Kurt Giidel maintained, and offered detailed proof, that in any arithmetic system, even in elementary parts of arithmetic, there are propositions which cannot be proved or disproved within the system. It is thus uncertain that the basic axioms of arithmetic will not give rise to contradictions. The repercussions of this discovery are still being felt and debated in 20th-century mathematics.The present volume reprints the first English translation of Giidel's far-reaching work. Not only does it make the argument more intelligible, but the introduction contributed by Professor R. B. Braithwaite (Cambridge University}, an excellent work of scholarship in its own right, illuminates it by paraphrasing the major part of the argument.This Dover edition thus makes widely available a superb edition of a classic work of original thought, one that will be of profound interest to mathematicians, logicians and anyone interested in the history of attempts to establish axioms that would provide a rigorous basis for all mathematics. Translated by B. Meltzer, University of Edinburgh. Preface. Introduction by R. B. Braithwaite.

    A timeless introduction to the field and a landmark in symbolic logic, showing that classical logic can be treated algebraically.

    Einstein's Heroes takes you on a journey of discovery about just such a miraculous language--the language of mathematics--one of humanity's mostamazing accomplishments. Blending science, history, and biography, this remarkable book reveals the mysteries of mathematics, focusing on the life and work of three of Albert Einstein's heroes: Isaac Newton, Michael Faraday, and especially James Clerk Maxwell, whose work directly inspired the theory of relativity. RobynArianrhod bridges the gap between science and literature, portraying mathematics as a language and arguing that a physical theory is a work of imagination involving the elegant and clever use of this language. The heart of the book illuminates how Maxwell, using the language of mathematics in a newand radical way, resolved the seemingly insoluble controversy between Faraday's idea of lines of force and Newton's theory of action-at-a-distance. In so doing, Maxwell not only produced the first complete mathematical description of electromagnetism, but actually predicted the existence of theradio wave, teasing it out of the mathematical language itself. Here then is a fascinating look at mathematics: its colorful characters, its historical intrigues, and above all its role as the uncannily accurate language of nature.

    Focused on groups, rings and fields, this text gives students a firm foundation for more specialized work by emphasizing an understanding of the nature of algebraic structures. KEY TOPICS: Sets and Relations; GROUPS AND SUBGROUPS; Introduction and Examples; Binary Operations; Isomorphic Binary Structures; Groups; Subgroups; Cyclic Groups; Generators and Cayley Digraphs; PERMUTATIONS, COSETS, AND DIRECT PRODUCTS; Groups of Permutations; Orbits, Cycles, and the Alternating Groups; Cosets and the Theorem of Lagrange; Direct Products and Finitely Generated Abelian Groups; Plane Isometries; HOMOMORPHISMS AND FACTOR GROUPS; Homomorphisms; Factor Groups; Factor-Group Computations and Simple Groups; Group Action on a Set; Applications of G-Sets to Counting; RINGS AND FIELDS; Rings and Fields; Integral Domains; Fermat's and Euler's Theorems; The Field of Quotients of an Integral Domain; Rings of Polynomials; Factorization of Polynomials over a Field; Noncommutative Examples; Ordered Rings and Fields; IDEALS AND FACTOR RINGS; Homomorphisms and Factor Rings; Prime and Maximal Ideas; Gr�bner Bases for Ideals; EXTENSION FIELDS; Introduction to Extension Fields; Vector Spaces; Algebraic Extensions; Geometric Constructions; Finite Fields; ADVANCED GROUP THEORY; Isomorphism Theorems; Series of Groups; Sylow Theorems; Applications of the Sylow Theory; Free Abelian Groups; Free Groups; Group Presentations; GROUPS IN TOPOLOGY; Simplicial Complexes and Homology Groups; Computations of Homology Groups; More Homology Computations and Applications; Homological Algebra; Factorization; Unique Factorization Domains; Euclidean Domains; Gaussian Integers and Multiplicative Norms; AUTOMORPHISMS AND GALOIS THEORY; Automorphisms of Fields; The Isomorphism Extension Theorem; Splitting Fields; Separable Extensions; Totally Inseparable Extensions; Galois Theory; Illustrations of Galois Theory; Cyclotomic Extensions; Insolvability of the Quintic; Matrix Algebra MARKET: For all readers interested in abstract algebra.

    Ten dimensions? Most of us have barely gotten used to the idea that there are four.Using simple geometry and an easygoing writing style, author Rob Bryanton starts with the lower dimensions that we are all familiar with, then uses those concepts to build one layer upon another, ultimately arriving at a way of imagining the tenth dimension.Part scientific exploration, part philosophy, this unique book touches upon such diverse topics as dark matter, Feynman's "sum over paths", the quantum observer, and the soul. It is aimed at anyone interested in leading-edge theories about cosmology and the nature of reality, but it is not about mainstream physics. Rather, Imagining the Tenth Dimension is a mind-expanding exercise that could change the way you view this incredible universe in which we live.

    In this book, William Cook takes readers on a mathematical excursion, picking up the salesman's trail in the 1800s when Irish mathematician W. R. Hamilton first defined the problem, and venturing to the furthest limits of today's state-of-the-art attempts to solve it. He also explores its many important applications, from genome sequencing and designing computer processors to arranging music and hunting for planets.In Pursuit of the Traveling Salesman travels to the very threshold of our understanding about the nature of complexity, and challenges you yourself to discover the solution to this captivating mathematical problem.

    His yearly gatherings of short and inventive problems were easily his most anticipated math columns. Loyal readers would savor the wit and elegance of his explorations in physics, probability, topology, and chess, among others. Grouped by subject and arrayed from easiest to hardest, the puzzles gathered here, which complement the lengthier, more involved problems in The Colossal Book of Mathematics, have been selected by Gardner for their illuminating; and often bewildering; solutions. Filled with over 300 illustrations, this new volume even contains nine new mathematical gems that Gardner, now ninety, has been gathering for the last decade. No amateur or expert math lover should be without this indispensable volume; a capstone to Gardner's seventy-year career.

    With his own research as well as that of some of the most distinguished scientists of our time, Schäfer moves us from a reality of Darwinian competition to cooperation, a meaningless universe to a meaningful one, and a disconnected, isolated existence to an interconnected one. In so doing, he shows us that our potential is infinite and calls us to live in accordance with the order of the universe, creating a society based on the cosmic principle of connection, emphasizing cooperation and community.

    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.

    G. Ballard has provided thoughtful remarks on the state of the world for decades. J.G. Ballard Conversations brings together several of Ballard's latest interviews and gives readers penetrating insight into the mind of one of the freshest thinkers at work today. Covering topics such at the 9/11 terrorist attacks, the evolution of sexual relationships, and our strange, immersive celebrity culture, this book is a fount of provocative takes on the things that matter. Rounded out with rare photographs of Ballard and supplemental resources, J.G. Ballard Conversations is a necessary item for anyone interested in the modern world.

    In Me, Myself, and Why, Jennifer Ouellette dives into the miniscule ranges of variation to understand just what sets us apart. She draws on cutting-edge research in genetics, neuroscience, and psychology-enlivened as always with her signature sense of humor-to explore the mysteries of human identity and behavior. Readers follow her own surprising journey of self-discovery as she has her genome sequenced, her brain mapped, her personality typed, and even samples a popular hallucinogen. Bringing together everything from Mendel's famous pea plant experiments and mutations in The X-Men to our taste for cilantro and our relationships with virtual avatars, Ouellette takes us on an endlessly thrilling and illuminating trip into the science of ourselves

    The ancient Greeks were so horrified by the implications of an endless number that they drowned the man who gave away the secret. And a German mathematician was driven mad by the repercussions of his discovery of transfinite numbers. Brian Clegg and Oliver Pugh’s brilliant graphic tour of infinity features a cast of characters ranging from Archimedes and Pythagoras to al-Khwarizmi, Fibonacci, Galileo, Newton, Leibniz, Cantor, Venn, Gödel and Mandelbrot, and shows how infinity has challenged the finest minds of science and mathematics. Prepare to enter a world of paradox.

    The book also provides a balanced presentation of time-varying and static fields, preparingstudents for employment in today's industrial and manufacturing sectors. Streamlined to facilitate student understanding, this edition features worked examples in every chapter that explain how to use the theory presented in the text to solve different kinds of problems. Numerical methods, including MATLAB and vector analysis, are also included to help students analyzesituations that they are likely to encounter in industry practice. Elements of Electromagnetics, Fifth Edition, is designed for introductory undergraduate courses in electromagnetics.

    But far more fundamentally, we live in a universe made of quanta. Many things are not made of atoms: light, radio waves, electric current, magnetic fields, Earth's gravitational field, not to mention exotica such a neutron stars, black holes, dark energy, and dark matter. But everything, including atoms, is made of highly unified or "coherent" bundles of energy called "quanta" that (like everything else) obey certain rules. In the case of the quantum, these rules are called "quantum physics." This is a book about quanta and their unexpected, some would say peculiar, behavior--tales, if you will, of the quantum.The quantum has developed the reputation of being capricious, bewildering, even impossible to understand. The peculiar habits of quanta are certainly not what we would have expected to find at the foundation of physical reality, but these habits are not necessarily bewildering and not at all impossible or paradoxical. This book explains those habits--the quantum rules--in everyday language, without mathematics or unnecessary technicalities. While most popular books about quantum physics follow the topic's scientific history from 1900 to today, this book follows the phenomena: wave-particle duality, fundamental randomness, quantum states, superpositions (being in two places at once), entanglement, non-locality, Schrodinger's cat, and quantum jumps, and presents the history and the scientists only to the extent that they illuminate the phenomena.

    Stimulating, illuminating, and always surprising, The Big Questions challenges readers to re-evaluate their most fundamental beliefs and reveals the relationship between the loftiest philosophical quests and our everyday lives.