Mathematics: The Loss of Certainty refutes that myth.

    But from the perspective of mathematics, groupings of ten are arbitrary, and can have serious shortcomings. Twelve would be better for divisibility, and eight is smaller and well suited to repeated halving. Grouping by two, as in binary code, has turned out to have its own remarkable advantages.Paul Lockhart reveals arithmetic not as the rote manipulation of numbers--a practical if mundane branch of knowledge best suited for balancing a checkbook or filling out tax forms--but as a set of ideas that exhibit the fascinating and sometimes surprising behaviors usually reserved for higher branches of mathematics. The essence of arithmetic is the skillful arrangement of numerical information for ease of communication and comparison, an elegant intellectual craft that arises from our desire to count, add to, take away from, divide up, and multiply quantities of important things. Over centuries, humans devised a variety of strategies for representing and using numerical information, from beads and tally marks to adding machines and computers. Lockhart explores the philosophical and aesthetic nature of counting and of different number systems, both Western and non-Western, weighing the pluses and minuses of each.A passionate, entertaining survey of foundational ideas and methods, Arithmetic invites readers to experience the profound and simple beauty of its subject through the eyes of a modern research mathematician.

    Now The Forgotten Ways Handbook moves beyond theory to practice, offering ways for any missionally minded person to apply the ideas contained in The Forgotten Ways to their life and ministry.This intensely practical handbook includes many helpful tools: summary sections encapsulating the ideas contained in each chapter in a popular way; suggested practices to help readers embed missional paradigms concretely; and adult learning-based techniques and examples from other churches and organizations that enable readers to process and assimilate the ideas in a group context. EXCERPTMake no mistake about it; the scope of the change that is required to shift to the kind of movement described in The Forgotten Ways is nothing less than paradigmatic. Every element of mDNA poses a direct challenge to the prevailing ways of doing church and mission. When taken together, all six elements of Apostolic Genius make the task seem enormous. But we don't think it is actually as difficult as it seems. And it is certainly not impossible. The Chinese church proves that a highly institutionalized form of Christianity can become a remarkable movement given the right circumstances. And we don't believe that we have to have persecution to activate Apostolic Genius. Less intense forms of adaptive challenges can, and do, force the church to respond. What we are witnessing in our own day indicates that. Because the church carries the gospel as well as the full coding of Apostolic Genius in her, the potential for world transformation is always present in us. We can always draw upon latent resources and instincts. God is able and very willing to stir his church up. In fact we see this as one of the very special works of the Holy Spirit--to awaken God's people to their calling and destiny as a movement that can and will change the world.

    For centuries, the power of zero savored of the demonic; once harnessed, it became the most important tool in mathematics. Zero follows this number from its birth as an Eastern philosophical concept to its struggle for acceptance in Europe and its apotheosis as the mystery of the black hole. Today, zero lies at the heart of one of the biggest scientific controversies of all time, the quest for the theory of everything. Elegant, witty, and enlightening, Zero is a compelling look at the strangest number in the universe and one of the greatest paradoxes of human thought.

    His book is a fresh look at the questions that occur to anyone who thinks deeply about these matters, questions about free will, self-determination, destiny, choice, and who are we anyway. I believe this is a “breakthrough book.” Robert’s style of writing about such ephemeral and difficult subjects as awareness and consciousness is honest, concise, and accurate. His ability to describe his experiences of living in a reality quite different from conventional ways of thinking is brilliantly unusual. On first encountering Robert Saltzman’s work, I am reminded of the same feelings of discovery, delight and excitement that I remember from meeting Alan Watts’ “The Wisdom of Insecurity”, Krishnamurti’s “Freedom from the Known,” and Chögyam Trungpa’s “Cutting Through Spiritual Materialism.” His clarity of mind shines brightly through every sentence in this book. His skill at making clear the most difficult ramifications and subtleties of awakened consciousness is so free of conventional cluttered thinking, so free of habitual phrases, so free of the taint of religious dogma and the conventional ways of speaking of such difficult matters, that this book stands out for me as an entirely fresh and illuminated exposition of awakened consciousness: an awakened understanding of what it is to be human. —Dr. Robert K. Hall

    It covered jazz, of course, but it also included Davis’s ruminations on race, politics and culture. Fascinated, Hef sent the writer—future Pulitzer-Prize-winning author Alex Haley, an unknown at the time—back to glean even more opinion and insight from Davis. The resulting exchange, published in the September 1962 issue, became the first official Playboy Interview and kicked off a remarkable run of public inquisition that continues today—and that has featured just about every cultural titan of the last half century.To celebrate the Interview’s 50th anniversary, the editors of Playboy have culled 50 of its most (in)famous Interviews and will publish them over the course of 50 weekdays (from September 4, 2012 to November 12, 2012) via Amazon’s Kindle Direct platform. Here is the interview with the novelist and philosopher Ayn Rand from the March 1964 issue.

    W. R. Dedekind. The first presents Dedekind's theory of the irrational number-the Dedekind cut idea-perhaps the most famous of several such theories created in the 19th century to give a precise meaning to irrational numbers, which had been used on an intuitive basis since Greek times. This paper provided a purely arithmetic and perfectly rigorous foundation for the irrational numbers and thereby a rigorous meaning of continuity in analysis.The second essay is an attempt to give a logical basis for transfinite numbers and properties of the natural numbers. It examines the notion of natural numbers, the distinction between finite and transfinite (infinite) whole numbers, and the logical validity of the type of proof called mathematical or complete induction.The contents of these essays belong to the foundations of mathematics and will be welcomed by those who are prepared to look into the somewhat subtle meanings of the elements of our number system. As a major work of an important mathematician, the book deserves a place in the personal library of every practicing mathematician and every teacher and historian of mathematics. Authorized translations by "Vooster " V. Beman.

    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.

    Composed of such luminaries as Kurt Gödel and Rudolf Carnap, and stimulated by the works of Ludwig Wittgenstein and Karl Popper, the Vienna Circle left an indelible mark on science.Exact Thinking in Demented Times tells the often outrageous, sometimes tragic, and never boring stories of the men who transformed scientific thought. A revealing work of history, this landmark book pays tribute to those who dared to reinvent knowledge from the ground up.

    Yet the questions remain; how does one practise Hinduism today under the neon glare of modernity? And is there any contradiction in the practice of Hinduism and our roles in modern life?Like the lotus that grows in water whose leaves and petals remain dry, the yogi derives nourishment from the material world but remains undefiled and untouched by its seductions. In this way, one who wishes to lead a spiritual life can simultaneously perform the role of a responsible member of modern society; neither necessarily compromises the other. With care, practice and knowledge one can reap the benefits of both worlds - to be spiritual you do not have to run away from the world. This book examines the philosophy, spiritual teachings, metaphysics and cultural practices of Hinduism- known collectively as the Sanatana Dharma - and attempts to give one a grasp of its true essence.ForewordHinduism is a many-splendoured and multi-faceted edifice, containing and reflecting an endless array of possibilities for spiritual growth and integration. Based essentially upon the sublime teachings of the Upanishads, the secret of Hinduism’s continued vitality down through the long and tortuous corridors of time, despite repeated invasions and repression, lies in its capacity for creative reinterpretation. Indeed, the whole history of Hinduism can be viewed as a series of challenges and creative responses, a process which continues down to the present day. In the emerging global society it is my conviction that the universal principles of Hinduism are becoming increasingly relevant as we hurtle headlong into the third millennium A.D. astride the irreversible arrow of time.Theoretical formulations apart, Hinduism involves treading a spiritual path, and there are many. It is these individual pathways involving a creative interaction between the Guru and the disciple that provide the circulatory life-blood of Hinduism, making it a powerful vehicle for inner growth and spiritual realisation. In this context, the experience of individuals who have trodden the path is of great value. The present book Jewel in the Lotus revolves around the life experience and teachings of Mumtaz Ali, better known as ‘M’. Some may find it surprising that a person born a Muslim should have such a deep insight into and experience with the Hindu tradition, but the real spiritual path knows no boundary of race or religion, sex or creed, language or nationality. And the mystics of all the world’s great religions – the Rishis, the Siddhas, the Tirthankaras, the Bodhisattvas, the Sufis, the Gurus and the saints have all illuminated one or other facets of the immeasurable resplendence of the Divine.I have had occasion to know ‘M’ over the last few years, and we have spoken together on several occasions including a three-day workshop on the Kena Upanishad over which I presided. He combines an excellent grasp of theUpanishadic teachings with deep insight into the heart of the spiritual tradition, based on his own remarkable experiences. This book in which he has dealt with some of the deeper aspects of Hinduism will, I am sure, be of great interest to students of contemporary religion, as well as seekers of truth around the world.— Dr. Karan Singh

    Part 1, on propositional logic, is the old Introduction, but contains much new material. Part 2 is entirely new, and covers quantification and identity for all the logics in Part 1. The material is unified by the underlying theme of world semantics. All of the topics are explained clearly using devices such as tableau proofs, and their relation to current philosophical issues and debates are discussed. Students with a basic understanding of classical logic will find this book an invaluable introduction to an area that has become of central importance in both logic and philosophy. It will also interest people working in mathematics and computer science who wish to know about the area.

    Our eyes and minds are drawn to symmetrical objects, from the pyramid to the pentagon. Of fundamental significance to the way we interpret the world, this unique, pervasive phenomenon indicates a dynamic relationship between objects. In chemistry and physics, the concept of symmetry explains the structure of crystals or the theory of fundamental particles; in evolutionary biology, the natural world exploits symmetry in the fight for survival; and symmetry—and the breaking of it—is central to ideas in art, architecture, and music.Combining a rich historical narrative with his own personal journey as a mathematician, Marcus du Sautoy takes a unique look into the mathematical mind as he explores deep conjectures about symmetry and brings us face-to-face with the oddball mathematicians, both past and present, who have battled to understand symmetry's elusive qualities. He explores what is perhaps the most exciting discovery to date—the summit of mathematicians' mastery in the field—the Monster, a huge snowflake that exists in 196,883-dimensional space with more symmetries than there are atoms in the sun.What is it like to solve an ancient mathematical problem in a flash of inspiration? What is it like to be shown, ten minutes later, that you've made a mistake? What is it like to see the world in mathematical terms, and what can that tell us about life itself? In Symmetry, Marcus du Sautoy investigates these questions and shows mathematical novices what it feels like to grapple with some of the most complex ideas the human mind can comprehend.

    We're all looking for new ways of thinking that can bring about real solutions to modern problems, from the pursuit of inner serenity to solving world conflicts. In Do You QuantumThink? author Dianne Collins shares her ingenious discovery that reveals a critical missing link to make sense of our changing times. Her discovery provides us with the understanding and methodology to rise above problems of today by laying the foundation for an entirely new way to think.Part science, part philosophy, part spirituality, Do You QuantumThink? draws on a wide spectrum of sources, from cutting edge innovations in the sciences to the insights of the world's greatest spiritual leaders. This book will make you laugh, free you from limiting ideas, and introduce you to the most advanced principles and practical methods for living. Do You QuantumThink? will rock your world in the best of ways as you experience one revelation after another.

    Imagine that you face three doors, behind one of which is a prize. You choose one but do not open it. The host--call him Monty Hall--opens a different door, alwayschoosing one he knows to be empty. Left with two doors, will you do better by sticking with your first choice, or by switching to the other remaining door? In this light-hearted yet ultimately serious book, Jason Rosenhouse explores the history of this fascinating puzzle. Using a minimum ofmathematics (and none at all for much of the book), he shows how the problem has fascinated philosophers, psychologists, and many others, and examines the many variations that have appeared over the years. As Rosenhouse demonstrates, the Monty Hall Problem illuminates fundamental mathematical issuesand has abiding philosophical implications. Perhaps most important, he writes, the problem opens a window on our cognitive difficulties in reasoning about uncertainty.

    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.