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Ten ... This number was of old held high in honor, for such is the number of fingers by which we count. OVID
INTENT ON SEEING THE SUN RISE FROM THE TOP OF MT. SNOWDON, the young William Wordsworth set out on a climb one evening two centuries ago with a friend and a shepherd guide. It was a close, warm summer night, the fog hanging low, air dripping with moisture. Beginning from a cottage at the mountain's base, the trio climbed in silence as the mists surrounded them. The poet's head was bent earthward, as if, he writes, it were set against an enemy. He was lost in thought, negotiating rocks and paths, panting breathlessly, leading the way through the midnight hours. Gradually, though dawn had not yet come, the ground at the poet's feet began to brighten. With each step the light increased. There was hardly time to ask or learn the cause, when suddenly—"Lo!" the poet cries in biblical fashion—he looked up and there was the moon
hung naked in a firmament
Of azure without cloud, and at my feet
Rested a silent sea of hoary mist.
From the mountain peak, the poet saw a vast sea of vapors below him, stretching out to the ocean, while the sky above was unclouded, the full moon illuminating the "ethereal vault." All was silent, save for a breach in the mist, a blue chasm not far off, a "breathing-place" whence came a "roar of waters, torrents, streams / Innumerable, roaring with one voice!" heard over the whole earth and sea and seemingly felt by the starry heavens.
When the scene dissolved and the poet thought about what he had seen, it seemed to him to be an image,
Of a majestic intellect, its acts
And its possessions, what it has and craves,
What in itself it is, and would become.
The moon hanging over the mists, the light above, the sound below, the dark abyss and the silent sky—"There I beheld," the poet writes, "the emblem of a mind." The poet spins out his image of the mind, the relations of its parts represented by the moon and the waters, its powers resembling those of human imagination, and exerting profound influence on its thoughts and creations. The mind's creations can possess such mastery, he asserts, that they can catch even the creators by surprise,
Like angels stopped upon the wing by sound
Of harmony from Heaven's remotest spheres.
The journey of the poet, the laborious climb through darkness and silence, should be familiar to anyone who has attempted to understand what seems first clouded in mist: the discomfort of the still air, the awkward pace on narrow paths, the isolated broodings of the climb. This book promises no less, but it hopes to provide something more, some hint of brightening by journey's end, some vision of the expanse and vistas that have opened to those who have made such journeys their lifework, some inkling of the powers and forms that compose these emblems of mind.
There are two paths to be negotiated here—each with its own twists and treacherous turns, each with its separate maps and resting places. The paths are those of music and mathematics, and the claim that they are similar, or at the very least related, has become a commonplace—as has the claim for the vast illumination they offer to those who pledge themselves to the climb. But it is a commonplace shrouded in mystery. Connections between the two have had almost no importance for the development either of math or of music; mostly any relationships have been irrelevant to their practitioners and creators, and mystically vague to everyone else. They are simply accepted without explanation or discussion, without even realizing what an unlikely pairing these two arts are. Why should there be any links at all? What do these activities really share? Do they share meanings or techniques or ideas? Why should we even suspect that they lead to a similar destination, let alone reveal similar visions? And if they are similar, is it simply because they are alike in the way all human creations are?
Music, after all, is amorphous: it shifts its texture and character from place to place and time to time. It can be crystalline or cloudy, sentimental or bombastic. It is transitory: when played it dissolves into memory. Mathematics, by contrast, is straightforward: it never alters its character, and it seems to soar above both place and time. Music is in the fray of things, played on grass reeds or gut strings, through brass tubes or hollowed bamboo, using all kinds of materials, natural and otherwise. Mathematics is, above all, spun from abstraction, not even requiring pencil and paper, which only record thought the way a tape recorder records music.
Unlike mathematics, music seems useless. A world without music would still provide food and clothing and shelter and uncounted luxuries; aside from the absence of such awkward and mechanically tortured contraptions as the piano and the clarinet, that world would be physically identical to our own. Music's main function seems to be as accompaniment for shamans and magicians and the sales pitches of Muzak contractors. But mathematics has left no part of our world untouched. It is used in drawing property lines, building submarines, predicting the curvature of space, solving algebra problems ("If two men can paint a room in three hours, how long will it take for three men to paint a room?"), and routing city traffic. A world without mathematics would be utterly different from our own.
Music seems steeped in affect; we commonly talk about music as sad or happy or angry or gentle. Music is spiritual, aesthetic, religious. Mathematicians couldn't care less about the emotions suggested by a theorem's proof. Has anybody ever encountered a "sad" theorem, or presented an "angry" proof, or inspired a courtship through abstract musings about topological spaces?
Mathematicians insist they are concerned only with the true, which under their glinting eyes increases in quantity with each generation. A problem unsolved in one generation is penetrated in the next; issues which gave Descartes sleepless nights are now tackled by high school students for homework. Music is another story. In what way is Stockhausen a step forward compared with Bach? When we learn a piece of music, what more do we learn other than the music itself? Musical knowledge, if it exists at all, is peculiar—incomprehensible from culture to culture, barely recognizable from time to time. For example, the sound of Greek music is lost to us completely; Gregorian chant seems unrelated to the Classical sonata. Music seems closer to language in its multiplicity and mystery than to mathematics; it is rooted in individual cultures. Play a South Sea fishing chant in Carnegie Hall, or a Beethoven piano sonata in the Australian bush, and you mix media and meanings; something is altered.
By contrast, mathematical truth stands as a rebuff to music and any of the arts: it seems as untied to space as to time. The Babylonians may have had a counting system based on the number 60 (whence comes our eccentric time measurements for minutes and seconds), but they reached the same results we do today whether we use a system based on the number 10 or, in our computers, a system based on the number 2. The prodigious mathematician Srinivasa Ramanujan could, in an isolated hut in India, teach himself mathematics from two mediocre textbooks, then send his brilliant results off to Cambridge, because what is true in one country's mathematical work is true in another. Mathematics seems free from cultural influence or constriction.
So can there be two more alien subjects with which to cultivate poetic metaphors? Even painting and music might be more likely mates. For the distinctions between math and music seem fundamental: they are between truth and beauty, timelessness and change, science and art.
But these distinctions are less fundamental than they seem at first. We need to step back and examine the notion of comparison itself, the ways in which we examine like and unlike things. Things can be different in so many ways; how do we know which ways are significant? A person is like a cabbage because both need air, and unlike a mackerel in choice of habitat; is a person then more like a cabbage than a mackerel? Whenever we compare two seemingly different things, we are caught in a web of questions about our meanings and our intentions, about what we consider important and why.
So before setting out, we must understand something about the ways in which we understand similarities. The most obvious way we show things are connected is to give them the same word: when children learn language, they learn concepts, abstractions that are embodied in a name. A nose is not a particular nose but a general one; it can even be recognized as a "nose" when almost all distinguishing characteristics, like shape and nostrils, are removed aside from its place on something recognized as a "face." A toy car is called "car" just as a real one is, even though they may share neither size nor shape, design or color. These notions are learned and applied in an utterly unselfconscious way. When a connection is made, it immediately specifies what is important, and what incidental. It is unimportant, for example, that a toy car have doors or small seats or even a windshield to be recognized as a car by a child; what is important is that it have four wheels and a bulge on top, and that it move. Making connections requires some sense of the essential; it requires abstraction.
When identifications and links are newly made, when they no longer depend on the given abstractions of language, then the project becomes still more difficult. We can no longer rely on words which represent connections previously confirmed. For instance, when we want to discuss how a car is like a doughnut, we need to find aspects of a car and a doughnut that are shared and give those characteristics some other name. When the notions are still more general, the challenge is daunting. In what way, to take a cue from Lewis Carroll, is a raven like a writing desk? Creating an abstract link between abstract concepts means understanding each thoroughly in itself, then understanding their relationship to each other. Thus, in comparing things so obviously different as mathematics and music, we must consider the essential aspects of each. We must abstract from mathematics and music their practices and concerns, discern their inner lives, then tease out any signs of shared patterns. We need to define what mathematicians call a mapping between one world and the other.
This is something we do any time we compare, even if the objects or concepts under consideration seem at first to have nothing in common. For example, we know that a raven has two feet and a writing desk none, that a raven is alive and a writing desk is not, that a raven flies and a desk sits. But we can also look at the raven and the desk in other ways. We can see that a raven's feet steady it on the ground and its wings steady it in the air, and that that natural stability is imitated by the manmade desk, which is meant to neither tip nor topple under pressure. We might even argue that a writing desk has a certain function decreed by its maker, and take a religious perspective for arguing the same about the raven. Or we may take the raven's symbolic and associative significance of darkness and foreboding, and connect that with the associations called to mind by any writer facing his desk preparing to write a first draft ("Quoth the raven ..."). The point is that a raven may indeed be like a writing desk, if our interests lie in certain directions. It depends on the context in which we look at them, on the domain in which we wish to compare them, and on what we consider important.
So too with mathematics and music. The links between them may lead us into profound regions we would never have stumbled on if our path were guided solely by one or the other; and our understanding of mathematics and music is bound to change based upon those connections. We may even come to see that the process by which we reason about them bears an uncanny resemblance to the processes at work within both—but that is anticipating what may lie at the end of a still tangled path. The beginnings are not in the gross categories of art and science and beauty and truth but in detail, in attitudes and approaches, in the human activities undertaken.
What exactly is "doing mathematics" or "making music"? Mathematics does not explore the structure of the physical universe as does physics or biology; it is something else entirely. But what? That simple question has puzzled philosophers of mathematics for several millennia. The working mathematician tends, generally, to avoid such questions about mathematics, just as most of us do about daily life. The mathematician works on problems, teaches students, and proceeds much as any other professional might. The musician does the same, caught up in the learning of repertoire or the composing of new works and the teaching of students but rarely approaching questions about what music itself might be or what its role is in culture. Music for the musician is like math for the mathematician: it simply presents itself as a mystery to be worked with and around. This is also why the philosophy of music has remained fairly primitive; what cannot be explained is passed over in silence.
Consider the most basic experience of "making" music. When I set out to learn to play Bach's D# Minor Fugue (from Book 1 of The Well-Tempered Clavier) on the piano, the music seemed frustratingly intricate. There is a theme that is simple enough: it begins with a leap upward, but it is felt less as a leap than an unfolding. It should be heard as if the second note grows out of the first, opposing it but also connected to it. The theme then turns with a plaintive caress and, as if taking a breath, gently echoes its own beginning before sadly returning, step by step, to its origins. The gesture's two parts have almost different characters—an excursion and a return—but the commanding spirit is melancholic, unsettled. When the theme reenters in another fugal voice, the restatement rises out of its own lingering sigh.
The problem in learning to play the fugue was not just training the fingers to create this voice so that it seemed to grow out of itself; it was to have each voice create a seamless line while two other such lines were proceeding at the same time. With practice, that could be done, the fingers passing notes among themselves, sliding one to the other, all the while keeping each voice intact. The problem was to hear these voices at the same time, to sense, when one voice entered with all its notes quickened, and another entered with all its notes slowed, and a third playfully combined the two timings; to play these properly required being able to focus attention above them all, so that at any instant any of these voices could be followed, while the integrity of the others was unaffected. One trick to ensure that no voice was slighted was to sing one voice while playing the others. This was not just a matter of learning to hear; it was a matter of comprehending, anticipating, so that the different layers of contrasting sound became strands in an integrated texture. Just as the theme needed to be felt as a single gesture rather than as a combination of notes and intervals, its combinations and varied statements needed to be bound together into a dramatic meditation, until the entire fugue could be seemingly exhaled in a single breath, its machinations becoming manifestations of natural forces, the melancholic theme building gradually to grandeur.
The learning of a composition combines physical, aural, and intellectual work, each feeding the others. Mathematics may leave out the physical and aural, but the intellectual work is daunting. At about the same time these labors with the Bach fugue were in progress, I was engaged in a more large-scale exercise. It involved not a single theme with a determined character but many disparate voices, each confusing in its own way. These voices had names. They were disciplines, regions of study: they were called analysis, algebra, set theory, and topology. Each one seemed to set up its own laws and then follow these rules wherever they led, spinning out theories of curves or numbers or spaces or objects which had no obvious meaning at all. My labor of understanding was divided here, attempting, for example, to comprehend how a theorem about measuring area under a curve was related to a theory of probability, then trying to understand why the notion that we can always choose individual objects from collections of objects is a mathematical notion—indeed, that it is one of the basic facts in an entire area of mathematical research.
Excerpted from EMBLEMS OF MIND by Edward Rothstein Copyright © 2006 by Edward Rothstein. Excerpted by permission of The University of Chicago Press. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Posted July 22, 2010
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