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"Tell the chef, the beer is on me."
I first published the novella A Clockwork Orange in 1962, which ought to be far enough in the past for it to be erased from the world’s literary memory. It refuses to be erased, however, and for this the film version of the book made by Stanley Kubrick may be held chiefly responsible. I should myself be glad to disown it for various reasons, but this is not permitted. I receive mail from students who try to write theses about it, or requests from Japanese dramaturges to turn it into a sort of Noh play. It seems likely to survive, while other works of mine that I value more bite the dust. This is not an unusual experience for an artist. Rachmaninoff used to groan because he was known mainly for a Prelude in C Sharp Minor which he wrote as a boy, while the works of his maturity never got into the programmes. Kids cut their pianistic teeth on a minuet in G which Beethoven composed only so that he could detest it. I have to go on living with A Clockwork Orange, and this means I have a sort of authorial duty to it. I have a very special duty to it in the United States, and I had better now explain what this duty is.
Let me put the situation baldly. A Clockwork Orange has never been published entire in America. The book I wrote is divided into three sections of seven chapters each. Take out your pocket calculator and you will find that these add up to a total of twenty-one chapters. 21 is the symbol of human maturity, or used to be, since at 21 you got the vote and assumed adult responsibility. Whatever its symbology, the number 21 was the number I started out with. Novelists of my stamp are interested in what is called arithmology, meaning that number has to mean something in human terms when they handle it. The number of chapters is never entirely arbitrary. Just as a musical composer starts off with a vague image of bulk and duration, so a novelist begins with an image of length, and this image is expressed in the number of sections and the number of chapters into which the work will be disposed. Those twenty-one chapters were important to me.
But they were not important to my New York publisher. The book he brought out had only twenty chapters. He insisted on cutting out the twenty-first. I could, of course, have demurred at this adn taken my book elsewhere, but it was considered that he was being charitable in accepting the work at all, and that all other New York, or Boston, publishers would kick out the manuscript on its dog-ear. I needed money back in 1961, even the pittance I was being offered as an advance, and if the condition of the book’s acceptance was also its truncation — well, so be it. So there is a profound difference between A Clockwork Orange as Great Britain knows it and the somewhat slimmer volume that bears the same name in the United States of America.
Let us go further. The rest of the world was sold the book out of Great Britain, and so most versions — certainly the French, Italian, Spanish, Catalan, Russian, Hebrew, Rumanian, and German translations — have the original twenty-one chapters. Now when Stanley Kubrick made his film — though he made it in England — he followed the American version and, so it seemed to his audiences outside America, ended the story somewhat prematurely. Audiences did not exactly clamour for their money back, but they wondered why Kubrick left out the dénouement. People wrote to me about this — indeed much of my later life has been expended on Xeroxing statements of intention and the frustration of intention — while both Kubrick and my New York publisher coolly bask in the rewards of their misdemeanour. Life is, of course, terrible.
What happens in that twenty-first chapter? You now have the chance to find out. Briefly, my young thuggish protagonist grows up. He grows bored with violence and recognises that human energy is better expended on creation than destruction. Senseless violence is a prerogative of youth, which has much energy but little talent for the constructive. Its dynamism has to find an outlet in smashing telephone kiosks, derailing trains, stealing cars and smashing them and, of course, in the much more satisfactory activity of destroying human beings. There comes a time, however, when violence is seen as juvenile and boring. It is the repartee of the stupid and ignorant. My young hoodlum comes to the revelation of the need to get something done in life — to marry, to beget children, to keep the orange of the world turning in the rookers of Bog, or hands of God, and perhaps even create something — music, say. After all, Mozart and Mendelssohn were composing deathless music in their teens or nadsats, and all my hero was doing was razrezzing and giving the old in-out. It is with a kind of shame that this growing youth looks back on his devastating past. He wants a different kind of future.
There is no hint of this change of intention in the twentieth chapter. The boy is conditioned, then deconditioned, and he forsees with glee a resumption of the operation of free and violent will. “I was cured all right,” he says, and so the American book ends. So the film ends too. The twenty-first chapter gives the novel the quality of genuine fiction, an art founded on the principle that human beings change. There is, in fact, not much point in writing a novel unless you can show the possibility of moral transformation, or an increase in wisdom, operating in your chief character or characters. Even trashy bestsellers show people changing. When a fictional work fails to show change, when it merely indicates that human character is set, stony, unregenerable, then you are out of the field of the novel and into that of the fable or the allegory. The American or Kubrickian Orange is a fable; the British or world one is a novel.
But my New York publisher believed that my twenty-first chapter was a sellout. It was veddy veddy British, don’t you know. It was bland and showed a Pelagian unwillingness to accept that a human being could be a model of unregenerable evil. The Americans, he said in effect, were tougher than the British and could face up to reality. Soon they would be facing up to it in Vietnam. My book was Kennedyan and accepted the notion of moral progress. What was really was a Nixonian book with no shred of optimism in it. Let us have evil prancing on the page and, up to the very last line, sneering in the face of all the inherited beliefs, Jewish, Christian, Muslim and Holy Roller, about people being able to make themselves better. Such a book would be sensational, and so it is. But I do not think it is a fair picture of human life.
I do not think so because, by definition, a human being is endowed with free will. He can use this to choose between good and evil. If he can only perform good or only perform evil, then he is a clockwork orange — meaning that he has the appearance of an organism lovely with colour and juice but is in fact only a clockwork toy to be wound up by God or the Devil or (since this is increasingly replacing both) the Almighty State. It is as inhuman to be totally good as it is to be totally evil. The important thing is moral choice. Evil has to exist along with good, in order that moral choice may operate. Life is sustained by the grinding opposition of moral entities. This is what the television news is all about. Unfortunately there is so much original sin in us all that we find evil rather attractive. To devastate is easier and more spectacular than to create. We like to have the pants scared off us by visions of cosmic destruction. To sit down in a dull room and compose the Missa Solemnis or The Anatomy of Melancholy does not make headlines or news flashes. Unfortunately my little squib of a book was found attractive to many because it was as odorous as a crateful of bad eggs with the miasma of original sin.
It seems priggish or pollyannaish to deny that my intention in writing the work was to titillate the nastier propensities of my readers. My own healthy inheritance of original sin comes out in the book and I enjoy raping and ripping by proxy. It is the novelist’s innate cowardice that makes him depute to imaginary personalities the sins that he is too cautious to commit for himself. But the book does also have a moral lesson, and it is the weary traditional one of the fundamental importance of moral choice. It is because this lesson sticks out like a sore thumb that I tend to disparage A Clockwork Orange as a work too didactic to be artistic. It is not the novelist’s job to preach; it is his duty to show. I have shown enough, though the curtain of an invented lingo gets in the way — another aspect of my cowardice. Nadsat, a Russified version of English, was meant to muffle the raw response we expect from pornography. It turns the book into a linguistic adventure. People preferred the film because they are scared, rightly, of language.
I don’t think I have to remind readers what the title means. Clockwork oranges don’t exist, except in the speech of old Londoners. The image is a bizarre one, always used for a bizarre thing. “He’s as queer as a clockwork orange” meant he was queer to the limit of queerness. It did not primarily denote homosexuality, though a queer, before restrictive legislation came in, was the term used for a member of the inverted fraternity. Europeans who translated the title as Arancia a Orologeria or Orange Mécanique could not understand its Cockney resonance and they assumed that it meant a hand grenade, a cheaper kind of explosive pineapple. I mean it to stand for the application of a mechanistic morality to a living organism oozing with juice and sweetness.
Readers of the twenty-first chapter must decide for themselves whether it enhances the book they presumably know or is really a discardable limb. I meant the book to end in this way, but my aesthetic judgments may have been faulty. Writers are rarely their own best critics, nor are critics. “Quod scripsi scripsi,” said Pontius Pilate when he made Jesus Christ the King of the Jews. “What I have written I have written.” We can destroy what we have written but we cannot unwrite it. I leave what I wrote with what Dr. Johnson called frigid indifference to the judgment of that .00000001 of the American population which cares about such things. Eat this sweetish segment or spit it out. You are free.
Anthony Burgess, November 1986
Cowboy skateboarding drinking hot coffee when it’s 100 degrees outside.
Source: Andrew Brown - Weird Stuff You’ll Only See in/near Tucson
From “Proofs Without Words” by Roger Nelsen.
Okay but like, there is so much more to be said about this. Like we can count the number of different ways to pack a hexagon with calissons. And it turns out the number of packings is also the same as drawing paths in the hexagon on the right, going from the left side to the right side of the hexagon, visiting only white and grey calissons, while avoiding the black ones and without crossing paths. So in the example above, we can draw 5 paths from left to right, where each one kind of wiggles up and down, and none of them overlap. The number of such paths is given by the determinant of a simple matrix, where entries come from Pascal’s Triangle.
For the case of a hexagon with side length 5, we’re looking at the determinant of the matrix below, which has entries from the 11th (or 10th, depending how you index) row of Pascal’s Triangle:
The determinant is this stupidly complicated formula for a bunch of numbers that are arranged in a square. You might have seen it briefly in high school algebra when studying systems of linear equations and Cramer’s rule. As it turns out, the determinant has a whole bunch of amazing uses for counting things, like how many ways can you put calissons on a hexagon. By the way, the answer for the above hexagon of size 5 is 267,227,532 packings.
Another thing these packings count is something called a plane partition. In genera, a plane partition is basically just stacking a bunch of blocks into the corner of a box with no gaps and everything is a smooth hill. Look below:
Now, the packings of hexagons do not count all plane partitions. Instead, they count the ones where the blocks cover a square at the bottom, and the height of each pile is no larger than the size of this square. In case that’s as clear as mud, check out this huge example:
Boy, that sure does look like a hexagon packing, doesn’t it? By the way, that hexagon has a side length of 30. Thanks to the power of math (and computers to do boring arithmetic), we can find that the above picture is just one of 4,165,457,904,340,919,485,420,942,168,075,326,611,541,373,543,430,798,671,358,047,428,426,506,925,547,438,251,610,506,720,329,700,499,354,829,616,209,137,772,863,162,211,410,372,528,376,990,867,545,521,482,362,075,056,103,720,140,710,647,411,783,904,290,355,185,482,877,153,598,793,526,763,431,130,749,282,194,115,544,782,656,143,919,020,194,290,255,761,694,152,890,975,799,253,442,233,181,825,888,394,674,176 such packings.
Now here’s another thing to knock your socks off. That determinant from way back? It has another, simpler form:
Translation: take every triple of three numbers i, j, and k from 1 to n, write out and evaluate that fraction, and multiple all the results together. So for a simple case of n = 2 we get:
Which then gives us:
I don’t know about you guys, but this kind of stuff is the coolest fucking shit ever to me.
By the way, if you wanna learn more about this stuff, I highly recommend Federico Ardila’s lectures on enumerative combinatorics. In particular, he talks about these packings in lectures 20 and 21 along with some other really cool stuff like Mayan diamond packings, the square ice model, and even a cameo from Lewis Carroll!
“ Mathematics is applied to the real world, and has proved fruitful. This suggests that the mathematical and empirical parts are in harmony and that the real world is also beautiful. Otherwise, mathematics would be just an ornament and the real world would be like an ugly body in beautiful clothing. ”—
5 Greatest a Discoveries in Mathematics #3
Godel’s Incompleteness Theorem
At the beginning of the 20th century, mathematician David Hilbert presented a list of 23 of the most important unsolved problems in mathematics. Second on his list was to prove that the axioms of arithmetic are consistent i.e. free from internal contradictions.
It may appear obviously true, yet in a groundbreaking 1931 paper that unified logic, mathematics and philosophy, Gödel is widely believed to have proven Hilbert’s problem in the negative. In fact, his incompleteness theorems went so far as to show that in any axiomatic system, such as arithmetic, there exist statements that are undecidable. An imperfect but useful analogy is found within the liar paradox. In this paradox we begin with a machine that can be fed any statement and outputs whether that statement is true or false with unfailing accuracy. Now consider inputting the statement:“this statement is false”. The machine could output neither “true” nor “false” without producing a contradiction - the equivalent of an undecidable proposition.
The results from Gödel’s theorems still resonate within many areas of mathematics and philosophy and have been extended, perhaps speciously at times, to speculate on the philosophical limitations of computational systems and even the human mind itself.
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