{"title":"Mathematical logic Books","description":"","products":[{"product_id":"the-foundations-of-mathematics-9780198706434","title":"The Foundations of Mathematics","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eThe transition from school mathematics to university mathematics is seldom straightforward. Students are faced with a disconnect between the algorithmic and informal attitude to mathematics at school, versus a new emphasis on proof, based on logic, and a more abstract development of general concepts, based on set theory. The authors have many years'' experience of the potential difficulties involved, through teaching first-year undergraduates and researching the ways in which students and mathematicians think. The book explains the motivation behind abstract foundational material based on students'' experiences of school mathematics, and explicitly suggests ways students can make sense of formal ideas.This second edition takes a significant step forward by not only making the transition from intuitive to formal methods, but also by reversing the process- using structure theorems to prove that formal systems have visual and symbolic interpretations that enhance mathematical thinking. This is exemplified by a new chapter on the theory of groups. While the first edition extended counting to infinite cardinal numbers, the second also extends the real numbers rigorously to larger ordered fields. This links intuitive ideas in calculus to the formal epsilon-delta methods of analysis. The approach here is not the conventional one of ''nonstandard analysis'', but a simpler, graphically based treatment which makes the notion of an infinitesimal natural and straightforward. This allows a further vision of the wider world of mathematical thinking in which formal definitions and proof lead to amazing new ways of defining, proving, visualising and symbolising mathematics beyond previous expectations.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003eThe writing is both rigorous and thorough, and the authors use compact presentations to support their explanations and proofs. Highly recommended. * N. W. Schillow, CHOICE *\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003eI: THE INTUITIVE BACKGROUND; II: THE BEGINNINGS OF FORMALISATION; III: THE DEVELOPMENT OF AXIOMATIC SYSTEMS; IV: USING AXIOMATIC SYSTEMS; V: STRENGTHENING THE FOUNDATIONS","brand":"Oxford University Press","offers":[{"title":"Default Title","offer_id":48732766929239,"sku":"9780198706434","price":23.99,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780198706434.jpg?v=1719998308"},{"product_id":"logic-9780198811701","title":"Logic","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eLogic is often perceived as having little to do with the rest of philosophy, and even less to do with real life. In this lively and accessible introduction, Graham Priest shows how wrong this conception is. He explores the philosophical roots of the subject, explaining how modern formal logic deals with issues ranging from the existence of God and the reality of time to paradoxes of probability and decision theory. Along the way, the basics of formal logic are explained in simple, non-technical terms, showing that logic is a powerful and exciting part of modern philosophy. In this new edition Graham Priest expands his discussion to cover the subjects of algorithms and axioms, and proofs in mathematics.ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003ePREFACE TO SECOND EDITION; PREFACE TO FIRST EDITION; GLOSSARY; PROBLEMS; PROBLEM SOLUTIONS; BIBLIOGRAPHY; GENERAL INDEX","brand":"Oxford University Press","offers":[{"title":"Default Title","offer_id":48732792717655,"sku":"9780198811701","price":9.49,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780198811701.jpg?v=1719998421"},{"product_id":"category-theory-9780199237180","title":"Category Theory","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eCategory theory is a branch of abstract algebra with incredibly diverse applications. This text and reference book is aimed not only at mathematicians, but also researchers and students of computer science, logic, linguistics, cognitive science, philosophy, and any of the other fields in which the ideas are being applied. Containing clear definitions of the essential concepts, illuminated with numerous accessible examples, and providing full proofs of all important propositions and theorems, this book aims to make the basic ideas, theorems, and methods of category theory understandable to this broad readership. Although assuming few mathematical pre-requisites, the standard of mathematical rigour is not compromised. The material covered includes the standard core of categories; functors; natural transformations; equivalence; limits and colimits; functor categories; representables; Yoneda''s lemma; adjoints; monads. An extra topic of cartesian closed categories and the lambda-calculus is also provided - a must for computer scientists, logicians and linguists!This Second Edition contains numerous revisions to the original text, including expanding the exposition, revising and elaborating the proofs, providing additional diagrams, correcting typographical errors and, finally, adding an entirely new section on monoidal categories. Nearly a hundred new exercises have also been added, many with solutions, to make the book more useful as a course text and for self-study.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003eThe book is well organised and very well written. The presentation of the material is from the concrete to the abstract, proofs are worked out in detail and the examples and the exercises spread throughout the text mark a pleasant rhythm for its reading. In all, Awodey's Category Theory is a very nice and recommendable introduction to the subject. * Pere Pascual, EMS Newsletter *\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003ePreface ; 1. Categories ; 2. Abstract Structures ; 3. Duality ; 4. Groups and Categories ; 5. Limits and Colimits ; 6. Exponentials ; 7. Naturality ; 8. Categories of Diagrams ; 9. Adjoints ; 10. Monads and Algrebras ; References ; Solutions to Selected Exercises ; Index","brand":"Oxford University Press","offers":[{"title":"Default Title","offer_id":48732837577047,"sku":"9780199237180","price":61.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780199237180.jpg?v=1719998611"},{"product_id":"essential-logic-for-computer-science-the-mit-press-9780262039185","title":"Essential Logic for Computer Science The MIT","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cb\u003eAn introduction to applying predicate logic to testing and verification of software and digital circuits that focuses on applications rather than theory.\u003c\/b\u003e\u003cp\u003eComputer scientists use logic for testing and verification of software and digital circuits, but many computer science students study logic only in the context of traditional mathematics, encountering the subject in a few lectures and a handful of problem sets in a discrete math course. This book offers a more substantive and rigorous approach to logic that focuses on applications in computer science. Topics covered include predicate logic, equation-based software, automated testing and theorem proving, and large-scale computation. \u003c\/p\u003e\u003cp\u003eFormalism is emphasized, and the book employs three formal notations: traditional algebraic formulas of propositional and predicate logic; digital circuit diagrams; and the widely used partially automated theorem prover, ACL2, which provides an accessible introduction to mechanized formalism\u003c\/p\u003e","brand":"MIT Press Ltd","offers":[{"title":"Default Title","offer_id":48733449617751,"sku":"9780262039185","price":43.2,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780262039185.jpg?v=1720000125"},{"product_id":"sheaves-in-geometry-and-logic-9780387977102","title":"Sheaves in Geometry and Logic","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eSheaves also appear in logic as carriers for models of set theory. Beginning with several examples, it explains the underlying ideas of topology and sheaf theory as well as the general theory of elementary toposes and geometric morphisms and their relation to logic.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e\u003cp\u003eFrom the reviews:\u003c\/p\u003e \u003cp\u003e\u003c\/p\u003e \u003cp\u003e\"A beautifully written book, a long and well motivated book packed with well chosen clearly explained examples. … authors have a rare gift for conveying an insider’s view of the subject from the start. This book is written in the best Mac Lane style, very clear and very well organized. … it gives very explicit descriptions of many advanced topics--you can learn a great deal from this book that, before it was published, you could only learn by knowing researchers in the field.\" (Wordtrade, 2008)\u003c\/p\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003ePreface; Prologue; Categorical Preliminaries; 1. Categories of Functors; 2. Sheaves of Sets; 3. Grothendieck Topologies and Sheaves; 4. First Properties of Elementary Topoi; 5. Basic Constructions of Topoi; 6. Topoi and Logic; 7. Geometric Morphisms; 8. Classifying Topoi; 9. Localic Topoi; 10. Geometric Logic and Classifying Topoi; Appendix: Sites for Topoi; Epilogue; Bibliography; Index of Notations; Index","brand":"Springer-Verlag New York Inc.","offers":[{"title":"Default Title","offer_id":48733727916375,"sku":"9780387977102","price":61.74,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780387977102.jpg?v=1720001409"},{"product_id":"foundations-and-fundamental-concepts-of-mathematics-dover-books-on-mathematics-9780486696096","title":"Foundations and Fundamental Concepts of","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eThird edition of popular undergraduate-level text offers historic overview, readable treatment of mathematics before Euclid, Euclid's Elements, non-Euclidean geometry, algebraic structure, formal axiomatics, sets, more. Problems, some with solutions. Bibliography.","brand":"Dover Publications Inc.","offers":[{"title":"Default Title","offer_id":48733800595799,"sku":"9780486696096","price":13.04,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780486696096.jpg?v=1720001754"},{"product_id":"geometry-9781071602973","title":"Geometry","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cdiv\u003eThis text is the fifth and final in the series of educational books written by Israel Gelfand with his colleagues for high school students. These books cover the basics of mathematics in a clear and simple format - the style Gelfand was known for internationally. Gelfand prepared these materials so as to be suitable for independent studies, thus allowing students to learn and practice the material at their own pace without a class.\u003c\/div\u003e\u003cdiv\u003e\u003cbr\u003e\u003c\/div\u003e\u003cdiv\u003e\n\u003ci\u003eGeometry\u003c\/i\u003e takes a different approach to presenting basic geometry for high-school students and others new to the subject.  Rather than following the traditional axiomatic method that emphasizes formulae and logical deduction, it focuses on geometric constructions. Illustrations and problems are abundant throughout, and readers are encouraged to draw figures and move them in the plane, allowing them to develop and enhance their geometrical vision, imagination, and creativity. Chapters are structured so that only certai\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e“This book is intended to engage the reader visually, tactilely, and kinesthetically. … It has a good set of material to enliven more traditional geometry instruction. … There are problems and exercises throughout. The exercises are accompanied by solutions.” (MAA Reviews, October 10, 2020)\u003c\/p\u003e\n\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003ePoints and Lines: A Look at Projective Geometry.- Parallel Lines: A Look at Affine Geometry.- Area: A Look at Symplectic Geometry.- Circles: A Look at Euclidean Geometry.\u003c\/div\u003e","brand":"Springer-Verlag New York Inc.","offers":[{"title":"Default Title","offer_id":48738192687447,"sku":"9781071602973","price":33.24,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9781071602973.jpg?v=1723811807"},{"product_id":"the-joy-of-abstraction-9781108477222","title":"The Joy of Abstraction","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eJourney through the world of abstract mathematics into category theory with popular science author Eugenia Cheng. Featuring humanizing examples and demystification of mathematical thought processes, this book is for fans of How to Bake Pi who want to dig deeper into mathematical concepts and build their mathematical background.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e'This book is an educational tour de force that presents mathematical thinking as a right-brained activity. Most 'left brain\/right brain' education-talk is at best a crude metaphor; but by putting the main focus on the process of (mathematical) abstraction, Eugenia Cheng supplies the reader (whatever their 'brain-type') with the mental tools to make that distinction precise and potentially useful. The book takes the reader along in small steps; but make no mistake, this is a major intellectual journey. Starting not with numbers, but everyday experiences, it develops what is regarded as a very advanced branch of abstract mathematics (category theory, though Cheng really uses this as a proxy for mathematical thinking generally). This is not watered-down math; it's the real thing. And it challenges the reader to think-deeply at times. We 'left-brainers' can learn plenty from it too.' Keith Devlin, Stanford University (Emeritus), author of The Joy of Sets\u003cbr\u003e'Eugenia Cheng loves mathematics—not the ordinary sort that most people encounter, but the most abstract sort that she calls 'the mathematics of mathematics.' And in this lovely excursion through her abstract world of Category Theory, she aims to give those who are willing to join her a glimpse of that world. The journey will change how they view mathematics. Cheng is a brilliant writer, with prose that feels like poetry. Her contagious enthusiasm makes her the perfect guide.' John Ewing, President, Math for America\u003cbr\u003e'Eugenia Cheng's singular contribution is in making abstract mathematics relevant to all through her great ingenuity in developing novel connections between logic and life. Her latest book, The Joy of Abstraction, provides a long awaited fully rigorous yet gentle introduction to the 'mathematics of mathematics,' allowing anyone to experience the joy of learning to think categorically.' Emily Riehl, Johns Hopkins University, author of Category Theory in Context\u003cbr\u003e'Archimedes is quoted as having said once: 'Mathematics reveals its secrets only to those who approach it with pure love, for its own beauty.' In this fascinating book, Eugenia Cheng approaches the abstract mathematical area of Category Theory with pure love, to reveal its beauty to anybody interested in learning something about contemporary mathematics.' Mario Livio, astrophysicist, author of The Golden Ratio and Brilliant Blunders\u003cbr\u003e'Eugenia Cheng's latest book will appeal to a remarkably broad and diverse audience, from non-mathematicians who would like to get a sense of what mathematics is really about, to experienced mathematicians who are not category theorists but would like a basic understanding of category theory. Speaking as one of the latter, I found it a real pleasure to be able to read the book without constantly having to stop and puzzle over the details. I have learnt a lot from it already, including what the famous Yoneda lemma is all about, and I look forward to learning more from it in the future.' Sir Timothy Gowers, Collège de France, Fields Medalist, main editor of The Princeton Companion to Mathematics\u003cbr\u003e'At last: a book that makes category theory as simple as it really is. Cheng explains the subject in a clear and friendly way, in detail, not relying on material that only mathematics majors learn. Category theory – indeed, mathematics as a whole – has been waiting for a book like this.' John Baez, University of California, Riverside\u003cbr\u003e'Many people speak derisively of category theory as the most abstract area of mathematics, but Eugenia Cheng succeeds in redeeming the word 'abstract'. This book is loquacious, conversational, and inviting. Reading this book convinced me I could teach category theory as an introductory course, and that is a real marvel, since it is a subject most people leave for experts.' Francis Su, Harvey Mudd College, author of Mathematics for Human Flourishing\u003cbr\u003e'Finally, a book about category theory that doesn't assume you already know category theory! In this inviting but rigorous introduction to what she calls 'the mathematics of mathematics', Eugenia Cheng brings the subject to us with insight, wit, and a point of view. Her story of finding joy-and advantage-in abstraction will inspire you to find it, too.' Patrick Honner, award-winning high school math teacher, columnist for Quanta Magazine, author of Painless Statistics\u003cbr\u003e'This higher category theory is the mathematics of the twenty-first century (at least my corner of it). If you'd like a taste of it, I recommend Dr. Cheng's book. The first half is an accessible and thought-provoking insight into categorical thinking. The second half climbs into the rarified air of theoretic math, but it is worth a read to get a feel for what some parts of modern mathematics look like.' Jonathan Kujawa, 3 Quarks Daily\u003cbr\u003e'… a successful addition to the literature that I am sure students will use in the future and I would be happy to recommend.' Constanze Roitzheim, Mathematische Semesterberichte\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003ePrologue; Part I. Building Up to Categories: 1. Categories: the idea; 2. Abstraction; 3. Patterns; 4. Context; 5. Relationships; 6. Formalism; 7. Equivalence relations; 8. Categories: the definition; Interlude: A Tour of Math: 9. Examples we've already seen, secretly; 10. Ordered sets; 11. Small mathematical structures; 12. Sets and functions; 13. Large worlds of mathematical structures; Part II. Doing Category Theory: 14. Isomorphisms; 15. Monics and epics; 16. Universal properties; 17. Duality; 18. Products and coproducts; 19. Pullbacks and pushouts; 20. Functors; 21. Categories of categories; 22. Natural transformations; 23. Yoneda; 24. Higher dimensions; 25. Epilogue: thinking categorically; Appendices: A. Background on alphabets; B. Background on basic logic; C. Background on set theory; D. Background on topological spaces; Glossary; Further reading; Acknowledgements; Index.","brand":"Cambridge University Press","offers":[{"title":"Default Title","offer_id":48738304360791,"sku":"9781108477222","price":19.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9781108477222.jpg?v=1723811906"},{"product_id":"mathematical-logic-through-python-9781108949477","title":"Mathematical Logic through Python","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eUsing a unique pedagogical approach, this text introduces mathematical logic by guiding students in implementing the underlying logical concepts and mathematical proofs via Python programming. This approach, tailored to the unique intuitions and strengths of the ever-growing population of programming-savvy students, brings mathematical logic into the comfort zone of these students and provides clarity that can only be achieved by a deep hands-on understanding and the satisfaction of having created working code. While the approach is unique, the text follows the same set of topics typically covered in a one-semester undergraduate course, including propositional logic and first-order predicate logic, culminating in a proof of Gödel''s completeness theorem. A sneak peek to Gödel''s incompleteness theorem is also provided. The textbook is accompanied by an extensive collection of programming tasks, code skeletons, and unit tests. Familiarity with proofs and basic proficiency in Python is assumed.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e'The authors transformed the first course in Mathematical Logic – an experience that many students view as daunting and technical – into an inspiring journey that sails playfully yet rigorously from logic's first principles to Gödel's Completeness Theorem. The secret sauce is making progress by writing many little Python programs instead of proving theorems, a hands-on approach that suits computer science students perfectly.' Shimon Schocken, Reichman University\u003cbr\u003e'Mathematical logic is all about expressions and syntactic operations, and many of its best ideas find a natural home in computer science. Gonczarowski and Nisan make the subject come alive by opening it up to computational implementation and exploration.' Jeremy Avigad, Carnegie Mellon University\u003cbr\u003e'Mathematical Logic through Python offers a refreshingly innovative approach that makes it stand out among several excellent books on mathematical logic. By building on readers' experience and intuition through programming, it naturally provides them with a deep understanding of the fundamental concepts of mathematical logic that underly computer science.' Yoram Moses, Technion - Israel Institute of Technology\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003ePreface; Introduction and Overview; Part I. Propositional Logic: 1. Propositional Logic Syntax; 2. Propositional Logic Semantics; 3. Logical Operators; 4. Proof by Deduction; 5. Working with Proofs; 6. The Tautology Theorem and the Completeness of Propositional Logic; Part II. Predicate Logic: 7. Predicate Logic Syntax and Semantics; 8. Getting Rid of Functions and Equality; 9. Deductive Proofs of Predicate Logic Formulas; 10. Working with Predicate Logic Proofs; 11. The Deduction Theorem and Prenex Normal Form; 12. The Completeness Theorem; 13. Sneak Peek at Mathematical Logic II: Godel's Incompleteness Theorem; Cheatsheet Axioms and Axiomatic Inference Rules Used in this Book; Notes; Index.","brand":"Cambridge University Press","offers":[{"title":"Default Title","offer_id":48738346893655,"sku":"9781108949477","price":21.84,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9781108949477.jpg?v=1723811956"},{"product_id":"how-to-expect-the-unexpected-the-science-of-making-predictions-and-the-art-of-knowing-when-not-to-9781529408676","title":"How to Expect the Unexpected: The Science of","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e\u003cb\u003eA Waterstones Best Popular Science Book of 2023\u003c\/b\u003e\u003cbr\u003e\u003cbr\u003e'Delightfully clear and vivid to read...A splendid book! \u003cb\u003ePhilip Pullman\u003c\/b\u003e\u003cbr\u003e\u003cbr\u003e'Absolutely fascinating' \u003cb\u003eJames O'Brien\u003c\/b\u003e\u003cbr\u003e\u003cbr\u003e'An exceptional book - readable, funny and more needed than ever' \u003cb\u003eDr Chris van Tulleken, bestselling author of \u003ci\u003eUltra-Processed People\u003c\/i\u003e\u003c\/b\u003e\u003cbr\u003e\u003cbr\u003eAre you more likely to become a professional footballer if your surname is Ball?\u003cbr\u003e· How can you be one hundred per cent sure you will win a bet?\u003cbr\u003e· Why did so many Pompeiians stay put while Mount Vesuvius was erupting?\u003cbr\u003e· How do you prevent a nuclear war?\u003cbr\u003e\u003cbr\u003eEver since the dawn of human civilisation, we have been trying to make predictions about what's in store for us. We do this on a personal level, so that we can get on with our lives efficiently (should I hang my laundry out to dry, or will it rain?). But we also have to predict on a much larger scale, often for the good of our broader society (how can we spot economic downturns or prevent terrorist attacks?).\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e For just as long, we have been getting it wrong. From religious oracles to weather forecasters, and from politicians to economists, we are subjected to poor predictions all the time. Our job is to separate the good from the bad. Unfortunately, the foibles of our own biology - the biases that ultimately make us human - can let us down when it comes to making rational inferences about the world around us. And that can have disastrous consequences.\u003cbr\u003e\u003cbr\u003e\u003cb\u003e\u003ci\u003eHow to Expect the Unexpected\u003c\/i\u003e\u003c\/b\u003e will teach you how and why predictions go wrong, help you to spot phony forecasts and give you a better chance of getting your own predictions correct.\u003c\/p\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003eA vivid, wide-ranging and delightful guide to the light and the dark side of prediction * Tim Harford, bestselling author of How to Make the World Add Up *\u003cbr\u003eKit Yates presents maths as it should be taught to everyone: accessible, fun, stimulating, and deeply relevant to our lives. Spend some time with this book and you're likely to make better judgements and decisions, to see through the charlatans and snake-oil salespeople - and perhaps even to fool yourself a little less. * Philip Ball, author of the award-winning Critical Mass *\u003cbr\u003eFascinating and fun. From the everyday to global challenges, Kit Yates explores how changing your mind - so often thought to be a weakness - is the best life skill we can all acquire. A brilliant book * Professor Alice Roberts *\u003cbr\u003eYates' writing is a beacon of clarity sorely needed in a complicated and confusing world. How do we overcome our biases, understand coincidences or tackle the unreliability of our intuition? With bountiful familiar examples, he effortlessly overturns so many of our deep-rooted wrong-headed notions gently and persuasively. I'll be quoting from this book * Jim Al-Khalili *\u003cbr\u003eI'm a Yates fan. His style is all-clarity-no-bullshit * Aperiodical *\u003cbr\u003eSeriously good * Caroline Lucas MP *\u003cbr\u003eAbsolutely fascinating * James O'Brien *\u003cbr\u003eAn exceptional book - readable, funny and more needed than ever * Dr Chris van Tulleken, bestselling author of Ultra-Processed People *\u003cbr\u003eYates' writing style imbues the subjects covered with an infectious enthusiasm, artfully dispelling the dry, stuffy perceptions many people have of maths * Physics World *\u003cbr\u003eHOW TO EXPECT THE UNEXPECTED is fascinating and (very much to the point) delightfully clear and vivid to read. Like many people, I like reading about maths without actually knowing how to do it, and part of the pleasure of reading this came from its many examples from everyday life. A splendid book! * Philip Pullman *","brand":"Quercus Publishing","offers":[{"title":"Default Title","offer_id":48740273619287,"sku":"9781529408676","price":18.75,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9781529408676.jpg?v=1720054281"},{"product_id":"numbers-10-things-you-should-know-9781841885636","title":"Numbers: 10 Things You Should Know","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e\u003cb\u003eUncover the language of our universe - numbers - in this wide-ranging whistle-stop tour of the history and majesty of mathematics.\u003c\/b\u003e\u003cbr\u003e\u003cbr\u003eOur world simply wouldn't function if we didn't have numbers. But where do they come from? Why do we cut cake the wrong way? How can there be different sizes of infinity?\u003cbr\u003e\u003cbr\u003eAll these questions and more are answered in this engaging romp through the history of numbers by acclaimed science writer, Colin Stuart. From the mathematicians who have (and haven't) shouted 'Eureka!' to the theories that affect and inform our everyday lives; \u003ci\u003eNumbers\u003c\/i\u003e shows us that maths was never boring - we were just being taught it in the wrong way.\u003cbr\u003e\u003cbr\u003eConsisting of ten bite-sized essays, there's no better guide to this fundamental science.\u003c\/p\u003e","brand":"Orion Publishing Co","offers":[{"title":"Default Title","offer_id":48742074909015,"sku":"9781841885636","price":9.49,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9781841885636.jpg?v=1720059926"},{"product_id":"metamaths-9781843545255","title":"Metamaths","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e\u003cb\u003eOne of the world's greatest mathematicians explains his revolutionary hypothesis about the enigma at the heart of maths: omega. \u003c\/b\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003e'Chaitin comes across as a kind of mathematical Richard Feynman, intuitive and high-spirited, irreverent and plain-spoken.' -- Peter Pesic, \u003ci\u003eTLS\u003c\/i\u003e\u003c\/b\u003e\u003cbr\u003e\u003cbr\u003e\u003ci\u003eMeta Maths\u003c\/i\u003e is Gregory Chaitin's exuberant account of his discovery of 'omega': the infinitely long, exquisitely complex and utterly incalculable representation of randomness and unknowability in mathematics. From Euclid to Gödel to Turing, Chaitin's infectious narrative guides us on a spellbinding journey through the historical advances in maths and science that led to his breakthrough discovery. Once there he takes us further, to the very frontiers of scientific thinking. \u003ci\u003eMeta Maths\u003c\/i\u003e shows that mathematics is as much art form as logic, as much science as pure reasoning, and sheds light on what we can ultimately hope to know about the universe and the very nature of life.\u003c\/p\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e\u003ci\u003eMeta Maths\u003c\/i\u003e is truly idiosyncratic. Informal, chatty and cerebral... it mixes mathematics with Chaitin's outlook on life and philosophy... Great fun. -- Alan Cane * Financial Times *","brand":"Atlantic Books","offers":[{"title":"Default Title","offer_id":48742085460311,"sku":"9781843545255","price":16.19,"currency_code":"GBP","in_stock":false}]},{"product_id":"understanding-proof-explanation-examples-and-solutions-for-a-level-mathematics-and-a-level-further-mathematics-9781911093787","title":"Understanding Proof: Explanation, Examples and","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e","brand":"Tarquin Publications","offers":[{"title":"Default Title","offer_id":48742597951831,"sku":"9781911093787","price":999.99,"currency_code":"GBP","in_stock":false}]},{"product_id":"sequents-and-trees-an-introduction-to-the-theory-and-applications-of-propositional-sequent-calculi-9783030571443","title":"Sequents and Trees: An Introduction to the Theory","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eThis textbook offers a detailed introduction to the methodology and applications of sequent calculi in propositional logic. Unlike other texts concerned with proof theory, emphasis is placed on illustrating how to use sequent calculi to prove a wide range of metatheoretical results.  The presentation is elementary and self-contained, with all technical details both formally stated and also informally explained.  Numerous proofs are worked through to demonstrate methods of proving important results, such as the cut-elimination theorem, completeness, decidability, and interpolation.  Other proofs are presented with portions left as exercises for readers, allowing them to practice techniques of sequent calculus.\u003cbr\u003eAfter a brief introduction to classical propositional logic, the text explores three variants of sequent calculus and their features and applications.  The remaining chapters then show how sequent calculi can be extended, modified, and applied to non-classical logics, including modal, intuitionistic, substructural, and many-valued logics.\u003cbr\u003e\u003ci\u003eSequents and Trees\u003c\/i\u003e is suitable for graduate and advanced undergraduate students in logic taking courses on proof theory and its application to non-classical logics.  It will also be of interest to researchers in computer science and philosophers.\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e“Each chapter of the book is structured in a similar way and contains the basic definitions, facts and necessary discussion regarding the key notions, accompanied with new ideas and a wide reference list, followed by the author's clear and approachable style. This book is self-contained, presenting an extensive survey of the applications and usefulness of cut elimination, and seems to be an extremely interesting source not only for logicians and philosophers, but also for researchers in computer science.” (Branislav Boričić, Mathematical Reviews, May, 2022)\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003eIntroduction.- Analytic Sequent Calculus for CPL.- Gentzen's Sequent Calculus LK.- Purely Logical Sequent Calculus.- Sequent Calculi for Modal Logics.- Alternatives to CPL.- Appendix.\u003cbr\u003e\u003cbr\u003e","brand":"Springer Nature Switzerland AG","offers":[{"title":"Default Title","offer_id":48743040811351,"sku":"9783030571443","price":41.24,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9783030571443.jpg?v=1720063851"},{"product_id":"modelling-puzzles-in-first-order-logic-9783030625467","title":"Modelling Puzzles in First Order Logic","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cp\u003eKeeping students involved and actively learning is challenging. Instructors in computer science are aware of the cognitive value of modelling puzzles and often use logical puzzles as an efficient pedagogical instrument to engage students and develop problem-solving skills. \u003c\/p\u003e  \u003cp\u003eThis unique book is a comprehensive resource that offers teachers and students fun activities to teach and learn logic. It provides new, complete, and running formalisation in Propositional and First Order Logic for over 130 logical puzzles, including Sudoku-like puzzles, zebra-like puzzles, island of truth, lady and tigers, grid puzzles, strange numbers, or self-reference puzzles.\u003c\/p\u003e  Solving puzzles with theorem provers can be an effective cognitive incentive to motivate students to learn logic. They will find a ready-to-use format which illustrates how to model each puzzle, provides running implementations, and explains each solution.\u003cp\u003e\u003c\/p\u003e  \u003cp\u003eThis concise and easy-to-follow textbook is a much-needed support tool for students willing to explore beyond the introductory level of learning logic and lecturers looking for examples to heighten student engagement in their computer science courses.  \u003c\/p\u003e  \u003cp\u003e \u003c\/p\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e“The purpose of this book is to introduce first-order logic (FOL) to newcomers. … The book is a treasure trove of puzzles like this. … All of these are motivated in an approachable, fun way. … the book is a hands-on guide to Prover9 and Mace4 … . It is quite valuable to have so many puzzles in a single book.” (Jesse Adam Alama, Mathematical Reviews, October, 2022)\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003ePreface.- Getting Started with Prover9 and Mace4.- Micro Arithmetic Puzzles.- Strange Numbers.- Practical Puzzles.- Lady and Tigers.- Einstein Puzzles.- Island of Truth.- Love and Marriage.- Grid Puzzles.- Japanese Puzzles.- Russian Puzzles.- Polyomino Puzzles.- Self-reference and Other Puzzles.- Epigraph in Natural Language","brand":"Springer Nature Switzerland AG","offers":[{"title":"Default Title","offer_id":48743042646359,"sku":"9783030625467","price":33.74,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9783030625467.jpg?v=1720063860"},{"product_id":"mathematical-logic-9783030738389","title":"Mathematical Logic","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cp\u003eThis introduction to first-order logic clearly works out the role of first-order logic in the foundations of mathematics, particularly the two basic questions of the range of the axiomatic method and of theorem-proving by machines. It covers several advanced topics not commonly treated in introductory texts, such as Fraïssé's characterization of elementary equivalence, Lindström's theorem on the maximality of first-order logic, and the fundamentals of logic programming.\u003c\/p\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e“This newest edition has been reclassified, fittingly, as a graduate text, and it is admirably suited to that role. … Those who are already well-versed in logic will find this text to be a valuable reference and a strong resource for teaching at the graduate level, while those who are new to the field will come to know not only how mathematical logic is studied but also, perhaps more importantly, why.” (Stephen Walk, MAA Reviews, January 6, 2023)\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003eA.- I Introduction.- II Syntax of First-Order Languages.- III Semantics of First-Order Languages.- IV A Sequent Calculus.- V The Completeness Theorem.- VI The Löwenheim–Skolem and the Compactness Theorem.- VII The Scope of First-Order Logic.- VIII Syntactic Interpretations and Normal Forms.- B.- IX Extensions of First-Order Logic.- X Computability and Its Limitations.- XI Free Models and Logic Programming.- XII An Algebraic Characterization of Elementary Equivalence.- XIII Lindström’s Theorems.- References.- List of Symbols.- Subject Index.","brand":"Springer Nature Switzerland AG","offers":[{"title":"Default Title","offer_id":48743047823703,"sku":"9783030738389","price":999.99,"currency_code":"GBP","in_stock":false}]},{"product_id":"computability-9783030832018","title":"Computability","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cp\u003eThis 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. \u003c\/p\u003e\u003cp\u003eAdvanced 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.\u003c\/p\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e“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)\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003eMathematical 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.","brand":"Springer Nature Switzerland AG","offers":[{"title":"Default Title","offer_id":48743052575063,"sku":"9783030832018","price":71.99,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9783030832018.jpg?v=1720063903"},{"product_id":"computability-9783030832049","title":"Computability","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cp\u003eThis 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. \u003c\/p\u003e\u003cp\u003eAdvanced 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.\u003c\/p\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e“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)\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003eMathematical 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.","brand":"Springer Nature Switzerland AG","offers":[{"title":"Default Title","offer_id":48743052673367,"sku":"9783030832049","price":52.24,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9783030832049.jpg?v=1720063903"},{"product_id":"the-logical-writings-of-karl-popper-9783030949259","title":"The Logical Writings of Karl Popper","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eThis open access book is the first ever collection of Karl Popper's writings on deductive logic.\u003cbr\u003eKarl R. Popper (1902-1994) was one of the most influential philosophers of the 20th century. His philosophy of science (\"falsificationism\") and his social and political philosophy (\"open society\") have been widely discussed way beyond academic philosophy. What is not so well known is that Popper also produced a considerable work on the foundations of deductive logic, most of it published at the end of the 1940s as articles at scattered places. This little-known work deserves to be known better, as it is highly significant for modern proof-theoretic semantics.\u003cbr\u003eThis collection assembles Popper's published writings on deductive logic in a single volume, together with all reviews of these papers. It also contains a large amount of unpublished material from the Popper Archives, including Popper's correspondence related to deductive logic and manuscripts that were (almost) finished, but did not reach the publication stage. All of these items are critically edited with additional comments by the editors. A general introduction puts Popper's work into the context of current discussions on the foundations of logic. This book should be of interest to logicians, philosophers, and anybody concerned with Popper's work.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003e Part I: Articles.- Chapter 1. Introduction to Popper’s Articles on Logic (David Binder, Thomas Piecha, and Peter Schroeder-Heister).- Chapter 2. Are Contradictions Embracing? (1943) (Karl R. Popper).- Chapter 3. Logic without Assumptions (1947) (Karl R. Popper).- Chapter 4. New Foundations for Logic (1947) (Karl R. Popper).- Chapter 5. Functional Logic without Axioms or Primitive Rules of Inference (1947)(Karl R. Popper).- Chapter 6. On the Theory of Deduction, Part I. Derivation and its Generalizations (1948) (Karl R. Popper).- Chapter 7. On the Theory of Deduction, Part II. The Definitions of Classical and Intuitionist Negation (1948) (Karl R. Popper).- Chapter 8. The Trivialization of Mathematical Logic (1949) (Karl R. Popper).- Chapter 9. A Note on Tarski’s Definition of Truth (1955) (Karl R. Popper).-Chapter 10. On a Proposed Solution of the Paradox of the Liar (1955) (Karl R. Popper).- Chapter 11. On Subjunctive Conditionals with Impossible Antecedents (1959) (Karl R. Popper).- Chapter 12. Lejewski’s Axiomatization of My Theory of Deducibility (1974) (Karl R. Popper).- Chapter 13. Reviews of Popper’s Articles on Logic (Wilhelm Ackermann et.al).- Part II: Manuscripts.- Chapter 14. Introduction to Popper’s Manuscripts on Logic (David Binder, Thomas Piecha, and Peter Schroeder-Heister).- Chapter 15. On Systems of Rules of Inference (Karl R. Popper and Paul Bernays).- Chapter 16. A General Theory of Inference (Karl R. Popper).- Chapter 17. On the Logic of Negation (Karl R. Popper).- Chapter 18. A Note on the Classical Conditional (Karl R. Popper).- Part III: Correspondence.- Chapter 19. Introduction to Popper’s Correspondence on Logic (David Binder, Thomas Piecha, and Peter Schroeder-Heister).- Chapter 20. Popper’s Correspondence with Paul Bernays (Karl R. Popper and Paul Bernays).- Chapter 21. Popper’s Correspondence with Luitzen Egbertus Jan Brouwer (Karl R. Popper and Luitzen E. J. Brouwer).- Chapter 22. Popper’s Correspondence with Rudolf Carnap (Karl R. Popper and Rudolf Carnap).- Chapter 23. Popper’s Correspondence with Alonzo Church (Karl R. Popper and Alonzo Church).- Chapter 24. Popper’s Correspondence with Kalman Joseph Cohen (Karl R. Popper and Kalman J. Cohen).- Chapter 25. Popper’s Correspondence with Henry George Forder (Karl R. Popper and Henry George Forder).- Chapter 26. Popper’s Correspondence with Harold Jeffreys (Karl R. Popper and Harold Jeffreys).- Chapter 27. Popper’s Correspondence with Stephen Cole Kleene (Karl R. Popper and Stephen C. Kleene).- Chapter 28. Popper’s Correspondence with William Calvert Kneale (Karl R. Popper and William C. Kneale).- Chapter 29. Popper’s Correspondence with Willard Van Orman Quine (Karl R. Popper and Willard V. O. Quine).- Chapter 30. Popper’s Correspondence with Heinrich Scholz (Karl R. Popper and Heinrich Scholz).- Chapter 31.  Popper’s Correspondence with Peter Schroeder-Heister (Karl R. Popper and Peter Schroeder-Heister).- Concordances.- Bibliography.- Index.","brand":"Springer Nature Switzerland AG","offers":[{"title":"Default Title","offer_id":48743059915095,"sku":"9783030949259","price":44.99,"currency_code":"GBP","in_stock":true}]},{"product_id":"the-real-numbers-an-introduction-to-set-theory-and-analysis-9783319015767","title":"The Real Numbers: An Introduction to Set Theory","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cp\u003eWhile most texts on real analysis are content to assume the real numbers, or to treat them only briefly, this text makes a serious study of the real number system and the issues it brings to light. Analysis needs the real numbers to model the line, and to support the concepts of continuity and measure. But these seemingly simple requirements lead to deep issues of set theory—uncountability, the axiom of choice, and large cardinals. In fact, virtually all the concepts of infinite set theory are needed for a proper understanding of the real numbers, and hence of analysis itself.\u003c\/p\u003e\u003cp\u003eBy focusing on the set-theoretic aspects of analysis, this text makes the best of two worlds: it combines a down-to-earth introduction to set theory with an exposition of the essence of analysis—the study of infinite processes on the real numbers. It is intended for senior undergraduates, but it will also be attractive to graduate students and professional mathematicians who, until now, have been content to \"assume\" the real numbers. Its prerequisites are calculus and basic mathematics.\u003c\/p\u003e\u003cp\u003eMathematical history is woven into the text, explaining how the concepts of real number and infinity developed to meet the needs of analysis from ancient times to the late twentieth century. This rich presentation of history, along with a background of proofs, examples, exercises, and explanatory remarks, will help motivate the reader. The material covered includes classic topics from both set theory and real analysis courses, such as countable and uncountable sets,  countable ordinals, the continuum problem, the Cantor–Schröder–Bernstein theorem, continuous functions, uniform convergence, Zorn's lemma, Borel sets, Baire functions, Lebesgue measure, and Riemann integrable functions.\u003c\/p\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e“This is a book of both analysis and set theory, and the analysis begins at an elementary level with the necessary treatment of completeness of the reals. … the analysis makes it valuable to the serious student, say a senior or first-year graduate student. … Stillwell’s book can work well as a text for the course in foundations, with its good treatment of the cardinals and ordinals. … This enjoyable book makes the connection clear.” (James M. Cargal, The UMAP Journal, Vol. 38 (1), 2017)\u003c\/p\u003e\u003cp\u003e“This book is an interesting introduction to set theory and real analysis embedded in properties of the real numbers. … The 300-plus problems are frequently challenging and will interest both upper-level undergraduate students and readers with a strong mathematical background. … A list of approximately 100 references at the end of the book will help students to further explore the topic. … Summing Up: Recommended. Lower-division undergraduates.” (D. P. Turner, Choice, Vol. 51 (11), August, 2014)\u003c\/p\u003e\u003cp\u003e“This is an informal look at the nature of the real numbers … . There are extensive historical notes about the evolution of real analysis and our understanding of real numbers. … Stillwell has deliberately set out to provide a different sort of construction where you understand what the foundation is supporting and why it is important. I think this is very successful, and his book … is much more informative and enjoyable.” (Allen Stenger, MAA Reviews, February, 2014)\u003c\/p\u003e\u003cp\u003e“This book will be fully appreciated by either professional mathematicians or those students, who already have passed a course in analysis or set theory. … The book contains a quantity of motivation examples, worked examples and exercises, what makes it suitable also for self-study.” (Vladimír Janiš, zbMATH, 2014)\u003c\/p\u003e\u003cp\u003e“The book offers a rigorous foundation of the real number system. It is intended for senior undergraduates who have already studied calculus, but a wide range of readers will find something interesting, new, or instructive in it. … This is an extremely reader-friendly book. It is full of interesting examples, very clear explanations, historical background, applications. Each new idea comes after proper motivation.” (László Imre Szabó, Acta Scientiarum Mathematicarum (Szeged), Vol. 80 (1-2), 2014)\u003c\/p\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003eThe Fundamental Questions.- From Discrete to Continuous.- Infinite Sets.- Functions and Limits.- Open Sets and Continuity.- Ordinals.- The Axiom of Choice.- Borel Sets.- Measure Theory.- Reflections.- Bibliography.- Index.","brand":"Springer International Publishing AG","offers":[{"title":"Default Title","offer_id":48743091241303,"sku":"9783319015767","price":999.99,"currency_code":"GBP","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9783319015767.jpg?v=1720064071"},{"product_id":"number-theory-an-introduction-via-the-density-of-primes-9783319438733","title":"Number Theory: An Introduction via the Density of","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eNow in its second edition, this textbook provides an introduction and overview of number theory based on the density and properties of the prime numbers. This unique approach offers both a firm background in the standard material of number theory, as well as an overview of the entire discipline. All of the essential topics are covered, such as the fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, and the distribution of primes. New in this edition are coverage of \u003ci\u003ep\u003c\/i\u003e-adic numbers, Hensel's lemma, multiple zeta-values, and elliptic curve methods in primality testing.\u003cbr\u003eKey topics and features include:\u003cul\u003e\n\u003cli\u003eA solid introduction to analytic number theory, including full proofs of Dirichlet's Theorem and the Prime Number Theorem\u003cbr\u003e\n\u003c\/li\u003e\n\u003cli\u003eConcise treatment of algebraic number theory, including a complete presentation of primes, prime factorizations in algebraic number fields, and unique factorization of ideals\u003c\/li\u003e\n\u003cli\u003eDiscussion of the AKS algorithm, which shows that primality testing is one of polynomial time, a topic not usually included in such texts\u003cbr\u003e\n\u003c\/li\u003e\n\u003cli\u003eMany interesting ancillary topics, such as primality testing and cryptography, Fermat and Mersenne numbers, and Carmichael numbers\u003c\/li\u003e\n\u003c\/ul\u003eThe user-friendly style, historical context, and wide range of exercises that range from simple to quite difficult (with solutions and hints provided for select exercises) make \u003ci\u003eNumber Theory: An Introduction via the Density of Primes\u003c\/i\u003e ideal for both self-study and classroom use. Intended for upper level undergraduates and beginning graduates, the only prerequisites are a basic knowledge of calculus, multivariable calculus, and some linear algebra. All necessary concepts from abstract algebra and complex analysis are introduced where needed.\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e“In this text, Fine (mathematics, Fairfield Univ.) and Rosenberger (Univ. of Hamburg, Germany) successfully present number theory from the inception of primes to recent developments in algebraic and analytic number theory and cryptography. … Numerous exercises and open problems are provided. The breadth and depth of topics covered are impressive, making this an excellent text for those interested in the field of number theory. Summing Up: Recommended. Upper-division undergraduates and graduate students.” (J. T. Zerger, Choice, Vol. 54 (9), May, 2017)\u003c\/p\u003e“The book is chatty and leisurely, with lots of historical notes and lots of worked examples. The exercises at the end of each chapter are good and there are a reasonable number of them. … a good text for an introductory course … .” (Allen Stenger, MAA Reviews, maa.org, November, 2016)\u003cp\u003e\u003c\/p\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003eIntroduction and Historical Remarks.- Basic Number Theory.- The Infinitude of Primes.- The Density of Primes.- Primality Testing: An Overview.- Primes and Algebraic Number Theory.- The Fields Q_\u003ci\u003ep\u003c\/i\u003e of \u003ci\u003ep\u003c\/i\u003e-adic Numbers: Hensel's Lemma.- References.- Index.","brand":"Birkhauser Verlag AG","offers":[{"title":"Default Title","offer_id":48743095566679,"sku":"9783319438733","price":44.99,"currency_code":"GBP","in_stock":true}]},{"product_id":"don-pigozzi-on-abstract-algebraic-logic-universal-algebra-and-computer-science-9783319747712","title":"Don Pigozzi on Abstract Algebraic Logic,","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cp\u003eThis book celebrates the work of Don Pigozzi on the occasion of his 80th birthday. In addition to articles written by leading specialists and his disciples, it presents Pigozzi’s scientific output and discusses his impact on the development of science. The book both catalogues his works and offers an extensive profile of Pigozzi as a person, sketching the most important events, not only related to his scientific activity, but also from his personal life.\u003c\/p\u003e  \u003cp\u003eIt reflects Pigozzi's contribution to the rise and development of areas such as abstract algebraic logic (AAL), universal algebra and computer science, and introduces new scientific results. Some of the papers also present chronologically ordered facts relating to the development of the disciplines he contributed to, especially abstract algebraic logic. The book offers valuable source material for historians of science, especially those interested in history of mathematics and logic.\u003c\/p\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003eA Mathematical Life; Pigozzi, Don.- Assertional logics, truth-equational logics, and the hierarchiesof abstract algebraic logic; Albuquerque, Hugo, Font, Josep Maria, Jansana, Ramon, and Moraschini, Tommaso.- Deduction-Detachment Theorem and Gentzen-Style Deductive Systems; Babenyshev, Sergey.- Introducing Boolean Semilattices; Bergman, Clifford.- The Equationally-Defined Commutator in Quasivarieties Generated by Two-Element Algebras; Czelakowski, Janusz.- A short overview of Hidden Logic; Ferreirim, Isabel and Martins, Manuel A.- Absorption and directed J´onsson terms; Kazda, Alexandr, Kozik, Marcin, McKenzie, Ralph and Moore, Matthew.- Relatively congruence modular quasivarieties of modules; Kearnes, Keith A. - The computational complexity of deciding whether a finite algebra generates a minimal variety; McNulty, George F.- Characterization of protoalgebraic \u003ci\u003ek\u003c\/i\u003e-deductive systems; Palasinska; Katarzyna.- Diagrammatic duality; Romanowska, Anna B. and Smith, Jonathan D.H.- Boolean product representations of algebras via binary polynomials; Salibra, Antonino, Ledda, Antonio, and Paoli, Francesco.- Paraconsistent constructive logic with strong negation as a contraction-free relevant logic; Spinks, Matthew and Veroff, Robert.- Possible classification of finite-dimensional compact Hausdorfftopological algebras; Taylor, Walter.- Categorical Abstract Algebraic Logic: Compatibility Operators and Correspondence Theorems; Voutsadakis; George.","brand":"Springer International Publishing AG","offers":[{"title":"Default Title","offer_id":48743104282967,"sku":"9783319747712","price":999.99,"currency_code":"GBP","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9783319747712.jpg?v=1720064130"},{"product_id":"the-times-mindgames-number-logic-puzzles-book-1-9780008190309","title":"The Times Mindgames Number  Logic Puzzles Book 1","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e500 number and logic puzzles to test your mental agility with this collection from the MindGames section of The Times, featuring 7 different types of puzzle challenge.The perfect gift for all number and logic puzzle enthusiasts who are looking for a varied challengeThis collection contains the favourites Suko, Brain Trainer, Cell Blocks, Futoshiki, Kakuro, Set Square and KenKen, all from the Times puzzles section.","brand":"HarperCollins Publishers","offers":[{"title":"Default Title","offer_id":48863899353431,"sku":"9780008190309","price":7.59,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780008190309.jpg?v=1722269528"},{"product_id":"the-times-super-fiendish-su-doku-book-6-9780008285487","title":"The Times Super Fiendish Su Doku Book 6","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eChallenge yourself at home with word and number puzzlesThese are previously unpublished quality Su Doku grids from The Times, and help to develop you to take on Extreme Su Doku.The 200 puzzles in this collection of treacherously difficult puzzles will stretch even the most advanced Su Doku enthusiast. You will need to use all of your best solving techniques to get to the end of this testing challenge.The puzzles in the collection are of the highest quality and are perfect for the advanced solver in need of a constant supply of ultra-difficult puzzles.Guaranteed to provide hours of mind-stretching entertainment.","brand":"HarperCollins Publishers","offers":[{"title":"Default Title","offer_id":48863931531607,"sku":"9780008285487","price":7.59,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780008285487.jpg?v=1722269666"},{"product_id":"limitless-mind-learn-lead-and-live-without-barriers-9780008305666","title":"Limitless Mind Learn Lead and Live Without","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eWhen we learn, we change what we believe and how we interact with the world. This changes who we are as people and what we can achieve.Many people grow up being told they are not a maths person' or perhaps not smart'. They come to believe their potential is limited.Now, however, the latest science has revealed that our identities are constantly in flux; when we learn new things, we can change our identities, increase our potential and broaden our capacity to receive new information.Drawing from the latest research, Professor Boaler followed thousands of school students, studied their learning practices and examined the most effective ways to transform pupils from low to high achievers. Throughout her study, Boaler has collaborated with Stanford University neuroscience experts, harnessing their expertise to reinforce her advanced understanding of learning and educational development.In Limitless Mind, Boaler presents original groundbreaking research that proves that limiting beliefs rea","brand":"HarperCollins Publishers","offers":[{"title":"Default Title","offer_id":48863940575575,"sku":"9780008305666","price":13.49,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780008305666.jpg?v=1722269698"},{"product_id":"the-times-killer-su-doku-book-16-9780008342913","title":"The Times Killer Su Doku Book 16","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eChallenge yourself at home with word and number puzzlesThe latest volume in the hugely popular Killer Su Doku series from The Times, featuring the highest-quality puzzles with an extra element of arithmetic.This addition to the successful Times Killer Su Doku series will test your skills to the limit, adding the challenge of arithmetic and taking Su Doku to a new and even deadlier level of difficulty.The puzzles use the same 9x9 grid as Su Doku but with an added mathematical challenge. The aim is not only to complete every row, column and cube so that it contains the numbers 1-9, it is also necessary to ensure that the outlined cubes add up to the same number as well.With 200 new Moderate, Tricky, Tough and Deadly Killer Su Doku puzzles, there is no chance to ease yourself in with simple puzzles. For those who like to live dangerously and pushbeyond their mental comfort zone, steel yourself for The Times'' next, terribly tough instalment.","brand":"HarperCollins Publishers","offers":[{"title":"Default Title","offer_id":48863954960727,"sku":"9780008342913","price":6.99,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780008342913.jpg?v=1722269756"},{"product_id":"the-times-killer-su-doku-book-17-9780008404338","title":"The Times Killer Su Doku Book 17","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eThe latest volume in the hugely popular Killer Su Doku series from The Times, featuring the highest-quality puzzles with an extra element of arithmetic.This addition to the successful Times Killer Su Doku series will test your skills to the limit, adding the challenge of arithmetic and taking Su Doku to a new and even deadlier level of difficulty.The puzzles use the same 9x9 grid as Su Doku but with an added mathematical challenge. The aim is not only to complete every row, column and cube so that it contains the numbers 1-9, it is also necessary to ensure that the outlined cubes add up to the same number as well.With 200 new Moderate, Tricky, Tough and Deadly Killer Su Doku puzzles, there is no chance to ease yourself in with simple puzzles. For those who like to live dangerously and push beyond their mental comfort zone, steel yourself for The Times'' next, terribly tough instalment.","brand":"HarperCollins Publishers","offers":[{"title":"Default Title","offer_id":48863971017047,"sku":"9780008404338","price":6.99,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780008404338.jpg?v=1722269823"},{"product_id":"on-formally-undecidable-propositions-of-principia-mathematica-and-related-systems-dover-books-on-mathematics-9780486669809","title":"On Formally Undecidable Propositions of Principia","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eFirst English translation of revolutionary paper (1931) that established that even in elementary parts of arithmetic, there are propositions which cannot be proved or disproved within the system. Introduction by R. B. Braithwaite.","brand":"Dover Publications Inc.","offers":[{"title":"Default Title","offer_id":48864740376919,"sku":"9780486669809","price":9.49,"currency_code":"GBP","in_stock":true}]},{"product_id":"agentzero-9780691158884","title":"AgentZero","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eIntroduces 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.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e\"\u003ci\u003eAgent Zero\u003c\/i\u003e 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.\"\u003cb\u003e—George Akerlof, Nobel Laureate in Economics\u003c\/b\u003e\u003cbr\u003e\"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.\"\u003cb\u003e—Paul Slovic, University of Oregon\u003c\/b\u003e\u003cbr\u003e\"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.\"\u003cb\u003e—Peyton Young, University of Oxford\u003c\/b\u003e\u003cbr\u003e\"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: \u003ci\u003eAgent_Zero\u003c\/i\u003e, a mathematically and computationally tractable agent whose inner workings are grounded in neuroscience. Much like you and me, \u003ci\u003eAgent_Zero\u003c\/i\u003e 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. \u003ci\u003eAgent_Zero\u003c\/i\u003e is a major advance.\"\u003cb\u003e—Scott Page, University of Michigan\u003c\/b\u003e\u003cbr\u003e\"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.\"\u003cb\u003e—W. Brian Arthur, Santa Fe Institute\u003c\/b\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003eForeword 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","brand":"Princeton University Press","offers":[{"title":"Default Title","offer_id":48865529135447,"sku":"9780691158884","price":46.75,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780691158884.jpg?v=1722274407"},{"product_id":"an-introduction-to-benfords-law-9780691163062","title":"An Introduction to Benfords Law","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e\u003c\/p\u003e\u003cp\u003eThis 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'\u003c\/p\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e\"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 Reviews\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003ePreface 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","brand":"Princeton University Press","offers":[{"title":"Default Title","offer_id":48865530839383,"sku":"9780691163062","price":67.5,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780691163062.jpg?v=1722274416"},{"product_id":"code-warriors-nsas-codebreakers-and-the-secret-intelligence-war-against-the-soviet-union-9780804170970","title":"Code Warriors Nsas Codebreakers and the Secret","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cb\u003eA sweeping, in-depth history of NSA, whose famous “cult of silence” has left the agency shrouded in mystery for decades \u003c\/b\u003e\u003cbr\u003e  \u003cbr\u003e The National Security Agency was born out of the legendary codebreaking programs of World War II that cracked the famed Enigma machine and other German and Japanese codes, thereby turning the tide of Allied victory. In the postwar years, as the United States developed a new enemy in the Soviet Union, our intelligence community found itself targeting not soldiers on the battlefield, but suspected spies, foreign leaders, and even American citizens. Throughout the second half of the twentieth century, NSA played a vital, often fraught and controversial role in the major events of the Cold War, from the Korean War to the Cuban Missile Crisis to Vietnam and beyond. \u003cbr\u003e\u003cbr\u003eIn \u003ci\u003eCode Warriors,\u003c\/i\u003e Stephen Budiansky—a longtime expert in cryptology—tells the fascinating story of how NSA came to be, from its roots in World War II","brand":"Random House USA Inc","offers":[{"title":"Default Title","offer_id":48865938899287,"sku":"9780804170970","price":17.1,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780804170970.jpg?v=1722276285"},{"product_id":"theres-something-about-godel-9781405197670","title":"Theres Something About Godel","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e\u003cb\u003eBerto''s highly readable and lucid guide introduces students and the interested reader to Gödel''s celebrated \u003ci\u003eIncompleteness Theorem\u003c\/i\u003e, and discusses some of the most famous - and infamous - claims arising from Gödel''s arguments.\u003c\/b\u003e\u003c\/p\u003e \u003cul\u003e \u003cli\u003eOffers a clear understanding of this difficult subject by presenting each of the key steps of the \u003ci\u003eTheorem\u003c\/i\u003e in separate chapters\u003c\/li\u003e \u003cli\u003eDiscusses interpretations of the \u003ci\u003eTheorem\u003c\/i\u003e made by celebrated contemporary thinkers\u003c\/li\u003e \u003cli\u003eSheds light on the wider extra-mathematical and philosophical implications of Gödel''s theories\u003c\/li\u003e \u003cli\u003eWritten in an accessible, non-technical style\u003c\/li\u003e \u003c\/ul\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e\"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)\u003c\/p\u003e \u003cp\u003e\"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)\u003c\/p\u003e \u003cp\u003e\"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.” (\u003ci\u003eTimes Higher Education Supplement\u003c\/i\u003e, February 2010)\u003c\/p\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003ePrologue.  \u003cp\u003eAcknowledgments.\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart I: The Gödelian Symphony.\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Foundations and Paradoxes.\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 \"This sentence is false\".\u003c\/p\u003e \u003cp\u003e2 The Liar and Gödel.\u003c\/p\u003e \u003cp\u003e3 Language and metalanguage.\u003c\/p\u003e \u003cp\u003e4 The axiomatic method, or how to get the non-obvious out of the obvious.\u003c\/p\u003e \u003cp\u003e5 Peano's axioms … .\u003c\/p\u003e \u003cp\u003e6 … and the unsatisfied logicists, Frege and Russell.\u003c\/p\u003e \u003cp\u003e7 Bits of set theory.\u003c\/p\u003e \u003cp\u003e8 The Abstraction Principle.\u003c\/p\u003e \u003cp\u003e9 Bytes of set theory.\u003c\/p\u003e \u003cp\u003e10 Properties, relations, functions, that is, sets again.\u003c\/p\u003e \u003cp\u003e11 Calculating, computing, enumerating, that is, the notion of algorithm.\u003c\/p\u003e \u003cp\u003e12 Taking numbers as sets of sets.\u003c\/p\u003e \u003cp\u003e13 It's raining paradoxes.\u003c\/p\u003e \u003cp\u003e14 Cantor's diagonal argument.\u003c\/p\u003e \u003cp\u003e15 Self-reference and paradoxes.\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Hilbert.\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Strings of symbols.\u003c\/p\u003e \u003cp\u003e2 \"… in mathematics there is no \u003ci\u003eignorabimus\u003c\/i\u003e\".\u003c\/p\u003e \u003cp\u003e3 Gödel on stage.\u003c\/p\u003e \u003cp\u003e4 Our first encounter with the Incompleteness Theorem … .\u003c\/p\u003e \u003cp\u003e5 … and some provisos.\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Gödelization, or Say It with Numbers!\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 \u003cb\u003eTNT.\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2 The arithmetical axioms of \u003cb\u003eTNT\u003c\/b\u003e and the \"standard model\" N.\u003c\/p\u003e \u003cp\u003e3 The Fundamental Property of formal systems.\u003c\/p\u003e \u003cp\u003e4 The Gödel numbering … .\u003c\/p\u003e \u003cp\u003e5 … and the arithmetization of syntax.\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Bits of Recursive Arithmetic … .\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Making algorithms precise.\u003c\/p\u003e \u003cp\u003e2 Bits of recursion theory.\u003c\/p\u003e \u003cp\u003e3 Church's Thesis.\u003c\/p\u003e \u003cp\u003e4 The recursiveness of predicates, sets, properties, and relations.\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 … And How It Is Represented in Typographical Number Theory.\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Introspection and representation.\u003c\/p\u003e \u003cp\u003e2 The representability of properties, relations, and functions … .\u003c\/p\u003e \u003cp\u003e3 … and the Gödelian loop.\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 \"I Am Not Provable\".\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Proof pairs.\u003c\/p\u003e \u003cp\u003e2 The property of being a theorem of \u003cb\u003eTNT\u003c\/b\u003e (is not recursive!)\u003c\/p\u003e \u003cp\u003e3 Arithmetizing substitution.\u003c\/p\u003e \u003cp\u003e4 How can a \u003cb\u003eTNT\u003c\/b\u003e sentence refer to itself?\u003c\/p\u003e \u003cp\u003e5 γ\u003c\/p\u003e \u003cp\u003e6 Fixed point.\u003c\/p\u003e \u003cp\u003e7 Consistency and omega-consistency.\u003c\/p\u003e \u003cp\u003e8 Proving G1.\u003c\/p\u003e \u003cp\u003e9 Rosser's proof.\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 The Unprovability of Consistency and the \"Immediate Consequences\" of G1 and G2.\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 G2.\u003c\/p\u003e \u003cp\u003e2 Technical interlude.\u003c\/p\u003e \u003cp\u003e3 \"Immediate consequences\" of G1 and G2.\u003c\/p\u003e \u003cp\u003e4 Undecidable1 and undecidable2.\u003c\/p\u003e \u003cp\u003e5 Essential incompleteness, or the syndicate of mathematicians.\u003c\/p\u003e \u003cp\u003e6 Robinson Arithmetic.\u003c\/p\u003e \u003cp\u003e7 How general are Gödel's results?\u003c\/p\u003e \u003cp\u003e8 Bits of Turing machine.\u003c\/p\u003e \u003cp\u003e9 G1 and G2 in general.\u003c\/p\u003e \u003cp\u003e10 Unexpected fish in the formal net.\u003c\/p\u003e \u003cp\u003e11 Supernatural numbers.\u003c\/p\u003e \u003cp\u003e12 The culpability of the induction scheme.\u003c\/p\u003e \u003cp\u003e13 Bits of truth (not too much of it, though).\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart II: The World after Gödel.\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Bourgeois Mathematicians! The Postmodern Interpretations.\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 What is postmodernism?\u003c\/p\u003e \u003cp\u003e2 From Gödel to Lenin.\u003c\/p\u003e \u003cp\u003e3 Is \"Biblical proof\" decidable?\u003c\/p\u003e \u003cp\u003e4 Speaking of the totality.\u003c\/p\u003e \u003cp\u003e5 Bourgeois teachers!\u003c\/p\u003e \u003cp\u003e6 (Un)interesting bifurcations.\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 A Footnote to Plato.\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Explorers in the realm of numbers.\u003c\/p\u003e \u003cp\u003e2 The essence of a life.\u003c\/p\u003e \u003cp\u003e3 \"The philosophical prejudices of our times\".\u003c\/p\u003e \u003cp\u003e4 From Gödel to Tarski.\u003c\/p\u003e \u003cp\u003e5 Human, too human.\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Mathematical Faith.\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 \"I'm not crazy!\"\u003c\/p\u003e \u003cp\u003e2 Qualified doubts.\u003c\/p\u003e \u003cp\u003e3 From Gentzen to the \u003ci\u003eDialectica\u003c\/i\u003e interpretation.\u003c\/p\u003e \u003cp\u003e4 Mathematicians are people of faith.\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Mind versus Computer: Gödel and Artificial Intelligence.\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Is mind (just) a program?\u003c\/p\u003e \u003cp\u003e2 \"Seeing the truth\" and \"going outside the system\".\u003c\/p\u003e \u003cp\u003e3 The basic mistake.\u003c\/p\u003e \u003cp\u003e4 In the haze of the transfinite.\u003c\/p\u003e \u003cp\u003e5 \"Know thyself\": Socrates and the inexhaustibility of mathematics.\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 Gödel versus Wittgenstein and the Paraconsistent Interpretation.\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 When geniuses meet … .\u003c\/p\u003e \u003cp\u003e2 The implausible Wittgenstein.\u003c\/p\u003e \u003cp\u003e3 \"There is no metamathematics\".\u003c\/p\u003e \u003cp\u003e4 Proof and prose.\u003c\/p\u003e \u003cp\u003e5 The single argument.\u003c\/p\u003e \u003cp\u003e6 But how can arithmetic be inconsistent?\u003c\/p\u003e \u003cp\u003e7 The costs and benefits of making Wittgenstein plausible.\u003c\/p\u003e \u003cp\u003eEpilogue.\u003c\/p\u003e \u003cp\u003eReferences.\u003c\/p\u003e \u003cp\u003eIndex.\u003c\/p\u003e","brand":"John Wiley and Sons Ltd","offers":[{"title":"Default Title","offer_id":48866718089559,"sku":"9781405197670","price":24.65,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9781405197670.jpg?v=1722279879"},{"product_id":"how-to-expect-the-unexpected-the-science-of-making-predictions-and-the-art-of-knowing-when-not-to-9781529408683","title":"How to Expect the Unexpected: The Science of","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e\u003cb\u003eA Waterstones Best Popular Science Book of 2023\u003c\/b\u003e\u003cbr\u003e\u003cbr\u003e'Delightfully clear and vivid to read...A splendid book! \u003cb\u003ePhilip Pullman\u003c\/b\u003e\u003cbr\u003e\u003cbr\u003e'Absolutely fascinating' \u003cb\u003eJames O'Brien\u003c\/b\u003e\u003cbr\u003e\u003cbr\u003e'An exceptional book - readable, funny and more needed than ever' \u003cb\u003eDr Chris van Tulleken, bestselling author of \u003ci\u003eUltra-Processed People\u003c\/i\u003e\u003c\/b\u003e\u003cbr\u003e\u003cbr\u003eAre you more likely to become a professional footballer if your surname is Ball?\u003cbr\u003e· How can you be one hundred per cent sure you will win a bet?\u003cbr\u003e· Why did so many Pompeiians stay put while Mount Vesuvius was erupting?\u003cbr\u003e· How do you prevent a nuclear war?\u003cbr\u003e\u003cbr\u003eEver since the dawn of human civilisation, we have been trying to make predictions about what's in store for us. We do this on a personal level, so that we can get on with our lives efficiently (should I hang my laundry out to dry, or will it rain?). But we also have to predict on a much larger scale, often for the good of our broader society (how can we spot economic downturns or prevent terrorist attacks?).\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e For just as long, we have been getting it wrong. From religious oracles to weather forecasters, and from politicians to economists, we are subjected to poor predictions all the time. Our job is to separate the good from the bad. Unfortunately, the foibles of our own biology - the biases that ultimately make us human - can let us down when it comes to making rational inferences about the world around us. And that can have disastrous consequences.\u003cbr\u003e\u003cbr\u003e\u003cb\u003e\u003ci\u003eHow to Expect the Unexpected\u003c\/i\u003e\u003c\/b\u003e will teach you how and why predictions go wrong, help you to spot phony forecasts and give you a better chance of getting your own predictions correct.\u003c\/p\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003eA vivid, wide-ranging and delightful guide to the light and the dark side of prediction * Tim Harford, bestselling author of How to Make the World Add Up *\u003cbr\u003eKit Yates presents maths as it should be taught to everyone: accessible, fun, stimulating, and deeply relevant to our lives. Spend some time with this book and you're likely to make better judgements and decisions, to see through the charlatans and snake-oil salespeople - and perhaps even to fool yourself a little less. * Philip Ball, author of the award-winning Critical Mass *\u003cbr\u003eFascinating and fun. From the everyday to global challenges, Kit Yates explores how changing your mind - so often thought to be a weakness - is the best life skill we can all acquire. A brilliant book * Professor Alice Roberts *\u003cbr\u003eYates' writing is a beacon of clarity sorely needed in a complicated and confusing world. How do we overcome our biases, understand coincidences or tackle the unreliability of our intuition? With bountiful familiar examples, he effortlessly overturns so many of our deep-rooted wrong-headed notions gently and persuasively. I'll be quoting from this book * Jim Al-Khalili *\u003cbr\u003eI'm a Yates fan. His style is all-clarity-no-bullshit * Aperiodical *\u003cbr\u003eSeriously good * Caroline Lucas MP *\u003cbr\u003eAbsolutely fascinating * James O'Brien *\u003cbr\u003eAn exceptional book - readable, funny and more needed than ever * Dr Chris van Tulleken, bestselling author of Ultra-Processed People *\u003cbr\u003eYates' writing style imbues the subjects covered with an infectious enthusiasm, artfully dispelling the dry, stuffy perceptions many people have of maths * Physics World *\u003cbr\u003eHOW TO EXPECT THE UNEXPECTED is fascinating and (very much to the point) delightfully clear and vivid to read. Like many people, I like reading about maths without actually knowing how to do it, and part of the pleasure of reading this came from its many examples from everyday life. A splendid book! * Philip Pullman *","brand":"Quercus Publishing","offers":[{"title":"Default Title","offer_id":48867512254807,"sku":"9781529408683","price":999.99,"currency_code":"GBP","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9781529408683.jpg?v=1722283636"},{"product_id":"combinators-a-centennial-view-9781579550431","title":"Combinators: A Centennial View","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e","brand":"Wolfram Media Inc","offers":[{"title":"Default Title","offer_id":48867617603927,"sku":"9781579550431","price":21.21,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9781579550431.jpg?v=1722284148"},{"product_id":"can-you-solve-my-problems-a-casebook-of-ingenious-perplexing-and-totally-satisfying-puzzles-9781783351152","title":"Can You Solve My Problems?: A casebook of","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e\u003ci\u003eAre you smarter than a Singaporean ten-year-old?\u003c\/i\u003e\u003cbr\u003e\u003ci\u003eCan you beat Sherlock Holmes?\u003c\/i\u003e\u003ci\u003e\u003cbr\u003eIf you think the answer is yes - I challenge you to solve my problems. \u003c\/i\u003e\u003cbr\u003e\u003cbr\u003eHere are 125 of the world's best brainteasers from the last two millennia, taking us from ancient China to medieval Europe, Victorian England to modern-day Japan, with stories of espionage, mathematical breakthroughs and puzzling rivalries along the way.\u003cbr\u003e\u003cbr\u003ePit your wits against logic puzzles and kinship riddles, pangrams and river-crossing conundrums. Some solutions rely on a touch of cunning, others call for creativity, others need mercilessly logical thought. Some can only be solved be 2 per cent of the population. All are guaranteed to sharpen your mind. Let's get puzzling!\u003c\/p\u003e","brand":"Guardian Faber Publishing","offers":[{"title":"Default Title","offer_id":48868279026007,"sku":"9781783351152","price":10.44,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9781783351152.jpg?v=1722287263"},{"product_id":"the-art-of-logic-how-to-make-sense-in-a-world-that-doesnt-9781788160391","title":"The Art of Logic: How to Make Sense in a World","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eFor thousands of years, mathematicians have used the timeless art of logic to see the world more clearly. In The Art of Logic, Royal Society Science Book Prize nominee Eugenia Cheng shows how anyone can think like a mathematician - and see, argue and think better.  Learn how to simplify complex decisions without over-simplifying them. Discover the power of analogies and the dangers of false equivalences. Find out how people construct misleading arguments, and how we can argue back.  Eugenia Cheng teaches us how to find clarity without losing nuance, taking a careful scalpel to the complexities of politics, privilege, sexism and dozens of other real-world situations. Her Art of Logic is a practical and inspiring guide to decoding the modern world.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003eMind-expanding ... a meaningful contribution to creating a better society as well as happier conversations and relationships * Guardian *\u003cbr\u003eA mathematician's thought-provoking attempt to lay out the tools of rational argument -- Michael Brooks * New Statesman Books of the Year *\u003cbr\u003eWith humour, grace, and a natural gift for making explanations seem fun, Eugenia Cheng has done it again. This is a book to savour, to consult, and to buy for all your friends. You'll think more clearly after reading this book, something that is unfortunately in short supply these days. I am buying several copies to send to heads of state. -- Daniel Levitin, bestselling author of The Organised Mind \u0026amp; A Field Guide to Lies and Statistics\u003cbr\u003eIn an era awash with conflict, exploitation, tribalism and fake news, the \"illuminating precision\" offered by logic is important. Cheng harnesses the power of abstraction to explore real-life phenomena such as sexism and white privilege. She walks us through the grand terrain of logic, from axioms to proofs. And she reveals how to build arguments as long chains of logical implications - a \"virtuosic and masterful\" skill that, combined with intelligent emotional engagement, can cut through pervasive irrationality * Nature *\u003cbr\u003eA perceptive analysis of logic and its limitations ... Cheng is successful not only in helping readers think more clearly, but in helping them understand why others sometimes appear to be illogical. This book has the potential to help understanding and avoid confrontational arguments that serve only to entrench opposing views * Times Higher Education *\u003cbr\u003eRadical and liberating * Emerald Street *\u003cbr\u003eWe're thankful that someone like Eugenia Cheng is here; someone to eloquently and efficiently expound on concepts like logic and truth at a time when their very basis seems to come under attack ... We're forever on the lookout for someone to make mathematics both fun and accessible, and it looks like we've found that person in Eugenia Cheng * How it Works Magazine *\u003cbr\u003eA concert pianist, mathematician, polyglot and YouTube star, Cheng has carved out quite a niche for herself ... she brings an ebullient enthusiasm that's infectious * Guardian *\u003cbr\u003eWitty, charming, and crystal clear. Eugenia Cheng's enthusiasm and carefully chosen metaphors and analogies carry us effortlessly through the mathematical landscape -- Ian Stewart\u003cbr\u003eClear, clever and friendly -- Alex Bellos","brand":"Profile Books Ltd","offers":[{"title":"Default Title","offer_id":48868430086487,"sku":"9781788160391","price":10.44,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9781788160391.jpg?v=1722287993"},{"product_id":"who-killed-professor-x-9783034808835","title":"Who Killed Professor X?","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cp\u003eThis graphic novel is both a historical novel as well as an entertaining way of using mathematics to solve a crime. The plot, the possible motive of every suspect, and the elements of his or her character are based on actual historical figures.\u003c\/p\u003e\u003cp\u003eThe 2nd International Congress of Mathematicians is being held in Paris in 1900. The main speaker, the renowned Professor X, is found dead in the hotel dining room. Foul play is suspected. The greatest mathematicians of all time (who are attending the Congress) are called in for questioning. Their statements to the police, however, take the form of mathematical problems. The Chief Inspector enlists the aid of a young mathematician to help solve the crime. Do numbers always tell the truth? Or don’t they?\u003c\/p\u003e\u003cp\u003e\u003c\/p\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e“It is a detective story in which several of the greatest historic mathematicians become all suspects for a murder on a colleague. … This is a wonderful booklet of fiction, but based on historical incidents. … It is a fantastic present that you can give to anybody between 9 and 99.” (Adhemar Bultheel, euro-math-soc.eu, June, 2015)\u003c\/p\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003eThe Crime.- The Suspects: Mathematicians.- Credits.- Examination of the Statements.","brand":"Birkhauser Verlag AG","offers":[{"title":"Default Title","offer_id":48869363777879,"sku":"9783034808835","price":14.25,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9783034808835.jpg?v=1722292387"},{"product_id":"an-introduction-to-fuzzy-sets-9781536180121","title":"An Introduction to Fuzzy Sets","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eAn Introduction to Fuzzy Sets provides a comparison of the quality of life in urban, intermediate and rural NUTS III regions in Portugal, with the main goal of identifying and analysing the necessary and conditions for a high quality of life in those different regions. The authors assess the necessary and sufficient conditions for higher Human Development Index levels, aiming to determine whether the same pattern could be used to explain the happiness index. In order to represent the applications of fuzzy set theory as well as neuro-fuzzy in industry, a literature review of these topics is carried out. As some researchers have efficiently utilized fuzzy logic and neuro-fuzzy, in-depth discussions are provided for stimulating future investigations. Following this, using the L. Zadeh theory of fuzzy sets, the authors consider all types of uncertainties in oil fields and oil production to make a decision as to what model is best in such a fuzzy environment. Additionally, several challenges are explored, such as: the fuzzy random finite difference numerical method, possibilistic uncertainty modelling, and the development of a fuzzy Wilks' theorem to model the hybrid structure of randomness and fuzziness modelling. In closing, a standard fuzzy arithmetic method is used for solving fuzzy equations, as well as for the optimization of fuzzy objectives. The fuzzy variable of the equation is fuzzified using a fuzzy set.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003ePreface; Quality of Life: Urban versus Rural Analysis Based on Fuzzy Sets Approach; Economically Speaking, Are Happiness and HDI the Same? The Fuzzy-Set Approach; Implementation of Fuzzy Logic and Neuro-Fuzzy in Industry; Lotfi Zadehs Theory of Fuzzy Sets in Decision-Making Process for Oil and Gas Production; Fuzziness-Randomness Modeling of Plasma Disruption in First Wall of Fusion Reactor Using Type I Fuzzy Random Set; Application of a Standard Fuzzy Arithmetic Method; Index.","brand":"Nova Science Publishers Inc","offers":[{"title":"Default Title","offer_id":48886173958487,"sku":"9781536180121","price":62.04,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9781536180121.jpg?v=1722539086"},{"product_id":"to-mock-a-mockingbird-and-other-logic-puzzles-9780192801425","title":"To Mock a Mockingbird and Other Logic Puzzles","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eIn this entertaining and challenging collection of logic puzzles, Raymond Smullyan-author of Forever Undecided-continues to delight and astonish us with his gift for making available, in the thoroughly pleasurable form of puzzles, some of the most important mathematical thinking of our time.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003ePART I - LOGIC PUZZLES; PART II - KNIGHTS, KNAVES, AND THE FOUNTAIN OF YOUTH; PART III - TO MOCK A MOCKINGBIRD; PART IV - SINGING BIRDS; PART V - THE MASTER FOREST; PART VI - THE GRAND QUESTION","brand":"Oxford University Press","offers":[{"title":"Default Title","offer_id":49083377844567,"sku":"9780192801425","price":12.34,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780192801425.jpg?v=1725548732"},{"product_id":"logical-methods-the-art-of-thinking-abstractly-and-mathematically-9783030637767","title":"Logical Methods: The Art of Thinking Abstractly","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eMany believe mathematics is only about calculations, formulas, numbers, and strange letters. But mathematics is much more than just crunching numbers or manipulating symbols. Mathematics is about discovering patterns, uncovering hidden structures, finding counterexamples, and thinking logically. Mathematics is a way of thinking. It is an activity that is both highly creative and challenging. \u003cp\u003eThis book offers an introduction to mathematical reasoning for beginning university or college students, providing a solid foundation for further study in mathematics, computer science, and related disciplines. Written in a manner that directly conveys the sense of excitement and discovery at the heart of doing science,  its 25 short and visually appealing chapters cover the basics of set theory, logic, proof methods, combinatorics, graph theory, and much more.\u003c\/p\u003e  \u003cp\u003eIn the book you will, among other things, find answers to:\u003c\/p\u003e\u003cp\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eWhat is a proof? What is a counterexample?\u003c\/li\u003e\n\u003cli\u003eWhat does it mean to say that something follows logically from a set of premises?\u003c\/li\u003e\n\u003cli\u003eWhat does it mean to abstract over something?\u003cbr\u003e\n\u003c\/li\u003e\n\u003cli\u003eHow can knowledge and information be represented and used in calculations?\u003c\/li\u003e\n\u003cli\u003eWhat is the connection between Morse code and Fibonacci numbers?\u003c\/li\u003e\n\u003cli\u003eWhy could it take billions of years to solve Hanoi's Tower?\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003e\u003c\/p\u003e  \u003cp\u003e\u003ci\u003eLogical Methods\u003c\/i\u003e is especially appropriate for students encountering such concepts for the very first time. Designed to ease the transition to a university or college level study of mathematics or computer science, it also provides an accessible and fascinating gateway to logical thinking for students of all disciplines.\u003cbr\u003e\u003c\/p\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e\"The definitions are followed by examples to help explain their meaning, along with counterexamples ... . Therefore, very little basic knowledge is required for this introduction to logical methods ... which is written in an accessible style ... . contained in the book are several hundred small figures; arrow, Venn, and Hasse diagrams; and simplifies visual representations ... . The author has also elected to use color to draw the reader's attention ... .\" \u003cbr\u003e“From personal teaching experience, knowledge of these mathematical areas is necessary for disparate fields of CS and informatics. These foundations are needed for many fields, from database theory to various domains of information systems applications. The book’s presentation of topics and incentives for problem-solving, along with its exercises, is very useful for university-level instructors and students. The compact chapters contain clear explanations, diagrams, and brief descriptions of interesting facts.” (Bálint Molnár, Computing Reviews, July 27, 2021)\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003ePreface.- 0 The Art of Thinking Abstractly and Mathematically.- 1 Basic Set Theory.- 2 Propositional Logic.- 3 Semantics from Propositional Logic.- 4 Concepts in Propositional Logic.- 5 Proofs, Conjectures, and Counterexamples.- 6 Relations.- 7 Functions.- 8 A Little More Set Theory.- 9 Closures and Inductively Defined Sets.- 10 Recursively Defined Functions.- 11 Mathematical Induction.- 12 Structural Induction.- 13 First-Order Languages.- 14 Representation of Quantified Statements.- 15 Interpretation in Models.- 16 Reasoning About Models.- 17 Abstraction with Equivalences and Partitions.- 18 Combinatorics.- 19 A Little More Combinatorics.- 20 A Bit of Abstract Algebra.- 21 Graph Theory.- 22 Walks in Graphs.- 23 Formal Languages and Grammars.- 24 Natural Deduction.- The Road Ahead.- Index. Symbols.","brand":"Springer Nature Switzerland AG","offers":[{"title":"Default Title","offer_id":49084751479127,"sku":"9783030637767","price":33.24,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9783030637767.jpg?v=1725553226"},{"product_id":"category-theory-invariances-and-symmetries-in-computer-science-9783111080567","title":"Category Theory: Invariances and Symmetries in","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cp\u003eThis 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. \u003c\/p\u003e","brand":"De Gruyter","offers":[{"title":"Default Title","offer_id":49084764356951,"sku":"9783111080567","price":129.67,"currency_code":"GBP","in_stock":true}]},{"product_id":"why-machines-learn-9780241586488","title":"Why Machines Learn","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e\u003cb\u003e''An invaluable companion for anyone who wants a deep understanding of what's under the hood of often inscrutable machines'' Melanie Mitchell  \u003c\/b\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eA rich, narrative explanation of the mathematics that has brought us machine learning and the ongoing explosion of artificial intelligence\u003c\/b\u003e\u003cbr\u003e\u003cbr\u003eMachine-learning systems are making life-altering decisions for us: approving mortgage loans, determining whether a tumour is cancerous, or deciding whether someone gets bail. They now influence discoveries in chemistry, biology and physics - the study of genomes, extra-solar planets, even the intricacies of quantum systems.\u003cbr\u003e\u003cbr\u003eWe are living through a revolution in artificial intelligence that is not slowing down. This major shift is based on simple mathematics, some of which goes back centuries: linear algebra and calculus, the stuff of eighteenth-century mathematics. Indeed by the mid-1850s, a lot of the groundwork was all done. It took the development of computer science and the kindling of 1990s computer chips designed for video games to ignite the explosion of AI that we see all around us today. In this enlightening book, Anil Ananthaswamy explains the fundamental maths behind AI, which suggests that the basics of natural and artificial intelligence might follow the same \u003ci\u003emathematical \u003c\/i\u003erules.\u003cbr\u003e\u003cbr\u003eAs Ananthaswamy resonantly concludes, to make the most of our most wondrous technologies we need to understand their profound limitations - the clues lie in the maths that makes AI possible.\u003c\/p\u003e","brand":"Penguin Books Ltd","offers":[{"title":"Default Title","offer_id":49400399003991,"sku":"9780241586488","price":24.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780241586488.jpg?v=1730470596"},{"product_id":"the-nuts-and-bolts-of-proofs-9780323990202","title":"The Nuts and Bolts of Proofs","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003e1. Getting Started 2. Basic Techniques to Prove If\/Then Statements 3. Special Kinds of Theorems 4. Some Mathematical Topics on Which to Practice Proof Techniques 5. Review Exercises","brand":"Elsevier Science \u0026 Technology","offers":[{"title":"Default Title","offer_id":49401787777367,"sku":"9780323990202","price":41.36,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780323990202.jpg?v=1730478520"},{"product_id":"fuzzy-expert-system-tools-d3-9780471958673","title":"Fuzzy Expert System Tools D3","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eFuzzy 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.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003eGetting Started.\u003cbr\u003e \u003cbr\u003e Fuzzy Set Theory.\u003cbr\u003e \u003cbr\u003e Possibility\/Probability Consistency Principle.\u003cbr\u003e \u003cbr\u003e Knowledge Representation.\u003cbr\u003e \u003cbr\u003e Imprecision and Fuzzy Logic.\u003cbr\u003e \u003cbr\u003e Knowledge Processing.\u003cbr\u003e \u003cbr\u003e Knowledge in FEST.\u003cbr\u003e \u003cbr\u003e Inference Engine.\u003cbr\u003e \u003cbr\u003e The Fuzzy Inference Engine.\u003cbr\u003e \u003cbr\u003e Fuzzy Inference in FEST.\u003cbr\u003e \u003cbr\u003e References.\u003cbr\u003e \u003cbr\u003e Index.","brand":"John Wiley \u0026 Sons Inc","offers":[{"title":"Default Title","offer_id":49402693615959,"sku":"9780471958673","price":199.76,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780471958673.jpg?v=1730481257"},{"product_id":"an-introduction-to-mathematical-logic-9780486497853","title":"An Introduction to Mathematical Logic","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eComprehensive overview, suitable for advanced undergraduates and graduate students, covers propositional logic; first-order languages and logic; incompleteness, undecidability, and indefinability; recursive functions; computability; and Hilbert's Tenth Problem. 1995 edition.","brand":"Dover Publications Inc.","offers":[{"title":"Default Title","offer_id":49402733592919,"sku":"9780486497853","price":30.39,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780486497853.jpg?v=1730481354"},{"product_id":"deduction-9780631227137","title":"Deduction","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eOffers 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.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e“\u003ci\u003eDeduction\u003c\/i\u003e 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.” \u003ci\u003eRick Benitez, University of Sydney\u003c\/i\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003e\u003cp\u003ePreface to the Second Edition viii\u003c\/p\u003e \u003cp\u003eAcknowledgments x\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Basic Concepts of Logic 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 Arguments 1\u003c\/p\u003e \u003cp\u003e1.2 Validity 16\u003c\/p\u003e \u003cp\u003e1.3 Implication and Equivalence 23\u003c\/p\u003e \u003cp\u003e1.4 Logical Properties of Sentences 27\u003c\/p\u003e \u003cp\u003e1.5 Satisfiability 31\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Sentences 36\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 The Language of Sentential Logic 36\u003c\/p\u003e \u003cp\u003e2.2 Truth Functions 40\u003c\/p\u003e \u003cp\u003e2.3 A Sentential Language 46\u003c\/p\u003e \u003cp\u003e2.4 Symbolization 49\u003c\/p\u003e \u003cp\u003e2.5 Validity 56\u003c\/p\u003e \u003cp\u003e2.6 Truth Tables 60\u003c\/p\u003e \u003cp\u003e2.7 Truth Tables for Formulas 63\u003c\/p\u003e \u003cp\u003e2.8 Truth Tables for Argument Forms 68\u003c\/p\u003e \u003cp\u003e2.9 Implication, Equivalence, and Satisfiability 71\u003c\/p\u003e \u003cp\u003e3 Truth Trees 76\u003c\/p\u003e \u003cp\u003e3.1 Thinking Backwards 76\u003c\/p\u003e \u003cp\u003e3.2 Constructing Truth Trees 80\u003c\/p\u003e \u003cp\u003e3.3 Negation, Conjunction, and Disjunction 84\u003c\/p\u003e \u003cp\u003e3.4 The Conditional and Biconditional 93\u003c\/p\u003e \u003cp\u003e3.5 Other Applications 101\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Natural Deduction 107\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Natural Deduction Systems 107\u003c\/p\u003e \u003cp\u003e4.2 Rules for Negation and Conjunction 110\u003c\/p\u003e \u003cp\u003e4.3 Rules for the Conditional and Biconditional 118\u003c\/p\u003e \u003cp\u003e4.4 Rules for Disjunction 122\u003c\/p\u003e \u003cp\u003e4.5 Derivable Rules 125\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Quantifiers 137\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Constants and Quantifiers 138\u003c\/p\u003e \u003cp\u003e5.2 Categorical Sentence Forms 144\u003c\/p\u003e \u003cp\u003e5.3 Polyadic Predicates 148\u003c\/p\u003e \u003cp\u003e5.4 The Language Q 153\u003c\/p\u003e \u003cp\u003e5.5 Symbolization 156\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Quantified Truth Trees 173\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Rules for Quantifiers 174\u003c\/p\u003e \u003cp\u003e6.2 Strategies 178\u003c\/p\u003e \u003cp\u003e6.3 Interpretations 189\u003c\/p\u003e \u003cp\u003e6.4 Constructing Interpretations from Trees 199\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Quantified Natural Deduction 206\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Deduction Rules for Quantifiers 206\u003c\/p\u003e \u003cp\u003e7.2 Universal Proof 214\u003c\/p\u003e \u003cp\u003e7.3 Derived Rules for Quantifiers 220\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Identity and Function Symbols 225\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Identity 225\u003c\/p\u003e \u003cp\u003e8.2 Truth Tree Rules for Identity 231\u003c\/p\u003e \u003cp\u003e8.3 Deduction Rules for Identity 235\u003c\/p\u003e \u003cp\u003e8.4 Function Symbols 238\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Necessity 249\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 If 249\u003c\/p\u003e \u003cp\u003e9.2 Modal Connectives 251\u003c\/p\u003e \u003cp\u003e9.3 Symbolization 256\u003c\/p\u003e \u003cp\u003e9.4 Modal Truth Trees 261\u003c\/p\u003e \u003cp\u003e9.5 Other Tree Rules 265\u003c\/p\u003e \u003cp\u003e9.6 World Travelling 268\u003c\/p\u003e \u003cp\u003e9.7 Modal Deduction 278\u003c\/p\u003e \u003cp\u003e9.8 Other Modal Systems 289\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Between Truth and Falsehood 295\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Vagueness and Presupposition 295\u003c\/p\u003e \u003cp\u003e10.2 Many-Valued Truth Tables 300\u003c\/p\u003e \u003cp\u003e10.3 Many-Valued Trees 314\u003c\/p\u003e \u003cp\u003e10.4 Many-Valued Deduction 325\u003c\/p\u003e \u003cp\u003e10.5 Fuzzy Logic 332\u003c\/p\u003e \u003cp\u003e10.6 Intuitionistic Logic 344\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Obligation 361\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1 Deontic Connectives 362\u003c\/p\u003e \u003cp\u003e11.2 Deontic Truth Trees 370\u003c\/p\u003e \u003cp\u003e11.3 Deontic Deduction 381\u003c\/p\u003e \u003cp\u003e11.4 Moral and Practical Reasoning 387\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 Counterfactuals 395\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e12.1 The Meaning of Counterfactuals 399\u003c\/p\u003e \u003cp\u003e12.2 Truth Tree Rules for Counterfactuals 402\u003c\/p\u003e \u003cp\u003e12.3 Deduction Rules for Counterfactuals 409\u003c\/p\u003e \u003cp\u003e12.4 Stalnaker’s Semantics: System CS 418\u003c\/p\u003e \u003cp\u003e12.5 Lewis’s Semantics: System CL 423\u003c\/p\u003e \u003cp\u003e\u003cb\u003e13 Common-Sense Reasoning 434\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e13.1 When Good Arguments Go Bad 435\u003c\/p\u003e \u003cp\u003e13.2 Truth Trees 439\u003c\/p\u003e \u003cp\u003e13.3 Defeasible Deduction 454\u003c\/p\u003e \u003cp\u003e13.4 Defeasible Deontic Logic 466\u003c\/p\u003e \u003cp\u003e\u003cb\u003e14 Quantifiers and Modality 475\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e14.1 Quantified S5 475\u003c\/p\u003e \u003cp\u003e14.2 Free Logic 487\u003c\/p\u003e \u003cp\u003eBibliography 504\u003c\/p\u003e \u003cp\u003eIndex 507\u003c\/p\u003e","brand":"John Wiley and Sons Ltd","offers":[{"title":"Default Title","offer_id":49403446755671,"sku":"9780631227137","price":37.95,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780631227137.jpg?v=1730483498"},{"product_id":"selected-logic-papers-9780674798373","title":"Selected Logic Papers","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eSelected 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.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e[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 *\u003cbr\u003eWhat [Quine] is expert in is, of course, logic… What [this book offers] is a view of the expert at work. \u003ci\u003eSelected Logic Papers\u003c\/i\u003e 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 *\u003cbr\u003eThis 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 Technology\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003eWhitehead 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).","brand":"Harvard University Press","offers":[{"title":"Default Title","offer_id":49403611316567,"sku":"9780674798373","price":31.46,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780674798373.jpg?v=1730483984"},{"product_id":"fixing-frege-9780691122311","title":"Fixing Frege","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eSurveys 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.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003eCo-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 Logic\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003eAcknowledgments 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","brand":"Princeton University Press","offers":[{"title":"Default Title","offer_id":49403736523095,"sku":"9780691122311","price":63.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780691122311.jpg?v=1730484399"},{"product_id":"enlightening-symbols-a-short-history-of-mathematical-notation-and-its-hidden-powers-9780691154633","title":"Enlightening Symbols  A Short History of","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eWhat 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.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e\"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 Society\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003eIntroduction 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","brand":"Princeton University Press","offers":[{"title":"Default Title","offer_id":49403790819671,"sku":"9780691154633","price":31.5,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780691154633.jpg?v=1730484552"},{"product_id":"actionminimizing-methods-in-hamiltonian-dynamics-9780691164502","title":"Actionminimizing Methods in Hamiltonian Dynamics","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e\u003c\/p\u003e\u003cp\u003eJohn 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 posse\u003c\/p\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003ePreface 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","brand":"Princeton University Press","offers":[{"title":"Default Title","offer_id":49403812381015,"sku":"9780691164502","price":37.8,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780691164502.jpg?v=1730484619"}],"url":"https:\/\/bookcurl.com\/collections\/mathematical-logic.oembed?page=12","provider":"Book Curl","version":"1.0","type":"link"}