Mathematical logic Books

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Book Synopsisthis section gained proofs of the Schroeder–Bernstein theorem and the Trichotomy Law for Sets, and lost most of the material about finite and countable sets, which has now been moved to a new section devoted to those two types of sets.Trade Review“This is a well-written book, based on very sound pedagogical ideas. It would be an excellent choice as a textbook for a ‘transition’ course.” (Margret Höft, zbMATH 1012.00013, 2021)“The contents of the book is organized in three parts … . this is a nice book, which also this reviewer has used with profit in his teaching of beginner students. It is written in a highly pedagogical style and based upon valuable didactical ideas.” (R. Steinbauer, Monatshefte für Mathematik, Vol. 174, 2014)“Books in this category are meant to teach mathematical topics and techniques that will become valuable in more advanced courses. This book meets these criteria. … This book is well suited as a textbook for a transitional course between calculus and more theoretical courses. I also recommend it for academic libraries.” (Edgar R. Chavez, ACM Computing Reviews, February, 2012)“This is an improved edition of a good book that can serve in the undergraduate curriculum as a bridge between computationally oriented courses like calculus and more abstract courses like algebra.” (Teun Koetsier, Zentralblatt MATH, Vol. 1230, 2012)Table of ContentsPreface to the Second Edition Preface to the First Edition To the Student To the Instructor Part I. Proofs 1. Informal Logic 2. Strategies for Proofs Part II. Fundamentals 3. Sets 4. Functions 5. Relations 6. Finite and Infinite Sets Part III. Extras 7. Selected Topics 8. Explorations Appendix: Properties of Numbers Bibliography Index

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Book SynopsisThe Discrete.- Integers.- Natural Numbers and Induction.- Some Points of Logic.- Recursion.- Underlying Notions in Set Theory.- Equivalence Relations and Modular Arithmetic.- Arithmetic in Base Ten.- The Continuous.- Real Numbers.- Embedding Z in R.- Limits and Other Consequences of Completeness.- Rational and Irrational Numbers.- Decimal Expansions.- Cardinality.- Final Remarks.- Further Topics.- Continuity and Uniform Continuity.- Public-Key Cryptography.- Complex Numbers.- Groups and Graphs.- Generating Functions.- Cardinal Number and Ordinal Number.- Remarks on Euclidean Geometry.Trade ReviewFrom the reviews:"The Art of Proof is a surprising union of rigor with taste and wit. The authors take a hard-core axiomatic approach, but the writing is never dry. Instead, topics are carefully chosen and meticulously developed with grace and humor, careful attention to detail, and just the right number of skill-building exercises and thought-provoking problems."The text is spare—well under two hundred pages—but contains a thorough axiomatic development of the integers and the reals, along with non-standard optional topics such as Cayley graphs and generating functions. Instead of the standard scattershot "symbolic logic-set theory-functions-proof by contradiction-zzzz..." books, this text keeps its focus on just a few fundamental ideas, of which induction is the most important. This helps my students to feel that they are participants in a grand undertaking—the construction of a number system—rather than passive victims of one proof technique after another." —Paul Zeitz (Mathematics Professor at the University of San Francisco)“This qualitative transition presents a most acute pedagogical challenge. … This book does feature definite mathematical content, contrasting with works that aim at decoupling purely logical apparatus from strictly mathematical concerns. … The authors write with the authority of research mathematicians and clearly mean to open that avenue to students. Summing Up: Recommended. Upper-division undergraduates through professionals.” (D. V. Feldman, Choice, Vol. 48 (8), April, 2011)“This book offers an approach well-balanced between rigor and clarifying simplification. Dilbert and Foxtrot cartoons with philosophical quotes presage the introduction of axioms and preliminary propositions. This graceful and witty blend succeeds well in a textbook for a post-calculus course transitioning a student to higher mathematics. The Art of Proof can also well serve independent readers looking for a solitary path to a vista on higher mathematics.” (Tom Schulte, The Mathematical Association of America, November, 2010)“This is an undergraduate text to extend, in a deeper and formal way, the usual initial knowledge of mathematics. The book deals with classical topics like integers, induction, algorithms, real numbers, rational numbers, modular arithmetic, limits, uncountable sets … . The publication may be useful for people using the book to teach a course on the above mentioned topics. … The aim behind this textbook is teaching how to read and write mathematics as well as understanding key methods and concepts.” (Claudi Alsina, Zentralblatt MATH, Vol. 1198, 2010)Table of ContentsPreface.- Notes for the Student.- Notes for Instructors.- Part I: The Discrete.- 1 Integers.- 2 Natural Numbers and Induction.- 3 Some Points of Logic.- 4 Recursion.- 5 Underlying Notions in Set Theory.- 6 Equivalence Relations and Modular Arithmetic.- 7 Arithmetic in Base Ten.- Part II: The Continuous.- 8 Real Numbers.- 9 Embedding Z in R.- 10. Limits and Other Consequences of Completeness.- 11 Rational and Irrational Numbers.- 12 Decimal Expansions.- 13 Cardinality.- 14 Final Remarks.- Further Topics.- A Continuity and Uniform Continuity.- B Public-Key Cryptography.- C Complex Numbers.- D Groups and Graphs.- E Generating Functions.- F Cardinal Number and Ordinal Number.- G Remarks on Euclidean Geometry.- List of Symbols.- Index.

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Book SynopsisInverse problems arise in many disciplines and hold great importance to practical applications. However, sound new methods are needed to solve these problems. Over the past few years, Japanese and Korean mathematicians have obtained a number of very interesting and unique results in inverse problems.Inverse Problems and Related Topics compiles papers authored by some of the top researchers in Korea and Japan. It presents a number of original and useful results and offers a unique opportunity to explore the current trends of research in inverse problems in these countries. Highlighting the existence and active work of several Japanese and Korean groups, it also serves as a guide to those seeking future scientific exchange with researchers in these countries.Trade Review"The aim of this book is to fill the gap between high-school mathematics and mathematics taught at university…the reader is shown what it means to prove something rigourously…This book is easy to read for anyone with a high-school mathematics background." - European Mathematical Society NewsletterTable of ContentsA Finite Difference Model for Calderón's Boundary Inverse Problem. Inverse Problems for Equations with Memory. Parameter Estimation of Elastic Media. The Probe Method and its Applications. Recent Progress in the Inverse Conductivity Problem with Single Measurement. A Moment Method on Inverse Problems for the Heat Equation. Some Remarks on Free Boundaries of Recirculation Euler Flows with Constant Vorticity. Algorithms for the Identification of Spatially Varying/Invariant Stiffness and Dampings in Flexible Beams. Numerical Solutions of the Cauchy Problem in Potential and Elastostatics. Inverse Source Problems in the Helmholtz Equations. A Numerical Method for a Magnetostatic Inverse Problem using the Edge Element. Exact Controllability Method and Multidimensional Linear Inverse Problems. Impedance Computed Tomo-Electrocardiography. An Inverse Problem for Free Channel Scattering. Surface Impedance Tensor and Boundary Value Problem. Aysmptotics for the Spectral and Weyl Functions of the Operator-Value Sturm-Liouville Problem. Exact Controllability Method and Multidimensional Linear Inverse Problems

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Book SynopsisMild Cognitive Impairment (MCI) has been identified as an important clinical transition between normal aging and the early stages of Alzheimer's disease (AD). Since treatments for AD are most likely to be most effective early in the course of the disease, MCI has become a topic of great importance and has been investigated in different populations of interest in many countries. This book brings together these differing perspectives on MCI for the first time. This volume provides a comprehensive resource for clinicians, researchers, and students involved in the study, diagnosis, treatment, and rehabilitation of people with MCI. Clinical investigators initially defined mild cognitive impairment (MCI) as a transitional condition between normal aging and the early stages of Alzheimer’s disease (AD). Because the prevalence of AD increases with age and very large numbers of older adults are affected worldwide, these clinicians saw a pressing need to identify AD as early as possible. It is at this very early stage in the disease course that treatments to slow the progress and control symptoms are likely to be most effective.Since the first introduction of MCI, research interest has grown exponentially, and the utility of the concept has been investigated from a variety of perspectives in different populations of interest (e.g., clinical samples, volunteers, population-based screening) in many different countries. Much variability in findings has resulted. Although it has been acknowledged that the differences observed between samples may be ‘legitimate variations’, there has been no attempt to understand what it is we have learned about MCI (i.e., common features and differences) from each of these perspectives.This book brings together information about MCI in different populations from around the world. Mild Cognitive Impairment will be an important resource for any clinician, researcher, or student involved in the study, detection, treatment, and rehabilitation of people with MCI.Trade Review"This valuable volume brings the kind of broad perspective to mild cognitive impairment that has long been needed. Rather than basing conclusions on a single sample or framework, the editors have pulled together articles from leading research groups around the world. This is the kind of comprehensive approach that is needed for developing systematic and valid definitions of MCI and identifying better tools that make it possible to differentiate between benign memory changes in later life and the early signs of pathological processes." - Steven H. Zarit, Department of Human Development and Family Studies, The Pennsylvania State University"This volume provides the most comprehensive overview of mild cognitive impairment currently available. The conceptual and methodological challenges for studying MCI are tackled with rigor, and the complexities of defining the syndrome are not underestimated. This book is certain to become a classic text for those studying or researching cognitive agin, MCI and dementia, and for clinicians seeking an authoritative reference on the clinical manifestations of MCI." - Kaarin J. Anstey, Centre for Mental Health Research, Australian National University"The editors of this book have done a great job. The description of the issues is laid out in a well-written introduction, making the descriptions of the research papers very accessible, even to the less well-informed reader. The conclusion likewise pulled together the various strands, including defining what still needs to be done to further refine the concept of MCI." - Graham A. Jackson, Laverndale Hospital, Scotland. In Dementia, August, 2008"This valuable volume brings the kind of broad perspective to mild cognitive impairment that has long been needed. Rather than basing conclusions on a single sample or framework, the editors have pulled together articles from leading research groups around the world. This is the kind of comprehensive approach that is needed for developing systematic and valid definitions of MCI and identifying better tools that make it possible to differentiate between benign memory changes in later life and the early signs of pathological processes." - Steven H. Zarit, Department of Human Development and Family Studies, The Pennsylvania State University"This volume provides the most comprehensive overview of mild cognitive impairment currently available. The conceptual and methodological challenges for studying MCI are tackled with rigor, and the complexities of defining the syndrome are not underestimated. This book is certain to become a classic text for those studying or researching cognitive agin, MCI and dementia, and for clinicians seeking an authoritative reference on the clinical manifestations of MCI." - Kaarin J. Anstey, Centre for Mental Health Research, Australian National University"The editors of this book have done a great job. The description of the issues is laid out in a well-written introduction, making the descriptions of the research papers very accessible, even to the less well-informed reader. The conclusion likewise pulled together the various strands, including defining what still needs to be done to further refine the concept of MCI." - Graham A. Jackson, Laverndale Hospital, Scotland. In Dementia, August, 2008Table of ContentsPart 1. Introduction. H. Tuokko, I. McDowell, An Overview of Mild Cognitive Impairment. Part 2. General Population Research on MCI. C. Fabrigoule, P. Barberger-Gateau, J.-F. Dartigues, The PAQUID Study. K. Palmer, L. Bäckman, B.J. Small, L. Fratiglioni, Cognitive Impairment in Elderly Persons without Dementia: Findings from the Kungsholmen Project. J. Fleming, F.E. Matthews, M. Chatfield, C. Brayne, Population Levels of Mild Cognitive Impairment in England and Wales. A. Collie, P. Maruff, D.G. Darby, C. Masters, J. Currie, The Melbourne Aging Study. Part 3. Specific Samples. R. Wilson, N.T. Aggarwal, D.A. Bennett, Mild Cognitive Impairment in the Religious Orders Study. G. Smith, M. Machulda, K. Kantarci, A Perspective from the Mayo Clinic. M.C. Tierney, Prediction of Probable Alzheimer's Disease: The Sunnybrook Memory Study. H. Wolf, H.-J. Gertz, Studies in the Leipzig Memory Clinic: Contribution to the Concept of Mild Cognitive Impairment. Part 4. Interventions. H. Chertkow, Emerging Pharmacological Therapies for Mild Cognitive Impairment. B. Woods, L. Clare, Cognition-based Therapies and Mild Cognitive Impairment. K. Peters, G. Winocur, Combined Therapies in Mild Cognitive Impairment. Part 5. Summary and Future Directions. H. Tuokko, D.F. Hultsch, The Future of Mild Cognitive Impairment.

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Book SynopsisThe book explores how build a mechanical inferences by making use of arithmetic operations on a string of numbers representing statements. In this way logic is reduced to a branch of the combinatory calculus. It covers the field of traditional logic by showing that any kind of inference can be mechanically reduced to three-variables and two-premise inferences. Meriological inferences can also be easily treated in this way. The book covers the following subjects: structural description of space; three-variable inferences through products, sums, subtractions, and divisions; generalization to n variables; relations; and applications.Table of ContentsStructural Description. Product Inferences. Sums. Subtractions. Divisions. Assessment of All the Previous Inferences. Generalized Representation and Structural Relations. Generalized Inferences. Applications. Conclusions. Bibliography. Author Index. Subject Index.

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Book SynopsisFuzzy set theory is a mathematical structure for representing uncertainty. Modern intelligent systems must combine knowledge based on techniques for gathering and processing information with methods of approximate reasoning. This enables an intelligent system to better emulate human decision-making in uncertain environments.Table of ContentsGetting Started. Fuzzy Set Theory. Possibility/Probability Consistency Principle. Knowledge Representation. Imprecision and Fuzzy Logic. Knowledge Processing. Knowledge in FEST. Inference Engine. The Fuzzy Inference Engine. Fuzzy Inference in FEST. References. Index.

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Book SynopsisOffers a presentation of classical first-order logic. This book presents a truth tree system based on the work of Jeffrey, as well as a natural deduction system inspired by that of Kalish and Montague.Trade Review“Deduction is the best logic textbook on the market. It is modern, clean, elegant, sharp and direct. It is a perfect accompaniment to the most recent developments in philosophy and logic; in every sense the logic textbook for the twenty-first century.” Rick Benitez, University of SydneyTable of ContentsPreface to the Second Edition viii Acknowledgments x 1 Basic Concepts of Logic 1 1.1 Arguments 1 1.2 Validity 16 1.3 Implication and Equivalence 23 1.4 Logical Properties of Sentences 27 1.5 Satisfiability 31 2 Sentences 36 2.1 The Language of Sentential Logic 36 2.2 Truth Functions 40 2.3 A Sentential Language 46 2.4 Symbolization 49 2.5 Validity 56 2.6 Truth Tables 60 2.7 Truth Tables for Formulas 63 2.8 Truth Tables for Argument Forms 68 2.9 Implication, Equivalence, and Satisfiability 71 3 Truth Trees 76 3.1 Thinking Backwards 76 3.2 Constructing Truth Trees 80 3.3 Negation, Conjunction, and Disjunction 84 3.4 The Conditional and Biconditional 93 3.5 Other Applications 101 4 Natural Deduction 107 4.1 Natural Deduction Systems 107 4.2 Rules for Negation and Conjunction 110 4.3 Rules for the Conditional and Biconditional 118 4.4 Rules for Disjunction 122 4.5 Derivable Rules 125 5 Quantifiers 137 5.1 Constants and Quantifiers 138 5.2 Categorical Sentence Forms 144 5.3 Polyadic Predicates 148 5.4 The Language Q 153 5.5 Symbolization 156 6 Quantified Truth Trees 173 6.1 Rules for Quantifiers 174 6.2 Strategies 178 6.3 Interpretations 189 6.4 Constructing Interpretations from Trees 199 7 Quantified Natural Deduction 206 7.1 Deduction Rules for Quantifiers 206 7.2 Universal Proof 214 7.3 Derived Rules for Quantifiers 220 8 Identity and Function Symbols 225 8.1 Identity 225 8.2 Truth Tree Rules for Identity 231 8.3 Deduction Rules for Identity 235 8.4 Function Symbols 238 9 Necessity 249 9.1 If 249 9.2 Modal Connectives 251 9.3 Symbolization 256 9.4 Modal Truth Trees 261 9.5 Other Tree Rules 265 9.6 World Travelling 268 9.7 Modal Deduction 278 9.8 Other Modal Systems 289 10 Between Truth and Falsehood 295 10.1 Vagueness and Presupposition 295 10.2 Many-Valued Truth Tables 300 10.3 Many-Valued Trees 314 10.4 Many-Valued Deduction 325 10.5 Fuzzy Logic 332 10.6 Intuitionistic Logic 344 11 Obligation 361 11.1 Deontic Connectives 362 11.2 Deontic Truth Trees 370 11.3 Deontic Deduction 381 11.4 Moral and Practical Reasoning 387 12 Counterfactuals 395 12.1 The Meaning of Counterfactuals 399 12.2 Truth Tree Rules for Counterfactuals 402 12.3 Deduction Rules for Counterfactuals 409 12.4 Stalnaker’s Semantics: System CS 418 12.5 Lewis’s Semantics: System CL 423 13 Common-Sense Reasoning 434 13.1 When Good Arguments Go Bad 435 13.2 Truth Trees 439 13.3 Defeasible Deduction 454 13.4 Defeasible Deontic Logic 466 14 Quantifiers and Modality 475 14.1 Quantified S5 475 14.2 Free Logic 487 Bibliography 504 Index 507

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Book SynopsisSelected Logic Papers, long out of print and now reissued with eight additional essays, includes much of the author's important work on mathematical logic and the philosophy of mathematics from the past sixty years.Trade Review[Quine] is at once the most elegant expounder of systematic logic in the older, pre-Gödelian style of Frege and Russell, the most distinguished American recruit to logical empiricism, probably the contemporary American philosopher most admired in the profession, and an original philosophical thinker of the first rank… This is an amazing feat of condensation with something solid to say in its brief scope about every major topic of interest in modern formal logic. * New York Review of Books *What [Quine] is expert in is, of course, logic… What [this book offers] is a view of the expert at work. Selected Logic Papers shows him actually doing logic… Logic is not a guide to life, but then Quine has never maintained that it was. It is a powerful adjunct to empirical inquiry, whose proper use requires prior discipline; its virtue lies in the fact that if we supply it with truth, it will never yield falsehood. Few have shown the manner of its use with more authority. * Partisan Review *This book is of continuing, not just historical interest. Quine is the greatest American philosopher of the twentieth century. His work in logic is inseparable from his work in other parts of philosophy. -- George Boolos, Massachusetts Institute of TechnologyTable of ContentsWhitehead and the Rise of Modern Logic (1941); Logic, Symbolic (1954); A Method of Generating Part of Arithmetic Without Use of Intuitive Logic (1934); Definition of Substitution (1936); Concatenation as a Basis for Arithmetic (1946); Set-theoretic Foundations for Logic (1936); Logic Based on Inclusion and Abstraction (1937); On Ordered Pairs and Relations (1945-46); On w-Inconsistency and a So-called Axiom of Infinity (1952); Element and Number (1941); On an Application of Tarski's Theory of Truth (1952); On Frege's Way Out (1954); Completeness of the Propositional Calculus (1937); On Cores and Prime Implicants of Truth Functions (1958); Two Theorems about Truth Functions (1951); On Boolean Functions (1949); On the Logic of Quantification (1945); A Proof Procedure for Quantification Theory (1954); Interpretations of Sets of Conditions (1953); Church's Theorem on the Decision Problem (1954); Quantification and the Empty Domain (1953); Reduction to a Dyadic Predicate (1953); Variables Explained Away (1960); Truth, Paradox, and Godel's Theorem (1992); Immanence and Validity (1991); MacHale on Boole (1985); Peirce's Logic (1989); Peano as Logician (1982); Free Logic, Description, and Virtual Classes (1994); The Inception of "New Foundations" (1987); Pythagorean Triples and Fermat's Last Theorem (1992).

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Book SynopsisPresents the history of a critical period in mathematics that includes accounts of the two principal influences upon Russell around 1900: the set theory of Cantor and the mathematical logic of Peano and his followers. This work provides surveys of many related topics and figures of the late nineteenth century.Trade Review"Grattan-Guiness's uniformly interesting and valuable account of the interwoven development of logic and related fields of mathematics ... between 1870 and 1940 presents a significantly revised analysis of the history of the period... [His] book is important because it supplies what has been lacking: a full account of the period from a primary mathematical perspective."--James W. Van Evra, IsisTable of ContentsCHAPTER 1 Explanations 1.1 Sallies 3 1.2 Scope and limits of the book 3 1.2.1 An outline history 3 1.2.2 Mathematical aspects 4 1.2.3 Historical presentation 6 1.2.4 Other logics, mathematics and philosophies 7 1.3 Citations, terminology and notations 1.3.1 References and the bibliography 9 1.3.2 Translations, quotations and notations 10 1.4 Permissions and acknowledgements 11 CHAPTER 2 Preludes: Algebraic Logic and Mathematical Analysis up to 1870 2.1 Plan of the chapter 14 2.2 'Logique' and algebras in French mathematics 14 2.2.1 The 'logique' and clarity of 'ideologie' 14 2.2.2 Lagrange's algebraic philosophy 15 2.2.3 The many senses of 'analysis' 17 2.2.4 Two Lagrangian algebras: functional equations and differential operators 17 2.2.5 Autonomy for the new algebras 19 2.3 Some English algebraists and logicians 20 2.3.1 A Cambridge revival: the 'Analytical Society, Lacroix, and the professing of algebras 20 2.3.2 The advocacy of algebras by Babbage, Herschel and Peacock 20 2.3.3 An Oxford movement: Whately and the professing of logic 22 2.4 A London pioneer: De Morgan on algebras and logic 25 2.4.1 Summary of his life 25 2.4.2 De Morgan's philosophies of algebra 25 2.4.3 De Morgan's logical career 26 2.4.4 De Morgan's contributions to the foundations of logic 27 2.4.5 Beyond the syllogism 29 2.4.6 Contretemps over 'the quantification of the predicate' 30 2.4.7 The logic of two place relations, 1860 32 2.4.8 Analogies between logic and mathematics 35 2.4.9 De Morgan's theory of collections 36 2.5 A Lincoln outsider: Boole on logic as applied mathematics 37 2.5.1 Summary of his career 37 2.5.2 Boole's 'general method in analysis' 1844 39 2.5.3 The mathematical analysis of logic, 1847. 'elective symbols' and laws 40 2.5.4 'Nothing' and the 'Universe' 42 2.5.5 Propositions, expansion theorems, and solutions 43 2.5.6 The laws of thought, 1854: modified principles and extended methods 46 2.5.7 Boole's new theory of propositions 49 2.5.8 The character of Boole's system 50 2.5.9 Boole's search for mathematical roots 53 2.6 The semi-followers of Boole 54 2.6.1 Some initial reactions to Boole's theory 54 2.6.2 The reformulation by Jevons 56 2.6.3 Jevons versus Boole 59 2.6.4 Followers of Boole and/or Jevons 60 2.7 Cauchy, Weierstrass and the rise of mathematical analysis 63 2.7.1 Different traditions in the calculus 63 2.7.2 Cauchy and the Ecole Polytechnique 64 2.7.3 The gradual adoption and adaptation of Cauchy's new tradition 67 2.7.4 The refinements of Weierstrass and his followers 68 2.8 Judgement and supplement 70 2.8.1 Mathematical analysis versus algebraic logic 70 2.8.2 The places of Kant and Bolzano 71 CHAPTER 3 Cantor: Mathematics as Mengenlehre 3.1 Prefaces 75 3.1.1 Plan of the chapter 75 3.1.2 Cantor's career 75 3.2 The launching of the Mengenlehre, 1870-1883 79 3.2.1 Riemann's thesis: the realm of discontinuous functions 79 3.2.2 Heine on trigonometric series and the real line, 1870-1872 81 3.2.3 Cantor's extension of Heine's findings, 1870-1872 83 3.2.4 Dedekind on irrational numbers, 1872 85 3.2.5 Cantor on line and plane, 1874-1877 88 3.2.6 Infinite numbers and the topology of linear sets, 1878-1883 89 3.2.7 The Grundlagen, 1883: the construction of number-classes 92 3.2.8 The Grundlagen: the definition of continuity 95 3.2.9 The successor to the Grundlagen, 1884 96 3.3 Cantor's Acta mathematica phase, 1883-1885 97 3.3.1 Mittag-Lefler and the French translations, 1883 97 3.3.2 Unpublished and published 'communications' 1884-1885 98 3.3.3 Order-types and partial derivatives in the 'communications' 100 3.3.4 Commentators on Cantor, 1883-1885 102 3.4 The extension of the Mengenlehre, 1886-1897 103 3.4.1 Dedekind's developing set theory, 1888 103 3.4.2 Dedekind's chains of integers 105 3.4.3 Dedekind's philosophy of arithmetic 107 3.4.4 Cantor's philosophy of the infinite, 1886-1888 109 3.4.5 Cantor's new definitions of numbers 110 3.4.6 Cardinal exponentiation: Cantor's diagonal argument, 1891 110 3.4.7 Transfinite cardinal arithmetic and simply ordered sets, 1895 112 3.4.8 Transfinite ordinal arithmetic and well-ordered sets, 1897 114 3.5 Open and hidden questions in Cantor's Mengenlehre 114 3.5.1 Well-ordering and the axioms of choice 114 3.5.2 What was Cantor's 'Cantor's continuum problem'? 116 3.5.3 "Paradoxes" and the absolute infinite 117 3.6 Cantor's philosophy of mathematics 119 3.6.1 A mixed position 119 3.6.2 (No) logic and metamathematics 120 3.6.3 The supposed impossibility of infinitesimals 121 3.6.4 A contrast with Kronecker 122 3.7 Concluding comments: the character of Cantor's achievements 124 CHAPTER 4 Parallel Processes in Set Theory, Logics and Axiomatics, 1870s-1900s 4.1 Plans for the chapter 126 4.2 The splitting and selling of Cantor's Mengenlehre 126 4.2.1 National and international support 126 4.2.2 French initiatives, especially from Borel 127 4.2.3 Couturat outlining the infinite, 1896 129 4.2.4 German initiatives from Mein 130 4.2.5 German proofs of the Schroder-Bernstein theorem 132 4.2.6 Publicity from Hilbert, 1900 134 4.2.7 Integral equations and functional analysis 135 4.2.8 Kempe on 'mathematical form' 137 4.2.9 Kempe-who? 139 4.3 American algebraic logic: Peirce and his followers 140 4.3.1 Peirce, published and unpublished 141 4.3.2 Influences on Peirre's logic: father's algebras 142 4.3.3 Peirce's first phase: Boolean logic and the categories, 1867-1868 144 4.3.4 Peirce's virtuoso theory of relatives, 1870 145 4.3.5 Peirce's second phase, 1880: the propositional calculus 147 4.3.6 Peirre's second phase, 1881: finite and infinite 149 4.3.7 Peirce's students, 1883: duality, and 'Quantifying' a proposition 150 4.3.8 Peirre on 'icons' and the order of 'quantifiers; 1885 153 ~~~ 4.3.9 The Peirceans in the 1890s 154 4.4 German algebraic logic: from the Grassmanns to Schr6der 156 4.4.1 The Grassmanns on duality 156 4.4.2 Schroder's Grassmannian phase 159 4.4.3 Schroder's Peirrean 'lectures' on logic 161 4.4.4 Schrrider's first volume, 1890 161 4.4.5 Part of the second volume, 1891 167 4.4.6 Schroder's third volume, 1895: the 'logic of relatives' 170 4.4.7 Peirce on and against Schroder in The monist, 1896-1897 172 4.4.8 Schroder on Cantorian themes, 1898 174 4.4.9 The reception and publication of Schroder in the 1900s 175 4.5 Frege: arithmetic as logic 177 4.5.1 Frege and Frege' 177 4.5.2 The 'concept-script' calculus of Frege's 'pure thought; 1879 179 4.5.3 Frege's arguments for logicising arithmetic, 1884 183 4.5.4 Keny's conception of Fregean concepts in the mid 1880s 187 4.5.5 Important new distinctions in the early 1890s 187 4.5.6 The 'fundamental laws' of logicised arithmetic, 1893 191 4.5.7 Frege's reactions to others in the later 1890s 194 4.5.8 More 'fundamental laws' of arithmetic, 1903 195 4.5.9 Frege, Korselt and Thomae on the foundations of arithmetic 197 4.6 Husserl: logic as phenomenology 199 4.6.1 A follower of Weierstrass and Cantor 199 4.6.2 The phenomenological 'philosophy of arithmetic; 1891 201 4.6.3 Reviews by Frege and others 203 4.6.4 Husserl's 'logical investigations; 1900-1901 204 4.6.5 Husserl's early talks in Gottingen, 1901 206 4.7 Hilbert: early proof and model theory, 1899-1905 207 4.7.1 Hilbert's growing concern with axiomatics 207 4.7.2 Hilbert's diferent axiom systems for Euclidean geometry, 1899-1902 208 4.7.3 From German completeness to American model theory 209 4.7.4 Frege, Hilbert and Korselt on the foundations of geometries 212 4.7.5 Hilbert's logic and proof theory, 1904-1905 213 4.7.6 Zermelo's logic and set theory, 1904-1909 216 CHAPTER 5 Peano: the Formulary of Mathematics 5.1 Prefaces 219 5.1.1 Plan of the chapter 219 5.1.2 Peano's career 219 5.2 Formalising mathematical analysis 221 5.2.1 Improving Genocchi, 1884 221 5.2.2 Developing Grassmann's 'geometrical calculus; 1888 223 5.2.3 The logistic of arithmetic, 1889 225 5.2.4 The logistic of geometry, 1889 229 5.2.5 The logistic of analysis, 1890 230 5.2.6 Bettazzi on magnitudes, 1890 232 5.3 The Rivista: Peano and his school, 1890-1895 232 5.3.1 The 'society of mathematicians' 232 5.3.2 'Mathematical logic, 1891 234 5.3.3 Developing arithmetic, 1891 235 5.3.4 Infinitesimals and limits, 1892-1895 236 5.3.5 Notations and their range, 1894 237 5.3.6 Peano on definition by equivalence classes 239 5.3.7 Burali-Forti's textbook, 1894 240 5.3.8 Burali-Forti's research, 1896-1897 241 5.4 The Formulaire and the Rivista, 1895-1900 242 5.4.1 The first edition of the Formulaire, 1895 242 5.4.2 Towards the second edition of the Formulaire, 1897 244 5.4.3 Peano on the eliminability of 'the' 246 5.4.4 Frege versus Peano on logic and definitions 247 5.4.5 Schroder's steamships versus Peano's sailing boats 249 5.4.6 New presentations of arithmetic, 1898 251 5.4.7 - Padoa on classhoody 1899 253 5.4.8 Peano's new logical summary, 1900 254 5.5 Peanists in Paris, August 1900 255 5.5.1 An Italian Friday morning 255 5.5.2 Peano on definitions 256 5.5.3 Burali-Forti on definitions of numbers 257 5.5.4 Padoa on definability and independence 259 5.5.5 Pieri on the logic of geometry 261 5.6 Concluding comments: the character of Peano's achievements 262 5.6.1 Peano's little dictionary, 1901 262 5.6.2 Partly grasped opportunities 264 5.6.3 Logic without relations 266 CHAPTER 6 Russell's Way In: From Certainty to Paradoxes, 1895-1903 6.1 Prefaces 268 6.1.1 Plans for two chapters 268 6.1.2 Principal sources 269 6.1.3 Russell as a Cambridge undergraduate, 1891-1894 271 6.1.4 Cambridge philosophy in the 1890s 273 6.2 Three philosophical phases in the foundation of mathematics, 1895-1899 274 6.2.1 Russell's idealist axiomatic geometries 275 6.2.2 The importance of axioms and relations 276 6.2.3 A pair of pas de deux with Paris: Couturat and Poincare on geometries 278 6.2.4 The emergence of "itehead, 1898 280 6.2.5 The impact of G. E. Moore, 1899 282 6.2.6 Three attempted books, 1898-1899 283 6.2.7 Russell's progress with Cantor's Mengenlehre, 1896-1899 285 6.3 From neo-Hegelianism towards 'Principles', 1899-1901 286 6.3.1 Changing relations 286 6.3.2 Space and time, absolutely 288 6.3.3 'Principles of Mathematics, 1899-1900 288 6.4 The first impact of Peano 290 6.4.1 The Paris Congress of Philosophy, August 1900: Schroder versus Peano on 'the' 290 6.4.2 Annotating and popularising in the autumn 291 6.4.3 Dating the origins of Russell's logicism 292 6.4.4 Drafting the logic of relations, October 1900 296 6.4.5 Part 3 of The principles, November 1900: quantity and magnitude 298 6.4.6 Part 4, November 1900: order and ordinals 299 6.4.7 Part 5, November 1900: the transfinite and the continuous 300 6.4.8 Part 6, December 1900: geometries in space 301 6.4.9 Whitehead on 'the algebra of symbolic logic, 1900 302 6.5 Convoluting towards logicism, 1900-1901 303 6.5.1 Logicism as generalised metageometry, January 1901 303 6.5.2 The first paper for Peano, February 1901: relations and numbers 305 6.5.3 Cardinal arithmetic with "itehead and Russell, June 1901 307 6.5.4 The second paper for Peano, March August 1901: set theory with series 308 6.6 From 'fallacy' to 'contradiction', 1900-1901 310 6.6.1 Russell on Cantor's 'fallacy; November 1900 310 6.6.2 Russell's switch to a 'contradiction' 311 6.6.3 Other paradoxes: three too large numbers 312 6.6.4 Three passions and three calamities, 1901-1902 314 6.7 Refining logicism, 1901-1902 315 6.7.1 Attempting Part 1 of The principles, May 1901 315 6.7.2 Part 2, June 1901: cardinals and classes 316 6.7.3 Part 1 again, April-May 1902: the implicational logicism 316 6.7.4 Part 1: discussing the indefinables 318 6.7.5 Part 7, June 1902: dynamics without statics; and within logic? 322 6.7.6 Sort-of finishing the book 323 6.7.7 The first impact of Frege, 1902 323 6.7.8 AppendixA on Frege 326 6.7.9 Appendix B: Russell's first attempt to solve the paradoxes 327 6.8 The roots of pure mathematics? Publishing The principles at last, 1903 328 6.8.1 Appearance and appraisal 328 6.8.2 A gradual collaboration with Whitehead 331 CHAPTER 7 Russell and Whitehead Seek the Principia Mathematica, 1903-1913 7.1 Plan of the chapter 333 7.2 Paradoxes and axioms in set theory, 1903-1906 333 7.2.1 Uniting the paradoxes of sets and numbers 333 7.2.2 New paradoxes, mostly of naming 334 7.2.3 The paradox that got away: heterology 336 7.2.4 Russell as cataloguer of the paradoxes 337 7.2.5 Controversies over axioms of choice, 1904 339 7.2.6 Uncovering Russell's 'multiplicative axiom, 1904 340 7.2.7 Keyser versus Russell over infinite classes, 1903-1905 342 7.3 The perplexities of denoting, 1903-1906 342 7.3.1 First attempts at a general system, 1903-1905 342 7.3.2 Propositional functions, reducible and identical 344 7.3.3 The mathematical importance of definite denoting functions 346 7.3.4 'On denoting' and the complex, 1905 348 7.3.5 Denoting, quantification and the mysteries of existence 350 7.3.6 Russell versus MacColl on the possible, 1904-1908 351 7.4 From mathematical induction to logical substitution, 1905-1907 354 7.4.1 Couturat's Russellian principles 354 7.4.2 A second pas de deux with Paris: Boutroux and Poincare on logicism 355 7.4.3 Poincare on the status of mathematical induction 356 7.4.4 Russell's position paper, 1905 357 7.4.5 Poincare and Russell on the vicious circle principle, 1906 358 7.4.6 The rise of the substitutional theory, 1905-1906 360 7.4.7 The fall of the substitutional theory, 1906-1907 362 7.4.8 Russell's substitutional propositional calculus 364 7.5 Reactions to mathematical logic and logicism, 1904-1907 366 7.5.1 The International Congress of Philosophy, 1904 366 7.5.2 German philosophers and mathematicians, especially Schonflies 368 7.5.3 Activities among the Peanists 370 7.5.4 American philosophers: Royce and Dewey 371 7.5.5 American mathematicians on classes 373 7.5.6 Huntington on logic and orders 375 7.5.7 Judgements fiom Shearman 376 7.6 Whitehead's role and activities, 1905-1907 377 7.6.1 Whitehead's construal of the 'material world' 377 7.6.2 The axioms of geometries 379 7.6.3 Whitehead's lecture course, 1906-1907 379 7.7 The sad compromise: logic in tiers 380 7.7.1 Rehabilitating propositional functions, 1906-1907 380 7.7.2 Two reflective pieces in 1907 382 7.7.3 Russell's outline of 'mathematical logic, 1908 383 7.8 The forming of Principia mathematica 384 7.8.1 Completing and funding Principia mathematica 384 7.8.2 The Organisation of Principia mathematica 386 7.8.3 The propositional calculus, and logicism 388 7.8.4 The predicate calculus, and descriptions 391 7.8.5 Classes and relations, relative to propositional functions 392 7.8.6 The multiplicative axiom: some uses and avoidance 395 7.9 Types and the treatment of mathematics in Principia mathematica 396 7.9.1 7~pes in orders 396 7.9.2 Reducing the edifice 397 7.9.3 Individuals, their nature and number 399 7.9.4 Cardinals and their finite arithmetic 401 7.9.5 The generalised ordinals 403 7.9.6 The ordinals and the alephs 404 7.9.7 The odd small ordinals 406 7.9.8 Series and continuity 406 7.9.9 Quantity with ratios 408 CHAPTER 8 The Influence and Place of Logicism, 1910-1930 8.1 Plans for two chapters 411 8.2 Whitehead's and Russell's transitions from logic to philosophy, 1910-1916 412 8.2.1 The educational concerns of "itehead, 1910-1916 412 8.2.2 Whitehead on the principles of geometry in the 1910s 413 8.2.3 British reviews of Principia mathematica 415 8.2.4 Russell and Peano on logic, 1911-1913 416 8.2.5 Russell's initial problems with epistemology, 1911-1912 417 8.2.6 Russell's first interactions with Wittgenstein, 1911-1913 418 8.2.7 Russell's confrontation with Wiener, 1913 419 8.3 Logicism and epistemology in America and with Russell, 1914-1921 421 8.3.1 Russell on logic and epistemology at Harvard, 1914 421 8.3.2 Two long American reviews 424 8.3.3 Reactions from Royce students: Sheffer and Lewis 424 8.3.4 Reactions to logicism in New York 428 8.3.5 OtherAmerican estimations 429 8.3.6 Russell's 'logical atomism' and psychology, 1917-1921 430 8.3.7 Russell's 'introduction'to logicism, 1918-1919 432 8.4 Revising logic and logicism at Cambridge, 1917-1925 434 8.4.1 New Cambridge authors, 1917-1921 434 8.4.2 Wittgenstein's 'Abhandlung' and Tractatus, 1921-1922 436 8.4.3 The limitations of Wittgenstein's logic 437 8.4.4 Towards extensional logicism: Russell's revision of Principia mathematica, 1923-1924 440 8.4.5 Ramsey's entry into logic and philosophy, 1920-1923 443 8.4.6 Ramsey's recasting of the theory of types, 1926 444 8.4.7 Ramsey on identity and comprehensive extensionality 446 8.5 Logicism and epistemology in Britain and America, 1921-1930 448 8.5.1 Johnson on logic, 1921-1924 448 8.5.2 Other Cambridge authors, 1923-1929 450 8.5.3 American reactions to logicism in mid decade 452 8.5.4 Groping towards metalogic 454 8.5.5 Reactions in and around Columbia 456 8.6 Peripherals: Italy and France 458 8.6.1 The occasional Italian survey 458 8.6.2 New French attitudes in the Revue 459 8.6.3 Commentaries in French, 1918-1930 461 8.7 German-speaking reactions to logicism, 1910-1928 463 8.7.1 (Neo-)Kantians in the 1910s 463 8.7.2 Phenomenologists in the 1910s 467 8.7.3 Frege's positive and then negative thoughts 468 8.7.4 Hilbert's definitive 'metamathematics; 1917-1930 470 8.7.5 Orders of logic and models of set theory: Lowenheim and Skolem, 1915-1923 475 8.7.6 Set theory and Mengenlehre in various forms 476 8.7.7 Intuitionistic set theory and logic: Brouwer and Weyl, 1910-1928 480 8.7.8 (Neo-)Kantians in the 1920s 484 8.7.9 Phenomenologists in the 1920s 487 8.8 The rise of Poland in the 1920s: the Lvnv-Warsaw school 489 8.8.1 From Lv6v to Warsaw: students of Twardowski 489 8.8.2 Logics with Lukasiewicz and Tarski 490 8.8.3 Russell's paradox and Lesniewski's three systems 492 8.8.4 Pole apart: Chwistek's 'semantic' logicism at Cracov 495 8.9 The rise of Austria in the 1920s: the Schlick circle 497 8.9.1 Formation and influence 497 8.9.2 The impact of Russell, especially upon Camap 499 8.9.3 'Logicism ' in Camap's Abriss, 1929 500 8.9.4 Epistemology in Camap's Aufbau, 1928 502 8.9.5 Intuitionism and proof theory: Brouwer and Godel, 1928-1930 504 CHAPTER 9 Postludes: Mathematical Logic and Logicism in the 1930s 9.1 Plan of the chapter 506 9.2 Godel's incompletability theorem and its immediate reception 507 9.2.1 The consolidation of Schlick's 'Vienna' Circle 507 9.2.2 News from G6del: the Konigsberg lectures, September 1930 508 9.2.3 G6del's incompletability theorem, 1931 509 9.2.4 Effects and reviews of G6del's theorem 511 9.2.5 Zermelo against Godeb the Bad Elster lectures, September 1931 512 9.3 Logic(ism) and epistemology in and around Vienna 513 9.3.1 Carnap for 'metalogic' and against metaphysics 513 9.3.2 Carnap's transformed metalogic: the 'logical syntax of language; 1934 515 9.3.3 Carnap on incompleteness and truth in mathematical theories, 1934-1935 517 9.3.4 Dubislav on definitions and the competing philosophies of mathematics 519 9.3.5 Behmann's new diagnosis of the paradoxes 520 9.3.6 Kaufmann and Waismann on the philosophy of mathematics 521 9.4 Logic(ism) in the U.S.A. 523 9.4.1 Mainly Eaton and Lewis 523 9.4.2 Mainly Weiss and Langer 525 9.4.3 Whitehead's new attempt to ground logicism, 1934 527 9.4.4 The debut of Quine 529 9.4.5 Two journals and an encyclopaedia, 1934-1938 531 9.4.6 Carnap's acceptance of the autonomy of semantics 533 9.5 The battle of Britain 535 9.5.1 The campaign of Stebbing for Russell and Carnap 535 9.5.2 Commentary from Black and Ayer 538 9.5.3 Mathematicians-and biologists 539 9.5.4 Retiring into philosophy: Russell's return, 1936-1937 542 9.6 European, mostly northern 543 9.6.1 Dingler and Burkamp again 543 9.6.2 German proof theory after Godel 544 9.6.3 Scholz's little circle at Munster 546 9.6.4 Historical studies, especially by Jorgensen 547 9.6.5 History philosophy, especially Cavailles 548 9.6.6 Other Francophone figures, especially Herbrand 549 9.6.7 Polish logicians, especially Tarski 551 9.6.8 Southern Europe and its former colonies 553 CHAPTER 10 The Fate of the Search 10.1 Influences on Russell, negative and positive 556 10.1.1 Symbolic logics: living together and living apart 556 10.1.2 The timing and origins of Russell's logicism 557 10.1.3 (Why) was Frege (so) little read in his lifetime? 558 10.2 The content and impact of logicism 559 10.2.1 Russell's obsession with reductionist logic and epistemology 560 10.2.2 The logic and its metalogic 562 10.2.3 The fate of logicism 563 10.2.4 Educational aspects, especially Piaget 566 10.2.5 The role of the U.S.A.: judgements in the Schi1pp series 567 10.3 The panoply of foundations 569 10.4 Sallies 573 CHAPTER 11 Transcription of Manuscripts 11.1 Couturat to Russell, 18 December 1904 574 11.2 Veblen to Russell, 13 May 1906 577 11.3 Russell to Hawtrey, 22 January 1907 (or 1909?) 579 11.4 Jourdain's notes on Wittgenstein's first views on Russell's paradox, April 1909 580 11.5 The application of Whitehead and Russell to the Royal Society, late 1909 581 11.6 Whitehead to Russell, 19 January 1911 584 11.7 Oliver Strachey to Russell, 4 January 1912 585 11.8 Quine and Russell, June-July 1935 586 11.8.1 Russell to Quine, 6 June 1935 587 11.8.2 Quine to Russell, 4 July 1935 588 11.9 Russell to Henkin, 1 April 1963 592 BIBLIOGRAPHY 594 INDEX 671

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Book SynopsisPresents an area of mathematical research that combines topology, geometry, and logic. This book seeks to explain and illustrate the implications of the general principle, first emphasized by Alex Nabutovsky, that logical complexity engenders geometric complexity.Trade Review"This is a terrific book. It does no less than introduce an entire new field of mathematics - a truly astounding development. It will be widely read, I think, as much because of the masterful exposition as for the beautiful mathematics. Weinberger gives very clear and accessible descriptions of all the relevant tools from computability, topology, and geometry, in a friendly and engaging style. He has done the mathematical community a great service indeed." - Robin Forman, Rice University; "This book represents a very exciting new area of research at the interface of topology and logic. Written in a quite readable style, and presenting the more accessible cases in detail while giving references for the more involved results, it is a book whose methods and ideas will surely have many more significant applications over the next several years." - Kevin M. Whtye, University of Illinois at Chicago"

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Book SynopsisSurveys the assortment of methods put forth for fixing Frege's system, in an attempt to determine just how much of mathematics can be reconstructed in each. This work considers every proposed fix, each with its distinctive philosophical advantages and drawbacks.Trade ReviewCo-Winner of the 2007 Shoenfield Prize, Association for Symbolic Logic "Fixing Frege fills a serious gap in the Frege's literature (always increasing but perhaps with an excessive attention paid to semantics and the philosophy of language) and should remain for a long time a necessary reference for scholars in the field."--Ignacio Angelelli, Review of Modern LogicTable of ContentsAcknowledgments ix CHAPTER 1: Frege, Russell, and After 1 CHAPTER 2: Predicative Theories 86 CHAPTER 3: Impredicative Theories 146 Tables 215 Notes 227 References 241 Index 249

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Book SynopsisWhat did mathematicians rely on for their work before then? And how did mathematical notations evolve into what we know today? This book explains the history behind the development of our mathematical notation system.Trade Review"Mazur (Euclid in the Rainforest) gives readers the fascinating history behind the mathematical symbols we use, and completely take for granted, every day. Mathematical notation turns numbers into sentences--or, to the uninitiated, a mysterious and impenetrable code. Mazur says the story of math symbols begins some 3,700 years ago, in ancient Babylon, where merchants incised tallies of goods on cuneiform tablets, along with the first place holder--a blank space. Many early cultures used letters for both numbers and an alphabet, but convenient objects like rods, fingers, and abacus beads, also proved popular. Mazur shows how our 'modern' system began in India, picking up the numeral 'zero' on its way to Europe, where it came into common use in the 16th century, thanks to travelers and merchants as well as mathematicians like Fibonacci. Signs for addition, subtraction, roots, and equivalence followed, but only became standardized through the influence of scientists and mathematicians like Rene Descartes and Gottfried Leibniz. Mazur's lively and accessible writing makes what could otherwise be a dry, arcane history as entertaining as it is informative."--Publishers Weekly "[A] fascinating narrative... This is a nuanced, intelligently framed chronicle packed with nuggets--such as the fact that Hindus, not Arabs, introduced Arabic numerals. In a word: enlightening."--George Szpiro, Nature "Mazur begins by illustrating how the ancient Incas and Mayans managed to write specific, huge numbers. Then, for more than 200 pages, he traces the history of division signs, square roots, pi, exponents, graph axes and other symbols in the context of cognition, communication, and analysis."--Washington Post "Mazur delivers a solid exposition of an element of mathematics that is fundamental to its history."--Library Journal "Mazur treats only a subset of F. Cajori's monumental A History of Mathematical Notation (Dover, 1993 first edition 1922) and there is overlap with many other mathematical history books, but Mazur adds new findings and insights and it is so much more entertaining ... and these features make it an interesting addition to the existing literature for anybody with only a slight interest in mathematics or its history."--European Mathematical Society "Symbols like '+' and '=' are so ingrained that it's hard to conceive of math without them. But a new book, Enlightening Symbols: A Short History of Mathematical Notation and its Hidden Power, offers a surprising reminder: Until the early 16th century, math contained no symbols at all."--Kevin Hartnett, Boston Globe "Enlightening Symbols retraces the winding road that has led to the way we now teach, study, and conceive mathematics... Thanks to Mazur's playful approach to the subject, Enlightening Symbols offers an enjoyable read."--Gaia Donati, Science "If you enjoy reading about history, languages and science, then you'll enjoy this book... The best part is the writing is compelling enough that you don't have to be a mathematician to enjoy this informative book."--Guardian.com's GrrlScientist blog "[I]nformative, highly readable and scholarly."--Brian Rotman, Literary Review "[T]his insightful account of the historical development of a highly characteristic feature of the mathematical enterprise also represents a valuable contribution to our understanding of the nature of mathematics."--Eduard Glas, Mathematical Reviews Clippings "Joseph Mazur's beautiful book Enlightening Symbols tells the story of human civilization through the development of mathematical notation. Surprises abound... The book is visually exquisite, great care having been taken with illustrations and figures. Mazur's discussion of the emergence of particular symbols affords the reader an overview of the often difficult primary literature."--Donal O'Shea, Sarasota Herald-Tribune "At whatever depth one chooses to read it, Enlightening Symbols has something for everyone. It is entertaining and eclectic, and Mazur's personal and easy style helps connect us with those who led the long and winding search for the best ways to quantify and analyze our world. Their success has liberated us from 'the shackles of our physical impressions of space'--and of the particular and the concrete--'enabling imagination to wander far beyond the tangible world we live in, and into the marvels of generality.'"--Robyn Arianrhod, Notices of the Notices of the American Mathematical Society "Mazur introduces the reader to major characters, weaves in relevant aspects of wider culture and gives a feel for the breadth of mathematical history. It is a useful book for both student and interested layperson alike."--Mark McCartney, London Mathematical Society "[T]his is a good book. It is well written by an experienced author and is full of interesting facts about how the symbols used in mathematics have arisen. It would certainly interest anyone who studies the history of mathematics."--Phil Dyke, Leonardo "Mazur is a master story teller."--John Stillwell, Bulletin of the American Mathematical SocietyTable of ContentsIntroduction ix Definitions xxi Note on the Illustrations xxiii Part 1 Numerals 1 1. Curious Beginnings 3 2. Certain Ancient Number Systems 10 3. Silk and Royal Roads 26 4. The Indian Gift 35 5. Arrival in Europe 51 6. The Arab Gift 60 7. Liber Abbaci 64 8. Refuting Origins 73 Part 2 Algebra 81 9. Sans Symbols 85 10. Diophantus's Arithmetica 93 11. The Great Art 109 12. Symbol Infancy 116 13. The Timid Symbol 127 14. Hierarchies of Dignity 133 15. Vowels and Consonants 141 16. The Explosion 150 17. A Catalogue of Symbols 160 18. The Symbol Master 165 19. The Last of the Magicians 169 Part 3 The Power of Symbols 177 20. Rendezvous in the Mind 179 21. The Good Symbol 189 22. Invisible Gorillas 192 23. Mental Pictures 210 24. Conclusion 216 Appendix A Leibniz's Notation 221 Appendix B Newton's Fluxion of xn 223 Appendix C Experiment 224 Appendix D Visualizing Complex Numbers 228 Appendix E Quaternions 230 Acknowledgments 233 Notes 235 Index 269

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Book SynopsisIntroduces a theoretical entity: Agent_Zero. This title weaves a computational tapestry with threads from Plato, Hume, Darwin, Pavlov, Smith, Tolstoy, Marx, James, and Dostoevsky, among others.Trade Review"Agent Zero offers a solution to some of social science's great puzzles. Its behavioral basis is the interplay of emotion, cognition, and network contagion effects. It elegantly explains why so many human actions are so manifestly dysfunctional, and why some are downright evil."—George Akerlof, Nobel Laureate in Economics"Rarely has a book stimulated me intellectually as much as this one. Particularly exciting is the incorporation of agents who feel (affect) and deliberate, as well as influence one another through social interaction. Epstein is a brilliantly creative scholar and the range of applications showcased here is stunning. In sum, this is a pathbreaking book."—Paul Slovic, University of Oregon"Joshua Epstein proposes a parsimonious but powerful model of individual behavior that can generate an extraordinary range of group behaviors, including mob violence, manias and financial panics, rebellions, network dynamics, and a host of other complex social phenomena. This is a highly original, beautifully conceived, and important book."—Peyton Young, University of Oxford"In social science generally and most notably in economics, the rational actor model has long been the benchmark for policy analysis and institutional design. Epstein now offers a worthy alternative: Agent_Zero, a mathematically and computationally tractable agent whose inner workings are grounded in neuroscience. Much like you and me, Agent_Zero is influenced by emotion, reason, and social pressures. Epstein demonstrates that collections of Agent Zeros perform amazingly like real groups, teams, and societies and can therefore serve as the fundamental building blocks for what he calls Generative Social Science. The rational actor now has a true competitor. Agent_Zero is a major advance."—Scott Page, University of Michigan"This is social science based on how our brains actually work. Epstein's computerized 'agents' can feel passion and fear, and can influence each other emotionally. And when they interact, we see many of the realities of social life, from the dynamics of juries to racist violence to Arab springs. A remarkable and original piece of work."—W. Brian Arthur, Santa Fe InstituteTable of ContentsForeword xi Preface xiii Acknowledgments xv INTRODUCTION 1 MOTIVATION 1 Generate Social Dynamics 2 A Core Target 2 THE MODEL COMPONENTS 5 Model Overview 6 Skeletal Equation 8 Specific Components 9 ORGANIZATION 10 Part I: Mathematical Model 10 Part II: Agent-Based Model 11 Part III: Extensions 13 Replicability and Research Resources on the Princeton University Press Website 16 Part IV: Future Research and Conclusions 17 PART 1. MATHEMATICAL MODEL 19 I.1. THE PASSIONS: FEAR CONDITIONING 19 Fear Circuitry and the Perils of Fitness 20 Nomenclature of Conditioning 29 The Rescorla-Wagner Model 33 Social Examples 37 Fear Extinction 41 I.2. REASON: THE COGNITIVE COMPONENT 46 I.3. THE SOCIAL COMPONENT 51 Simple Version of the Core Target 55 Examples of Fear Contagion 57 Mechanisms of Fear Contagion 59 Conformist Empirical Estimates 63 Generalizing Rescorla-Wagner 67 The Central Case 69 Tolstoy: The First Agent Modeler 71 A Mathematical Aside on Social Norms as Vector Fields 74 Extinction of Majorities 78 I.4. INTERIM CONCLUSIONS 80 PART II. AGENT-BASED COMPUTATIONAL MODEL 81 Affective Component 84 "Rational" Component 85 Social Component 88 Action 89 Pseudocode 89 II.1. COMPUTATIONAL PARABLES 90 Parable 1: The Slaughter of Innocents through Dispositional Contagion 90 Parable 2: Agent_Zero Initiates: Leadership as Susceptibility to Dispositional Contagion 94 Run 3. Information Cuts Both Ways 96 Run 4. A Day in the Life of Agent_Zero: How Affect and Probability Can Change on Different Time Scales 98 Run 5. Lesion Studies 102 PART III. EXTENSIONS 107 III.1. ENDOGENOUS DESTRUCTIVE RADIUS 107 III.2. AGE AND IMPULSE CONTROL 109 III.3. FIGHT VS. FLIGHT 110 Case 1: Fight 111 Case 2: Flight 112 Capital Flight 114 III.4. REPLICATING THE Latane-DARLEY EXPERIMENT 114 Threshold Imputation 115 The Dialogue 118 III.5. MEMORY 118 III.6. COUPLINGS: ENTANGLEMENT OF PASSION AND REASON 122 Mathematical Treatment 124 III.7.ENDOGENOUS DYNAMICS OF CONNECTION STRENGTH 128 Affective Homophily 128 General Setup 130 Agent-Based Model: Nonequlibrium Dynamics 135 III.8. GROWING THE 2011 ARAB SPRING 138 III.9. JURY PROCESSES 143 Phase 1. Public Phase 143 Phase 2. Courtroom Trial Phase 145 Phase 3. Jury Phase 147 III.10. EMERGENT DYNAMICS OF NETWORK STRUCTURE 152 Network Structure Dynamics as a Poincare Map 153 Relation to Literature 159 III.11. MULTIPLE SOCIAL LEVELS 160 Agent_Zero as Witness to History 161 III.12. THE 18TH BRUMAIRE OF AGENT_ZERO 165 III.13. INTRODUCTION OF PRICES AND SEASONAL ECONOMIC CYCLES 168 Prices 168 A Christmas Story 173 III.14. SPIRALS OF MUTUAL ESCALATION 176 PART IV. FUTURE RESEARCH AND CONCLUSION 181 IV.1. FUTURE RESEARCH 181 IV.2. CONCLUSION 187 Civil Violence 187 Economics 188 Health Behavior 189 Psychology 190 Jury Dynamics 191 The Formation and Dynamics of Networks 191 Mutual Escalation Dynamics 192 Birth and Intergenerational Transmission 192 IV.3. TOWARD NEW GENERATIVE FOUNDATIONS 192 Appendix I. Threshold Imputation Bounds 195 Appendix II. Mathematica Code 197 Appendix III. Agent_Zero NetLogo Source Code 213 Appendix IV. Parameter Settings for Model Runs 221 References 227 Index 243

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Book SynopsisThis book provides the first comprehensive treatment of Benford's law, the surprising logarithmic distribution of significant digits discovered in the late nineteenth century. Establishing the mathematical and statistical principles that underpin this intriguing phenomenon, the text combines up-to-date theoretical results with overviews of the law'Trade Review"This is a marvelous and excellent introduction."--Adhemar Bultheel, European Mathematical Society Bulletin "A must-read for novices and experts alike. It can be used for a graduate-level topics course or as a reference text for researchers in the field. The exposition is outstanding, with hundreds of carefully chosen examples, figures and diagrams to illustrate the theory. For those who are up for a challenge, the book contains several open problems as well. An Introduction to Benford's Law will surely be the go-to text on the subject for years to come."--Pieter C. Allaart, Mathematical ReviewsTable of ContentsPreface vii 1 Introduction 1 1.1 History 3 1.2 Empirical evidence 4 1.3 Early explanations 6 1.4 Mathematical framework 7 2 Significant Digits and the Significand 11 2.1 Significant digits 11 2.2 The significand 12 2.3 The significand sigma-algebra 14 3 The Benford Property 22 3.1 Benford sequences 23 3.2 Benford functions 28 3.3 Benford distributions and random variables 29 4 The Uniform Distribution and Benford's Law 43 4.1 Uniform distribution characterization of Benford's law 43 4.2 Uniform distribution of sequences and functions 46 4.3 Uniform distribution of random variables 54 5 Scale-, Base-, and Sum-Invariance 63 5.1 The scale-invariance property 63 5.2 The base-invariance property 74 5.3 The sum-invariance property 80 6 Real-valued Deterministic Processes 90 6.1 Iteration of functions 90 6.2 Sequences with polynomial growth 93 6.3 Sequences with exponential growth 97 6.4 Sequences with super-exponential growth 101 6.5 An application to Newton's method 111 6.6 Time-varying systems 116 6.7 Chaotic systems: Two examples 124 6.8 Differential equations 127 7 Multi-dimensional Linear Processes 135 7.1 Linear processes, observables, and difference equations 135 7.2 Nonnegative matrices 139 7.3 General matrices 145 7.4 An application to Markov chains 162 7.5 Linear difference equations 165 7.6 Linear differential equations 170 8 Real-valued Random Processes 180 8.1 Convergence of random variables to Benford's law 180 8.2 Powers, products, and sums of random variables 182 8.3 Mixtures of distributions 202 8.4 Random maps 213 9 Finitely Additive Probability and Benford's Law 216 9.1 Finitely additive probabilities 217 9.2 Finitely additive Benford probabilities 219 10 Applications of Benford's Law 223 10.1 Fraud detection 224 10.2 Detection of natural phenomena 225 10.3 Diagnostics and design 226 10.4 Computations and Computer Science 228 10.5 Pedagogical tool 230 List of Symbols 231 Bibliography 234 Index 245

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Book SynopsisJohn Mather's seminal works in Hamiltonian dynamics represent some of the most important contributions to our understanding of the complex balance between stable and unstable motions in classical mechanics. His novel approach--known as Aubry-Mather theory--singles out the existence of special orbits and invariant measures of the system, which posseTable of ContentsPreface vii 1 Tonelli Lagrangians and Hamiltonians on Compact Manifolds 1 1.1 Lagrangian Point of View 1 1.2 Hamiltonian Point of View 4 2 From KAM Theory to Aubry-Mather Theory 8 2.1 Action-Minimizing Properties of Measures and Orbits on KAM Tori 8 3 Action-Minimizing Invariant Measures for Tonelli Lagrangians 18 3.1 Action-Minimizing Measures and Mather Sets 18 3.2 Mather Measures and Rotation Vectors 24 3.3 Mather's a-and B-Functions 28 3.4 The Symplectic Invariance of Mather Sets 35 3.5 An Example: The Simple Pendulum (Part I) 39 3.6 Holonomic Measures and Generic Properties of Tonelli Lagrangians 45 4 Action-Minimizing Curves for Tonelli Lagrangians 48 4.1 Global Action-Minimizing Curves: Aubry and Mane Sets 48 4.2 Some Topological and Symplectic Properties of the Aubry and Mane Sets 66 4.3 An Example: The Simple Pendulum (Part II) 68 4.4 Mather's Approach: Peierls' Barrier 71 5 The Hamilton-Jacobi Equation and Weak KAM Theory 76 5.1 Weak Solutions and Subsolutions of Hamilton-Jacobi and Fathi's Weak KAM theory 76 5.2 Regularity of Critical Subsolutions 85 5.3 Non-Wandering Points of the Mane Set 87 Appendices A On the Existence of Invariant Lagrangian Graphs 89 A.1 Symplectic Geometry of the Phase Space 89 A.2 Existence and Nonexistence of Invariant Lagrangian Graphs 91 B Schwartzman Asymptotic Cycle and Dynamics 97 B.1 Schwartzman Asymptotic Cycle 97 B.2 Dynamical Properties 99 Bibliography 107 Index 113

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Book SynopsisBetween inventing the concept of a universal computer in 1936 and breaking the German Enigma code during World War II, Alan Turing (1912-1954), the British founder of computer science and artificial intelligence, came to Princeton University to study mathematical logic. Some of the greatest logicians in the world--including Alonzo Church, Kurt GodeTrade Review"This book presents the story of Turing's work at Princeton University and includes a facsimile of his doctoral dissertation, 'Systems of Logic Based on Ordinals,' which he completed in 1936. The author includes a detailed history of Turing's work in computer science and the attempts to ground the field in formal logic."--Mathematics Teacher "This book is not for the faint hearted, as with the great masters of painting it will insist that some thought goes into appreciating it... I love the book as a book. It is a collectors item and after all what better pursuit can one have than collecting books!"--Patrick Fogarty, Mathematics TodayTable of ContentsPreface ix The Birth of Computer Science at Princeton in the 1930s Andrew W. Appel 1 Turing's Thesis Solomon Feferman 13 Notes on the Manuscript 27 Systems of Logic Based on Ordinals Alan Turing 31 A Remarkable Bibliography 141 Contributors 143

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Book SynopsisTable of Contents*Frontmatter, pg. i*Contents, pg. vii*Chapter 1. Introduction, pg. 1*Chapter 2. Auxiliary Results, pg. 29*Chapter 3. Reduction to Restriction Estimates near the Principal Root Jet, pg. 50*Chapter 4. Restriction for Surfaces with Linear Height below 2, pg. 57*Chapter 5. Improved Estimates by Means of Airy-Type Analysis, pg. 75*Chapter 6. The Case When hlin(PHI) => 2: Preparatory Results, pg. 105*Chapter 7. How to Go beyond the Case hlin(PHI) => 5, pg. 131*Chapter 8. The Remaining Cases Where m = 2 and B = 3 or B = 4, pg. 181*Chapter 9. Proofs of Propositions 1.7 and 1.17, pg. 244*Bibliography, pg. 251*Index, pg. 257

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Book SynopsisTrade Review"An important contribution to the study of the history of mathematics, and any student, educator, or practitioner of mathematics or computer science, would benefit from reading this work."---Mark Causapin, MAA Reviews"In reading von Plato’s book the attention of the scholarly reader will be always captured."---L. Bellotti, History and Philosophy of Logic"This book presents an informed and informative hisotry of a crucially important part of mathematics. . . . a valuable addition to our corporate understanding."---Rob Ashmore, Mathematics Today

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Book SynopsisTrade Review"Fascinating... Chamberland offers enticing explanations that will leave readers hungry to know more. This wonderful book never loses its focus or momentum."--Publishers Weekly "[B]oth amateur and professional mathematicians alike will find new items of interest here... [A] welcome, splendid, fruitful addition to my math bookshelf."--Math Tango blog "The collection is outright delightful. It will agitate the minds of students and shake the sense of know-all off many a professional and most of the amateurs."--Alexander Bogomolny, Cut the Knot blog "Boring deep into the innocuous-looking number one, Chamberland opens an unexpected entry point into a dizzying maze of infinities... A bracing mathematical adventure."--Booklist "The exotics like pi and e have gotten their share of attention in the world of popular mathematical writing. Now it's time to give proper attention to the integers 1 through 9... [Single Digits] is consistently entertaining and well-written."--MAA Reviews "Chamberland takes readers on a fascinating exploration of small numbers, from one to nine, looking at their history, applications, and connections to various areas of mathematics, including number theory, geometry, chaos theory, numerical analysis, and mathematical physics... Appealing to high-school and college students, professional mathematicians, and those mesmerized by patterns, this book shows that single digits offer a plethora of possibilities that readers can count on."--DVD, Lunar and Planetary Information Bulletin "Chamberland makes this an entertaining and historical exposition, using wit and humor throughout."--Math Horizons "To put it simply, this book is a delight. Chamberland has assembled a fascinating collection of vignettes, each tied to a digit from one to nine, that inform, entertain, and intrigue... This wide spectrum of ideas is consistently interesting, and the author's skill in mining each nugget is worthy of great respect."--Choice "The range of topics included virtually guarantees that any reader will find new and unfamiliar material to enjoy... [Single Digits] is a very enjoyable book which, at many points, makes some very deep mathematics quite accessible. Highly recommended."--Keith Johnson, CMS Notes "For instructors of math courses of all levels, the vignettes in Single Digits can provide a very readable introduction or jumping-off point for discussions and projects... In an introductory group theory course, it would be a good exercise for students to consider perfect riffle shuffles in decks of size other than 52. Finally, a statistics class collecting and analyzing real-world data sets could consider whether Benford's Law applies in their situation."--Matthew Welz, MAA Focus "I highly recommend Single Digits: In Praise of Small Numbers. It would be a fine addition to any high school or math department library. As a carefully curated set of interesting topics, it would serve as a good place to start exploring the ocean of ideas in mathematics."--Bruce Cohen, NCTMTable of Contents*Frontmatter, pg. i*Contents, pg. v*Preface, pg. xi*Chapter 1. The Number One, pg. 1*Chapter 2. The Number Two, pg. 24*Chapter 3. The Number Three, pg. 69*Chapter 4. The Number Four, pg. 111*Chapter 5. The Number Five, pg. 132*Chapter 6. The Number Six, pg. 156*Chapter 7. The Number Seven, pg. 170*Chapter 8. The Number Eight, pg. 191*Chapter 9. The Number Nine, pg. 205*Chapter 10. Solutions, pg. 216*Further reading, pg. 219*Credits for illustrations, pg. 223*Index, pg. 225

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Book SynopsisTo help readers become better at solving real-world problems, this enlightening, entertaining, and inspiring book teaches simple, effective thinking techniques. The goal is not to quickly solve each challenge but to come up with as many different ways of thinking about it as possible.Trade Review"[Making Up Your Own Mind] is an elegant blend of entertainment and enlightenment."---Tom Schulte, MAA Reviews

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Book SynopsisThis book explains, in lay terms, the surprisingly simple system of mathematical logic used in digital computer circuitry. Anecdotal in its style and often funny, it follows the development of this logic system from its origins in Victorian England to its rediscovery in this century as the foundation of all modern computing machinery. ONES AND ZEROS will be enjoyed by anyone who has a general interest in science and technology.Table of ContentsBefore We Begin. Number Systems and Counting. The Basic Functions of Boolean Algebra: And, Or, And Not. Combinational Logic. The Algebra of Sets and Venn Diagrams. Other Boolean Functions. Realizing Any Boolean Function with And, Or, And Not. More Digital Circuits. Laws of Boolean Algebra. Boolean Logic. Appendix A: Counting in Base 2. Appendix B: Powers of 2. Appendix C: Summary of Boolean Functions. Further Reading. Answers to Exercises. Index. About the Author.

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Book SynopsisBerto''s highly readable and lucid guide introduces students and the interested reader to Gödel''s celebrated Incompleteness Theorem, and discusses some of the most famous - and infamous - claims arising from Gödel''s arguments. Offers a clear understanding of this difficult subject by presenting each of the key steps of the Theorem in separate chapters Discusses interpretations of the Theorem made by celebrated contemporary thinkers Sheds light on the wider extra-mathematical and philosophical implications of Gödel''s theories Written in an accessible, non-technical style Trade Review"There's Something about G¨odel is a bargain: two books in one. The first half is a gentle but rigorous introduction to the incompleteness theorems for the mathematically uninitiated. The second is a survey of the philosophical, psychological, and sociological consequences people have attempted to derive from the theorems, some of them quite fantastical." (Philosophia Mathematica, 2011) “There is a story that in 1930 the great mathematician John von Neumann emerged from a seminar delivered by Kurt Gödel saying: ‘It's all over.’ Gödel had just proved the two theorems about the logical foundations of mathematics that are the subject of this valuable new book by Francesco Berto. Berto's clear exposition and his strategy of dividing the proof into short, easily digestible chunks make it pleasant reading ... .Berto is lucid and witty in exposing mistaken applications of Gödel's results ... [and] has provided a thoroughly recommendable guide to Gödel's theorems and their current status within, and outside, mathematical logic.” (Times Higher Education Supplement, February 2010)Table of ContentsPrologue. Acknowledgments. Part I: The Gödelian Symphony. 1 Foundations and Paradoxes. 1 "This sentence is false". 2 The Liar and Gödel. 3 Language and metalanguage. 4 The axiomatic method, or how to get the non-obvious out of the obvious. 5 Peano's axioms … . 6 … and the unsatisfied logicists, Frege and Russell. 7 Bits of set theory. 8 The Abstraction Principle. 9 Bytes of set theory. 10 Properties, relations, functions, that is, sets again. 11 Calculating, computing, enumerating, that is, the notion of algorithm. 12 Taking numbers as sets of sets. 13 It's raining paradoxes. 14 Cantor's diagonal argument. 15 Self-reference and paradoxes. 2 Hilbert. 1 Strings of symbols. 2 "… in mathematics there is no ignorabimus". 3 Gödel on stage. 4 Our first encounter with the Incompleteness Theorem … . 5 … and some provisos. 3 Gödelization, or Say It with Numbers! 1 TNT. 2 The arithmetical axioms of TNT and the "standard model" N. 3 The Fundamental Property of formal systems. 4 The Gödel numbering … . 5 … and the arithmetization of syntax. 4 Bits of Recursive Arithmetic … . 1 Making algorithms precise. 2 Bits of recursion theory. 3 Church's Thesis. 4 The recursiveness of predicates, sets, properties, and relations. 5 … And How It Is Represented in Typographical Number Theory. 1 Introspection and representation. 2 The representability of properties, relations, and functions … . 3 … and the Gödelian loop. 6 "I Am Not Provable". 1 Proof pairs. 2 The property of being a theorem of TNT (is not recursive!) 3 Arithmetizing substitution. 4 How can a TNT sentence refer to itself? 5 γ 6 Fixed point. 7 Consistency and omega-consistency. 8 Proving G1. 9 Rosser's proof. 7 The Unprovability of Consistency and the "Immediate Consequences" of G1 and G2. 1 G2. 2 Technical interlude. 3 "Immediate consequences" of G1 and G2. 4 Undecidable1 and undecidable2. 5 Essential incompleteness, or the syndicate of mathematicians. 6 Robinson Arithmetic. 7 How general are Gödel's results? 8 Bits of Turing machine. 9 G1 and G2 in general. 10 Unexpected fish in the formal net. 11 Supernatural numbers. 12 The culpability of the induction scheme. 13 Bits of truth (not too much of it, though). Part II: The World after Gödel. 8 Bourgeois Mathematicians! The Postmodern Interpretations. 1 What is postmodernism? 2 From Gödel to Lenin. 3 Is "Biblical proof" decidable? 4 Speaking of the totality. 5 Bourgeois teachers! 6 (Un)interesting bifurcations. 9 A Footnote to Plato. 1 Explorers in the realm of numbers. 2 The essence of a life. 3 "The philosophical prejudices of our times". 4 From Gödel to Tarski. 5 Human, too human. 10 Mathematical Faith. 1 "I'm not crazy!" 2 Qualified doubts. 3 From Gentzen to the Dialectica interpretation. 4 Mathematicians are people of faith. 11 Mind versus Computer: Gödel and Artificial Intelligence. 1 Is mind (just) a program? 2 "Seeing the truth" and "going outside the system". 3 The basic mistake. 4 In the haze of the transfinite. 5 "Know thyself": Socrates and the inexhaustibility of mathematics. 12 Gödel versus Wittgenstein and the Paraconsistent Interpretation. 1 When geniuses meet … . 2 The implausible Wittgenstein. 3 "There is no metamathematics". 4 Proof and prose. 5 The single argument. 6 But how can arithmetic be inconsistent? 7 The costs and benefits of making Wittgenstein plausible. Epilogue. References. Index.

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Book SynopsisBerto''s highly readable and lucid guide introduces students and the interested reader to Gödel''s celebrated Incompleteness Theorem, and discusses some of the most famous - and infamous - claims arising from Gödel''s arguments. Offers a clear understanding of this difficult subject by presenting each of the key steps of the Theorem in separate chapters Discusses interpretations of the Theorem made by celebrated contemporary thinkers Sheds light on the wider extra-mathematical and philosophical implications of Gödel''s theories Written in an accessible, non-technical style Trade Review"This is a beautifully clear and accurate presentation of the material, with no technical demands beyond what is required for accuracy, and filled with interesting philosophical suggestions." (John Woods, University of British Columbia) "There's Something about G¨odel is a bargain: two books in one. The first half is a gentle but rigorous introduction to the incompleteness theorems for the mathematically uninitiated. The second is a survey of the philosophical, psychological, and sociological consequences people have attempted to derive from the theorems, some of them quite fantastical." (Philosophia Mathematica, 2011) "There is a story that in 1930 the great mathematician John von Neumann emerged from a seminar delivered by Kurt Gödel saying: ‘It's all over.’ Gödel had just proved the two theorems about the logical foundations of mathematics that are the subject of this valuable new book by Francesco Berto. Berto's clear exposition and his strategy of dividing the proof into short, easily digestible chunks make it pleasant reading ... .Berto is lucid and witty in exposing mistaken applications of Gödel's results ... [and] has provided a thoroughly recommendable guide to Gödel's theorems and their current status within, and outside, mathematical logic.” (Times Higher Education Supplement, February 2010)Table of ContentsPrologue. Acknowledgments. Part I: The Gödelian Symphony. 1 Foundations and Paradoxes. 1 "This sentence is false". 2 The Liar and Gödel. 3 Language and metalanguage. 4 The axiomatic method, or how to get the non-obvious out of the obvious. 5 Peano's axioms … . 6 … and the unsatisfied logicists, Frege and Russell. 7 Bits of set theory. 8 The Abstraction Principle. 9 Bytes of set theory. 10 Properties, relations, functions, that is, sets again. 11 Calculating, computing, enumerating, that is, the notion of algorithm. 12 Taking numbers as sets of sets. 13 It's raining paradoxes. 14 Cantor's diagonal argument. 15 Self-reference and paradoxes. 2 Hilbert. 1 Strings of symbols. 2 "… in mathematics there is no ignorabimus". 3 Gödel on stage. 4 Our first encounter with the Incompleteness Theorem … . 5 … and some provisos. 3 Gödelization, or Say It with Numbers! 1 TNT. 2 The arithmetical axioms of TNT and the "standard model" N. 3 The Fundamental Property of formal systems. 4 The Gödel numbering … . 5 … and the arithmetization of syntax. 4 Bits of Recursive Arithmetic … . 1 Making algorithms precise. 2 Bits of recursion theory. 3 Church's Thesis. 4 The recursiveness of predicates, sets, properties, and relations. 5 … And How It Is Represented in Typographical Number Theory. 1 Introspection and representation. 2 The representability of properties, relations, and functions … . 3 … and the Gödelian loop. 6 "I Am Not Provable". 1 Proof pairs. 2 The property of being a theorem of TNT (is not recursive!) 3 Arithmetizing substitution. 4 How can a TNT sentence refer to itself? 5 γ 6 Fixed point. 7 Consistency and omega-consistency. 8 Proving G1. 9 Rosser's proof. 7 The Unprovability of Consistency and the "Immediate Consequences" of G1 and G2. 1 G2. 2 Technical interlude. 3 "Immediate consequences" of G1 and G2. 4 Undecidable1 and undecidable2. 5 Essential incompleteness, or the syndicate of mathematicians. 6 Robinson Arithmetic. 7 How general are Gödel's results? 8 Bits of Turing machine. 9 G1 and G2 in general. 10 Unexpected fish in the formal net. 11 Supernatural numbers. 12 The culpability of the induction scheme. 13 Bits of truth (not too much of it, though). Part II: The World after Gödel. 8 Bourgeois Mathematicians! The Postmodern Interpretations. 1 What is postmodernism? 2 From Gödel to Lenin. 3 Is "Biblical proof" decidable? 4 Speaking of the totality. 5 Bourgeois teachers! 6 (Un)interesting bifurcations. 9 A Footnote to Plato. 1 Explorers in the realm of numbers. 2 The essence of a life. 3 "The philosophical prejudices of our times". 4 From Gödel to Tarski. 5 Human, too human. 10 Mathematical Faith. 1 "I'm not crazy!" 2 Qualified doubts. 3 From Gentzen to the Dialectica interpretation. 4 Mathematicians are people of faith. 11 Mind versus Computer: Gödel and Artificial Intelligence. 1 Is mind (just) a program? 2 "Seeing the truth" and "going outside the system". 3 The basic mistake. 4 In the haze of the transfinite. 5 "Know thyself": Socrates and the inexhaustibility of mathematics. 12 Gödel versus Wittgenstein and the Paraconsistent Interpretation. 1 When geniuses meet … . 2 The implausible Wittgenstein. 3 "There is no metamathematics". 4 Proof and prose. 5 The single argument. 6 But how can arithmetic be inconsistent? 7 The costs and benefits of making Wittgenstein plausible. Epilogue. References. Index.

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Book SynopsisPreface.- Introduction.- Propositional Logic: Formulas, Models, Tableaux.- Propositional Logic: Deductive Systems.- Propositional Logic: Resolution.- Propositional Logic: Binary Decision Diagrams.- Propositional Logic: SAT Solvers.- First-Order Logic: Formulas, Models, Tableaux.- First-Order Logic: Deductive Systems.- First-Order Logic: Terms and Normal Forms.- First-Order Logic: Resolution.- First-Order Logic: Logic Programming.- First-Order Logic: Undecidability and Model Theory.- Temporal Logic: Formulas, Models, Tableaux.- Temporal Logic: A Deductive System.- Verification of Sequential Programs.- Verification of Concurrent Programs.- Set Theory.- Index of Symbols.- Index of Names.- Subject Index.Trade ReviewAsst. Prof. Manoj Raut, Dhirubhai Ambani Institute of Information and Communication Technology, IndiaExcerpts from full review posted Jan 15 2013 to Computing Reviews [Review #: CR140831]I have used the second edition of this book for my class. I find this new third edition more interesting and more elaborately written; I like it very much, and applaud the author for his work.Table of ContentsPreface.- Introduction.- Propositional Logic: Formulas, Models, Tableaux.- Propositional Logic: Deductive Systems.- Propositional Logic: Resolution.- Propositional Logic: Binary Decision Diagrams.- Propositional Logic: SAT Solvers.- First-Order Logic: Formulas, Models, Tableaux.- First-Order Logic: Deductive Systems.- First-Order Logic: Terms and Normal Forms.- First-Order Logic: Resolution.- First-Order Logic: Logic Programming.- First-Order Logic: Undecidability and Model Theory.- Temporal Logic: Formulas, Models, Tableaux.- Temporal Logic: A Deductive System.- Verification of Sequential Programs.- Verification of Concurrent Programs.- Set Theory.- Index of Symbols.- Index of Names.- Subject Index.

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Book SynopsisGallery of the Infinite is a mathematician's unique view of the infinitely many sizes of infinity. Written in a playful yet informative style, it introduces important concepts from set theory (including the Cantor Diagonalization Method and the Cantor-Bernstein Theorem) using colourful pictures, with little text and almost no formulas.Trade ReviewThis is a beautiful book. The pictures keep the reader engaged in a colourful mathematical journey. It is written in an engaging style suitable for over 11’s but also contains ideas that are likely to interest most adults (without the need for a refresher course, since the book does a good job of being self-contained). [...] Although a mathematician would likely be aware of many of the concepts the book presents, I would still recommend it both as a tool to intrigue others (it makes a great ‘coffee table’ book) and also since it contains many imaginative explanations and original arguments. The illustrations and narrative keep the reader entertained and make the book hard to put down." - London Mathematical Society Newsletter"This is a lovely book… Although a certain affinity with mathematical reasoning is needed the book can be read by almost anyone." - Teun Koetsier, Zentralblatt Math

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Book SynopsisThe second of a three-volume set of books on the theory of algebras, a study that provides a consistent framework for understanding algebraic systems, including groups, rings, modules, semigroups and lattices.Table of Contents The classification of varieties Equational logic Rudiments of model theory Bibliography Index

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Book SynopsisThe third of a three-volume set of books on the theory of algebras, a study that provides a consistent framework for understanding algebraic systems, including groups, rings, modules, semigroups and lattices.Table of Contents Finite algebras and their clones Abstract clone theory Commutator theory Bibliography Index

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Book SynopsisCircular analyses of philosophical, linguistic, or computational phenomena have been attacked on the assumption that they conflict with mathematical rigour. Barwise and Moss have undertaken to prove this assumption false. This volume is concerned with extending the modelling capabilities of set theory to provide a uniform treatment of circular phenomena. As a means of guiding the reader through the concrete examples of the theory, the authors have included many exercises and solutions: these exercises range in difficulty and ultimately stimulate the reader to come up with new results. Vicious Circles is intended for use by researchers who want to use hypersets; although some experience in mathematics is necessary, the book is accessible to people with widely differing backgrounds and interests.Trade Review' ... a book to learn from.' L'Enseignement MathématiqueTable of ContentsPart I. Background: 1. Introduction; 2. Background on set theory; Part II. Vicious Circles: 3. Circularity in computer science; 4. Circularity in philosophy; 5. Circularity and paradox; Part III. Basic Theory: 6. The solution dilemma; 7. Bisimulation; Part IV. Elementary applications: 8. Graphs; 9. Modal logic; 10. Streams; 11. Games; 12. Modeling the semantic paradoxes; Part V. Further Theory: 13. Greatest fixed points; 14. Uniform operators; 15. Corecursion; Part VI. Further Applications: 16. Some applications; 17. Modeling partial information; 18. Circularity and the notion of set; 19. Conclusions and future directions.

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Book SynopsisDonald Knuth's influence in computer science ranges from the invention of literate programming to the development of the TeX programming language. One of the foremost figures in the field of mathematical sciences, his papers are widely referenced and stand as milestones of development over a wide range of topics. This volume assembles more than three dozen of Professor Knuth's pioneering contributions to discrete mathematics. It includes a variety of topics in combinatorial mathematics (finite geometries, graph theory, enumeration, partitions, tableaux, matroids, codes); discrete algebra (finite fields, groupoids, closure operators, inequalities, convolutions, Pfaffians); and concrete mathematics (recurrence relations, special numbers and notations, identities, discrete probability). Of particular interest are two fundamental papers in which the evolution of random graphs is studied by means of generating functions.Table of Contents1. Discussion of Mr. Riordan's paper 'Abel identities and inverse relations'; 2. Duality in addition chains; 3. Combinatorial analysis and computers; 4. Tables of finite fields; 5. Finite semifields and projective planes; 6. A class of projective planes; 7. Construction of a random sequence; 8. Oriented subtrees of an arc digraph; 9. Another enumeration of trees; 10. Notes on central groupoids; 11. Permutations, matrices, and generalized Young tableaux; 12. A note on solid partitions; 13. Subspaces, subsets, and partitions; 14. Enumeration of plane partitions; 15. Complements and transitive closures; 16. Permutations with nonnegative partial sums; 17. Wheels within wheels; 18. The asymptotic number of geometries; 19. Random matroids; 20. Identities from partition involutions; 21. Huffman's algorithm via algebra; 22. A permanent inequality; 23. Efficient balanced codes; 24. The power of a prime that divides a generalized binomial coefficient; 25. The first cycles in an evolving graph; 26. The birth of the giant component; 27. Polynomials involving the floor function; 28. The sandwich theorem; 29. Aztec diamonds, checkerboard graphs, and spanning trees.

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Book Synopsis

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Book Synopsis

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Book SynopsisSince its inception, fuzzy logic has attracted an incredible amount of interest, and this interest continues to grow at an exponential rate. As such, scientists, researchers, educators and practitioners of fuzzy logic continue to expand on the applicability of what and how fuzzy can be utilised in the real-world. In this book, the authors present key application areas where fuzzy has had significant success. The chapters cover a plethora of application domains, proving credence to the versatility and robustness of a fuzzy approach. A better understanding of fuzzy will ultimately allow for a better appreciation of fuzzy. This book provides the reader with a varied range of examples to illustrate what fuzzy logic can be capable of and how it can be applied. The text will be ideal for individuals new to the notion of fuzzy, as well as for early career academics who wish to further expand on their knowledge of fuzzy applications. The book is also suitable as a supporting text for advanced undergraduate and graduate-level modules on fuzzy logic, soft computing, and applications of AI.Table of ContentsRecognising Handwritten Digits Using a Fuzzy Neural Network Joshua Reynolds and Tianhua Chen Fuzzy Assessment of Student Academic Performances Shangen Yang and Tianhua Chen A Hybrid Fuzzy Neural Network for Image Recognition Samaresh Nayak and Tianhua Chen A Fuzzy Diagnostic System for Heart Disease Siyue Song, Tianhua Chen, and Grigoris Antoniou Analysing Medical Notes using Fuzzy Logic Siyue Song, Tianhua Chen, and Grigoris Antoniou Fostering Positive Personalisation through Fuzzy Clustering Raymond Moodley Fuzzy Logic in Modern Information Retrieval Steve Wade Fuzzy Applied to Sentiment Analysis Orestes Appel Fuzzy Logic, a Logicians Perspective Patrick Fogarty Applications of Fuzzy Logic in an Automated Warehouse Patrick Fogarty Can Fuzzy Systems Assist with Project Planning? Daniel Maia and Arjab Khuman Fuzzy Logic in Autonomous Vehicles David McDougall and Arjab Khuman AI Spawning Fuzzy Logic Fuzzy Inference System Reece Carey and Arjab Khuman The Application of Fuzzy Logic on Intelligent Transportation Systems Nath Lloyd and Arjab Khuman Fuzzy Logic Applied to Water Processes Will Chapman and Arjab Khuman Applications of Fuzzy Logic in Autonomous Vehicles Sam Asquith and Arjab Khuman Predicting Cyber Threats using Fuzzy Logic Jarrad Morden and Arjab Khuman Implementations of Fuzzy Logic in Camera Systems Sophie Hughes and Arjab Khuman Application of a Fuzzy Logic Control System for Stock Market Prediction Based on Technical Indicators and Fundamental Analysis Humza Nazir and Arjab Khuman The Application of Fuzzy Logic in Determining Outcomes of Sporting Events Spencer Deane and Arjab Khuman Using Fuzzy Logic to Educate People on Phishing Harry Taylor and Arjab Khuman

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Book SynopsisThis volume is dedicated to Hiroakira Ono life’s work on substructural logics. Chapters, written by well-established academics, cover topics related to universal algebra, algebraic logic and the Full Lambek calculus; the book includes a short biography about Hiroakira Ono. The book starts with detailed surveys on universal algebra, abstract algebraic logic, topological dualities, and connections to computer science.It further contains specialised contributions on connections to formal languages (recognizability in residuated lattices and connections to the finite embedding property), covering systems for modal substructural logics, results on the existence and disjunction properties and finally a study of conservativity of expansions. This book will be primarily of interest to researchers working in algebraic and non-classical logic.Table of ContentsChapter 1. A scientific autobiography (Hiroakira Ono).- Part I: Expository and survey chapters.- Chapter 2. Universal algebraic methods for non-classical logics (James G. Raftery).- Chapter 3. Abstract algebraic logic - An introductory chapter (Josep Maria Font).- Chapter 4. Topological duality and algebraic completions (Mai Gehrke).- Chapter 5. An algebraic glimpse at bunched implications and separation logic (Peter Jipsen and Tadeusz Litak).- Part II: Special topics.- Chapter 6. Recognizability in Residuated Lattices (José Gil-Férez and Constantine Tsinakis).- Chapter 7. Finite embeddability property for residuated lattices via regular languages (Rostislav Horčík). Chapter 8. Cover systems for the modalities of linear logic (Robert Goldblatt).- Chapter 9. A negative solution to Ono’s Problem P52: Existence and disjunction properties in intermediate predicate logic (Nobu-Yuki Suzuki).- Chapter 10. Conservative expansions of substructural logics (Jacopo Amidei, Rodolfo C. Ertola-Biraben and Franco Montagna).

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Book SynopsisThis survey of computability theory offers the techniques and tools that computer scientists (as well as mathematicians and philosophers studying the mathematical foundations of computing) need to mathematically analyze computational processes and investigate the theoretical limitations of computing. Beginning with an introduction to the mathematisation of “mechanical process” using URM programs, this textbook explains basic theory such as primitive recursive functions and predicates and sequence-coding, partial recursive functions and predicates, and loop programs. Advanced chapters cover the Ackerman function, Tarski’s theorem on the non-representability of truth, Goedel’s incompleteness and Rosser’s incompleteness theorems, two short proofs of the incompleteness theorem that are based on Lob's deliverability conditions, Church’s thesis, the second recursion theorem and applications, a provably recursive universal function for the primitive recursive functions, Oracle computations and various classes of computable functionals, the Arithmetical hierarchy, Turing reducibility and Turing degrees and the priority method, a thorough exposition of various versions of the first recursive theorem, Blum’s complexity, Hierarchies of primitive recursive functions, and a machine-independent characterisation of Cobham's feasibly computable functions.Trade Review“This textbook is suited for self-study … . As a second reading however a reader interested in rigorous proofs and/or different approaches to known concepts will benefit from this wealth of material.” (Dieter Riebesehl, zbMATH 1507.03002, 2023)Table of ContentsMathematical Background; a Review.- A Theory of Computability.- Primitive Recursive Functions.- Loop Programs.-The Ackermann Function.- (Un)Computability via Church's Thesis.- Semi-Recursiveness.- Yet another number-theoretic characterisation of P.- Godel's Incompleteness Theorem via the Halting Problem.- The Recursion Theorem.- A Universal (non-PR) Function for PR.- Enumerations of Recursive and Semi-Recursive Sets.- Creative and Productive Sets Completeness.- Relativised Computability.- POSSIBILITY: Complexity of P Functions.- Complexity of PR Functions.- Turing Machines and NP-Completeness.

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Book SynopsisThis book explores the premise that a physical theory is an interpretation of the analytico–canonical formalism. Throughout the text, the investigation stresses that classical mechanics in its Lagrangian formulation is the formal backbone of theoretical physics. The authors start from a presentation of the analytico–canonical formalism for classical mechanics, and its applications in electromagnetism, Schrödinger's quantum mechanics, and field theories such as general relativity and gauge field theories, up to the Higgs mechanism.The analysis uses the main criterion used by physicists for a theory: to formulate a physical theory we write down a Lagrangian for it. A physical theory is a particular instance of the Lagrangian functional. So, there is already an unified physical theory. One only has to specify the corresponding Lagrangian (or Lagrangian density); the dynamical equations are the associated Euler–Lagrange equations. The theory of Suppes predicates as the main tool in the axiomatization and examples from the usual theories in physics. For applications, a whole plethora of results from logic that lead to interesting, and sometimes unexpected, consequences.This volume looks at where our physics happen and which mathematical universe we require for the description of our concrete physical events. It also explores if we use the constructive universe or if we need set–theoretically generic spacetimes.Trade Review“This book is a compilation, ‘an essay’, of the bulk of their work from 1990 to the present. This 191 page essay includes some historical background and lots of snippets and parts of da Costa and Doria’s work on the meta-mathematics of mathematical physics. It starts with a primer on graduate-level basic physics … ending with a consideration of hypercomputation.” (Deborah Konkowski, zbMATH 1494.00005, 2022)Table of ContentsForeword1. PreliminaryPart I. Physics: A Primer2. Classical mechanics3. Variational calculus4. Lagrangian formulation5. Hamilton’s equations6. Hamilton–Jacobi theory7. Where the action is8. From classical to quantum9. Field theory10. Electromagnetism11. Special relativity12. General relativity13. Gauge field theoriesPart II. Axiomatics14. Axiomatizations in ZFCPart III. Technicalities15. HierarchiesPart IV. More applications16. Arnol’d’s 1974 problems17. Forcing and gravitation18. Economics and ecology.Part V. Computer science19. Fast–growing functionsPart VI. Hypercomputation20. HypercomputationReferences

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Book SynopsisThis textbook covers topics of undergraduate mathematics in abstract algebra, geometry, topology and analysis with the purpose of connecting the underpinning key ideas. It guides STEM students towards developing knowledge and skills to enrich their scientific education. In doing so it avoids the common mechanical approach to problem-solving based on the repetitive application of dry formulas. The presentation preserves the mathematical rigour throughout and still stays accessible to undergraduates. The didactical focus is threaded through the assortment of subjects and reflects in the book’s structure.Part 1 introduces the mathematical language and its rules together with the basic building blocks. Part 2 discusses the number systems of common practice, while the backgrounds needed to solve equations and inequalities are developed in Part 3. Part 4 breaks down the traditional, outdated barriers between areas, exploring in particular the interplay between algebra and geometry. Two appendices form Part 5: the Greek etymology of frequent terms and a list of mathematicians mentioned in the book. Abundant examples and exercises are disseminated along the text to boost the learning process and allow for independent work.Students will find invaluable material to shepherd them through the first years of an undergraduate course, or to complement previously learnt subject matters. Teachers may pick’n’mix the contents for planning lecture courses or supplementing their classes.Trade Review“The book being reviewed is a collection of what the author considers to be essential material for undergraduates … . it has to be said that many students will find that there is plenty to learn from this well-written book, which would also be a useful reference text had there been a properly compiled index.” (Peter Shiu, The Mathematical Gazette, Vol. 107 (570), November, 2023)Table of ContentsPart I: Basic Objects and Formalisation - Round-up of Elementary Logic.- Naive Set Theory.- Functions.- More Set Theory and Logic.- Boolean Algebras. Part 2: Numbers and Structures - Intuitive Arithmetics.- Real Numbers.- Totally Ordered Spaces.- Part 3: Elementary Real Functions - Real Polynomials.- Real Functions of One Real Variables.- Algebraic Functions.- Elementary Transcendental Functions.- Complex Numbers.- Enumerative Combinatorics.- Part 4: Geometry through Algebra - Vector Spaces.- Orthogonal Operators.- Actions & Representations.- Elementary Plane Geometry.- Metric Spaces.- Part 5: Appendices - Etymologies.- Index of names.- Main figures.- Glossary.- References.

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Book SynopsisThis open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems.

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Book SynopsisThis volume explores A.P. Morse’s (1911-1984) development of a formal language for writing mathematics, his application of that language in set theory and mathematical analysis, and his unique perspective on mathematics. The editor brings together a variety of Morse’s works in this compilation, including Morse's book A Theory of Sets, Second Edition (1986), in addition to material from another of Morse’s publications, Web Derivatives, and notes for a course on analysis from the early 1950's. Because Morse provided very little in the way of explanation in his written works, the editor’s commentary serves to outline Morse’s goals, give informal explanations of Morse’s formal language, and compare Morse’s often unique approaches to more traditional approaches. Minor corrections to Morse’s previously published works have also been incorporated into the text, including some updated axioms, theorems, and definitions. The editor’s introduction thoroughly details the corrections and changes made and provides readers with valuable insight on Morse’s methods.A.P. Morse’s Set Theory and Analysis will appeal to graduate students and researchers interested in set theory and analysis who also have an interest in logic. Readers with a particular interest in Morse’s unique perspective and in the history of mathematics will also find this book to be of interest.Table of ContentsPreface.- Editor's Introduction.- Language and Inference.- Logic.- Set Theory.- Elementary Analysis.- Metrics.- Measure.- Linear Measure and Total Variation.- Integration.- Product Measures.- Web Derivatives.- Classical Differentiation.- The Construction of Definition.- The Consistency of the Axiom of Size.- Suggested Reading.- Publications of A.P. Morse.- Errata to A Theory of Sets, Second Edition.- Integration with Respect to Addor Functions.- The Henstock-Kurzweil Integral.

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Book SynopsisThis book constitutes revised selected papers from the refereed proceedings of the 4th International Workshop on Dynamic Logic, DaLí 2022, held in Haifa, Israel, in July/August 2022.The 8 full papers presented in this volume were carefully reviewed and selected from 22 submissions. They deal with new trends and applications in the area of Dynamic Logic. Table of ContentsFirst steps in updating knowing how.- Parametrized modal logic II: the unidimensional case.- Relating Kleene algebras.- Dynamic epistemic logic for budget-constrained agents.- Action models for coalition logic.- Quantum logic for observation of physical quantities.- Cautious distributed belief.- A STIT logic of intentionality.

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Book SynopsisEdited in collaboration with FoLLI, this book constitutes the refereed proceedings of the 10th Indian Conference on Logic and Its Applications, ICLA 2023, which was held in Indore, India, in March 2023.Besides 6 invited papers presented in this volume, there are 9 contributed full papers which were carefully reviewed and selected from 18 submissions. The volume covers a wide range of topics. These topics are related to modal and temporal logics, intuitionistic connexive and imperative logics, systems for reasoning with vagueness and rough concepts, topological quasi-Boolean logic and quasi-Boolean based rough set models, and first-order definability of path functions of graphs.Table of ContentsA Note on the Ontology of Mathematics.- Boolean Functional Synthesis: From Under the Hood of Solvers.- Labelled Calculi for Lattice-based Modal Logics.- Two Ways to Scare a Gruffalo.- Determinacy Axioms and Large Cardinals.- Big ideas from logic for mathematics and computing education.- Modal Logic of Generalized Separated Topological Spaces.- Multiple-valued Semantics for Metric Temporal Logic.- Segment transit function of the induced path function of graphs and its first-order definability.- Fuzzy Free Logic with Dual Domain Semantics.- A New Dimension of Imperative Logic. -Quasi-Boolean based models in Rough Set theory: A case of Covering.- Labelled calculi for the logics of rough concepts.- An Infinity of Intuitionistic Connexive Logics.- Relational Semantics for Normal Topological Quasi-Boolean Logic.

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Book SynopsisThis book constitutes the proceedings of the 5th International Workshop on Formal Methods Teaching, FMTea 2023, which was held in Lübeck, Germany, in March 2023.The 7 full papers presented in this volume were carefully reviewed and selected from 10 submissions. FMTea 2023 aim is to support a worldwide improvement in learning Formal Methods, mainly by teaching but also via self-learning.Table of ContentsAutomated Exercise Generation for Satisfiability Checking.- Graphical Loop Invariant Based Programming.- A Gentle Introduction to Verification of Parameterized Reactive Systems.- Model Checking Concurrent Programs for Autograding in pseuCo Book.- Teaching TLA+ to Engineers at Microsoft.- Teaching and Training in Formalisation with B.- Teaching low-code Formal Methods with Coloured Petri Nets.

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Book SynopsisFrom the ‘punctuated equilibrium' of Eldrege and Gould, through Lewontin's ‘triple helix' and the various visions and revisions of the Extended Evolutionary Synthesis (EES) of Laland and others, both data and theory have demanded an opening-up of the 1950's Evolutionary Synthesis that so firmly wedded evolutionary theory to the mathematics of gene frequency analysis. It can, however, be argued that a single deep and comprehensive mathematical theory may simply not be possible for the almost infinite varieties of evolutionary process active at and across the full range of scales of biological, social, institutional, and cultural phenomena. Indeed, the case history of 'meme theory' should have raised a red flag that narrow gene-centered models of evolutionary process may indeed have serious limitations. What is attempted here is less grand, but still broader than a gene-centered analysis. Following the instruction of Maturana and Varela that all living systems are cognitive, in a certain sense, and that living as a process is a process of cognition, the asymptotic limit theorems of information and control theories that bound all cognition provide a basis for constructing an only modestly deep but wider-ranging series of probability models that might be converted into useful statistical tools for the analysis of observational and experimental data related to evolutionary process. The line of argument in this series of interrelated essays proves to be surprisingly direct.Table of Contents1 Onthemajortransitions1.1 Introduction1.2 Symmetryandsymmetry-breaking1.3 Resources1.4 Cognitioninnonergodicsystems1.5 Theprebiotic`bigbang'1.6 Biological`recombinationtransparency'1.7 Asimpleapplication1.8 Specializationandcooperation:multipleworkspaces1.9 Discussion1. MathematicalAppendix1. References2 OntheExtendedEvolutionarySynthesis2.1 Introduction2.2 Firstnotions2.3 Thebasictheory2.4 Examples2.5 Moretheory:selectionpressureasshadowprice2.6 Extendingthemodels2.7 Discussion2.8 MathematicalAppendix2.9 References3O nregulation3.1 Introduction3.2 Theory3.3 Applications3.4 Discussion3.5 MathematicalAppendix3.6 References4 Punctuatedregulationasanevolutionarymechanism4.1 Introduction4.2 FisherZerosreconsidered4.3 ExtinctionI:Simplenoise-inducedtransitions4.4 ExtinctionII:Morecomplicatednoise-inducedtransitions4.5 ExtinctionIII:Environmentalshadowprice4.6 Discussion4.7 MathematicalAppendix4.8 References5 Institutionaldynamicsunderselectionpressureanduncertainty5.1 Introduction5.2 ARateDistortionTheoremmodelofcontrol5.3 Selectionpressuredynamics5.4 Destabilizationbydelay5.5 ExtendingtheDataRateTheorem5.6 Movingon5.7 Reconsideringcognition\textit{AnSich5.8 Changingtheviewpoint5.9 Discussion5. References6O n`Speciation':Fragmentsizeininformationsystemphasetransitions6.1 Introduction6.2`Simple'phasetransition6.3 Phasetransitionsinnetworksofinformation-exchangemodules6.4 Discussion6.5 MathematicalAppendix:`Biological'renormalizations6.6 References7 Adaptingcognitionmodelstobiomolecularcondensatedynamics7.1 Introduction7.2 Resources7.3 Cognition7.4 PhasetransitionsI:Fisherzeros7.5 Cognitive`reactionrate'7.6 PhasetransitionsII:Signaltransductionandnoise7.7 Discussion7.8 MathematicalAppendix:Groupoids7.9 References8 EvolutionaryExaptation:Sharedinterbrainactivityinsocialcommunication8.1 Introduction8.2 Correlation8.3 Cognition8.4 Dynamics8.5 Cognitionrate8.6 Anexample8.7 Cooperation:Multipleworkspaces8.8 Networktopologyisimportant8.9 Timeandresourceconstraintsareimportant8.10 Furthertheoreticaldevelopment8.11 Discussion8.12 MathematicalAppendix8.13 References9 Afterward

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Book SynopsisThe starting point for this monograph is the previously unknown connection between the Continuum Hypothesis and the saturation of the non-stationary ideal on ω1; and the principle result of this monograph is the identification of a canonical model in which the Continuum Hypothesis is false. This is the first example of such a model and moreover the model can be characterized in terms of maximality principles concerning the universal-existential theory of all sets of countable ordinals. This model is arguably the long sought goal of the study of forcing axioms and iterated forcing but is obtained by completely different methods, for example no theory of iterated forcing whatsoever is required. The construction of the model reveals a powerful technique for obtaining independence results regarding the combinatorics of the continuum, yielding a number of results which have yet to be obtained by any other method. This monograph is directed to researchers and advanced graduate students in Set Theory. The second edition is updated to take into account some of the developments in the decade since the first edition appeared, this includes a revised discussion of Ω-logic and related matters.

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Book SynopsisThis book analyzes the generation of the arrow-categories of a given category, which is a foundational and distinguishable Category Theory phenomena, in analogy to the foundational role of sets in the traditional set-based Mathematics, for defi nition of natural numbers as well. This inductive transformation of a category into the infinite hierarchy of the arrowcategories is extended to the functors and natural transformations. The author considers invariant categorial properties (the symmetries) under such inductive transformations. The book focuses in particular on Global symmetry (invariance of adjunctions) and Internal symmetries between arrows and objects in a category (in analogy to Field Theories like Quantum Mechanics and General Relativity). The second part of the book is dedicated to more advanced applications of Internal symmetry to Computer Science: for Intuitionistic Logic, Untyped Lambda Calculus with Fixpoint Operators, Labeled Transition Systems in Process Algebras and Modal logics as well as Data Integration Theory.

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