Philosophy of mathematics Books

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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 SynopsisMath and Art: An Introduction to Visual Mathematics explores the potential of mathematics to generate visually appealing objects and reveals some of the beauty of mathematics. It includes numerous illustrations, computer-generated graphics, photographs, and art reproductions to demonstrate how mathematics can inspire or generate art.Focusing on accessible, visually interesting, and mathematically relevant topics, the text unifies mathematics subjects through their visual and conceptual beauty. Sequentially organized according to mathematical maturity level, each chapter covers a cross section of mathematics, from fundamental Euclidean geometry, tilings, and fractals to hyperbolic geometry, platonic solids, and topology. For art students, the book stresses an understanding of the mathematical background of relatively complicated yet intriguing visual objects. For science students, it presents various elegant mathematical theories and notions.Features Provides an accessible introduction to mathematics in art Supports the narrative with a self-contained mathematical theory, with complete proofs of the main results (including the classification theorem for similarities) Presents hundreds of figures, illustrations, computer-generated graphics, designs, photographs, and art reproductions, mainly presented in full color Includes 21 projects and approximately 280 exercises, about half of which are fully solved Covers Euclidean geometry, golden section, Fibonacci numbers, symmetries, tilings, similarities, fractals, cellular automata, inversion, hyperbolic geometry, perspective drawing, Platonic and Archimedean solids, and topology New to the Second Edition New exercises, projects and artworks Revised, reorganized and expanded chapters More use of color throughout Trade Review"A beautiful book that brings out a wide range of mathematics, ancient to modern, with rich and often unexpected connections to the visual arts."– Catherine A. Gorini, Maharishi International University"Kalajdzievski takes us on a fascinating journey through the most visual subjects in mathematics. This book has the rare quality of not only organizing topics in a sequence that reveals how geometric concepts build upon one another, but also presenting each topic in a compact and self-contained manner for readers who prefer to browse for different entry points into the text. Although verbal explanations and mathematical formulae abound here, it is the colorful diagrams and photographs that capture the attention and enchant the eye. "– James Mai, Professor of Art, Illinois State University"The book presents mathematical and geometrical topics which can be expressed as the artistic pieces and serve to inspiring the artists to explore visual beauty and power of mathematics. In comparison with the first edition (of 2008), this book is noticeably extended to 280 exercises (from 190 originally) with solutions given to a half of them, 740 figures and artworks (from 556 previously), and 21 projects suggested for students.[. . . ] The book contains various illustrations and computer-generated graphics, photographs and art reproductions almost in each page, revealing an astonishing interaction of mathematics and artistic findings in human civilization and culture. [. . . ] The book can be useful to instructors and students, and interesting to any readers wishing to extend their knowledge and understanding of the esthetics and science of the visual math and mathematical art."– Technometrics"There are many books about mathematics and art; this one distinguishes itself as an “unorthodox geometry textbook,” with exercises and fun art projects. The book is based on 20 years of offering a course to more than 10,000 students. It stops short of covering some of the mathematics (groups are mentioned but not defined), though one theorem (classification of similarities) is proved in an appendix. Topics are Euclidean geometry, transformations of the plane, similarities and fractals, hyperbolic geometry, perspective, three-dimensional objects, and topology. The book averages two figures per page, with many utterly beautiful in color. You might be surprised at the sophisticated mathematical content of some crop circles (no doubt made by aliens!), and amazed by some of the illustrations of artworks."– Mathematics Magazine, MAAPraise for the First Edition"This delightful book grew out of set of teaching notes for an interdisciplinary course called Math in Art that was co-taught by a mathematician and an artist or architect. … The mathematical ideas are presented visually in a way that seems quite natural, and it engages the reader through explorations with lots of hands-on exercises. The mathematical presentation is solid, and the choice of topics puts the focus on the visual presentation of mathematical concepts. The illustrations are beautiful! … This text is very readable. The mathematics is accessible to those with little mathematical background, and yet the presentation is still engaging for those with more background."—MAA Reviews, March 2009"All in all, this work offers an excellent account of art inspired by mathematics and art generated by mathematics, and it should interest readers in both fields. Summing Up: Highly Recommended."– R.M. Davis, emeritus, Albion College, in Choice: Current Review for Academic Libraries, February 2009, Vol. 46, No. 6Table of ContentsChapter 1. Euclidean Geometry. 1.0. Introduction. 1.1. The Five Axioms of Euclidean Geometry. 1.2. Ruler and Compass Constructions. 1.3. The Golden Ratio. 1.4. Fibonacci Numbers. Chapter 2. Plane Transformations. 2.1. Plane Symmetries. 2.2.* Plane Symmetries, Vectors, and Matrices (Optional). 2.3. Groups of Symmetries Of Planar Objects. 2.4. Frieze Patterns. 2.5. Wallpaper Designs and Tilings of the Plane. 2.6. Tilings and Art. Chapter 3. Similarities, Fractals, and Cellular Automata. 3.1. Similarities and some other Planar Transformations. 3.2.* Complex Numbers (Optional). 3.3. Fractals: Definition and Some Examples. 3.4. Julia Sets. 3.5. Cellular Automata. Chapter 4. Hyperbolic Geometry. 4.1. Non-Euclidean Geometries: Background and Some History. 4.2. Inversion. 4.3. Hyperbolic Geometry. 4.4. Some Basic Constructions in the Poincaré Model. 4.5. Tilings of the Hyperbolic Plane. Chapter 5. Perspective. 5.1. Perspective: A brief overview of the Evolution of the rules of perspective. 5.2. Perspective Drawing and Constructions of Some Two-Dimensional (Planar) Objects. 5.3. Perspective Images of Three-Dimensional Objects. 5.4.* Mathematics of Perspective Drawing: A Brief Overview (Optional). Chapter 6. Some Three-Dimensional Objects. 6.1. Regular and Other Polyhedra. 6.2. Sphere, Cylinder, Cone, and Conic Sections. 6.3. Geometry, Tilings, Fractals, and Cellular Automata in Three Dimensions. Chapter 7. Topology. 7.1. Homotopy of Spaces: An Informal Introduction. 7.2. Two-Manifolds and The Euler Characteristic. 7.3. Non-Orientable Two-Manifolds and Three-Manifolds. Appendix: Classification Theorem for Similarities. Solutions.

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Using a complete interpretation of Whitehead’s philosophical and mathematical writings, this book argues that Whitehead has never been properly understood. It applies Whitehead’s philosophy to problems in the interpretation of science, empirical knowledge, and nature, and develops a new account of philosophical naturalism.

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Book SynopsisThe Unconscious as Space explores the experience of being and the practice of psychoanalysis by thinking of the unconscious in mathematical terms.Anca Carrington introduces mathematical models of space, from dimension theory to algebraic topology and knot theory, and considers their immediate psychoanalytic relevance. The hypothesis that the unconscious is structured like a space marked by impossibility is then examined. Carrington considers the clinical implications, with particular focus on the interplay between language and the unconscious as related topological spaces in which movement takes place along knot-like pathways.The Unconscious as Space will be of appeal to psychotherapists, psychoanalysts and mental health professionals in practice and in training.

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Book SynopsisOriginally published in 1966. An introduction to current studies of kinds of inference in which validity cannot be determined by ordinary deductive models. In particular, inductive inference, predictive inference, statistical inference, and decision making are examined in some detail. The last chapter discusses the relationship of these forms of inference to philosophical notions of rationality. Special features of the monograph include a discussion of the legitimacy of various criteria for successful predictive inference, the development of an intuitive model which exhibits the difficulties of choosing probability measures over infinite sets, and a comparison of rival views on the foundations of probability in terms of the amount of information which the members of these schools believe suitable for fruitful formalization. The bibliographies include articles by statisticians accessible to students of symbolic logic. Table of Contents1. Inductive and Predictive Inference 2. Hypothesis and Predictive Inference 3. Probability and Predictive Inference 4. Statistical Inference 5. Bayesian Statistical Inference 6. Statistical Decision and Utility 7. Theories and Rationality 8. Bibliography

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Book SynopsisThe Equation of Knowledge: From Bayes'' Rule to a Unified Philosophy of Science introduces readers to the Bayesian approach to science: teasing out the link between probability and knowledge. The author strives to make this book accessible to a very broad audience, suitable for professionals, students, and academics, as well as the enthusiastic amateur scientist/mathematician. This book also shows how Bayesianism sheds new light on nearly all areas of knowledge, from philosophy to mathematics, science and engineering, but also law, politics and everyday decision-making.Bayesian thinking is an important topic for research, which has seen dramatic progress in the recent years, and has a significant role to play in the understanding and development of AI and Machine Learning, among many other things. This book seeks to act as a tool for proselytising the benefits and limits of Bayesianism to a wider public. Features Trade ReviewLê Nguyên Hoang takes us on a fascinating intellectual journey into Bayesianism, cutting across many social and natural sciences. The Equation of Knowledge: From Bayes' Rule to a Unified Philosophy of Science is a real page turner.—George Zaccour, HEC Montréal and co-author of Handbook of Dynamic Game Theory"Each chapter is opened with a fascinating epigraph quoting famous persons, and is completed by the most recent references. There are multiple illustrations, and the Bayes’ formulae are many times presented via various funny symbols of emoji kind. The book is addressed to a wide audience of students, professionals, and actually any reader interested to be better acquainted with modern ideas in various sciences and philosophy of science, and their Bayesian statistical description and interpretation."— Stan Lipovetsky, Technometrics (Volume 63, 2021 - Issue 1)"[. . . ] Trained in the hard school of online videos, Le Nguyen Hoang has found a new tone to talk about science, a tone that is both rigorous and narrative, where examples illuminate the most abstract questions."— From the Foreword by Gilles Dowek, Professor at École Polytechnique and researcher at the Laboratoire d'Informatique de l'École Polytechnique and the Institut National de Recherche en Informatique et en Automatique (INRIA).Lê Nguyên Hoang takes us on a fascinating intellectual journey into Bayesianism, cutting across many social and natural sciences. The Equation of Knowledge: From Bayes' Rule to a Unified Philosophy of Science is a real page turner.— George Zaccour, HEC Montréal and co-author of Handbook of Dynamic Game Theory"Making math accessible to everyone, showing its connections with dozens of different domains, narrating scientific discovery as a personal human adventure, and sharing impressive enthusiasm: there is definitely something of Greg Chaitin's Meta Math! in Lê Nguyên Hoang's book!" — Rémi Peyre, École des Mines de Nancy"A remarkable piece of work, broad and insightful at the same time. This book is unique in that it gives an accessible journey from subtle probabilistic puzzles to the most advanced concepts at the heart of the machine learning revolution; with unrivalled clarity, it exposes deep ideas that have remained very confidential outside of specialized circles, and that yet are becoming fundamental in the way we understand our world."— Clément Hongler, Associate Professor and Chair of Statistical Field Theory, EPFL "As someone who practices research and publishes academic papers, it is frustrating to note how little we, scientists, are trained in epistemology. ‘How do we know that we know?’ This question is often neglected or taken for granted. The recent controversies about reproducibility of scientific publishing might be one of the tips of a larger iceberg. This book will, in my opinion, be remembered as one of those that helped melt the iceberg." — El Mahdi El Mhamdi, École Polytechnique Fédérale de Lausanne."The book has a lively writing style, rather like you are listening to an inspiring lecturer. Indeed the author has a French YouTube channel and is clearly enthusiastic about exposition. It is overtly an account of what the author personally finds interesting. [. . .] In teaching a basic college course, focused on the mathematical setup and on the analysis of data, I often find there is one student who comes to office hours and is interested in seeing connections with broad scientific fields, or in conceptual issues of the philosophy of science. I could certainly recommend this book to such a student. Similarly, for the MAA community it could be an innovative basis for an undergraduate seminar course, in which students would choose a topic from the book and delve deeper into it."— David Aldous, Mathematical Association of AmericaTable of ContentsSction I. Pure Bayesianism. 1. On A Transformative Journey. 2. Bayes Theorem. 3. Logically Speaking... 4. Let’s Generalize! 5. All Hail Prejudices. 6. The Bayesian Prophets. 7. Solomonoff’s Demon. Section II. Applied Bayesianism. 8. Can You Keep A Secret? 9. Game, Set and Math. 10. Will Darwin Select Bayes? 11. Exponentially Counter-Intuitive. 12. Ockham Cuts to the Chase. 13. Facts Are Misleading. Section III. Pragmatic Bayesianism. 14. Quick And Not Too Dirty. 15. Wish Me Luck. 16. Down Memory Lane. 17. Let’s Sleep On It. 18. The Unreasonable Effectiveness of Abstraction. 19. The Bayesian Brain. Section IV. Beyond Bayesianism. 20. It’s All Fiction. 21. Exploring The Origins Of Beliefs. 22. Beyond Bayesianism.

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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 SynopsisThis text seeks to get students actively involved in problem-solving, especially of a geometric nature. The approach highlights the mathematical connections between concepts, and aims to enhance students' geometrical intuition.Table of ContentsGetting Started: Strategies for Solving Problems. Episodes in the Measurement of Length, Area, and Volume. Polyhedra. Shortest Path Problems. Kaleidoscopes. Symmetry. What Shapes Are Best? Beehives and Other Packing Problems. Where to Go From Here?: Project Ideas. Credits. Index.

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Book SynopsisCasti Tours offers the most spectacular vistas of modern applied mathematics a Nature Mathematical modeling is about rulesa the rules of reality. Reality Rules explores the syntax and semantics of the language in which these rules are written, the language of mathematics. Characterized by the clarity and vision typical of the author''s previous books, Reality Rules is a window onto the competing dialects of this languagea in the form of mathematical models of real-world phenomenaa that researchers use today to frame their views of reality. Moving from the irreducible basics of modeling to the upper reaches of scientific and philosophical speculation, Volumes 1 and 2, The Fundamentals and The Frontier, are ideal complements, equally matched in difficulty, yet unique in their coverage of issues central to the contemporary modeling of complex systems. Engagingly written and handsomely illustrated, Reality Rules is a fascinating journey into the conceptual underpinninTable of ContentsThe Ways of Modelmaking: Natural Systems and Formal MathematicalRepresentations. Catastrophes, Dynamics and Life: The Singularities of Ecologicaland Natural Resource Systems. Pattern and the Emergence of Living Forms: Cellular Automata andDiscrete Dynamics. Order in Chaos: Variety and Pattern in the Flow of Fluids,Populations and Money. Index.

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Book SynopsisCasti Tours offers the most spectacular vistas of modern appliedmathematics.-- Nature Mathematical modeling is about rules--the rules of reality. RealityRules explores the syntax and semantics of the language in whichthese rules are written, the language of mathematics. Characterizedby the clarity and vision typical of the author''s previous books,Reality Rules is a window onto the competing dialects of thislanguage--in the form of mathematical models of real-worldphenomena--that researchers use today to frame their views ofreality. Moving from the irreducible basics of modeling to the upper reachesof scientific and philosophical speculation, Volumes 1 and 2, TheFundamentals and The Frontier, are ideal complements, equallymatched in difficulty, yet unique in their coverage of issuescentral to the contemporary modeling of complex systems. Engagingly written and handsomely illustrated, Reality Rules is afascinating journey into the conceptual underpinnings of rTable of ContentsStrategies for Survival: Competition, Games and the Theory ofEvolution. The Analytical Engine: A System-Theoretic View of Brains, Minds andMechanisms. Taming Nature and Man: Control, Anticipation and Adaptation inSocial and Biological Processes. The Geometry of Human Affairs: Connective Structure in Art,Literature and Games of Chance. The Mystique of Mechanism: Computation, Complexity and the Limitsto Reason. How Do We Know?: Myths, Models and Paradigms in the Creation ofBeliefs. Index.

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Book SynopsisExploring the syntax and semantics of the language in which mathematical modelling rules are written, this text moves from the basics of modelling to the upper reaches of scientific and philosophical speculation.Table of ContentsStrategies for Survival: Competition, Games and the Theory of Evolution. The Analytical Engine: A System-Theoretic View of Brains, Minds and Mechanisms. Taming Nature and Man: Control, Anticipation and Adaptation in Social and Biological Processes. The Geometry of Human Affairs: Connective Structure in Art, Literature and Games of Chance. The Mystique of Mechanism: Computation, Complexity and the Limits to Reason. How Do We Know?: Myths, Models and Paradigms in the Creation of Beliefs. Index.

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Book SynopsisThis 'state-of-the-art' collection of essays presents some of the best contemporary research into the dynamical processes underlying the formation, maintenance, metamorphosis and dissolution of norms. The volume combines formal modelling with more traditional analysis.Table of Contents1. The evolution of strategies in the iterated prisoner's dilemma Robert Axelrod; 2. Learning to co-operate Cristina Bicchieri; 3. On the dynamics of social norms Pier Luigi Sacco; 4. Learning and efficiency in common interest signalling games David Canning; 5. Learning on a Torus Luca Anderlini and Antonella Ianni; 6. Evolutive vs. naive Bayesian learning Immanuel M. Bomze and Jurgen Eichberger; 7. Learning and mixed strategy equilibria in evolutionary games Vincent P. Crawford; 8. Bayesian learning in games: a non-Bayesian perspective J. S. Jordan; 9. Savage-Bayesian agents play a repeated game Yaw Nyarko; 10. Chaos and the explanatory significance of equilibrium: strange attractors in evolutionary game theory Brian Skyrms.

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Book SynopsisThis is Volume I of the first authoritative translation of Archimedes' works into English. Also provided are a scientific edition of the diagrams, a translation of the ancient commentator Eutocius and a commentary, where attention is paid to the cognitive and aesthetic nature of Archimedes' mathematical practice.Trade ReviewReview of the hardback: ' … this translation is certainly an event of great importance concerning the edition of ancient Greek mathematics. the forthcoming volumes are awaited with impatience!' Zentralblatt MATHTable of ContentsIntroduction; Translation and Commentary: On Sphere and Cylinder Book I; On Sphere and Cylinder Book II; Eutocius' Commentary to On Sphere and Cylinder Book I; Eutocius' Commentary to On Sphere and Cylinder Book II; Bibliography; Index.

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Book SynopsisPhenomenology, Logic, and the Philosophy of Mathematics, first published in 2005, is about logic, mathematical knowledge and mathematical objects. It is concerned with the role of reason and intuition in the exact sciences and it analyzes many of the central positions in the philosophy of logic and philosophy of mathematics: platonism, nominalism, intuitionism, formalism, pragmatism, and others.Table of ContentsPart I. Reason, Science, and Mathematics: 1. Science as a triumph of the human spirit and science in crisis: Husserl and the Fortunes of Reason; 2. Mathematics and transcendental phenomenology; Part II. Kurt Godel, Phenomenology and the Philosophy of Mathematics: 3. Kurt Godel and phenomenology; 4. Godel's philosophical remarks on mathematics and logic; 5. Godel's path from the incompleteness theorems (1931) to Phenomenology (1961); 6. Godel and the intuition of concepts; 7. Godel and Quine on meaning and mathematics; 8. Maddy on realism in mathematics; 9. Penrose and the view that minds are not machines; Part III. Constructivism, Fulfilled Intentions, and Origins: 10. Intuitionism, meaning theory and cognition; 11. The philosophical background of Weyl's mathematical constructivism; 12. What is a proof?; 13. Phenomenology and mathematical knowledge; 14. Logicism, impredicativity, formalism; 15. The philosophy of arithmetic: Frege and Husserl.

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Book SynopsisThis book looks to change our thinking about thinking by looking at communication and cognition (commognition). The explanatory power of the commognitive framework and the manner in which it contributes to our understanding of human development is illustrated through commognitive analysis of mathematical discourse accompanied by vignettes from mathematics classrooms.Trade Review'Sfard has provided us with one of the most impressive, unified, homogenous theories of learning …' Computer-Supported Collaborative LearningTable of ContentsIntroduction; Part I. Discourse on Thinking: 1. Puzzling about (mathematical) thinking; 2. Objectification; 3. Commognition: thinking as communicating; 4. Thinking in language; Part II. Mathematics as Discourse: 5. Mathematics as a form of communication; 6. Objects of mathematical discourse: what mathematizing is all about; 7. Routines: how we mathematize; 8. Explorations, deeds, and rituals: what we mathematize for; 9. Looking back and ahead: solving old quandaries and facing new ones.

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Book SynopsisIn fourth-century Greece Archytas of Tarentum solved a famous mathematical puzzle, saved Plato from the tyrant of Syracuse and led a powerful Greek city state. This 2005 book presents an interpretation of his significance for fourth-century Greek thought and provides a full commentary on all the fragments and testimonia.Trade ReviewReview of the hardback: 'Huffman's book offers expert discussion of a variety of difficult topics … a much needed and authoritative commentary … Lucid argued, helpfully organised, and impressive in its scholarship, this book sets a high standard indeed … a rich volume of over 600 pages … there is much of real value here … a detailed and scholarly treatment of dauntingly difficulty material. Scholars owe Huffman a debt for undertaking this task, and executing it with such authority. It is a work to emulate'. Sylvia Berryman, The University of British ColumbiaReview of the hardback: 'We have here another blockbuster offering from Carl Huffman who has already put us in his debt by a definite study of Philolaus. This work will serve in turn to establish Archytas as a philosopher in his own right, and not simply a footnote to Pythagoras, as has all too often been the case hitherto.' Bryn Mawr Classical ReviewTable of ContentsPart I. Introductory Essays: 1. Life, writings and reception; 2. The philosophy of Archytas; 3. The authenticity question; Part II. Genuine Fragments: 1. Fragment 1; 2. Fragment 2; 3. Fragment 3; 4. Fragment 4; Part III: Genuine Testimonia: 1. Life and writings (A1–A6, B5–B8); 2. Moral philosophy and character; 3. Geometry: the duplication of the cube (A14 and A15); 4. Music; 5. Metaphysics; 6. Physics; 7. Miscellaneous testimonia; Appendix: Spurious writings and testimonia; Appendix: Archytas' name.

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Book SynopsisTopics covered include the realism/anti-realism debate in mathematics, the limits of mathematics, the significance of mathematical notation, inconsistent mathematics and the applications of mathematics. Intended as a primary text for an introductory undergraduate course in the philosophy of mathematics.Trade Review'The present book is like a warm breeze after a cold winter in the rarefied atmosphere of the philosophy of mathematics … the philosophical discussions are always clear, provocative and stimulating. One of the challenges an instructor will face by adopting this book will undoubtedly be to contain the desire of students to discuss in depth some of the issues presented and to curb their enthusiasm and desire to know more or find answers to the questions.' Mathematical ReviewsTable of Contents1. Mathematics and its philosophy; 2. The limits of mathematics; 3. Plato's heaven; 4. Fiction, metaphor, and partial truths; 5. Mathematical explanation; 6. The applicability of mathematics; 7. Who's afraid of inconsistent mathematics?; 8. A rose by any other name; 9. Epilogue: desert island theorems.

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Book SynopsisGottlob Frege (18481925) was unquestionably one of the most important philosophers of all time. He trained as a mathematician, and his work in philosophy started as an attempt to provide an explanation of the truths of arithmetic, but in the course of this attempt he not only founded modern logic but also had to address fundamental questions in the philosophy of language and philosophical logic. Frege is generally seen (along with Russell and Wittgenstein) as one of the fathers of the analytic method, which dominated philosophy in English-speaking countries for most of the twentieth century. His work is studied today not just for its historical importance but also because many of his ideas are still seen as relevant to current debates in the philosophies of logic, language, mathematics and the mind. The Cambridge Companion to Frege provides a route into this lively area of research.Trade Review'Central to this end were Frege's insights on quantification, the notation that expressed it, the logicist project, and the extension of mathematical notions like function and argument to natural language. The long-awaited Cambridge Companion to Frege is a compendium of Fregean scholarship that rigorously explores these and similar topics; editors Thomas Ricketts and Michael Potter have compiled a comprehensive collection of fourteen essays that individually provide focused appraisals of a number of Frege's most substantial insights.' Alexander Bozzo, University of Milwaukee'The long-awaited publication of The Cambridge Companion to Frege is a major event in Frege scholarship … Every serious reader of Frege should read it.' Notre Dame Philosophical Reviews (ndpr.nd.edu)Table of ContentsPreface; Note on translations; Chronology; 1. Introduction Michael Potter; 2. Understanding Frege's project Joan Weiner; 3. Frege's conception of logic Warren Goldfarb; 4. Dummett's Frege Peter Sullivan; 5. What is a predicate? Alex Oliver; 6. Concepts, objects, and the context principle Thomas Ricketts; 7. Sense and reference Michael Kremer; 8. On sense and reference: a critical reception William Taschek; 9. Frege and semantics Richard Heck; 10. Frege's mathematical setting Mark Wilson; 11. Frege and Hilbert Michael Hallett; 12. Frege's folly Peter Milne; 13. Frege and Russell Peter Hylton; 14. Inheriting from Frege: the work of reception, as Wittgenstein did it Cora Diamond.

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Book SynopsisIn this ambitious study, David Corfield sets out a variety of approaches to new thinking about the philosophy of mathematics, and challenges both philosophers and mathematicians to develop the broadest and richest philosophical resources for work in their disciplines.Trade Review'Corfield's book as a whole is itself a fine specimen of a philosophical approach to mathematics that takes its questions and its resources from 'real' mathematics, showing convincingly the richness and fruitfulness of such an approach.' Philosophia Mathematica'I found this book interesting and it is certainly worth looking at if only to increase one's sense of the possibilities for the philosophy of mathematics.' Metascience'What is really special about the book under review is that it demonstrates a philosopher struggling to grapple with modern mathematics as it is actually carried out by practitioners. This is what the author means by 'real mathematics' as quoted in the book title.' Zentralblatt MATHTable of ContentsPreface; 1. Introduction: a role for history; Part I. Human and Artificial Mathematicians: 2. Communicating with automated theorem provers; 3. Automated conjecture formation; 4. The role of analogy in mathematics; Part II. Plausibility, Uncertainty and Probability: 5. Bayesianism in mathematics; 6. Uncertainty in mathematics and science; Part III. The Growth of Mathematics: 7. Lakatos's philosophy of mathematics; 8. Beyond the methodology of mathematical research programmes; 9. The importance of mathematical conceptualisation; Part IV. The Interpretation of Mathematics: 10. Higher dimensional algebra; Appendix; Bibliography; Index.

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Book SynopsisAlone in a cube that's glowing in the darkness, X is content within its little universe of infinite thought. This solitude is disturbed by the appearance of Y, who insists on exposing X to the richness of the physical world. Each begins to long for what the other has, luring them into a strange loop.In this play for two variables, Marcus du Sautoy and Victoria Gould use mathematics and theatre to navigate the furthest reaches of our world. Through a series of surreal episodes, X and Y tackle some of life's greatest questions: where did the universe come from, does time have an end, do we have free will?I is a Strange Loop was first performed by the authors at the Barbican Pit, London, in March 2019.''I is a Strange Loop is a play that plays with ideas, concepts, abstractions and relationships that are, usually, hidden from the sight of ordinary mortals, articulating the ineffable, incarnating the incorporeal, revealing the inconceivable. It makeTrade Review'I is a Strange Loop is a play that plays with ideas, concepts, abstractions andrelationships that are, usually, hidden from the sight of ordinary mortals, articulatingthe ineffable, incarnating the incorporeal, revealing the inconceivable.It makes us feel we know a great deal more than we do. It is also very funny,utterly compelling and marvellously human.' - Simon McBurney'[An] ambitious and stimulating piece.' - Financial Times'Tackles what it means to be human at a time when advances in technologyand scientific research are hurtling forward with unprecedented speed.' - British Theatre Guide

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Book SynopsisA philosophical discussion of the concept of numberIn the book, The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number, Gottlob Frege explains the central notions of his philosophy and analyzes the perspectives of predecessors and contemporaries. The book is the first philosophically relevant discussion of the concept of number in Western civilization. The work went on to significantly influence philosophy and mathematics. Frege was a German mathematician and philosopher who published the text in 1884, which seeks to define the concept of a number. It was later translated into English. This is the revised second edition.Table of Contents I. Views of certain writers on the nature of arithmetical propositions II. Views of certain writers on the concept of Number III. View on unity and one IV. The concept of number V. Conclusions

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Book SynopsisExplores the central problems and the most intriguing new directions in the philosophy of mathematics. The papers are organized thematically, rather than chronologically, to give the best overview of philosophical issues connected with mathematics and the development of mathematical knowledge.Trade Review"For breadth of coverage, Jacquette's anthology of recent work in philosophy of mathematics has few if any rivals. Many of Jacquette's selections are important for understanding current debates, and he provides helpful introductory discussions. This collection will very likely become a standard resource for students and teachers of this field." Sanford Shieh, Wesleyan University Table of ContentsPreface. Acknowledgments. Introduction: Mathematics and Philosophy of Mathematics: Dale Jacquette. Part I: The Realm of Mathematics:. 1. What is Mathematics About?: Michael Dummett. 2. Mathematical Explanation: Mark Steiner. 3. Frege versus Cantor and Dedekind: On the Concept of Number: William W. Tait. 4. The Present Situation in Philosophy of Mathematics: Henry Mehlberg. Part II: Ontology of Mathematics and the Nature and Knowledge of Mathematical Truth:. 5. What Numbers Are: N.P. White. 6. Mathematical Truth: Paul Benacerraf. 7. Ontology and Mathematical Truth: Michael Jubien. 8. An Anti-Realist Account of Mathematical Truth: Graham Priest. 9. What Mathematical Knowledge Could Be: Jerrold J. Katz. 10. The Philosophical Basis of our Knowledge of Number: William Demonpoulos. Part III: Models and Methods of Mathematical Proof:. 11. Mathematical Proof: G.H. Hardy. 12. What Does a Mathematical Proof Prove?: Imre Lakatos. 13. The Four-Color Problem: Kenneth Appel and Wolfgang Haken. 14. Knowledge of Proofs: Peter Pagin. 15. The Phenomenology of Mathematical Proof: Gian-Carlo Rota. 16. Mechanical Procedures and Mathematical Experience: Wilfried Sieg. Part IV: Intuitionism:. 17. Intuitionism and Formalism: L.E.J. Brouwer. 18. Mathematical Intuition: Charles Parsons. 19. Brouwerian Intuitionism: Michael Detlefsen. 20. A Problem for Intuitionism: The Apparent Possibility of Performing Infinitely Many Tasks in a Finite Time: A.W. Moore. 21. A Pragmatic Analysis of Mathematical Realism and Intuitionism: Michel J. Blais. Part V: Philosophical Foundations of Set Theory:. 22. Sets and Numbers: Penelope Maddy. 23. Sets, Aggregates, and Numbers: Palle Yourgrau. 24. The Approaches to Set Theory: John Lake. 25. Where Do Sets Come From? Harold T. Hodes. 26. Conceptual Schemes in Set Theory: Robert McNaughton. 27. What is Required of a Foundation for Mathematics? John Mayberry. Index.

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Book SynopsisWhat is so special about the number 30? How many colors are needed to color a map? Do the prime numbers go on forever? And are there more whole numbers than even numbers? This title explores these and other mathematical puzzles. It leads the reader into some of the fundamental ideas of mathematics, the ideas that make the subject interesting.Trade Review"A thoroughly enjoyable sampler of fascinating mathematical problems and their solutions."--Science "Each chapter is a gem of mathematical exposition... [The book] will not only stretch the imagination of the amateur, but it will also give pleasure to the sophisticated mathematician."--American Mathematical MonthlyTable of ContentsPreface v Introduction 5 1. The Sequence of Prime Numbers 9 2. Traversing Nets of Curves 13 3. Some Maximum Problems 17 4. Incommensurable Segments and Irrational Numbers 22 5. A Minimum Property of the Pedal Triangle 27 6. A Second Proof of the Same Minimum Property 30 7. The Theory of Sets 34 8. Some Combinatorial Problems 43 9. On Waring's Problem 52 10. On Closed Self-Intersecting Curves 61 11. Is the Factorization of a Number into Prime Factors Unique?66 12. The Four-Color Problem 73 13. The Regular Polyhedrons 82 14. Pythagorean Numbers and Fermat's Theorem 88 15. The Theorem of the Arithmetic and Geometric Means 95 16. The Spanning Circle of a Finite Set of Points 103 17. Approximating Irrational Numbers by Means of Rational Numbers ill 18. Producing Rectilinear Motion by Means of Linkages 119 19. Perfect Numbers 129 20. Euler's Proof of the Infinitude of the Prime Numbers 135 21. Fundamental Principles of Maximum Problems 139 22. The Figure of Greatest Area with a Given Perimeter 142 23. Periodic Decimal Fractions 147 24. A Characteristic Property of the Circle 160 25. Curves of Constant Breadth 163 26. The Indispensability of the Compass for the Constructions of Elementary Geometry 177 27. A Property of the Number 30 187 28. An Improved Inequality 192 Notes and Remarks 197

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Book SynopsisNow available in a one-volume paperback, this book traces the development of the most important mathematical concepts, giving special attention to the lives and thoughts of such mathematical innovators as Pythagoras, Newton, Poincare, and Godel. Beginning with a Sumerian short story--ultimately linked to modern digital computers--the author clearly introduces concepts of binary operations; point-set topology; the nature of post-relativity geometries; optimization and decision processes; ergodic theorems; epsilon-delta arithmetization; integral equations; the beautiful ideals of Dedekind and Emmy Noether; and the importance of purifying mathematics. Organizing her material in a conceptual rather than a chronological manner, she integrates the traditional with the modern, enlivening her discussions with historical and biographical detail.Trade Review"Here is one of the clearest expositions of our age's fundamental science, its history and controversies, put together in a volume that manages to bring to its difficult subject all the can't-put-it-down suspense of a good thriller."--Wall Street Journal

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

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Book SynopsisPresenting mathematical ideas of people from a variety of small-scale and traditional cultures, this book humanizes our view of mathematics and expands our conception of what is mathematical. It demonstrates that traditional cultures have mathematical ideas that are far more substantial and sophisticated than is generally acknowledged.Trade ReviewHonorable Mention for the 2003 Award for Best Professional/Scholarly Book in Mathematics and Statistics, Association of American Publishers "A useful reminder of how universal mathematical and logical structures are in any culture. Mathematicians will enjoy seeing the subject they love cropping up in apparently unexpected contexts. Non-mathematicians should be encouraged to realize that some of the processes that seem to appear naturally in everyday life do in fact have a mathematical content."--John O'Connor, Nature "For a mathematician, Mathematics Elsewhere will expand the universe; for a non-mathematician, the expansion will just take a little more time. The book succeeds well in presenting and explaining very different ways of doing math both within particular cultural contexts and in terms of modern mathematics... The author is clearly an excellent teacher and a wonderful explainer. Every time I felt a bit lost, the next sequence would present the same concept in different words or with another example. She is adept at moving from the general to the specific, from narrative to figurative."--Helaine Selin, Science "This interesting book is a fundamental work in the area of ethnomathematics... [T]he author opens numerous doors and directions in which one finds interesting, nontrivial mathematics. Persons interested in investigating the mathematics of non-Western cultures can use this book as a motivation to look beyond the obvious."--Thomas E. Golsdorf, Mathematical Reviews "Ascher illustrates that non-Western cultures have developed sophisticated mathematical ideas often without having any formal concept of mathematics. This stimulating book deserves a wide audience, especially among those involved in teaching the subject."--Andrew Bowler, New Scientist "In a follow-up to Ascher's highly recommended Ethnomathematics, this scholarly work describes the anthropology of mathematical ideas in traditional societies and shows how the same ideas might be expressed by standard mathematical expressions... It is particularly interesting to see how people with no separate mathematical language made practical use of sophisticated mathematical ideas."--Library Journal "All throughout the book, I was struck by how many uses human cultures have found for modular arithmetic... [I]t appears that mathematics may be an essential survival skill for the human species rather than an extraneous one. The descriptions in this book describe so many different applications, that it becomes hard to deny that something more fundamental is responsible for the many ways we find to person mathematical operations."--Charles Ashbacher, MAA Online "Ascher's spendid book is rich in possibilities for raising readers' horizons: anthropological, educational, mathematical, and philosophical."--Philip J. Davis, SIAM News "Ascher's book is at once a scholarly progress report and an introduction for the curious general reader to a relatively new area of study known as ethnomathematics... Ascher offers a new way of understanding the customs and traditions of non-Western people, adding the lens of mathematics to those of literature, anthropology, and sociology... [She] proves adept at illuminating the connections between local and global mathematics... Part of what makes the volume accessible to the general reader ... is Ascher's evident love for her subject. The mathematics she includes clearly serves a larger purpose: to enhance and illuminate the anecdotes that are the foundation of genuine cultural understanding."--James V. Rauff, Natural HistoryTable of ContentsPreface ix Introduction 1 CHAPTER 1: The Logic of Divination 5 CHAPTER 2: Marking Time 39 CHAPTER 3: Cycles of Time 59 CHAPTER 4: Models and Maps 89 CHAPTER 5: Systems of Relationships 127 CHAPTER 6: Figures on the Threshold 161 CHAPTER 7: Epilogue 191 Index 205

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Book SynopsisPresents excerpts from the travel diary of a Japanese mathematician, Yamaguchi Kanzan, who journeyed on foot throughout Japan to collect temple geometry problems. This book explains the sacred and devotional aspects of sangaku, and reveals how Japanese folk mathematicians discovered many theorems independently of mathematicians in the West.Trade ReviewWinner of the 2008 PROSE Award in Mathematics, Association of American Publishers "Now Fukagawa Hidetoshi, a mathematics teacher, and writer Tony Rothman present a collection of Sangaku problems in their book, Sacred Mathematics. The puzzles range from simple algebra within the grasp of any intermediate-school student, to challenging problems that require graduate-school mathematics to solve. Copious illustrations and many detailed solutions show the scope, complexity, and beauty of what was tackled in Japan during the Tokugawa shogunate."--Peter J. Lu, Nature "Fascinating and beautiful book."--Physics World "This book is the most thorough (and beautiful) account of Japanese temple geometry (sangaku) available."--Paul J. Campbell, Mathematics Magazine "The difficult problems with complete solutions and rich commentary that comprise the heart of this book will interest every mathematics student."--Choice "This is a marvelous book. Good books are not just written or compiled, they are crafted. Sacred Mathematics is a well crafted work that combines mathematics, history and cultural considerations into an intriguing narrative... The writing style is appealing and the organization of material excellent. Princeton University Press must be congratulated on producing this quality publication and offering it at an agreeable price. This book is highly recommended for personal reading and library acquisition. It should be especially appealing to problem solvers."--Frank J. Swetz, Convergence "A unique book in every respect. Sacred Mathematics demonstrates how mathematical thinking can vary by culture yet transcend cultural and geographic boundaries."--International Institute for Asian Studies NewsletterTable of ContentsForeword by Freeman Dyson ix Preface by Fukagawa Hidetoshi xiii Preface by Tony Rothman xv Acknowledg ments xix What Do I Need to Know to Read This Book? xxi Notation xxv Chapter 1: Japan and Temple Geometry 1 Chapter 2: The Chinese Foundation of Japanese Mathematics 27 Chapter 3: Japa nese Mathematics and Mathematicians of the Edo Period 59 Chapter 4: Easier Temple Geometry Problems 89 Chapter 5: Harder Temple Geometry Problems 145 Chapter 6: Still Harder Temple Geometry Problems 191 Chapter 7: The Travel Diary of Mathematician Yamaguchi Kanzan 243 Chapter 8: East and West 283 Chapter 9: The Mysterious Enri 301 Chapter 10: Introduction to Inversion 313 For Further Reading 337 Index 341

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Book SynopsisConcentrates on thirty highlights of pure and applied mathematics. This book opens by discussing the four main philosophical foundations of mathematics of the nineteenth century and ends by describing the four important open mathematical problems of the twenty-first century.Trade Review"Odifreddi's overview is of course a personal one, but it is hard to argue with either his choices or his organization. This is a perfect handle on an otherwise bewildering proliferation of ideas."--Ben Longstaff, New Scientist "Odifreddi clearly and concisely describes important 20th-century developments in pure and applied mathematics... Unlike similar volumes, this book keeps descriptions general and contains a short section on the philosophical foundations of mathematics to help non-mathematicians easily navigate the material."--Library Journal "This is an astonishingly readable, succinct, and wonderful account of twentieth-century mathematics! It is a great book for mathematics majors, students in liberal-arts courses in mathematics, and the general public. I am amazed at how easily the author has set out the achievements in a broad array of mathematical fields. The writing appears effortless."--Paul Campbell, Mathematics Magazine "Piergiogio Odifreddi's book successfully portrays the major developments in 20th century mathematics by an examination of the mathematical problems that have gained prominence during the past 100 years... [T]he literary style is such that the contents are made accessible to a very wide readership, but with no hint of oversimplification."--P.N. Ruane, MathDL "Odifreddi ... has an engaging and effective style and a knack for compact but comprehensible summaries, making his presentation seem effortless. The Mathematical Century can be dabbled in, read through, or perhaps even used as a quick reference."--Danny Yee, Danny ReviewsTable of ContentsForeword xi Acknowledgments xvii Introduction 1 CHAPTER 1: THE FOUNDATIONS 8 1.1. The 1920s: Sets 10 1.2. The 1940s: Structures 14 1.3. The 1960s: Categories 17 1.4. The 1980s: Functions 21 CHAPTER TWO: PURE MATHEMATICS 25 2.1. Mathematical Analysis: Lebesgue Measure (1902) 29 2.2. Algebra: Steinitz Classification of Fields (1910) 33 2.3. Topology: Brouwer's Fixed-Point Theorem (1910) 37 2.4. Number Theory: Gelfand Transcendental Numbers (1929) 39 2.5. Logic: Godel's Incompleteness Theorem (1931) 43 2.6. The Calculus of Variations: Douglas's Minimal Surfaces (1931) 47 2.7. Mathematical Analysis: Schwartz's Theory of Distributions (1945) 52 2.8. Differential Topology: Milnor's Exotic Structures (1956) 56 2.9. Model Theory: Robinson's Hyperreal Numbers (1961) 59 2.10. Set Theory: Cohen's Independence Theorem (1963) 63 2.11. Singularity Theory: Thom's Classification of Catastrophes (1964) 66 2.12. Algebra: Gorenstein's Classification of Finite Groups (1972) 71 2.13. Topology: Thurston's Classification of 3-Dimensional Surfaces (1982) 78 2.14. Number Theory: Wiles's Proof of Fermat's Last Theorem (1995) 82 2.15. Discrete Geometry: Hales's Solution of Kepler's Problem (1998) 87 CHAPTER THREE: APPLIED MATHEMATICS 92 3.1. Crystallography: Bieberbach's Symmetry Groups (1910) 98 3.2. Tensor Calculus: Einstein's General Theory of Relativity (1915) 104 3.3. Game Theory: Von Neumann's Minimax Theorem (1928) 108 3.4. Functional Analysis: Von Neumann's Axiomatization of Quantum Mechanics (1932) 112 3.5. Probability Theory: Kolmogorov's Axiomatization (1933) 116 3.6. Optimization Theory: Dantzig's Simplex Method (1947) 120 3.7. General Equilibrium Theory: The Arrow-Debreu Existence Theorem (1954) 122 3.8. The Theory of Formal Languages: Chomsky's Classification (1957) 125 3.9. Dynamical Systems Theory: The KAM Theorem (1962) 128 3.10. Knot Theory: Jones Invariants (1984) 132 CHAPTER FOUR: MATHEMATICS AND THE COMPUTER 139 4.1. The Theory of Algorithms: Turing's Characterization (1936) 145 4.2. Artificial Intelligence: Shannon's Analysis of the Game of Chess (1950) 148 4.3. Chaos Theory: Lorenz's Strange Attractor (1963) 151 4.4. Computer-Assisted Proofs: The Four-Color Theorem of Appel and Haken (1976) 154 4.5. Fractals: The Mandelbrot Set (1980) 159 CHAPTER FIVE: OPEN PROBLEMS 165 5.1. Arithmetic: The Perfect Numbers Problem (300 BC) 166 5.2. Complex Analysis: The Riemann Hypothesis (1859) 168 5.3. Algebraic Topology: The Poincare Conjecture (1904) 172 5.4. Complexity Theory: The P=NP Problem (1972) 176 Conclusion 181 References and Further Reading 187 Index 189

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Book SynopsisBritish mathematician Alan Turing, credited with cracking the German Enigma code during World War II, he died in 1954 after eating a cyanide-laced apple - his death was ruled a suicide. This book reveals the author's personal reflections on Turing and other fellow mathematicians.Trade Review"The text is enlivened by many unusual mathematical examples, and by Ruelle's reflections on his own and other famous mathematicians' experiences...If mathematics is what mathematicians do, are there any psychological traits or personalities that characterize mathematics? Ruelle addresses this lightly with some illuminating insights...Mathematicians and theoretical physicists will enjoy Ruelle."--Donal O'Shea, Nature "The mathematician David Ruelle is well known for his work on nonlinear dynamics and turbulence, and his new book, The Mathematician's Brain, is a book about mathematics and what it all means... The book's value lies in Mr. Ruelle's description of the curious inner life of mathematicians."--David Berlinski, New York Sun "[David Ruelle], a mathematical physicist, reflects on how the mathematician works and how mathematics sheds light on the nature of knowledge. Ruelle also examines the anatomy of mathematical texts, looks at processes by which mathematical concepts are developed, and explores ideas such as infinity, the circle theorem, and algebraic geometry."--Science News "After a lifetime of research and teaching, [Ruelle argues] that mathematical breakthroughs do not come from simply manipulating symbols according to strict rules. His chapters on individual mathematicians work very well, and allow the reader...a real sense of what it is like to work at the forefront of the discipline."--Andrew Robinson, Physics World "An idiosyncratic, oddly intriguing work."--J. Mayer, Choice "David Ruelle is a mathematical physicist who tries to explain to the general reader what mathematics is and how mathematicians go about their work... The book is well organized, clearly written and gives a fair impression of the working mathematician."--Michael Atiyah, Brain "For any reader interested as much in what being a mathematician is like as in what mathematics is, this book offers the inside scoop... It is only a very good book that stimulates discussion of foundational issues at all, and The Mathematician's Brain does that and much else beside. One finds a rich, multi-textured, human account of mathematics and mathematical life here, an account that makes one wish to spend an afternoon with the author, in pleasant conversation about whatever captures one's fancy at the moment."--Tim Maudlin, Journal of Statistical Physics "The Mathematician's Brain takes you inside the world--and heads--of mathematicians. It is a journey you won't soon forget."--L'Enseignement Mathematique "The Mathematician's Brain is a very readable tour through the landscape of contemporary mathematics. David Ruelle locates mathematics as a human practice, subject to social and political pressures as well as the limitations of human brains, without losing site of its status as an objective, rule-governed discipline. The book is packed with personal anecdotes and speculative comments on the nature of mathematics which display the author's clear enthusiasm for his subject... As an accessible run-through of one mathematician's love-affair with his subject, The Mathematician's Brain is an inviting presentation which introduces readers to the fascinating realm of mathematics and its philosophy."--Mary C. Leng, Mathematical Reviews "It has an intimate, personal definitions flavor, inviting the reader to get to know Ruelle himself, not only the mathematics he cares to expound. He turns out do be no dry, scholar, but a humane, opinionated, deeply thoughtful fellow human. The mathematics he chooses to present is and well explained. The philosophical and aesthetic issues he explores are important and often neglected."--Reuben Hersh, Siam Review "There is an enormous amount to admire in the book... The range of topics treated is very generous."--David Corfield, NoticesTable of ContentsPreface vii Chapter 1: Scientific Thinking 1 Chapter 2: What Is Mathematics? 5 Chapter 3: The Erlangen Program 11 Chapter 4: Mathematics and Ideologies 17 Chapter 5: The Unity of Mathematics 23 Chapter 6: A Glimpse into Algebraic Geometry and Arithmetic 29 Chapter 7: A Trip to Nancy with Alexander Grothendieck 34 Chapter 8: Structures 41 Chapter 9: The Computer and the Brain 46 Chapter 10: Mathematical Texts 52 Chapter 11: Honors 57 Chapter 12: Infinity: The Smoke Screen of the Gods 63 Chapter 13: Foundations 68 Chapter 14: Structures and Concept Creation 73 Chapter 15: Turing's Apple 78 Chapter 16: Mathematical Invention: Psychology and Aesthetics 85 Chapter 17: The Circle Theorem and an Infinite-Dimensional Labyrinth 91 Chapter 18: Mistake! 97 Chapter 19: The Smile of Mona Lisa 103 Chapter 20: Tinkering and the Construction of Mathematical Theories 108 Chapter 21: The Strategy of Mathematical Invention 113 Chapter 22: Mathematical Physics and Emergent Behavior 119 Chapter 23: The Beauty of Mathematics 127 Notes 131 Index 157

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Book SynopsisShows how life often works at the extremes - with values becoming as small (or as large) as possible - and how mathematicians over the centuries have struggled to calculate these problems of minima and maxima. This is the book on optimization for math enthusiasts of all backgrounds.Trade Review"This book was terrific fun to read! I thought I would skim the chapters to write my review, but I was hooked by the preface, and read through the first 100 pages in one sitting... [Nahin shows] obvious delight and enjoyment--he is having fun and it is contagious."--Bonnie Shulman, MAA Online "When Least is Best is clearly the result of immense effort... [Nahin] just seems to get better and better... The book is really a popular book of mathematics that touches on a broad range of problems associated with optimization."--Dennis S. Bernstein, IEEE Control Systems Magazine "[When Least is Best is] a wonderful sourcebook from projects and is just plain fun to read."--Choice "This book is highly recommended."--Clark Kimberling, Mathematical Intelligener "A valuable and stimulating introduction to problems that have fascinated mathematicians and physicists for millennia."--D.R. Wilkins, Contemporary Physics "Nahin delivers maximal mathematical enjoyment with minimal perplexity and boredom... [He lets] general readers in on the thrill of riding high-school geometry and algebra to breakthrough insights... A refreshingly lucid and humanizing approach to mathematics."--Booklist "Anyone with a modest command of calculus, a curiosity about how mathematics developed, and a pad of paper for calculations will enjoy Nahin's lively book. His enthusiasm is infectious, his writing style is active and fluid, and his examples always have a point... [H]e loves to tell stories, so even the familiar is enjoyably refreshed."--Donald R. Sherbert, SIAM ReviewTable of ContentsPreface xiii 1. Minimums, Maximums, Derivatives, and Computers 1 1.1 Introduction 1 1.2 When Derivatives Don't Work 4 1.3 Using Algebra to Find Minimums 5 1.4 A Civil Engineering Problem 9 1.5 The AM-GM Inequality 13 1.6 Derivatives from Physics 20 1.7 Minimizing with a Computer 24 2. The First Extremal Problems 37 2.1 The Ancient Confusion of Length and Area 37 2.2 Dido' Problem and the Isoperimetric Quotient 45 2.3 Steiner '"Solution" to Dido' Problem 56 2.4 How Steiner Stumbled 59 2.5 A "Hard "Problem with an Easy Solution 62 2.6 Fagnano' Problem 65 3. Medieval Maximization and Some Modern Twists 71 3.1 The Regiomontanus Problem 71 3.2 The Saturn Problem 77 3.3 The Envelope-Folding Problem 79 3.4 The Pipe-and-Corner Problem 85 3.5 Regiomontanus Redux 89 3.6 The Muddy Wheel Problem 94 4. The Forgotten War of Descartes and Fermat 99 4.1 Two Very Different Men 99 4.2 Snell' Law 101 4.3 Fermat, Tangent Lines, and Extrema 109 4.4 The Birth of the Derivative 114 4.5 Derivatives and Tangents 120 4.6 Snell' Law and the Principle of Least Time 127 4.7 A Popular Textbook Problem 134 4.8 Snell' Law and the Rainbow 137 5. Calculus Steps Forward, Center Stage 140 5.1 The Derivative:Controversy and Triumph 140 5.2 Paintings Again, and Kepler' Wine Barrel 147 5.3 The Mailable Package Paradox 149 5.4 Projectile Motion in a Gravitational Field 152 5.5 The Perfect Basketball Shot 158 5.6 Halley Gunnery Problem 165 5.7 De L' Hospital and His Pulley Problem, and a New Minimum Principle 171 5.8 Derivatives and the Rainbow 179 6. Beyond Calculus 200 6.1 Galileo'Problem 200 6.2 The Brachistochrone Problem 210 6.3 Comparing Galileo and Bernoulli 221 6.4 The Euler-Lagrange Equation 231 6.5 The Straight Line and the Brachistochrone 238 6.6 Galileo' Hanging Chain 240 6.7 The Catenary Again 247 6.8 The Isoperimetric Problem, Solved (at last!) 251 6.9 Minimal Area Surfaces, Plateau' Problem, and Soap Bubbles 259 6.10 The Human Side of Minimal Area Surfaces 271 7. The Modern Age Begins 279 7.1 The Fermat/Steiner Problem 279 7.2 Digging the Optimal Trench, Paving the Shortest Mail Route, and Least-Cost Paths through Directed Graphs 286 7.3 The Traveling Salesman Problem 293 7.4 Minimizing with Inequalities (Linear Programming) 295 7.5 Minimizing by Working Backwards (Dynamic Programming) 312 Appendix A. The AM-GM Inequality 331 Appendix B. The AM-QM Inequality, and Jensen' Inequality 334 Appendix C. "The Sagacity of the Bees" 342 Appendix D. Every Convex Figure Has a Perimeter Bisector 345 Appendix E. The Gravitational Free-Fall Descent Time along a Circle 347 Appendix F. The Area Enclosed by a Closed Curve 352 Appendix G. Beltrami 'Identity 359 Appendix H. The Last Word on the Lost Fisherman Problem 361 Acknowledgments 365 Index 367

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Book SynopsisA student in class asks the math teacher: "Shouldn't minus times minus make minus?" Teachers soon convince most students that it does not. Yet the innocent question brings with it a germ of mathematical creativity. What happens if we encourage that thought, odd and ungrounded though it may seem? Few books in the field of mathematics encourage suchTrade Review"Alberto A. Martinez ... shows that the concept of negative numbers has perplexed not just young students but also quite a few notable mathematicians... The rule that minus times minus makes plus is not in fact grounded in some deep and immutable law of nature. Martinez shows that it's possible to construct a fully consistent system of arithmetic in which minus times minus makes minus. It's a wonderful vindication for the obstinate smart-aleck kid in the back of the class."--Greg Ross, American Scientist "Alberto Martinez ... has written an entire book about the fact that the product of two negative numbers is considered positive. He begins by reminding his readers that it need not be so... The book is written in a relaxed, conversational manner... It can be recommended to anyone with an interest in the way algebra was developed behind the scenes, at a time when calculus and analytic geometry were the main focus of mathematical interest."--James Case, SIAM News "[Negative Math] is very readable and the style is entertaining. Much is done through examples rather than formal proofs. The writer avoids formal mathematical logic and the more esoteric abstract algebras such as group theory."--Mathematics MagazineTable of ContentsFigures ix Chapter 1: Introduction 1 Chapter 2: The Problem 10 Chapter 3: History: Much Ado About Less than Nothing 18 The Search for Evident Meaning 36 Chapter 4: History: Meaningful and Meaningless Expressions 43 Impossible Numbers? 66 Chapter 5: History: Making Radically New Mathematics 80 From Hindsight to Creativity 104 Chapter 6: Math Is Rather Flexible 110 Sometimes -1 Is Greater than Zero 112 Traditional Complications 115 Can Minus Times Minus Be Minus? 131 Unity in Mathematics 166 Chapter 7: Making a Meaningful Math 174 Finding Meaning 175 Designing Numbers and Operations 186 Physical Mathematics? 220 Notes 235 Further Reading 249 Acknowledgments 259 Index 261

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Book SynopsisA collection of the author's general writings on philosophy, mathematics, and physics. It includes the author's exposition of his synthesis of electromagnetism and gravitation.Trade Review"This work edited by Pesic is an interesting collection of Hermann Weyl's essays, letters, and manuscripts. Though Weyl certainly made important contributions to mathematics and physics, this collection gives a broader picture of his work and thinking. Including many previously unpublished works and photographs, Mind and Nature presents what Weyl saw as the connections between mathematics, physics, and metaphysics."--Choice "Weyl's Mind and Nature: Selected Writings on Philosophy, Mathematics, and Physics is of more interest to historians or to philosophers like me, but I still heartily recommend it to physicists and mathematicians. A selection of philosophical writings from the period 1921-55, it is beautifully edited with an introduction and scholarly endnotes by Peter Pesic. Mind and Nature includes several treasures... What a pleasure, what a privilege, to read and contemplate Hermann Weyl's monumental achievements."--Jeremy Butterfield, Physics Today "Edited by Peter Pesic ... these works show a side of Weyl deeply concerned about the nature of infinity, knowledge, and truth... Rarely are we given insights into the thinking of great mathematicians of the past. Even more rarely are we given their thinking presented with such thought and obvious care. This book should produce a reflective response among the teachers of mathematics who read it."--Mathematics Teacher "[T]hese books have much to stimulate the philosopher of science today, perhaps because of, rather than in spite of the heavily idealistic leanings. More importantly, they still have much to offer the philosophically minded physicist and mathematician."--Jeremy Gray, MAA Reviews "Pesic has collected here a selection of Weyl's writings that beautifully present the underlying synthesis of philosophy, mathematics and physics in his work, and in particular those that have remained unpublished, untranslated or that have simply fallen out if print... One must be content with dipping into these essays and relishing their fecundity and insight."--Steven French, MetascienceTable of ContentsIntroduction 1 1921 Ch 1. Electricity and Gravitation 20 1922 Ch 2. Two Letters by Einstein and Weyl on a Metaphysical Question 25 1927 Ch 3. Time Relations in the Cosmos, Proper Time, Lived Time, and Metaphysical Time 29 1932 Ch 4. The Open World: Three Lectures on the Metaphysical Implications of Science 34 1934 Ch 5. Mind and Nature 83 1946 Ch 6. Address at the Princeton Bicentennial Conference 162 ca.1949 Ch 7. Man and the Foundations of Science 175 1954 Ch 8. The Unity of Knowledge 194 1955 Ch 9. Insight and Reflection 204 Notes 223 References 241 Acknowledgments 253 Index 255

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Book SynopsisTraces the growth of mathematical modernism from its roots in problem solving and theory to its interactions with physics, philosophy, theology, psychology, and ideas about real and artificial languages. This book is suitable for mathematicians and historians.Trade ReviewOne of Choice's Outstanding Academic Titles for 2009 "In Plato's Ghost, he has ... present[ed] us with an ambitious and in many respects remarkable synthesis of the modern transformation of mathematics via structural and set-theoretic notions, together not only with its logic and philosophy but also with related developments in artificial languages and psychology... I can certainly recommend Plato's Ghost highly as a rich resource and point of departure for readers who want to learn more about this exciting period in the development of modern mathematics."--Solomon Feferman, American Scientist "This accessible, rigorous volume belongs in every serious library."--J. McCleary, Choice "In a book aimed at the educated public, the author presents an impressive amount of data--both of the kind mathematicians with some awareness of the history of their subject may be aware of, and of an entirely different kind, coming from the outskirts of mathematics, from philosophy, from physics, or from the popularization of mathematics, which will likely be new even to historians of mathematics."--Victor V Pambuccian, Mathematical Reviews "It is ... no small assertion to say that the book under review, Plato's Ghost, is [Gray's] most far-reaching and ambitious work to date... [T]here is a wealth of valuable data here which, if not fully processed and pigeonholed, is at least tagged and cataloged in a helpful way. Plato's Ghost provides an insightful and informative resource for anyone doing mathematics today who has wondered how (and perhaps why) the subject has come to possess the features it has today. The book gives us a lot to think about, which is exactly what a good history should do."--Jeremy Avigad, Mathematical Intelligencer "In this book Jeremy Gray offers us the fruit of more than a decade reading and thinking about modernism in mathematics. He presents it, in very well written form, to a broad audience interested in mathematics, its history and philosophy."--Erhard Scholz, Metascience "What we have here ... is an excellent and detailed survey of how modernism took root in mathematics. Plato's Ghost provides the launching pad for future ruminations on the modernist thesis."--Calvin Jongsma, Perspectives on Science and Christian Faith "I commend Gray for writing an extraordinarily detailed and fascinating history of modernist mathematics, whose philosophical fruits remain ripe for the picking. The sections on geometry shine with clarity and convey the drama of modernism in a compelling and page-turning way. The treatments of less-studied actors are fascinating and promise to be of much use in incorporating their work into ongoing scholarship. The book could be fruitfully used as a supplement to a variety of courses in philosophy, including philosophy of mathematics and logic, history of analytic philosophy, and philosophy of science. It is a monument of scholarship and will reward careful study."--Andrew Arana, Philosophia Mathematica "In the course of this study Gray uncovers many new and unexpected things... Gray's book offers a rich and ... balanced account of how modernist ideas gradually gained inroads within pure mathematics."--David E. Rowe, Bulletin of the American Mathematical SocietyTable of ContentsIntroduction 1 I.1 Opening Remarks 1 I.2 Some Mathematical Concepts 16 CHAPTER 1: Modernism and Mathematics 18 1.1 Modernism in Branches of Mathematics 18 1.2 Changes in Philosophy 24 1.3 The Modernization of Mathematics 32 CHAPTER 2: Before Modernism 39 2.1 Geometry 39 2.2 Analysis 58 2.3 Algebra 75 2.4 Philosophy 78 2.5 British Algebra and Logic 101 2.6 The Consensus in 1880 112 CHAPTER 3: Mathematical Modernism Arrives 113 3.1 Modern Geometry: Piecemeal Abstraction 113 3.2 Modern Analysis 129 3.3 Algebra 148 3.4 Modern Logic and Set Theory 157 3.5 The View from Paris and St. Louis 170 CHAPTER 4: Modernism Avowed 176 4.1 Geometry 176 4.2 Philosophy and Mathematics in Germany 196 4.3 Algebra 213 4.4 Modern Analysis 216 4.5 Modernist Objects 235 4.6 American Philosophers and Logicians 239 4.7 The Paradoxes of Set Theory 247 4.8 Anxiety 266 4.9 Coming to Terms with Kant 277 CHAPTER 5: Faces of Mathematics 305 5.1 Introduction 305 5.2 Mathematics and Physics 306 5.3 Measurement 328 5.4 Popularizing Mathematics around 1900 346 5. Writing the History of Mathematics 365 CHAPTER 6: Mathematics, Language, and Psychology 374 6.1 Languages Natural and Artificial 374 6.2 Mathematical Modernism and Psychology 388 CHAPTER 7: After the War 406 7.1 The Foundations of Mathematics 406 7.2 Mathematics and the Mechanization of Thought 430 7.3 The Rise of Mathematical Platonism 440 7.4 Did Modernism'"Win"? 452 7.5 The Work Is Done 458 Appendix: Four Theorems in Projective Geometry 463 Glossary 467 Bibliography 473 Index 503

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Book SynopsisWhy do so many gamblers risk it all when they know the odds of winning are against them? Why do they believe dice are 'hot' in a winning streak? Why do we expect heads on a coin toss after several flips have turned up tails? This book takes a look at the mathematics, history, and psychology of gambling to reveal various misconceptions about luck.Trade Review"From the dice-playing of Neolithic peoples to modern lotteries and casino capitalism, he tracks the history of placing bets. He explains both the mathematics of chance and the psychological and emotional factors that entice some people to risk it all to win that improbable jackpot."--Joanne Baker, Nature "In What's Luck Got to Do With It?, mathematician Joseph Mazur explores these misconceptions, taking the reader on an entertaining and accessible tour of the history of gambling, the way mathematicians quantify luck and the psychology that keeps gamblers returning to the table. A book worth taking a chance on."--New Scientist "Doubtless aimed at the interested gambler, the frequent cultural references, anecdotes and intervention of psychology nevertheless make the book appealing reading."--Times Higher Education "Both an analysis of the idea of luck, the gambling impulse, and a history of it, stretching back to Neolithic times, the Renaissance (Francis Drake and Ben Johnson often played hazard--an early form of dice) up to the age of one-arm bandits."--Steven Carroll, The Age "Because Mazur's not judgmental about luck and gambling, but is analytical, the book is a winner. It's not just a mathematician telling us that we'll never hit a million-dollar jackpot--it's a mathematician looking at why we continue to hope to hit that jackpot. This book should be required reading for anyone in the casino business, and anyone who spends more than a fraction of their disposable income on gambling should find it informative, if nothing else. It's a reasoned, but also passionate, search for the meaning of luck that may change the way you look at a pair of dice--or your mortgage."--dieiscast.com "What's Luck Got to Do with It? is an entertaining and informative history of gambling beginning with the Ice Age... Anyone who has an interest in probability will enjoy Mazur's ideas and insights."--Mathematics Teacher "Readers will find many an unexpected treat in Mazur's exploration of luck, or, as Mazur might say, the likelihood of long runs of desired outcomes within the purview of the law of large numbers."--Andrew James Simpson, Mathematical Reviews Clippings "Mazur's book is appealing to virtually anyone with an interest in the human psyche. It should certainly be given out to anyone arriving for work on their first day on Wall Street. Perhaps it would help to avoid a few more disasters."--Sam Marsden, Jackpot Gaming LimitedTable of ContentsIntroduction xi Part I: The History Chapter 1. Pits, Pebbles, and Bones Rolling to Discover Fate 3 Chapter 2. The Professionals Luck Becomes Measurable 19 Chapter 3. From Coffeehouses to Casinos Gaming Becomes Big Business 37 Chapter 4. There's No Stopping It Now From Bans to Bookies 46 Chapter 5. Betting with Trillions The 2008 World Economic Calamity 58 Part II : The Mathematics Chapter 6. Who's Got a Royal Flush? One Deal as Likely as Another 75 Chapter 7. The Behavior of a Coin Making Predictions with Probability 83 Chapter 8. Someone Has to Win Betting against Expectation 101 Chapter 9. A Truly Astonishing Result The Weak Law of Large Numbers 118 Chapter 10. The Skill/Luck Spectrum Even Great Talent Needs Some Good Fortune 131 Part III : The Analysis Chapter 11. Let It Ride The House Money Effect 157 Chapter 12. Knowing When to Quit Psychomanaging Risk 168 Chapter 13. The Theories What Makes a Gambler? 182 Chapter 14. Hot Hands Expecting Long Runs of the Same Outcome 202 Chapter 15. Luck The Dicey Illusion 209 Acknowledgments 217 Appendix A. Descriptions of the Games Used in This Book 219 Appendix B. Glossary of Gambling Terms Used in This Book 224 Appendix C. The Weak Law of Large Numbers 227 Appendix D. Glossary of Mathematical Definitions 229 Appendix E. Callouts 236 Notes 249 Further Reading 265 Index 267

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