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Overview
This book is their story. Ivar Ekeland here takes the reader on a journey through scientific attempts to envision the best of all possible worlds. He begins with the French physicist Maupertuis, whose least action principle asserted that everything in nature occurs in the way that requires the least possible action. This idea, Ekeland shows, was a pivotal breakthrough in mathematics, because it was the first expression of the concept of optimization, or the creation of systems that are the most efficient or functional. Although the least action principle was later elaborated on and overshadowed by the theories of Leonhard Euler and Gottfried Leibniz, the concept of optimization that emerged from it is an important one that touches virtually every scientific discipline today.
Tracing the profound impact of optimization and the unexpected ways in which it has influenced the study of mathematics, biology, economics, and even politics, Ekeland reveals throughout how the idea of optimization has driven some of our greatest intellectual breakthroughs. The result is a dazzling display of erudition—one that will be essential reading for popular-science buffs and historians of science alike.
Product Details
| ISBN-13: | 9780226199955 |
|---|---|
| Publisher: | University of Chicago Press |
| Publication date: | 10/31/2007 |
| Edition description: | Reprint |
| Pages: | 214 |
| Product dimensions: | 6.00(w) x 9.00(h) x 0.80(d) |
About the Author
Read an Excerpt
The Best of All Possible Worlds MATHEMATICS AND DESTINY
By Ivar Ekeland
The University of Chicago Press Copyright © 2006 The University of Chicago
All right reserved.
ISBN: 978-0-226-19994-8
Chapter One Keeping the Beat
"BEFORE WE PROCEED, we must be aware that every pendulum keeps its beat so well defined and fixed that it is not possible to have it move according to any other period than the only natural one." Thus speaks Galileo in his Discourses and Mathematical Proofs Concerning Two New Sciences, the last book he ever published (1638). He died four years later, leaving a rich scientific legacy, of which this simple statement may be the most important part: as a fact, it quickly turned out to be false, but it changed our ideas about physical motion, and inspired a new technology to measure time.
A pendulum is simply a small load suspended to a string or to a rod fixed at one end. If left alone it ends up hanging vertically, and if we push it away from the vertical, it starts beating. Galileo found that all beats last the same time, called the period, which depends on the length of the pendulum, but not on the amplitude of the beats or on the weight of the load. It also states that the period varies as the square root of the length: to double its period, one should make the pendulum four times as long. Making it heavier, or pushing it farther away from the vertical, has no effect. This property is known as isochrony, and it is the main reason why we are able to measure time with accuracy.
It has been said that Galileo discovered that law during a service in the Pisa cathedral, by comparing the oscillations of the great lantern hanging in the nave to the beats of his own pulse. What a beautiful symbol! The great cosmic cycles, the succession of night and day, the phases of the moon, the coming of the tide, the return of the seasons, have always been the background against which history is staged. But there is also for each of us a smaller companion, which measures not cosmic, but biological, even personal, time: our own pulse is a natural wristwatch. Comparing the rhythm of nature to that of our blood gives us the idea of a standard time, which should be both universal, valid for everyone at once, like the oscillations of the lantern, and homogeneous, reproducing itself regularly, like the beats of our heart. It is a truly revolutionary idea, contrary to all the experience which humankind has gathered since the earliest times: all natural rhythms are variable and irregular. The pulse beat is not the same across individuals, and it is affected by emotional or physical strains. Daylight varies according to the latitude and the season, the lunar month changes as well, and defining correctly the year is a major astronomical problem. Coordinating all these rhythms, to keep Christmas in midwinter for instance, required the invention of the Gregorian calendar with its complicated rules about leap years. It still is not good enough, because these rhythms change: the rotation of the Earth is slowing down, so that the day lengthens slightly, and once in a while the atomic clocks that keep standard time have to be pushed forward one second.
Today hours are constant: an hour is an hour, anywhere in the world, at any time, just as a meter is a meter and a pound is a pound. But this is quite a modern idea: for our ancestors, hours were uneven. In classical antiquity, there were twelve hours between sunrise and sunset, and twelve hours between sunset and sunrise. So day hours and night hours (aka vigils) had different durations, except at the spring and fall equinoxes. Coming to work in the fields at the eleventh hour meant that most of the day had gone by; little wonder that those who had been around since dawn found it unfair to be paid no more than the latecomer, as in the evangelical parable. The duration of hours varied with season and location: summer hours were different from winter hours, Florence hours were different from Rome hours (not that there was any direct way to compare them in those times).
Not so with the pendulum beat. The great lantern of the Pisa cathedral beats for all to see, and each and every one of its oscillations has the same duration. They slowly dampen, and eventually the pendulum will stop, but a whiff of air or a pull on the ropes will start it again, always with the same period, measuring out equal intervals of time. Bring it to Rome, and it will keep the same beat as in Pisa, by day as by night, in summer as in winter. This is Galileo's great discovery: the pendulum provides us with a natural way to measure time in a universal and homogeneous way. It divides time into intervals of constant duration, unlike the day, the month, or the year, which are difficult to carry around and vary according to place and date.
Between the fourth century BC, and the fourth century AD, over a period stretching eight hundred years, there flourished in Alexandria an extraordinary school of Greek mathematicians, starting with Euclid, the legendary founder of geometry, and ending with Hypatia, probably the first woman to leave a name in mathematics. Their work was familiar to Galileo and to all the scientists of his time: they had explored essentially all the possibilities offered by the ruler and the compass, and no better instruments were available. The basic shapes of geometry still were those which could be constructed using only ruler and compass: lines and circles, of course, but also the three conics, ellipse, parabola, and hyperbola, on which no progress had been made since the comprehensive treatise of Apollonius, written in Alexandria in the course of the third century BC. At about the same time, another great scientist, Archimedes, showed how to compute the areas of these curves, and also the volume of the bodies they generate by rotating around an axis. The technology in Alexandria was impressive as well, probably better than that which was available to Galileo. Many treatises on architecture and engineering have survived, and the renown of some of their realizations have crossed the bridge of centuries. The war machines Archimedes built kept at bay for three years the Roman army besieging Syracuse; the great beacon of Alexandria harbor could be seen from thirty miles at sea.
Galileo does to time what the great geometers of antiquity did to space: he turns it into a homogeneous and measurable quantity. Whereas the Greeks had a well-established theory of space, which remained fruitful and essentially unchanged until the non-Euclidian geometries were discovered in the nineteenth century, they did not have a corresponding theory for time. They had a grasp of statics, not of dynamics. Every kind of motion, be it of an arrow flying toward its aim, a runner catching up with a tortoise, or a stone thrown in the air, was a problem for them. What force is driving the stone, once it has left the thrower's hand? How can the runner catch up with the tortoise? Mark the position of the tortoise and wait till the runner has reached it; but the tortoise has progressed in the meantime, so there is a new position to be marked, and an additional time to wait before the runner reaches it; but the tortoise has moved again, so that it will always be slightly ahead, and the runner should never catch up. This is Zeno's paradox, clearly a question posed by someone who has a better grasp of space than of time.
Unlike geometers, the Greek physicists did not worry about the possibility of motion-they just took it as a fact-but they looked for its causes. The most influential work on the subject was the Physics of Aristotle, written in the fourth century BC. This was the main influence that Galileo would have to fight to establish the "new science," as he called it. Aristotle's physics is plain good sense: whenever something moves, something else must be driving it, and whenever the driving force stops, the driven object must stop. It is not without its problems: how come the stone I throw does not fall directly to the ground as it leaves my hand? Why does it rise first, and then fall back? It is quite interesting to see trajectories of projectiles drawn in Galileo's times: the projectile is shown to rise in an arc, and then to fall steeply, almost vertically, as if it had been dropped from the high point of the trajectory. This is in accordance with Aristotle's teaching, but it is not what actually happens: the second part of the trajectory is an arc, symmetric to the first one. It was clear enough why the stone would eventually fall, but much ingenuity was spent in explaining why it would have to rise first, and the air was suspected to play a role in carrying it. In short, the Greeks did not develop a theory of time and motion analogous to the theory they developed for space and shape.
No wonder: in Greek philosophy, motion is synonymous with change, and hence with imperfection. Something truly perfect would not change, it would neither grow nor decay, it would be unalterable and eternal. In Platonic philosophy, perfect objects do exist; they constitute the only true reality: what we see during our lives are only poor reflections of these ideal objects, mere shadows on a wall. After our death, however, we will be allowed to contemplate the originals, to see good everlasting, truth everlasting, beauty everlasting, and we will carry some memory of them in our later lives. We do not discover mathematical truths; we remember them from our passages through this world outside our own. There is a famous scene in the dialogue Meno, where Socrates leads an uneducated slave into "remembering" that the diagonal c of a rectangle is related to its sides a and b by the famous theorem of Pythagoras: the square of c is equal to the sum of the squares of a and b. It is well worth reading, and is an example of good teaching. Socrates never tells the slave anything; he simply asks him the right questions, in the right order, and lets him grope around until he suddenly sees the theorem, sees it as truly and self-evidently as if he had always known it. In fact, says Plato, Meno knew that theorem because he had already seen it, in the world of eternal truths which his soul had visited before it was sent back to Earth in the body of a slave. Socrates was wont to say that he held the same profession as his mother, a midwife, because he delivered souls of the burdens they did not know they carried, just as she delivered pregnant women of their unseen progeny.
In the Platonic tradition, truth is never discovered; it is remembered. Between two successive lives, the soul journeys through the realm of the dead and the unborn, to contemplate one more time the Ideas, perfect, immutable, and eternal, which are the blueprints of everything it will meet during its travels on Earth. Even the word "theory" is a witness to that conception of knowledge: in Greek, theorein means "to see," and theoreia means "the things which have been seen." Anything transient, such as physical motion, has no place among the Ideas; we can have no "theory" for it because we cannot have seen it before our lifetime. Only immobility can carry some kinship with the perfection of Ideas, and there is indeed in Greek physics a well-developed theory of rest, or equilibrium, in more scientific terms: the most famous instance is the theory of equilibrium of fluids, which allegedly sent Archimedes running naked through the streets of Syracuse in the first joy of discovery.
If an object is left at equilibrium, on its own, it will stay there forever. To drive it away from equilibrium, we must exert some force on it, preferably by direct contact; this force is the cause of the motion, and as soon as the cause disappears, the motion should stop. That is the intellectual framework in which Aristotle and his successors try to understand the various kinds of motions that surround us in the real world. This is not without difficulties; to explain the motion of the stars, for instance, which they imagine as luminous dots pinned on a gigantic sphere surrounding us and on which the Sun travels daily, they have to call in legions of angels or demons which push the exterior of the heavenly sphere to make it rotate. During antiquity and the Middle Ages, the world is seen as full of a bewildering variety of motions, of objects scrambling back to equilibrium. There is no general theory; for each motion, some reason must be found why that particular object should have fallen out of equilibrium at that particular time, and how it will proceed to reach anew a state of rest. This is not an easy task, and some answers had evaded scientists for centuries.
For instance, since Roman times it had been noticed that water could be pumped no more than ten meters high at one time; if higher reaches were sought, more pumps were needed, each one pumping water into a basin for the next one to pump from, but each pump could do no better than ten meters. The explanation given was that nature had some kind of distaste for vacuum, and would therefore tend to fill every empty space in the universe before reaching an equilibrium. Why this particular distaste would stop at ten meters, or why the universe would be content with filling the pumps up to ten meters' height, was beyond the reach of the most imaginative explanations. In short, until Galileo, physical motions are seen as perturbing the fundamental order of the universe, which is mirrored by classical geometry. Motion is disorder. The natural state of a physical object is to be immobile.
That day in the Pisa Duomo, Galileo sees the opposite: back and forth swings the great lantern, back and forth. It goes through the vertical position, runs up the other side, hesitates a moment, and then swings back. Eventually it will slow down; its swing will gradually wind down, keeping the same beat, until it finally hangs motionless, the smoke from the candles rising vertically to the gilded ceiling. Why should this position be more natural than this symmetric motion, back and forth, back and forth, with a majestic regularity? What is there to prevent it going on forever? Is it slowing down of itself, or are we witnessing the effect of friction, exerted by the surrounding air and the suspending ropes? Would these not count as imperfections, against the perfection of an oscillatory motion, indefinitely going through the same positions at regular intervals? Certainly the air does not sustain the motion, as we can see from the trailing smoke of the candles: it must be that the motion sustains itself, and it is slowed down by its surroundings. If these could be corrected, the pendulum would beat forever, like the pulse of this great cathedral. And it would spin out, forever and ever, intervals of equal duration, which could thenceforth be used to measure time, just like folding rules are used to measure lengths.
Galileo's theory of the pendulum-and we may use that word as the ancient Greeks did, equating theory to vision, because Galileo actually saw it that day in the cathedral, and all his subsequent work was to remember and understand what he had seen-consists first in the basic intuition that motion need not be from one equilibrium to another, that a pendulum will swing forever, pausing briefly twice a beat as it reaches the top of its trajectory before falling back again. If it eventually slows down and stops, it is because of various imperfections which have to be taken into account, the correction of which will lead, if not to perpetual motion, at least to prolonged life. The second great idea was that all oscillations of the same pendulum, large or small, have the same duration, depending only on its length (this is isochrony, as I mentioned earlier). For the first time in history, humankind had found a chronometer, an instrument which measured time with accuracy and was easy to carry about. Two pendulums of equal length, one in Paris and one in Rome, would have the same beat, regardless of the amplitude of their swing. A piece of string ten inches long is a simple chronometer. Just attach a weight to one end and let it swing from the other. One full beat lasts almost one second; there are sixty beats to the minute, and, if you are patient enough, 3,600 to the hour. A pendulum which is four times as long will be twice as slow: a string of one meter will have a half beat of one second.
(Continues...)
Excerpted from The Best of All Possible Worlds by Ivar Ekeland Copyright © 2006 by The University of Chicago. Excerpted by permission.
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Table of Contents
Introduction1. Keeping the Beat
2. The Birth of Modern Science
3. The Least Action Principle
4. From Computations to Geometry
5. Poincaré and Beyond
6. Pandora's Box
7. May the Best One Win
8. The End of Nature
9. The Common Good
10. A Personal Conclusion
Appendix 1. Finding the Second Diameter of a Convex Table
Appendix 2. The Stationary Action Principle for General Systems
Bibliographical Notes
Index