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Michael Guillen author of Five Equations That Changed the World How often can you say that a book on math — on math! — is a real page-turner? Well, this one is. As engaging as a soap opera, as fascinating as a whodunit, as funny as the Sunday comics, Mlodinow's book is storytelling at its best.
Brian Greene author of The Elegant Universe There is perhaps no better way to prepare for the scientific breakthroughs of tomorrow than to learn the language of geometry, and Euclid's Window makes this task lively and enjoyable.
Euclid was a man who possibly did not discover even one significant law of geometry. Yet he is the most famous geometer ever known and for good reason: for millennia it has been his window that people first look through when they view geometry. Here and now, he is our poster boy for the first great revolution in the concept of space — the birth of abstraction, and the idea of proof.
The concept of space began, naturally enough, as a concept of place, our place, earth. It began with a development the Egyptians and Babylonians called "earth measurement." The Greek word for that is geometry, but the subjects are not at all alike. The Greeks were the first to realize that nature could be understood employing mathematics — that geometry could be applied to reveal, not merely to describe. Evolving geometry from simple descriptions of stone and sand, the Greeks extracted the ideals of point, line, and plane. Stripping away the window-dressing of matter, they uncovered a structure possessing a beauty civilization had never before seen. At the climax of this struggle to invent mathematics stands Euclid. The story of Euclid is a story of revolution. It is the story of the axiom, the theorem, the proof, the story of the birth of reason itself.
Copyright © 2001 by Leonard Mlodinow
Twenty-four centuries ago, a Greek man stood at the sea's edge watching ships disappear in the distance. Aristotle must have passed much time there, quietly observing many vessels, for eventually he was struck by a peculiar thought. All ships seemed to vanish hull first, then masts and sails. He wondered, how could that be? On a flat earth, ships should dwindle evenly until they disappear as a tiny featureless dot. That the masts and sails vanish first, Aristotle saw in a flash of genius, is a sign that the earth is curved. To observe the large-scale structure of our planet, Aristotle had looked through the window of geometry.
Today we explore space as millennia ago we explored the earth. A few people have traveled to the moon. Unmanned ships have ventured to the outer reaches of the solar system. It is feasible that within this millennium we will reach the nearest star — a journey of about fifty years at the probably-some-day-attainable speed of one-tenth the speed of light. But measured even in multiples of the distance to Alpha Centauri, the outer reaches of the universe are several billion measuring sticks away. It is unlikely that we will ever be able to watch a vessel approach the horizon of space as Aristotle did on earth. Yet we have discerned much about the nature and structure of the universe as Aristotle did, by observing, employing logic, and staring blankly into space an awful lot. Over the centuries, genius and geometry have helped us gaze beyond our horizons. What can you prove about space? How do you know where you are? Can space be curved? How many dimensions are there? How does geometry explain the natural order and unity of the cosmos? These are the questions behind the five geometric revolutions of world history.
It started with a little scheme hatched by Pythagoras: to employ mathematics as the abstract system of rules that can model the physical universe. Then came a concept of space removed from the ground we trod upon, or the water we swam through. It was the birth of abstraction and proof. Soon the Greeks seemed to be able to find geometric answers to every scientific question, from the theory of the lever to the orbits of the heavenly bodies. But Greek civilization declined and the Romans conquered the Western world. One day just before Easter in A.D. 415, a woman was pulled from a chariot and killed by an ignorant mob. This scholar, devoted to geometry, to Pythagoras, and to rational thought, was the last famous scholar to work in the library at Alexandria before the descent of civilization into the thousand years of the Dark Ages.
Soon after civilization reemerged, so did geometry, but it was a new kind of geometry. It came from a man most civilized — he liked to gamble, sleep until the afternoon, and criticize the Greeks because he considered their method of geometric proof too taxing. To save mental labor, René Descartes married geometry and number. With his idea of coordinates, place and shape could be manipulated as never before, and number could be visualized geometrically. These techniques enabled calculus and the development of modern technology. Thanks to Descartes, geometric concepts such as coordinates and graphs, sines and cosines, vectors and tensors, angles and curvature, appear in every context of physics from solid state electronics to the large-scale structure of space-time, from the technology of transistors and computers to lasers and space travel. But Descartes's work also enabled a more abstract — and revolutionary — idea, the idea of curved space. Do all triangles really have angle sums of 180 degrees, or is that only true if the triangle is on a flat piece of paper? It is not just a question of origami. The mathematics of curved space caused a revolution in the logical foundations, not only of geometry but of all of mathematics. And it made possible Einstein's theory of relativity. Einstein's geometric theory of space and that extra dimension, time, and of the relation of space-time to matter and energy, represented a paradigm change of a magnitude not seen in physics since Newton. It sure seemed radical. But that was nothing, compared to the latest revolution.
One day in June 1984, a scientist announced that he had made a breakthrough in the theory that would explain everything from why subatomic particles exist, and how they interact, to the large-scale structure of space-time and the nature of black holes. This man believed that the key to understanding the unity and order of the universe lies in geometry — geometry of a new and rather bizarre nature. He was carried off the stage by a group of men in white uniforms.
It turned out the event was staged. But the sentiment and genius were real. John Schwarz had been working for a decade and a half on a theory, called string theory, that most physicists reacted to in much the same way they would react to a stranger with a crazed expression asking for money on the street. Today, most physicists believe that string theory is correct: the geometry of space is responsible for the physical laws governing that which exists within it.
The manifesto of the seminal revolution in geometry was written by a mystery man named Euclid. If you don't recall much of that deadly subject called Euclidean Geometry, it is probably because you slept through it. To gaze upon geometry the way it is usually presented is a good way to turn a young mind to stone. But Euclidean geometry is actually an exciting subject, and Euclid's work a work of beauty whose impact rivaled that of the Bible, whose ideas were as radical as those of Marx and Engels. For with his book, Elements, Euclid opened a window through which the nature of our universe has been revealed. And as his geometry has passed through four more revolutions, scientists and mathematicians have shattered theologians' beliefs, destroyed philosophers' treasured worldviews, and forced us to reexamine and reimagine our place in the cosmos. These revolutions, and the prophets and stories behind them, are the subject of this book.
Copyright © 2001 by Leonard Mlodinow
I went on to get an advanced degree and conduct research in mathematical physics. I pretty much gave up baseball and started writing stories when I wasn't doing mathematical physics (or doing my laundry). To me, telling stories and doing science never seemed that different. One is phrased in language, the other in mathematics; but the thrill of each resides in creating or exploring new worlds. Eventually, I got to merge writing and science when I was offered a job writing for Star Trek: the Next Generation. I ended up writing for numerous shows, even sitcoms such as Night Court, in which I was prone to building plots around mad scientists and baseball.
Then kids came, and a responsible job as a vice president with an office in downtown New York. A couple of years ago I decided to write Euclid's Window for the child I hoped still lurked somewhere inside me. Could I recapture that excitement about the way geometry underlies everything? From standing on that baseball diamond to arguing physics with Richard Feynman at Cal Tech to dreaming up a Star Trek story to discussing math with my two boisterous boys, it has always seemed to me that geometry -- just understanding the space around us near and far -- is at the heart of much of human civilization. The best way to convey my vision of this wonderful art was to tell the stories of the five people I see as the poster boys of the great revolutions that occurred over the last 3,000 years or so: Euclid, Descartes, Gauss, Einstein, and Witten -- the last of whom is still very much alive, wasn't happy about being set up alongside these hall of famers, and will probably never really forgive me for doing it anyway.
My plan was ambitious: to take the reader on a voyage of 3,000 years, through all the revolutions in thought that brought us from Euclid to today's twisted 11-dimensional world of string theory, and to do it without letting the mathematics interfere with the story, which really is a page-turner. It was a far bigger project than I imagined. But I'm still alive and look forward to the time when, in a few years, my eldest will be able to understand my book. While I hope that it will inspire him as I was inspired, I know one thing is certain: To find it he won't have to go searching through any bins at the rummage sale. (Leonard Mlodinow)
Anonymous
Posted April 26, 2006
Mlodinow has given an interesting twist to ordinary history. I am not partial to history books, however, the information presented in this novel has some humor to it. It describes the geniuses who formed our current mathematical ideas and explains why some mathematical elements are called what they are. Unlike the critics who responded with their utmost praise, I did not enjoy this book because it dragged along. I love math but do not consider this book to be a pleasurable read.
1 out of 2 people found this review helpful.
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Posted April 29, 2013
Very Good Book about the history of mathematics and its basic ideas. Presented in a way that even a non-math savvy reader can understand. Very practical, without complex formulas of theorems. Awesome examples for laymen. Presents the story like a sort of drama! Very funny at times. Def must read!
Was this review helpful? Yes NoThank you for your feedback. Report this reviewThank you, this review has been flagged.Buy this one in paper unless you enjoy looking at weird character errors. Every umlauted O has been changed to an umlauted A, for example. Doesn't anyone proofread these?
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Posted December 28, 2011
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