"Ah, I'm Pingree. We meet again. Splendid. Won't you sit down?" I looked around David's room. Short of the library stacks, I had never seen so many books piled into a single room. Where could I sit down? Every square inch of horizontal surface was covered. Books, papers, notes, manuscripts-all congregated in random and chaotic disorder. This small encounter and the snapshot of the protagonist on the cover of this book introduce the reader to David E. Pingree, the eminent classicist, Orientalist, historian of ancient science, and member of the Department of the History of Mathematics at Brown University. This is a book of his stories, retold by Phil Davis, award-winning author and raconteur par excellence, who reconstructs them from letters and many conversations with his friend Pingree. The stories trace connections between ancient characters, historical and mythical, and recreate a world in which the pursuit of knowledge for its own sake leads to unexpected pleasures and associations. They capture a world best described by Saul Lieberman's quip about Gershom Scholem's great work on the Kabala: "Trash is trash; but the study of trash is scholarship," and David Pingree's imagined response, "Yes, but there's always something of value to be learned." The book is dedicated to preserving and promoting the specialized knowledge and thoughts of David Pingree, a truly remarkable person and to inspire readers to follow academic tradition and at the same time explore unusual connections.

This book is a specialized monograph on the development of the mathematical and computational metatheory of reductive logic and proof-search, areas of logic that are becoming important in computer science. A systematic foundational text on these emerging topics, it includes proof-theoretic, semantic/model-theoretic and algorithmic aspects. The scope ranges from the conceptual background to reductive logic, through its mathematical metatheory, to its modern applications in the computational sciences. Suitable for researchers and graduate students in mathematical, computational and philosophical logic, and in theoretical computer science and artificial intelligence, this is the latest in the prestigous world-renowned Oxford Logic Guides, which contains Michael Dummet's Elements of intuitionism (2nd Edition), Dov M. Gabbay, Mark A. Reynolds, and Marcelo Finger's Temporal Logic Mathematical Foundations and Computational Aspects , J. M. Dunn and G. Hardegree's Algebraic Methods in Philosophical Logic, H. Rott's Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning , and P. T. Johnstone's Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2 .

Computer arithmetic has become so fundamentally embedded into digital design that many engineers are unaware of the many research advances in the area. As a result, they are losing out on emerging opportunities to optimize its use in targeted applications and technologies. In many cases, easily available standard arithmetic hardware might not necessarily be the most efficient implementation strategy. Multiple-Base Number System: Theory and Applications stands apart from the usual books on computer arithmetic with its concentration on the uses and the mathematical operations associated with the recently introduced multiple-base number system (MBNS). The book identifies and explores several diverse and never-before-considered MBNS applications (and their implementation issues) to enhance computation efficiency, specifically in digital signal processing (DSP) and public key cryptography. Despite the recent development and increasing popularity of MBNS as a specialized tool for high-performance calculations in electronic hardware and other fields, no single text has compiled all the crucial, cutting-edge information engineers need to optimize its use. The authors' main goal was to disseminate the results of extensive design research-including much of their own-to help the widest possible audience of engineers, computer scientists, and mathematicians. Dedicated to helping readers apply discoveries in advanced integrated circuit technologies, this single reference is packed with a wealth of vital content previously scattered throughout limited-circulation technical and mathematical journals and papers-resources generally accessible only to researchers and designers working in highly specialized fields. Leveling the informational playing field, this resource guides readers through an in-depth analysis of theory, architectural techniques, and the latest research on the subject, subsequently laying the groundwork users require to begin applying MBNS.

For many civilian, security, and military applications, distributed and networked coordination offers a more promising alternative to centralized command and control in terms of scalability, flexibility, and robustness. It also introduces its own challenges. Distributed Networks: Intelligence, Security, and Applications brings together scientific research in distributed network intelligence, security, and novel applications. The book presents recent trends and advances in the theory and applications of network intelligence and helps you understand how to successfully incorporate them into distributed systems and services. Featuring contributions by leading scholars and experts from around the world, this collection covers: Approaches for distributed network intelligence Distributed models for distributed enterprises, including forecasting and performance measurement models Security applications for distributed enterprises, including intrusion tackling and peer-to-peer traffic detection Future wireless networking scenarios, including the use of software sensors instead of hardware sensors Emerging enterprise applications and trends such as the smartOR standard and innovative concepts for human-machine interaction in the operating room Several chapters use a tutorial style to emphasize the development process behind complex distributed networked systems and services, which highlights the difficulties of knowledge engineering of such systems. Delving into novel concepts, theories, and advanced technologies, this book offers inspiration for further research and development in distributed computing and networking, especially related to security solutions for distributed environments.

Beginning with a short biography of Kurt Godel, "Godel's Theorem in Focus" provides the reader with a clear guide to the mechanics of Godel's proof in a format intelligible to the non-mathematician. The book moves on to explanations of the mechanics of Godel's proof and its significance for mathematical logic and the philosophy of mathematics. In the final section, S. G. Shanker presents a major new critique of Godel's theorem.

This book deals with the determinants of linear operators in Euclidean, Hilbert and Banach spaces. Determinants of operators give us an important tool for solving linear equations and invertibility conditions for linear operators, enable us to describe the spectra, to evaluate the multiplicities of eigenvalues, etc. We derive upper and lower bounds, and perturbation results for determinants, and discuss applications of our theoretical results to spectrum perturbations, matrix equations, two parameter eigenvalue problems, as well as to differential, difference and functional-differential equations.

This book brings together the scattered literature associated with the seemingly unrelated regression equations (SURE) model used by econometricians and others. It focuses on the theoretical statistical results associated with the SURE model.

Fuzzy knowledge and fuzzy systems affect our lives today as systems enter the world of commerce. Fuzzy systems are incorporated in domestic appliances (washing machine, air conditioning, microwave, telephone) and in transport systems (a pilotless helicopter has recently completed a test flight). Future applications are expected to have dramatic implications for the demand for labor, among other things. It was with such thoughts in mind that this first international survey of future applications of fuzzy logic has been undertaken. The results are likely to be predictive for a decade beyond the millenium. The predictive element is combined with a bibliography which serves as an historical anchor as well as being both extensive and extremely useful. Analysis and Evaluation of Fuzzy Systems is thus a milestone in the development of fuzzy logic and applications of three representative subsystems: Fuzzy Control, Fuzzy Pattern Recognition and Fuzzy Communications.

This book includes articles on denotational semanitcs, recursion theoretic aspects of computer science, model theory and algebra, automath and automated reasoning, stability theory, topoi and mathematics, and topoi and logic. It is intended for mathematical logicians and computer scientists.

This book explores the limits of our knowledge. The author shows how uncertainty and indefiniteness not only define the borders confining our understanding, but how they feed into the process of discovery and help to push back these borders. Starting with physics the author collects examples from economics, neurophysiology, history, ecology and philosophy. The first part shows how information helps to reduce indefiniteness. Understanding rests on our ability to find the right context, in which we localize a problem as a point in a network of connections. New elements must be combined with the old parts of the existing complex knowledge system, in order to profit maximally from the information. An attempt is made to quantify the value of information by its ability to reduce indefiniteness. The second part explains how to handle indefiniteness with methods from fuzzy logic, decision theory, hermeneutics and semiotics. It is not sufficient that the new element appears in an experiment, one also has to find a theoretical reason for its existence. Indefiniteness becomes an engine of science, which gives rise to new ideas.

First published in 1990, this book consists of a detailed exposition of results of the theory of "interpretation" developed by G. Kreisel - the relative impenetrability of which gives the elucidation contained here great value for anyone seeking to understand his work. It contains more complex versions of the information obtained by Kreisel for number theory and clustering around the no-counter-example interpretation, for number-theorectic forumulae provide in ramified analysis. It also proves the omega-consistency of ramified analysis. The author also presents proofs of Schutte's cut-elimination theorems which are based on his consistency proofs and essentially contain them - these went further than any published work up to that point, helping to squeeze the maximum amount of information from these proofs.

Professor Merrie Bergmann presents an accessible introduction to the subject of many-valued and fuzzy logic designed for use on undergraduate and graduate courses in non-classical logic. Bergmann discusses the philosophical issues that give rise to fuzzy logic - problems arising from vague language - and returns to those issues as logical systems are presented. For historical and pedagogical reasons, three-valued logical systems are presented as useful intermediate systems for studying the principles and theory behind fuzzy logic. The major fuzzy logical systems - Lukasiewicz, Goedel, and product logics - are then presented as generalisations of three-valued systems that successfully address the problems of vagueness. A clear presentation of technical concepts, this book includes exercises throughout the text that pose straightforward problems, that ask students to continue proofs begun in the text, and that engage students in the comparison of logical systems.

Published with the aid of a grant from the National Endowment for the Humanities. Contains the only complete English-language text of The Concept of Truth in Formalized Languages. Tarski made extensive corrections and revisions of the original translations for this edition, along with new historical remarks. It includes a new preface and a new analytical index for use by philosophers and linguists as well as by historians of mathematics and philosophy.

Das Buch behandelt Matrizengleichungen und -funktionen sowie die computergerechte Darstellung und Losung der Bewegungsgleichungen von Schwingungssystemen mit endlich vielen Freiheitsgraden und fuhrt in die Grundlagen der Naherungsmethoden von Rayleigh und Ritz ein. Das Eigenwertproblem wird, anders als sonst ublich, von einem allgemeinen Standpunkt aus betrachtet. Dadurch gewinnt die Darstellung an Verstandlichkeit und an Anwendungsbreite. Das Buch ist sowohl fur Studierende als auch fur Physiker und Ingenieure in der Praxis geschrieben.

The Only Undergraduate Textbook to Teach Both Classical and Virtual Knot Theory An Invitation to Knot Theory: Virtual and Classical gives advanced undergraduate students a gentle introduction to the field of virtual knot theory and mathematical research. It provides the foundation for students to research knot theory and read journal articles on their own. Each chapter includes numerous examples, problems, projects, and suggested readings from research papers. The proofs are written as simply as possible using combinatorial approaches, equivalence classes, and linear algebra. The text begins with an introduction to virtual knots and counted invariants. It then covers the normalized f-polynomial (Jones polynomial) and other skein invariants before discussing algebraic invariants, such as the quandle and biquandle. The book concludes with two applications of virtual knots: textiles and quantum computation.

Category theory is unmatched in its ability to organize and layer abstractions and to find commonalities between structures of all sorts. No longer the exclusive preserve of pure mathematicians, it is now proving itself to be a powerful tool in science, informatics, and industry. By facilitating communication between communities and building rigorous bridges between disparate worlds, applied category theory has the potential to be a major organizing force. This book offers a self-contained tour of applied category theory. Each chapter follows a single thread motivated by a real-world application and discussed with category-theoretic tools. We see data migration as an adjoint functor, electrical circuits in terms of monoidal categories and operads, and collaborative design via enriched profunctors. All the relevant category theory, from simple to sophisticated, is introduced in an accessible way with many examples and exercises, making this an ideal guide even for those without experience of university-level mathematics.

The present study is an extension of the topic introduced in Dr. Hailperin's Sentential Probability Logic, where the usual true-false semantics for logic is replaced with one based more on probability, and where values ranging from 0 to 1 are subject to probability axioms. Moreover, as the word "sentential" in the title of that work indicates, the language there under consideration was limited to sentences constructed from atomic (not inner logical components) sentences, by use of sentential connectives ("no," "and," "or," etc.) but not including quantifiers ("for all," "there is"). An initial introduction presents an overview of the book. In chapter one, Halperin presents a summary of results from his earlier book, some of which extends into this work. It also contains a novel treatment of the problem of combining evidence: how does one combine two items of interest for a conclusion-each of which separately impart a probability for the conclusion-so as to have a probability for the conclusion based on taking both of the two items of interest as evidence? Chapter two enlarges the Probability Logic from the first chapter in two respects: the language now includes quantifiers ("for all," and "there is") whose variables range over atomic sentences, not entities as with standard quantifier logic. (Hence its designation: ontological neutral logic.) A set of axioms for this logic is presented. A new sentential notion-the suppositional-in essence due to Thomas Bayes, is adjoined to this logic that later becomes the basis for creating a conditional probability logic. Chapter three opens with a set of four postulates for probability on ontologically neutral quantifier language. Many properties are derived and a fundamental theorem is proved, namely, for any probability model (assignment of probability values to all atomic sentences of the language) there will be a unique extension of the probability values to all closed sentences of the language.

This volume, which presents the cumulation of the authors' research in the field, deals with Lidstone, Hermite, Abel-Gontscharoff, Birkhoff, piecewise Hermite and Lidstone, spline and Lidstone-spline interpolating problems. Explicit representations of the interpolating polynomials and associated error functions are given, as well as explicit error inequalities in various norms. Numerical illustrations are provided of the importance and sharpness of the various results obtained. Also demonstrated are the significance of these results in the theory of ordinary differential equations such as maximum principles, boundary value problems, oscillation theory, disconjugacy and disfocality. The book should be useful for mathematicians, numerical analysts, computer scientists and engineers.

This book presents a philosophy of science, based on panenmentalism: an original modal metaphysics, which is realist about individual pure (non-actual) possibilities and rejects the notion of possible worlds. The book systematically constructs a new and novel way of understanding and explaining scientific progress, discoveries, and creativity. It demonstrates that a metaphysics of individual pure possibilities is indispensable for explaining and understanding mathematics and natural sciences. It examines the nature of individual pure possibilities, actualities, mind-dependent and mind-independent possibilities, as well as mathematical entities. It discusses in detail the singularity of each human being as a psychical possibility. It analyses striking scientific discoveries, and illustrates by means of examples of the usefulness and vitality of individual pure possibilities in the sciences.

Fuzzy social choice theory is useful for modeling the uncertainty and imprecision prevalent in social life yet it has been scarcely applied and studied in the social sciences. Filling this gap, Application of Fuzzy Logic to Social Choice Theory provides a comprehensive study of fuzzy social choice theory. The book explains the concept of a fuzzy maximal subset of a set of alternatives, fuzzy choice functions, the factorization of a fuzzy preference relation into the "union" (conorm) of a strict fuzzy relation and an indifference operator, fuzzy non-Arrowian results, fuzzy versions of Arrow's theorem, and Black's median voter theorem for fuzzy preferences. It examines how unambiguous and exact choices are generated by fuzzy preferences and whether exact choices induced by fuzzy preferences satisfy certain plausible rationality relations. The authors also extend known Arrowian results involving fuzzy set theory to results involving intuitionistic fuzzy sets as well as the Gibbard-Satterthwaite theorem to the case of fuzzy weak preference relations. The final chapter discusses Georgescu's degree of similarity of two fuzzy choice functions.

Aimed at graduate students and research logicians and mathematicians, this much-awaited text covers over forty years of work on relative classification theory for non-standard models of arithmetic. With graded exercises at the end of each chapter, the book covers basic isomorphism invariants: families of types realized in a model, lattices of elementary substructures and automorphism groups. Many results involve applications of the powerful technique of minimal types due to Haim Gaifman, and some of the results are classical but have never been published in a book form before.

This volume, first published in 2000, presents a classical approach to the foundations and development of the geometry of vector fields, describing vector fields in three-dimensional Euclidean space, triply-orthogonal systems and applications in mechanics. Topics covered include Pfaffian forms, systems in n-dimensional space, and foliations and their Godbillion-Vey invariant. There is much interest in the study of geometrical objects in n-dimensional Euclidean space and this volume provides a useful and comprehensive presentation.

This text explores the many transformations that the mathematical proof has undergone from its inception to its versatile, present-day use, considering the advent of high-speed computing machines. Though there are many truths to be discovered in this book, by the end it is clear that there is no formalized approach or standard method of discovery to date. Most of the proofs are discussed in detail with figures and equations accompanying them, allowing both the professional mathematician and those less familiar with mathematics to derive the same joy from reading this book.

This book argues for a view in which processes of dialogue and interaction are taken to be foundational to reasoning, logic, and meaning. This is both a continuation, and a substantial modification, of an inferentialist approach to logic. As such, the book not only provides a critical introduction to the inferentialist view, but it also provides an argument that this shift in perspective has deep and foundational consequences for how we understand the nature of logic and its relationship with meaning and reasoning. This has been upheld by several technical results, including, for example a novel approach to logical paradox and logical revision, and an account of the internal justification of logical rules. The book shows that inferentialism is greatly strengthened, such that it can answer the most stringent criticisms of the view. This leads to a view of logic that emphasizes the dynamics of reasoning, provides a novel account of the justification and normativity of logical rules, thus leading to a new, attractive approach to the foundations of logic. The book addresses readers interested in philosophy of language, philosophical and mathematical logic, theories of reasoning, and also those who actively engage in current debates involving, for example, logical revision, and the relationship between logic and reasoning, from advanced undergraduates, to professional philosophers, mathematicians, and linguists.

The Annual European Meeting of the Association for Symbolic Logic, generally known as the Logic Colloquium, is the most prestigious annual meeting in the field. Many of the papers presented there are invited surveys of developments, and the rest of the papers are chosen to complement the invited talks. This 2007 volume includes surveys, tutorials, and selected research papers from the 2005 meeting. Highlights include three papers on different aspects of connections between model theory and algebra; a survey of major advances in combinatorial set theory; a tutorial on proof theory and modal logic; and a description of Bernay's philosophy of mathematics.