Philosophy of mathematics Books

Book SynopsisDescribes the developments in number theory by looking at one of the most exciting unsolved problems in contemporary mathematics - the Birch and Swinnerton-Dyer Conjecture. The Clay Mathematics Institute is offering a prize of $1 million to anyone who can discover a general solution to the problem.Trade Review"The authors present their discussion in an informal, sometimes playful manner and with detail that will appeal to an audience with a basic understanding of calculus. This book will captivate math enthusiasts as well as readers curious about an intriguing and still unanswered question."--Margaret Dominy, Library Journal "Minimal prerequisites and its clear writing make this book (which even has a few exercises) a great choice for a seminar for mathematics majors, who at some point should have such an excursion to one of the frontiers of mathematics."--Mathematics Magazine "The authors of Elliptic Tales do a superb job in demonstrating the approach that mathematicians take when they confront unsolved problems involving elliptic curves."--Sungkon Chang, Times Higher Education "One cannot help being impressed, in reading the book and pursuing a few of the references, by the magnitude of the enterprise it chronicles."--James Case, SIAM News "Ash and Gross thoroughly explain the statement and significance of the linchpin Birch and Swinnerton-Dyer conjection... [A]sh and Gross deliver ample and current intellectual and technical substance."--Choice "I would envision this book as an excellent text for an undergraduate 'capstone' course in mathematics; the book lends itself to independent reading, but topics may be explored in much greater depth and rigor in the classroom. Additionally, the book indeed brings together ideas from calculus, complex variables and algebra, showing how a single mathematical research question may require an integrated understanding of the various branches of mathematics. Thus, it encourages students to reinforce their understanding of these various fields, while simultaneously introducing them to an open question in mathematics and a vibrant field of study."--Lisa A. Berger, Mathematical Reviews Clippings "The book is very pleasantly written, and in my opinion, the authors have done an admirable job in giving an idea to non-experts what the Birch-Swinnerton Dyer conjecture is about."--Jan-Hendrik Evertse, Zentralblatt MATH "The book's most important contributions ... are the sense of discovery, invention, and insight into the habits of mind used by mathematicians on this journey. I would recommend this book to anyone who wants to be challenged mathematically or who wants to experience mathematics as creative and exciting."--Jacqueline Coomes, Mathematics Teacher "[T]his book is a wonderful introduction to what is arguably one of the most important mathematical problems of our time and for that reason alone it deserves to be widely read. Another reason to recommend this book is the opportunity to share in the readily apparent joy the authors have for their subject and the beauty they see in it, not least because ... joy and beauty are the most important reasons for doing mathematics, irrespective of its dollar value."--Rob Ashmore, Mathematics Today "This book has many nice aspects. Ash and Gross give a truly stimulating introduction to elliptic curves and the BSD conjecture for undergraduate students. The main achievement is to make a relative easy exposition of these so technical topics."--Jonathan Sanchez-Hernandez, Mathematical SocietyTable of ContentsPreface xiii Acknowledgments xix Prologue 1 PART I. DEGREE Chapter 1. Degree of a Curve 13 1.Greek Mathematics 13 2.Degree 14 3.Parametric Equations 20 4.Our Two Definitions of Degree Clash 23 Chapter 2. Algebraic Closures 26 1.Square Roots of Minus One 26 2.Complex Arithmetic 28 3.Rings and Fields 30 4.Complex Numbers and Solving Equations 32 5.Congruences 34 6.Arithmetic Modulo a Prime 38 7.Algebraic Closure 38 Chapter 3. The Projective Plane 42 1.Points at Infinity 42 2.Projective Coordinates on a Line 46 3.Projective Coordinates on a Plane 50 4.Algebraic Curves and Points at Infinity 54 5.Homogenization of Projective Curves 56 6.Coordinate Patches 61 Chapter 4. Multiplicities and Degree 67 1.Curves as Varieties 67 2.Multiplicities 69 3.Intersection Multiplicities 72 4.Calculus for Dummies 76 Chapter 5. B'ezout's Theorem 82 1.A Sketch of the Proof 82 2.An Illuminating Example 88 PART II. ELLIPTIC CURVES AND ALGEBRA Chapter 6. Transition to Elliptic Curves 95 Chapter 7. Abelian Groups 100 1.How Big Is Infinity? 100 2.What Is an Abelian Group? 101 3.Generations 103 4.Torsion 106 5.Pulling Rank 108 Appendix: An Interesting Example of Rank and Torsion 110 Chapter 8. Nonsingular Cubic Equations 116 1.The Group Law 116 2.Transformations 119 3.The Discriminant 121 4.Algebraic Details of the Group Law 122 5.Numerical Examples 125 6.Topology 127 7.Other Important Facts about Elliptic Curves 131 5.Two Numerical Examples 133 Chapter 9. Singular Cubics 135 1.The Singular Point and the Group Law 135 2.The Coordinates of the Singular Point 136 3.Additive Reduction 137 4.Split Multiplicative Reduction 139 5.Nonsplit Multiplicative Reduction 141 6.Counting Points 145 7.Conclusion 146 Appendix A: Changing the Coordinates of the Singular Point 146 Appendix B: Additive Reduction in Detail 147 Appendix C: Split Multiplicative Reduction in Detail 149 Appendix D: Nonsplit Multiplicative Reduction in Detail 150 Chapter 10. Elliptic Curves over Q 152 1.The Basic Structure of the Group 152 2.Torsion Points 153 3.Points of Infinite Order 155 4.Examples 156 PART III. ELLIPTIC CURVES AND ANALYSIS Chapter 11. Building Functions 161 1.Generating Functions 161 2.Dirichlet Series 167 3.The Riemann Zeta-Function 169 4.Functional Equations 171 5.Euler Products 174 6.Build Your Own Zeta-Function 176 Chapter 12. Analytic Continuation 181 1.A Difference that Makes a Difference 181 2.Taylor Made 185 3.Analytic Functions 187 4.Analytic Continuation 192 5.Zeroes, Poles, and the Leading Coefficient 196 Chapter 13. L-functions 199 1.A Fertile Idea 199 2.The Hasse-Weil Zeta-Function 200 3.The L-Function of a Curve 205 4.The L-Function of an Elliptic Curve 207 5.Other L-Functions 212 Chapter 14. Surprising Properties of L-functions 215 1.Compare and Contrast 215 2.Analytic Continuation 220 3.Functional Equation 221 Chapter 15. The Conjecture of Birch and Swinnerton-Dyer 225 1.How Big Is Big? 225 2.Influences of the Rank on the Np's 228 3.How Small Is Zero? 232 4.The BSD Conjecture 236 5.Computational Evidence for BSD 238 6.The Congruent Number Problem 240 Epilogue 245 Retrospect 245 Where DoWe Go from Here? 247 Bibliography 249 Index 251

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Book SynopsisTraces the origins and development of mathematics in the ancient Middle East, from its earliest beginnings in the fourth millennium BCE to the end of indigenous intellectual culture in the second century BCE when cuneiform writing was gradually abandoned.Trade ReviewWinner of the 2011 Pfizer Award for Best Scholarly Book, History of Science Society One of Choice's Outstanding Academic Titles for 2009 Honourable Mention in the British-Kuwait Friendship Society Prize in Middle Eastern Studies 2009, British Society for Middle Eastern Studies "[F]ascinating."--Edward Rothstein, New York Times "Robson brings both a profound erudition in cuneiform and a nondogmatic constructionist view of mathematics to tell the history of Mesopotamian mathematics over the three millennia before the Common Era, connecting as she does the mathematical accomplishments to the cultural and societal norms of the day... A magisterial work, lucidly written, certain to endure."--M. Schiff, Choice "Author Robson deals admirably with an enormous scope (more than 3,000 years, with roughly equal space devoted to each 500-year epoch); numerous sources (950 published clay tablets, all of which are available at a simple Website); and the cultural context (social history, an ethnomathematical approach)."--Mathematics Magazine "Robson's book is a wonderful summary of what we know so far, and will be the standard for this generation, but the potential is there for far more research to teach us even more about mathematics in ancient Iraq."--Victor J. Katz, Mathematical Reviews "For archaeologists and archaeologically-minded historians ... Robson provide[s] significant new insights into the mathematics of ancient civilisations, while challenging us to consider how language, material culture, and socio-technical practices are integrated, not only in mathematics, but in many domains."--Stephen Chrisomalis, Antiquity "The wealth of detail and breadth of scope make this an excellent resource for a wide variety of readership. It can be read as one great narrative sweep, or one can bear down on a particular facet. The work is a huge advance in the presentation of modern scholarship on ancient mathematics to interested readers, specialist and non-specialist alike."--Duncan J. Melville, Historia Mathematica "Nothing comparable has been done before, and it has been a great pleasure to read the book, from which I have learned much."--Jens Hoyrup, Mathematical Intelligencer "Eleanor Robson's book Mathematics in Ancient Iraq is presently unique and will surely become a classic in the history of early mathematics. Despite the meticulous and detailed presentation of a representative selection of available sources, the book is very readable and captures the attention of the interested reader from the first to the last page. I recommend it to anyone who would like to learn something about the fascinating story of the development of mathematical activities in Mesopotamia."--Peter Damerow, Notices of the AMS "[Mathematics in Ancient Iraq] is argued passionately, persuasively and, I am pleased to add, enjoyably."--Bob Berghout, Australian Mathematical Society Gazette "Mathematics in Ancient Iraq fills a gap that has existed for a very long time."--Annette Imhausen, British Society for the History of Maths "Robson displays a confidence, familiarity, and breadth of scholarship that is impressive and inspiring. She epitomizes a new wave of research in the history of mathematics. She provides context, setting, and interpretative themes for generations of scholars to come, whether they will embrace them or resist them. Indeed, Robson's work is more than just a social history--it is emblematic of a new approach to this discipline. The details will excite specialists, the generalities will delight the uninitiated. 'Sparkling' indeed, this work is guaranteed to be an influential and foundational reference book, indispensable to the collections of the many disciplines it draws from."--Clemency Montelle, Journal of the American Oriental Society "Robson, as a professional assyriologist, is preeminently well positioned to write a history that situates Mesopotamian mathematics in its ancient social and intellectual context; and whether or not one always agrees with her interpretations of the mathematics, her competence in these aspects is nowhere in doubt."--Alexander Jones, British Journal for the History of Science "[T]he book is a very significant contribution to the history of mathematics. It is well written, solidly founded and argued, and easy to understand. It is a fine and important addition to the literature on Babylonian mathematics, and it will be very useful to readers from both inside and outside the field. The book is warmly recommended to everyone who is interested in mathematics and its history, in ancient cultures, or in science seen as an integrated part of culture, and to the broader public of historians of early science or Mesopotamian culture."--Lis Brack-Bernsen, Journal of World History "The book contains numerous charts, tables, images and databases that help us understand the issues addressed. It is excellently documented and it contains a comprehensive and up to date bibliography. Eleanor Robson is a scholar who commands the field that she investigates."--Piedad Yuste, Metascience "[T]he publication of a book of this kind is very welcome. Nothing like it has been published before, and it is going to be immensely helpful to both writers and readers of future articles and books about the subject."--Joran Friberg, Archive Fur OrientforschungTable of ContentsList of Figures xi List of Tables xvii Preface xxi Acknowledgments xxv Chapter One: Scope, Methods, Sources 1 1.1 The Subject: Ancient Iraq and Its Mathematics 1 1.2 The Artefacts: Assyriological and Mathematical Analysis 8 1.3 The Contexts: Textuality, Materiality, and Social History 17 Chapter Two: Before the Mid-Third Millennium 27 2.1 Background and Evidence 28 2.2 Quantitative Management and Emerging Statehood 33 2.3 Enumeration and Abstraction 40 2.4 Symmetry, Geometry, and Visual Culture 45 2.5 Conclusions 51 Chapter Three: The Later Third Millennium 54 3.1 Background and Evidence 55 3.2 Maps, Plans, and Itineraries: Visual and Textual Representations of Spatial Relationships 60 3.3 Accounting for Time and Labour: Approximation, Standardisation, Prediction 67 3.4 The Development of the Sexagesimal Place Value System (SPVS) 75 3.5 Conclusions 83 Chapter Four: The Early Second Millennium 85 4.1 Background and Evidence 86 4.2 Metrology, Multiplication, Memorisation: Elementary Mathematics Education 97 4.3 Words and Pictures, Reciprocals and Squares 106 4.4 Measurement, Justice, and the Ideology of Kingship 115 4.5 Conclusions 123 Chapter Five: Assyria 125 5.1 Background and Evidence 126 5.2 Palatial and Mercantile Numeracy in Early Assyria 129 5.3 Counting Heads, Marking Time: Quantifi cations in Royal Inscriptions and Records 136 5.4 Aru: Number Manipulation in Neo-Assyrian Scholarship 143 5.5 Conclusions 149 Chapter Six: The Later Second Millennium 151 6.1 Background and Evidence 151 6.2 Tabular Accounting in Southern Babylonia 157 6.3 Land Surveyors and Their Records in Northern Babylonia 166 6.4 Quantifi cation as Literary Device in the Epic of Gilgames 177 6.5 Conclusions 181 Chapter Seven: The Early First Millennium 183 7.1 Background and Evidence 184 7.2 Libraries and Schools: The Formalisation of the First-Millennium Scribal Curriculum 192 7.3 Home Economics: Numeracy in a Mid-First-Millennium Urban Household 198 7.4 Measuring Houses, Maintaining Professionalism 206 7.5 Conclusions 212 Chapter Eight: The Later First Millennium 214 8.1 Background and Evidence 215 8.2 Babylon: Mathematics in the Service of Astronomy? 220 8.3 Achaemenid Uruk: The Sangu-Ninurta and Ekur-z?kir Families 227 8.4 Seleucid Uruk: The Hunzu and Sin-leqi-unninni Families 240 8.5 Conclusions 260 Chapter Nine: Epilogue 263 9.1 The Big Picture: Three Millennia of Mathematics in Ancient Iraq 263 9.2 Ancient Mathematics in the Modern World 268 9.3 Inside Ancient Mathematics: Translation, Representation, Interpretation 274 9.4 The Worlds of Ancient Mathematics: History, Society, Community 284 9.5 Conclusions 288 Appendix A: Metrological Systems 291 Appendix B: Published Mathematical Tablets 299 Notes 345 Bibliography 373 Index of Tablets 409 Subject Index 425

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Book SynopsisTrade Review“[A] jaunty book.”—James Ryerson, New York Times Book Review“Devlin leads a cheerful pursuit to rediscover the hero of 13th-century European mathematics, taking readers across centuries and through the back streets of medieval and modern Italy in this entertaining and surprising history.”—Publishers Weekly“Finding Fibonacci showcases Devlin’s writerly flair.”—Davide Castelvecchi, Nature“[Devlin] talks his way into Italian research libraries in search of early manuscripts, photographs all 11 street signs on Via Leonardo Fibonacci in Florence and strives to cultivate a love for numbers in his readers.”—Andrea Marks, Scientific American“Engaging and entertaining.”—Library Journal“Personal and lively.”—Adhemar Bultheel, European Mathematical Society“Devlin’s enthusiasm for his subject is infectious.”—Tony Mann, Times Higher Education

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Book SynopsisTrade Review"One of Choice's Outstanding Academic Titles for 2013""Masterly. . . . Gray encapsulates Poincaré's multiple dimensions; his intellectual biography is both a tour de force and a triumph of readability."---George Szpiro, Nature"Gray shows us the full dazzling sweep of what Poincaré accomplished, including the work on dynamical systems and chaos that only came into its own in recent years. A tour de force, Gray's masterful treatment will long remain an invaluable resource for all who want to understand Poincaré, so embedded within his times and yet so far ahead of them."---Peter Pesic, Science"[A] comprehensive but uncluttered guide to Poincaré's extensive oeuvres."---Madeline Muntersbjorn, Times Higher Education"Full of the mathematical, physical and metaphysical ideas of a man who was not only a dispassionate observer of the world around us, but of our way of understanding it."---Mark Ronan, Standpoint Magazine"[A] comprehensive assessment of Poincare's work and its importance, essential for anyone interested in Poincare's scholarship or the history of mathematics."---Laura Tarwater Scharp, Sacramento Book Review"Comprehensive." * Science News *"A fundamental study of the scientific work of one of the greatest mathematicians and mathematical physicists of the three decades straddling the 19th and 20th centuries. . . . Chapters are organized topically, not chronologically. Each illuminates in depth one or other of Poincaré's works but all are set in context both historical and temathic such that each can serve as an introduction into the many subjects to which Poincaré made a contribution."---Alexander Bogomolny, CTK Insights"Poincare's work is fully alive in science today. This biography is one of the first thorough introductions to his work, it should get the attention of mathematicians, natural scientists and philosophers."---Ferdinand Verhulst, European Legacy"Gray, a mathematics historian and scholar on the life and work of Henry Poincaré, has, with the support of a Leverhulme Research Fellowship, produced this comprehensive and definitive 'scientific biography.' Gray offers abundant rich information on Poincaré's ideas and scientific process, the evolution and maturity of his mathematics including missteps, the dexterity of his reasoning, and the influences that shaped his thought." * Choice *"I recommend [this] book highly."---Robert E. O'Malley, Jr., SIAM Review"Jeremy Gray's book on Poincaré's mathematics, physics, and philosophy is an important contribution to the literature and a huge step towards a full biography of this pioneer ofmodern science."---Reinhard Siegmund-Schultze, Zentralblatt MATH"Gray's book is a comprehensive scientific biography of Poincare. It embraces the broad scope of Poincare's work, from his philosophical speculations to his popular writing, and gives a thorough overview of his extensive mathematical researches."---Peter Lynch, Irish Mathematical Society Bulletin"[T]he author does not simply give platitudes when writing about Poincare's ideas: mathematicians will enjoy reading about his discoveries concerning the three-body problem, the theory of functions, topology, number theory, Lie theory, algebraic geometry, and probability. This scientific biography is the first to comprehensively cover all of Poincare's main contributions to mathematics, philosophy, and physics."---Alan S. McRae, Mathemematical Reviews Clippings"Jeremy Gray has done a marvelous job of exposition and of binding together the many different cognitive, social and biographical strands into the coherent whole of a challenging, but highly rewarding, 'scientific biography'."---Klaus Hentschel, British Journal for the History of Science"A good intellectual biography of an artist should help the reader see how a particular worldview shapes the pursuit of art. Gray's book does that most admirably."---Daniel S. Alexander, H-France Review"Henry Poincaré is likely to remain the standard by which scientific biographies, at least those that concern physicists and mathematicians, are judged for some time."---Christopher Cumo, Canadian Journal of History"I warmly recommend the book to anyone with an interest in the development of modern mathematics. It will surely be the definitive scientific biography of Poincare for the foreseeable future."---John Stillwell, Notices of the AMS"Gray describes Poincaré's scientific epoch in a beautiful way. Due attention is paid to the mathematical and further scientific aspects of his life, and the intellectual complexity of his achievements, both in their range and their depth, are amply discussed. Gray displays a mastery of his material that is rare even among historians of mathematics and science, and his biography is richly rewarding, engrossing, and informative. He deserves our congratulations."---H. W. Broer, Journal of the British Society for the History of Mathematics"Gray succeeds admirably in presenting both the conceptual and the historical context necessary to appreciate Poincaré's contributions. Gray's masterful biography may well serve as a standard example for future endeavors of this kind."---Tilman Sauer, Isis"The obvious virtue of this book is its comprehensiveness. The deeper virtue is to connect Poincaré's views of all the parts of his work and to encourage more of that. Gray gives us Poincaré's view of Science as a whole."---Colin McLarty, Mathematical Intelligencer"The book is an endless source of interesting insights by Poincaré. . . . I would recommend the book for mathematicians, mathematics educators, and philosophers in higher education who want a rich understanding of Poincaré, his work, and his times."---Mary L. Garner, Mathematics Teacher

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Book SynopsisFirst published in 1919, Introduction to Mathematical Philosophy shows Russell drawing on his formidable knowledge of philosophy and mathematics to write a brilliant introduction to the subject. This Routledge Classics edition includes a new Foreword by Michael Potter.Table of ContentsForeword to the Routledge Classics Edition Michael Potter Preface 1. The Series of Natural Numbers 2. Definition of Number 3. Finitude and Mathematical Induction 4. The Definition of Order 5. Kinds of Relations 6. Similarity of Relations 7. Rational, Real, and Complex Numbers 8. Infinite Cardinal Numbers 9. Infinite Series and Ordinals 10. Limits and Continuity 11. Limits and Continuity of Functions 12. Selections and the Multiplicative Axiom 13. The Axiom of Infinity and Logical Types 14. Incompatibility and the Theory of Deduction 15. Propositional Functions 16. Descriptions 17. Classes 18. Mathematics and Logic. Index

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Book SynopsisSpherical Geometry and Its Applications introduces spherical geometry and its practical applications in a mathematically rigorous form. The text can serve as a course in spherical geometry for mathematics majors. Readers from various academic backgrounds can comprehend various approaches to the subject. The book introduces an axiomatic system for spherical geometry and uses it to prove the main theorems of the subject. It also provides an alternate approach using quaternions. The author illustrates how a traditional axiomatic system for plane geometry can be modified to produce a different geometric world but a geometric world that is no less real than the geometric world of the plane. Features: A well-rounded introduction to spherical geometry Provides several proofs of some theorems to appTable of ContentsReview of three-dimensional geometry Geometry in a plane Geometry in space Plane trigonometry Coordinates and vectors The sphere in space Great circles Distance and angles Area Spherical coordinates Axiomatic spherical geometry Basic axioms Angles Triangles Congruence Inequalities Area Trigonometry Spherical Pythagorean theorem and law of sines Spherical law of cosines and analogue formula Right triangles The four-parts and half angle formulas Dualization Solution of triangles Astronomy The celestial sphere Changing coordinates Rise and set of objects in the sky The measurement of time Rise and set times in standard time Polyhedra Regular solids Crystals Spherical mappings Rotations and reflections Spherical projections Quaternions Review of complex numbers Quaternions: Definitions and basic properties Application to the sphere Triangles Rotations and Reflections Selected solutions to exercises

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Book SynopsisWhy is 7 such a lucky number and 13 so unlucky? Why does a jury traditionally have `12 good men and true', and why are there 24 hours in the day and 60 seconds in a minute? This fascinating new book explores the world of numbers from pin numbers to book titles, and from the sixfold shape of snowflakes to the way our roads, houses and telephone numbers are designated in fact and fiction. Using the numbers themselves as its starting point it investigates everything from the origins and meaning of counting in early civilizations to numbers in proverbs, myths and nursery rhymes and the ancient `science' of numerology. It also focuses on the quirks of odds and evens, primes, on numbers in popular sports - and much, much more. So whether you've ever wondered why Heinz has 57 varieties, why 999 is the UK's emergency phone number but 911 is used in America, why Coco Chanel chose No. 5 for her iconic perfume, or how the title Catch 22 was chosen, then this is the book for you. Dip in anywhere and you'll find that numbers are not just for adding and measuring but can be hugely entertaining and informative whether you're buying a diamond or choosing dinner from the menu.Table of Contents1 Introduction 8 2 Numbers of many sorts 10 3 Numbers and counting 12 4 1 - The number of unity 17 5 2 - The duality 19 6 3 - The trinity of perfection 21 7 4 - Truth and justice 23 8 5 - The number of nature 26 9 6 - Without a fault Six and the snowflake 29 10 7 - The symbol of fortune 32 11 The seven wonders of the world 35 12 Puzzles to solve 37 13 8 - Harmony and balance 38 14 9 - Unbounded 40 15 The nine muses 43 16 10 - On our fingers and toes 44 17 11 - The final hour 46 18 12 - A number in time 47 19 The twelve labours of Hercules 49 20 13 - And other teens 53 21 0 - The story of zero 54 22 Lucky - and unlucky - numbers 56 23 Odds and evens 59 24 Many favourite numbers 60 25 The secrets of numerology 62 26 Big numbers 66 27 Small numbers 68 28 The appeal of primes 69 29 Making shapes with numbers 72 30 Numbers - more different types 74 31 - the most famous number 75 32 Fibonacci - the brilliant number sequence 77 33 The golden ratio 79 34 Numbers in use 81 35 The world we live in 84 36 Our planet earth 86 37 Lines on the map 89 38 Measuring the world 94 39 A matter of weight 98 40 By volume 101 41 More about money 104 42 Inventing the calendar 106 43 Time and the circle 109 44 The living world 110 45 The human body 114 46 A number for your home 120 47 Addresses in fiction 121 48 A dark history 123 49 The streets of power 124 50 Numbers for the post 126 51 Roads to take - navigating by numbers 127 52 The number to call 130 53 PIN - what's your number? 132 54 Edible connections 134 55 Sizing up our drinks 137 56 Of yarns, fabrics and clothes 139 57 All that glitters 140 58 Beauty by numbers 142 59 For our leisure and entertainment 144 60 Proverbs and sayings 146 61 Bingo lingo 147 62 Books with numbers 149 63 In the film title 156 64 Counting in song 161 65 Jazz numbers 163 66 A musical miscellany 165 67 Poetry's secrets revealed 167 68 The beautiful game 170 69 The oval ball game - rugby football 174 70 The game of golf 176 71 Throwing darts 178 72 Cricket - bat on ball 179 73 On court - the game of tennis 182 74 Snooker and other cue games 185 75 The game of baseball 187 76 Chancing your luck 189 77 Throwing dice 192 78 Playing dominoes 194 79 Solving the square 195 80 Index 200 81 About the author 208

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Book SynopsisTrade Review"I greatly enjoyed Richeson's Tales of Impossibility. It deserves to become a classic and can be highly recommended."---Robin Wilson, Times Higher Education"Even if you never read a single proof through to its conclusion, you’ll enjoy the many entertaining side trips into a geometry far beyond what you learned in high school."---Jim Stein, New Books in Mathematics"The whole book, both informative and amusing, is a highly recommended read."---Adhemar Bulteel, European Mathematical Society"This book was a pleasure to read and I would recommend it for anybody who wants a lovely overview of many areas of the history of mathematics, with a focus on some very easy to understand problems."---Jonathan Shock, Mathemafrica"Richeson clearly explains what it means to be impossible to solve a problem, cites other impossibility results, goes into detail about geometric constructions with various instruments, and discusses the defective proofs and the cranks that have turned up along the way." * Mathematics Magazine *"This fascinating text will appeal to all those interested in the history of mathematics, not leasy because of its helpful notes on each chapter and its two dozen pages of references for further reading"---Laurence E. Nicholas CMath FIMA, Mathematics Today"A fact-filled, insightful, panoramic view of how mathematics developed to what it is today transformed by folks thinking both inside and outside of G so as to resolve the impossible."---Andrew J. Simoson, Mathematical Intelligencer

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Book SynopsisGottfried Wilhelm Leibniz (1646-1716) was a man of extraordinary intellectual creativity who lived an exceptionally rich and varied intellectual life in troubled times. More than anything else, he was a man who wanted to improve the life of his fellow human beings through the advancement of all the sciences and the establishment of a stable and just political order. In this Very Short Introduction Maria Rosa Antognazza outlines the central features of Leibniz''s philosophy in the context of his overarching intellectual vision and aspirations. Against the backdrop of Leibniz''s encompassing scientific ambitions, she introduces the fundamental principles of Leibniz''s thought, as well as his theory of truth and theory of knowledge. Exploring Leibniz''s contributions to logic, mathematics, physics, and metaphysics, she considers how his theories sat alongside his concerns with politics, diplomacy, and a broad range of practical reforms: juridical, economic, administrative, technological, medical, and ecclesiastical. Discussing Leinbniz''s theories of possible worlds, she concludes by looking at what is ultimately real in this actual world that we experience, the good and evil there is in it, and Leibniz''s response to the problem of evil through his theodicy. 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.Table of ContentsCONCLUSION; REFERENCES; FURTHER READING; INDEX

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a huge range and FREE tracked UK delivery on ALL orders.

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a huge range and FREE tracked UK delivery on ALL orders.

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a huge range and FREE tracked UK delivery on ALL orders.

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Book SynopsisMajor shifts in the field of model theory in the twentieth century have seen the development of new tools, methods, and motivations for mathematicians and philosophers. In this book, John T. Baldwin places the revolution in its historical context from the ancient Greeks to the last century, argues for local rather than global foundations for mathematics, and provides philosophical viewpoints on the importance of modern model theory for both understanding and undertaking mathematical practice. The volume also addresses the impact of model theory on contemporary algebraic geometry, number theory, combinatorics, and differential equations. This comprehensive and detailed book will interest logicians and mathematicians as well as those working on the history and philosophy of mathematics.Trade Review'The book under review has a lot to offer at many levels. First of all, it may serve as a guide to recent advances in pure and applied model theory. Such a guide may be useful not only to novices, but also to old hands. Secondly, Baldwin summarizes several trends in contemporary philosophy of mathematics, and his insights should be of interest to philosophers as well as to mathematicians.' Roman Kossak, The Mathematical IntelligencerTable of ContentsPart I. Refining the Notion of Categoricity: 1. Formalization; 2. The context of formalization; 3. Categoricity; Part II. The Paradigm Shift: 4. What was model theory about?; 5. What is contemporary model theory about?; 6. Isolating tame mathematics; 7. Infinitary logic; 8. Model theory and set theory; Part III. Geometry: 9. Axiomatization of geometry; 10. π, area, and circumference of circles; 11. Complete: the word for all seasons; Part IV. Methodology: 12. Formalization and purity in geometry; 13. On the nature of definition: model theory; 14. Formalism-freeness; 15. Summation.

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Book SynopsisSets are central to mathematics and its foundations, but what are they? In this book Luca Incurvati provides a detailed examination of all the major conceptions of set and discusses their virtues and shortcomings, as well as introducing the fundamentals of the alternative set theories with which these conceptions are associated. He shows that the conceptual landscape includes not only the naïve and iterative conceptions but also the limitation of size conception, the definite conception, the stratified conception and the graph conception. In addition, hepresents a novel, minimalist account of the iterative conception which does not require the existence of a relation of metaphysical dependence between a set and its members. His book will be of interest to researchers and advanced students in logic and the philosophy of mathematics.Trade Review'Incurvati provides a veritable handbook for researchers and practitioners in the domain of logic and the foundations of mathematics … Each chapter raises significant foundational questions, fertile ground for further research.' R. L. Pour, ChoiceTable of Contents1. Concepts and conceptions; 2. The iterative conception; 3. Challenges to the iterative conception; 4. The naïve conception; 5. The limitation of size conception; 6. The stratified conception; 7. The graph conception.

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Book SynopsisMathematical Explorations follows on from the author''s previous book, Creative Mathematics, in the same series, and gives the reader experience in working on problems requiring a little more mathematical maturity. The author''s main aim is to show that problems are often solved by using mathematics that is not obviously connected to the problem, and readers are encouraged to consider as wide a variety of mathematical ideas as possible. In each case, the emphasis is placed on the important underlying ideas rather than on the solutions for their own sake. To enhance understanding of how mathematical research is conducted, each problem has been chosen not for its mathematical importance, but because it provides a good illustration of how arguments can be developed. While the reader does not require a deep mathematical background to tackle these problems, they will find their mathematical understanding is enriched by attempting to solve them.Trade Review'The broad array of mathematics that is covered in this text might provide a thought-provoking summary of topics and give students an opportunity to expand their horizons.' Mark Hunacek, MAA ReviewsTable of ContentsPreface; How to use this book; 1. Paying for parking; 2. Lengths and angles; 3. Magic squares; 4. Intersecting chords; 5. Crossing squares; 6. Repeated vector products; 7. A rolling disc; 8. Sums of powers of digits; 9. The metric dimension; 10. Primes and irreducible elements; 11. The symmetries of a quadrilateral; 12. Removing a vertex; 13. Squares within squares; 14. Catalan numbers; References; Index.

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Book SynopsisAn electrifying biography of one of the most extraordinary scientists of the twentieth century and the world he made. Trade Review"[Bhattacharya's] crystal-clear prose…mak[es] for a tour de force of enjoyable science writing….[A] marvelously bracing biography of the ideas of John von Neumann, ideas that continue to grow and flourish with a life of their own." -- Stephen Budiansky - Wall Street Journal"Vivid…[The Man From the Future is] devoted to exploring the ideas and technological inquiries [von Neumann] inspired." -- Jennifer Szalai - New York Times"Lucid and rewarding….Bhattacharya composes a rich intellectual map of von Neumann’s pursuits, shading in their histories and evolutions, and tracing the routes and connections between them." -- Samanth Subramanian - The New Republic"Examines the tremendous impact von Neumann had on various scientific disciplines in eight exceptional chapters." -- Dov Greenbaum and Mark Gerstein - Science"Rather like the books of Stephen Hawking or Carlo Rovelli…this one is rewarding on different levels. Everyone can grasp the significance of the puzzles posed, and if readers want to follow the genius through the steps of his solutions then Bhattacharya is a clear and authoritative guide." -- The Economist"Offers us a striking portrait of a man who contributed as much to the technological transformation of the world as any other scientist of the 20th century…[A]lways engaging and generally illuminating." -- David Nirenberg - The Nation"Non-Euclidean geometry, set theory, the prisoner’s dilemma, Gödel’s incompleteness theorems, self-replicating machines, game theory and nonlocality are among the astonishing range of topics that science journalist Ananyo Bhattacharya covers as he takes us on a whistle-stop tour through Von Neumann’s restless mind…[A] splendid new biography." -- Manjit Kumar - Guardian"Bhattacharya both begins and concludes this impressive biography of John von Neumann by celebrating his contribution to the 'march of ideas.'" -- Francis P. Sempa - New York Journal of Books"Bhattacharya tells the story tremendously well, situating von Neumann’s work—in fields from quantum mechanics to game theory to cellular automata—as comfortably as I’ve ever seen it done. He’s also good at deadpan humor." -- David Bodanis - Financial Times"Bhattacharya is a first-class science writer with an impeccable pedigree and he does the best job I have seen of explaining the significance of von Neumann's work across many different fields… A fine tribute to von Neumann's genius and his contributions to science." -- John Gribbin - Literary Review"[An] agile, intelligent, intellectually enraptured account of Von Neumann’s life." -- Simon Ings - Sunday Telegraph"Any future intelligence capable of sending a representative back in time to help invent itself will be intelligent enough to conceal this from us. Ananyo Bhattacharya’s The Man from the Future is therefore unable to confirm this suggestion, but much else about John von Neumann’s presence in the twentieth century is revealed along the way." -- George Dyson, author of Turing's Cathedral"Despite his central contributions to the theory of computation, economics, logic, complexity, and quantum physics, somehow John von Neumann never became a household name to rival Einstein and Feynman. Ananyo Bhattacharya’s biography deserves to change that. Consistently clear and careful without sacrificing elegance or accessibility, it does full justice to this legendary figure of twentieth-century science." -- Philip Ball, author of Beyond Weird"An engaging and fascinating book that blends science and history. I loved it." -- Paul Davies, author of The Demon in the Machine"This is a sparkling book, with an intoxicating mix of pen-portraits and grand historical narrative. Above all, it fizzes with a dizzying mix of deliciously vital ideas. The Man from the Future is a staggering achievement." -- Tim Harford, author of How to Make the World Add Up"More than just a biography, The Man from the Future elucidates the breath-taking scientific progress in the mid-20th century, skillfully woven together in the story of one man, John von Neumann." -- Sabine Hossenfelder, author of Lost in Math"A gripping tale of the most significant mathematical, scientific and geopolitical events of the early 20th century. Bhattacharya’s storytelling seamlessly weaves together the science, the vibrant social and historical context, and the private idiosyncrasies of John von Neumann and the fascinating geniuses around him, without mythologizing." -- Andrew Steele, author of Ageless"Sharp, expansive….A salient portrait of one of the most electrifying and productive scientists of the past century." -- Kirkus Reviews

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Book SynopsisThis wordless collection of strips by renowned artist/designer Rian Hughes reveals the lighter side of our obsession with social rankings.When everyone has a number, everyone knows their place. Lower numbers are better, higher numbers are less important, and that''s just the way it is. But what if that number could change? You might try to buck the system and assert your individuality... or you might end up with a big fat zero.Big questions are explored and unexpected answers found in the first solo comics collection from award-winning designer & illustrator Rian Hughes. His whimsical, witty, and insightful strips will make you both smile and consider. Where do you stand in the pecking order? Is your number up? 2018 Pubwest Design Awards - Gold Winner for Graphic Album, New Material

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Book SynopsisProfessor Michael Edgeworth McIntyre is an eminent scientist who has also had a part-time career as a musician. In this book he offers an extraordinary synthesis, revealing the many deep connections between science, music, and mathematics. He avoids equations and technical jargon. The connections are deep in the sense of being embedded in our very nature, rooted in biological evolution over hundreds of millions of years.Michael guides us through biological evolution, perception psychology, and even unconscious science and mathematics, all the way to the scientific uncertainties about the climate crisis.He also has a message of hope for the future. Contrary to popular belief, he holds that biological evolution has given us not only the nastiest, but also the most compassionate and cooperative parts of human nature. This insight comes from recognizing that biological evolution is far more than a simple competition between selfish genes. Instead, he argues, in some ways it is more like the turbulent, eddying flow in a river or in an atmospheric jet stream, a complex process spanning a vast range of timescales.Professor McIntyre is a Fellow of the Royal Society of London (FRS) and has long been interested in how different branches of science can better communicate with each other, and with the public. His work harnesses aspects of neuroscience and psychology that point toward the deep 'lucidity principles' that underlie skilful communication, principles related to the way music works — music of any genre.This Second Edition sharpens the previous discussion of communication skills and their importance for today's great problems, ranging from the widely discussed climate crisis to the need to understand the strengths and weaknesses of artificial intelligence.

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Book SynopsisWhy is the number seven lucky - even holy - in almost every culture? Why do we speak of the four corners of the earth? Why do cats have nine lives (except in Iran, where they have seven)? From literature to folklore to private superstitions, numbers play a conspicuous role in our daily lives. In The Mystery of Numbers Annemarie Schimmel conducts an illuminating tour of the mysteries attributed to numbers over the centuries. She covers the origins of numbers, the symbolism of numbers, the source of this symbolism, and examines individual numbers from one to ten thousand. Using examples ranging from the Bible to Shakespeare, this engaging account uncovers the roots of a phenomenon we all feel every Friday the thirteenth.Trade ReviewA delightful, cross-cultural romp, through the history of number mysticism... * The New York Times Book Review *Table of ContentsIntroduction; Number and number systems; The heritage of the Pythagoreans; Gnosis and Cabala; Islamic mysticism; Medieval and baroque number symbolism; Superstitions; Number games and magic squares; Little dictionary of numbers; Bibliography; Illustration credits.

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Book SynopsisDo numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic problems. As a way out of this dilemma, Shapiro articulates a structuralist approach. On this view, the subject matter of arithmetic, for example, is not a fixed domain of numbers independent of each other, but rather is the natural number structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle. Using this framework, realism in mathematicTrade ReviewExtremely interesting and deserves the attention of anyone with a serious interest in the field ... a careful study of the book will be enormously rewarding to anyone with some prior exposure to the field. * Philosophia Mathematica *

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Book SynopsisThis is a charming and insightful contribution to an understanding of the Science Wars between postmodernist humanism and science, driving toward a resolution of the mutual misunderstanding that has driven the controversy. It traces the root of postmodern theory to a debate on the foundations of mathematics, early in the 20th century then compares developments in mathematics to what took place in the arts and humanities, discussing issues as diverse as literary theory, arts, and artificial intelligence. This is a straight forward, easily understood presentation of what can be difficult theoretical concepts and demonstrates that a pattern of misreading mathematics can be seen on both the part of science and on the part of postmodern thinking. This is a humorous, playful yet deeply serious look at the intellectual foundations of mathematics for those in the humanities and is the perfect critical introduction to the bases of modernism and postmodernism for those in the sciences.Trade ReviewThe book makes pleasant and interesting reading. * Mathematical Reviews *Table of Contents1: Introduction 2: Around the Cartesian Circuit 2.1: Imagination 2.2: Intuition 2.3: Counting to One 3: Space Oddity and Linguistic Turn 4: Wound of Language 4.1: Being and Time Continuum 4.2: Language and Will 5: Beyond the Code 5.1: Medium of Free Becoming 5.2: Nonpresence of Identity 6: The Expired Subject 6.1: Empire of Signs 6.2: Mechanical Bride 7: The Vanishing Author 8: Say Hello to the Structure Bubble 8.1: Algebra of Language 8.2: Functionalism Chic 9: Don't Think, Look 9.1: Interpolating the Self 9.2: Language Games 9.3: Thermostats "R" Us 10: Postmodern Enigmas 10.1: Unspeakable Diffd'erance 10.2: Dysfunctionalism Chic Notes Select Bibliography Index

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Book SynopsisIn this deft and vigorous book, Mark Balaguer demonstrates that there are no good arguments for or against mathematical platonism (i.e., the view that abstract, or non-spatio-temporal, mathematical objects exist, and that mathematical theories are descriptions of such objects). Balaguer does this by establishing that both platonism and anti-platonism are defensible positions. In Part I, he shows that the former is defensible by introducing a novel version of platonism, which he calls full-blooded platonism, or FBP. He argues that if platonists endorse FBP, they can then solve all of the problems traditionally associated with their view, most notably the two Benacerrafian problems (that is, the epistemological problem and the non-uniqueness problem). In Part II, Balaguer defends anti-platonism (in particular, mathematical fictionalism) against various attacks, chief among them the Quine-Putnam indispensability argument. Balaguer''s version of fictionalism bears similarities to Hartry FiTrade Reviewexcellent book...exceptionally clear, insightful, and useful critical surve? * The Review of Modern Logic Platonism and anti-platonism in mathematics is an impressive work. Balaguer presents forceful arguments for the viability of both FBP and fictionalism, and against the feasibility of any substantially different Platonist or anti-Platonist position. ... an admirable achievement.The Bulletin of Symbolic Logic *excellent book...exceptionally clear, insightful, and useful critical survey. * The Review of Modern Logic *Platonism and anti-platonism in mathematics is an impressive work. Balaguer presents forceful arguments for the viability of both FBP and fictionalism, and against the feasibility of any substantially different Platonist or anti-Platonist position. ... an admirable achievement. * The Bulletin of Symbolic Logic *

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Book SynopsisKurt Gödel (1906 - 1978) was the most outstanding logician of the twentieth century, famous for his hallmark works on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum hypothesis. He is also noted for his work on constructivity, the decision problem, and the foundations of computability theory, as well as for the strong individuality of his writings on the philosophy of mathematics. He is less well known for his discovery of unusual cosmological models for Einstein''s equations, in theory permitting time travel into the past. The Collected Works is a landmark resource that draws together a lifetime of creative thought and accomplishment. The first two volumes were devoted to Gödel''s publications in full (both in original and translation), and the third volume featured a wide selection of unpublished articles and lecture texts found in Gödel''s Nachlass. These long-awaited final two volumes contain Gödel''s corTrade ReviewThe whole enterprise is superbly coordinated and assembled under the direction of Solomon Feferman ... The book is a tour de force and a labour of love. Superbly crafted and presented, what a bargain, given the many gems it contains! * The Mathematical Gazette *The books are carefully and beautifully produced and offer rich material, illuminating not only the outstanding work of Gödel, but also the whole mathematical logic of the twentieth century, including some philosophical and historical aspects. * EMS *Table of ContentsGödel's life and workSolomon Feferman: A Gödel chronologyJohn W. Dawson, Jr.: Gödel 1929: Introductory note to 1929, 1930 and 1930aBurton Dreben and Jean van Heijenoort: Über die Vollständigkeit des Logikkalküls On the completeness of the calculus of logic Gödel 1930: (See introductory note under Gödel 1929.) Die Vollständigkeit der Axiome des logischen Funktionenkalküls The completeness of the axioms of the functional calculus of logic Gödel 1930a: (See introductory note under Gödel 1929.) Über die Vollständigkeit des Logikkalküls On the completeness of the calculus of logic Gödel 1930b: Introductory note to 1930b, 1931 and 1932bStephen C. Kleene: Einige metamathematische Resultate über Entscheidungs-definitheit und Widerspruchsfreiheit Some metamathematical results on completeness and consistency Gödel 1931: (See introductory note under Gödel 1930b.) Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I On formally undecidable propositions of Principia mathematica and related systems I Gödel 1931a: Introductory note to 1931a, 1932e, f and gJohn W. Dawson, Jr.: Diskussion zur Grundlegung der Mathematik Discussion on providing a foundation for mathematics Gödel 1931b: Review of Neder 1931 Gödel 1931c: Introductory note to 1931cSolomon Feferman: Review of Hilbert 1931 Gödel 1931d: Review of Betsch 1926 Gödel 1931e: Review of Becker 1930 Gödel 1931f: Review of Hasse and Scholz 1928 Gödel 1931g: Review of von Juhos 1930 Gödel 1932: Introductory note to 1932A. S. Troelstra: Zum intuitionistischen aussagenkalkül On the intuitionistic propositional calculus Gödel 1932a: Introductory note to 1932a, 1933i and lWarren D. Goldfarb: Ein Spezialfall des Enscheidungsproblems der theoretischen Logik A special case of the decision problem for theoretical logic Gödel 1932b: (See introductory note under Gödel 1930b.) Über Vollständigkeit und Widerspruchsfreiheit On completeness and consistency Gödel 1932c: Introductory note to 1932cW. V. Quine: Eine Eigenschaft der Realisierungen des Aussagenkalküls A property of the realizations of the propositional calculus Gödel 1932d: Review of Skolem 1931 Gödel 1932e: (See introductory note under Gödel 1931a.) Review of Carnap 1931 Gödel 1932f: (See introductory note under Gödel 1931a.) Review of Heyting 1931 Gödel 1932g: (See introductory note under Gödel 1931a.) Review of von Neumann 1931 Gödel 1932h: Review of Klein 1931 Gödel 1932i: Review of Hoensbroech 1931 Gödel 1932j: Review of Klein 1932 Gödel 1932k: Introductory note to 1932k, 1934e and 1936bStephen C. Kleene: Review of Church 1932 Gödel 1932l: Review of Kalmár 1932 Gödel 1932m: Review of Huntington 1932 Gödel 1932n: Review of Skolem 1932 Gödel 1932o: Review of Dingler 1931 Gödel 1933: Introductory note to 1933W. V. Quine: [[Über die Parryschen Axiome]] [[On Parry's axioms]] Gödel 1933a: Introductory note to 1933aW. V. Quine: Über Unabhängigkeitsbeweise im Aussagenkalkül On independence proofs in the propositional calculus Gödel 1933b: Introductory note to 1933b, c, d, g and hJudson Webb: Über die metrische Einbettbarkeit der Quadrupel des R[3 in Kugelflächen On the isometric embeddability of quadruples of points of R[3 in the surface of a sphere Gödel 1933c: (See introductory note under Gödel 1933b.) Über die Waldsche Axiomatik des Zwichenbegriffes On Wald's axiomization of the notion of betweenness Gödel 1933d: (See introductory note under Gödel 1933b.) Zur Axiomatik der elementargeometrischen Verknüpfungs-relationen On the axiomatization of the relations of connection in elementary geometry Gödel 1933e: Introductory note to 1933eA. S. Troelstra: Zur institutionistischen Arithmetik und Zahlentheorie On intuitionistic arithmetic and number theory Gödel 1933f: Introductory note to 1933fA. S. Troelstra: Eine Interpretation des institutionistischen Aussagenkalküls An interpretation of the intuitionistic propositional calculus Gödel 1933g: (See introductory note under Gödel 1933b.) Bemerkung über projektive Abbildungen Remark concerning projective mappings Gödel 1933h: (See introductory note under Gödel 1933b.) Diskussion über koordinatenlose Differentialgeometrie Discussion concerning coordinate-free differential geometry Gödel 1933i: (See introductory note under Gödel 1932a.) Zum Enscheidungsproblem des logischen Funktionenkalküls On the decision probelm for the functional calculus of logic Gödel 1933j: Review of Kaczmarz 1932 Gödel 1933k: Review of Lewis 1932 Gödel 1933l: (See introductory note under Gödel 1932a.) Review of Kalmár 1933 Gödel 1933m: Review of Hahn 1932 Gödel 1934: Introductory note to 1934Stephen C. Kleene: On undecidable propositions of formal mathematical systems Gödel 1934a: Review of Skolem 1933 Gödel 1934b: Introductory note to 1934bW. V. Quine: Review of Quine 1933 Gödel 1934c: Introductory note to 1934c and 1935Robert L. Vaught: Review of Skolem 1933a Gödel 1934d: Review of Chen 1933 Gödel 1934e: (See introductory note under Gödel 1932k.) Review of Church 1933 Gödel 1934f: Review of Notcutt 1934 Gödel 1935: (See introductory note under Gödel 1934c.) Review of Skolem 1934 Gödel 1935a: Introductory note to 1935aW. V. Quine: Review of Huntington 1934 Gödel 1935b: Review of Carnap 1934 Gödel 1935c: Review of Kalmár 1934 Gödel 1936: Introductory note to 1936John W. Dawson, Jr.: Diskussionsbemerkung Discussion remark Gödel 1936a: Introductory note to 1936aRohit Parikh: Über die Länge von Beweisen On the length of proofs Gödel 1936b: (See introductory note under Gödel 1932k.) Review of Church 1935 Textual notes References Index

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Book SynopsisThe Quine-Putnam indispensability argument in the philosophy of mathematics urges us to place mathematical entities on the same ontological footing as other theoretical entities essential to our best scientific theories. Recently, the argument has come under serious scrutiny, with many influential philosophers unconvinced of its cogency. This book not only outlines the indispensability argument in considerable detail but also defends it against various challenges.Trade ReviewOverall, the book presents a clear picture of the Quinean world view. * Mathematical Reviews *

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Book SynopsisWhen engaged in mathematics, most people tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism about mathematics raises serious questions: What are mathematical things? Where are they? How do we know about them? Penelope Maddy delineates and defends a novel version of mathematical realism that answers the traditional questions and refocuses philosophical attention on the pressing foundational issues of contemporary mathematics.Trade ReviewShe has ... clearly marked out an original and interesting position. * Times Higher Education Supplement *the book is written in a lively, engaging style. We hope that it serves to stimulate others to think seriously about issues in philosophy of mathematics because, as Maddy claims, these issues bear directly on mainstream philosophy. * Philosophy of Science *Table of ContentsRealism: Pre-theoretic realism; Realism in philosophy; Realism and truth; Realism in mathematics; Perception and intuition: What is the question?; Perception; Intuition; Godelian Platonism; Numbers: What numbers could not be; Numbers as properties; Frege numbers; Axioms: Reals and sets of reals; Axiomization; Open problems; Competing theories; The challenge; Monism and beyond: Monism; Field's nominalism; Structuralism; Summary; References; Index.

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Book SynopsisStewart Shapiro presents a distinctive and persuasive view of the foundations of mathematics, arguing controversially that second-order logic has a central role to play in laying these foundations. To support this contention, he first gives a detailed development of second-order and higher-order logic, in a way that will be accessible to graduate students. He then demonstrates that second-order notions are prevalent in mathematics as practised, and that higher-order logic is needed to codify many contemporary mathematical concepts. Throughout, he emphasizes philosophical and historical issues that the subject raises. Foundations without Foundationalism is a key contribution both to philosophy of mathematics and to mathematical logic. ''In this excellent treatise Shapiro defends the use of second-order languages and logic as framework for mathematics. His coverage of the wide range of logical and philosophical topics required for understanding the controversy over second-order logic is Trade ReviewContains more on second-order logic than is readily available in any other textbook or survey. Philosophically, the book also contains many words of wisdom. * Journal of Symbolic Logic *Table of ContentsPART I: ORIENTATION; 1. TERMS AND QUESTIONS; 2. FOUNDATIONALISM AND FOUNDATIONS OF MATHEMATICS; PART II: LOGIC AND MATHEMATICS; 3. THEORY; 4. METATHEORY; 5. SECOND-ORDER LOGIC AND MATHEMATICS; 6. ADVANCED METATHEORY; PART III: HISTORY AND PHILOSOPHY; 7. THE HISTORICAL 'TRIUMPH' OF FIRST-ORDER LANGUAGES; 8. SECOND-ORDER LOGIC AND RULE-FOLLOWING; 9. THE COMPETITION; REFERENCES; INDEX

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Book SynopsisPaolo Mancosu presents a series of innovative studies in the history and the philosophy of logic and mathematics in the first half of the twentieth century. The Adventure of Reason is divided into five main sections: history of logic (from Russell to Tarski); foundational issues (Hilbert''s program, constructivity, Wittgenstein, Gödel); mathematics and phenomenology (Weyl, Becker, Mahnke); nominalism (Quine, Tarski); semantics (Tarski, Carnap, Neurath). Mancosu exploits extensive untapped archival sources to make available a wealth of new material that deepens in significant ways our understanding of these fascinating areas of modern intellectual history. At the same time, the book is a contribution to recent philosophical debates, in particular on the prospects for a successful nominalist reconstruction of mathematics, the nature of finitist intuition, the viability of alternative definitions of logical consequence, and the extent to which phenomenology can hope to account for the exaTrade ReviewThis book contains an enormous amount of material that historians will wish to consult. Mancosu convincingly demonstrates that there is a great deal more that we can still learn about the origins of modern mathematical logic. * Michael Potter, Philosophia Mathematica *Table of ContentsPART 1: HISTORY OF LOGIC; OART 2: FOUNDATIONS OF MATHEMATICS; PART 3: PHENOMENOLOGY AND MATHEMATICS; PART 4: NOMINALISM; PART 5: THE EMERGENCE OF SEMANTICS: TRUTH AND LOGICAL CONSEQUENCE

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Book SynopsisThis is the story behind the idea of number, from the Pythagoreans, up until the turn of the 20th century, through Greek, Islamic & European mathematics.Trade ReviewCorry has compiled a readable account of the history of mathematics focusing on numbers, although for most of the period in question, arithmetic and geometry are not easily separable. The required level of sophistication of the reader is not great, it is certainly at the level of a first-year undergraduate, or a keen sixth-former who is studying mathematics. Even as an experienced university mathematician, the reviewer learnt many interesting things, and has some misconceptions remedied, on reading Corry's Brief History. * Robin Chapman, LMS Newsletter *This fine book gives what its title promises ... a well-written treatment of the subject. * Underwood Dudley, MAA Reviews *It is a highly recommended and pleasant read, not pedantic, but not casual either ... The book is written with great care ... * Adhemar Bultheel, European Mathematical Society *A Brief History of Numbers is a meticulously researched and carefully crafted look at how mathematicians have explored the concept of number. Corry's prose is clear and engaging, and the mathematical content is uniformly accessible to his audience. ... I highly recommend A Brief History of Numbers to mathematics teachers who wish to know more about how our current edifice of natural, rational, real, complex, and infinite numbers came to be built. * James V. Rauff, Mathematics Teacher *Table of Contents1. The System of Numbers: An Overview ; 2. Writing Numbers: Now and Back Then ; 3. Numbers and Magnitudes in the Greek Mathematical Tradition ; 4. Construction Problems and Numerical Problems in the Greek Mathematical Tradition ; 5. Numbers in the Tradition of Medieval Islam ; 6. Numbers in Europe from the 12th to the 16th Centuries ; 7. Number and Equations at the Beginning of the Scientific Revolution ; 8. Number and Equations in theWorks of Descartes, Newton, and their Contemporaries ; 9. New Definitions of Complex Numbers in the Early 19th Century ; 10. "What are numbers and what should they be?" Understanding Numbers in the Late 19th Century ; 11. Exact Definitions for the Natural Numbers: Dedekind, Peano and Frege ; 12. Numbers, Sets and Infinity. A Conceptual Breakthrough at the Turn of the Twentieth Century ; 13. Epilogue: Numbers in Historical Perspective

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Book SynopsisGrounding Concepts tackles the issue of arithmetical knowledge, developing a new position which respects three intuitions which have appeared impossible to satisfy simultaneously: a priorism, mind-independence realism, and empiricism.Drawing on a wide range of philosophical influences, but avoiding unnecessary technicality, a view is developed whereby arithmetic can be known through the examination of empirically grounded concepts. These are concepts which, owing to their relationship to sensory input, are non-accidentally accurate representations of the mind-independent world. Examination of such concepts is an armchair activity, but enables us to recover information which has been encoded in the way our concepts represent. Emphasis on the key role of the senses in securing this coding relationship means that the view respects empiricism, but without undermining the mind-independence of arithmetic or the fact that it is knowable by means of a special armchair method called conceptual Trade ReviewAnyone with the slightest interest in the nature of mathematics should give [Jenkins] serious study. * James Robert Brown and James Davies. Philosophical Quarterly *offers and original treatment of arithmetic that is clearly articulated and carefully argued... It is a book that should be read by anyone with an interest in these topics, and will repay careful study. * Albert Casullo, Mind *I think highly of this book. Grounding Concepts adds a genuinely new option to the philosophical landscape. The central idea - that sense experience may be relevant to the epistemic status of concepts and thus play a non-evidential role in explaining knowledge - is both sensible and clever. The book is sophisticated and accessible, both extremely careful and extremely clear... Grounding Concepts is an excellent book. It provides a sophisticated and clear discussion of a difficult nest of issues in the philosophy of mathematics, epistemology, philosophy of mind, and metaphysics. By developing a new theoretical option, it makes a significant contribution to the literature on the epistemology of the a priori. Anyone interested in the epistemology of arithmetic or the nature of a priori knowledge would profit from reading it. * Joshua Schechter, Notre Dame Philosophical Reviews *Table of ContentsPART 1 - REALISM AND KNOWLEDGE; PART 2 - AN EPISTEMOLOGY FOR ARITHMETIC; PART 3 - OBJECTIONS

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Book SynopsisAddresses fundamental and interrelated philosophical issues concerning modality and identity, issues that were pivotal to the development of analytic philosophy in the twentieth century. This work is intended for graduate students in the subject and professional philosophers.Trade ReviewThe essays make important contributions to contemporary debated concerning modality, individuation, mathmatical structuralism and personal identity. The collection is tus warmly recommended to anyone interested in these areas. * Oystein Linnebo MIND *the volume . . . is of high quality and contains important contributions to many areas of contemporary metaphysics * Matti Eklund, Notre Dame Philosophical Review *all in all, this is an impressive volume, of significant interest to anyone who wants to stay abreast of developments in contemporary metaphysics * Matti Eklund, Notre Dame Philosophical Review *Table of ContentsI. MODALITY ; II. IDENTITY AND INDIVIDUATION ; III. PERSONAL IDENTITY

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Book SynopsisTruth Through Proof defends an anti-platonist philosophy of mathematics derived from game formalism. Classic formalists claimed implausibly that mathematical utterances are truth-valueless moves in a game. Alan Weir aims to develop a more satisfactory successor to game formalism utilising a widely accepted, broadly neo-Fregean framework, in which the proposition expressed by an utterance is a function of both sense and background circumstance. This framework allows for sentences whose truth-conditions are not representational, which are made true or false by conditions residing in the circumstances of utterances but not transparently in the sense. Applications to projectivism and fiction pave the way for the claim that mathematical utterances are made true or false by the existence of concrete proofs or refutations, though these truth-making conditions form no part of their sense or informational content. The position is compared with rivals, an account of the applicability of mathematics developed, and a new account of the nature of idealisation proffered in which it is argued that the finitistic limitations Gödel placed on proofs are without rational justification. Finally a non-classical logical system is provided in which excluded middle fails, yet enough logical power remains to recapture the results of standard mathematics.Trade ReviewIn this fascinating book, Weir defends a new account of what makes mathematical assertions objectively true or false. * Julian C. Cole, Philosophy in Review *Table of ContentsIntroduction ; 1. Metaphysics ; 2. Ontological Reduction ; 3. Neo-formalism ; 4. Objections and Comparisons ; 5. Applying Mathematics ; 6. Proof Set in Concrete ; 7. Idealisation Naturalised ; 8. Logic ; Conclusion ; Appendix

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Book SynopsisThis Handbook explores the history of mathematics under a series of themes which raise new questions about what mathematics has been and what it has meant to practise it. It addresses questions of who creates mathematics, who uses it, and how. A broader understanding of mathematical practitioners naturally leads to a new appreciation of what counts as a historical source. Material and oral evidence is drawn upon as well as an unusual array of textual sources. Further, the ways in which people have chosen to express themselves are as historically meaningful as the contents of the mathematics they have produced. Mathematics is not a fixed and unchanging entity. New questions, contexts, and applications all influence what counts as productive ways of thinking. Because the history of mathematics should interact constructively with other ways of studying the past, the contributors to this book come from a diverse range of intellectual backgrounds in anthropology, archaeology, art history, pTrade ReviewReview from previous edition "wonderful food for thought for any practitioner" * Times Higher Education Supplement *"a splendid, something-for-everybody treasure-trove of interesting, informative, challenging, well written testaments to the variety and vigor of history of mathematics in our time" * Historia Mathematica *"Well written, well edited and well rounded... a healthy contribution to a burgeoning field of newly self-aware research." * British Journal for the History of Science *Table of ContentsINTRODUCTION; GEOGRAPHIES AND CULTURES: GLOBAL; GEOGRAPHIES AND CULTURES: REGIONAL; GEOGRAPHIES AND CULTURES: LOCAL; PEOPLE AND PRACTICES: LIVES; PEOPLE AND PRACTICES: PRACTICES; PEOPLE AND PRACTICES: PRESENTATION; INTERACTIONS AND INTERPRETATIONS: INTELLECTUAL; INTERACTIONS AND INTERPRETATIONS: MATHEMATICAL; INTERACTIONS AND INTERPRETATIONS: HISTORICAL; ABOUT THE CONTRIBUTORS; INDEX

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Book SynopsisMary Leng offers a defense of mathematical fictionalism, according to which we have no reason to believe that there are any mathematical objects. Perhaps the most pressing challenge to mathematical fictionalism is the indispensability argument for the truth of our mathematical theories (and therefore for the existence of the mathematical objects posited by those theories). According to this argument, if we have reason to believe anything, we have reason to believe that the claims of our best empirical theories are (at least approximately) true. But since claims whose truth would require the existence of mathematical objects are indispensable in formulating our best empirical theories, it follows that we have good reason to believe in the mathematical objects posited by those mathematical theories used in empirical science, and therefore to believe that the mathematical theories utilized in empirical science are true. Previous responses to the indispensability argument have focussed on Trade ReviewMathematics and Reality is to be recommended highly ... it presents a distinctive new version of fictionalism to throw into the contemporary mix that will repay close attention by all philosophers of mathematics. * Alan Weir, British Journal for the Philosophy of Science *this book has the potential to serve as a source of productive disagreement that would significantly advance the realism-anti-realism debate in mathematics. * Jeffrey W. Rowland, Mind *Table of Contents1. Introduction ; 2. Naturalism and Ontology ; 3. The Indispensability of Mathematics ; 4. Naturalism and Mathematical Practice ; 5. Naturalism and Scientific Practice ; 6. Naturalized Ontology ; 7. Mathematics and Make-Believe ; 8. Mathematical Fictionalism and Constructive Empiricism ; 9. Explaining the Success of Mathematics ; 10. Conclusion

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Book SynopsisRichard Tieszen presents an analysis, development, and defense of a number of central ideas in Kurt Gödel''s writings on the philosophy and foundations of mathematics and logic. Tieszen structures the argument around Gödel''s three philosophical heroes - Plato, Leibniz, and Husserl - and his engagement with Kant, and supplements close readings of Gödel''s texts on foundations with materials from Gödel''s Nachlass and from Hao Wang''s discussions with Gödel. As well as providing discussions of Gödel''s views on the philosophical significance of his technical results on completeness, incompleteness, undecidability, consistency proofs, speed-up theorems, and independence proofs, Tieszen furnishes a detailed analysis of Gödel''s critique of Hilbert and Carnap, and of his subsequent turn to Husserl''s transcendental philosophy in 1959. On this basis, a new type of platonic rationalism that requires rational intuition, called ''constituted platonism'', is developed and defended. Tieszen showTrade ReviewTieszen has long been one of the bridge builders in contemporary philosophy, who is engaged by the philosophical issues and studies them with a broad background and an open mind. There is much to be learned by this, and I am eagerly looking forward to Tieszen's continuation of this interesting and very valuable work. * Dagfinn Follesda, Philosophia MathematicaJuliette Kennedy, Notre Dame Philosophical Reviews *Table of ContentsPreface ; 1. Setting the Stage ; 2. Consistency, and the Ascent to Platonic Rationalism ; 3. Godel's Path From Hilbert and Carnap to Husserl ; 4. A New Kind of Platonism ; 5. Consciousness, Reason, and Intentionality ; 6. Constituted Platonism, Reason, and Mathematical Knowledge ; 7. Minds and Machines ; 8. Reason, Science, and Evidence ; Bibliography ; Index

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Book SynopsisSeries Editor's Foreword Preface Introduction 1. Judgement and the Emergence of Logical Realism in Britain 2. From Descriptive Psychology to Analytic Philosophy (1888-1899) 3. Psychologism and the Problem of Error (1899-1907) 4. Judgement, Propositional Attitudes and the Proposition (1908-1944) 5. Tropes and Predication Conclusion Bibliography IndexTrade Review“This book is one recent product of her work on this subject, which first saw light as a dissertation, then in a series of papers, and now appears in a revised and expanded version of her early work for the History of Analytic Philosophy series … . The perspective van der Schaar brings here is … a valuable addition to the detailed account of the early development of analytic philosophy at Cambridge.” (Consuelo Preti, Journal of the History of Analytical Philosophy, Vol. 4 (3), 2016)Table of ContentsSeries Editor's Foreword Preface Introduction 1. Judgement and the Emergence of Logical Realism in Britain 2. From Descriptive Psychology to Analytic Philosophy (1888-1899) 3. Psychologism and the Problem of Error (1899-1907) 4. Judgement, Propositional Attitudes and the Proposition (1908-1944) 5. Tropes and Predication Conclusion Bibliography Index

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Book SynopsisMax Tegmark leads us on an astonishing journey through past, present and future, and through the physics, astronomy and mathematics that are the foundation of his work, most particularly his hypothesis that our physical reality is a mathematical structure and his theory of the ultimate multiverse. In a dazzling combination of both popular and groundbreaking science, he not only helps us grasp his often mind-boggling theories, but he also shares with us some of the often surprising triumphs and disappointments that have shaped his life as a scientist. Fascinating from first to last—this is a book that has already prompted the attention and admiration of some of the most prominent scientists and mathematicians.

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Book SynopsisThe amazing story of one of the greatest math problems of all time and the reclusive genius who solved itIn the tradition of Fermat’s Enigma and Prime Obsession, George Szpiro brings to life the giants of mathematics who struggled to prove a theorem for a century and the mysterious man from St. Petersburg, Grigory Perelman, who fi nally accomplished the impossible. In 1904 Henri Poincaré developed the Poincaré Conjecture, an attempt to understand higher-dimensional space and possibly the shape of the universe. The problem was he couldn’t prove it. A century later it was named a Millennium Prize problem, one of the seven hardest problems we can imagine. Now this holy grail of mathematics has been found.Accessibly interweaving history and math, Szpiro captures the passion, frustration, and excitement of the hunt, and provides a fascinating portrait of a contemporary noble-genius.

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Book SynopsisLearn about the most important mathematical ideas, theorems, and movements in The Math Book.Part of the fascinating Big Ideas series, this book tackles tricky topics and themes in a simple and easy to follow format. Learn about Math in this overview guide to the subject, brilliant for novices looking to find out more and experts wishing to refresh their knowledge alike! The Math Book brings a fresh and vibrant take on the topic through eye-catching graphics and diagrams to immerse yourself in. This captivating book will broaden your understanding of Math, with:- More than 85 ideas and events key to the development of mathematics- Packed with facts, charts, timelines and graphs to help explain core concepts- A visual approach to big subjects with striking illustrations and graphics throughout- Easy to follow text makes topics accessible for people at any level of understandingThe Math Book is a captivati

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Book SynopsisThroughout history, thinkers from mathematicians to theologians have pondered the mysterious relationship between numbers and the nature of reality. In this fascinating book, Mario Livio tells the tale of a number at the heart of that mystery: phi, or 1.6180339887...This curious mathematical relationship, widely known as The Golden Ratio, was discovered by Euclid more than two thousand years ago because of its crucial role in the construction of the pentagram, to which magical properties had been attributed. Since then it has shown a propensity to appear in the most astonishing variety of places, from mollusk shells, sunflower florets, and rose petals to the shape of the galaxy. Psychological studies have investigated whether the Golden Ratio is the most aesthetically pleasing proportion extant, and it has been asserted that the creators of the Pyramids and the Parthenon employed it. It is believed to feature in works of art from Leonardo da Vinci's Mona

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a huge range and FREE tracked UK delivery on ALL orders.

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a huge range and FREE tracked UK delivery on ALL orders.

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