|
Chapter 9--Mathematics Jack
was happy to agree to read Mishima's Forbidden Colors with
Charlie and Henry, both just beginning their reading of Mishima.
Charlie had read Jack's Mishima to Henry one afternoon in
the boat for his own first introduction to the author, and, though Henry
already knew this play, he was pleased with the enthusiasm in Charlie's
reading--which only took about an hour. Then, as all three got together,
Jack and Charlie could take turns reading from the novel, so it was not
so demanding on a single reader, before discussing what they read.
97
|
|
"I could read you one of those novellas," Charlie said.
"But no, let's do Forbidden Colors," Jack said. "It will take us a while, but I know it quite well, after having worked on the film version, and why not start with the best--that's the main reason some of us chose Lady Murasaki's The Tale of Genji when we decided to read it two years ago." Charlie said, "When I told Ralph I read Henry your Mishima the other day, he told me that, not only had Jordan played Mishima, and Ben the Student, but he himself had been Saigo Takamori, when you did the play in New York. He said he learned to do a kendo dance with Jordan that might be his best memory of him--the closest he'd ever worked with him on stage. What made you so enthusiastic about Mishima? And why tell his story as a Noh play?" Jack said, "I picked up the countess's enthusiasm for Mishima when he was the first of the Japanese authors she started me reading that year she was introducing me to the whole new world of Japanese literature. I liked him. I was particularly interested in what he did in those Modern Noh Plays, using the old stories of some of the best-known medieval Noh plays, but presenting them more like modern French plays, with their themes adjusted accordingly." "But you were also studying the classical Noh form." "Yes. And was impressed by what he did with it. Then Mishima committed seppuku in that sensational fashion while I was in Japan the first time working as script writer on a film Randall had arranged for me, and I got caught up in the furor, I suppose. By the time I got back I had sketched out the play, and was trying to do just the opposite of what Mishima had done, taking a modern story, his own, and trying to treat it the way Zeami would have, using two famous medieval plays as my models, Sotoba Komachi, which Mishima had used as the story for one of his, and Atsumori, a war story with a young samurai hero that is frequently performed--and using things 98
|
| from other plays I was reading as well. As you
know, I have Mishima coming back as a ghost, to visit a modern student
at the Kinkakuji, the setting for one of his novels--one we should read--and
appealing to the student to affirm traditional samurai values. I
even tried to write the play in traditional verse form, a twelve syllable
line with caesura off center, 5-7, or 7-5, the units we know in the haiku
or tanka."
"And Jordan Simms became a Mishima enthusiast, too, as he worked with you on the play, didn't he?" "Oh, he was already an enthusiast--or he wouldn't have done the play. But he was very effective on stage as Mishima, had found a kindred spirit, an actor as inherently dramatic as Jordan was himself, as he had proved in dramatizing his own death that way. As Christine told us, Jordan was alternating between reading the last of the Mishima novels that had been translated, Runaway Horses, and a biography of Mishima written by John Nathan, in the car earlier that day he was shot, as Christine drove the Ferrari across Arizona and California." So Henry had Jack and Charlie reading Mishima, Thomas Plato, and Shoko some of the shorter Mishima works. When enough were there, they might also read one of Mishima's Noh plays aloud, with Henry as the privileged audience. While still in New York, in that year after they had done The Tempest, Jack had begun writing The Bridge of Dreams, a novel based on his life with Betty. Each of the others that were part of the story, as he or she read parts of the book would talk to him about it--as an interested set of critics. Henry was also interested in the mathematics of the structure Jack had used, and asked, "In whatever you write you are always interested in form, aren't you Jack?" Jack said, "The only thing I knew for sure when I started that novel was that there'd be 24 chapters, each perhaps 10 pages in length, with its own beginning, middle, and end--on the model of Hawthorne's The Scarlet Letter. Hawthorne has 99
|
| a real sense of the curtain, knows how to close a scene.
Soon after that, I decided to open each chapter with a sonnet, probably
because I'd already written a sonnet on Pygmalion, and could adapt that
for what was then the first chapter."
"But why did you decide on 24 chapters?" Henry asked. "Because 24 divides by everything--2, 3, 4, 6, 8, 12. Where do we get the year, with its twelve months, the day with its 24 hours? We know night and day. We know the seasons. Both come out of earth's relationship to the sun--a miniscule planet's relationship to a minor star--but pretty fundamental for us. It's grounded in the mathematical structure of our universe, Henry," Jack said with a smile. "Since it's the number of hours in a day, if you read a chapter an hour, it lasts for a day--and a night. If you read a chapter a month it will last for two years. The sonnet opening each chapter is a personal thing--with its own mathematics, of course. Then each chapter is now exactly 20 pages long." "Why not 24--so you would have 24 times 24?" "As I got well into the composition I was considering 24 pages to each chapter, to square the book at 576--and liked that idea. But the truth is that I just like an established form, for the security that it gives me--like the chessboard does you. But when I then reformatted the novel to cut 1000 words from each chapter, it gave me close to 20 pages a chapter, 480 pages in all, and I settled for that." "Then the action of the book itself takes exactly 20 years--why is that?" Charlie asked. "Because that's how long I knew Betty, and how long it took for Christine to grow up to become the Betty I first knew--in this case a generation. As Thomas knows when he says Christine is his princess, she isn't yet what Betty became in those 20 years, or what the countess became in her 80." Henry laughed at this. "But 24 is using a base of 12 and 20 a base of 10--how do you reconcile those two units?" 100
|
|
"We deal with most things in units of 10, because we have 10 fingers,
I suppose. Is this why 10 syllables becomes the unit for Shakespeare,
when the sonnet and the standard line of verse had come over from France
and Italy as 12? Yet we use the England 'foot' to measure, which
is 12 inches, which poses problems for the decimal system--yet tend to
let feet accumulate in groups of ten--or three. So we use both.
It would be possible to use almost any number as base, like any alphabet
will work for the written form of a language. The Japanese started
with Chinese kanji, but can now type their language in Roman letters, you
know."
Christine had became most interested in the novel as the story of her life, from conception, to birth, to coming to California as a baby, to moving to New York as a 12-year-old, to going to France one summer, to her theatre experience. Then, she liked it as the story of her becoming her mother, as she first comes to look like Betty, and then begins to follow her in her commitment to the theatre as a career. Jack sent query letters on his novel out to about 40 publishers across the country to find out they were only interested in novels submitted by agents, and then that agents were only interested in novelists who had a track record. Jack had a record as a film writer, but had never been nominated for an Oscar, and had never written a novel before. He finally got a nibble from a small publisher in the Bay area who agreed to publish the work if Jack paid half the cost and cut it to about half its length. He did trim that 1000 words from each of the 24 chapters and reformatted it to book size. He then signed a contract with that publisher for 2500 copies at $10,000--supposedly half the cost of publication. Then there were delays. Six months later the publisher informed Jack that he was going out of business, considering bankruptcy--which he had evidently known when he took the money. Not having any luck getting the novel published, Jack thought he might do something with some of the pieces, and, 101
|
| among other things, began to work on a film script for
the experience with Marvin Crawford in making Hostages on Horseback,
chapter 12 in the novel. From the beginning, he had Charlie in mind.
Soon after Charlie had come back to California, with Miss Julie,
he’d decided that he'd lived with his parents long enough, and arranged
to rent the house Jack and Laura had lived in in North Hollywood, and still
owned. Then, when they had stayed in California, Marcella had moved
in with him, which had surprised most people. Now they were talking
with Randall Best about the two of them starring in a film together
for the first time--doing Jack’s script, with Charlie playing Marvin and
Marcella Charlene.
One day, when they were visiting Henry and Shoko at the lake, Henry suggested casting an I Ching on what he could expect in solving the shooting of Ben, and perhaps writing it up as a crime novel--mostly to show Charlie how the system worked. He explained that the I Ching is, in one sense, a method for exploring the subliminal knowledge you may have to bring to bear on a problem, reducing the frustrating complexity of the world of becoming and change to a system simple enough that you think you begin to understand it. Henry had been working most especially with Shoko on mathematics, on manipulating numbers, and using an abacus, which she'd used since she was a child in Japan. He'd been working with it long enough now that it was becoming second nature in computational problems. Henry liked the way numbers helped him to order his experience, "much as any toddler, like Hajime, begins to abstract numbers, using his fingers to count more than fingers." As chess provides a "Clean, Well-Lighted Place" for a blind man, all living things, all animals, all insects, need a framework, a pattern they understand, to act, within their range of competence, and the knowledge provided by their senses, with more confidence. "We can all use the help of tools like this," he said. 102
|
|
"For a blind man, the number 64 in both chess and the I Ching
provides
the same kind of ordered structure, a nicely defined quantity based not
on 10, or on 12, but on 2--you can go 2, 4, 8, 16, 32, 64, or squaring
2 to 4 to 16, 8 to 64.
"You have 8 x 8 so 64 positions on the chess board, and visualize the possible moves of the pieces in that network. It's no accident that that's the basic number in the I Ching, 82. But there the principle of multiplication is more obvious, as with human reproduction. We pair up, because that's the way reproduction works--by twos not threes. If you take the yin and yang--the symbol in Chinese philosophy of dichotomy, of black and white, man and woman--each has the seed of the other in it. Square that and you have the four seasons, cube it and you have the eight possible trigrams, to the 6th power and you have the 64 hexagrams, the number of squares on a chess board. The computer works on the same principle, as does the human race--now at 4 billion and climbing, I hear--based on a binary system. The Chinese language is very different from English, but their mathematics and ours has to be the same--for, like us, they have ten fingers, and, like us, are searching for abstract truths, trying to discover the eternal verities that we all recognize when we once see them--American, Chinese, or Afghanistani. That may be seen as the function of education." Charlie asked Henry, "When you say you expect the three elements of your study--Plato, Calculus, and the I Ching--to become one, allowing you look up and see the Idea of the Good, I understand the Republic to be the source of that mystical abstraction, but where in Plato is the connection with the mathematics that are common to the other two?" Henry said, "You found it right away! What attracted you to Plato, there in the Meno--the Pythagorean Theorem--which we have in geometric, and so visual, terms in that dialogue--which the slave boy can see, and which a blind 103
|
| man can see--which Socrates uses to show the relationship
of the mortal to the immortal--which I like. It is Socrates the Mystic
who most appeals to me--in this dialogue, in his explanation of Diotima's
theory of love in the Symposium, and in the faith he affirms
at the time of his death in the Phaedo. It is a kind
of Unified Theory for him--he affirms all Knowledge as One. I say
that Socrates is my primary role model, but I have also come to think of
myself, in this quest, as a Pythagorean, as, I would argue, Socrates was
himself."
"Then why take Socrates as your role model instead of his teacher, Pythagoras?" Marcella asked. "Because, thanks to Plato, I know Socrates much better than I know any other philosopher--as a literary experience, the way I know Huckleberry Finn, for example--much better than I do Plato or Aristotle. He seems a friend as well as the incarnation of the philosophical life. We know almost nothing about the historical Pythagoras, while we know more about Socrates than we do about our next door neighbors." "What do we know about Pythagoras?" Charlie asked. "Pythagoras himself has been credited with three great developments in mathematics: he may have been the first to consider the earth one of the planets in analyzing heavenly movements, the first to reduce harmonics to mathematical principles, the relationships of sounds dependent on the frequency of the vibrations, and gave us the Pythagorean theorem, which gathers together a lot of what is central in plane geometry--comes at the end of the first chapter of Euclid’s Elements, summing up the analysis of triangles." "But we have no historical evidence for this?" "Not much, but at least those developments are associated with his school, which he established at Croton, in Southern Italy, about 532 BC. It was entirely grounded in a mathematical curriculum, as a kind of mystical religion, so that harmony becomes 'the music of the spheres,' and, in the 104
|
| Meno, the Pythagorean Theorem becomes a
prime example of a universal truth. Most of the departments
in a modern university were first established as disciplines by the Greeks
in this classical period, but his school seems the first in history to
have an established curriculum--a quadrivium: arithmetic, geometry, astronomy,
and harmonics (or music)--so was an important model for Plato's Academy.
For the Pythagoreans it involved a commitment to the life of the mind,
so the members of the school had an Orphic dedication, as Socrates does
at his best--and is what I seek for myself."
"But what did he, or Plato, contribute to mathematics?" "Plato does seem to have modeled the Academy, founded roughly 388 BC, on the school of Pythagoras, and the Academy may be as important in the history of mathematics as the dialogues are in the history of philosophy. Over the gate of the Academy it is supposed to have read, 'Let no one without geometry enter here.' So Plato comes right in the center, in time and place, between Pythagoras and his school in Southern Italy, and Euclid, and the Elements, which summed up Greek mathematics, in the Alexandria that had become the intellectual center by 300 BC. Euclid may, in fact, have been a student at the Academy in his youth." "The curriculum Plato himself presents for his guardians in the Republic seems to have much in common with that of the Pythagorean school, in fact, moving through a hierarchy of mathematical disciplines, and he likes to engage in the mathematical calculations of things like the ideal number for the population for the ideal state," Marcella said. "And, as you have pointed out, in the later dialogues we meet a number of important mathematicians," Charlie remarked, "like Zeno and Theaetetus." "As with most of the characters we meet in all the dialogues--the sophists and visiting celebrities in the early dialogues--(of whom Diotima may have been one, but 105
|
| certainly Protagoras, Gorgias, Ion) all were historical
figures, as were all the other characters in the Symposium--Agathon,
Aristophanes, Alcibiades. So with Zeno and Theaetetus, and other
mathematicians, in the later dialogues. Theaetetus was a member of
the Academy, and would become an important link in the development of geometry
between the late Pythagoreans, and Parmenides and Zeno of the Eleatic School,
and Euclid and the Elements, the summing up of 300 years
of Greek geometry, written at Alexandria about 100 years after Socrates
drank the hemlock."
"The beauty of geometry is that it is a visual way of idealizing a set of values," Charlie said. "You start with that point, which you say has no dimension, though you think you see it there on the board, or on your sheet of paper. Then two points determine a straight line, which can evidently go on to infinity, which, again, you can imagine as an idea, and about which an infinite number of planes can be established, though once you have parallel lines you have established a plane as well. We are always working with these concepts in finite terms, the line from point a to point b in a single plane. Three points determine a plane, and a triangle, and a circle circumscribing that triangle on that plane." "We think of a sheet of paper as a plane," Marcella said, "but can't make it perfectly flat, or draw a perfect right triangle, or a perfect circle on it--but we do draw these figures, perhaps with a pair of dividers and a ruler, geometry's tools, to give us the illusion of creating those ideal figures, which are certainly then easier to visualize than the Idea of the Good. We see three intersecting lines determining a triangle, four equal lines, with four right angles, a square, five lines a pentagon, six a hexagon, eight an octagon, until as you approach infinity in number of lines you employ in creating a regular polygon, you also approach a circle--an important principle in the Calculus, as I have been told." 106
|
|
"And Theaetetus was important there," Henry said. "Shoko has
been doing my research, and I'll have her read what she found out about
his contributions to geometry."
Shoko took a spiral notebook from the shelf and read, "'Theaetetus is credited with being one of the first mathematicians to begin the systematic study of types of irrational numbers called 'quadratic surds,' and with completing the theory of the 'regular solids' by adding to the three known by the Pythagoreans (tetrahedron, cube, and dodecahedron) the remaining two (octahedron, icosahedron), which Euclid analyzes near the end of the Elements.' I could find the book and read more--maybe from Euclid--explaining those two contributions, if you'd like me to." Henry shook his head. "That would take more geometry than any of us probably brought to this meeting to understand, and there is none of that in Plato's dialogue the Theaetetus, where he just appears as a promising young man. And Zeno appears in the Parmenides without any technical analysis of his famous paradoxes, provoked by Eleatic school as a challenge aimed at Pythagorean principles, I believe." "And definitely over my head," Marcella said, with a smile. "But, if I may change the subject, I've been reading the Symposium, which you two also know much better, as you get ready to do Jack's dramatic adaptation in New York, and I'd like to know how a woman is given the most important speech at that men's party. I think Socrates' relationship to Diotima is the most interesting thing in that dialogue." "To characterize Diotima as well as Aristophanes is part of Plato's dramatic genius. She was evidently historical, too--and a mystic--who visited Athens when Socrates was young, a priestess helping to cure the plague, who may have taught something like the doctrine Socrates attributes to her. Socrates might have presented this as his own theory of love--but instead presents a fascinating character as his teacher." 107
|
|
"And certainly one who finds the desire of mortality for immortality
the force that generates love--which might also apply to your quest," Marcella
remarked.
"Mathematics starts with practical problems, counting and measuring things. But, in all this, we are working toward the Calculus, discovered, as Socrates would say, almost simultaneously by Newton and Leibniz, shortly after Descartes had discovered the use of the two-dimensional graph to bring algebra and geometry together in Analytical Geometry, making the beauty and wonder of mathematics even more available to the inner eye of a blind man." "In the Enlightenment. But what does that add?" "Well, Charlie, we generally accept that scientists are not inventing things--they discover what always was, and always will be, true, as with the slave boy in the Meno. The central problem of the Calculus, as with the I Ching, is to deal with change, in this case under two aspects--the differential and the integral--the one establishing limits in approaching the infinitesimal, going beyond the microscope--the other, going beyond the telescope in approaching infinity. The paradoxes of Zeno in this context function rather like Kant's antinomies--two contradictory things both seen to be true. Both Newton and Einstein assumed they could discover the Unified Field Theory to reconcile the microcosm with the macrocosm. Newton was himself a Pythagorean, I would say. He evidently thought Pythagoras had discovered the inverse square law of gravity, that the model he had developed for musical harmony was a metaphor or model for the universe." "And where does the I Ching come in?" Marcella asked. "The problem the ancient Chinese addressed in the I Ching, too, is how to come to terms with change, reconciling the chance elements in experience to the eternal cycles we observe, which form the parameters of our understanding. We'll cast an I Ching soon, and consider its mathematics." 108 |