In 1931 Kurt Gödel published his fundamental paper, "On Formally Undecidable Propositions of Principia Mathematica and Related Systems." This revolutionary paper challenged certain basic assumptions underlying much research in mathematics and logic. Gödel received public recognition of his work in 1951 when he was awarded the first Albert Einstein Award for achievement in the natural sciences—perhaps the highest award of its kind in the United States. The award committee described his work in mathematical logic as "one of the greatest contributions to the sciences in recent times." However, few mathematicians of the time were equipped to understand the young scholar's complex proof. Ernest Nagel and James Newman provide a readable and accessible explanation to both scholars and non-specialists of the main ideas and broad implications of Gödel's discovery. It offers every educated person with a taste for logic and philosophy the chance to understand a previously difficult and inaccessible subject. Marking the 50th anniversary of the original publication of Gödel's Proof , New York University Press is proud to publish this special anniversary edition of one of its bestselling and most frequently translated books. With a new introduction by Douglas R. Hofstadter, this book will appeal students, scholars, and professionals in the fields of mathematics, computer science, logic and philosophy, and science.

This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.

When Mr Ruche, a reclusive Parisian bookseller, receives a letter from a long lost friend in the Amazon bequesting him a vast library of mathematical books, he is propelled into a great exploration of the story of maths, from brilliant Greek thinkers, such as Archimedes and Pythagoras, to the modern-day genius Fermat. Meanwhile Max, a deaf boy whose dysfunctional family live with Mr Ruche, finds a voluble parrot in a local fleamarket. He turns out to be a bird who discusses maths with anyone who will listen. So when Mr Ruche learns of his friend's mysterious death in the rainforests of Brazil he decides that with the parrot's help he will use these books to teach Max and his twin brother and sister the mysteries and wonders of numbers and shapes. Charming, fresh, with a narrative which races along, the novel takes the reader on a delightful journey through the history of mathematics.

"Weinberg takes us across centuries from ancient Miletus to medieval Baghdad and Oxford, from Plato's Academy and the Museum of Alexandria to the cathedral school of Chartres and the Royal Society of London. He shows that the scientists of ancient and medieval times not only did not understand what we understand about the world--they did not understand what there is to understand, or how to understand it. Yet over the centuries, through the struggle to solve such mysteries as the curious backward movement of the planets and the rise and fall of the tides, the modern discipline of science eventually emerged"--Amazon.com ; Presents a commentary on the history of science that examines historic clashes and collaborations between science and the competing realms of religion, technology, poetry, mathematics, and philosophy

With his ability to make abstract ideas concrete and approachable, Berlinski tells an engrossing tale and introduces us to one of the greatest of all human endeavours.

"The invention of numerals is perhaps the greatest abstraction the human mind has ever created. Virtually everything in our lives is digital, numerical, or quantified. The story of how and where we got these numerals, which we so depend on, has for thousands of years been shrouded in mystery. Finding Zero is an adventure filled saga of Amir Aczel's lifelong obsession: to find the original sources of our numerals. Aczel has doggedly crisscrossed the ancient world, scouring dusty, moldy texts, cross examining so-called scholars who offered wildly differing sets of facts, and ultimately penetrating deep into a Cambodian jungle to find a definitive proof. Here, he takes the reader along for the ride. The history begins with the early Babylonian cuneiform numbers, followed by the later Greek and Roman letter numerals. Then Aczel asks the key question: where do the numbers we use today, the so-called Hindu-Arabic numerals, come from? It is this search that leads him to explore uncharted territory, to go on a grand quest into India, Thailand, Laos, Vietnam, and ultimately into the wilds of Cambodia. There he is blown away to find the earliest zero--the keystone of our entire system of numbers--on a crumbling, vine-covered wall of a seventh-century temple adorned with eaten-away erotic sculptures. While on this odyssey, Aczel meets a host of fascinating characters: academics in search of truth, jungle trekkers looking for adventure, surprisingly honest politicians, shameless smugglers, and treacherous archaeological thieves--who finally reveal where our numbers come from. "--

René Descartes (1596–1650) is one of the towering and central figures in Western philosophy and mathematics. His apothegm “ Cogito, ergo sum ” marked the birth of the mind-body problem, while his creation of so-called Cartesian coordinates have made our physical and intellectual conquest of physical space possible. But Descartes had a mysterious and mystical side, as well. Almost certainly a member of the occult brotherhood of the Rosicrucians, he kept a secret notebook, now lost, most of which was written in code. After Descartes’s death, Gottfried Leibniz, inventor of calculus and one of the greatest mathematicians in history, moved to Paris in search of this notebook—and eventually found it in the possession of Claude Clerselier, a friend of Descartes. Leibniz called on Clerselier and was allowed to copy only a couple of pages—which, though written in code, he amazingly deciphered there on the spot. Leibniz’s hastily scribbled notes are all we have today of Descartes’s notebook, which has disappeared. Why did Descartes keep a secret notebook, and what were its contents? The answers to these questions lead Amir Aczel and the reader on an exciting, swashbuckling journey, and offer a fascinating look at one of the great figures of Western culture.

Traces the development of mathematics from its beginnings in Babylonia and ancient Egypt to the work of Riemann and Godel in modern times.

The son of a prominent Japanese mathematician who came to the United States after World War II, Ken Ono was raised on a diet of high expectations and little praise. Rebelling against his pressure-cooker of a life, Ken determined to drop out of high school to follow his own path. To obtain his father's approval, he invoked the biography of the famous Indian mathematical prodigy Srinivasa Ramanujan, whom his father revered, who had twice flunked out of college because of his single-minded devotion to mathematics. Ono describes his rocky path through college and graduate school, interweaving Ramanujan's story with his own and telling how at key moments, he was inspired by Ramanujan and guided by mentors who encouraged him to pursue his interest in exploring Ramanujan's mathematical legacy. Picking up where others left off, beginning with the great English mathematician G.H. Hardy, who brought Ramanujan to Cambridge in 1914, Ono has devoted his mathematical career to understanding how in his short life, Ramanujan was able to discover so many deep mathematical truths, which Ramanujan believed had been sent to him as visions from a Hindu goddess. And it was Ramanujan who was ultimately the source of reconciliation between Ono and his parents. Ono's search for Ramanujan ranges over three continents and crosses paths with mathematicians whose lives span the globe and the entire twentieth century and beyond. Along the way, Ken made many fascinating discoveries. The most important and surprising one of all was his own humanity

A "skillful and literate" ( New York Times Book Review ) biography of the persecuted genius who helped create the modern computer. To solve one of the great mathematical problems of his day, Alan Turing proposed an imaginary computer. Then, attempting to break a Nazi code during World War II, he successfully designed and built one, thus ensuring the Allied victory. Turing became a champion of artificial intelligence, but his work was cut short. As an openly gay man at a time when homosexuality was illegal in England, he was convicted and forced to undergo a humiliating "treatment" that may have led to his suicide. With a novelist's sensitivity, David Leavitt portrays Turing in all his humanity?his eccentricities, his brilliance, his fatal candor?and elegantly explains his work and its implications.

Geometry defines the world around us, helping us make sense of everything from architecture to military science to fashion. Euclid's Elements is arguably the most influential book in the history of mathematics. Berlinski provides a concise homage to this elusive mathematician and his staggering achievements, and shows that, for centuries, scientists and thinkers have relied on Euclid's axiomatic system, a method of proof still taught in classrooms around the world.

Astrophysicist and author Mario Livio investigates perhaps the most human of all our characteristics—curiosity—in this “lively, expert, and definitely not dumbed-down account” ( Kirkus Reviews ) as he explores our innate desire to know why . Experiments demonstrate that people are more distracted when they overhear a phone conversation—where they can know only one side of the dialogue—than when they overhear two people talking and know both sides. Why does half a conversation make us more curious than a whole conversation? “Have you ever wondered why we wonder why? Mario Livio has, and he takes you on a fascinating quest to understand the origin and mechanisms of our curiosity. I thoroughly recommend it.” (Adam Riess, Nobel Prize Winner in Physics, 2011). Curiosity is not only at the heart of mystery and suspense novels, it is also essential to other creative endeavors, from painting to sculpture to music. It is the principal driver of basic scientific research. Even so, there is still no definitive scientific consensus about why we humans are so curious, or about the mechanisms in our brain that are responsible for curiosity. In the ever-fascinating Why? Livio interviewed scientists in several fields to explore the nature of curiosity. He examined the lives of two of history’s most curious geniuses, Leonardo da Vinci and Richard Feynman. He also talked to people with boundless curiosity: a superstar rock guitarist who is also an astrophysicist; an astronaut with degrees in computer science, biology, literature, and medicine. What drives these people to be curious about so many subjects? An astrophysicist who has written about mathematics, biology, and now psychology and neuroscience, Livio has firsthand knowledge of his subject which he explores in a lucid, entertaining way that will captivate anyone who is curious about curiosity.

"Bestselling popular science author Amir Aczel selects the most fascinating individuals and stories in the history of mathematics, presenting a colorful narrative that explores the quirky personalities behind some of the most profound, enduring theorems. Through such mathematical geniuses as Archimedes, Leonardo of Pisa (a.k.a. Fibonacci), Tartaglia ("the stutterer"), Descartes, Gottfried Leibniz, Carl Gauss, Joseph Fourier (Napoleon's mathematician), Evariste Galois, Georg Cantor, Ramanujan, and "Nicholas Bourbaki," we gather little known details about the alliances and rivalries that profoundly impacted the development of what the scheming doctor-turned-mathematician Geronimo Girolamo called "The Great Art." This story of mathematics is not your dry "college textbook" account; tales of duels, battlefield heroism, flamboyant arrogance, pranks, secret societies, imprisonment, feuds, theft, and even some fatal errors of judgment fill these pages (clearly, genius doesn't guarantee street smarts). Ultimately, readers will come away from this book entertained, with a newfound appreciation of the tenacity, complexity, eccentricity, and brilliance of the mathematical genius"--

To most graph theorists there are two outstanding landmarks in the history of their subject. One is Euler's solution of the Konigsberg Bridges Problem, dated 1736, and the other is the appearance of Denes Konig's textbook in 1936. "From Konigsberg to Konig's book" sings the poetess, "So runs the graphic tale . . . " [10]. There were earlier books that took note of graph theory. Veb len's Analysis Situs, published in 1931, is about general combinato rial topology. But its first two chapters, on "Linear graphs" and "Two-Dimensional Complexes," are almost exclusively concerned with the territory still explored by graph theorists. Rouse Ball's Mathematical Recreations and Essays told, usually without proofs, of the major graph-theoretical advances ofthe nineteenth century, of the Five Colour Theorem, of Petersen's Theorem on I-factors, and of Cayley's enumerations of trees. It was Rouse Ball's book that kindled my own graph-theoretical enthusiasm. The graph-theoretical papers of Hassler Whitney, published in 1931-1933, would have made an excellent textbook in English had they been collected and published as such. But the honour of presenting Graph Theory to the mathe matical world as a subject in its own right, with its own textbook, belongs to Denes Konig. Low was the prestige of Graph Theory in the Dirty Thirties. It is still remembered, with resentment now shading into amuse ment, how one mathematician scorned it as "The slums of Topol ogy.""

"From the time of Pythagoras, we have been tempted to treat numbers as the ultimate or only truth. This book tells the history of that habit of thought. But more, it argues that the logic of counting sacrifices much of what makes us human, and that we have a responsibility to match the objects of our attention to the forms of knowledge that do them justice. Humans have extended the insights and methods of number and mathematics to more and more aspects of the world, even to their gods and their religions.Today those powers are greater than ever, as computation is applied to virtually every aspect of human activity.But the rules of mathematics do not strictly apply to many things-from elementary particles to people-in the world.By subjecting such things to the laws of logic and mathematics, we gain some kinds of knowledge, but we also lose others. How do our choices about what parts of the world to subject to the logics of mathematics affect how we live and how we die?This question is rarely asked, but it is urgent, because the sciences built upon those laws now govern so much of our knowledge, from physics to psychology.Number and Knowledge sets out to ask it. In chapters proceeding chronologically from Ancient Greek philosophy and the rise of monotheistic religions to the emergence of modern physics and economics, the book traces how ideals, practices, and habits of thought formed over millennia have turned number into the foundation-stone of human claims to knowledge and certainty.But the book is also a philosophical and poetic exhortation to take responsibility for that history, for the knowledge it has produced, and for the many aspects of the world and of humanity that it ignores or endangers.To understand what can be counted and what can't is to embrace the ethics of purposeful knowing"--

The world around us is saturated with numbers. They are a fundamental pillar of our modern society, and accepted and used with hardly a second thought. But how did this state of affairs come to be? In this book, Leo Corry tells the story behind the idea of number from the early days of the Pythagoreans, up until the turn of the twentieth century. He presents an overview of how numbers were handled and conceived in classical Greek mathematics, in the mathematics of Islam, in European mathematics of the middle ages and the Renaissance, during the scientific revolution, all the way through to the. --

Out of the Ghetto is an account of the developing interrelationship between the Jews and their Gentile environment unique in its breadth and objectivity. He presents the story of Jewish emancipation as a whole, from both Jewish and non-Jewish points of view. If the results of the Jewish emancipation process differed from country to country, the forces effecting the changes were identical—the upheaval of the French Revolution, the loosening of bonds between church and state, and the ideas of the Enlightenment. It was those humanistic ideas which made possible the Jew's transition from the ghetto to partial inclusion in society at large and which attracted Jewish intellectuals to the "secular knowledge" of languages, mathematics, philosophy, and the wider world beyond their ancient learning.

This book gives a remarkably fine account of the influences mathematics has exerted on the development of philosophy, the physical sciences, religion, and the arts in Western life.

"Practical, scientific, philosophical & artistic problems have caused men to investigate mathematics. But there is one other motive which is as strong as any of these-the search for beauty. Mathematics is an art, & as such affords the pleasures which all the arts afford."--The Author Preface Why mathematics? A historical orientation Logic & mathematics Number: the fundamental concept Algebra, the higher arithmetic The nature & uses of Euclidean geometry Charting the earth & the heavens The mathematical order of nature The awakening of Europe Mathematics & painting in the Renaissance Projective geometry Coordinate geometry The simplest formulas in action Parametric equations & curvilinear motion The application of formulas to gravitation The differential calculus The integral calculus Trigonometric functions & oscillatory motion The trigonometric analysis of musical sounds Non-Euclidean geometries & their significance Arithmetics & their algebras The statistical approach to the social & biological sciences The theory of probability The nature & values of mathematics Table of Trigonometic Ratios Answers to Selected & Review Exercises Additional Answers & Solutions Index

This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.