Charming Proofs A Journey into Elegant Mathematics

Claudi Alsina e Roger B. Nelsen

Dolciani mathematical expositions, nº 42

The Mathematical Association of America | 2011 | 320 páginas | rar - pdf | 2,35 Mb

link (password : matav)

Theorems and their proofs lie at the heart of mathematics. In speaking of the purely aesthetic qualities of theorems and proofs, G. H. Hardy wrote that in beautiful proofs 'there is a very high degree of unexpectedness, combined with inevitability and economy'. Charming Proofs presents a collection of remarkable proofs in elementary mathematics that are exceptionally elegant, full of ingenuity, and succinct. By means of a surprising argument or a powerful visual representation, the proofs in this collection will invite readers to enjoy the beauty of mathematics, and to develop the ability to create proofs themselves. The authors consider proofs from topics such as geometry, number theory, inequalities, plane tilings, origami and polyhedra. Secondary school and university teachers can use this book to introduce their students to mathematical elegance. More than 130 exercises for the reader (with solutions) are also included.

Contents
A garden of integers
Distinguished numbers
Points in the plane
The polygonal playground
A treasury of triangle theorems
The enchantment of the equilateral triangle
The quadrilaterals' corner
Squares everywhere
Curves ahead
Adventures in tiling and coloring
Geometry in three dimensions
Additional theorems, problems, and proofs.

Uses of Infinity

Leo Zippin

(New Mathematical Library)

Mathematical Association of America | 1978 | 160 páginas | rar - pdf | Mb

link (password: matav)

edição de 1962

The word "infinity" usually elicits feelings of awe, wonder, and admiration; the concept infinity has fascinated philosophers and theologians. The author shows how professional mathematicians tame this unwieldy concept, come to terms with it, and use its various aspects as their most powerful tools of the trade.
The early chapters are descriptive and intuitive, full of examples that not only illustrate some infinite processes, but that are worth studying for their own sake. Many questions are raised in the beginning, partially answered in various contexts throughout the book, and finally treated with the precision necessary to give the reader an excellent grasp of the fundamental notions used in the calculus as well as in virtually all other mathematical disciplines. The text is peppered with challenging problems whose solutions appear at the end of the book.

Contents
Preface
Chapter 1 Popular and Mathematical Infinities
Chapter 2 From Natural Numbers to square root 2
Chapter 3 From square root 2 to the Transfinite
Chapter 4 Zig-Zags: To the Limit if the Limit Exists
Chapter 5 The Self Perpetuating Golden Rectangle
Chapter 6 Constructions and Proofs
Solutions to Problems
Bibliography

Infinity and the Mind: The Science and Philosophy of the Infinite

 Rudy Rucker

Princeton University Press | 2005 | 368 páginas | rar - epub | 8,4 Mb

link (password: matav)

djvu - 5,9  Mb
link
link1

pdf - 28,7 Mb - link 

In Infinity and the Mind, Rudy Rucker leads an excursion to that stretch of the universe he calls the "Mindscape," where he explores infinity in all its forms: potential and actual, mathematical and physical, theological and mundane. Rucker acquaints us with Gödel's rotating universe, in which it is theoretically possible to travel into the past, and explains an interpretation of quantum mechanics in which billions of parallel worlds are produced every microsecond. It is in the realm of infinity, he maintains, that mathematics, science, and logic merge with the fantastic. By closely examining the paradoxes that arise from this merging, we can learn a great deal about the human mind, its powers, and its limitations.
Using cartoons, puzzles, and quotations to enliven his text, Rucker guides us through such topics as the paradoxes of set theory, the possibilities of physical infinities, and the results of Gödel's incompleteness theorems. His personal encounters with Gödel the mathematician and philosopher provide a rare glimpse at genius and reveal what very few mathematicians have dared to admit: the transcendent implications of Platonic realism.

Contents

Preface to the 2005 Edition ix

Preface to the Paperback Edition xvii

Preface xix

Chapter One: Infinity 1

A Short History of Infinity
Physical Infinities; 
Temporal Infinities; Spatial Infinities; Infinities in the Small; Conclusion

Infinities in the Mindscape 35

The Absolute Infinite 44

Connections 49

Puzzles and Paradoxes 51
Chapter Two: All the Numbers 53
From Pythagoreanism to Cantorism 53
Transfinite Numbers 64
From Omega to Epsilon-Zero; The Alefs
Infinitesimals and Surreal Numbers 78
Higher Physical Infinities 87
Puzzles and Paradoxes 91
Chapter Three: The Unnameable 93
The Berry Paradox 93
Naming Numbers; Understanding Names
Random Reals 107
Constructing Reals; The Library of Babel ; Richard’s Paradox; Coding the World 
What is Truth? 143
Conclusion 152
Puzzles and Paradoxes 155
Chapter Four: Robots and Souls 157
Gödel’s Incompleteness Theorem 157
Conversations with Gödel 164
Towards Robot Consciousness171
Formal Systems and Machines;  The Liar Paradox and the Non-Mechanizability of Mathematics; Artificial Intelligence via Evolutionary Processes; Robot Consciousness
Beyond Mechanism?185
Puzzles and Paradoxes187
Chapter Five: The One and the Many189
The Classical One/Many Problem189
What is a Set?191
The Universe of Set Theory196
Pure Sets and the Physical Universe; Proper Classes and Metaphysical Absolutes
Interface Enlightenment206
One/Many in Logic and Set Theory; Mysticism and Rationality; Satori
Puzzles and Paradoxes219
Excursion One: The Transfinite Cardinals 221
On and Alef-One 221
Cardinality 226
The Continuum 238
Large Cardinals 253
Excursion Two: Gödel’s Incompleteness Theorems 267
Formal Systems 267
Self-Reference 280
Gödel’s Proof 285
A Technical Note on Man-Machine Equivalence 292
Answers to the Puzzles and Paradoxes 295
Notes 307
Bibliography 329

Numbers and Infinity: A Historical Account of Mathematical Concepts

Ernst Sondheimer, Alan Rogerson

Cambridge University Press | 1982 | 192 Páginas | DjVu | 1,3 Mb

link direto

link

This book is a history of the number concept (with emphasis on the concept of infinity) from ancient days to nearly the present. If I have one complaint about it, it is that the book is very thin and could have said more on a few topics that I think are important; for example, the treatment of Robinson's "non-standard analysis" is very brief and would benefit by expansion. Still, it is an interesting read, and belongs in the library of anyone interested in the subject matter.

Since much of the calculus depends on the concept of the infinite (and the infinitesimal) this book is particularly recommended for those interested in the origins of the calculus.

"Infinity and the Mind: The Science and Philosophy of the Infinite

Rudy Rucker

Princeton University Press | 2004 | 368 páginas | DJVU | 5,7 Mb

pdf - 27 Mb - link
scribd.com

In Infinity and the Mind, Rudy Rucker leads an excursion to that stretch of the universe he calls the "Mindscape," where he explores infinity in all its forms: potential and actual, mathematical and physical, theological and mundane. Rucker acquaints us with Gödel's rotating universe, in which it is theoretically possible to travel into the past, and explains an interpretation of quantum mechanics in which billions of parallel worlds are produced every microsecond. It is in the realm of infinity, he maintains, that mathematics, science, and logic merge with the fantastic. By closely examining the paradoxes that arise from this merging, we can learn a great deal about the human mind, its powers, and its limitations.

Using cartoons, puzzles, and quotations to enliven his text, Rucker guides us through such topics as the paradoxes of set theory, the possibilities of physical infinities, and the results of Gödel's incompleteness theorems. His personal encounters with Gödel the mathematician and philosopher provide a rare glimpse at genius and reveal what very few mathematicians have dared to admit: the transcendent implications of Platonic realism.

TABLE OF CONTENTS:
Preface to the Paperback Edition ix
Preface xi
Chapter One: Infinity 1
A Short History of Infinity 1
Physical Infinities 9
Temporal Infinities 10
Spatial Infinities 15
Infinities in the Small 24
Conclusion 34
Infinities in the Mindscape 35
The Absolute Infinite 44
Connections 49
Puzzles and Paradoxes 51
Chapter Two: All the Numbers 53
From Pythagoreanism to Cantorism 53
Transfinite Numbers 64
From Omega to Epsilon-Zero 65
The Alefs 73
lnfinitesimals and Surreal Numbers 78
Higher Physical Infinities 87
Puzzles and Paradoxes 91
Chapter Three: The Unnameable 93
The Berry Paradox 93
Naming Numbers 95
Understanding Names 100
Random Reals 107
Constructing Reals 108
The Library of Babel 120
Richard's Paradox 126
Coding the World 130
What is Truth? 143
Conclusion 152
Puzzles and Paradoxes 155
Chapter Four: Robots and Souls 157
Godel's incompleteness Theorem 157
Conversations with Godel 164
Towards Robot Consciousness 171
Formal Systems and Machines 172
The Liar Paradox and the Non-Mechanizability of Mathematics 175
Artificial Intelligence via Evolutionary Processes 180
Robot Consciousness 183
Beyond Mechanism? 185
Puzzles and Paradoxes 187
Chapter Five: The One and the Many 189
The Classical One/Many Problem 189
What is a Set? 191
The Universe of Set Theory 196
Pure Sets and the Physical Universe 196
Proper Classes and Metaphysical Absolutes 202
Interface Enlightenment 206
One/Many in Logic and Set Theory 207
Mysticism and Rationality 209
Satori 214
Puzzles and Paradoxes 219
Excursion One: The Transfinite Cardinals 221
On and Alef-One 221
Cardinality 226
The Continuum 238
Large Cardinals 253
Excursion Two: Godel's Incompleteness Theorems 267
Formal Systems 267
Self-Reference 280
Godel's Proof 285
A Technical Note on Man-Machine Equivalence 292
Answers to the Puzzles and Paradoxes 295
Notes 307
Bibliography 329
Index 339

Roads to Infinity: The Mathematics of Truth and Proof

John Stillwell

A K Peters | 2010 | 250 páginas | PDF | 1,3 Mb

link direto
link

This popular account of set theory and mathematical logic introduces the reader to modern ideas about infinity and their implications for mathematics. It unifies ideas from set theory and mathematical logic and traces their effects on mainstream mathematical topics of today, such as number theory and combinatorics.The treatment is historical and partly informal, but with due attention to the subtleties of the subject. Ideas are shown to evolve from natural mathematical questions about the nature of infinity and the nature of proof, set against a background of broader questions and developments in mathematics. A particular aim of the book is to acknowledge some important but neglected figures in the history of infinity, such as Post and Gentzen, alongside the recognized giants Cantor and Goedel.

Keys to Infinity

Clifford A. Pickover

Wiley | 1995 | 352 páginas

PDF - 5,94 Mb

link

djvu - 4 Mb
link
uploading.com

"An original and exciting exploration of how utterly weird, and utterly beautiful, the infinite can be."-Ian Stewart, author of Does God Play Dice?
What can we know about numbers too large to compute or even imagine? Do the tiny bubbles in the froth of a milkshake actually form an infinite fractal pattern? What are apocalyptic numbers and recursive worlds? These and dozens of equally beguiling mathematical mysteries, problems, and paradoxes fill this mind-bending new book.
In each chapter, acclaimed author Clifford Pickover poses a delightful brain-teasing challenge that reveals the scope and splendor of the world of infinity. Try scaling the ladders to heaven, playing a game of infinite chess, or escaping from the land of Fractalia. Along the way you will encounter a myriad of intriguing topics from vampire numbers, to abduction algebra, to the infinity worms of Callisto.
Every problem and puzzle is presented in a remarkably accessible style requiring no specialized mathematical knowledge. Over one hundred illustrations enhance the text and help to explain the mathematical concepts, and stunning color images created by the author reveal the breathtaking beauty of the patterns of infinity. A variety of computer programs offer additional ways to penetrate the enigma of infinity.
For anyone who has ever wondered just how big infinity really is, or just how small, this book will provide an endless source of insight, creativity, and fun.
Advance praise for
KEYS TO INFINITY
"In this the latest of Dr. Pickover's marvelous books, he breaks all finite chains to soar into the transcendental, mind-boggling regions of mathematical infinity. Written in the author's informal, clear style, it is a treasure trove of recreational problems, many published here for the first time, with special emphasis on computer programs and riveting graphics. As you soar, fasten your seat belt."-Martin Gardner, author of The Magic Numbers of Dr. Matrix
"Inventive, quirky, fun! Pickover presents an engaging, inspiring romp in the realm of number and mathematical thought."-Ivars Peterson, author of The Mathematical Tourist
"Join Pickover on his wonderful merry-go-round of ideas, and reach for the infinite. Keys to Infinity is an engaging book. . .a must for those wishing to explore the infinite in all its manifestations."-Theoni Pappas, author of The Joy of Mathematics
"Keys to Infinity contains a near infinity of absorbing themes: from stepladders to the moon and spiral earths, to worm worlds, random chords, and self-similar curlicues. Fascinating!"-Manfred Schroeder, author of Fractals, Chaos, Power Laws
"What could be more appropriate to the subject of infinity than a book like this one, so dense with wonderful puzzles, anecdotes, images, and computer programs that you could pore over it forever? In Keys to Infinity, Pickover has once again assembled a mathematical feast."-Carl Zimmer, Senior Editor Discover
"Cliff Pickover has produced yet another book of mathematical puzzles, weird facts, computer art, and simple programs to challenge our minds and enthrall us with the beauty of the infinite mathematical world in which we live."-Dr. Julien C. Sprott, author of Strange Attractors

Uses of Infinity

 (New mathematical library)

Leo Zippin

Yale University | 1962 | 158 páginas | djvu | 1,14 Mb

link direto
link
uploading.com

pdf - link

This intriguing, accessible work leads readers to an excellent grasp of the fundamental notions of infinity used in the calculus and in virtually all other mathematical disciplines. Each chapter's teachings are supplemented by challenging problems, with solutions at the end of the book

Playing with Infinity: Mathematical Explorations and Excursions

Rozsa Peter

Dover Publications | 1976 | 268 páginas | Djvu | 2,3 MB

link direto
link
pdf - link

This book is written for intellectually minded people who are not mathematicians. It is written for men ofliterature, of art, of the humanities. I have received a great deal from the arts and I would now like in my turn to present mathematics and let everyone see that mathematics and the arts are not so different from each other. I love mathematics not only for its technical applications, but principally because it is beautiful; because man has breathed his spirit of play into it, and because it has given him his greatest game-the encompassing of the infinite. Mathematics can give to the world such worthwhile things-about ideas, about infinity; and yet how essentially human it is-unlike the dull multiplication table, it bears on it for ever the stamp of man's handiwork.

In Search of Infinity

N.Ya. Vilenkin

Birkhäuser Boston | 1995 | 145 páginas  |

PDF - link

djvu - link direto
link
link1

Referência em: MathEduc

The concept of infinity has been for hundreds of years one of the most fascinating and elusive ideas to tantalize the minds of scholars and lay people alike. The theory of infinite sets lies at the heart of much of mathematics, yet is has produced a series of paradoxes that have led many scholars to doubt the soundness of it foundations. The author of this book presents a popular-level account of the roads followed by human thought in attempts to understand the idea of the infinite in mathematics and physics. In doing so, he brings to the general reader a deep insight into the nature of the problem and its importance to an understanding of our world.

Infinity: Beyond the Beyond the Beyond

Lillian R. Lieber

Paul Dry Books | 2007 | 359 páginas

djvu - link
link1

mediafire.com (PDF | 3,27 MB)

Descrição: "Another excellent book for the lay reader of mathematics . . . In explaining [infinity], the author introduces the reader to a good many other mathematical terms and concepts that seem unintelligible in a formal text but are much less formidable when presented in the author's individual and very readable style."-Library Journal

"The interpolations tying mathematics into human life and thought are brilliantly clear."-Booklist

"Mrs. Lieber, in this text illustrated by her husband, Hugh Gray Lieber, has tackled the formidable task of explaining infinity in simple terms, in short line, short sentence technique popularized by her in The Education of T.C. MITS."-Chicago Sunday Tribune

Infinity, another delightful mathematics book from the creators of The Education of T.C. MITS, offers an entertaining, yet thorough, explanation of the concept of, yes, infinity. Accessible to non-mathematicians, this book also cleverly connects mathematical reasoning to larger issues in society. The new foreword by Harvard mathematics professor Barry Mazur is a tribute to the Liebers' influence on generations of mathematicians.

Lillian Lieber was a professor and head of the Department of Mathematics at Long Island University. She wrote a series of light-hearted (and well-respected) math books, many of them illustrated by her husband.

Barry Mazur is the Gerhard Gade University Professor of Mathematics at Harvard University and is the author of Imagining Numbers.

To Infinity and Beyond A cultural history of the Infinite

Eli Maor

Princeton University Press | 1991 | 304 páginas | DjVu | 15 MB

link direto
link

pdf - link

Referência em: MathEduc

Eli Maor examines the role of infinity in mathematics and geometry and its cultural impact on the arts and sciences. He evokes the profound intellectual impact the infinite has exercised on the human mind--from the "horror infiniti" of the Greeks to the works of M. C. Escher; from the ornamental designs of the Moslems, to the sage Giordano Bruno, whose belief in an infinite universe led to his death at the hands of the Inquisition. But above all, the book describes the mathematician's fascination with infinity--a fascination mingled with puzzlement. "Maor explores the idea of infinity in mathematics and in art and argues that this is the point of contact between the two, best exemplified by the work of the Dutch artist M. C. Escher, six of whose works are shown here in beautiful color plates."--Los Angeles Times "[Eli Maor's] enthusiasm for the topic carries the reader through a rich panorama."--Choice "Fascinating and enjoyable.... places the ideas of infinity in a cultural context and shows how they have been espoused and molded by mathematics."--Science