Including a selection of exercises, adjusted for this edition, at the end of each chapter, it offers a new and simpler treatment of the representability of recursive functions, a traditional stumbling block for students on the way to the Godel incompleteness theorems.

    A lecture class taught by Wittgenstein, however, hardly resembled a lecture. He sat on a chair in the middle of the room, with some of the class sitting in chairs, some on the floor. He never used notes. He paused frequently, sometimes for several minutes, while he puzzled out a problem. He often asked his listeners questions and reacted to their replies. Many meetings were largely conversation. These lectures were attended by, among others, D. A. T. Gasking, J. N. Findlay, Stephen Toulmin, Alan Turing, G. H. von Wright, R. G. Bosanquet, Norman Malcolm, Rush Rhees, and Yorick Smythies. Notes taken by these last four are the basis for the thirty-one lectures in this book. The lectures covered such topics as the nature of mathematics, the distinctions between mathematical and everyday languages, the truth of mathematical propositions, consistency and contradiction in formal systems, the logicism of Frege and Russell, Platonism, identity, negation, and necessary truth. The mathematical examples used are nearly always elementary.

    The author has made this edition more accessible to better meet the needs of today's undergraduate mathematics and philosophy students. It is intended for the reader who has not studied logic previously, but who has some experience in mathematical reasoning. Material is presented on computer science issues such as computational complexity and database queries, with additional coverage of introductory material such as sets.

    It presents in a unified manner an introduction to the mathematical theory underlying computer graphic applications. It covers topics of keen interest to students in engineering and computer science: transformations, projections, 2-D and 3-D curve definition schemes, and surface definitions. It also includes techniques, such as B-splines, which are incorporated as part of the software in advanced engineering workstations. A basic knowledge of vector and matrix algebra and calculus is required.

    This well-respected text offers an accessible introduction to functional programming concepts and techniques for students of mathematics and computer science. The treatment is as nontechnical as possible, and it assumes no prior knowledge of mathematics or functional programming. Cogent examples illuminate the central ideas, and numerous exercises appear throughout the text, offering reinforcement of key concepts. All problems feature complete solutions.

    

    More than 40 million students have trusted Schaum's to help them succeed in the classroom and on exams. Schaum's is the key to faster learning and higher grades in every subject. Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills.This Schaum's Outline gives you:Practice problems with full explanations that reinforce knowledgeCoverage of the most up-to-date developments in your course fieldIn-depth review of practices and applicationsFully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time-and get your best test scores!Schaum's Outlines-Problem Solved.

    He was especially careful to present new and unfamiliar puzzles that had not been included in such classic collections as those by Sam Loyd and Henry Dudeney. Later, these puzzles were published in book collections, incorporating reader feedback on alternate solutions or interesting generalizations.The present volume contains a rich selection of 70 of the best of these brain teasers, in some cases including references to new developments related to the puzzle. Now enthusiasts can challenge their solving skills and rattle their egos with such stimulating mind-benders as The Returning Explorer, The Mutilated Chessboard, Scrambled Box Tops, The Fork in the Road, Bronx vs. Brooklyn, Touching Cigarettes, and 64 other problems involving logic and basic math. Solutions are included.

    GRAPH THEORY WITH APPLICATIONS TO ENGINEERING AND COMPUTER SCIENCE-PHI-DEO, NARSINGH-1979-EDN-1

    Mathematics: The Loss of Certainty refutes that myth.

    Originally written to define the relation between the theories of transfinite numbers and mathematical games, the resulting work is a mathematically sophisticated but eminently enjoyable guide to game theory. By defining numbers as the strengths of positions in certain games, the author arrives at a new class, the surreal numbers, that includes both real numbers and ordinal numbers. These surreal numbers are applied in the author's mathematical analysis of game strategies. The additions to the Second Edition present recent developments in the area of mathematical game theory, with a concentration on surreal numbers and the additive theory of partizan games.

    Assuming a minimum of mathematical preparation, Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Four case studies illustrate applications of category theory to programming language design, semantics, and the solution of recursive domain equations. A brief literature survey offers suggestions for further study in more advanced texts.

    Written by two of the best-known names in categorical logic, Conceptual Mathematics is the first book to apply categories to the most elementary mathematics. It thus serves two purposes: first, to provide a key to mathematics for the general reader or beginning student; and second, to furnish an easy introduction to categories for computer scientists, logicians, physicists, and linguists who want to gain some familiarity with the categorical method without initially committing themselves to extended study.

    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.

    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).