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© 2016,In 1936, when he was just twentyfour years old, Alan Turing wrote a remarkable paper in which he outlined the theory of computation, laying out the ideas that underlie all modern computers. This groundbreaking and powerful theory now forms the basis of computer science. In Turing's Vision , Chris Bernhardt explains the theory, Turing's most important contribution, for the general reader. Bernhardt argues that the strength of Turing's theory is its simplicity, and that, explained in a straightforward manner, it is eminently understandable by the nonspecialist. As Marvin Minsky writes, "The sheer simplicity of the theory's foundation and extraordinary short path from this foundation to its logical and surprising conclusions give the theory a mathematical beauty that alone guarantees it a permanent place in computer theory." Bernhardt begins with the foundation and systematically builds to the surprising conclusions. He also views Turing's theory in the context of mathematical history, other views of computation (including those of Alonzo Church), Turing's later work, and the birth of the modern computer. In the paper, "On Computable Numbers, with an Application to the Entscheidungsproblem ," Turing thinks carefully about how humans perform computation, breaking it down into a sequence of steps, and then constructs theoretical machines capable of performing each step. Turing wanted to show that there were problems that were beyond any computer's ability to solve; in particular, he wanted to find a decision problem that he could prove was undecidable. To explain Turing's ideas, Bernhardt examines three wellknown decision problems to explore the concept of undecidability; investigates theoretical computing machines, including Turing machines; explains universal machines; and proves that certain problems are undecidable, including Turing's problem concerning computable numbers.

© 2016,The Math Myth expands on Andrew Hacker's scrutiny of some widely held assumptions: that mathematics broadens our minds; that mastery of arcane concepts  cosine, logarithms, the area of a sphere  will be needed for most jobs; that the Common Core's single format should be required of every student. He worries that a frenzied emphasis on STEM (science, technology, engineering, and mathematics) is diverting resources from other pursuits and subverting the spirit of the country.

© 2016,Elements of Mathematics takes readers on a fascinating tour that begins in elementary mathematicsbut, as John Stillwell shows, this subject is not as elementary or straightforward as one might think. Not all topics that are part of today's elementary mathematics were always considered as such, and great mathematical advances and discoveries had to occur in order for certain subjects to become "elementary." Stillwell examines elementary mathematics from a distinctive twentyfirstcentury viewpoint and describes not only the beauty and scope of the discipline, but also its limits. From Gaussian integers to propositional logic, Stillwell delves into arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, and logic. He discusses how each area ties into more advanced topics to build mathematics as a whole. Through a rich collection of basic principles, vivid examples, and interesting problems, Stillwell demonstrates that elementary mathematics becomes advanced with the intervention of infinity. Infinity has been observed throughout mathematical history, but the recent development of "reverse mathematics" confirms that infinity is essential for proving wellknown theorems, and helps to determine the nature, contours, and borders of elementary mathematics. Elements of Mathematics gives readers, from high school students to professional mathematicians, the highlights of elementary mathematics and glimpses of the parts of math beyond its boundaries.

© 2015,Following hot on the heels of The Imitation Game , this is the first modern biography of Alan Turing by a member of the familyAlan's nephew, Sir Dermot Turing Alan Turing was an extraordinary man who crammed into a life of only 42 years the careers of mathematician, codebreaker, computer scientist, and biologist. He is widely regarded as a war hero grossly mistreated by his unappreciative country and it has become hard to disentangle the real man from the story. It is easy to cast him as a misfit, the stereotypical professor. But actually Alan Turing was never a professor, and his nickname "Prof" was given by his codebreaking friends at Bletchley Park. Now, Alan Turing's nephew, Dermot Turing, has taken a fresh look at the influences on Alan Turing's life and creativity, and the later creation of a legend. Dermot's vibrant and entertaining approach to the life and work of a true genius makes this a fascinating read. This unique family perspective features insights from secret documents only recently released to the UK National Archives and other sources not tapped by previous biographers, looks into the truth behind Alan's conviction for gross indecency, and includes previously unpublished photographs from the Turing family album.

© 2015,When Critical Multiculturalism Meets Mathematics details the development and outcomes of a teacher professional development project that merged multiculturalism and mathematics. In six compact chapters the authors describe the impetus for their multiyear project and present rich case studies of nine teacher participants. The cases stand alone as compelling reading, yet Marshall et al. extend beyond their distinctiveness to explain the statistical data related to the project s broader impact. Emphasizing both qualitative and quantitative findings makes this book ideal for novice researchers interested in mixed method study. Likewise, the authors unveil the anatomy and a few complexities of conducting research in the real world contexts of schools including participant recruitment and resolution of unanticipated matters that can arise within research teams. A unique twist in the final chapter is Marshall et al. s critique of their own missteps as researchers, which are used skillfully and unobtrusively to proffer tips for future studies. They conclude by theorizing affirmed intersectionality, identified as the critical element that facilitated teachers recognition and acceptance of the compatibility between the study s two components."

© 2015,What do pure mathematicians do, and why do they do it? Looking beyond the conventional answersfor the sake of truth, beauty, and practical applicationsthis book offers an eclectic panorama of the lives and values and hopes and fears of mathematicians in the twentyfirst century, assembling material from a startlingly diverse assortment of scholarly, journalistic, and pop culture sources. Drawing on his personal experiences and obsessions as well as the thoughts and opinions of mathematicians from Archimedes and Omar Khayy#65533;m to such contemporary giants as Alexander Grothendieck and Robert Langlands, Michael Harris reveals the charisma and romance of mathematics as well as its darker side. In this portrait of mathematics as a community united around a set of common intellectual, ethical, and existential challenges, he touches on a wide variety of questions, such as: Are mathematicians to blame for the 2008 financial crisis? How can we talk about the ideas we were born too soon to understand? And how should you react if you are asked to explain number theory at a dinner party? Disarmingly candid, relentlessly intelligent, and richly entertaining, Mathematics without Apologies takes readers on an unapologetic guided tour of the mathematical life, from the philosophy and sociology of mathematics to its reflections in film and popular music, with detours through the mathematical and mystical traditions of Russia, India, medieval Islam, the Bronx, and beyond.

© 2015,This monograph considers several wellknown mathematical theorems and asks the question, "Why prove it again?" while examining alternative proofs. It explores the different rationales mathematicians may have for pursuing and presenting new proofs of previously established results, as well as how they judge whether two proofs of a given result are different. While a number of books have examined alternative proofs of individual theorems, this is the first that presents comparative case studies of other methods for a variety of different theorems. The author begins by laying out the criteria for distinguishing among proofs and enumerates reasons why new proofs have, for so long, played a prominent role in mathematical practice. He then outlines various purposes that alternative proofs may serve. Each chapter that follows provides a detailed case study of alternative proofs for particular theorems, including the Pythagorean Theorem, the Fundamental Theorem of Arithmetic, Desargues' Theorem, the Prime Number Theorem, and the proof of the irreducibility of cyclotomic polynomials. Why Prove It Again? will appeal to a broad range of readers, including historians and philosophers of mathematics, students, and practicing mathematicians. Additionally, teachers will find it to be a useful source of alternative methods of presenting material to their students.

© 2016,This annual anthology brings together the year's finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, The Best Writing on Mathematics 2015 makes available to a wide audience many articles not easily found anywhere elseand you don't need to be a mathematician to enjoy them. These writings offer surprising insights into the nature, meaning, and practice of mathematics today. They delve into the history, philosophy, teaching, and everyday occurrences of math, and take readers behind the scenes of today's hottest mathematical debates. Here David Hand explains why we should actually expect unlikely coincidences to happen; Arthur Benjamin and Ethan Brown unveil techniques for improvising custommade magic number squares; Dana Mackenzie describes how mathematicians are making essential contributions to the development of synthetic biology; Steven Strogatz tells us why it's worth writing about math for people who are alienated from it; Lisa Rougetet traces the earliest written descriptions of Nim, a popular game of mathematical strategy; Scott Aaronson looks at the unexpected implications of testing numbers for randomness; and much, much more. In addition to presenting the year's most memorable writings on mathematics, this musthave anthology includes a bibliography of other notable writings and an introduction by the editor, Mircea Pitici. This book belongs on the shelf of anyone interested in where math has taken usand where it is headed.

© 2016,Instant New York Times Bestseller The phenomenal true story of the black female mathematicians at NASA at the leading edge of the feminist and civil rights movement, whose calculations helped fuel some of America's greatest achievements in spacea powerful, revelatory contribution that is as essential to our understanding of race, discrimination, and achievement in modern America as Between the World and Me and The Immortal Life of Henrietta Lacks. Soon to be a major motion picture starring Taraji P. Henson, Octavia Spencer, Janelle Monae, Kirsten Dunst, and Kevin Costner. Before John Glenn orbited the earth, or Neil Armstrong walked on the moon, a group of dedicated female mathematicians known as "human computers" used pencils, slide rules and adding machines to calculate the numbers that would launch rockets, and astronauts, into space. Among these problemsolvers were a group of exceptionally talented African American women, some of the brightest minds of their generation. Originally relegated to teaching math in the South's segregated public schools, they were called into service during the labor shortages of World War II, when America's aeronautics industry was in dire need of anyone who had the right stuff. Suddenly, these overlooked math whizzes had a shot at jobs worthy of their skills, and they answered Uncle Sam's call, moving to Hampton, Virginia and the fascinating, highenergy world of the Langley Memorial Aeronautical Laboratory. Even as Virginia's Jim Crow laws required them to be segregated from their white counterparts, the women of Langley's allblack "West Computing" group helped America achieve one of the things it desired most: a decisive victory over the Soviet Union in the Cold War, and complete domination of the heavens. Starting in World War II and moving through to the Cold War, the Civil Rights Movement and the Space Race, Hidden Figures follows the interwoven accounts of Dorothy Vaughan, Mary Jackson, Katherine Johnson and Christine Darden, four African American women who participated in some of NASA's greatest successes. It chronicles their careers over nearly three decades they faced challenges, forged alliances and used their intellect to change their own lives, and their country's future.

© 2016,Mathematics in Ancient Egypt traces the development of Egyptian mathematics, from the end of the fourth millennium BCand the earliest hints of writing and number notationto the end of the pharaonic period in GrecoRoman times. Drawing from mathematical texts, architectural drawings, administrative documents, and other sources, Annette Imhausen surveys three thousand years of Egyptian history to present an integrated picture of theoretical mathematics in relation to the daily practices of Egyptian life and social structures. Imhausen shows that from the earliest beginnings, pharaonic civilization used numerical techniques to efficiently control and use their material resources and labor. Even during the Old Kingdom, a variety of metrological systems had already been devised. By the Middle Kingdom, procedures had been established to teach mathematical techniques to scribes in order to make them proficient administrators for their king. Imhausen looks at counterparts to the notation of zero, suggests an explanation for the evolution of unit fractions, and analyzes concepts of arithmetic techniques. She draws connections and comparisons to Mesopotamian mathematics, examines which individuals in Egyptian society held mathematical knowledge, and considers which scribes were trained in mathematical ideas and why. Of interest to historians of mathematics, mathematicians, Egyptologists, and all those curious about Egyptian culture, Mathematics in Ancient Egypt sheds new light on a civilization's unique mathematical evolution.

© 2015,One of the traditional ways mathematical ideas and even new areas of mathematics are created is from experiments. One of the bestknown examples is that of the Fermat hypothesis, which was conjectured by Fermat in his attempts to find integer solutions for the famous Fermat equation. This hypothesis led to the creation of a whole field of knowledge, but it was proved only after several hundred years. This book, based on the author's lectures, presents several new directions of mathematical research. All of these directions are based on numerical experiments conducted by the author, which led to new hypotheses that currently remain open, i.e., are neither proved nor disproved. The hypotheses range from geometry and topology (statistics of plane curves and smooth functions) to combinatorics (combinatorial complexity and random permutations) to algebra and number theory (continuous fractions and Galois groups). For each subject, the author describes the problem and presents numerical results that led him to a particular conjecture. In the majority of cases there is an indication of how the readers can approach the formulated conjectures (at least by conducting more numerical experiments). Written in Arnold's unique style, the book is intended for a wide range of mathematicians, from high school students interested in exploring unusual areas of mathematics on their own, to college and graduate students, to researchers interested in gaining a new, somewhat nontraditional perspective on doing mathematics. In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession.

© 2016,This book presents the first algebraic treatment of quasitruth fuzzy logic and covers the algebraic foundations of manyvalued logic. It offers a comprehensive account of basic techniques and reports on important results showing the pivotal role played by perfect manyvalued algebras (MValgebras). It is well known that the firstorder predicate ukasiewicz logic is not complete with respect to the canonical set of truth values. However, it is complete with respect to all linearly ordered MV algebras. As there are no simple linearly ordered MValgebras in this case, infinitesimal elements of an MValgebra are allowed to be truth values. The book presents perfect algebras as an interesting subclass of local MValgebras and provides readers with the necessary knowledge and tools for formalizing the fuzzy concept of quasi true and quasi false. All basic concepts are introduced in detail to promote a better understanding of the more complex ones. It is an advanced and inspiring referenceguide for graduate students and researchers in the field of nonclassical manyvalued logics."

© 2015,Where did math come from? Who thought up all those algebra symbols, and why? What is the story behind π? ... negative numbers? ... the metric system? ... quadratic equations? ... sine and cosine? ... logs? Including five new independent historical sketches, each complete with added questions and projects, this second edition answers these questions and many others in an informal, easygoing style, accessible to teachers, students and anyone who is curious about the history of mathematical ideas. The 30 short stories are preceded by a 58page bird'seye overview of the entire panorama of mathematical history, a whirlwind tour of the most important people, events, and trends that shaped the mathematics we know today. The 'Books You Ought to Read' section and new bibliography also ensure that uptodate starting points are provided for readers who want to pursue a topic further.

© 2015,What do pure mathematicians do, and why do they do it? Looking beyond the conventional answersfor the sake of truth, beauty, and practical applicationsthis book offers an eclectic panorama of the lives and values and hopes and fears of mathematicians in the twentyfirst century, assembling material from a startlingly diverse assortment of scholarly, journalistic, and pop culture sources. Drawing on his personal experiences and obsessions as well as the thoughts and opinions of mathematicians from Archimedes and Omar Khayyám to such contemporary giants as Alexander Grothendieck and Robert Langlands, Michael Harris reveals the charisma and romance of mathematics as well as its darker side. In this portrait of mathematics as a community united around a set of common intellectual, ethical, and existential challenges, he touches on a wide variety of questions, such as: Are mathematicians to blame for the 2008 financial crisis? How can we talk about the ideas we were born too soon to understand? And how should you react if you are asked to explain number theory at a dinner party? Disarmingly candid, relentlessly intelligent, and richly entertaining, Mathematics without Apologies takes readers on an unapologetic guided tour of the mathematical life, from the philosophy and sociology of mathematics to its reflections in film and popular music, with detours through the mathematical and mystical traditions of Russia, India, medieval Islam, the Bronx, and beyond.

© 2015,What is math? How exactly does it work? And what do three siblings trying to share a cake have to do with it? In How to Bake Pi , math professor Eugenia Cheng provides an accessible introduction to the logic and beauty of mathematics, powered, unexpectedly, by insights from the kitchen: we learn, for example, how the béchamel in a lasagna can be a lot like the number 5, and why making a good custard proves that math is easy but life is hard. Of course, it's not all about cooking; we'll also run the New York and Chicago marathons, take a closer look at St. Paul's Cathedral, pay visits to Cinderella and Lewis Carroll, and even get to the bottom of why we think of a tomato as a vegetable. At the heart of it all is Cheng's work on category theory, a cuttingedge "mathematics of mathematics," that is about figuring out how math works. This is not the math of our high school classes: seen through category theory, mathematics becomes less about numbers and formulas and more about how we know, believe, and understand anything, including whether our brother took too much cake. Many of us think that math is hard, but, as Cheng makes clear, math is actually designed to make difficult things easier. Combined with her infectious enthusiasm for cooking and a true zest for life, Cheng's perspective on math becomes this singular book: a funny, lively, and clear journey through a vast territory no popular book on math has explored before. How to Bake Pi offers a whole new way to think about a field all of us think we know; it will both dazzle the constant reader of popular mathematics and amuse and enlighten even the most hardened mathphobe. So, what is math? Let's look for the answer in the kitchen.

© 2015,The transition from school mathematics to university mathematics is seldom straightforward. Students are faced with a disconnect between the algorithmic and informal attitude to mathematics at school, versus a new emphasis on proof, based on logic, and a more abstract development of general concepts, based on set theory. The authors have many years' experience of the potential difficulties involved, through teaching firstyear undergraduates and researching the ways in which students and mathematicians think. The book explains the motivation behind abstract foundational material based on students' experiences of school mathematics, and explicitly suggests ways students can make sense of formal ideas. This second edition takes a significant step forward by not only making the transition from intuitive to formal methods, but also by reversing the process using structure theorems to prove that formal systems have visual and symbolic interpretations that enhance mathematical thinking. This is exemplified by a new chapter on the theory of groups. While the first edition extended counting to infinite cardinal numbers, the second also extends the real numbers rigorously to larger ordered fields. This links intuitive ideas in calculus to the formal epsilondelta methods of analysis. The approach here is not the conventional one of 'nonstandard analysis', but a simpler, graphically based treatment which makes the notion of an infinitesimal natural and straightforward. This allows a further vision of the wider world of mathematical thinking in which formal definitions and proof lead to amazing new ways of defining, proving, visualising and symbolising mathematics beyond previous expectations.

© 2015,Can a theatrical play or a visual image capture the beauty and excitement of mathematics? Some of the world's top mathematicians are also accomplished artists: musicians, photographers, painters, dancers, writers, filmmakers. In this volume, they share some of their work and reflect on the roles that mathematics and art have played in their lives. Avi Wigderson, Institute for Advanced Study, Princeton Much has been written about the close relationship of mathematics and music. It is a rather onesided affair because mathematicians are usually very musical while musiciansincluding myselfare considerably less knowledgeable on matters of science. However we are all eager to learn from each other and the present volume is a most valuable contribution to this interesting dialogue. Sir Andras Schiff, pianist and conductor Creativity, Originality and Imagination are three common features of mathematicians and artists. This inspiring book contains much more about the similarities between art and mathematics, as described by outstanding mathematicians with a wide range of artistic interests. Noga Alon, Tel Aviv University Mathematics and art are intertwined on so many levels ... It is exciting to see such a broad treatment of this important theme. Charles Simonyi, Chairman of the Board, Institute for Advanced Study, Princeton Why are mathematicians drawn to art? How do they perceive it? What motivates them to pursue excellence in music or painting? Do they view their art as a conveyance for their mathematics or an escape from it? What are the similarities between mathematical talent and creativity and their artistic equivalents? What are the differences? Can a theatrical play or a visual image capture the beauty and excitement of mathematics? Some of the world's top mathematicians are also accomplished artists: musicians, photographers, painters, dancers, writers, filmmakers. In this volume, they share some of their work and reflect on the roles that mathematics and art have played in their lives. They write about creativity, communication, making connections, negotiating successes and failures, and navigating the vastly different professional worlds of art and mathematics.

© 2015,In Professor Povey's Perplexing Problems , Thomas Povey shares 109 of his favorite problems in physics and maths. A tour de force of imagination and exposition, he guides us through uncompromisingly challenging territory that expands our minds and encourages a playful and exploratory approach to study. "The puzzles," he says, "are like toys. We should pick up the one we most enjoy, and play with it." Whether you are an aspiring scientist or an oldhand, pitting yourself against these problems will test your ability to think, and inspire you with curiosity and enthusiasm for physics. Presented with charm and wit, the questions span the gap between highschool and universityentrance standard material. Detailed answers are lightened with a fascinating and refreshing blend of scientific history, application and personal anecdote. On this delightful and idiosyncratic romp through preuniversity maths and physics, the author shows us that behind every single one of these questions lies a new way of thinking about subjects we thought we had understood. He argues that engaging with the unfamiliar is key to forming deeper insights and developing intellectual independence. Professor Povey's Perplexing Problems is a manifesto that science should be playful, and a celebration of the curious.