The road to reality :

by Penrose, Roger.
Published by : Jonathan Cape, (London :) Physical details: xxviii, 1094 p. : ill. ; 25 cm. ISBN:9780224044479. Year: 2004 Fiction notes: Click to open in new window
Item type Current location Call number Copy number Status Date due
Reference Items British Council Library
BLUE ZONE
500.2 (Browse shelf) 1 Not for loan

Enhanced descriptions from Syndetics:

The Road to Reality, some 1000 pages long, aims to provide a comprehensive account of our present understanding of the physical universe, and the essentials of its underlying mathematical theory. No particular mathematical knowledge on the part of the reader is assumed - the early chapters providing the essential mathematical background for the physical theories described in the remainder of the book. The aim is to convey something of an overall understanding - a feeling for the deep beauty and philosophical connotations of the subject, as well as of its intricate logical interconnections. Clearly, a work of this nature is challenging, but there is enough descriptive material to carry the less mathematically inclined reader through, as well as some 450-500, mostly hand-drawn, figures. The book provides a feeling for all the key issues and deep current controversies, and counters the common complaint that cutting-edge science is fundamentally inaccessible.

Formerly CIP. Uk

Includes bibliographical references and index.

Table of contents provided by Syndetics

  • Preface
  • Acknowledgements
  • Notation
  • Prologue
  • 1 The roots of science
  • 1.1 The quest for the forces that shape the world
  • 1.2 Mathematical truth
  • 1.3 Is Plato's mathematical world 'real'?
  • 1.4 Three worlds and three deep mysteries
  • 1.5 The Good, the True, and the Beautiful
  • 2 An ancient theorem and a modern question
  • 2.1 The Pythagorean theorem
  • 2.2 Euclid's postulates
  • 2.3 Similar-areas proof of the Pythagorean theorem
  • 2.4 Hyperbolic geometry: conformal picture
  • 2.5 Other representations of hyperbolic geometry
  • 2.6 Historical aspects of hyperbolic geometry
  • 2.7 Relation to physical space
  • 3 Kinds of number in the physical world
  • 3.1 A Pythagorean catastrophe?
  • 3.2 The real-number system
  • 3.3 Real numbers in the physical world
  • 3.4 Do natural numbers need the physical world?
  • 3.5 Discrete numbers in the physical world
  • 4 Magical complex numbers
  • 4.1 The magic number 'i'
  • 4.2 Solving equations with complex numbers
  • 4.3 Convergence of power series
  • 4.4 Caspar Wessel's complex plane
  • 4.5 How to construct the Mandelbrot set
  • 5 Geometry of logarithms, powers, and roots
  • 5.1 Geometry of complex algebra
  • 5.2 The idea of the complex logarithm
  • 5.3 Multiple valuedness, natural logarithms
  • 5.4 Complex powers
  • 5.5 Some relations to modern particle physics
  • 6 Real-number calculus
  • 6.1 What makes an honest function?
  • 6.2 Slopes of functions
  • 6.3 Higher derivatives; C1-smooth functions
  • 6.4 The 'Eulerian' notion of a function?
  • 6.5 The rules of differentiation
  • 6.6 Integration
  • 7 Complex-number calculus
  • 7.1 Complex smoothness; holomorphic functions
  • 7.2 Contour integration
  • 7.3 Power series from complex smoothness
  • 7.4 Analytic continuation
  • 8 Riemann surfaces and complex mappings
  • 8.1 The idea of a Riemann surface
  • 8.2 Conformal mappings
  • 8.3 The Riemann sphere
  • 8.4 The genus of a compact Riemann surface
  • 8.5 The Riemann mapping theorem
  • 9 Fourier decomposition and hyperfunctions
  • 9.1 Fourier series
  • 9.2 Functions on a circle
  • 9.3 Frequency splitting on the Riemann sphere
  • 9.4 The Fourier transform
  • 9.5 Frequency splitting from the Fourier transform
  • 9.6 What kind of function is appropriate?
  • 9.7 Hyperfunctions
  • 10 Surfaces
  • 10.1 Complex dimensions and real dimensions
  • 10.2 Smoothness, partial derivatives
  • 10.3 Vector Fields and 1-forms
  • 10.4 Components, scalar products
  • 10.5 The Cauchy-Riemann equations
  • 11 Hypercomplex numbers
  • 11.1 The algebra of quaternions
  • 11.2 The physical role of quaternions?
  • 11.3 Geometry of quaternions
  • 11.4 How to compose rotations
  • 11.5 Clifford algebras
  • 11.6 Grassmann algebras
  • 12 Manifolds of n dimensions
  • 12.1 Why study higher-dimensional manifolds?
  • 12.2 Manifolds and coordinate patches
  • 12.3 Scalars, vectors, and covectors
  • 12.4 Grassmann products
  • 12.5 Integrals of forms
  • 12.6 Exterior derivative
  • 12.7 Volume element; summation convention
  • 12.8 Tensors; abstract-index and diagrammatic notation
  • 12.9 Complex manifolds
  • 13 Symmetry groups
  • 13.1 Groups of transformations
  • 13.2 Subgroups and simple groups
  • 13.3 Linear transformations and matrices
  • 13.4 Determinants and traces
  • 13.5 Eigenvalues and eigenvectors
  • 13.6 Representation theory and Lie algebras
  • 13.7 Tensor representation spaces; reducibility
  • 13.8 Orthogonal groups
  • 13.9 Unitary groups
  • 13.10 Symplectic groups
  • 14 Calculus on manifolds
  • 14.1 Differentiation on a manifold?
  • 14.2 Parallel transport
  • 14.3 Covariant derivative
  • 14.4 Curvature and torsion
  • 14.5 Geodesics, parallelograms, and curvature
  • 14.6 Lie derivative
  • 14.7 What a metric can do for you
  • 14.8 Symplectic manifolds

Excerpt provided by Syndetics

Prologue Am-tep was the King's chief craftsman, an artist of consummate skills. It was night, and he lay sleeping on his workshop couch, tired after a handsomely productive evening's work. But his sleep was restless - perhaps from an intangible tension that had seemed to be in the air. Indeed, he was not certain that he was asleep at all when it happened. Daytime had come - quite suddenly - when his bones told him that surely it must still be night. He stood up abruptly. Something was odd. The dawn's light could not be in the north; yet the red light shone alarmingly through his broad window that looked out northwards over the sea. He moved to the window and stared out, incredulous in amazement. The Sun had never before risen in the north! In his dazed state, it took him a few moments to realize that this could not possibly be the Sun. It was a distant shaft of a deep fiery red light that beamed vertically upwards from the water into the heavens. As he stood there, a dark cloud became apparent at the head of the beam, giving the whole structure the appearance of a distant giant parasol, glowing evilly, with a smoky flaming staff. The parasol's hood began to spread and darken - a daemon from the underworld. The night had been clear, but now the stars disappeared one by one, swallowed up behind this advancing monstrous creature from Hell. Though terror must have been his natural reaction, he did not move, transfixed for several minutes by the scene's perfect symmetry and awesome beauty. But then the terrible cloud began to bend slightly to the east, caught up by the prevailing winds. Perhaps he gained some comfort from this and the spell was momentarily broken. But apprehension at once returned to him as he seemed to sense a strange disturbance in the ground beneath, accompanied by ominous-sounding rumblings of a nature quite unfamiliar to him. He began to wonder what it was that could have caused this fury. Never before had he witnessed a God's anger of such magnitude. His first reaction was to blame himself for the design on the sacrificial cup that he had just completed - he had worried about it at the time. Had his depiction of the Bull-God not been sufficiently fearsome? Had that god been offended? But the absurdity of this thought soon struck him. The fury he had just witnessed could not have been the result of such a trivial action, and was surely not aimed at him specifically. But he knew that there would be trouble at the Great Palace. The Priest-King would waste no time in attempting to appease this Daemon-God. There would be sacrifices. The traditional offerings of fruits or even animals would not suffice to pacify an anger of this magnitude. The sacrifices would have to be human. Quite suddenly, and to his utter surprise, he was blown backwards across the room by an impulsive blast of air followed by a violent wind. The noise was so extreme that he was momentarily deafened. Many of his beautifully adorned pots were whisked from their shelves and smashed to pieces against the wall behind. As he lay on the floor in a far corner of the room where he had been swept away by the blast, he began to recover his senses, and saw that the room was in turmoil. He was horrified to see one of his favourite great urns shattered to small pieces, and the wonderfully detailed designs, which he had so carefully crafted, reduced to nothing. Am-tep arose unsteadily from the floor and after a while again approached the window, this time with considerable trepidation, to re-examine that terrible scene across the sea. Now he thought he saw a disturbance, illuminated by that far-off furnace, coming towards him. This appeared to be a vast trough in the water, moving rapidly towards the shore, followed by a cliff-like wall of wave. He again became transfixed, watching the approaching wave begin to acquire gigantic proportions. Eventually the disturbance reached the shore and the sea immediately before him drained away, leaving many ships stranded on the newly formed beach. Then the cliff-wave entered the vacated region and struck with a terrible violence. Without exception the ships were shattered, and many nearby houses instantly destroyed. Though the water rose to great heights in the air before him, his own house was spared, for it sat on high ground a good way from the sea. The Great Palace too was spared. But Am-tep feared that worse might come, and he was right - though he knew not how right he was. He did know, however, that no ordinary human sacrifice of a slave could now be sufficient. Something more would be needed to pacify the tempestuous anger of this terrible God. His thoughts turned to his sons and daughters, and to his newly born grandson. Even they might not be safe. Am-tep had been right to fear new human sacrifices. A young girl and a youth of good birth had been soon apprehended and taken to a nearby temple, high on the slopes of a mountain. The ensuing ritual was well under way when yet another catastrophe struck. The ground shook with devastating violence, whence the temple roof fell in, instantly killing all the priests and their intended sacrificial victims. As it happened, they would lie there in mid-ritual - entombed for over three-and-a-half millennia! The devastation was frightful, but not final. Many on the island where Am-tep and his people lived survived the terrible earthquake, though the Great Palace was itself almost totally destroyed. Much would be rebuilt over the years. Even the Palace would recover much of its original splendour, constructed on the ruins of the old. Yet Am-tep had vowed to leave the island. His world had now changed irreparably. In the world he knew, there had been a thousand years of peace, prosperity, and culture where the Earth-Goddess had reigned. Wonderful art had been allowed to flourish. There was much trade with neighbouring lands. The magnificent Great Palace was a huge luxurious labyrinth, a virtual city in itself, adorned by superb frescoes of animals and flowers. There was running water, excellent drainage, and flushed sewers. War was almost unknown and defences unnecessary. Now, Am-tep perceived the Earth-Goddess overthrown by a Being with entirely different values. It was some years before Am-tep actually left the island, accompanied by his surviving family, on a ship rebuilt by his youngest son, who was a skilled carpenter and seaman. Am-tep's grandson had developed into an alert child, with an interest in everything in the world around. The voyage took some days, but the weather had been supremely calm. One clear night, Am-tep was explaining to his grandson about the patterns in the stars, when an odd thought overtook him: The patterns of stars had been disturbed not one iota from what they were before the Catastrophe of the emergence of the terrible daemon. Am-tep knew these patterns well, for he had a keen artist's eye. Surely, he thought, those tiny candles of light in the sky should have been blown at least a little from their positions by the violence of that night, just as his pots had been smashed and his great urn shattered. The Moon also had kept her face, just as before, and her route across the star-filled heavens had changed not one whit, as far as Am-tep could tell. For many moons after the Catastrophe, the skies had appeared different. There had been darkness and strange clouds, and the Moon and Sun had sometimes worn unusual colours. But this had now passed, and their motions seemed utterly undisturbed. The tiny stars, likewise, had been quite unmoved. If the heavens had shown such little concern for the Catastrophe, having a stature far greater even than that terrible Daemon, Am-tep reasoned, why should the forces controlling the Daemon itself show concern for what the little people on the island had been doing, with their foolish rituals and human sacrifice? He felt embarrassed by his own foolish thoughts at the time, that the daemon might be concerned by the mere patterns on his pots. Yet Am-tep was still troubled by the question 'why?' What deep forces control the behaviour of the world, and why do they sometimes burst forth in violent and seemingly incomprehensible ways? He shared his questions with his grandson, but there were no answers. . . . A century passed by, and then a millennium, and still there were no answers. . . . Amphos the craftsman had lived all his life in the same small town as his father and his father before him, and his father's father before that. He made his living constructing beautifully decorated gold bracelets, earrings, ceremonial cups, and other fine products of his artistic skills. Such work had been the family trade for some forty generations - a line unbroken since Am-tep had settled there eleven hundred years before. But it was not just artistic skills that had been passed down from generation to generation. Am-tep's questions troubled Amphos just as they had troubled Am-tep earlier. The great story of the Catastrophe that destroyed an ancient peaceful civilization had been handed down from father to son. Am-tep's perception of the Catastrophe had also survived with his descendants. Amphos, too, understood that the heavens had a magnitude and stature so great as to be quite unconcerned by that terrible event. Nevertheless, the event had had a catastrophic effect on the little people with their cities and their human sacrifices and insignificant religious rituals. Thus, by comparison, the event itself must have been the result of enormous forces quite unconcerned by those trivial actions of human beings. Yet the nature of those forces was as unknown in Amphos's day as it was to Am-tep. Amphos had studied the structure of plants, insects and other small animals, and crystalline rocks. His keen eye for observation had served him well in his decorative designs. He took an interest in agriculture and was fascinated by the growth of wheat and other plants from grain. But none of this told him 'why?', and he felt unsatisfied. He believed that there was indeed reason underlying Nature's patterns, but he was in no way equipped to unravel those reasons. One clear night, Amphos looked up at the heavens, and tried to make out from the patterns of stars the shapes of those heroes and heroines who formed constellations in the sky. To his humble artist's eye, those shapes made poor resemblances. He could himself have arranged the stars far more convincingly. He puzzled over why the gods had not organized the stars in a more appropriate way? As they were, the arrangements seemed more like scattered grains randomly sowed by a farmer, rather than the deliberate design of a god. Then an odd thought overtook him: Do not seek for reasons in the specific patterns of stars, or of other scattered arrangements of objects; look, instead, for a deeper universal order in the way that things behave. Amphos reasoned that we find order, after all, not in the patterns that scattered seeds form when they fall to the ground, but in the miraculous way that each of those seeds develops into a living plant having a superb structure, similar in great detail to one another. We would not try to seek the meaning in the precise arrangement of seeds sprinkled on the soil; yet, there must be meaning in the hidden mystery of the inner forces controlling the growth of each seed individually, so that each one follows essentially the same wonderful course. Nature's laws must indeed have a superbly organized precision for this to be possible. Amphos became convinced that without precision in the underlying laws, there could be no order in the world, whereas much order is indeed perceived in the way that things behave. Moreover, there must be precision in our ways of thinking about these matters if we are not to be led seriously astray. It so happened that word had reached Amphos of a sage who lived in another part of the land, and whose beliefs appeared to be in sympathy with those of Amphos. According to this sage, one could not rely on the teachings and traditions of the past. To be certain of one's beliefs, it was necessary to form precise conclusions by the use of unchallengeable reason. The nature of this precision had to be mathematical - ultimately dependent on the notion of number and its application to geometric forms. Accordingly, it must be number and geometry, not myth and superstition, that governed the behaviour of the world. As Am-tep had done a century and a millennium before, Amphos took to the sea. He found his way to the city of Croton, where the sage and his brotherhood of 571 wise men and 28 wise women were in search of truth. After some time, Amphos was accepted into the brotherhood. The name of the sage was Pythagoras . Excerpted from The Road to Reality: A Complete Guide to the Laws of the Universe by Roger Penrose All rights reserved by the original copyright owners. Excerpts are provided for display purposes only and may not be reproduced, reprinted or distributed without the written permission of the publisher.

Reviews provided by Syndetics

Library Journal Review

From an emeritus professor of mathematics at Oxford: everything you ever wanted to know about the laws that govern the universe-and what makes them so appealing. With a six-city author tour. (c) Copyright 2010. Library Journals LLC, a wholly owned subsidiary of Media Source, Inc. No redistribution permitted.

Publishers Weekly Review

At first, this hefty new tome from Oxford physicist Penrose (The Emperor's New Mind) looks suspiciously like a textbook, complete with hundreds of diagrams and pages full of mathematical notation. On a closer reading, however, one discovers that the book is something entirely different and far more remarkable. Unlike a textbook, the purpose of which is purely to impart information, this volume is written to explore the beautiful and elegant connection between mathematics and the physical world. Penrose spends the first third of his book walking us through a seminar in high-level mathematics, but only so he can present modern physics on its own terms, without resorting to analogies or simplifications (as he explains in his preface, "in modern physics, one cannot avoid facing up to the subtleties of much sophisticated mathematics"). Those who work their way through these initial chapters will find themselves rewarded with a deep and sophisticated tour of the past and present of modern physics. Penrose transcends the constraints of the popular science genre with a unique combination of respect for the complexity of the material and respect for the abilities of his readers. This book sometimes begs comparison with Stephen Hawking's A Brief History of Time, and while Penrose's vibrantly challenging volume deserves similar success, it will also likely lie unfinished on as many bookshelves as Hawking's. For those hardy readers willing to invest their time and mental energies, however, there are few books more deserving of the effort. 390 illus. (Feb. 24) (c) Copyright PWxyz, LLC. All rights reserved

CHOICE Review

Science has two primary goals: to find explanations of myriad natural phenomena characterizing the world of experience and to unravel from these explanations the ultimate nature of reality. These goals are interlinked in that the mode of explanation will be the road by which reality will be unveiled. The scientific mode, finding spectacular expression in physics, is through mathematics. Thus, mathematics becomes the road to reality for scientists and especially for physicists. Physicist Penrose guides the reader through that complex and exciting road with profound erudition, deep insight, and great flair. He begins with a clear, succinct exposition of the interrelationships among mental, mathematical, and physical worlds. All through, there is blending of technical mathematics with keen observations and historical asides. Penrose reflects on his own views on current physics, confessing only partial trust in string theories to provide the ultimate key to unlocking the terminus. Though the book is for "the serious lay leader," not many will fathom much beyond the first chapter. Those with graduate work in theoretical physics will be enormously enriched. More than anything else, this book will be a great resource for graduate seminars. A must for all libraries serving physics departments. ^BSumming Up: Essential. Lower-division undergraduates through professionals. V. V. Raman emeritus, Rochester Institute of Technology

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