Arithmetic Sayings And Quotes

One ought not to judge her: all children are Heartless. They have not grown a heart yet, which is why they can climb high trees and say shocking things and leap so very high grown-up hearts flutter in terror. Hearts weigh quite a lot. That is why it takes so long to grow one. But, as in their reading and arithmetic and drawing, different children proceed at different speeds. (It is well known that reading quickens the growth of a heart like nothing else.) Some small ones are terrible and fey, Utterly Heartless. Some are dear and sweet and Hardly Heartless At All. September stood very generally in the middle on the day the Green Wind took her, Somewhat Heartless, and Somewhat Grown.

I would say that the five most important skills are of course, reading, writing, arithmetic, and then as you’re adding in, persuasion, which is talking. And then finally, I would add computer programming just because it’s an applied form of arithmetic that just gets you so much leverage for free in any domain that you operate in. If you’re good with computers, if you’re good at basic mathematics, if you’re good at writing, if you’re good at speaking, and if you like reading, you’re set for life.

We say to the confused, Know thyself, as if knowing yourself was not the fifth and most difficult of human arithmetical operations, we say to the apathetic, Where there's a will, there's a way, as if the brute realities of the world did not amuse themselves each day by turning that phrase on its head, we say to the indecisive, Begin at the beginning, as if beginning were the clearly visible point of a loosely wound thread and all we had to do was to keep pulling until we reached the other end, and as if, between the former and the latter, we had held in our hands a smooth, continuous thread with no knots to untie, no snarls to untangle, a complete impossibility in the life of a skein, or indeed, if we may be permitted one more stock phrase, in the skein of life.

The guns-for-everyone advocates hate that statistic, and dispute it, but as Bill Clinton likes to say, it’s not opinion. It’s arithmetic, honey.

It is by now proverbial that every proverb has its opposite. For every Time is money there is a Stop and smell the roses. When someone says You never stand in the same river twice someone else has already replied There is nothing new under the sun. In the mind's arithmetic, 1 plus -1 equals 2. Truths are not quantities but scripts: Become for a moment the mind in which this is true.

In One Dimensions, did not a moving Point produce a Line with two terminal points? In two Dimensions, did not a moving Line produce a Square wit four terminal points? In Three Dimensions, did not a moving Square produce - did not the eyes of mine behold it - that blessed being, a Cube, with eight terminal points? And in Four Dimensions, shall not a moving Cube - alas, for Analogy, and alas for the Progress of Truth if it be not so - shall not, I say the motion of a divine Cube result in a still more divine organization with sixteen terminal points? Behold the infallible confirmation of the Series, 2, 4, 8, 16: is not this a Geometrical Progression? Is not this - if I might qupte my Lord's own words - "Strictly according to Analogy"? Again, was I not taught by my Lord that as in a Line there are two bonding points, and in a Square there are four bounding Lines, so in a Cube there must be six bounding Squares? Behold once more the confirming Series: 2, 4, 6: is not this an Arithmetical Progression? And consequently does it not of necessity follow that the more divine offspring of the divine Cube in the Land of Four Dimensions, must have eight bounding Cubes: and is not this also, as my Lord has taught me to believe, "strictly according to analogy"?

There’s a very simple point about arithmetic—if you happen to be in a swing state, a state where the outcome is indefinite, and you don’t vote for, say, Clinton, that’s equivalent to voting for Trump. That’s arithmetic.

The most elementary book on arithmetic contains more knowledge and insight about reality than the collected sayings of every guru who ever lived.

Here we come to the central question of this book: What, precisely, does it mean to say that our sense of morality and justice is reduced to the language of a business deal? What does it mean when we reduce moral obligations to debts? What changes when the one turns into the other? And how do we speak about them when our language has been so shaped by the market? On one level the difference between an obligation and a debt is simple and obvious. A debt is the obligation to pay a certain sum of money. As a result, a debt, unlike any other form of obligation, can be precisely quantified. This allows debts to become simple, cold, and impersonal-which, in turn, allows them to be transferable. If one owes a favor, or one’s life, to another human being-it is owed to that person specifically. But if one owes forty thousand dollars at 12-percent interest, it doesn’t really matter who the creditor is; neither does either of the two parties have to think much about what the other party needs, wants, is capable of doing-as they certainly would if what was owed was a favor, or respect, or gratitude. One does not need to calculate the human effects; one need only calculate principal, balances, penalties, and rates of interest. If you end up having to abandon your home and wander in other provinces, if your daughter ends up in a mining camp working as a prostitute, well, that’s unfortunate, but incidental to the creditor. Money is money, and a deal’s a deal. From this perspective, the crucial factor, and a topic that will be explored at length in these pages, is money’s capacity to turn morality into a matter of impersonal arithmetic-and by doing so, to justify things that would otherwise seem outrageous or obscene. The factor of violence, which I have been emphasizing up until now, may appear secondary. The difference between a “debt” and a mere moral obligation is not the presence or absence of men with weapons who can enforce that obligation by seizing the debtor’s possessions or threatening to break his legs. It is simply that a creditor has the means to specify, numerically, exactly how much the debtor owes.

What will that mean to each of you? “It will mean that those of you who might have lived to be seventy-one must die at seventy. Some of you who might have lived to be eighty-six must cough up your ghost at eighty-five. That’s a great age. A year more or less doesn’t sound like much. When the time comes, boys, you may regret. But, you will be able to say, this year I spent well, I gave for Pip, I made a loan of life for sweet Pipkin, the fairest apple that ever almost fell too early off the harvest tree. Some of you at forty-nine must cross life off at forty-eight. Some at fifty-five must lay them down to Forever’s Sleep at fifty-four. Do you catch the whole thing intact now, boys? Do you add the figures? Is the arithmetic plain? A year! Who will bid three hundred and sixty-five entire days from out his own soul, to get old Pipkin back? Think, boys. Silence. Then, speak.

Boy everyone in this country is running around yammering about their fucking rights. "I have a right, you have no right, we have a right." Folks I hate to spoil your fun, but... there's no such thing as rights. They're imaginary. We made 'em up. Like the boogie man. Like Three Little Pigs, Pinocio, Mother Goose, shit like that. Rights are an idea. They're just imaginary. They're a cute idea. Cute. But that's all. Cute...and fictional. But if you think you do have rights, let me ask you this, "where do they come from?" People say, "They come from God. They're God given rights." Awww fuck, here we go again...here we go again. The God excuse, the last refuge of a man with no answers and no argument, "It came from God." Anything we can't describe must have come from God. Personally folks, I believe that if your rights came from God, he would've given you the right for some food every day, and he would've given you the right to a roof over your head. GOD would've been looking out for ya. You know that. He wouldn't have been worried making sure you have a gun so you can get drunk on Sunday night and kill your girlfriend's parents. But let's say it's true. Let's say that God gave us these rights. Why would he give us a certain number of rights? The Bill of Rights of this country has 10 stipulations. OK...10 rights. And apparently God was doing sloppy work that week, because we've had to ammend the bill of rights an additional 17 times. So God forgot a couple of things, like...SLAVERY. Just fuckin' slipped his mind. But let's say...let's say God gave us the original 10. He gave the british 13. The british Bill of Rights has 13 stipulations. The Germans have 29, the Belgians have 25, the Sweedish have only 6, and some people in the world have no rights at all. What kind of a fuckin' god damn god given deal is that!?...NO RIGHTS AT ALL!? Why would God give different people in different countries a different numbers of different rights? Boredom? Amusement? Bad arithmetic? Do we find out at long last after all this time that God is weak in math skills? Doesn't sound like divine planning to me. Sounds more like human planning . Sounds more like one group trying to control another group. In other words...business as usual in America. Now, if you think you do have rights, I have one last assignment for ya. Next time you're at the computer get on the Internet, go to Wikipedia. When you get to Wikipedia, in the search field for Wikipedia, i want to type in, "Japanese-Americans 1942" and you'll find out all about your precious fucking rights. Alright. You know about it. In 1942 there were 110,000 Japanese-American citizens, in good standing, law abiding people, who were thrown into internment camps simply because their parents were born in the wrong country. That's all they did wrong. They had no right to a lawyer, no right to a fair trial, no right to a jury of their peers, no right to due process of any kind. The only right they had was...right this way! Into the internment camps. Just when these American citizens needed their rights the most...their government took them away. and rights aren't rights if someone can take em away. They're priveledges. That's all we've ever had in this country is a bill of TEMPORARY priviledges; and if you read the news, even badly, you know the list get's shorter, and shorter, and shorter. Yeup, sooner or later the people in this country are going to realize the government doesn't give a fuck about them. the government doesn't care about you, or your children, or your rights, or your welfare or your safety. it simply doesn't give a fuck about you. It's interested in it's own power. That's the only thing...keeping it, and expanding wherever possible. Personally when it comes to rights, I think one of two things is true: either we have unlimited rights, or we have no rights at all.

The first time I heard the saying 'Live every day like you are going to live forever and every day like is your last.' I thought it was one of those unsolvable story problems from my fifth grade arithmetic books, but it turned out to be the truest thing about my year.

My father says that in our part of the world this idea of jihad was very much encouraged by the CIA. Children in the refugee camps were even given school textbooks produced by an American university which taught basic arithmetic through fighting. They had examples like “If out of 10 Russian infidels, 5 are killed by one Muslim, 5 would be left” or “15 bullets – 10 bullets = 5 bullets.

My father says that in our part of the world this idea of jihad was very much encouraged by the CIA. Children in the refugee camps were even given school textbooks produced by an American university which taught basic arithmetic through fighting. They had examples like, ‘If out of 10 Russian infidels, 5 are killed by one Muslim, 5 would be left’ or ‘15 bullets – 10 bullets = 5 bullets’.

American Arithmetic (excerpt) We are Americans, and we are less than 1 percent of Americans. We do a better job of dying by police than we do existing. --- At the National Museum of the American Indian, 68 percent of the collection is from the United States. I am doing my best to not become a museum of myself. I am doing my best to breathe in and out. I am begging: Let me be lonely but not invisible. But in an American room of one hundred people, I am Native American—less than one, less than whole—I am less than myself. Only a fraction of a body, let’s say, I am only a hand- and when I slip it beneath the shirt of my lover I disappear completely.

Mathematicians call it “the arithmetic of congruences.” You can think of it as clock arithmetic. Temporarily replace the 12 on a clock face with 0. The 12 hours of the clock now read 0, 1, 2, 3, … up to 11. If the time is eight o’clock, and you add 9 hours, what do you get? Well, you get five o’clock. So in this arithmetic, 8 + 9 = 5; or, as mathematicians say, 8 + 9 ≡ 5 (mod 12), pronounced “eight plus nine is congruent to five, modulo twelve.

It was a mistake in the system; perhaps it lay in the precept which until now he had held to be uncontestable, in whose name he had sacrificed others and was himself being sacrificed: in the precept, that the end justifies the means. . . . Perhaps later, much later, the new movement would arise—with new flags, a new spirit knowing of both: of economic fatality and the “oceanic sense.” Perhaps the members of the new party will wear monks’ cowls, and preach that only purity of means can justify the ends. Perhaps they will teach that the tenet is wrong which says that a man is the quotient of one million divided by one million, and will introduce a new kind of arithmetic based on multiplication: on the joining of a million individuals to form a new entity which, no longer an amorphous mass, will develop a consciousness and an individuality of its own, with an “oceanic feeling” increased a millionfold, in unlimited yet self-contained space. Rubashov broke off his pacing and listened. The sound of muffled drumming came down the corridor.

Three can eat as cheaply as two, the well-known arithmetic of resignation in any family where a child is expected, now one can say with even greater authority, Ten million can eat as cheaply as five, and with a quiet smile, A nation is nothing but a great big family.

Schools normally schedule one subject, for example, Japanese, the first period, when you just do Japanese; then, say, arithmetic the second period, when you just do arithmetic. But here it was quite different. At the beginning of the first period, the teacher made a list of all the problems and questions in the subjects to be studied that day. Then she would say, “Now, start with any of these you like.” […] This method of teaching enabled the teachers to observe - as the children progressed to higher grades - what they were interested in as well as their way of thinking and their character. It was an ideal way of teachers to really get to know their pupils.

There is a mathematical underpinning that you must first acquire, mastery of each mathematical subdiscipline leading you to the threshold of the next. In turn you must learn arithmetic, Euclidian geometry, high school algebra, differential and integral calculus, ordinary and partial differential equations, vector calculus, certain special functions of mathematical physics, matrix algebra, and group theory. For most physics students, this might occupy them from, say, third grade to early graduate school—roughly 15 years. Such a course of study does not actually involve learning any quantum mechanics, but merely establishing the mathematical framework required to approach it deeply.

The cabby left, muttering under his nose. "What's he muttering about?" Mr. Goliadkin thought through his tears. "I hired him for the evening, I'm sort of...within my rights nows...so there! I hired him for the evening, and that's the end of the matter. Even if he just stands there, it's all the same. It's as I will. I'm free to go, and free not to go. And that I'm now standing behind the woodpile--that, too, is quite all right...and don't you dare say anything; I say, the gentleman wants to stand behind the woodpile, so he stands behind the woodpile...and it's no taint to anybody's honor--so there! So there, lady mine, if you'd like to know. Thus and so, I say, but in our age, lady mine, nobody lives in a hut. So there! In our industrial age, lady mine, you can't get anywhere without good behavior, of which you yourself serve as a pernicious example...You say one must serve as a chief clerk and live in a hut on the seashore. First of all, lady mine, there are no chief clerks on the seashore, and second, you and I can't possible get to be a chief clerk. For, to take an example, suppose I apply, I show up--thus and so, as a chief clerk, say, sort of...and protect me from my enemy...and they'll tell you, my lady, say, sort of...there are lots of chief clerks, and here you're not at some émigrée Falbala's, where you learned good behavior, of which you yourself serve as a pernicious example. Good behavior, my lady, means sitting at home, respecting your father, and not thinking of any little suitors before it's time. Little suitors, my lady, will be found in due time! So there! Of course, one must indisputably have certain talents, to wit: playing the piano on occasion, speaking French, some history, geography, catechism, and arithmetic--so there!--but not more. Also cooking; cooking should unfailingly be part of every well-behaved girl's knowledge!

Anything could be true. The so-called laws of Nature were nonsense. The law of gravity was nonsense. 'If I wished,' O'Brien had said, 'I could float off this floor like a soap bubble.' Winston worked it out. 'If he thinks he floats off the floor, and if I simultaneously think I see him do it, then the thing happens.' Suddenly, like a lump of submerged wreckage breaking the surface of water, the thought burst into his mind: 'It doesn't really happen. We imagine it. It is hallucination.' He pushed the thought under instantly. The fallacy was obvious. It presupposed that somewhere or other, outside oneself, there was a 'real' world where 'real' things happened. But how could there be such a world? What knowledge have we of anything, save through our own minds? All happenings are in the mind. Whatever happens in all minds, truly happens. He had no difficulty in disposing of the fallacy, and he was in no danger of succumbing to it. He realized, nevertheless, that it ought never to have occurred to him. The mind should develop a blind spot whenever a dangerous thought presented itself. The process should be automatic, instinctive. Crimestop, they called it in Newspeak. He set to work to exercise himself in crimestop. He presented himself with propositions -- 'the Party says the earth is flat', 'the party says that ice is heavier than water' -- and trained himself in not seeing or not understanding the arguments that contradicted them. It was not easy. It needed great powers of reasoning and improvisation. The arithmetical problems raised, for instance, by such a statement as 'two and two make five' were beyond his intellectual grasp. It needed also a sort of athleticism of mind, an ability at one moment to make the most delicate use of logic and at the next to be unconscious of the crudest logical errors. Stupidity was as necessary as intelligence, and as difficult to attain.

The earliest known work in Arabic arithmetic was written by al-Khowârizmî, a mathematician who lived around 825, some four hundred years before Fibonacci.11 Although few beneficiaries of his work are likely to have heard of him, most of us know of him indirectly. Try saying “al-Khowârizmî” fast. That’s where we get the word “algorithm,” which means rules for computing.12 It was al-Khowârizmî who was the first mathematician to establish rules for adding, subtracting, multiplying, and dividing with the new Hindu numerals. In another treatise, Hisâb al-jabr w’ almuqâbalah, or “Science of transposition and cancellation,” he specifies the process for manipulating algebraic equations. The word al-jabr thus gives us our word algebra, the science of equations.13

I know perfectly well that it is impossible, according to arithmetic and scholarly books, to live in a far valley off a handful of ewes and two low yield cows. But we live, I say. You children all lived; your sisters now have sturdy children in far-off districts. And what you are now carrying under your heart will also live and be welcome, little one, despite arithmetic and scholarly books.

Your Lordship tempts his servant to see whether he remembers the revelations imparted to him. Trifle not with me, my Lord; I crave, I thirst, for more knowledge. Doubtless we cannot see that other higher Spaceland now, because we have no eye in our stomachs. But, just as there was the realm of Flatland, though that poor puny Lineland Monarch could neither turn to left nor right to discern it, and just as there was close at hand, and touching my frame, the land of Three Dimensions, though I, blind senseless wretch, had no power to touch it, no eye in my interior to discern it, so of a surety there is a Fourth Dimension, which my Lord perceives with the inner eye of thought. And that it must exist my Lord himself has taught me. Or can he have forgotten what he himself imparted to his servant? In One Dimension, did not a moving Point produce a Line with two terminal points? In Two Dimensions, did not a moving Line produce a Square with four terminal points? In Three Dimensions, did not a moving Square produce—did not this eye of mine behold it—that blessed Being, a Cube, with eight terminal points? And in Four Dimensions shall not a moving Cube—alas, for Analogy, and alas for the Progress of Truth, if it be not so—shall not, I say, the motion of a divine Cube result in a still more divine Organization with sixteen terminal points? Behold the infallible confirmation of the Series, 2, 4, 8, 16: is not this a Geometrical Progression? Is not this—if I might quote my Lord’s own words—“strictly according to Analogy”? Again, was I not taught by my Lord that as in a Line there are two bounding Points, and in a Square there are four bounding Lines, so in a Cube there must be six bounding Squares? Behold once more the confirming Series, 2, 4, 6: is not this an Arithmetical Progression? And consequently does it not of necessity follow that the more divine offspring of the divine Cube in the Land of Four Dimensions, must have 8 bounding Cubes: and is not this also, as my Lord has taught me to believe, “strictly according to Analogy”? O, my Lord, my Lord, behold, I cast myself in faith upon conjecture, not knowing the facts; and I appeal to your Lordship to confirm or deny my logical anticipations. If I am wrong, I yield, and will no longer demand a fourth Dimension; but, if I am right, my Lord will listen to reason. I ask therefore, is it, or is it not, the fact, that ere now your countrymen also have witnessed the descent of Beings of a higher order than their own, entering closed rooms, even as your Lordship entered mine, without the opening of doors or windows, and appearing and vanishing at will? On the reply to this question I am ready to stake everything. Deny it, and I am henceforth silent. Only vouchsafe an answer.

Kurt Gödel, who was able to prove in 1940 that given any axiomatic system that can produce arithmetic, we have to choose between completeness and coherence. Completeness means that the truth value of every statement in the system is determinable—that is that all statements can be assigned the appropriate truth value (usually true or false, for us). Coherence means that there are no contradictory statements, which is to say no paradoxes, within the system. We can have one or the other, but except in very special cases that have little applicability, we cannot have both.

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!

If you object (as some of us did to Dr. Goris) that Cantor's transfinite numbers aren't really numbers at all but rather sets, then be apprised that what, say, 'P(Infinity to the Infinity +n), really is is a symbol for the number of members in a given set, the same way '3' is a symbol for the number of members in the set {1,2,3}. And since the transfinites are provably distinct and compose an infinite ordered sequence just like the integers,they really are numbers, symbolizable (for now) by Cantor's well-known system of alephs or '(Aleph symbol's). And, as true numbers, transfinites turn out to be susceptible to the same kinds of arithmetical relations and operations as regular numbers-although, just as with 0, the rules for these operations are very different in the case of (Alephs) and have to be independently established and proved.

Pythagoras, as everyone knows, said that 'all things are numbers'. This statement, interpreted in a modern way, is logically nonsense, but what he meant was not exactly nonsense. He discovered the importance of numbers in music, and the connection which he established between music and arithmetic survives in the mathematical terms 'harmonic mean' and 'harmonic progression'. He thought of numbers as shapes, as they appear on dice or playing cards. We still speak of squares and cubes of numbers, which are terms that we owe to him. He also spoke of oblong numbers, triangular numbers, pyramidal numbers, and so on. These were the numbers of pebbles (or, as we should more naturally say, shot) required to make the shapes in question. He presumably thought of the world as atomic, and of bodies as built up of molecules composed of atoms arranged in various shapes. In this way he hoped to make arithmetic the fundamental study in physics as in aesthetics.

If we say God is Light, Love, Truth, Power, Goodness, Law, Principle, we confound attributes with existence. If we say God is a Spirit, God is space, we merely fill the imagination, not satisfy the understanding: it is feeding the thoughts with air, and leaving the intellect hungry. A Trinitarian Deity is one of the scholastic perplexities of the intellect. The first rule of arithmetic is against it. If it means three Gods in one, it is an enigma. If it means three doctrinal aspects of God, it confuses all simplicity of feeling. In the simple, moral heart of man, God is one, and his name is Love; not a weak, vapoury sentimentality, but an austere, healthy love, whose expression is strength, purity, truth, justice, service, and tenderness. But this conception of Deity belongs to the empire of the emotions, it is a matter of feeling, not of proof, and can authorise no intolerance towards others, itself existing only by the sufferance of the intellect, which has chastened its expression, and is supreme over it.

He would place his mouth, still full of sleep, on hers, and perhaps pull her back into the bedroom and down into the bed with him, into that liquid pool of flesh, his mouth sliding over her, furry pleasure, the covers closing over them as they sank into weightlessness. But he hadn't done that for some time. He had been waking earlier and earlier; she, on the other hand, had been having trouble getting out of bed. She was losing that compulsion, that joy, whatever had nagged her out into the cold morning air, driven her to fill all those notebooks, all those printed pages. Instead, she would roll herself up in the blankets after Bernie got up, tucking in all the corners, muffling herself in wool. She had begun to have the feeling that nothing was waiting for her outside the bed's edge. No emptiness but nothing, the zero with legs in the arithmetic book. 'I'm off,' he'd say to her groggy bundled back. She'd be awake enough to hear this; then she would lapse back into a humid sleep. His absence was one more reason for not getting up.

Two mathematicians were having dinner. One was complaining: ‘The average person is a mathematical idiot. People cannot do arithmetic correctly, cannot balance a checkbook, cannot calculate a tip, cannot do percents, …’ The other mathematician disagreed: ‘You’re exaggerating. People know all the math they need to know.’ Later in the dinner the complainer went to the men’s room. The other mathematician beckoned the waitress to his table and said, ‘The next time you come past our table, I am going to stop you and ask you a question. No matter what I say, I want you to answer by saying “x squared.”‘ She agreed. When the other mathematician returned, his companion said, ‘I’m tired of your complaining. I’m going to stop the next person who passes our table and ask him or her an elementary calculus question, and I bet the person can solve it.’ Soon the waitress came by and he asked: ‘Excuse me, Miss, but can you tell me what the integral of 2x with respect to x is?’ The waitress replied: ‘x squared.’ The mathematician said, ‘See!’ His friend said, ‘Oh … I guess you were right.’ And the waitress said, ‘Plus a constant.

Result: 325 ccs of water, which weighs 325 grams! Therefore Rocky’s ball also weighs 325 grams. I return to the tunnel to tell Rocky all about how smart I am. He balls a fist at me as I enter. “You left! Bad!” “I measured the mass! I made a very smart experiment.” He holds up a string with beads on it. “Twenty-six.” The beaded string is just like the ones he sent me back when we talked about our atmospheres— “Oh,” I say. It’s an atom. That’s how he talks about atoms. I count the beads. There are twenty-six in all. He’s talking about element 26—one of the most common elements on Earth. “Iron,” I say. I point at the necklace. “Iron.” He points at the necklace and says, “♫♩♪♫♫.” I record the word in my dictionary. “Iron,” he says again, pointing at the necklace. “Iron.” He points to the ball in my hand. “Iron.” It takes a second to sink in. Then I slap my forehead. “You are bad.” It was a fun experiment, but a total waste of time. Rocky was giving me all the information I needed. Or trying to, at least. I know how dense iron is, and I know how to calculate the volume of a sphere. Getting to mass from there is just a little arithmetic.

Now to picture the mechanism of this process of construction and not merely its progressive extension, we must note that each level is characterized by a new co-ordination of the elements provided—already existing in the form of wholes, though of a lower order—by the processes of the previous level. The sensori-motor schema, the characteristic unit of the system of pre-symbolic intelligence, thus assimilates perceptual schemata and the schemata relating to learned action (these schemata of perception and habit being of the same lower order, since the first concerns the present state of the object and the second only elementary changes of state). The symbolic schema assimilates sensori-motor schemata with differentiation of function; imitative accommodation is extended into imaginal significants and assimilation determines the significates. The intuitive schema is both a co-ordination and a differentiation of imaginal schemata. The concrete operational schema is a grouping of intuitive schemata, which are promoted, by the very fact of their being grouped, to the rank of reversible operations. Finally, the formal schema is simply a system of second-degree operations, and therefore a grouping operating on concrete groupings. Each of the transitions from one of these levels to the next is therefore characterized both by a new co-ordination and by a differentiation of the systems constituting the unit of the preceding level. Now these successive differentiations, in their turn, throw light on the undifferentiated nature of the initial mechanisms, and thus we can conceive both of a genealogy of operational groupings as progressive differentiations, and of an explanation of the pre-operational levels as a failure to differentiate the processes involved. Thus, as we have seen (Chap. 4), sensori-motor intelligence arrives at a kind of empirical grouping of bodily movements, characterized psychologically by actions capable of reversals and detours, and geometrically by what Poincaré called the (experimental) group of displacement. But it goes without saying that, at this elementary level, which precedes all thought, we cannot regard this grouping as an operational system, since it is a system of responses actually effected; the fact is therefore that it is undifferentiated, the displacements in question being at the same time and in every case responses directed towards a goal serving some practical purpose. We might therefore say that at this level spatio-temporal, logico-arithmetical and practical (means and ends) groupings form a global whole and that, in the absence of differentiation, this complex system is incapable of constituting an operational mechanism. At the end of this period and at the beginning of representative thought, on the other hand, the appearance of the symbol makes possible the first form of differentiation: practical groupings (means and ends) on the one hand, and representation on the other. But this latter is still undifferentiated, logico-arithmetical operations not being distinguished from spatio-temporal operations. In fact, at the intuitive level there are no genuine classes or relations because both are still spatial collections as well as spatio-temporal relationships: hence their intuitive and pre-operational character. At 7–8 years, however, the appearance of operational groupings is characterized precisely by a clear differentiation between logico-arithmetical operations that have become independent (classes, relations and despatialized numbers) and spatio-temporal or infra-logical operations. Lastly, the level of formal operations marks a final differentiation between operations tied to real action and hypothetico-deductive operations concerning pure implications from propositions stated as postulates.

I turned the dial up into the phone portion of the band on my ham radio in order to listen to a Saturday morning swap net. Along the way, I came across an older sounding chap, with a tremendous signal and a golden voice. You know the kind, he sounded like he should be in the broadcasting business. He was telling whoever he was talking with something about “a thousand marbles.” I was intrigued and stopped to listen to what he had to say. “Well, Tom, it sure sounds like you’re busy with your job. I’m sure they pay you well but it’s a shame you have to be away from home and your family so much. Hard to believe a young fellow should have to work sixty or seventy hours a week to make ends meet. Too bad you missed your daughter’s dance recital.” He continued, “Let me tell you something, Tom, something that has helped me keep a good perspective on my own priorities.” And that’s when he began to explain his theory of a “thousand marbles.” “You see, I sat down one day and did a little arithmetic. The average person lives about seventy-five years. I know, some live more and some live less, but on average, folks live about seventy-five years. “Now then, I multiplied 75 times 52 and I came up with 3,900 which is the number of Saturdays that the average person has in their entire lifetime. Now stick with me Tom, I’m getting to the important part. “It took me until I was fifty-five years old to think about all this in any detail,” he went on, “and by that time I had lived through

Gadgetry will continue to relieve mankind of tedious jobs. Kitchen units will be devised that will prepare ‘automeals,’ heating water and converting it to coffee; toasting bread; frying, poaching or scrambling eggs, grilling bacon, and so on. Breakfasts will be ‘ordered’ the night before to be ready by a specified hour the next morning. Communications will become sight-sound and you will see as well as hear the person you telephone. The screen can be used not only to see the people you call but also for studying documents and photographs and reading passages from books. Synchronous satellites, hovering in space will make it possible for you to direct-dial any spot on earth, including the weather stations in Antarctica. [M]en will continue to withdraw from nature in order to create an environment that will suit them better. By 2014, electroluminescent panels will be in common use. Ceilings and walls will glow softly, and in a variety of colors that will change at the touch of a push button. Robots will neither be common nor very good in 2014, but they will be in existence. The appliances of 2014 will have no electric cords, of course, for they will be powered by long- lived batteries running on radioisotopes. “[H]ighways … in the more advanced sections of the world will have passed their peak in 2014; there will be increasing emphasis on transportation that makes the least possible contact with the surface. There will be aircraft, of course, but even ground travel will increasingly take to the air a foot or two off the ground. [V]ehicles with ‘Robot-brains’ … can be set for particular destinations … that will then proceed there without interference by the slow reflexes of a human driver. [W]all screens will have replaced the ordinary set; but transparent cubes will be making their appearance in which three-dimensional viewing will be possible. [T]he world population will be 6,500,000,000 and the population of the United States will be 350,000,000. All earth will be a single choked Manhattan by A.D. 2450 and society will collapse long before that! There will, therefore, be a worldwide propaganda drive in favor of birth control by rational and humane methods and, by 2014, it will undoubtedly have taken serious effect. Ordinary agriculture will keep up with great difficulty and there will be ‘farms’ turning to the more efficient micro-organisms. Processed yeast and algae products will be available in a variety of flavors. The world of A.D. 2014 will have few routine jobs that cannot be done better by some machine than by any human being. Mankind will therefore have become largely a race of machine tenders. Schools will have to be oriented in this direction…. All the high-school students will be taught the fundamentals of computer technology will become proficient in binary arithmetic and will be trained to perfection in the use of the computer languages that will have developed out of those like the contemporary “Fortran". [M]ankind will suffer badly from the disease of boredom, a disease spreading more widely each year and growing in intensity. This will have serious mental, emotional and sociological consequences, and I dare say that psychiatry will be far and away the most important medical specialty in 2014. [T]he most glorious single word in the vocabulary will have become work! in our a society of enforced leisure.

Hume begins by distinguishing seven kinds of philosophical relation: resemblance, identity, relations of time and place, proportion in quantity or number, degrees in any quality, contrariety, and causation. These, he says, may be divided into two kinds: those that depend only on the ideas, and those that can be changed without any change in the ideas. Of the first kind are resemblance, contrariety, degrees in quality, and proportions in quantity or number. But spatio-temporal and causal relations are of the second kind. Only relations of the first kind give certain knowledge; our knowledge concerning the others is only probable. Algebra and arithmetic are the only sciences in which we can carry on a long chain of reasoning without losing certainty. Geometry is not so certain as algebra and arithmetic, because we cannot be sure of the truth of its axioms. It is a mistake to suppose, as many philosophers do, that the ideas of mathematics 'must be comprehended by a pure and intellectual view, of which the superior faculties of the soul are alone capable'. The falsehood of this view is evident, says Hume, as soon as we remember that 'all our ideas are copied from our impressions'. The three relations that depend not only on ideas are identity, spatio-temporal relations, and causation. In the first two, the mind does not go beyond what is immediately present to the senses. (Spatio-temporal relations, Hume holds, can be perceived, and can form parts of impressions.) Causation alone enables us to infer some thing or occurrence from some other thing or occurrence: "'Tis only causation, which produces such a connexion, as to give us assurance from the existence or action of one object, that 'twas followed or preceded by any other existence or action.

this I say,—we must never forget that all the education a man's head can receive, will not save his soul from hell, unless he knows the truths of the Bible. A man may have prodigious learning, and yet never be saved. He may be master of half the languages spoken round the globe. He may be acquainted with the highest and deepest things in heaven and earth. He may have read books till he is like a walking cyclopædia. He may be familiar with the stars of heaven,—the birds of the air,—the beasts of the earth, and the fishes of the sea. He may be able, like Solomon, to "speak of trees, from the cedar of Lebanon to the hyssop that grows on the wall, of beasts also, and fowls, and creeping things, and fishes." (1 King iv. 33.) He may be able to discourse of all the secrets of fire, air, earth, and water. And yet, if he dies ignorant of Bible truths, he dies a miserable man! Chemistry never silenced a guilty conscience. Mathematics never healed a broken heart. All the sciences in the world never smoothed down a dying pillow. No earthly philosophy ever supplied hope in death. No natural theology ever gave peace in the prospect of meeting a holy God. All these things are of the earth, earthy, and can never raise a man above the earth's level. They may enable a man to strut and fret his little season here below with a more dignified gait than his fellow-mortals, but they can never give him wings, and enable him to soar towards heaven. He that has the largest share of them, will find at length that without Bible knowledge he has got no lasting possession. Death will make an end of all his attainments, and after death they will do him no good at all. A man may be a very ignorant man, and yet be saved. He may be unable to read a word, or write a letter. He may know nothing of geography beyond the bounds of his own parish, and be utterly unable to say which is nearest to England, Paris or New York. He may know nothing of arithmetic, and not see any difference between a million and a thousand. He may know nothing of history, not even of his own land, and be quite ignorant whether his country owes most to Semiramis, Boadicea, or Queen Elizabeth. He may know nothing of the affairs of his own times, and be incapable of telling you whether the Chancellor of the Exchequer, or the Commander-in-Chief, or the Archbishop of Canterbury is managing the national finances. He may know nothing of science, and its discoveries,—and whether Julius Cæsar won his victories with gunpowder, or the apostles had a printing press, or the sun goes round the earth, may be matters about which he has not an idea. And yet if that very man has heard Bible truth with his ears, and believed it with his heart, he knows enough to save his soul. He will be found at last with Lazarus in Abraham's bosom, while his scientific fellow-creature, who has died unconverted, is lost for ever. There is much talk in these days about science and "useful knowledge." But after all a knowledge of the Bible is the one knowledge that is needful and eternally useful. A man may get to heaven without money, learning, health, or friends,—but without Bible knowledge he will never get there at all. A man may have the mightiest of minds, and a memory stored with all that mighty mind can grasp,—and yet, if he does not know the things of the Bible, he will make shipwreck of his soul for ever. Woe! woe! woe to the man who dies in ignorance of the Bible! This is the Book about which I am addressing the readers of these pages to-day. It is no light matter what you do with such a book. It concerns the life of your soul. I summon you,—I charge you to give an honest answer to my question. What are you doing with the Bible? Do you read it? HOW READEST THOU?

Thinking About Ecstasy " Gradually he could hear her. Stop, she was saying, stop! And found the bed full of glass, his ankles bleeding, driven through the window of her cupola. California summer. That was pleasure. He knows about that: stained glass of the body lit by our lovely chemistry and neural ghost. Pleasure as fruit and pleasure as ambush. Excitement a wind so powerful, we cannot find a shape for it, so our apparatus cannot hold on to the brilliant pleasure for long. Enjoyment is different. It understands and keeps. The having of the having. But ecstasy is a question. Doubling sensation is merely arithmetic. If ecstasy means we are taken over by something, we become an occupied country, the audience to an intensity we are only the proscenium for. The man does not want to know rapture by standing outside himself. He wnats to know delight as the native land he is.

I was intrigued and stopped to listen to what he had to say. “Well, Tom, it sure sounds like you’re busy with your job. I’m sure they pay you well but it’s a shame you have to be away from home and your family so much. Hard to believe a young fellow should have to work sixty or seventy hours a week to make ends meet. Too bad you missed your daughter’s dance recital.” He continued, “Let me tell you something, Tom, something that has helped me keep a good perspective on my own priorities.” And that’s when he began to explain his theory of a “thousand marbles.” “You see, I sat down one day and did a little arithmetic. The average person lives about seventy-five years. I know, some live more and some live less, but on average, folks live about seventy-five years. “Now then, I multiplied 75 times 52 and I came up with 3,900 which is the number of Saturdays that the average person has in their entire lifetime. Now stick with me Tom, I’m getting to the important part. “It took me until I was fifty-five years old to think about all this in any detail,” he went on, “and by that time I had lived through over twenty-eight hundred Saturdays. I got to thinking that if I lived to be seventy-five, I only had about a thousand of them left to enjoy. “So I went to a toy store and bought every single marble they had. I ended up having to visit three toy stores to round up 1,000 marbles. I took them home and put them inside of a large, clear plastic container right here . . . next to my gear. Every Saturday since then, I have taken one marble out and thrown it away. “I found that by watching the marbles diminish, I focused more on the really important things in life. There is nothing like watching your time here on this earth run out to help get your priorities straight. “Now let me tell you one last thing before I sign off with you and take my lovely wife out for breakfast. This morning, I took the very last marble out of the container. I figure if I make it until next Saturday then I have been given a little extra time. And the one thing we can all use is a little more time. “It was nice to meet you, Tom. I hope you spend more time with your family, and I hope to meet you again here on the band.” You could have heard a pin drop on the band when this fellow signed off. I guess he gave us all a lot to think about.

Science might maintain the quantitative constancy of matter, but the so-called matter is mere abstraction. To say matter is changeless is as much as to say 2 is always 2, changeless and constant, because the arithmetical number is not more abstract than the physiological matter.

Small, Medium, Large, Extra Large Think of these plants as if they were shirt sizes. Shirts come in all four sizes: small, medium, large, and extra large, and so do our plants. It’s that simple. The extra large, of course, are those that take up the entire square foot—plants like cabbages, peppers, broccoli, cauliflower, and geraniums. Next are the large plants—those that can be planted four to a square foot, which equals 6 inches apart. Large plants include leaf lettuce, dwarf marigolds, Swiss chard, and parsley. Several crops could be one per square foot if you let it grow to its full size or it can be planted four per square foot if you harvest the outer leaves throughout the season. This category includes parsley, basil, and even the larger heads of leaf lettuce and Swiss chard. Using the SFG method, you snip and constantly harvest the outer leaves of edible greens, so they don’t take up as much space as in a conventional garden. Medium plants come next. They fit nine to every square foot, which equals 4 inches apart. Medium plants include bush beans, beets, and large turnips. Another way to get the proper spacing and number per square foot is to be a little more scientific and do a little arithmetic as shown below. You can see that one, four, nine, or sixteen plants should be spaced an equivalent number of inches apart. This is the same distance the seed packet will say “thin to.” Of course we don’t have to “thin to” because we don’t plant a whole packet of seeds anymore. So if you’re planting seeds, or even putting in transplants that you purchased or grew from seed, just find the seed packet or planting directions to see what the distance is for thinning. This distance then determines whether you’re going to plant one, four, nine, or sixteen plants.

Apart from minor grounds on which Kant's philosophy may be criticized, there is one main objection which seems fatal to any attempt to deal with the problem of a priori knowledge by his method. The thing to be accounted for is our certainty that the facts must always conform to logic and arithmetic. To say that logic and arithmetic are contributed by us does not account for this. Our nature is as much a fact of the existing world as anything, and there can be no certainty that it will remain constant. It might happen, if Kant is right, that to-morrow our nature would so change as to make two and two become five. This possibility seems never to have occurred to him, yet it is one which utterly destroys the certainty and universality which he is anxious to vindicate for arithmetical propositions. It is true that this possibility, formally, is inconsistent with the Kantian view that time itself is a form imposed by the subject upon phenomena, so that our real Self is not in time and has no to-morrow. But he will still have to suppose that the time-order of phenomena is determined by characteristics of what is behind phenomena, and this suffices for the substance of our argument.

He set to work to exercise himself in crimestop. He presented himself with propositions —‘the Party says the earth is flat’, ‘the party says that ice is heavier than water’— and trained himself in not seeing or not understanding the arguments that contradicted them. It was not easy. It needed great powers of reasoning and improvisation. The arithmetical problems raised, for instance, by such a statement as ‘two and two make five’ were beyond his intellectual grasp. It needed also a sort of athleticism of mind, an ability at one moment to make the most delicate use of logic and at the next to be unconscious of the crudest logical errors. Stupidity was as necessary as intelligence, and as difficult to attain.

Early education,” says President Thwing, “occupies itself with description (geometry, space, arithmetic, time, science, the world of nature). Later education with comparison and relations.

Early education,” says President Thwing, “occupies itself with description (geometry, space, arithmetic, time, science, the world of nature). Later education with comparison and relations.” If one asks, “Why not both together? Why learn facts at one time and their relations at another? Is it not the most vital possible way to learn facts to learn them in their relations?”—the answer that would be generally made reveals that most teachers are pessimists, that they have very small faith in what can be expected of the youngest pupils. The theory is that interpretative minds must not be expected of them. Some of us find it very hard to believe as little as this, in any child. Most children have such an incorrigible tendency for putting things together that they even put them together wrong rather than not put them together at all. Under existing educational conditions a child is more of a philosopher at six than he is at twenty-six. The third stage of education for which Dr. Thwing partitions off the human mind is the “stage in which a pupil becomes capable of original research, a discoverer of facts and relations” himself. In theory this means that when a man is thirty years old and all possible habits of originality have been trained out of him, he should be allowed to be original. In practice it means removing a man’s brain for thirty years and then telling him he can think.

Early education,” says President Thwing, “occupies itself with description (geometry, space, arithmetic, time, science, the world of nature). Later education with comparison and relations.” If one asks, “Why not both together? Why learn facts at one time and their relations at another? Is it not the most vital possible way to learn facts to learn them in their relations?”—the answer that would be generally made reveals that most teachers are pessimists, that they have very small faith in what can be expected of the youngest pupils. The theory is that interpretative minds must not be expected of them. Some of us find it very hard to believe as little as this, in any child. Most children have such an incorrigible tendency for putting things together that they even put them together wrong rather than not put them together at all. Under existing educational conditions a child is more of a philosopher at six than he is at twenty-six.

The notion of the continuum which is the only well-known example of a non-denumerable set, that is to say a set of which the mathematicians have a clear idea in common (or believe to have in common which in practice amounts to the same thing). I regard that notion as acquired from the geometric intuition of the continuum; it is well-known that the complete arithmetical notion of the continuum requires that one admits the legitimacy of an infinity of countably many successive arbitrary choices.

Weakness—the very word seems to be condemnatory. But what is weakness? A flower is weak. A rock by the side of the flower is very strong. Would you like to be like a rock, or would you like to be like a flower? A flower is weak, remember, very weak—just a small wind and the flower will be gone. The petals will fall to the earth. A flower is a miracle; it is a miracle how the flower exists. So weak, so soft! Seems to be impossible—how is it possible? Rocks seem to be okay; they exist, they have their arithmetic to exist. But the flower? It seems to be completely unsupported—but still a flower exists; that’s the miracle. Would you like to be like a flower? If you ask, deep down your ego will say, “Be like a rock.” And even if you insist, because a rock looks ugly, then the ego will say, “If you want to be a flower, be at least a plastic flower. Be at least strong! Winds won’t disturb you, rains won’t destroy you, and you can remain forever and ever.” A real flower comes in the morning, laughs for a moment, spreads its fragrance, and is gone. An unreal flower, a plastic flower, can remain forever and ever. But it is unreal, and it is strong because it is unreal. Reality is soft and weak. And the higher the reality is, the softer.

A good example is overflow detection on arithmetic, or providing bignums instead of just letting 32-bit integers wrap around. Now, implementing those is more expensive but I believe that providing full-blown bignums is a little less error-prone for some kinds of programming. A trap that I find systems programmers and designers of operating-systems algorithms constantly falling into is they say, "Well, we need to synchronize some phases here so we're going to use a take-a-number strategy. Every time we enter a new phase of the computation we'll increment some variable and that'll be the new number and then the different participants will make sure they're all working on the same phase number before a certain operation happens." And that works pretty well in practice, but if you use a 32-bit integer it doesn't take that long to count to four billion anymore. What happens if that number wraps around? Will you still be OK or not? It turns out that a lot of such algorithms in the literature have that lurking bug. What if some thread stalls for 2 to the 32nd iterations? That's highly unlikely in practice, but it's a possibility. And one should either mitigate that correctness problem or else do the calculation to show that, yeah, it's sufficiently unlikely that I don't want to worry about it. Or maybe you're willing to accept one glitch every day. But the point is you should do the analysis rather than simply ignoring the issue. And the fact that counters can wrap around is a lurking pitfall

Livingston: What do you think makes a good hacker? Spolsky: I think what makes a good hack is the observation that you can do without something that everybody else thinks you need. To me, the most elegant hack is when somebody says, "These 2,000 lines of code end up doing the same thing as those 2 lines of code would do. I know it seems complicated, but arithmetically it's really the same." When someone cuts through a lot of crap and says, "You know, it doesn't really matter." For example, Ruby on Rails is a framework that you can use with the Ruby programming language to access databases. It is the first framework that you can use from any programming language for accessing databases to realize that it's OK to require that the names of the columns in the database have a specific format. Everybody else thought, "You need to be allowed to use whatever name you want in the database and whatever name you want in the application." Therefore you have to create all this code to map between the name in the database and the name in the application. Ruby on Rails finally said, "It's no big deal if you're just forced to use the same name in both places. You know, it doesn't really matter." And suddenly it becomes much simpler and much cleaner. To me, that is an elegant hack—saying, "This particular distinction that we used to fret over, just throw it away.

2,000–3,000 PEOPLE, NOT GENERAL FAME This is one of the messages Eric burned into my brain last year, and it’s guided many decisions since. We were sitting in a large soaking tub talking about the world (as mathematicians and human guinea pigs do in San Francisco), and he said: “General fame is overrated. You want to be famous to 2,000 to 3,000 people you handpick.” I’m paraphrasing, but the gist is that you don’t need or want mainstream fame. It brings more liabilities than benefits. However, if you’re known and respected by 2–3K high-caliber people (e.g., the live TED audience), you can do anything and everything you want in life. It provides maximal upside and minimal downside. GOOD QUESTION TO ASK YOURSELF WHEN TACKLING INCUMBENT COMPANIES (OR IDEAS) “How is their bread buttered?” “What is it that they can’t afford to say or think?” “CONSENSUS” SHOULD SET OFF YOUR SPIDEY SENSE “Somehow, people have to learn that consensus is a huge problem. There’s no ‘arithmetic consensus’ because it doesn’t require a consensus. But there is a Washington consensus. There is a climate consensus. In general, consensus is how we bully people into pretending that there’s nothing to see. ‘Move along, everyone.’ I think that, in part, you should learn that people don’t naturally come to high levels of agreement unless something is either absolutely clear, in which case consensus isn’t present, or there’s an implied threat of violence to livelihood or self.” TF: I start nearly every public presentation I give with a slide that contains one quote: “Whenever you find yourself on the side of the majority, it’s time to pause and reflect.” —Mark Twain. This isn’t just for my audience. It’s also a reminder for me.

[...] These observations will allow us to understand more precisely in what sense one can say, as we did at the beginning, that the limits of the indefinite can never be reached through any analytical procedure, or, in other words, that the indefinite, while not absolutely and in every way inexhaustible, is at least analytically inexhaustible. In this regard, we must naturally consider those procedures analytical which ,in order to reconstitute a whole, consist in taking its elements distinctly and successively; such is the procedure for the formation of an arithmetical sum, and it is precisely in this regard that it differs essentially from integration. This is particularly interesting from our point of view, for one can see in it, as a very clear example, the true relationship between analysis and synthesis: contrary to current opinion, accordng to which analysis is as it were a preparation for synthesis, or again something leading to it, so much so that one must always begin with analysis, even when one does not intend to stop there, the truth is that one can never actually arrive at synthesis through analysis. All synthesis, in the true sense of the word, is something immediate, so to speak, something that is not preceded by any analysis and is entirely indfependent of it, just as integration is an operation carried out in a single stroke, by no means presupposing the consideration of elements comparable to those of an arithmetical sum; and as this arithmetical sum can yield no means of attaining and exhausting the indefinite, this latter must, in every domain, be one of those things that by their very nature resist analysis and can be known only through synthesis.[3] [3]Here, and in what follows, it should be understood that we take the terms 'analysis' and 'synthesis' in their true and original sense, and one must indeed take care to distinguish this sense from the completely different and quite improper sense in which one currently speaks of 'mathematical analysis', according to which integration itself, despite its essentially synthetic character, is regarded as playing a part in what one calls 'infinitesimal analysis'; it is for this reason, moreorever, that we prefer to avoid using this last expression, availing ourselves only of those of 'the infinitesimal calculus' and 'the infinitesimal method', which lead to no such equivocation.

He set to work to exercise himself in crimestop. He presented himself with propositions - "the Party says the earth is flat," "the Party says that ice is heavier than water" - and trained himself in not seeing or not understand the arguments that contradicted them. It was not easy. It needed great powers of reasoning and improvisation. The arithmetical problems raised, for instance, by such a statement as "two and two make five" were beyond his intellectual grasp. It needed also a sort of athleticism of the mind, an ability at one moment to make the most delicate use of logic and at the next to be unconscious of the crudest logical errors. Stupidity was as necessary as intelligence, and as difficult to attain.

What I’m trying to say is that his calculation of π was heroic, both logically and arithmetically. By using a 96-gon inside the circle and a 96-gon outside the circle, he ultimately proved that π is greater than 3 + 10/71 and less than 3 + 10/70.

What is at issue here is not the familiar construct of formal mathematics, but a belief in the existence of ω (the set of natural numbers) prior to all mathematical constructions. What is the origin of this belief? The famous saying by Kronecker that God created the numbers, all the rest is the work of Man, presumably was not meant to be taken seriously. Nowhere in the Book of Genesis do we find the passage: And God said, let there be numbers, and there were numbers; odd and even created he them, and he said unto them be fruitful and multiply; and he commanded them to keep the laws of induction. No, the belief in ω stems from the speculations of Greek philosophy on the existence of ideal entities or the speculations of German philosophy on a priori categories of thought.

The important point here is that music for the Greeks and the wider classical tradition was not so much understood as something performed, composed, practiced, or played; rather, music was interpreted as a mathematical discipline that sought to discover and formalize the symmetrical relations between sounds.6 It was an integral component to the mathematical disciplines that comprised the quadrivium: arithmetic, geometry, music, and astronomy. For the classical mind, arithmetic revealed “number in itself,” geometry revealed “number in space,” music revealed “number in time,” and astronomy revealed “number in space and time.” In this sense, music was an integral part of the Greek educational curriculum which functioned as a metaphor for this whole cosmic chain of interrelationships and harmonies. Indeed, Plato could say: “The whole choral art is also in our view the whole of education” (Laws, Bk II). The Greeks understood the nature of reality and its systems of relations in musical terms.

As he spoke and she listened, the sounds of people talking, of children playing, became faint. The girl and he were alone under the great sailing moon. . . He told a story he was amazed to hear. What he had to say about horses seemed to have meanings pertinent to the whole world. He was clearing up mysteries for himself as he went along. If you got to the bottom of one subject, did the truth about all other subjects lie there, too? If you knew one thing fully, did you, in a way, know all? Was that the reason old farmers and coon hunters were so wise? Once before in his life he had been drunk. At the age of sixteen, he had sampled a jug of raw corn whisky. He had felt a kind of power at the time: as if he had transcended himself, were suspended above himself. This enabled him to see a lot of the world ordinarily not visible; he saw also his own smallness in this world. Now he was drunk again, but in an entirely different way. He was more himself than he had ever been before; and this was happening at the very minute when he was also more aware of another person than he had ever been before. How could this be? It contradicted all the rules of arithmetic. To give himself away and to have more left. He felt like saying his own name over and over again. . .that was who he had been, but might never be again; for this girl was making him over by listening to him. . . .it was not a one-sided conversation. . .he could never have done it without her. She taught him all his powers, showed him all his meanings. Until she asked her questions, he didn't know his answers. He had never in his life felt so radiant. She looked at him, she asked. He spoke. Something towered upward out of the interchange; together they opened up meaning he had never glimpsed before. . .

Among socialists only Saint Simon realized to some extent the position of the entrepreneurs in the capitalistic economy. As a result he is often denied the name of Socialist. The others completely fail to realize that the functions of entrepreneurs in the capitalist order must be performed in a socialist community also. This is reflected most clearly in the writings of Lenin. According to him the work performed in a capitalist order by those whom he refused to designate as ‘working’ can be boiled down to ‘Auditing of Production and Distribution’ and ‘keeping the records of labour and products’. This could easily be attended to by the armed workers, ‘by the whole of the armed people’.1 Lenin quite rightly separates these functions of the ‘capitalists and clerks’ from the work of the technically trained higher personnel, not however missing the opportunity to take a side thrust at scientifically trained people by giving expression to that contempt for all highly skilled work which is characteristic of Marxian proletarian snobbishness. ‘This recording, this exercise of audit,’ he says, ‘Capitalism has simplified to the utmost and has reduced to extremely simple operations of superintendence and book-entry within the grasp of anyone able to read and write. To control these operations a knowledge of elementary arithmetic and the drawing of correct receipts is sufficient.’2 It is therefore possible straisrhtwav to enable all members of society to do these things for themselves.3 This is all, absolutely all that Lenin had to say on this problem; and no socialist has a word more to say. They have no greater perception of the essentials of economic life than the errand boy, whose only idea of the work of the entrepreneur is that he covers pieces of paper with letters and figures.

What, precisely, does it mean to say that our sense of morality and justice is reduced to the language of a business deal? What does it mean when we reduce moral obligations to debts? What changes when the one turns into the other? And how do we speak about them when our language has been so shaped by the market? On one level the difference between an obligation and a debt is simple and obvious. A debt is the obligation to pay a certain sum of money. As a result, a debt, unlike any other form of obligation, can be precisely quantified. This allows debts to become simple, cold, and impersonal—which, in turn, allows them to be transferable. If one owes a favor, or one’s life, to another human being, it is owed to that person specifically. But if one owes forty thousand dollars at 12-percent interest, it doesn’t really matter who the creditor is; neither does either of the two parties have to think much about what the other party needs, wants, is capable of doing—as they certainly would if what was owed was a favor, or respect, or gratitude. One does not need to calculate the human effects; one need only calculate principal, balances, penalties, and rates of interest. If you end up having to abandon your home and wander in other provinces, if your daughter ends up in a mining camp working as a prostitute, well, that’s unfortunate, but incidental to the creditor. Money is money, and a deal’s a deal. From this perspective, the crucial factor, and a topic that will be explored at length in these pages, is money’s capacity to turn morality into a matter of impersonal arithmetic—and by doing so, to justify things that would otherwise seem outrageous or obscene. The factor of violence, which I have been emphasizing up until now, may appear secondary. The difference between a “debt” and a mere moral obligation is not the presence or absence of men with weapons who can enforce that obligation by seizing the debtor’s possessions or threatening to break his legs. It is simply that a creditor has the means to specify, numerically, exactly how much the debtor owes.

You buy a 6% bond at 100, convertible into stock at 25—that is, at the rate of 40 shares for each $1,000 bond. The stock goes to 30, which makes the bond worth at least 120, and so it sells at 125. You either sell or hold. If you hold, hoping for a higher price, you are pretty much in the position of a common shareholder, since if the stock goes down your bond will go down too. A conservative person is likely to say that beyond 125 his position has become too speculative, and therefore he sells and makes a gratifying 25% profit. So far, so good. But pursue the matter a bit. In many cases where the holder sells at 125 the common stock continues to advance, carrying the convertible with it, and the investor experiences that peculiar pain that comes to the man who has sold out much too soon. The next time, he decides to hold for 150 or 200. The issue goes up to 140 and he does not sell. Then the market breaks and his bond slides down to 80. Again he has done the wrong thing. Aside from the mental anguish involved in making these bad guesses—and they seem to be almost inevitable—there is a real arithmetical drawback to operations in convertible issues.

Nowhere in all this elaborate brain circuitry, alas, is there the equivalent of the chip found in a five-dollar calculator. This deficiency can make learning that terrible quartet—“Ambition, Distraction, Uglification, and Derision,” as Lewis Carroll burlesqued them—a chore. It’s not so bad at first. Our number sense endows us with a crude feel for addition, so that, even before schooling, children can find simple recipes for adding numbers. If asked to compute 2 + 4, for example, a child might start with the first number and then count upward by the second number: “two, three is one, four is two, five is three, six is four, six.” But multiplication is another matter. It is an “unnatural practice,” Dehaene is fond of saying, and the reason is that our brains are wired the wrong way. Neither intuition nor counting is of much use, and multiplication facts must be stored in the brain verbally, as strings of words. The list of arithmetical facts to be memorized may be short, but it is fiendishly tricky: the same numbers occur over and over, in different orders, with partial overlaps and irrelevant rhymes. (Bilinguals, it has been found, revert to the language they used in school when doing multiplication.) The human memory, unlike that of a computer, has evolved to be associative, which makes it ill-suited to arithmetic, where bits of knowledge must be kept from interfering with one another: if you’re trying to retrieve the result of multiplying 7 X 6, the reflex activation of 7 + 6 and 7 X 5 can be disastrous. So multiplication is a double terror: not only is it remote from our intuitive sense of number; it has to be internalized in a form that clashes with the evolved organization of our memory. The result is that when adults multiply single-digit numbers they make mistakes ten to fifteen per cent of the time. For the hardest problems, like 7 X 8, the error rate can exceed twenty-five per cent. Our inbuilt ineptness when it comes to more complex mathematical processes has led Dehaene to question why we insist on drilling procedures like long division into our children at all. There is, after all, an alternative: the electronic calculator. “Give a calculator to a five-year-old, and you will teach him how to make friends with numbers instead of despising them,” he has written. By removing the need to spend hundreds of hours memorizing boring procedures, he says, calculators can free children to concentrate on the meaning of these procedures, which is neglected under the educational status quo.

We are already acquainted with the phenomena of the growing sensitiveness of conscience. We know how we come to see sin, where we saw none before, and what a feeling of insecurity about the past that new vision has often given us. Yet death is a sudden stride into the light. Even in our General Confessions, the past was discernible in a kind of soft twilight; now it will be dragged out into unsheltered splendour. The dawn of the judgment, mere dawn though it will be, is brighter than any terrestrial noon; and it is a light which magnifies more than any human microscope. There lie fifty crowded years, or more. O, such an interminable-seeming waste of life, with actions piled on actions, and all swarming with minutest incredible life, and an element of eternity in every nameless moving point of that teeming wilderness! How colossal will appear the sins we know of, so gigantic now that we hardly know them again! How big our little sins! How full of malice our faults that seemed but half-sins, if they were sins at all. Then again, the forgotten sins, who can count them? Who believed they were half so many or half so serious? The unsuspected sins, and the sinfulness of our many ignorances, and the deliberateness of our indeliberations, and the rebellions of our self-will, and the culpable recklessness of our precipitations, and the locust-swarms of our thought-peopled solitudes, and the incessant persevering cataracts of our poisoned tongues, and the inconceivable arithmetic of our multiplied omissions—and a great solid neglected grace lying by the side of each one of these things—and each one of them as distinct, and quiet, and quietly compassed, and separately contemplated, and overpoweringly light-girdled, in the mind of God, as if each were the grand sole truth of His self-sufficing unity! Who will dare to think that such a past will not be a terrific pain, a light from which there is no terrified escape? Or who will dare to say that his past will not look such to him, when he lies down to die? Surely it would be death itself to our entrapped and amazed souls, if we did not see the waters of the great flood rising far off, and sweeping onward with noiseless, but resistless, inundation, the billows of that Red Sea of our salvation, which takes away the sins of the world, and under which all those Egyptians of our own creation, those masters whom we ourselves appointed over us, with their living hosts, their men, their horses, their chariots, and their incalculable baggage, will look in the morning- light of eternity, but a valley of sunlit waters.

...he will accompany us on a hike in the hills, leaping and whizzing back and forth, and coming when called as well as a dog. It is just that the organism, the whole pattern of nerve and muscle, is more complex and intelligent than logical systems of arithmetic, geometry and grammar - which are in fact nothing but inferior ritual. Life itself dances, for what else are trees, ferns, butterflies, and snakes but elaborate forms of dancing? Even wood and bones show, in their structure, the characteristic patterns of flowing water, which (as Lao-tzu pointed out in 400 B.C.) derives its incredible power by following gravity and seeking that "lowest level which all men abhor." When dance I do not think-count my steps, and some women say I have no sense of rhythm, but I have a daughter who (without ever having taken lessons in dancing) can follow me as if she were my shadow or I were hers. The whole secret of life and of creative energy consists in flowing with gravity. Even when he leaps and bounces our cat is going with it. This is the way the whole earth and everything in the universe beehives.* But man is making a mess of the earth be- *Harrumph! Excuse the pun, but it is important, because bees live in hexagonal as distinct from quadrilateral structures, and this is the natural way in which all things, such as bubbles and pebbles, congregate, nestling into each other by gravity. It will follow, because 2 x 6 is 12, that - as Buckminster Fuller has pointed out - as number-system to the base 12 (duodecimal) is closer to nature than one to the base of 10 (decimal). For 12 is divisible by both 2 and 3, whereas 10 is not. After all, we use the base 12 for measuring circles and spheres and time, and so can "think circles" around people who use only meters. The world is better duodecimal than decimated.

Peggy, to say, “Hey, Peg, let’s stop asking Wanda how many dresses she has.” When she finished her arithmetic, she did start a note to Peggy. Suddenly she paused and shuddered. She pictured herself in the school yard, a new target for Peggy and the girls. Peggy might ask her where she got the dress she had on, and Maddie would have to say that it was one of Peggy’s old ones that Maddie’s mother had tried to disguise with new trimmings so that no one in Room 13 would recognize it. If only Peggy would decide of her own accord to stop having fun with Wanda. Oh, well! Maddie ran her hand through her short blond hair as though to push the uncomfortable thoughts away. What difference did it make? Slowly Maddie tore the note she had started into bits. She was Peggy’s best friend, and Peggy was the best-liked girl in the whole room. Peggy could not possibly do anything that was really wrong, she thought.