The Pea and the Sun

A Mathematical Paradox

Author: Leonard M. Wapner

Publisher: A K Peters/CRC Press

ISBN: 9781568813271

Category: Mathematics

Page: 232

View: 9016


Take an apple and cut it into five pieces. Would you believe that these five pieces can be reassembled in such a fashion so as to create two apples equal in shape and size to the original? Would you believe that you could make something as large as the sun by breaking a pea into a finite number of pieces and putting it back together again? Neither did Leonard Wapner, author of The Pea and the Sun, when he was first introduced to the Banach-Tarski paradox, which asserts exactly such a notion. Written in an engaging style, The Pea and the Sun catalogues the people, events, and mathematics that contributed to the discovery of Banach and Tarski's magical paradox. Wapner makes one of the most interesting problems of advanced mathematics accessible to the non-mathematician.

The Pea and the Sun

A Mathematical Paradox

Author: Leonard M. Wapner

Publisher: A K Peters/CRC Press

ISBN: 9781568812137

Category: Mathematics

Page: 232

View: 7461


Take an apple and cut it into five pieces. Would you believe that these five pieces can be reassembled in such a fashion so as to create two apples equal in shape and size to the original? Would you believe that you could make something as large as the sun by breaking a pea into a finite number of pieces and putting it back together again? Neither did Leonard Wapner, author of The Pea and the Sun, when he was first introduced to the Banach-Tarski paradox, which asserts exactly such a notion. Written in an engaging style, The Pea and the Sun catalogues the people, events, and mathematics that contributed to the discovery of Banach and Tarski's magical paradox. Wapner makes one of the most interesting problems of advanced mathematics accessible to the non-mathematician.

Unexpected Expectations

The Curiosities of a Mathematical Crystal Ball

Author: Leonard M. Wapner

Publisher: CRC Press

ISBN: 1568817215

Category: Mathematics

Page: 220

View: 4857


Unexpected Expectations: The Curiosities of a Mathematical Crystal Ball explores how paradoxical challenges involving mathematical expectation often necessitate a reexamination of basic premises. The author takes you through mathematical paradoxes associated with seemingly straightforward applications of mathematical expectation and shows how these unexpected contradictions may push you to reconsider the legitimacy of the applications. The book requires only an understanding of basic algebraic operations and includes supplemental mathematical background in chapter appendices. After a history of probability theory, it introduces the basic laws of probability as well as the definition and applications of mathematical expectation/expected value (E). The remainder of the text covers unexpected results related to mathematical expectation, including: The roles of aversion and risk in rational decision making A class of expected value paradoxes referred to as envelope problems Parrondo’s paradox—how negative (losing) expectations can be combined to give a winning result Problems associated with imperfect recall Non-zero-sum games, such as the game of chicken and the prisoner’s dilemma Newcomb’s paradox—a great philosophical paradox of free will Benford’s law and its use in computer design and fraud detection While useful in areas as diverse as game theory, quantum mechanics, and forensic science, mathematical expectation generates paradoxes that frequently leave questions unanswered yet reveal interesting surprises. Encouraging you to embrace the mysteries of mathematics, this book helps you appreciate the applications of mathematical expectation, "a statistical crystal ball." Listen to an interview with the author on

The Banach-Tarski Paradox

Author: Stan Wagon

Publisher: Cambridge University Press

ISBN: 9780521457040

Category: Mathematics

Page: 253

View: 6863


This volume explores the consequences of the paradox for measure theory and its connections with group theory, geometry, and logic. It unifies the results of contemporary research on the paradox and presents several new results including some unusual paradoxes in hyperbolic space. It also provides up to date proofs and discusses many unsolved problems.

The Outer Limits of Reason

What Science, Mathematics, and Logic Cannot Tell Us

Author: Noson S. Yanofsky

Publisher: MIT Press

ISBN: 0262316781

Category: Science

Page: 424

View: 1936


Many books explain what is known about the universe. This book investigates what cannot be known. Rather than exploring the amazing facts that science, mathematics, and reason have revealed to us, this work studies what science, mathematics, and reason tell us cannot be revealed. In The Outer Limits of Reason, Noson Yanofsky considers what cannot be predicted, described, or known, and what will never be understood. He discusses the limitations of computers, physics, logic, and our own thought processes.Yanofsky describes simple tasks that would take computers trillions of centuries to complete and other problems that computers can never solve; perfectly formed English sentences that make no sense; different levels of infinity; the bizarre world of the quantum; the relevance of relativity theory; the causes of chaos theory; math problems that cannot be solved by normal means; and statements that are true but cannot be proven. He explains the limitations of our intuitions about the world -- our ideas about space, time, and motion, and the complex relationship between the knower and the known.Moving from the concrete to the abstract, from problems of everyday language to straightforward philosophical questions to the formalities of physics and mathematics, Yanofsky demonstrates a myriad of unsolvable problems and paradoxes. Exploring the various limitations of our knowledge, he shows that many of these limitations have a similar pattern and that by investigating these patterns, we can better understand the structure and limitations of reason itself. Yanofsky even attempts to look beyond the borders of reason to see what, if anything, is out there.

Things to Make and Do in the Fourth Dimension

A Mathematician's Journey Through Narcissistic Numbers, Optimal Dating Algorithms, at Least Two Kinds of Infinity, and More

Author: Matt Parker

Publisher: Farrar, Straus and Giroux

ISBN: 0374710376

Category: Mathematics

Page: 464

View: 3626


A book from the stand-up mathematician that makes math fun again! Math is boring, says the mathematician and comedian Matt Parker. Part of the problem may be the way the subject is taught, but it's also true that we all, to a greater or lesser extent, find math difficult and counterintuitive. This counterintuitiveness is actually part of the point, argues Parker: the extraordinary thing about math is that it allows us to access logic and ideas beyond what our brains can instinctively do—through its logical tools we are able to reach beyond our innate abilities and grasp more and more abstract concepts. In the absorbing and exhilarating Things to Make and Do in the Fourth Dimension, Parker sets out to convince his readers to revisit the very math that put them off the subject as fourteen-year-olds. Starting with the foundations of math familiar from school (numbers, geometry, and algebra), he reveals how it is possible to climb all the way up to the topology and to four-dimensional shapes, and from there to infinity—and slightly beyond. Both playful and sophisticated, Things to Make and Do in the Fourth Dimension is filled with captivating games and puzzles, a buffet of optional hands-on activities that entices us to take pleasure in math that is normally only available to those studying at a university level. Things to Make and Do in the Fourth Dimension invites us to re-learn much of what we missed in school and, this time, to be utterly enthralled by it.

Science in the Looking Glass

What Do Scientists Really Know?

Author: Edward Brian Davies

Publisher: Oxford University Press on Demand

ISBN: 0198525435

Category: Science

Page: 295

View: 8030


In this wide-ranging book, Brian Davies discusses the basis for scientists' claims to knowledge about the world. He looks at science historically, emphasizing not only the achievements of scientists from Galileo onwards, but also their mistakes. He rejects the claim that all scientific knowledge is provisional, by citing examples from chemistry, biology and geology. A major feature of the book is its defense of the view that mathematics was invented rather than discovered. A large number of examples are used to illustrate these points, and many of the deep issues in today's world discussed-from psychology and evolution to quantum theory, consciousness and even religious belief. Disentangling knowledge from opinion and aspiration is a hard task, but this book provided a clear guide to the difficulties.

Why Beliefs Matter

Reflections on the Nature of Science

Author: E. Brian Davies

Publisher: Oxford University Press

ISBN: 0199586209

Category: Mathematics

Page: 250

View: 2559


`It is a brilliant work, beautifully written, and brimming with surprising information and stimulating philosophical speculations.' Notices of teh American Mathematical Society --

Number Theory Through Inquiry

Author: David C. Marshall,Edward Odell,Michael Starbird

Publisher: MAA

ISBN: 0883857510

Category: Mathematics

Page: 140

View: 6313


This innovative textbook leads students on a carefully guided discovery of introductory number theory. The book has two equally significant goals. The first is to help students develop mathematical thinking skills, particularly theorem-proving skills. The other goal is to help students understand some of the wonderfully rich ideas in the mathematical study of numbers. This book is appropriate for a proof transitions course, for independent study, or for a course designed as an introduction to abstract mathematics. It is designed to be used with an instructional technique variously called guided discovery or Modified Moore Method or Inquiry Based Learning (IBL). Instructors' materials explain the instructional method, which gives students a totally different experience compared to a standard lecture course. Students develop an attitude of personal reliance and a sense that they can think effectively about difficult problems; goals that are fundamental to the educational enterprise within and beyond mathematics.

How to Prove It

A Structured Approach

Author: Daniel J. Velleman

Publisher: Cambridge University Press

ISBN: 1139450972

Category: Mathematics

Page: N.A

View: 6329


Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. The author shows how complex proofs are built up from these smaller steps, using detailed 'scratch work' sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. To give students the opportunity to construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. No background beyond standard high school mathematics is assumed. This book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and of course mathematicians.

The Book Thief

Author: Markus Zusak

Publisher: Knopf Books for Young Readers

ISBN: 9780307433848

Category: Young Adult Fiction

Page: 592

View: 2339


DON’T MISS BRIDGE OF CLAY, MARKUS ZUSAK’S FIRST NOVEL SINCE THE BOOK THIEF. The extraordinary #1 New York Times bestseller that is now a major motion picture, Markus Zusak's unforgettable story is about the ability of books to feed the soul. When Death has a story to tell, you listen. It is 1939. Nazi Germany. The country is holding its breath. Death has never been busier, and will become busier still. Liesel Meminger is a foster girl living outside of Munich, who scratches out a meager existence for herself by stealing when she encounters something she can’t resist–books. With the help of her accordion-playing foster father, she learns to read and shares her stolen books with her neighbors during bombing raids as well as with the Jewish man hidden in her basement. In superbly crafted writing that burns with intensity, award-winning author Markus Zusak, author of I Am the Messenger, has given us one of the most enduring stories of our time. “The kind of book that can be life-changing.” —The New York Times “Deserves a place on the same shelf with The Diary of a Young Girl by Anne Frank.” —USA Today

The Biologist's Imagination

Innovation in the Biosciences

Author: William Hoffman,William R. Hoffman,Leo Furcht

Publisher: Oxford University Press (UK)

ISBN: 0199974594

Category: Science

Page: 284

View: 8598


"Scholars and policymakers alike agree that innovation in the biosciences is key to future growth. The field continues to shift and expand, and it is certainly changing the way people live their lives in a variety of ways. But despite the lion's share offederal research dollars being devoted to innovation in the biosciences, the field has yet to live up to its billing as a source of economic productivity and growth. With vast untapped potential to imagine and innovate in the biosciences, adaptation of the innovative model is needed. In The Biologist's Imagination, William Hoffman and Leo Furcht examine the history of innovation in the biosciences, tracing technological innovation from the late eighteenth century to the present and placing special emphasis on how and where technology evolves. Place is key to innovation, from the early industrial age to the rise of the biotechnology industry in the second half of the twentieth century. The book uses the distinct history of bioscientific innovation to discuss current trends as they relate to medicine, agriculture, biofuels, stem-cell research, neuroscience, and more. Ultimately, Hoffman and Furcht argue that, as things currently stand, we fall short in our efforts to innovate in the biosciences; our system of innovation is itself in need of innovation. It needs to adapt to the massive changes brought about by converging technologies, globalization in higher education as well as in finance, and increases in entrepreneurship. The Biologist's Imagination is both an analysis of past models for bioscience innovation and a forward-looking, original argument for how future models should be developed"--

The Language of Mathematics

Making the Invisible Visible

Author: Keith Devlin

Publisher: Macmillan

ISBN: 9780805072549

Category: Mathematics

Page: 352

View: 3229


"The great book of nature," said Galileo, "can be read only by those who know the language in which it was written. And this language is mathematics." In The Language of Mathematics, award-winning author Keith Devlin reveals the vital role mathematics plays in our eternal quest to understand who we are and the world we live in. More than just the study of numbers, mathematics provides us with the eyes to recognize and describe the hidden patterns of life—patterns that exist in the physical, biological, and social worlds without, and the realm of ideas and thoughts within. Taking the reader on a wondrous journey through the invisible universe that surrounds us—a universe made visible by mathematics—Devlin shows us what keeps a jumbo jet in the air, explains how we can see and hear a football game on TV, allows us to predict the weather, the behavior of the stock market, and the outcome of elections. Microwave ovens, telephone cables, children's toys, pacemakers, automobiles, and computers—all operate on mathematical principles. Far from a dry and esoteric subject, mathematics is a rich and living part of our culture. An exploration of an often woefully misunderstood subject, The Language of Mathematics celebrates the simplicity, the precision, the purity, and the elegance of mathematics.

The Grapes of Wrath

Author: John Steinbeck

Publisher: Penguin

ISBN: 0670016918

Category: Fiction

Page: 496

View: 2478


Depicts the hardships and suffering endured by the Joads as they journey from Oklahoma to California during the Depression.

Oppositions and Paradoxes

Philosophical Perplexities in Science and Mathematics

Author: John L. Bell

Publisher: Broadview Press

ISBN: 1554813026

Category: Philosophy

Page: 216

View: 5353


Since antiquity, opposed concepts such as the One and the Many, the Finite and the Infinite, and the Absolute and the Relative, have been a driving force in philosophical, scientific, and mathematical thought. Yet they have also given rise to perplexing problems and conceptual paradoxes which continue to haunt scientists and philosophers. In Oppositions and Paradoxes, John L. Bell explains and investigates the paradoxes and puzzles that arise out of conceptual oppositions in physics and mathematics. In the process, Bell not only motivates abstract conceptual thinking about the paradoxes at issue, but he also offers a compelling introduction to central ideas in such otherwise-difficult topics as non-Euclidean geometry, relativity, and quantum physics. These paradoxes are often as fun as they are flabbergasting. Consider, for example, the famous Tristram Shandy paradox: an immortal man composing an autobiography so slowly as to require a year of writing to describe each day of his life — he would, if he had infinite time, presumably never complete the work, although no individual part of it would remain unwritten. Or think of an office mailbox labelled “mail for those with no mailbox”—if this is a person’s mailbox, how can they possibly have “no mailbox”? These and many other paradoxes straddle the boundary between physics and metaphysics, and demonstrate the hidden difficulty in many of our most basic concepts.

Pathological Altruism

Author: Barbara Oakley

Publisher: OUP USA

ISBN: 0199738572

Category: Medical

Page: 465

View: 6291


Pathological Altruism is a groundbreaking new book - the first to explore the negative aspects of altruism and empathy, seemingly uniformly positive traits. In fact, pathological altruism, in the form of an unhealthy focus on others to the detriment of one's own needs, may underpin some personality disorders. Hyperempathy - an excess of concern for what others think and how they feel - helps explain popular but poorly defined concepts such as codependency. The contributing authors of this book provide a scientific, social, and cultural foundation for the subject of pathological altruism, creating a new field of inquiry. Each author's approach points to one disturbing truth: what we value so much, the altruistic "good" side of human nature, can also have a dark side that we ignore at our peril.

The Shape of Inner Space

String Theory and the Geometry of the Universe's Hidden Dimensions

Author: Shing-Tung Yau,Steve Nadis

Publisher: Basic Books

ISBN: 0465022669

Category: Science

Page: 400

View: 5303


String theory says we live in a ten-dimensional universe, but that only four are accessible to our everyday senses. According to theorists, the missing six are curled up in bizarre structures known as Calabi-Yau manifolds. In The Shape of Inner Space, Shing-Tung Yau, the man who mathematically proved that these manifolds exist, argues that not only is geometry fundamental to string theory, it is also fundamental to the very nature of our universe. Time and again, where Yau has gone, physics has followed. Now for the first time, readers will follow Yau’s penetrating thinking on where we’ve been, and where mathematics will take us next. A fascinating exploration of a world we are only just beginning to grasp, The Shape of Inner Space will change the way we consider the universe on both its grandest and smallest scales.

The Book of Khalid

Author: Ameen Rihani

Publisher: BoD – Books on Demand

ISBN: 3732680789

Category: Fiction

Page: 208

View: 7374


Reproduction of the original: The Book of Khalid by Ameen Rihani

In Bluebeard's Castle

Some Notes Towards the Redefinition of Culture

Author: George Steiner

Publisher: Yale University Press

ISBN: 9780300017106

Category: History

Page: 141

View: 692


The author presents a penetrating analysis of the collapse of Western culture during the last half of the twentieth century.

Divided Spheres

Geodesics and the Orderly Subdivision of the Sphere

Author: Edward S. Popko

Publisher: CRC Press

ISBN: 1466504307

Category: Mathematics

Page: 532

View: 6243


This well-illustrated book—in color throughout—presents a thorough introduction to the mathematics of Buckminster Fuller’s invention of the geodesic dome, which paved the way for a flood of practical applications as diverse as weather forecasting and fish farms. The author explains the principles of spherical design and the three main categories of subdivision based on geometric solids (polyhedra). He illustrates how basic and advanced CAD techniques apply to spherical subdivision and covers modern applications in product design, engineering, science, games, and sports balls.