The book represents the first philosophically sound discussion of the concept of number in Western civilization. It profoundly influenced developments in the philosophy of mathematics and in general ontology.

    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.New York University Press is proud to publish this special edition of one of its bestselling 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.

    In it, Russell offers a nontechnical, undogmatic account of his philosophical criticism as it relates to arithmetic and logic. Rather than an exhaustive treatment, however, the influential philosopher and mathematician focuses on certain issues of mathematical logic that, to his mind, invalidated much traditional and contemporary philosophy.In dealing with such topics as number, order, relations, limits and continuity, propositional functions, descriptions, and classes, Russell writes in a clear, accessible manner, requiring neither a knowledge of mathematics nor an aptitude for mathematical symbolism. The result is a thought-provoking excursion into the fascinating realm where mathematics and philosophy meet — a philosophical classic that will be welcomed by any thinking person interested in this crucial area of modern thought.

    A lecture class taught by Wittgenstein, however, hardly resembled a lecture. He sat on a chair in the middle of the room, with some of the class sitting in chairs, some on the floor. He never used notes. He paused frequently, sometimes for several minutes, while he puzzled out a problem. He often asked his listeners questions and reacted to their replies. Many meetings were largely conversation. These lectures were attended by, among others, D. A. T. Gasking, J. N. Findlay, Stephen Toulmin, Alan Turing, G. H. von Wright, R. G. Bosanquet, Norman Malcolm, Rush Rhees, and Yorick Smythies. Notes taken by these last four are the basis for the thirty-one lectures in this book. The lectures covered such topics as the nature of mathematics, the distinctions between mathematical and everyday languages, the truth of mathematical propositions, consistency and contradiction in formal systems, the logicism of Frege and Russell, Platonism, identity, negation, and necessary truth. The mathematical examples used are nearly always elementary.

    In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably (but in different ways) with Bertrand Russell, W. V. Quine, and Godel himself, and which remains at the focus of Anglo-Saxon philosophical discussion. The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers. It is a substantially revised version of the edition first published in 1964 and includes a revised bibliography. The volume will be welcomed as a major work of reference at this level in the field.

    Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations.

    This book contains my answer to that question. The purpose of the book is to tell the beginning student of advanced mathematics the basic set- theoretic facts of life, and to do so with the minimum of philosophical discourse and logical formalism. The point of view throughout is that of a prospective mathematician anxious to study groups, or integrals, or manifolds. From this point of view the concepts and methods of this book are merely some of the standard mathematical tools; the expert specialist will find nothing new here. Scholarly bibliographical credits and references are out of place in a purely expository book such as this one. The student who gets interested in set theory for its own sake should know, however, that there is much more to the subject than there is in this book. One of the most beautiful sources of set-theoretic wisdom is still Hausdorff's Set theory. A recent and highly readable addition to the literature, with an extensive and up-to-date bibliography, is Axiomatic set theory by Suppes.

    Athos, Greece, to haul off monks engaged in a dangerously heretical practice known as Name Worshipping. Exiled to remote Russian outposts, the monks and their mystical movement went underground. Ultimately, they came across Russian intellectuals who embraced Name Worshipping--and who would achieve one of the biggest mathematical breakthroughs of the twentieth century, going beyond recent French achievements.Loren Graham and Jean-Michel Kantor take us on an exciting mathematical mystery tour as they unravel a bizarre tale of political struggles, psychological crises, sexual complexities, and ethical dilemmas. At the core of this book is the contest between French and Russian mathematicians who sought new answers to one of the oldest puzzles in math: the nature of infinity. The French school chased rationalist solutions. The Russian mathematicians, notably Dmitri Egorov and Nikolai Luzin--who founded the famous Moscow School of Mathematics--were inspired by mystical insights attained during Name Worshipping. Their religious practice appears to have opened to them visions into the infinite--and led to the founding of descriptive set theory.The men and women of the leading French and Russian mathematical schools are central characters in this absorbing tale that could not be told until now. Naming Infinity is a poignant human interest story that raises provocative questions about science and religion, intuition and -creativity.

    Providing a concise introduction to abstract algebra, this work unfolds some of the fundamental systems with the aim of reaching applicable, significant results.

    Ian Stewart presents us with a wealth of magical puzzles, each one spun around an amazing tale, including Counting the Cattle of the Sun, The Great Drain Robbery, and Preposterous Piratical Predicaments. Fully illustrated with explanatory diagrams, each tale is told with engaging wit, sure to amuse everyone with an interest in puzzles and mathematics. Along the way, we also meet many curious characters. Containing twenty specially-commissioned cartoons, this book will delight all who are familiar with Stewart's many other books, such as What Shape is a Snowflake? and Flatterland and anyone interested in mathematical problems. In short, these stories are engaging, challenging, and lots of fun!

    He shows that there are many ways of achieving happiness; for example, there is the inherent happiness that comes with the love of a child; the competitive happiness of triumphing over your opponents; the sensual happiness of the hedonist. Rather than preaching a particular behavior or way of life, Morris provides knowledge that we can use, if we wish, to make ourselves happier.

    A modern classic by an accomplished mathematician and best-selling author has been updated to encompass and explain the recent headline-making advances in the field in non-technical terms.

    More and more people are discovering the power of their minds to shape the world around them faster than ever before. The question is: how do you create the reality of your design?Brian Scott wants to help you find the answer. After walking away unscathed from a near-fatal shooting in his home, Brian began a fanatical search for answers. He deepened his research into parallel realities, quantum mechanics, and consciousness to uncover what happened in his close call with death. Along the way, he developed a series of techniques capable of creating profound transformations.In The Reality Revolution: The Mind-Blowing Movement to Hack Your Reality, Brian introduces you to the techniques that have helped his clients find lasting love, create wealth, and revitalize health. You'll learn how to surf through parallel realities and unlock the power of your mind through a mix of researched and science-backed techniques like qi gong, meditation, quantum jumping, energy work, and reality transurfing. If you're ready to create an incredible reality for yourself, this book shows you the way.

    Mathematics, Applied & Natural Sciences

    Being Mortal by Atul Gawande - A 20-minute Summary Inside this Instaread Summary: • Overview of the entire book• Introduction to the important people in the book• Summary and analysis of all the chapters in the book• Key Takeaways of the book• A Reader's Perspective Preview of this summary: Chapter 1 Gawande grew up in Ohio. His parents were immigrants from India and both were doctors. His grandparents stayed in India, and there were few older people in his neighborhood, so he had little experience with aging or death until he met his wife’s grandmother, Alice Hobson. Hobson was seventy-seven and living on her own in Virginia. She was a spirited widow who fixed her own plumbing and volunteered with Meals On Wheels. However, Hobson was losing strength and height steadily each year as her arthritis worsened.Gawande’s father enthusiastically adopted the customs of his new country, but he could not understand the way in which seniors were treated in the US. In India, the elderly were treated with great respect and lived out their lives with family.In the United States, Sitaram Gawande, Gawande’s grandfather, likely would have been sent to a nursing home like most of the elderly who cannot handle the basics of daily living by themselves. However, in India, Sitaram Gawande was able to live in his own home and manage his own affairs, with family constantly around him. He died at the age of one hundred and ten when he fell off a bus during a business trip.Until recently, most elderly people stayed with their families. Even as the nuclear family unit became predominant, replacing the multi-generational family unit, people cared for their elderly relatives. Families were large and one child, usually a daughter, would not marry in order to take care of the parents.This has changed in much of the world, where elderly people end up struggling to live alone, like Hobson, rather than living with dignity amid family, like Sitaram Gawande.One cause of this change can be found in the nature of knowledge. When few people lived to be very old, elders were honored. Their store of knowledge was greatly useful. People often portrayed themselves as older to command respect. Modern society’s emphasis on youth is a complete reversal of this attitude. Technological advances are perceived as the territory of the young, and everyone wants to be younger. High-tech job opportunities are all over the world, and young people do not hesitate to leave their parents behind to pursue them.In developed countries, parents embrace the concept of a retirement filled with leisure activities. Parents are happy to begin living for themselves once children are grown. However, this system only works for young, healthy retirees, but not for those who cannot continue to be independent. Hobson, for example, was falling frequently and suffering memory lapses. Her doctor did tests and wrote prescriptions, but did not know what to do about her deteriorating condition. Neither did her family… About the Author With Instaread Summaries, you can get the summary of a book in 30 minutes or less. We read every chapter, summarize and analyze it for your convenience.