveolar fricatives in Russia and the Auvergne, promised to submit the irresistible and etrnal movement of languages no longer to mere chance, but to something that closely resembled calculation; - and that, secondly, and consequently, the noblest aspect of linguistics (and if I had been familiar with Trouetzkoy's phonology and with Jakobson, this conclusion would have been even more obvious) was its power of deduction -- but that there remained something even nobler, which was the terrain of pure deduction, in other words, mathematics. And that it is why I absolutely had to become a mathematician.

In 1684 Dr Halley came to visit at Cambridge [and] after they had some time together the Dr asked him what he thought the curve would be that would be described by the Planets supposing the force of attraction towards the Sun to be reciprocal to the square of their distance from it. This was a reference to a piece of mathematics known as the inverse square law, which Halley was convinced lay at the heart of the explanation, though he wasn’t sure exactly how. Sr Isaac replied immediately that it would be an [ellipse]. The Doctor, struck with joy & amazement, asked him how he knew it. ‘Why,’ saith he, ‘I have calculated it,’ whereupon Dr Halley asked him for his calculation without farther delay. Sr Isaac looked among his papers but could not find it. This was astounding – like someone saying he had found a cure for cancer but couldn’t remember where he had put the formula. Pressed by Halley, Newton agreed to redo the calculations and produce a paper. He did as promised, but then did much more. He retired for two years of intensive reflection and scribbling, and at length produced his masterwork: the Philosophiae Naturalis Principia Mathematica or Mathematical Principles of Natural Philosophy, better known as the Principia.

On a flat surface with just the normal x and y coordinates, any high school algebra student, with the help of old Pythagoras, can calculate the distance between points. But imagine a flat map (of the world, for example) that represents locations on what is actually a curved globe. Things get stretched out near the poles, and measurement gets more complex. Calculating the actual distance between two points on the map in Greenland is different from doing so for points near the equator. Riemann worked out ways to determine mathematically the distance between points in space no matter how arbitrarily it curved and contorted.

An electronic machine can carry out mathematical calculations, remember historical facts, play chess and translate books from one language to another. It is able to solve mathematical problems more quickly than man and its memory is faultless. Is there any limit to progress, to its ability to create machines in the image and likeness of man? It seems the answer is no. It is not impossible to imagine the machine of future ages and millennia. It will be able to listen to music and appreciate art; it will even be able to compose melodies, paint pictures and write poems. Is there a limit to its perfection? Can it be compared to man? Will it surpass him? Childhood memories… tears of happiness … the bitterness of parting… love of freedom … feelings of pity for a sick puppy … nervousness … a mother’s tenderness … thoughts of death … sadness … friendship … love of the weak … sudden hope … a fortunate guess … melancholy … unreasoning joy … sudden embarrassment… The machine will be able to recreate all of this! But the surface of the whole earth will be too small to accommodate this machine – this machine whose dimensions and weight will continually increase as it attempts to reproduce the peculiarities of mind and soul of an average, inconspicuous human being. Fascism annihilated tens of millions of people.

And that is not all: even if man really were nothing but a piano-key, even if this were proved to him by natural science and mathematics, even then he would not become reasonable, but would purposely do something perverse out of simple ingratitude, simply to gain his point. And if he does not find means he will contrive destruction and chaos, will contrive sufferings of all sorts, only to gain his point! He will launch a curse upon the world, and as only man can curse (it is his privilege's, the primarily distinction between him and other animas), may be by his curse alone he will attain his object- that is, convince himself that he is a man and not a piano key! If you say all this too, can be calculated and tabulated-chaos and darkness and curses, so that the mere possibility of calculating it all beforehand would stop it all, the reason would reassert itself, then man would purposely go mad in order to be rid of reason and gain his point! I believe in it, I answer for it, for the whole work of man really seems to consist in nothing but proving to himself every minute that he is a man and not a piano-key! Good heavens, gentleman, what sort of free will is left when we come to tabulation and arithmetic, when it will all be a case of twice two make four? Twice two makes four without my will. As if free will meant that!

Countries measured their success by the size of their territory, the increase in their population and the growth of their GDP – not by the happiness of their citizens. Industrialised nations such as Germany, France and Japan established gigantic systems of education, health and welfare, yet these systems were aimed to strengthen the nation rather than ensure individual well-being. Schools were founded to produce skilful and obedient citizens who would serve the nation loyally. At eighteen, youths needed to be not only patriotic but also literate, so that they could read the brigadier’s order of the day and draw up tomorrow’s battle plans. They had to know mathematics in order to calculate the shell’s trajectory or crack the enemy’s secret code. They needed a reasonable command of electrics, mechanics and medicine in order to operate wireless sets, drive tanks and take care of wounded comrades. When they left the army they were expected to serve the nation as clerks, teachers and engineers, building a modern economy and paying lots of taxes. The same went for the health system. At the end of the nineteenth century countries such as France, Germany and Japan began providing free health care for the masses. They financed vaccinations for infants, balanced diets for children and physical education for teenagers. They drained festering swamps, exterminated mosquitoes and built centralised sewage systems. The aim wasn’t to make people happy, but to make the nation stronger. The country needed sturdy soldiers and workers, healthy women who would give birth to more soldiers and workers, and bureaucrats who came to the office punctually at 8 a.m. instead of lying sick at home. Even the welfare system was originally planned in the interest of the nation rather than of needy individuals. When Otto von Bismarck pioneered state pensions and social security in late nineteenth-century Germany, his chief aim was to ensure the loyalty of the citizens rather than to increase their well-being. You fought for your country when you were eighteen, and paid your taxes when you were forty, because you counted on the state to take care of you when you were seventy.30 In 1776 the Founding Fathers of the United States established the right to the pursuit of happiness as one of three unalienable human rights, alongside the right to life and the right to liberty. It’s important to note, however, that the American Declaration of Independence guaranteed the right to the pursuit of happiness, not the right to happiness itself. Crucially, Thomas Jefferson did not make the state responsible for its citizens’ happiness. Rather, he sought only to limit the power of the state.

The fundamental problem with learning mathematics is that while the number sense may be genetic, exact calculation requires cultural tools—symbols and algorithms—that have been around for only a few thousand years and must therefore be absorbed by areas of the brain that evolved for other purposes. The process is made easier when what we are learning harmonizes with built-in circuitry. If we can’t change the architecture of our brains, we can at least adapt our teaching methods to the constraints it imposes. For nearly three decades, American educators have pushed “reform math,” in which children are encouraged to explore their own ways of solving problems. Before reform math, there was the “new math,” now widely thought to have been an educational disaster. (In France, it was called les maths modernes and is similarly despised.) The new math was grounded in the theories of the influential Swiss psychologist Jean Piaget, who believed that children are born without any sense of number and only gradually build up the concept in a series of developmental stages. Piaget thought that children, until the age of four or five, cannot grasp the simple principle that moving objects around does not affect how many of them there are, and that there was therefore no point in trying to teach them arithmetic before the age of six or seven.

When an official report in the UK was commissioned to examine the mathematics needed in the workplace, the investigator found that estimation was the most useful mathematical activity. Yet when children who have experienced traditional math classes are asked to estimate, they are often completely flummoxed and try to work out exact answers, then round them off to look like an estimate. This is because they have not developed a good feel for numbers, which would allow them to estimate instead of calculate, and also because they have learned, wrongly, that mathematics is all about precision, not about making estimates or guesses. Yet both are at the heart of mathematical problem solving.

Computational models of the mind would make sense if what a computer actually does could be characterized as an elementary version of what the mind does, or at least as something remotely like thinking. In fact, though, there is not even a useful analogy to be drawn here. A computer does not even really compute. We compute, using it as a tool. We can set a program in motion to calculate the square root of pi, but the stream of digits that will appear on the screen will have mathematical content only because of our intentions, and because we—not the computer—are running algorithms. The computer, in itself, as an object or a series of physical events, does not contain or produce any symbols at all; its operations are not determined by any semantic content but only by binary sequences that mean nothing in themselves. The visible figures that appear on the computer’s screen are only the electronic traces of sets of binary correlates, and they serve as symbols only when we represent them as such, and assign them intelligible significances. The computer could just as well be programmed so that it would respond to the request for the square root of pi with the result “Rupert Bear”; nor would it be wrong to do so, because an ensemble of merely material components and purely physical events can be neither wrong nor right about anything—in fact, it cannot be about anything at all. Software no more “thinks” than a minute hand knows the time or the printed word “pelican” knows what a pelican is. We might just as well liken the mind to an abacus, a typewriter, or a library. No computer has ever used language, or responded to a question, or assigned a meaning to anything. No computer has ever so much as added two numbers together, let alone entertained a thought, and none ever will. The only intelligence or consciousness or even illusion of consciousness in the whole computational process is situated, quite incommutably, in us; everything seemingly analogous to our minds in our machines is reducible, when analyzed correctly, only back to our own minds once again, and we end where we began, immersed in the same mystery as ever. We believe otherwise only when, like Narcissus bent above the waters, we look down at our creations and, captivated by what we see reflected in them, imagine that another gaze has met our own.

What we can imagine as plausible is a narrow band in the middle of a much broader spectrum of what is actually possible. [O]ur eyes are built to cope with a narrow band of electromagnetic frequencies. [W]e can't see the rays outside the narrow light band, but we can do calculations about them, and we can build instruments to detect them. In the same way, we know that the scales of size and time extend in both directions far outside the realm of what we can visualize. Our minds can't cope with the large distances that astronomy deals in or with the small distances that atomic physics deals in, but we can represent those distances in mathematical symbols. Our minds can't imagine a time span as short as a picosecond, but we can do calculations about picoseconds, and we can build computers that can complete calculations within picoseconds. Our minds can't imagine a timespan as long as a million years, let alone the thousands of millions of years that geologists routinely compute. Just as our eyes can see only that narrow band of electromagnetic frequencies that natural selection equipped our ancestors to see, so our brains are built to cope with narrow bands of sizes and times. Presumably there was no need for our ancestors to cope with sizes and times outside the narrow range of everyday practicality, so our brains never evolved the capacity to imagine them. It is probably significant that our own body size of a few feet is roughly in the middle of the range of sizes we can imagine. And our own lifetime of a few decades is roughly in the middle of the range of times we can imagine.

To do so he used something called a tensor. In Euclidean geometry, a vector is a quantity (such as of velocity or force) that has both a magnitude and a direction and thus needs more than a single simple number to describe it. In non-Euclidean geometry, where space is curved, we need something more generalized—sort of a vector on steroids—in order to incorporate, in a mathematically orderly way, more components. These are called tensors. A metric tensor is a mathematical tool that tells us how to calculate the distance between points in a given space. For two-dimensional maps, a metric tensor has three components. For three-dimensional space, it has six independent components. And once you get to that glorious four-dimensional entity known as spacetime, the metric tensor needs ten independent components.

I can imagine some other world in which a conference of learned, and totally blind, bat-like creatures is flabbergasted to be told of animals called humans that are actually capable of using the newly discovered inaudible rays called "light" for finding their way about. These otherwise humble humans are almost totally deaf (well, they can hear after a fashion and even utter a few ponderously slow, deep drawling growls, but they only use these sounds for rudimentary purposes like communicating with each other; they don't seem capable of using them to detect even the most massive objects). They have, instead, highly specialized organs called "eyes" for exploiting "light" rays. The sun is the main source of light rays, and humans, remarkably, manage to exploit the complex echoes that bounce off objects when light rays from the sun hit them. They have an ingenious device called a "lens", whose shape appears to be mathematically calculated so that it bends these silent rays in such a way that there is an exact one-to-one mapping between objects in the world and an "image" on a sheet of cells called the "retina". Theses retinal cells are capable of, in some mysterious way, of rendering the light "audible" (one might say), and they relay their information to the brain. Our mathematicians have shown that it is theoretically possible, by doing the right highly complex calculations, to navigate safely through the world using these light rays, just as effectively as one can in the ordinary way using ultrasound -- in some respects even more effectively! But who would have thought that a humble human could do these calculations?

In fact, Hinduism�s pervading influence seems to go much earlier than Christianity. American mathematician, A. Seindenberg, has for example shown that the Sulbasutras, the ancient Vedic science of mathematics, constitute the source of mathematics in the Antic world, from Babylon to Greece : � the arithmetic equations of the Sulbasutras he writes, were used in the observation of the triangle by the Babylonians, as well as in the edification of Egyptian pyramids, in particular the funeral altar in form of pyramid known in the vedic world as smasana-cit (Seindenberg 1978: 329). In astronomy too, the "Indus" (from the valley of the Indus) have left a universal legacy, determining for instance the dates of solstices, as noted by 18th century French astronomer Jean-Sylvain Bailly : � the movement of stars which was calculated by Hindus 4500 years ago, does not differ even by a minute from the tables which we are using today". And he concludes: "the Hindu systems of astronomy are much more ancient than those of the Egyptians - even the Jews derived from the Hindus their knowledge �. There is also no doubt that the Greeks heavily borrowed from the "Indus". Danielou notes that the Greek cult of Dionysos, which later became Bacchus with the Romans, is a branch of Shivaism : � Greeks spoke of India as the sacred territory of Dionysos and even historians of Alexander the Great identified the Indian Shiva with Dionysos and mention the dates and legends of the Puranas �. French philosopher and Le Monde journalist Jean-Paul Droit, recently wrote in his book "The Forgetfulness of India" that � the Greeks loved so much Indian philosophy, that Demetrios Galianos had even translated the Bhagavad Gita �.

To prove to an indignant questioner on the spur of the moment that the work I do was useful seemed a thankless task and I gave it up. I turned to him with a smile and finished, 'To tell you the truth we don't do it because it is useful but because it's amusing.' The answer was thought of and given in a moment: it came from deep down in my mind, and the results were as admirable from my point of view as unexpected. My audience was clearly on my side. Prolonged and hearty applause greeted my confession. My questioner retired shaking his head over my wickedness and the newspapers next day, with obvious approval, came out with headlines 'Scientist Does It Because It's Amusing!' And if that is not the best reason why a scientist should do his work, I want to know what is. Would it be any good to ask a mother what practical use her baby is? That, as I say, was the first evening I ever spent in the United States and from that moment I felt at home. I realised that all talk about science purely for its practical and wealth-producing results is as idle in this country as in England. Practical results will follow right enough. No real knowledge is sterile. The most useless investigation may prove to have the most startling practical importance: Wireless telegraphy might not yet have come if Clerk Maxwell had been drawn away from his obviously 'useless' equations to do something of more practical importance. Large branches of chemistry would have remained obscure had Willard Gibbs not spent his time at mathematical calculations which only about two men of his generation could understand. With this trust in the ultimate usefulness of all real knowledge a man may proceed to devote himself to a study of first causes without apology, and without hope of immediate return.

It is the best of times in physics. Physicists are on the verge of obtaining the long-sought theory of everything. In a few elegant equations, perhaps concise enough to be emblazoned on a T-shirt, this theory will reveal how the universe began and how it will end. The key insight is that the smallest constituents of the world are not particles, as had been supposed since ancient times, but “strings”—tiny strands of energy. By vibrating in different ways, these strings produce the essential phenomena of nature, the way violin strings produce musical notes. String theory isn’t just powerful; it’s also mathematically beautiful. All that remains to be done is to write down the actual equations. This is taking a little longer than expected. But, with almost the entire theoretical-physics community working on the problem—presided over by a sage in Princeton, New Jersey—the millennia-old dream of a final theory is sure to be realized before long. It is the worst of times in physics. For more than a generation, physicists have been chasing a will-o’-the-wisp called string theory. The beginning of this chase marked the end of what had been three-quarters of a century of progress. Dozens of string-theory conferences have been held, hundreds of new Ph.D.’s have been minted, and thousands of papers have been written. Yet, for all this activity, not a single new testable prediction has been made; not a single theoretical puzzle has been solved. In fact, there is no theory so far—just a set of hunches and calculations suggesting that a theory might exist. And, even if it does, this theory will come in such a bewildering number of versions that it will be of no practical use: a theory of nothing. Yet the physics establishment promotes string theory with irrational fervor, ruthlessly weeding dissenting physicists from the profession. Meanwhile, physics is stuck in a paradigm doomed to barrenness.

and, finally, they knew the value of using what he calls “spiritual and physical terror.” This third lesson, though it was surely based on faulty observation and compounded of his own immense prejudices, intrigued the young Hitler. Within ten years he would put it to good use for his own ends. I understood the infamous spiritual terror which this movement exerts, particularly on the bourgeoisie, which is neither morally nor mentally equal to such attacks; at a given sign it unleashes a veritable barrage of lies and slanders against whatever adversary seems most dangerous, until the nerves of the attacked persons break down… This is a tactic based on precise calculation of all human weaknesses, and its result will lead to success with almost mathematical certainty… I achieved an equal understanding of the importance of physical terror toward the individual and the masses… For while in the ranks of their supporters the victory achieved seems a triumph of the justice of their own cause, the defeated adversary in most cases despairs of the success of any further resistance.

Nine months later, on September 1, 1939, Oppenheimer and a different collaborator—yet another student, Hartland Snyder—published a paper titled “On Continued Gravitational Contraction.” Historically, of course, the date is best known for Hitler’s invasion of Poland and the start of World War II. But in its quiet way, this publication was also a momentous event. The physicist and science historian Jeremy Bernstein calls it “one of the great papers in twentieth-century physics.” At the time, it attracted little attention. Only decades later would physicists understand that in 1939 Oppenheimer and Snyder had opened the door to twenty-first-century physics. They began their paper by asking what would happen to a massive star that has begun to burn itself out, having exhausted its fuel. Their calculations suggested that instead of collapsing into a white dwarf star, a star with a core beyond a certain mass—now believed to be two to three solar masses—would continue to contract indefinitely under the force of its own gravity. Relying on Einstein’s theory of general relativity, they argued that such a star would be crushed with such “singularity” that not even light waves would be able to escape the pull of its all-encompassing gravity. Seen from afar, such a star would literally disappear, closing itself off from the rest of the universe. “Only its gravitation field persists,” Oppenheimer and Snyder wrote. That is, though they themselves did not use the term, it would become a black hole. It was an intriguing but bizarre notion—and the paper was ignored, with its calculations long regarded as a mere mathematical curiosity.

Such was the ugly face of the Middle Ages. It is not surprising that mathematics made little progress; toward the Renaissance, European mathematics reached a level that, roughly, the Babylonians had attained some 2,000 years earlier and much of the progress made was due to the knowledge that filtered in from the Arabs, the Moors and other Muslim peoples, who themselves were in contact with the Hindus, and they, in turn, with the Far East. The history of Pi in the Middle Ages bears this out. No significant progress in the method of determining Pi was made until Viete discovered an infinite product of square roots in 1593, and what little progress there was in the calculation of its numerical value, by various modifications of the Archimedean method, was due to the decimal notation which began to infiltrate from the East through the Muslims in the 12th century. Arab mathematicians came to Europe through the trade in the Mediterranean, mainly via Italy; ironically, the other stream of mathematics was the Church itself. Not only because the mediaeval priests had a near monopoly of learning, but also because they needed mathematics and astronomy as custodians of the calendar. Like the Soviet High Priests who publish Pravda for other but read summaries of the New York Times themselves, sot he mediaeval Church condemned mathematics as devilish for others, but dabbled quite a lot in it itself. Gerbert d'Aurillac, who ruled as Pope Sylvester II from 999 to 1003, was quite a mathematician; so was Cardinal Nicolaus Cusanus (1401-1464); and much of the work done on Pi was done behind thick cloister walls. And just like the Soviets did not hesitate to spy on the atomic secrets of bourgeois pseudo-science, so the mediaeval Church did not hesitate to spy on the mathematics of the Muslim infidels.

When I threw the stick at Jamie, I hadn't intended to hit him with it. But the moment it left my hand, I knew that's what was going to happen. I didn't yet know any calculus or geometry, but I was able to plot, with some degree of certainty, the trajectory of that stick. The initial velocity, the acceleration, the impact. The mathematical likelihood of Jamie's bloody cheek. It had good weight and heft, that stick. It felt nice to throw. And it looked damn fine in the overcast sky, too, flying end over end, spinning like a heavy, two-pronged pinwheel and (finally, indifferently, like math) connecting with Jamie's face. Jamie's older sister took me by the arm and she shook me. Why did you do that? What were you thinking? The anger I saw in her eyes. Heard in her voice. The kid I became to her then, who was not the kid I thought I was. The burdensome regret. I knew the word "accident" was wrong, but I used it anyway. If you throw a baseball at a wall and it goes through a window, that is an accident. If you throw a stick directly at your friend and it hits your friend in the face, that is something else. My throw had been something of a lob and there had been a good distance between us. There had been ample time for Jamie to move, but he hadn't moved. There had been time for him to lift a hand and protect his face from the stick, but he hadn't done that either. He just stood impotent and watched it hit him. And it made me angry: That he hadn't tried harder at a defense. That he hadn't made any effort to protect himself from me. What was I thinking? What was he thinking? I am not a kid who throws sticks at his friends. But sometimes, that's who I've been. And when I've been that kid, it's like I'm watching myself act in a movie, reciting somebody else's damaging lines. Like this morning, over breakfast. Your eyes asking mine to forget last night's exchange. You were holding your favorite tea mug. I don't remember what we were fighting about. It doesn't seem to matter any more. The words that came out of my mouth then, deliberate and measured, temporarily satisfying to throw at the bored space between us. The slow, beautiful arc. The spin and the calculated impact. The downward turn of your face. The heavy drop in my chest. The word "accident" was wrong. I used it anyway.

there are continually turning up in life moral and rational persons, sages and lovers of human- ity who make it their object to live all their lives as morally and rationally as possible, to be, so to speak, a light to their neighbours simply in order to show them that it is possible to live morally and rationally in this world. And yet we all know that those very people sooner or later have been false to themselves, playing some queer trick, o en a most un- seemly one. Now I ask you: what can be expected of man since he is a being endowed with strange qualities? Show- er upon him every earthly blessing, drown him in a sea of happiness, so that nothing but bubbles of bliss can be seen Free eBooks at Planet eBook.com on the surface; give him economic prosperity, such that he should have nothing else to do but sleep, eat cakes and busy himself with the continuation of his species, and even then out of sheer ingratitude, sheer spite, man would play you some nasty trick. He would even risk his cakes and would deliberately desire the most fatal rubbish, the most uneco- nomical absurdity, simply to introduce into all this positive good sense his fatal fantastic element. It is just his fantastic dreams, his vulgar folly that he will desire to retain, simply in order to prove to himself—as though that were so neces- sary— that men still are men and not the keys of a piano, which the laws of nature threaten to control so completely that soon one will be able to desire nothing but by the cal- endar. And that is not all: even if man really were nothing but a piano-key, even if this were proved to him by natural science and mathematics, even then he would not become reasonable, but would purposely do something perverse out of simple ingratitude, simply to gain his point. And if he does not nd means he will contrive destruction and chaos, will contrive su erings of all sorts, only to gain his point! He will launch a curse upon the world, and as only man can curse (it is his privilege, the primary distinction be- tween him and other animals), may be by his curse alone he will attain his object—that is, convince himself that he is a man and not a piano-key! If you say that all this, too, can be calculated and tabulated—chaos and darkness and curses, so that the mere possibility of calculating it all be- forehand would stop it all, and reason would reassert itself, then man would purposely go mad in order to be rid of rea- 0 Notes from the Underground son and gain his point! I believe in it, I answer for it, for the whole work of man really seems to consist in nothing but proving to himself every minute that he is a man and not a piano-key! It may be at the cost of his skin, it may be by can- nibalism! And this being so, can one help being tempted to rejoice that it has not yet come o , and that desire still de- pends on something we don’t know?