This volume is based on the talks given at the Workshop on Infinity and Truth held at the Institute for Mathematical Sciences, National University of Singapore, from 25 to 29 July 2011. The chapters cover topics in mathematical and philosophical logic that examine various aspects of the foundations of mathematics. The theme of the volume focuses on two basic foundational questions: (i) What is the nature of mathematical truth and how does one resolve questions that are formally unsolvable within the Zermelo-Fraenkel Set Theory with the Axiom of Choice, and (ii) Do the discoveries in mathematics provide evidence favoring one philosophical view over others? These issues are discussed from the vantage point of recent progress in foundational studies.The final chapter features questions proposed by the participants of the Workshop that will drive foundational research. The wide range of topics covered here will be of interest to students, researchers and mathematicians concerned with issues in the foundations of mathematics.

This edited collection casts light on central issues within contemporary philosophy of mathematics such as the realism/anti-realism dispute; the relationship between logic and metaphysics; and the question of whether mathematics is a science of objects or structures. The discussions offered in the papers involve an in-depth investigation of, among other things, the notions of mathematical truth, proof, and grounding; and, often, a special emphasis is placed on considerations relating to mathematical practice. A distinguishing feature of the book is the multicultural nature of the community that has produced it. Philosophers, logicians, and mathematicians have all contributed high-quality articles which will prove valuable to researchers and students alike.

Set theory is concerned with the foundation of mathematics. In the original formulations of set theory, there were paradoxes contained in the idea of the "set of all sets". Current standard theory (Zermelo-Fraenkel) avoids these paradoxes by restricting the way sets may be formed by other sets, specifically to disallow the possibility of forming the set of all sets. In the 1930s, Quine proposed a different form of set theory in which the set of all sets - the universal set - is allowed, but other restrictions are placed on these axioms. Since then, the steady interest expressed in these non-standard set theories has been boosted by their relevance to computer science. The second edition still concentrates largely on Quine's New Foundations, reflecting the author's belief that this provides the richest and most mysterious of the various systems dealing with set theories with a universal set. Also included is an expanded and completely revised account of the set theories of Church-Oswald and Mitchell, with descriptions of permutation models and extensions that preserve power sets. Dr Foster here presents the reader with a useful and readable introduction for those interested in this topic, and a reference work for those already involved in this area.

All current methods of secure communication such as public-key cryptography can eventually be broken by faster computing. At the interface of physics and computer science lies a powerful solution for secure communications: quantum cryptography. Because eavesdropping changes the physical nature of the information, users in a quantum exchange can easily detect eavesdroppers. This allows for totally secure random key distribution, a central requirement for use of the one-time pad. Since the one-time pad is theoretically proven to be undecipherable, quantum cryptography is the key to perfect secrecy. Quantum Communications and Cryptography is the first comprehensive review of the past, present, and potential developments in this dynamic field. Leading expert contributors from around the world discuss the scientific foundations, experimental and theoretical developments, and cutting-edge technical and engineering advances in quantum communications and cryptography. The book describes the engineering principles and practical implementations in a real-world metropolitan network as well as physical principles and experimental results of such technologies as entanglement swapping and quantum teleportation. It also offers the first detailed treatment of quantum information processing with continuous variables. Technologies include both free-space and fiber-based communications systems along with the necessary protocols and information processing approaches. Bridging the gap between physics and engineering, Quantum Communications and Cryptography supplies a springboard for further developments and breakthroughs in this rapidly growing area.

Following developments in modern geometry, logic and physics, many scientists and philosophers in the modern era considered Kanta (TM)s theory of intuition to be obsolete. But this only represents one side of the story concerning Kant, intuition and twentieth century science. Several prominent mathematicians and physicists were convinced that the formal tools of modern logic, set theory and the axiomatic method are not sufficient for providing mathematics and physics with satisfactory foundations. All of Hilbert, GAdel, PoincarA(c), Weyl and Bohr thought that intuition was an indispensable element in describing the foundations of science. They had very different reasons for thinking this, and they had very different accounts of what they called intuition. But they had in common that their views of mathematics and physics were significantly influenced by their readings of Kant. In the present volume, various views of intuition and the axiomatic method are explored, beginning with Kanta (TM)s own approach. By way of these investigations, we hope to understand better the rationale behind Kanta (TM)s theory of intuition, as well as to grasp many facets of the relations between theories of intuition and the axiomatic method, dealing with both their strengths and limitations; in short, the volume covers logical and non-logical, historical and systematic issues in both mathematics and physics.

The Asian Logic Conference is the most significant logic meeting outside of North America and Europe, and this volume represents work presented at, and arising from the 12th meeting. It collects a number of interesting papers from experts in the field. It covers many areas of logic.

For computer scientists, especially those in the security field, the use of chaos has been limited to the computation of a small collection of famous but unsuitable maps that offer no explanation of why chaos is relevant in the considered contexts. Discrete Dynamical Systems and Chaotic Machines: Theory and Applications shows how to make finite machines, such as computers, neural networks, and wireless sensor networks, work chaotically as defined in a rigorous mathematical framework. Taking into account that these machines must interact in the real world, the authors share their research results on the behaviors of discrete dynamical systems and their use in computer science. Covering both theoretical and practical aspects, the book presents: Key mathematical and physical ideas in chaos theory Computer science fundamentals, clearly establishing that chaos properties can be satisfied by finite state machines Concrete applications of chaotic machines in computer security, including pseudorandom number generators, hash functions, digital watermarking, and steganography Concrete applications of chaotic machines in wireless sensor networks, including secure data aggregation and video surveillance Until the authors' recent research, the practical implementation of the mathematical theory of chaos on finite machines raised several issues. This self-contained book illustrates how chaos theory enables the study of computer security problems, such as steganalysis, that otherwise could not be tackled. It also explains how the theory reinforces existing cryptographically secure tools and schemes.

This volume is based on the successful 6th China-Japan Seminar on number theory that was held in Shanghai Jiao Tong University in August 2011. It is a compilation of survey papers as well as original works by distinguished researchers in their respective fields. The topics range from traditional analytic number theory - additive problems, divisor problems, Diophantine equations - to elliptic curves and automorphic L-functions. It contains new developments in number theory and the topics complement the existing two volumes from the previous seminars which can be found in the same book series.

Features Provides a uniquely historical perspective on the mathematical underpinnings of a comprehensive list of games Suitable for a broad audience of differing mathematical levels. Anyone with a passion for games, game theory, and mathematics will enjoy this book, whether they be students, academics, or game enthusiasts Covers a wide selection of topics at a level that can be appreciated on a historical, recreational, and mathematical level.

This fourth volume in the series of yearbooks by the Association of Mathematics Educators in Singapore entitled Reasoning, Communication and Connections in Mathematics is unique in that it focuses on a single theme in mathematics education. The objective is to encourage teachers and researchers to advance reasoning, communication and connections in mathematics classrooms.Several renowned international researchers in the field have published their work in this volume. The fifteen chapters of the book illustrate evidence-based practices that school teachers and researchers can experiment with in their own classrooms to bring about meaningful learning outcomes. Three major themes: mathematical tasks, classroom discourse, and connectivity within and beyond mathematics, shape the ideas underpinning reasoning, communication and connections in these chapters. The book makes a significant contribution towards mathematical processes essential for learners of mathematics. It is a good resource for mathematics educators and research students.

This penultimate volume contains numerous original, elegant, and surprising results in 1-dimensional cellular automata. Perhaps the most exciting, if not shocking, new result is the discovery that only 82 local rules, out of 256, suffice to predict the time evolution of any of the remaining 174 local rules from an arbitrary initial bit-string configuration. This is contrary to the well-known folklore that 256 local rules are necessary, leading to the new concept of quasi-global equivalence.Another surprising result is the introduction of a simple, yet explicit, infinite bit string called the super string S, which contains all random bit strings of finite length as sub-strings. As an illustration of the mathematical subtlety of this amazing discrete testing signal, the super string S is used to prove mathematically, in a trivial and transparent way, that rule 170 is as chaotic as a coin toss.Yet another unexpected new result, among many others, is the derivation of an explicit basin tree generation formula which provides an analytical relationship between the basin trees of globally-equivalent local rules. This formula allows the symbolic, rather than numerical, generation of the time evolution of any local rule corresponding to any initial bit-string configuration, from one of the 88 globally-equivalent local rules.But perhaps the most provocative idea is the proposal for adopting rule 137, over its three globally-equivalent siblings, including the heretofore more well-known rule 110, as the prototypical universal Turing machine.

Computational Intelligence Assisted Design framework mobilises computational resources, makes use of multiple Computational Intelligence (CI) algorithms and reduces computational costs. This book provides examples of real-world applications of technology. Case studies have been used to show the integration of services, cloud, big data technology and space missions. It focuses on computational modelling of biological and natural intelligent systems, encompassing swarm intelligence, fuzzy systems, artificial neutral networks, artificial immune systems and evolutionary computation. This book provides readers with wide-scale information on CI paradigms and algorithms, inviting readers to implement and problem solve real-world, complex problems within the CI development framework. This implementation framework will enable readers to tackle new problems without difficulty through a few tested MATLAB source codes

"This engagingly written text provides a useful pedagogical introduction to an extensive class of geometrical phenomena in the optics of polarization and phase, including simple explanations of much of the underlying mathematics." -Michael Berry, University of Bristol, UK "The author covers a vast number of topics in great detail, with a unifying mathematical treatment. It will be a useful reference for both beginners and experts...." -Enrique Galvez, Charles A. Dana Professor of Physics and Astronomy, Colgate University "a firm and comprehensive grounding both for those looking to acquaint themselves with the field and those of us that need reminding of the things we thought we knew, but hitherto did not understand: an essential point of reference." -Miles Padgett, Kelvin Chair of Natural Philosophy and Vice Principal (Research), University of Glasgow This book focuses on the various forms of wavefield singularities, including optical vortices and polarization singularities, as well as orbital angular momentum and associated applications. It highlights how an understanding of singular optics provides a completely different way to look at light. Whereas traditional optics focuses on the shape and structure of the non-zero portions of the wavefield, singular optics describes a wave's properties from its null regions. The contents cover the three main areas of the field: the study of generic features of wavefields, determination of unusual properties of vortices and wavefields that contain singularities, and practical applications of vortices and other singularities.

This book provides a broad introduction to some of the most fascinating and beautiful areas of discrete mathematical structures. It starts with a chapter on sets and goes on to provide examples in logic, applications of the principle of inclusion and exclusion and finally the pigeonhole principal. Computational techniques including the principle of mathematical introduction are provided, as well as a study on elementary properties of graphs, trees and lattices. Some basic results on groups, rings, fields and vector spaces are also given, the treatment of which is intentionally simple since such results are fundamental as a foundation for students of discrete mathematics. In addition, some results on solutions of systems of linear equations are discussed.

This book provides a broad introduction to some of the most fascinating and beautiful areas of discrete mathematical structures. It starts with a chapter on sets and goes on to provide examples in logic, applications of the principle of inclusion and exclusion and finally the pigeonhole principal. Computational techniques including the principle of mathematical introduction are provided, as well as a study on elementary properties of graphs, trees and lattices. Some basic results on groups, rings, fields and vector spaces are also given, the treatment of which is intentionally simple since such results are fundamental as a foundation for students of discrete mathematics. In addition, some results on solutions of systems of linear equations are discussed.

Algebraic logic is a subject in the interface between logic, algebra and geometry, it has strong connections with category theory and combinatorics. Tarski s quest for finding structure in logic leads to cylindric-like algebras as studied in this book, they are among the main players in Tarskian algebraic logic. Cylindric algebra theory can be viewed in many ways: as an algebraic form of definability theory, as a study of higher-dimensional relations, as an enrichment of Boolean Algebra theory, or, as logic in geometric form ( cylindric in the name refers to geometric aspects). Cylindric-like algebras have a wide range of applications, in, e.g., natural language theory, data-base theory, stochastics, and even in relativity theory. The present volume, consisting of 18 survey papers, intends to give an overview of the main achievements and new research directions in the past 30 years, since the publication of the Henkin-Monk-Tarski monographs. It is dedicated to the memory of Leon Henkin. "

Kurt Godel (1906-1978) was an Austrian-American mathematician, who is best known for his incompleteness theorems. He was the greatest mathematical logician of the 20th century, with his contributions extending to Einstein s general relativity, as he proved that Einstein s theory allows for time machines.
The Godel incompleteness theorem - the usual formal mathematical systems cannot prove nor disprove all true mathematical sentences - is frequently presented in textbooks as something that happens in the rarefied realms of mathematical logic, and that has nothing to do with the real world. Practice shows the contrary though; one can demonstrate the validity of the phenomenon in various areas, ranging from chaos theory and physics to economics and even ecology. In this lively treatise, based on Chaitin s groundbreaking work and on the da Costa-Doria results in physics, ecology, economics and computer science, the authors show that the Godel incompleteness phenomenon can directly bear on the practice of science and perhaps on our everyday life.
This accessible book gives a new, detailed and elementary explanation of the Godel incompleteness theorems and presents the Chaitin results and their relation to the da Costa-Doria results, which are given in full, but with no technicalities. Besides theory, the historical report and personal stories about the main character and on this book s writing process, make it appealing leisure reading for those interested in mathematics, logic, physics, philosophy and computer sciences.

See also: http: //www.youtube.com/watch?v=REy9noY5Sg8

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Convergence of Blockchain, AI and IoT: A Digital Platform discusses the convergence of three powerful technologies that play into the digital revolution and blur the lines between biological, digital, and physical objects. This book covers novel algorithms, solutions for addressing issues in applications, security, authentication, and privacy. Discusses innovative technological upgradation and significant challenges in the current era Gives an overview of clinical scientific research that enables smart diagnosis through artificial intelligence Provides an insight into how disruptive technology enabled with the self-running devices and protection mechanism is involved in an augmented reality with blockchain mechanism Talks about neural science being capable of enhancing deep brain waves to predict an overall improvement in human thoughts and behaviours Covers the digital currency mechanism in detail Enhances the knowledge of the readers about smart contract and ledger mechanism with artificial intelligence and blockchain mechanism Targeted audiences range from those interested in the technical revolution of blockchain, big data and the Internet of Things, to research scholars and the professional market.

The Asian Logic Conference is part of the series of logic conferences inaugurated in Singapore in 1981. It is normally held every three years and rotates among countries in the Asia-Pacific region. The 11th Asian Logic Conference is held in the National University of Singapore, in honour of Professor Chong Chitat on the occasion of his 60th birthday. The conference is on the broad area of logic, including theoretical computer science. It is considered a major event in this field and is regularly sponsored by the Association of Symbolic Logic. This volume contains papers from this meeting.

First developed in the early 1980s by Lenstra, Lenstra, and Lov sz, the LLL algorithm was originally used to provide a polynomial-time algorithm for factoring polynomials with rational coefficients. It very quickly became an essential tool in integer linear programming problems and was later adapted for use in cryptanalysis. This book provides an introduction to the theory and applications of lattice basis reduction and the LLL algorithm. With numerous examples and suggested exercises, the text discusses various applications of lattice basis reduction to cryptography, number theory, polynomial factorization, and matrix canonical forms.

This work provides a posteriori error analysis for mathematical idealizations in modeling boundary value problems, especially those arising in mechanical applications, and for numerical approximations of numerous nonlinear var- tional problems. An error estimate is called a posteriori if the computed solution is used in assessing its accuracy. A posteriori error estimation is central to m- suring, controlling and minimizing errors in modeling and numerical appr- imations. In this book, the main mathematical tool for the developments of a posteriori error estimates is the duality theory of convex analysis, documented in the well-known book by Ekeland and Temam ( 49]). The duality theory has been found useful in mathematical programming, mechanics, numerical analysis, etc. The book is divided into six chapters. The first chapter reviews some basic notions and results from functional analysis, boundary value problems, elliptic variational inequalities, and finite element approximations. The most relevant part of the duality theory and convex analysis is briefly reviewed in Chapter 2.

Combinatory logic is one of the most versatile areas within logic that is tied to parts of philosophical, mathematical, and computational logic. Functioning as a comprehensive source for current developments of combinatory logic, this book is the only one of its kind to cover results of the last four decades. Using a reader-friendly style, the author presents the most up-to-date research studies. She includes an introduction to combinatory logic before progressing to its central theorems and proofs. The text makes intelligent and well-researched connections between combinatory logic and lambda calculi and presents models and applications to illustrate these connections.

Fuzzy Cluster Analysis presents advanced and powerful fuzzy clustering techniques. This thorough and self-contained introduction to fuzzy clustering methods and applications covers classification, image recognition, data analysis and rule generation. Combining theoretical and practical perspectives, each method is analysed in detail and fully illustrated with examples. Features include:

• Sections on inducing fuzzy if-then rules by fuzzy clustering and non-alternating optimization fuzzy clustering algorithms
• Discussion of solid fuzzy clustering techniques like the fuzzy c-means, the Gustafson-Kessel and the Gath-and-Geva algorithm for classification problems
• Focus on linear and shell clustering techniques used for detecting contours in image analysis
• Accompanying software and data sets pertaining to the examples presented, enabling the reader to learn through experimentation
• Examination of the difficulties involved in evaluating the results of fuzzy cluster analysis and of determining the number of clusters with analysis of global and local validity measures
• Description of different fuzzy clustering techniques allowing the user to select the method most appropriate to a particular problem

Kurt Godel, the greatest logician of our time, startled the world of mathematics in 1931 with his Theorem of Undecidability, which showed that some statements in mathematics are inherently "undecidable." His work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum theory brought him further worldwide fame. In this introductory volume, Raymond Smullyan, himself a well-known logician, guides the reader through the fascinating world of Godel's incompleteness theorems. The level of presentation is suitable for anyone with a basic acquaintance with mathematical logic. As a clear, concise introduction to a difficult but essential subject, the book will appeal to mathematicians, philosophers, and computer scientists.

First published in 1982, this reissue contains a critical exposition of the views of Frege, Dedekind and Peano on the foundations of arithmetic. The last quarter of the 19th century witnessed a remarkable growth of interest in the foundations of arithmetic. This work analyses both the reasons for this growth of interest within both mathematics and philosophy and the ways in which this study of the foundations of arithmetic led to new insights in philosophy and striking advances in logic.

This historical-critical study provides an excellent introduction to the problems of the philosophy of mathematics - problems which have wide implications for philosophy as a whole. This reissue will appeal to students of both mathematics and philosophy who wish to improve their knowledge of logic.