Today, unfortunately, the traditional place of mathematics in education is in grave danger. The teaching and learning of mathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ability but does not lead to real understanding or to greater intellectual independence. This new edition of Richard Courant's and Herbert Robbins's classic work seeks to address this problem. Its goal is to put the meaning back into mathematics.Written for beginners and scholars, for students and teachers, for philosophers and engineers, What is Mathematics? Second Edition is a sparkling collection of mathematical gems that offers an entertaining and accessible portrait of the mathematical world. Covering everything from natural numbers and the number system to geometrical constructions and projective geometry, from topology and calculus to matters of principle and the Continuum Hypothesis, this fascinating survey allows readers to delve into mathematics as an organic whole rather than an empty drill in problem solving. With chapters largely independent of one another and sections that lead upward from basic to more advanced discussions, readers can easily pick and choose areas of particular interest without impairing their understanding of subsequent parts.Brought up to date with a new chapter by Ian Stewart, What is Mathematics? Second Edition offers new insights into recent mathematical developments and describes proofs of the Four-Color Theorem and Fermat's Last Theorem, problems that were still open when Courant and Robbins wrote this masterpiece, but ones that have since been solved.Formal mathematics is like spelling and grammar - a matter of the correct application of local rules. Meaningful mathematics is like journalism - it tells an interesting story. But unlike some journalism, the story has to be true. The best mathematics is like literature - it brings a story to life before your eyes and involves you in it, intellectually and emotionally. What is Mathematics is like a fine piece of literature - it opens a window onto the world of mathematics for anyone interested to view.

    But as anyone who has ever opened up a spreadsheet packed with seemingly infinite lines of data knows, numbers aren't enough: we need to know how to make those numbers talk. In The Model Thinker, social scientist Scott E. Page shows us the mathematical, statistical, and computational models—from linear regression to random walks and far beyond—that can turn anyone into a genius. At the core of the book is Page's "many-model paradigm," which shows the reader how to apply multiple models to organize the data, leading to wiser choices, more accurate predictions, and more robust designs. The Model Thinker provides a toolkit for business people, students, scientists, pollsters, and bloggers to make them better, clearer thinkers, able to leverage data and information to their advantage.

    These topics lie at the heart of many exciting areas of contemporary science and engineering - communication, signal processing, data mining, machine learning, pattern recognition, computational neuroscience, bioinformatics, and cryptography. This textbook introduces theory in tandem with applications. Information theory is taught alongside practical communication systems, such as arithmetic coding for data compression and sparse-graph codes for error-correction. A toolbox of inference techniques, including message-passing algorithms, Monte Carlo methods, and variational approximations, are developed alongside applications of these tools to clustering, convolutional codes, independent component analysis, and neural networks. The final part of the book describes the state of the art in error-correcting codes, including low-density parity-check codes, turbo codes, and digital fountain codes -- the twenty-first century standards for satellite communications, disk drives, and data broadcast. Richly illustrated, filled with worked examples and over 400 exercises, some with detailed solutions, David MacKay's groundbreaking book is ideal for self-learning and for undergraduate or graduate courses. Interludes on crosswords, evolution, and sex provide entertainment along the way. In sum, this is a textbook on information, communication, and coding for a new generation of students, and an unparalleled entry point into these subjects for professionals in areas as diverse as computational biology, financial engineering, and machine learning.

    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.

    Remarkable for his range of thought and his mastery of treatment, Archimedes addressed such topics as the famous problems of the ratio of the areas of a cylinder and an inscribed sphere; the measurement of a circle; the properties of conoids, spheroids, and spirals; and the quadrature of the parabola. This edition offers an informative introduction with many valuable insights into the ancient mathematician's life and thought as well as the views of his contemporaries. Modern mathematicians, physicists, science historians, and logicians will find this volume a source of timeless fascination.

    In this book Daniel Wegner offers a novel understanding of the issue. Like actions, he argues, the feeling of conscious will is created by the mind and brain. Yet if psychological and neural mechanisms are responsible for all human behavior, how could we have conscious will? The feeling of conscious will, Wegner shows, helps us to appreciate and remember our authorship of the things our minds and bodies do. Yes, we feel that we consciously will our actions, Wegner says, but at the same time, our actions happen to us. Although conscious will is an illusion, it serves as a guide to understanding ourselves and to developing a sense of responsibility and morality.Approaching conscious will as a topic of psychological study, Wegner examines the issue from a variety of angles. He looks at illusions of the will?-those cases where people feel that they are willing an act that they are not doing or, conversely, are not willing an act that they in fact are doing. He explores conscious will in hypnosis, Ouija board spelling, automatic writing, and facilitated communication, as well as in such phenomena as spirit possession, dissociative identity disorder, and trance channeling. The result is a book that sidesteps endless debates to focus, more fruitfully, on the impact on our lives of the illusion of conscious will.Selected as a Finalist in the category of Psychology/Mental Health in the 2002 Independent Publisher Book Awards (IPPYs) presented by Independent Publisher Magazine., Silver Award Winner for Philosophy in the 2002 ForeWord Magazine Book of the Year Awards. and Selected as an Outstanding Academic Book for 2002 by Choice Magazine

    It is difficult to think about law, politics, religion, public policy, intimate relationships, morality—as well as feelings of remorse or personal achievement—without first imagining that every person is the true source of his or her thoughts and actions. And yet the facts tell us that free will is an illusion.In this enlightening book, Sam Harris argues that this truth about the human mind does not undermine morality or diminish the importance of social and political freedom, but it can and should change the way we think about some of the most important questions in life.

    Its aim is to deduce all the fundamental propositions of logic and mathematics from a small number of logical premises and primitive ideas, establishing that mathematics is a development of logic. This abridged text of Volume I contains the material that is most relevant to an introductory study of logic and the philosophy of mathematics (more advanced students will of course wish to refer to the complete edition). It contains the whole of the preliminary sections (which present the authors' justification of the philosophical standpoint adopted at the outset of their work); the whole of Part I (in which the logical properties of propositions, propositional functions, classes and relations are established); section A of Part II (dealing with unit classes and couples); and Appendices A and C (which give further developments of the argument on the theory of deduction and truth functions).

    Edited by the Sustainability Institute’s Diana Wright, this essential primer brings systems thinking out of the realm of computers and equations and into the tangible world, showing readers how to develop the systems-thinking skills that thought leaders across the globe consider critical for 21st-century life.Some of the biggest problems facing the world—war, hunger, poverty, and environmental degradation—are essentially system failures. They cannot be solved by fixing one piece in isolation from the others, because even seemingly minor details have enormous power to undermine the best efforts of too-narrow thinking.While readers will learn the conceptual tools and methods of systems thinking, the heart of the book is grander than methodology. Donella Meadows was known as much for nurturing positive outcomes as she was for delving into the science behind global dilemmas. She reminds readers to pay attention to what is important, not just what is quantifiable, to stay humble, and to stay a learner.In a world growing ever more complicated, crowded, and interdependent, Thinking in Systems helps readers avoid confusion and helplessness, the first step toward finding proactive and effective solutions.

    Yet brain scientists have uncovered details every business leader, parent, and teacher should know—like the need for physical activity to get your brain working its best.How do we learn? What exactly do sleep and stress do to our brains? Why is multi-tasking a myth? Why is it so easy to forget—and so important to repeat new knowledge? Is it true that men and women have different brains?In Brain Rules, Dr. John Medina, a molecular biologist, shares his lifelong interest in how the brain sciences might influence the way we teach our children and the way we work. In each chapter, he describes a brain rule—what scientists know for sure about how our brains work—and then offers transformative ideas for our daily lives.Medina’s fascinating stories and infectious sense of humor breathe life into brain science. You’ll learn why Michael Jordan was no good at baseball. You’ll peer over a surgeon’s shoulder as he proves that most of us have a Jennifer Aniston neuron. You’ll meet a boy who has an amazing memory for music but can’t tie his own shoes.You will discover how:Every brain is wired differentlyExercise improves cognitionWe are designed to never stop learning and exploringMemories are volatileSleep is powerfully linked with the ability to learnVision trumps all of the other sensesStress changes the way we learnIn the end, you’ll understand how your brain really works—and how to get the most out of it.

    In How Not to Be Wrong, Jordan Ellenberg shows us how terribly limiting this view is: Math isn’t confined to abstract incidents that never occur in real life, but rather touches everything we do—the whole world is shot through with it.Math allows us to see the hidden structures underneath the messy and chaotic surface of our world. It’s a science of not being wrong, hammered out by centuries of hard work and argument. Armed with the tools of mathematics, we can see through to the true meaning of information we take for granted: How early should you get to the airport? What does “public opinion” really represent? Why do tall parents have shorter children? Who really won Florida in 2000? And how likely are you, really, to develop cancer?How Not to Be Wrong presents the surprising revelations behind all of these questions and many more, using the mathematician’s method of analyzing life and exposing the hard-won insights of the academic community to the layman—minus the jargon. Ellenberg chases mathematical threads through a vast range of time and space, from the everyday to the cosmic, encountering, among other things, baseball, Reaganomics, daring lottery schemes, Voltaire, the replicability crisis in psychology, Italian Renaissance painting, artificial languages, the development of non-Euclidean geometry, the coming obesity apocalypse, Antonin Scalia’s views on crime and punishment, the psychology of slime molds, what Facebook can and can’t figure out about you, and the existence of God.Ellenberg pulls from history as well as from the latest theoretical developments to provide those not trained in math with the knowledge they need. Math, as Ellenberg says, is “an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength.” With the tools of mathematics in hand, you can understand the world in a deeper, more meaningful way. How Not to Be Wrong will show you how.

    Whether we're buying a pair of jeans, ordering a cup of coffee, selecting a long-distance carrier, applying to college, choosing a doctor, or setting up a 401K, everyday decisions have become increasingly complex due to the overwhelming abundance of choice with which we are presented. In The Paradox of Choice, Barry Schwartz explains why too much of a good thing has proven detrimental to our psychological and emotional well-being. In accessible, engaging, and anecdotal prose, Schwartz explains how a culture that thrives on the availability of constantly evolving options can also foster profound dissatisfaction and self-blame in individuals, which can lead to a paralysis in decision making and, in some cases, depression. With the latest studies on how we make choices in our personal and professional lives, Schwartz offers practical advice on how to focus on the right choices, and how to derive greater satisfaction from choices that we do make.

    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.

    Instead, we find that the wealthiest societies of the world have become depressed, anxious, sleep deprived, and overmedicated.Max Strom, author of A Life Worth Breathing and global teacher of personal transformation, reveals that we each have internal, human technology capable of empowering our lives and leading us to deeper levels of happiness. In his new book, There Is No App for Happiness, Strom illustrates three imperatives to take back control of our lives.Imperative One: Self-study. Overcoming our negative presets.Imperative Two: Live as if your time and your lifespan were the same thing.Imperative Three: Learn a daily regimen that heals and empowers you, and practice it one hour a day.Learn that joy and fulfillment require us to be active participants and that we should not strive for a virtual life—but a life truly lived. There Is No App for Happiness will propel you into a new and more meaningful experience of living.

     Based on the precepts of cognitive science and drawing on a half century of interdisciplinary studies, Smart Thinking is the first book to reveal a three-part formula that distinguishes Smart Thinking from innate intelligence and shows how memory works, how to learn effectively, and how to use knowledge when you need to get things done. Beginning with defining the difference between Smart Thinking and innate or raw intelligence, cognitive psychologist Art Markman demonstrates how it is possible to learn Smart Thinking that you can apply to the real world. This engaging and practical book introduces a three-part formula for Smart Thinking, which demonstrates how anyone can: � Develop Smart Habits � Acquire High-Quality Knowledge � Use High-Quality Knowledge when needed Smart Thinking explores each part of the Smart Thinking formula and provides: � An understanding of how the mind works and the means to replace self-limiting habits with those that foster Smart Thinking � Insights into how memory functions and how to improve the quality of what you learn � Ways to present new information effectively � Specific techniques for improving your understanding of how the world works � The ability to define and solve problems by finding the relevant knowledge from any area of expertise and applying it effectively Drawing on multiple research disciplines, including psychology, artificial intelligence, philosophy, neuroscience, learning sciences, linguistics, anthropology, sociology, and education, Markman provides insights into the functioning of the mind and synthesizes this understanding into practical tools and exercises that develop new skills and achieve personal goals. The book culminates in tips for creating a Culture of Smart to make everyone in an organization more effective.