Haskell emerged in the last decade as a standard for lazy functional programming, a programming style where arguments are evaluated only when the value is actually needed. Haskell is a marvellous demonstration tool for logic and maths because its functional character allows implementations to remain very close to the concepts that get implemented, while the laziness permits smooth handling of infinite data structures.This book does not assume the reader to have previous experience with either programming or construction of formal proofs, but acquaintance with mathematical notation, at the level of secondary school mathematics is presumed. Everything one needs to know about mathematical reasoning or programming is explained as we go along. After proper digestion of the material in this book the reader will be able to write interesting programs, reason about their correctness, and document them in a clear fashion. The reader will also have learned how to set up mathematical proofs in a structured way, and how to read and digest mathematical proofs written by others.

    As a member of the Los Alamos National Laboratory from 1944 on, Ulam helped to precipitate some of the most dramatic changes of the postwar world. He was among the first to use and advocate computers for scientific research, originated ideas for the nuclear propulsion of space vehicles, and made fundamental contributions to many of today's most challenging mathematical projects. With his wide-ranging interests, Ulam never emphasized the importance of his contributions to the research that resulted in the hydrogen bomb. Now Daniel Hirsch and William Mathews reveal the true story of Ulam's pivotal role in the making of the "Super," in their historical introduction to this behind-the-scenes look at the minds and ideas that ushered in the nuclear age. An epilogue by Françoise Ulam and Jan Mycielski sheds new light on Ulam's character and mathematical originality.

    It discusses new results, along with applications of probability theory to a variety of problems. The book contains many exercises and is suitable for use as a textbook on graduate-level courses involving data analysis. Aimed at readers already familiar with applied mathematics at an advanced undergraduate level or higher, it is of interest to scientists concerned with inference from incomplete information.

    In this Very Short Introduction, Kenneth Falconer explains the basic concepts of fractal geometry, which produced a revolution in our mathematical understanding of patterns in the twentieth century, and explores the wide range of applications in science, and in aspects of economics.About the Series: Oxford's Very Short Introductions series offers concise and original introductions to a wide range of subjects--from Islam to Sociology, Politics to Classics, Literary Theory to History, and Archaeology to the Bible. Not simply a textbook of definitions, each volume in this series provides trenchant and provocative--yet always balanced and complete--discussions of the central issues in a given discipline or field. Every Very Short Introduction gives a readable evolution of the subject in question, demonstrating how the subject has developed and how it has influenced society. Eventually, the series will encompass every major academic discipline, offering all students an accessible and abundant reference library. Whatever the area of study that one deems important or appealing, whatever the topic that fascinates the general reader, the Very Short Introductions series has a handy and affordable guide that will likely prove indispensable.

    Illustrations.

    It was studied from antiquity to the Renaissance as a way of glimpsing the nature of reality. Geometry is number in space; music is number in time; and comology expresses number in space and time. Number, music, and geometry are metaphysical truths: life across the universe investigates them; they foreshadow the physical sciences.Quadrivium is the first volume to bring together these four subjects in many hundreds of years. Composed of six successful titles in the Wooden Books series-Sacred Geometry, Sacred Number, Harmonograph, The Elements of Music, Platonic & Archimedean Solids, and A Little Book of Coincidence-it makes ancient wisdom and its astonishing interconnectedness accessible to us today.Beautifully produced in six different colors of ink, Quadrivium will appeal to anyone interested in mathematics, music, astronomy, and how the universe works.

    "More concretely," the authors explain, "it is the controlled manipulation of mathematical formulas, using a collection of techniques for solving problems."

    Russell's classic The Principles of Mathematics sets forth his landmark thesis that mathematics and logic are identical―that what is commonly called mathematics is simply later deductions from logical premises.His ideas have had a profound influence on twentieth-century work on logic and the foundations of mathematics.

    And the beauty of it is that it's all approximate!Using Enrico Fermi's theory of approximation, Santos brings the world of numbers into perspective. For puzzle junkies and trivia fanatics, these 70 word puzzles will show the reader how to take a bit of information, add what they already know, and extrapolate an answer.Santos has done the impossible: make math and the multiple possibilities of numbers fun and informative. Can you really cry a river? Is it possible to dig your way out of jail with just a teaspoon and before your life sentence is up?Taking an academic subject and using it as the prism to view everyday off-the-wall questions as math problems to be solved is a natural step for the lovers of sudoku, cryptograms, word puzzles, and other thought-provoking games.

    Luenberger, known for his ability to make complex ideas simple, presents essential ideas of investments and their applications, offering students the most comprehensive treatment of the subject available.

    Now, in The Unimaginable Mathematics of Borges' Library of Babel, William Goldbloom Bloch takes readers on a fascinating tour of the mathematical ideas hiddenwithin one of the classic works of modern literature.Written in the vein of Douglas R. Hofstadter's Pulitzer Prize-winning G�del, Escher, Bach, this original and imaginative book sheds light on one of Borges' most complex, richly layered works. Bloch begins each chapter with a mathematical idea--combinatorics, topology, geometry, informationtheory--followed by examples and illustrations that put flesh on the theoretical bones. In this way, he provides many fascinating insights into Borges' Library. He explains, for instance, a straightforward way to calculate how many books are in the Library--an easily notated but literallyunimaginable number--and also shows that, if each book were the size of a grain of sand, the entire universe could only hold a fraction of the books in the Library. Indeed, if each book were the size of a proton, our universe would still not be big enough to hold anywhere near all the books.Given Borges' well-known affection for mathematics, this exploration of the story through the eyes of a humanistic mathematician makes a unique and important contribution to the body of Borgesian criticism. Bloch not only illuminates one of the great short stories of modern literature but alsoexposes the reader--including those more inclined to the literary world--to many intriguing and entrancing mathematical ideas.

    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).

    From the early Greek influences to the Middle Ages and the Renaissance to the end of the 19th century, trace the fascinating foundation of mathematics as it developed through the ages. Aristotle, Galileo, Kepler, Newton: you know the names. Now here's what they really did, and the effect their discoveries had on our culture, all explained in a way the layperson can understand. Begin with the basis of arithmetic (Plato and the introduction of geometry), and discover why the use of Arabic numerals was critical to the development of both commerce and science. The development of calculus made space travel a reality, while the abacus prefigured the computer. The greats examined in depth include Leonardo da Vinci, a brilliant mathematician as well as artist; Pascal, who laid out the theory of probabilities; and Fermat, whose intriguing theory has only recently been solved.

    Bridging the gap from geometry to the latest work in observational cosmology, the book illustrates the connection between geometry and the behavior of the physical universe and explains how radiation remaining from the big bang may reveal the actual shape of the universe.

    a personal confession of its creator and a kind of involuntary and unperceived memoir.". Stewart affirms this maxim in his colorful reinterpretation of the lives and works of 17th-century philosophers Spinoza and Leibniz. In November 1676, the foppish courtier Leibniz, "the ultimate insider... an orthodox Lutheran from conservative Germany," journeyed to The Hague to visit the self-sufficient, freethinking Spinoza, "a double exile... an apostate Jew from licentious Holland." A prodigious polymath, Leibniz understood Spinoza's insight that "science was in the process of rendering the God of revelation obsolete; that it had already undermined the special place of the human individual in nature." Spinoza embraced this new world. Seeing the orthodox God as a "prop for theocratic tyranny," he articulated the basic theory for the modern secular state. Leibniz, on the other hand, spent the rest of his life championing God and theocracy like a defense lawyer defending a client he knows is guilty. He elaborated a metaphysics that was, at bottom, a reaction to Spinoza and collapses into Spinozism, as Stewart deftly shows. For Stewart, Leibniz's reaction to Spinoza and modernity set the tone for "the dominant form of modern philosophy"—a category that includes Kant, Hegel, Bergson, Heidegger and "the whole 'postmodern' project of deconstructing the phallogocentric tradition of western thought." Readers of philosophy may find much to disagree with in these arguments, but Stewart's wit and profluent prose make this book a fascinating read.