This fault is not a minor academic quibble: it underlies a reproducibility crisis now threatening entire disciplines. In an increasingly statistics-reliant society, this same deeply rooted error shapes decisions in medicine, law, and public policy with profound consequences. The foundation of the problem is a misunderstanding of probability and its role in making inferences from observations.Aubrey Clayton traces the history of how statistics went astray, beginning with the groundbreaking work of the seventeenth-century mathematician Jacob Bernoulli and winding through gambling, astronomy, and genetics. Clayton recounts the feuds among rival schools of statistics, exploring the surprisingly human problems that gave rise to the discipline and the all-too-human shortcomings that derailed it. He highlights how influential nineteenth- and twentieth-century figures developed a statistical methodology they claimed was purely objective in order to silence critics of their political agendas, including eugenics.Clayton provides a clear account of the mathematics and logic of probability, conveying complex concepts accessibly for readers interested in the statistical methods that frame our understanding of the world. He contends that we need to take a Bayesian approach--that is, to incorporate prior knowledge when reasoning with incomplete information--in order to resolve the crisis. Ranging across math, philosophy, and culture, Bernoulli's Fallacy explains why something has gone wrong with how we use data--and how to fix it.

    Yet unlike Einstein, with whom he formed a warm and abiding friendship, Gödel has long escaped all but the most casual scrutiny of his life.Stephen Budiansky’s Journey to the Edge of Reason is the first biography to fully draw upon Gödel’s voluminous letters and writings—including a never-before-transcribed shorthand diary of his most intimate thoughts—to explore Gödel’s profound intellectual friendships, his moving relationship with his mother, his troubled yet devoted marriage, and the debilitating bouts of paranoia that ultimately took his life. It also offers an intimate portrait of the scientific and intellectual circles in prewar Vienna, a haunting account of Gödel’s and Jewish intellectuals’ flight from Austria and Germany at the start of the Second World War, and a vivid re-creation of the early days of Princeton’s Institute for Advanced Study, where Gödel and Einstein both worked.Eloquent and insightful, Journey to the Edge of Reason is a fully realized portrait of the odd, brilliant, and tormented man who has been called the greatest logician since Aristotle, and illuminates the far-reaching implications of Gödel’s revolutionary ideas for philosophy, mathematics, artificial intelligence, and man’s place in the cosmos.

    Rammed full of practical ideas – all of which are beautifully articulated and backed by research – this is a truly wonderful book.-Craig Barton , Author of How I Wish I'd Taught MathsI can’t recommend highly enough this very do-able and high-leverage approach to any math teacher or curriculum developer who wants to level up their task design, or just try out something new.-Kate Nowak , Former high school math teacher and currently Vice President of Product Strategy for Illustrative Mathematics

    This book upends the conventional approach to math, inviting you to think creatively about shape and dimension, the infinite and infinitesimal, symmetries, proofs, and how these concepts all fit together. What awaits readers is a freewheeling tour of the inimitable joys and unsolved mysteries of this curiously powerful subject.Like the classic math allegory Flatland, first published over a century ago, or Douglas Hofstadter's Godel, Escher, Bach forty years ago, there has never been a math book quite like Math Without Numbers. So many popularizations of math have dwelt on numbers like pi or zero or infinity. This book goes well beyond to questions such as: How many shapes are there? Is anything bigger than infinity? And is math even true? Milo Beckman shows why math is mostly just pattern recognition and how it keeps on surprising us with unexpected, useful connections to the real world.The ambitions of this book take a special kind of author. An inventive, original thinker pursuing his calling with jubilant passion. A prodigy. Milo Beckman completed the graduate-level course sequence in mathematics at age sixteen, when he was a sophomore at Harvard; while writing this book, he was studying the philosophical foundations of physics at Columbia under Brian Greene, among others.

    Informative, readable, and entertaining, Joe provides clues you won't find anywhere else and surprising insights sure to generate lively debate.

    

    Thus it has often been considered as the "purest" branch of mathematics in the sense that it was the furthest from any specific application. However, nowadays number theory is considered to be one of the most important branches of mathematics for application in cryptography. This book is based on teaching materials from an undergraduate course Elementary Number Theory and graduate courses Diophantine equations and Diophantine Approximations and Application. The book also provides an insight into subjects that are at the centre of research interest of the author of the book as well as other members of the Croatian number theory research group. This book is primarily intended for undergraduate students who attend courses in number theory and its application. Also, it can be useful for advanced high school students who are preparing for competition in mathematics in which at all levels, from the school level competition to the international competition number theory has a significant role. In addition, it is intended for graduate students and researchers in the fields of number theory, algebra and cryptography.

    This bestselling book helps students fill in the gaps in their knowledge. Thomas A. Garrity explains the basic points and a few key results of all the most important undergraduate topics in mathematics, emphasizing the intuitions behind the subject. The explanations are accompanied by numerous examples, exercises and suggestions for further reading that allow the reader to test and develop their understanding of these core topics. Featuring four new chapters and many other improvements, this second edition of All the Math You Missed is an essential resource for advanced undergraduates and beginning graduate students who need to learn some serious mathematics quickly.

    Are you prepared? Rivalry is shifting from well-defined industries to broader ecosystems: automobiles to mobility platforms; banking to fintech; television broadcasting to video streaming. Your competitors are coming from new directions and pursuing different goals from those of your familiar rivals. In this world, succeeding with the old rules can mean losing the new game. Winning the Right Game introduces the concepts, tools, and frameworks necessary to confront the threat of ecosystem disruption and to develop the strategies that will let your organization play ecosystem offense.To succeed in this world, you need to change your perspective on competition, growth, and leadership. In this book, strategy expert Ron Adner offers a new way of thinking, illustrating breakthrough ideas with compelling cases. How did a strategy of ecosystem defense save Wayfair and Spotify from being crushed by giants Amazon and Apple? How did Oprah Winfrey redraw industry boundaries to transition from television host to multimedia mogul? How did a shift to an alignment mindset enable Microsoft's cloud-based revival? Each was rooted in a new approach to competitors, partners, and timing that you can apply to your own organization. For today's leaders the difference between success and failure is no longer simply winning, but rather being sure that you are winning the right game.

    This book addresses the question: how far are we from such discovery? Over the last few decades, multiple roads to finding a quantum theory of gravity have been proposed but no obvious description of nature has emerged in this domain. What is to be made of this situation? This volume probes the state-of-the art in this daunting quest of theoretical physics by collecting critical interviews with nearly forty leading theorists in this field. These broad-ranging conversations give important insights and candid opinions on the various approaches to quantum gravity, including string theory, loop quantum gravity, causal set theory and asymptotic safety. This unique, readable overview provides a gateway into cutting edge research for students and others who wish to engage with the open problem of quantum gravity.

    The text emphasizes a modern perspective on the subject, with gentle mentions of the unifying categorical principles underlying the various constructions and the role of universal properties. A key feature is the treatment of modules, including a proof of the classification theorem for finitely generated modules over Euclidean domains. Noetherian modules and some of the language of exact complexes are introduced. In addition, standard topics - such as the Chinese Remainder Theorem, the Gauss Lemma, the Sylow Theorems, simplicity of alternating groups, standard results on field extensions, and the Fundamental Theorem of Galois Theory - are all treated in detail. Students will appreciate the text's conversational style, 400+ exercises, an appendix with complete solutions to around 150 of the main text problems, and an appendix with general background on basic logic and na�ve set theory.

    Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. The proofs are not terse, and aim for understanding over economy. Furthermore, dozens of proofs are preceded by "scratch work" or a proof sketch to give students a big-picture view and an explanation of how they would come up with it on their own. This book covers intuitive proofs, direct proofs, sets, induction, logic, the contrapositive, contradiction, functions and relations. The text aims to make the ideas visible, and contains over 200 illustrations. The writing is relaxed and conversational, and includes periodic attempts at humor. This text is also an introduction to higher mathematics. This is done in-part through the chosen examples and theorems. Furthermore, following every chapter is an introduction to an area of math. These include Ramsey theory, number theory, topology, sequences, real analysis, big data, game theory, cardinality and group theory. After every chapter are "pro-tips," which are short thoughts on things I wish I had known when I took my intro-to-proofs class. They include finer comments on the material, study tips, historical notes, comments on mathematical culture, and more. Also, after each chapter's exercises is an introduction to an unsolved problem in mathematics. In the first appendix we discuss some further proof methods, the second appendix is a collection of particularly beautiful proofs, and the third is some writing advice. ~Jay Cummings

    He treats philosophical issues as they arise organically in mathematics, discussing such topics as platonism, realism, logicism, structuralism, formalism, infinity, and intuitionism in mathematical contexts. He organizes the book by mathematical themes--numbers, rigor, geometry, proof, computability, incompleteness, and set theory--that give rise again and again to philosophical considerations.

    A trail of clues leads from scene to scene, presenting Molly with a number of challenges. But who is leaving the clues, and where will they lead? This interactive mystery shows math isn't just about numbers—it’s about imagination! An explorative and creative approach to the world of mathematics.

    He emphasizes throughout elegant deduction and practical applications, and argues that geometry can offer the quickest route to the whole spirit of mathematics at itsbest. Along the way, we encounter the quirky and the unexpected, meet the great personalities involved, and uncover some of the loveliest surprises in mathematics.

    This book includes separate sections of answers, hints, and full solutions. Prerequisites include multiplying expressions with square roots, systems of equations, the quadratic formula, the equation for a straight line, power rules, factoring, and other standard algebra techniques. A variety of problems are included, such as: * systems of equations (many are nonstandard, including a quadratic term or a reciprocal, for example) * simplifying expressions or solving equations that feature square roots * applying algebra to derive equations * variables in the denominator * rules for exponents * inequalities * the equation for a straight line * multiplying, distributing, or factoring expressions * applications of algebra (such as in classic physics problems) * transformations of variables * exposure to techniques such as completing the square, partial fractions, or separation of variables * cross multiplying * ratios * rationalizing the denominator and multiplying by the conjugate. This book is NOT indented to “teach” algebra (though the solutions may be instructive), but is designed to offer practice with a variety of algebra skills (which most students could benefit from) for students who are familiar with the skills listed. The author, Chris McMullen, PhD, has over twenty years of experience teaching math skills to physics students. He prepared this workbook of the Improve Your Math Fluency series to share his strategies for solving algebra problems.

    Châtelet was not only a philosopher, political theorist, theorist of individuation and of the magnification of human freedoms, but also a talented mathematician and an original theorist of the virtual, the diagram, and the gesture.With their characteristic ebullience and speculative agility in transporting concepts between different fields, Châtelet's polymath interrogations were an acknowledged inspiration to his fellow philosophers, including Gilles Deleuze and Alain Badiou.In the essays collected in this volume, Châtelet explores the articulation between mathematics and physical reality, algebra and geometry, romanticism and science, finite beings and the infinite manifestations of nature, and gesture and abstraction. The book also offers interviews with Châtelet and review articles in which he reckons with contemporaries including Badiou, Deleuze, Roger Penrose, and René Thom.The extensive introduction by Châtelet's former colleague Charles Alunni outlines the life and career of this “last romantic philosopher” and the continuing importance of his work for our understanding of the relationships between the mathematical and the physical, the abstract and the concrete, and scientific thinking and the politics of liberation.

    This book and its companion explore various new notions of space, including both formal and conceptual points of view, as presented by leading experts at the New Spaces in Mathematics and Physics workshop held at the Institut Henri Poincar� in 2015. This volume covers a broad range of topics in mathematical physics, including noncommutative geometry, supergeometry, derived symplectic geometry, higher geometric quantization, intuitionistic quantum logic, problems with the continuum description of spacetime, twistor theory, loop quantum gravity, and geometry in string theory. It is addressed primarily to mathematical physicists and mathematicians, but also to historians and philosophers of these disciplines.

    

    How is this remarkable result proved? This short book explains. It also discusses Gödel's Second Incompleteness Theorem. Based on lecture notes for a course given in Cambridge for many years, the aim is to make the Theorems available, clearly and accessibly, even to those with a quite limited formal background.

    PLT's multi-disciplinary activities are designed to develop students' critical thinking, problem-solving, and decision-making skills. We teach students how to think, not what to think about the environment. Our hands-on activities make teaching and learning fun! Lesson plans are flexible and can be easily incorporated into existing curriculum or nonformal education programs. The activities are inquiry-based with an emphasis on science, reading, writing, mathematics, and social studies and correlated to state and national academic standards. Activities are practical, easy to do, and require minimal preparation; no special supplies are needed. They include background information and science content for teachers. Supporting features include literature connections, technology extensions, differentiated instruction, enrichment suggestions, and student assessment tools.

    It is separated into two halves, each with different approaches to that subject. The first half goes through the concepts of causality, with very little in the way of estimation. It introduces the concept of identification thoroughly and clearly and discusses it as a process of trying to isolate variation that has a causal interpretation. Subjects include a heavy emphasis on data-generating processes and causal diagrams.Concepts are demonstrated with a heavy emphasis on graphical intuition and the question of what we do to data. When we “add a control variable” what does that actually do?Key Features:• Extensive code examples in R, Stata, and Python• Chapters on overlooked topics in econometrics classes: heterogeneous treatment effects, simulation and power analysis, new cutting-edge methods, and uncomfortable ignored assumptions• An easy-to-read conversational tone• Up-to-date coverage of methods with fast-moving literature like difference-in-differences

    

    Informed by his work on the computational universe of possible programs and on computational language design, Wolfram explains new and existing ideas about combinators with unique clarity and stunning visualizations, as well as provides insights on their historical connections and the curious story of Moses Sch�onfinkel, inventor of combinators. Though invented well before Turing machines, combinators have often been viewed as an inaccessibly abstract approach to computation. This book brings them to life as never before in a thought-provoking and broadly accessible exposition of interest across mathematics and computer science, as well as to those concerned with the foundations of formal and computational thinking, and with the history of ideas"--