The most fundamental differences are philosophical, and readers of this book will emerge with a clearer understandingof paradoxical-sounding concepts such as infinity, curved space, and imaginary numbers. The first few chapters are about general aspects of mathematical thought. These are followed by discussions of more specific topics, and the book closes with a chapter answering common sociological questionsabout the mathematical community (such as Is it true that mathematicians burn out at the age of 25?) It is the ideal introduction for anyone who wishes to deepen their understanding of mathematics.About the Series: Combining authority with wit, accessibility, and style, Very Short Introductions offer an introduction to some of life's most interesting topics. Written by experts for the newcomer, they demonstrate the finest contemporary thinking about the central problems and issues in hundredsof key topics, from philosophy to Freud, quantum theory to Islam.

    Indeed, numbers are part of every discipline in the sciences and the arts.With 350 illustrations, including diagrams, photographs and computer imagery, the book chronicles the centuries-long search for the meaning of numbers by famous and lesser-known mathematicians, and explains the puzzling aspects of the mathematical world. Topics include:The earliest ideas of numbers and counting Patterns, logic, calculating Natural, perfect, amicable and prime numbers Numerology, the power of numbers, superstition The computer, the Enigma Code Infinity, the speed of light, relativity Complex numbers The Big Bang and Chaos theories The Philosopher's Stone. The Book of Numbers shows enthusiastically that numbers are neither boring nor dull but rather involve intriguing connections, rivalries, secret documents and even mysterious deaths.

     Now Roy Sorensen offers the first narrative history of paradoxes, a fascinating and eye-opening account that extends from the ancient Greeks, through the Middle Ages, the Enlightenment, and into the twentieth century. When Augustine asked what God was doing before He made the world, he wastold: Preparing hell for people who ask questions like that. A Brief History of the Paradox takes a close look at questions like that and the philosophers who have asked them, beginning with the folk riddles that inspired Anaximander to erect the first metaphysical system and ending with suchthinkers as Lewis Carroll, Ludwig Wittgenstein, and W.V. Quine. Organized chronologically, the book is divided into twenty-four chapters, each of which pairs a philosopher with a major paradox, allowing for extended consideration and putting a human face on the strategies that have been taken towardthese puzzles. Readers get to follow the minds of Zeno, Socrates, Aquinas, Ockham, Pascal, Kant, Hegel, and many other major philosophers deep inside the tangles of paradox, looking for, and sometimes finding, a way out. Filled with illuminating anecdotes and vividly written, A Brief History of the Paradox will appeal to anyone who finds trying to answer unanswerable questions a paradoxically pleasant endeavor.

    Illustrations.

    Now, in The Unimaginable Mathematics of Borges' Library of Babel, William Goldbloom Bloch takes readers on a fascinating tour of the mathematical ideas hiddenwithin one of the classic works of modern literature.Written in the vein of Douglas R. Hofstadter's Pulitzer Prize-winning G�del, Escher, Bach, this original and imaginative book sheds light on one of Borges' most complex, richly layered works. Bloch begins each chapter with a mathematical idea--combinatorics, topology, geometry, informationtheory--followed by examples and illustrations that put flesh on the theoretical bones. In this way, he provides many fascinating insights into Borges' Library. He explains, for instance, a straightforward way to calculate how many books are in the Library--an easily notated but literallyunimaginable number--and also shows that, if each book were the size of a grain of sand, the entire universe could only hold a fraction of the books in the Library. Indeed, if each book were the size of a proton, our universe would still not be big enough to hold anywhere near all the books.Given Borges' well-known affection for mathematics, this exploration of the story through the eyes of a humanistic mathematician makes a unique and important contribution to the body of Borgesian criticism. Bloch not only illuminates one of the great short stories of modern literature but alsoexposes the reader--including those more inclined to the literary world--to many intriguing and entrancing mathematical ideas.

    A Key Strength Of This Text Is Zill'S Emphasis On Differential Equations As Mathematical Models, Discussing The Constructs And Pitfalls Of Each. The Third Edition Is Comprehensive, Yet Flexible, To Meet The Unique Needs Of Various Course Offerings Ranging From Ordinary Differential Equations To Vector Calculus. Numerous New Projects Contributed By Esteemed Mathematicians Have Been Added. Key Features O The Entire Text Has Been Modernized To Prepare Engineers And Scientists With The Mathematical Skills Required To Meet Current Technological Challenges. O The New Larger Trim Size And 2-Color Design Make The Text A Pleasure To Read And Learn From. O Numerous NEW Engineering And Science Projects Contributed By Top Mathematicians Have Been Added, And Are Tied To Key Mathematical Topics In The Text. O Divided Into Five Major Parts, The Text'S Flexibility Allows Instructors To Customize The Text To Fit Their Needs. The First Eight Chapters Are Ideal For A Complete Short Course In Ordinary Differential Equations. O The Gram-Schmidt Orthogonalization Process Has Been Added In Chapter 7 And Is Used In Subsequent Chapters. O All Figures Now Have Explanatory Captions. Supplements O Complete Instructor'S Solutions: Includes All Solutions To The Exercises Found In The Text. Powerpoint Lecture Slides And Additional Instructor'S Resources Are Available Online. O Student Solutions To Accompany Advanced Engineering Mathematics, Third Edition: This Student Supplement Contains The Answers To Every Third Problem In The Textbook, Allowing Students To Assess Their Progress And Review Key Ideas And Concepts Discussed Throughout The Text. ISBN: 0-7637-4095-0

    By 1949, Godel had produced a remarkable proof: In any universe described by the Theory of Relativity, time cannot exist. Einstein endorsed this result reluctantly but he could find no way to refute it, since then, neither has anyone else. Yet cosmologists and philosophers alike have proceeded as if this discovery was never made. In A World Without Time, Palle Yourgrau sets out to restore Godel to his rightful place in history, telling the story of two magnificent minds put on the shelf by the scientific fashions of their day, and attempts to rescue the brilliant work they did together.

    In the Fourth Edition CALCULUS, EARLY TRANSCENDENTALS these functions are introduced in the first chapter and their limits and derivatives are found in Chapters 2 and 3 at the same time as polynomials and other elementary functions. In this Fourth Edition, Stewart retains the focus on problem solving, the meticulous accuracy, the patient explanations, and the carefully graded problems that have made these texts word so well for a wide range of students. All new and unique features in CALCULUS, FOURTH EDITION have been incorporated into these revisions also.

    Ariew's introduction places the correspondence in historical context and discusses the vibrant philosophical climate of the times. Appendices provide those selections from the works of Newton that Clarke frequently refers to in the correspondence. A bibliography is also included.

    Connecting games to math, science, and psychology, GameTek has grown to be one of the most popular parts of the show.This volume commemorates the anniversary with a collection of over seventy of the best segments, many with annotations and illustrations.With chapters on everything from Rock, Paper, Scissors to the Prisoner’s Dilemma to Player Engagement to Quasicrystals to Buddha’s Forbidden Games, GameTek is sure to delight not just game designers and players, but anyone who wants to learn about the world from a new perspective.Sections:• Game Theory• Math• Psychology• Science• Game Mechanics• Psychology Games• HistoryFrom the first time I heard it, the GameTek segment in The Dice Tower podcast became my favorite part of the show. Listening to Geoff is like going to your favorite lesson with your favorite teacher. He teaches about games (yay!) and does it in a very interesting way with lots of examples. He does amazing stuff. He knows about the construction of games, he knows the theory, he knows all that stuff behind the scenes that we gamers do not see when just playing a game and having fun.Ignacy Trzewiczek, Portal GamesThere are many hobby game 'experts' out there, dying to give you their opinion on how the industry works, how games work, what types of games are best, and so on. Geoff Engelstein is the expert that requires your attention. He is a scholar of games, and his research on games and other principles that apply to gaming is matched by none.Stephen Buonocore, Stronghold GamesOver the years, I’ve listened to a lot of people talk about board games, yet the short snippets that Geoff puts out are the ones that I find myself thinking about in the quiet of the night. His are the segments that you laugh at and say, “I have NO idea what you are talking about” — but later on use to show people just how intellectual you are.Tom Vasel, The Dice Tower

    . . a most valuable contribution." — Douglas R. Hofstadter, author of Gödel, Escher, BachThe foundations of game theory were laid by John von Neumann, who in 1928 proved the basic minimax theorem, and with the 1944 publication of the Theory of Games and Economic Behavior, the field was established. Since then, game theory has become an enormously important discipline because of its novel mathematical properties and its many applications to social, economic, and political problems.Game theory has been used to make investment decisions, pick jurors, commit tanks to battle, allocate business expenses equitably — even to measure a senator's power, among many other uses. In this revised edition of his highly regarded work, Morton Davis begins with an overview of game theory, then discusses the two-person zero-sum game with equilibrium points; the general, two-person zero-sum game; utility theory; the two-person, non-zero-sum game; and the n-person game.A number of problems are posed at the start of each chapter and readers are given a chance to solve them before moving on. (Unlike most mathematical problems, many problems in game theory are easily understood by the lay reader.) At the end of the chapter, where solutions are discussed, readers can compare their "common sense" solutions with those of the author. Brimming with applications to an enormous variety of everyday situations, this book offers readers a fascinating, accessible introduction to one of the most fruitful and interesting intellectual systems of our time.

    As a member of the Los Alamos National Laboratory from 1944 on, Ulam helped to precipitate some of the most dramatic changes of the postwar world. He was among the first to use and advocate computers for scientific research, originated ideas for the nuclear propulsion of space vehicles, and made fundamental contributions to many of today's most challenging mathematical projects. With his wide-ranging interests, Ulam never emphasized the importance of his contributions to the research that resulted in the hydrogen bomb. Now Daniel Hirsch and William Mathews reveal the true story of Ulam's pivotal role in the making of the "Super," in their historical introduction to this behind-the-scenes look at the minds and ideas that ushered in the nuclear age. An epilogue by Françoise Ulam and Jan Mycielski sheds new light on Ulam's character and mathematical originality.

    This volume provides a comprehensible guide to his work by a wide range of experts who are actively engaged in new work on Wittgenstein. The essays, which are both expository and original, address central themes in his philosophy of mind, language, logic, and mathematics and clarify the connections among the different stages in the development of his work.

    a personal confession of its creator and a kind of involuntary and unperceived memoir.". Stewart affirms this maxim in his colorful reinterpretation of the lives and works of 17th-century philosophers Spinoza and Leibniz. In November 1676, the foppish courtier Leibniz, "the ultimate insider... an orthodox Lutheran from conservative Germany," journeyed to The Hague to visit the self-sufficient, freethinking Spinoza, "a double exile... an apostate Jew from licentious Holland." A prodigious polymath, Leibniz understood Spinoza's insight that "science was in the process of rendering the God of revelation obsolete; that it had already undermined the special place of the human individual in nature." Spinoza embraced this new world. Seeing the orthodox God as a "prop for theocratic tyranny," he articulated the basic theory for the modern secular state. Leibniz, on the other hand, spent the rest of his life championing God and theocracy like a defense lawyer defending a client he knows is guilty. He elaborated a metaphysics that was, at bottom, a reaction to Spinoza and collapses into Spinozism, as Stewart deftly shows. For Stewart, Leibniz's reaction to Spinoza and modernity set the tone for "the dominant form of modern philosophy"—a category that includes Kant, Hegel, Bergson, Heidegger and "the whole 'postmodern' project of deconstructing the phallogocentric tradition of western thought." Readers of philosophy may find much to disagree with in these arguments, but Stewart's wit and profluent prose make this book a fascinating read.

    In Are Numbers Real?, Brian Clegg explores the way that math has become more and more detached from reality, and yet despite this is driving the development of modern physics. From devising a new counting system based on goats, through the weird and wonderful mathematics of imaginary numbers and infinity, to the debate over whether mathematics has too much influence on the direction of science, this fascinating and accessible book opens the reader’s eyes to the hidden reality of the strange yet familiar entities that are numbers.