Philosophy of mathematics Books

Book SynopsisDoes the universe embody beautiful ideas?   Artists as well as scientists throughout human history have pondered this “beautiful question.” With Nobel laureate Frank Wilczek as your guide, embark on a voyage of related discoveries, from Plato and Pythagoras up to the present. Wilczek’s groundbreaking work in quantum physics was inspired by his intuition to look for a deeper order of beauty in nature. This is the deep logic of the universe—and it is no accident that it is also at the heart of what we find aesthetically pleasing and inspiring.   Wilczek is hardly alone among great scientists in charting his course using beauty as his compass. As he reveals in A Beautiful Question, this has been the heart of scientific pursuit from Pythagoras and the ancient belief in the music of the spheres to Galileo, Newton, Maxwell, Einstein, and into the deep waters of twentieth-century physics. Wilczek brings us right to the edge of knowledge today, where the core insights of even the craziest quantum ideas apply principles we all understand. The equations for atoms and light are almost the same ones that govern musical instruments and sound; the subatomic particles that are responsible for most of our mass are determined by simple geometric symmetries.   Gorgeously illustrated, A Beautiful Question is a mind-shifting book that braids the age-old quest for beauty and the age-old quest for truth into a thrilling synthesis. It is a dazzling and important work from one of our best thinkers, whose humor and infectious sense of wonder animate every page. Yes: The world is a work of art, and its deepest truths are ones we already feel, as if they were somehow written in our souls.

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Book SynopsisFundamentals of Bayesian Epistemology provides an accessible introduction to the key concepts and principles of the Bayesian formalism. This volume introduces degrees of belief as a concept in epistemology and the rules for updating degrees of belief derived from Bayesian principles.Table of ContentsQuick Reference Preface I Our Subject 1: Beliefs and Degrees of Belief II The Bayesian Formalism 2: Probability Distributions 3: Conditional Credences 4: Updating by Conditionalization 5: Further Rational Constraints

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Book Synopsis"The book that inspired the film The imitation game."Trade ReviewA New York Times Bestseller The Imitation Game, Winner of the 2015 Academy Award for Best Adapted Screenplay Winner of the 2015 (27th) USC Libraries Scripter Award, University of Southern California Libraries One of The Guardian's Best Popular Physical Science Books of 2014, chosen by GrrlScientist "Scrupulous and enthralling."--A. O. Scott, New York Times "One of the finest scientific biographies ever written."--Jim Holt, New Yorker "Andrew Hodges' 1983 book Alan Turing: The Enigma, is the indispensable guide to Turing's life and work and one of the finest biographies of a scientific genius ever written."--Michael Hiltzik, Los Angeles Times "Turing's rehabilitation from over a quarter-century's embarrassed silence was largely the result of Andrew Hodges's superb biography, Alan Turing: The Enigma (1983; reissued with a new introduction in 2012). Hodges examined available primary sources and interviewed surviving witnesses to elucidate Turing's multiple dimensions. A mathematician, Hodges ably explained Turing's intellectual accomplishments with insight, and situated them within their wider historical contexts. He also empathetically explored the centrality of Turing's sexual identity to his thought and life in a persuasive rather than reductive way."--Michael Saler, Times Literary Supplement "On the face of it, a richly detailed 500-page biography of a mathematical genius and analysis of his ideas, might seem a daunting proposition. But fellow mathematician and author Hodges has acutely clear and often extremely moving insight into the humanity behind the leaping genius that helped to crack the Germans' Enigma codes during World War II and bring about the dawn of the computer age... This melancholy story is transfigured into something else: an exploration of the relationship between machines and the soul and a full-throated celebration of Turing's brilliance, unselfconscious quirkiness and bravery in a hostile age."--Sinclair McKay, Wall Street Journal "A first-class contribution to history and an exemplary work of biography."--I. J. Good, Nature "An almost perfect match of biographer and subject... [A] great book."--Ray Monk, Guardian "A superb biography... Written by a mathematician, it describes in plain language Turing's work on the foundations of computer science and how he broke the Germans' Enigma code in the Second World War. The subtle depiction of class rivalries, personal relationships, and Turing's tragic end are worthy of a novel. But this was a real person. Hodges describes the man, and the science that fascinated him--which once saved, and still influences, our lives."--Margaret Boden, New Scientist "Andrew Hodges's magisterial Alan Turing: The Enigma ... is still the definitive text."--Joshua Cohen, Harper's "Andrew Hodges's biography is a meticulously researched and written account detailing every aspect of Turing's life... This account of Turing's life is a definitive scholarly work, rich in primary source documentation and small-grained historical detail."--Mathematics Teacher "Tells a powerful story that combines professional success and personal tragedy."--Nancy Szokan, Washington Post "[A] really excellent biography... The great thing about this book is that the author is a mathematician and can explain the details of Turing's work--as a scientist, mathematician, and a code breaker--in a way that is easy to understand. He is also wonderful at the emotional nuance of Alan's life, who was a somewhat odd--a student was assigned to him in school to help him maintain a semblance of tidiness in his appearance, rooms and school work and at Bletchley Park he was known for chaining his tea mug to a pipe--but he was also charming and intelligent and Hodges brings all the aspects of his personality and life into sharp focus."--Off the Shelf "This book is an incredibly detailed and meticulously researched biography of Alan Turing. Reading it is a melancholy experience, since you know from the outset that the ending is a tragic one and that knowledge overshadows you throughout. While the author divides the text into two parts, it actually reads like a play in four acts... This book is Turing's memorial, and one that does justice to the subject."--Katherine Safford-Ramus, MAA Reviews "The new paperback edition of the 1983 book that inspired the film, with an updated introduction by Oxford mathematics professor Andrew Hodges, is an exhilarating, compassionate and detailed biography of a complicated man."--Jane Ciabattari, BBC "If [The Imitation Game] does nothing else but send you, as it did me, to Alan Hodges's Alan Turing: The Enigma (1983, newly prefaced in the 2014 Princeton University Press edition) it more than justifies its existence. A great read, Hodges's intellectual biography depicts Turing as a brilliant mathematician; a crucial pioneering figure in the theorization and engineering of digital computing; and the biggest brain in Bletchley Park's Hut #8."--Amy Taubin, Artforum "It is indeed the ultimate biography of Alan Turing. It will bring you as close as possible to his enigmatic personality."--Adhemar Bultheel, European Mathematical Society "A book whose time has finally come. I found it to be a page-turner in spite of the occasionally esoteric explanations of mathematical theories that reminded of why Brooklyn Technical High School was not the wisest choice for me."--Terrance, Paris Readers Circle "Thanks to the movie The Imitation Game, Alan Turing has emerged from history's shadows, where his memory had languished for decades. For anyone whose interest in the pioneering computer scientist, mathematician, and logician was piqued by the film, the book that served as the film's source material, Andrew Hodges's exhaustive biography Alan Turing: The Enigma, has the answers."--Frank Caso, Simply CharlyTable of ContentsList of Plates ix Foreword by Douglas Hofstadter xi Preface xv PART ONE: THE LOGICAL 1 Esprit de Corps to 13 February 1930 3 2 The Spirit of Truth to 14 April 1936 60 3 New Men to 3 September 1939 141 4 The Relay Race to 10 November 1942 202 BRIDGE PASSAGE to 1 April 1943 305 PART TWO: THE PHYSICAL 5 Running Up to 2 September 1945 325 6 Mercury Delayed to 2 October 1948 394 7 The Greenwood Tree to 7 February 1952 491 8 On the Beach to 7 June 1954 574 Postscript 665 Author's Note 666 Notes 680 Acknowledgements 714 Index 715

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Book SynopsisThomas Reid was an intellectual polymath interested in all aspects of Enlightenment thought. Paul Wood reconstructs Reid's career as a mathematician and natural philosopher and shows how he grappled with Sir Isaac Newton's scientific legacy.

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Book SynopsisNumeracy has shaped human history as much as literacy: mathematics has enabled us to measure the cosmos, control the Earth, and create all technological change. A Cultural History of Mathematics presents the first comprehensive and global history from antiquity to today. The work is divided into 6 volumes, with each volume covering the same topics, so readers can either study a period/volume or follow a topic across history. The 6 volumes cover: Antiquity (c.3000 BCE-500 CE); the Medieval Age (500-1400); the Early Modern Age (1450-1687); the Eighteenth Century (1687-1800); the Nineteenth Century (1800-1914); the Modern Age (1914-present).Themes (and chapter titles) are: everyday numeracy; practice & profession; inventing mathematics; mathematics & worldviews; describing & understanding the world; mathematics & technological change; representing mathematics.The page extent for the pack is approximately 15

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Book SynopsisSpanning the period between Wittgenstein's return to Cambridge in 1929 and the first version of Philosophical Investigations in 1936, Piotr Dehnel explores the middle stage in Ludwig Wittgenstein's philosophical development and identifies the major issues which engrossed him, including phenomenology, philosophy of mathematics and philosophy of language. Contrary to the dominant perspective, Dehnel argues that this period was intrinsically different from the early and late stages and should not be viewed as a mere transitional phase. The distinctiveness of Wittgenstein's middle work can be seen in his philosophical thinking as it unfolds in a non-linear trajectory: thoughts do not follow upon each other, ideas do not appear sequentially one by one, and insights do not form a straight chain. Dehnel portrays the diffused and multifarious quality of Wittgenstein's middle thinking, enabling readers to form a more comprehensive view of his entire philosophy and acquire a better grasp Trade ReviewThe book sheds an interesting new light on interpretations of Ludwig Wittgenstein’s philosophy as it offers one of the first explorations of his concepts between the Tractatus and the Philosophical Investigations. The author argues that, rather than developing in a linear sequence from insight to insight and from idea to idea, Wittgenstein’s thought in the middle period expands radially, unfolding in several directions at the same time. A must-read for Wittgenstein researchers, the book is certainly of profound interest to humanities scholars and social scientists alike. * Leszek Koczanowicz, Professor of Philosophy and Cultural Studies, SWPS University of Social Sciences and Humanities, Poland *This book offers broad hermeneutic explanations of Wittgenstein’s writings from 1929 to 1936. They are based on a thorough knowledge of the source material, which they place in the context of his thought and its philosophical environment. I am impressed with the scientific merit of the present work. * Herbert Hrachovec, Associate Professor at the Institute for Philosophy, University of Vienna, Austria *Table of ContentsIntroduction 1. The Phenomenological Turn 2. Verification: 1929-1932 3. Wittgenstein’s Critique of Frege in the Notes of 1929-1932 4. ‘A Clever Man got Caught in this Net of Language’: Wittgenstein’s Attack on Set Theory 5. The Big Typescript as a Work of the Middle Period 6. P.S. Understanding, Expecting, Wishing 7. Magic, Rituals and Philosophy: Wittgenstein on Frazer’s The Golden Bough 8. Wittgenstein as a Philosopher of Culture Bibliography Index

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Book SynopsisGary Kemp is Senior Lecturer in Philosophy at the University of Glasgow, UK.Trade ReviewThis is a superb volume written with a light style. It will engage and inform students, and be a go-to text for experts wanting a summary of the major themes of Quine’s work (Kemp has the knack of finding just the right passage for the purposes at hand). It not only expertly discusses Quine himself, but offers insight into a range of related topics, especially naturalism, ontology, and the engagement of Chomsky with Quine’s work. I know of no better volume of its kind. * John Collins, Professor, University of East Anglia, UK *Gary Kemp’s earlier Quine, A Guide for the Perplexed was one of the best entry points into Quine’s naturalist conception of philosophy. Now in this newly titled and revised version of that book, Kemp offers important updates to the original that will further help those studying Quine’s philosophy for the first time. * Robert Sinclair, Professor of Philosophy, Soka University, Tokyo, Japan *Quine is one of the most influential philosophers of our time, and Gary Kemp’s book is a perfect introduction to his thought. It spells out his major views in a rigorous and reader-friendly way that will be most useful for anyone seeking for a way into the complexities of this fascinating philosophy. * Rogério Severo, Professor of Philosophy, Federal University of Rio Grande do Sul, Brazil *This is an excellent introduction to Quine’s thought. Kemp is approachable while also preserving rigor. The historical and further reading notes are especially helpful to those wanting to pursue Quine’s thought further. Experts will also find the work stimulating, particularly the discussions of Quine and Chomsky. * Sean Morris, Professor of Philosophy, Metropolitan State University of Denver, USA *This is an excellent overview of Quine’s philosophy. Kemp introduces Quine’s most prominent views, shows how they hang together, and demonstrates their continued importance. An accessible and up-to-date guide for both students and academics, written by a leading Quine scholar. * Sander Verhaegh, Associate Professor, Tilburg University, The Netherlands *Table of ContentsPreface Abbreviations 1. Philosophy as Quine Found it 2. Convention, Analyticity and Holism 3. The Indeterminacy of Translation 4. Naturalized Epistemology and the Roots of Reference 5. Ontology I: Truth, Physical Objects, and the Language of Science 6. Ontology II: Extensionality and Abstract Objects 7. Science, Philosophy and Empiricism Notes Bibliography Index

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Book SynopsisTwo veteran math educators demonstrate how some "magnificent mistakes" had profound consequences for our understanding of mathematics' key concepts. In the nineteenth century, English mathematician William Shanks spent fifteen years calculating the value of pi, setting a record for the number of decimal places. Later, his calculation was reproduced using large wooden numerals to decorate the cupola of a hall in the Palais de la Decouverte in Paris. However, in 1946, with the aid of a mechanical desk calculator that ran for seventy hours, it was discovered that there was a mistake in the 528th decimal place. Today, supercomputers have determined the value of pi to trillions of decimal places. This is just one of the amusing and intriguing stories about mistakes in mathematics in this layperson's guide to mathematical principles. In another example, the authors show that when we "prove" that every triangle is isosceles, we are violating a concept not even known to Euclid - that of "betweenness." And if we disregard the time-honored Pythagorean theorem, this is a misuse of the concept of infinity. Even using correct procedures can sometimes lead to absurd - but enlightening - results. Requiring no more than high-school-level math competency, this playful excursion through the nuances of math will give you a better grasp of this fundamental, all-important science.

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Book SynopsisBeginning with a review of formal languages and their syntax and semantics, Logic, Proof and Computation conducts a computer assisted course in formal reasoning and the relevance of logic to mathematical proof, information processing and philosophy. Topics covered include formal grammars, semantics of formal languages, sequent systems, truth-tables, propositional and first order logic, identity, proof heuristics, regimentation, set theory, databases, automated deduction, proof by induction, Turing machines, undecidability and a computer illustration of the reasoning underpinning Gödel's incompleteness proof. LPC is designed as a multidisciplinary reader for students in computing, philosophy and mathematics.

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Book SynopsisThere is a wonderfully weird but real world out there, and we are a part of it. It is time for physics to take life seriously.Can we ever truly comprehend the universe before we fully understand consciousness and the wonders, and limits, of the mind? Ulf Danielsson, an acclaimed theoretical physicist who has dedicated his career to probing the deepest mysteries of nature, thinks not. As he dismantles the arguments of esteemed mathematicians and scientists, who would substitute their mathematical models for reality and equate the mind to a computer, he makes a lucid and passionate case that it is nature, full of beauty and meaning, which must compel us. In challenging established worldviews, he also takes a fresh look at major philosophical debates, including the notion of free will.Fearless, provocative, and witty, The World Itself is essential reading for anyone curious about the profound questions surrounding life, the universe, and everything.Trade Review“[A] thought-provoking treatment of an array of issues at the frontier of science and philosophy. . . . Well worth our attention.” —PopMatters“Engaging and varied. . . . Books like this invite us to direct our curiosities—both as groups and individuals—in useful ways.” —North of Oxford“Engrossing. . . . Danielsson’s clarity of thought and expression and his use of illuminating literary and historical references are equal to the quality of his writing. Science ‘popularizing’ doesn’t get much more comprehensible, or provocative, than this.” —Kirkus Reviews (starred review)“Danielsson takes readers on an odyssey through the width and depth of his field, and it is truly a fascinating journey. Touching on subjects as diverse as evolutionary biology, philosophy, and even popular culture, Danielsson makes his topics both appreciably substantial and approachable.” —Library Journal (starred review)“There are some mind-bending ideas and the philosophical reflections on math and physics are stimulating. . . . This pensive take on physics has much to offer.” —Publishers Weekly“The World Itself offers a bold perspective on mathematics, physics, and the nature of reality. There’s much I agree with and less that I don’t, but Ulf Danielsson, a leading theoretical physicist, proves himself an insightful and patient guide through some of the universe’s deepest mysteries.” —Brian Greene, author of The Fabric of the Cosmos and Until the End of Time“Danielsson is Sweden’s most important public writer on the implications of natural science. His lucid, powerful, passionate, and engaging work advances original arguments of great importance. The World Itself is destined to become a modern classic as it upends many of the received wisdoms about the scientific worldview.” —Martin Hägglund, author of This Life: Secular Faith and Spiritual Freedom“Danielsson displays a remarkably broad understanding of science and philosophy, and dispenses with false notions about the world in this brief, yet provocative book. I hope it stimulates lots of discussion and debate, as it should. For those who have thought about these issues, there is much of interest here. For those who haven’t, this is a great place to start.” —Lawrence M. Krauss, author of A Universe from Nothing and The Known Unknowns“In this accessible and beautifully written book, Danielsson argues for views diametrically opposite to mine on the nature of intelligence, consciousness, and physical reality—I highly recommend it!” —Max Tegmark, author of Our Mathematical Universe and Life 3.0: Being Human In the Age of Artificial Intelligence

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Book SynopsisMit dem Namen Euler wird der Beginn der modernen Mathematik verknüpft. Ausgehend von Eulers Leben und seiner wissenschaftlichen Arbeit illustriert der Autor im 2. Teil der mathematisch-kulturhistorischen Zeitreise den Werdegang der heutigen Mathematik. Dabei konzentriert er sich angesichts der hoch komplexen und fragmentierten Entwicklung der Mathematik im ausgehenden 20. Jahrhundert auf wichtige und exemplarische Entwicklungen. Ein spannendes Lesevergnügen für Mathematiker und alle, die sich für die Kulturgeschichte der Mathematik interessieren.Trade ReviewAus den Rezensionen:"… Bei Springer erschien Hans Wußings bedeutende kulturgeschichtliche Zeitreise durch die Geschichte der Mathematik, deren erster Band in dieser Zeitung schon besprochen worden ist. Noch rechtzeitig vor Jahresende wird nun auch der zweite Band, von Euler bis zur Gegenwart, erscheinen, auf den schon jetzt aufmerksam gemacht werden soll ..." (Günter Kröber, in: Neues Deutschland, 29.-30. Nov. 2008, S. 16) "Das zweibändige Springer-Lehrbuch … von Hans Wußing, der seit 1957 in Leipzig Geschichte der Mathematik lehrt, versprach schon vor seinem Erscheinen ein Klassiker zu werden, der in keiner gut sortierten, allgemein bildenden Bibliothek Fehlen sollte. Auf insgesamt 1204 Seiten wurden diese Erwartungen nach einem Gesamtüberblick über die Geschichte der Mathematik von den Anfängen bis heute voll und ganz erfüllt." (in: fachbuch journal, 2009, Vol. 1, Issue 1, S. 65) "Zwei Bücher mit Garantie: Wer auch nur irgendeine Seite aufschlägt, wird sich sofort festlesen und, gefangengenommen von der anschaulichen Darstellung, fasziniert im Zaubergarten der Mathematik umherstreifen." (in: c´t 2009, Heft 8) "… Abgerundet wird das Buch … mit Gedanken und einem Ausblick zur Mathematik, den Eberhard Zeidler geschrieben hat. … Das … Buch bietet einen guten Überblick über die verschiedenen Gebiete des Fachs … Wie im ersten Band überzeugt Wußings Werk erneut durch viele farbige Abbildungen … und dem mit voller Freude geschriebenen Text. Insgesamt kann beide Bände jedem ans Herz legen, der einen detaillierten Gesamtüberblick über die kulturgeschichtliche Entwicklung der Mathematik … bekommen möchte und dabei Wert auf Anschauung und lebendige Sprache legt. Insgesamt ein fantastisches Werk." (http://www.spektrumdirekt.de/artikel/988679) Aus den Rezensionen:"Mit dem Band ‘Von Euler bis zur Gegenwart‘ setzt Wußing seine kulturgeschichtliche Reise durch ‘6000 Jahre Mathematik‘ … fort. … Es entstehen wichtige Teildisziplinen der Mathematik … Zur Fortsetzung. Grundlegendes Werk zur Mathematikgeschichte …" (Olaf Kaptein, in: ekz-Informationsdienst Einkaufszentrale für öffentliche Bibliotheken, ID 16/2009 - BA 5/2009) "... Positiv anzumerken ist ... die Prägnanz. Erwähnenswert sind ... die sorgsam ausgewählten und ... zum Nachdenken anregenden Zitate. Viele prachtvolle und farbige Abbildungen lassen den optischem [sic] Eindruck dem erzählerischen in nichts nach stehen. ... Die Motivation zur Entwicklung mathematischer Theorien wird hier meist besser als in den meisten Lehrbüchern vollbracht. Für mich ist ‘6000 Jahre Mathematik‘ auch deshalb vor allem eine Geschichte der mathematischen Ideen, die mit diesem zweiten Band ein geglücktes Ende gefunden hat." (in: Rho, July/2009) "... Die Texte von Wußing sind informations- und zitatenreich, halten geschickt das Gleichgewicht zwischen der Darstellung mathematischer Probleme und Inhalte, historischen Hintergründen und Biographischem, wobei gelegentlich auch Anekdotisches wohl ausgewogen zur Sprache kommt. Sie beziehen auch kulturhistorische Facetten, z. B. einige Gedichte über Mathematik und Mathematiker, ein. ... Der Text endet wie schon im Titel angekündigt mit einem Ausblick auf die aktuelle und zukünftige Entwicklung der Mathematik ... das schöne Buch ..." (Peter Schreiber, in: Mathematische Semesterberichte, 28/July/2009) "Nach dem begrifflichen Unterschied zwischen Geschichte der Mathematik und Historiographie ... verdeutlichte Hans Wußing sein Vorhaben: ‘ ... die Idee, eine die Fächer übergreifende Historiographie der Mathematik ins Auge zu fassen, leicht lesbar, mit wenigen Formeln, dafür ... reichlich kulturellen, philosophischen und historischen Bezügen, alle Zeiten und Kulturen berührend‘ ... Man kann ihm zum Gelingen dieser Absicht gratulieren: In zwei Bänden, betitelt 6000 Jahre Mathematik, ist ihm dies wahrlilch gelungen! ... Wer bereits gewohnt, lockert er die Lesbarkeit durch eine große Anzahl von Abbilgungen auf ..." (W. Kaunzner, in: Zentralblatt MATH, 2009, Vol. 1167)“... Diese erfreulich flüssig zu lesende Werk ist in der Lage, Historiker der Naturwissenschaften sowie andere, kulturhistoriche interessierte Historiker zur Mathematikgeschischte hinzuführen. Auch für alle mathematikhistorisch interessierten Philosophen, Mathematiker (z.B. Studenten und Lehrer), Naturwissenschaftler, Ingenieure kann es als solide Einführung dienen.“ (Uta Lindgren, in: Sudhoffs Archiv, 2011, Vol. 95, Issue 1, S. 125 f.)Table of ContentsMathematik im Zeitalter des Absolutismus und der Aufklärung.- Mathematik während der Industriellen Revolution.- Globalisierung der Mathematik seit dem Ende des 19. Jahrhunderts.- Gedanken zur Zukunft der Mathematik – Ein Ausblick von Eberhard Zeidler.

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Book Synopsis„Was ist Mathematik?” – auf diese Frage gibt dieses dicke Buch zahllose Antworten. Mathematik ist eben viel mehr als ein Schul- und Studienfach oder Rechnen: Es ist Teil der menschlichen Kultur, ein riesiges aktives Forschungsgebiet und ein nützlicher Werkzeugkasten. „Was ist Mathematik?” – statt einer einzelnen Antwort zeichnen die Autoren ein Panorama, bunt und vielfältig. Da geht es um Philosophie, Beweise, große und kleine Probleme, fundamentale Konzepte, Teilgebiete, Forschungspraxis, Anwendungen der Mathematik. Und um Geschichten aus der Geschichte. Das Buch richtet sich an alle, die wissen und darüber nachdenken wollen, was Mathematik ist, insbesondere auch an Studierende der Mathematik. Es begleitet eine Vorlesung, die an der Freien Universität Berlin jährlich vor allem für Lehramtsstudierende angeboten wird.Table of ContentsWas ist Mathematik?- Mathematische Forschung.- Beweise.- Formeln, Zeichnungen und Bilder.- Philosophie der Mathematik.- Primzahlen.- Zahlenbereiche.- Unendlichkeit.- Dimensionen.- Zufall – Wahrscheinlichkeiten – Statistik.- Funktionen.- Anwendungen.- Rechnen.- Algorithmen und Komplexität.- Mathematik in der Öffentlichkeit.

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Book SynopsisBayes''s theorem is a tool for assessing how probable evidence makes some hypothesis. The papers in this volume consider the worth and applicability of the theorem. Richard Swinburne sets out the philosophical issues. Elliott Sober argues that there are other criteria for assessing hypotheses. Colin Howson, Philip Dawid and John Earman consider how the theorem can be used in statistical science, in weighing evidence in criminal trials, and in assessing evidence for the occurrence of miracles. David Miller argues for the worth of the probability calculus as a tool for measuring propensities in nature rather than the strength of evidence. The volume ends with the original paper containing the theorem, presented to the Royal Society in 1763.Trade ReviewReview from previous edition This is a high quality, concise collection of articles on the foundations of probability and statistics. ... The volume closes with an Appendix containing a very polished reproduction of Bayes's classic 'An Essay Towards the Solving a Problem in the Doctrine of Chances'. The Essay still reads very well, and it should be on every probabilist's 'must read' list. I feel quite comfortable saying something almost as glowing about this entire volume. I found this book very edifying and clear, and the debates and issues it encompasses are of great importance for contemporary philosophy of probability, statistics, and decision-making. I highly recommend this book to anyone with interests in these areas, and I commend Swinburne for putting together this neat little book. * Notre Dame Philosophical Review *Table of ContentsIntroduction ; Bayesianism - its scopes and limits ; Bayesianism in Statistics ; Bayes's Theorem and Weighing Evidence by Juries ; Bayes, Hume, Price, and Miracles ; Propensities May Satisfy Bayes's Theorem ; 'An Essay Towards Solving a Problem in the Doctrine of Chances' by Thomas Bayes, presented to the Royal Society by Richard Price. Preceded by a historical introduction by G A Barnard.

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Book SynopsisIn this first modern, critical assessment of the place of mathematics in Berkeley's philosophy and Berkeley's place in the history of mathematics, Douglas M. Jesseph provides a bold reinterpretation of Berkeley's work. Jesseph challenges the prevailing view that Berkeley's mathematical writings are peripheral to his philosophy and argues that mathematics is in fact central to his thought, developing out of his critique of abstraction. Jesseph's argument situates Berkeley's ideas within the larger historical and intellectual context of the Scientific Revolution. Jesseph begins with Berkeley's radical opposition to the received view of mathematics in the philosophy of the late seventeenth and early eighteenth centuries, when mathematics was considered a science of abstractions. Since this view seriously conflicted with Berkeley's critique of abstract ideas, Jesseph contends that he was forced to come up with a nonabstract philosophy of mathematics. Jesseph examines Berkeley's unique tr

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Book SynopsisThis text presents an account of the argument between Thomas Hobbes and John Wallis, from the core mathematics to the broader issues. Their battle of the books illuminates the relationship between science and 17th-century debates over the limits of sovereign power and the existence of God.

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Book SynopsisRanging from math to literature to philosophy, Uncountable explains how numbers triumphed as the basis of knowledge—and compromise our sense of humanity.Trade Review"Ricardo and David Nirenberg, father and son scholars of mathematics and history, have teamed up in a breathtaking voyage examining the foundations and limits of knowledge in western thought. Not content with secondary sources, they have translated from the literature in their original languages: Arabic, French, German, Greek, Hebrew, Italian, Latin, and Spanish. In particular, they target mathematics and the natural sciences, and the way the concepts of sameness and differences affect our understanding of the natural world. But in the process, the authors touch upon many other facets of human endeavor, all named after their Greek roots: poetry, philosophy, psychology, economy. Along this wildly entertaining journey, we meet dozens of erudite thinkers, scientists, and writers such as Anaximander, Al-Farabi, Fyodor Dostoevsky, Ludwig Wittgenstein, Werner Heisenberg, and Reiner Maria Rilke. The book arrives just in time to give us ammunition as attempts are being made to put truth itself into the supercollider. It is a source of inspiration and comfort to learn how the far-flung ideas about numbers, our existence, and the world we live in have been debated in the past."--Joachim Frank, Columbia University, Nobel Prize in ChemistryTable of ContentsIntroduction: Playing with Pebbles 1 World War Crisis 2 The Greeks: A Protohistory of Theory 3 Plato, Aristotle, and the Future of Western Thought 4 Monotheism’s Math Problem 5 From Descartes to Kant: An Outrageously Succinct History of Philosophy 6 What Numbers Need: Or, When Does 2 + 2 = 4? 7 Physics (and Poetry): Willing Sameness and Difference 8 Axioms of Desire: Economics and the Social Sciences 9 Killing Time 10 Ethical Conclusions Acknowledgments Notes Bibliography Index of Names

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Book SynopsisThe role of gender in making and shaping mathematicians.Trade Review'Mathematicians do their best work in their youth'; 'mathematicians work in complete isolation'; 'mathematics and politics don't mix.'These and other myths are discussed and debunked—in both theoretical and concrete terms—in the particular context of the role of women in mathematics. Henrion studies the nature of the participation of women in mathematical research and surrounding issues of gender and race by weaving her narrative around detailed profiles of nine respected women mathematicians (including two African American women). The individual biographies themselves make for enthralling, often inspiring, reading; combined with Henrion's careful, generally evenhanded, and tightly conceived commentary, this volume should be compelling reading for women mathematics students and professionals. A fine addition to the literature on women in science and, as it is written by a mathematical 'insider,' it is all the more likely to receive attention by the mathematics community. Highly recommended. Undergraduates through faculty. -- S. J. Colley * Choice *

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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 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 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 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 SynopsisProvides a collection of English translations of mathematical texts from five important ancient and medieval non-Western mathematical cultures, and puts them into historical and mathematical context. This book is intended for mathematics teachers who want to use non-Western mathematical ideas in the classroom.Trade ReviewJoseph Warren Dauben, Winner of the 2012 Albert Leon Whiteman Memorial Prize, American Mathematical Society "This pioneering work provides English translations of mathematical texts from each of these regions and cultures, and a better understanding of their contributions to mathematics. There are nuggets of information difficult to find elsewhere. The use of non-mathematical sources, particularly letters and administrative documents from Egypt and Mesopotamia, reveals the practical applications of mathematics and the scribes who composed and used the documents...An essential resource for anyone wishing to know more about how the mathematics of the different regions influenced and shaped the development of world mathematics."--George Gheverghese Joseph, Nature "We're aware that the ancient cultures were mathematically advanced. Now translations of early texts from five key regions are available together for the first time, and put into context by experts."--Nature Physics "The corrections to the Eurocentrism that understandably characterized Western assays of the intellectual history of the planet early on must inevitably be applied to the history of mathematics. Editor Katz and his scholarly coauthors have greatly advanced the process with this one-volume sourcebook...The introductory essays that precede each section are lucidly written, well within the reach of an undergraduate math major. Katz asks more or less rhetorically 'how much effect the mathematics of these civilizations had on what is now world mathematics of the twenty-first century.' This invaluable book will help significantly in formulating an answer."--M. Schiff, Choice "This book is an essential resource for anyone with at least an undergraduate degree in mathematics who wants to learn about non-Western mathematical developments and how they helped shape and enrich world mathematics. The book is also an indispensable guide for mathematics teachers who want to use non-Western mathematical ideas in the classroom."--L'Enseignement Mathematique "The Mathematics of Egypt, Mesopotamia, China, India, and Islam is a wonderful collection, for which Victor Katz is to be commended. This book is a one-stop source for numerous original mathematical texts in translation. I cannot overemphasize how wonderful it is to have this large, exquisite selection of ... mathematical texts together in one volume... Every history of mathematics teacher will want a copy of this book in their personal library as well as in the library of their college or university."--James V. Rauff, Mathematics and Computer Education "What we have here is a useful selection, one that should be of interest to specialists in world history or in the history of the sciences in any of these culture areas and, in particular, to scholars who are engaged with the history of mathematics as specialists or because of its role as a tool."--Tom Archibald, Isis "[This] is the biggest sourcebook containing the newest fruit of historical research and that is why the book can replace older sources for the history of mathematics."--EMS NewsletterTable of ContentsPreface ix Permissions xi Introduction 1 Chapter 1: Egyptian Mathematics Annette Imhausen Preliminary Remarks 7 I. Introduction 9 a. Invention of writing and number systems 13 b. Arithmetic 14 c.Metrology 17 II. Hieratic Mathematical Texts 17 a. Table texts 18 b. Problem texts 24 III. Mathematics in Administrative Texts 40 a. Middle Kingdom texts: The Reisner papyri 40 b. New Kingdom texts: Ostraca from Deir el Medina 44 IV. Mathematics in the Graeco-Roman Period 46 a. Context 46 b. Table texts 47 c. Problem texts 48 V. Appendices 52 a. Glossary of Egyptian terms 52 b. Sources 52 c. References 54 Chapter 2: Mesopotamian Mathematics Eleanor Robson I. Introduction 58 a. Mesopotamian mathematics through Western eyes 58 b.Mathematics and scribal culture in ancient Iraq 62 c. From tablet to translation 65 d. Explananda 68 II. The Long Third Millennium, c. 3200-2000 BCE 73 a. Uruk in the late fourth millennium 73 b. Shuruppag in the mid-third millennium 74 c. Nippur and Girsu in the twenty-fourth century BCE 76 d. Umma and Girsu in the twenty-first century BCE 78 III. The Old Babylonian Period, c. 2000-1600 BCE 82 a. Arithmetical and metrological tables 82 b. Mathematical problems 92 c. Rough work and reference lists 142 IV. Later Mesopotamia, c. 1400-150 BCE 154 V. Appendices 180 a. Sources 180 b. References 181 Chapter 3: Chinese Mathematics Joseph W. Dauben Preliminary Remarks 187 I. China: The Historical and Social Context 187 II. Methods and Procedures: Counting Rods, The "Out-In" Principle 194 III. Recent Archaeological Discoveries: The Earliest Yet-Known Bamboo Text 201 IV. Mathematics and Astronomy: The Zhou bi suan jing and Right Triangles (The Gou-gu or "Pythagorean" Theorem) 213 V. The Chinese "Euclid", Liu Hui 226 a. The Nine Chapters 227 b. The Sea Island Mathematical Classic 288 VI. The "Ten Classics" of Ancient Chinese Mathematics 293 a. Numbers and arithmetic: The Mathematical Classic of Master Sun 295 b. The Mathematical Classic of Zhang Qiujian 302 VII. Outstanding Achievements of the Song and Yuan Dynasties (960-1368 CE) 308 a. Qin Jiushao 309 b. Li Zhi (Li Ye) 323 c. Yang Hui 329 d. Zhu Shijie 343 VIII. Matteo Ricci and Xu Guangxi, "Prefaces" to the First Chinese Edition of Euclid's Elements (1607) 366 IX. Conclusion 375 X. Appendices 379 a. Sources 379 b. Bibliographic guides 379 c. References 380 Chapter 4: Mathematics in India Kim Plofker I. Introduction: Origins of Indian Mathematics 385 II. Mathematical Texts in Ancient India 386 a. The Vedas 386 b. The S'ulbasutras 387 c. Mathematics in other ancient texts 393 d. Number systems and numerals 395 III. Evolution of Mathematics in Medieval India 398 a.Mathematics chapters in Siddhanta texts 398 b. Transmission of mathematical ideas to the Islamic world 434 c. Textbooks on mathematics as a separate subject 435 d. The audience for mathematics education 477 e. Specialized mathematics: Astronomical and cosmological problems 478 IV. The Kerala School 480 a. Madhava, his work, and his school 480 b. Infinite series and the role of demonstrations 481 c. Other mathematical interests in the Kerala school 493 V. Continuity and Transition in the Second Millennium 498 a. The ongoing development of Sanskrit mathematics 498 b. Scientific exchanges at the courts of Delhi and Jaipur 504 c. Assimilation of ideas from Islam; mathematical table texts 506 VI. Encounters with Modern Western Mathematics 507 a. Early exchanges with European mathematics 507 b. European versus "native" mathematics education in British India 508 c. Assimilation into modern global mathematics 510 VII. Appendices 511 a. Sources 511 b. References 512 Chapter 5: Mathematics in Medieval Islam J. Lennart Berggren I. Introduction 515 II. Appropriation of the Ancient Heritage 520 III. Arithmetic 525 IV. Algebra 542 V. Number Theory 560 VI. Geometry 564 a. Theoretical geometry 564 b. Practical geometry 610 VII. Trigonometry 621 VIII. Combinatorics 658 IX. On mathematics 666 X. Appendices 671 a. Sources 671 b. References 674 Contributors 677 Index 681

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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 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 SynopsisRevealing the mathematical side of Benjamin Franklin, this book explains the mathematics behind Franklin's popular "Poor Richard's Almanac", which featured such things as population estimates and a host of mathematical digressions. It includes optional math problems that challenge readers to match wits with the Founding Father himself.Trade Review"Pasles...speculates gleefully on the oft-denied mathematical genius of Benjamin Franklin...Drawing on Franklin's letters and journals as well as modern-day reconstructions of his library, Pasles touches on Franklin's fondness for magazines of mathematical diversions; publication of arithmetic problems in Poor Richard's Almanac; startlingly accurate projections of population growth and cost-benefit arguments against slavery."--Publisher's Weekly "In Franklin's Numbers, a book mixing intellectual history and mathematical puzzles (with solutions appended), Paul Pasles brings out a less-celebrated sphere of Franklin's intellect. He makes the case for the founding father as a mathematician."--Jared Wunsch, Nature "Pasles delivers surprising news to Sudoku lovers: Benjamin Franklin once shared their passion...Pasles illuminates Franklin's innovative use of mathematical logic in settling moral questions and in assessing population trends. Franklin's mathematical pursuits thus emerge as a complement to his much-lauded work in politics and science. An unexpected but welcome perspective on the genial genius of Philadelphia."--Bryce Christensen, Booklist "There is hardly a discipline on which Franklin did not stamp his mark during the 18th century. But the role that mathematics played in his life has been overlooked, argues Paul Pasles. Franklin, for instance, was fascinated with magic squares, and this book provides plenty of background to help the reader admire his interest."--New Scientist "[This is] a book that is an easy read for the innumerate but which also provides nourishment for those more skilled in the niceties of math...Also included are some contemporary puzzles that offer the reader the chance to contest skills with Franklin himself."--James Srodes, The Washington Times "Making frequent use of Franklin's writings as well as mathematical brainteasers of the type that Franklin enjoyed, Benjamin Franklin's Numbers is an engaging and thoroughly unique biography of a singular figure in American history."--Ray Bert, Civil Engineering "I thoroughly enjoyed reading this book. It is written in a pleasant, conversational style and the author's enthusiasm for his subject is infectious. The text is richly embroidered with colorful details, both mathematical and historical."--Eugene Boman, Convergence: A Magazine of the Mathematical Association of America "Pasles has succeeded in writing a book dealing with mathematics that is accessible to readers at all levels, yet thoroughly referenced and scholarly enough to satisfy researchers. His endeavor was eased by the fact that the bulk of the material concerns Franklin's magic squares and circles, which only require that the reader have the ability to add. Unexpectedly, Pasles contributes much that is new; he corrects the errors of previous authors and presents new ideas through literary sleuthing and mathematical analysis."--C. Bauer, Choice "Pasles makes a convincing case for Franklin as the last true Renaissance man in what is an entertaining and informative book that will even appeal to readers with only limited knowledge of mathematics."--Physics World "With seven years of diligent study, by going through a vast amount of archive material, references including primary sources and books and research papers, the author has produced a carefully documented and fascinating account to substantiate the theme he makes, namely, that Franklin 'possessed a mathematical mind.'"--Man Keung Siu, Mathematical Reviews "[Paul C. Pasles] and the publisher should ... be commended for producing a highly aesthetically pleasing book, with a color centerpiece showing many of Franklin's beloved magic squares in their full glory."--Eli Maor, SIAM Review "This book will appeal to readers with an interdisciplinary interest in both history and mathematics. Teachers who enjoy showing students the many ways in which they can draw on mathematics to construct logical, real-world arguments will find useful examples for the classroom. The book also includes a variety of number puzzles that can be used to challenge students."--Michelle Cirillo, Mathematics Teacher "I found Benjamin Franklin's Numbers a delightful book. I enjoyed studying and playing with the magic squares and patterns, and I was fascinated by the biographical tidbits about Franklin. This book is very well written, and I highly recommend it to anyone with an interest in mathematics or in Benjamin Franklin."--James V. Rauff, Mathematics and Computer EducationTable of ContentsPreface ix Chapter 1: The Book Franklin Never Wrote 1 Chapter 2: A Brief History of Magic 20 Chapter 3: Almanacs and Assembly 61 Interlude: Philomath Math 83 Chapter 4: Publisher, Theorist, Inventor, Innovator 87 Chapter 5: A Visit to the Country 117 Chapter 6: The Mutation Spreads (Adventures Among the English) 141 Chapter 7: Circling the Square 158 Chapter 8: Newly Unearthed Discoveries 191 Chapter 9: Legacy 226 Acknowledgements 243 Appendix 245 Index 253

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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 SynopsisPlotting humankind's efforts to visualize data, this book discusses atheoretical plotting of data to reveal suggestive patterns. It includes chapters illustrating the uses and abuses of this invention (plotting), from a murder trial in Connecticut to the Vietnam War's effect on college admissions.Trade ReviewOne of Choice's Outstanding Academic Titles for 2005 "Well written and innovative... The book is fascinating with its wide view, including introductions to historical personalities, analyses of statistical paradoxes, and well-documented discussions of actual uses of visual data to mislead viewers."--Choice "During a dairyman's strike in 19th century New England, when there was suspicion of milk being watered down, Henry David Thoreau wrote, 'Sometimes circumstantial evidence can be quite convincing; like when you find a trout in the milk.' Howard Wainer uses this as a metaphor in his entertaining, informative, and persuasive book on graphs, or the visual communication of information. Sometimes a well-designed graph tells a very convincing story."--Raymond N. Greenwell, MAA Online "Wainer's wit and broad intellect make this a very entertaining book."--Linda Pickle, ,American Statistician "[A] personalized and readable jaunt through the history of charting."--The Economist "This book may be seen as a chronology of graphic date presentation beginning with Playfair to the present and pointing toward the future... It is a remarkable value that every practitioner of statistics can afford."--Malcolm James Ree, Personnel Psychology "Graphic Discovery is a welcome addition to the literature on investigation and effective communication through graphic display. It contains a wealth of information and opinions, which are motivated and illustrated through a plethora of real life examples which can be easily incorporated into any educational setting: classroom, seminar, self-enhancement... This book will be useful to and it can be mastered by a diverse readership."--Thomas E. Bradstreet, Computational StatisticsTable of ContentsPreface xiii Introduction 1 In the sixteenth century, the bubonic plague provided the motivation for the English to begin gathering data on births, marriages, and deaths. These data, the Bills of Mortality, were the grist that Dr. John Arbuthnot used to prove the existence of God. Unwittingly, he also provided strong evidence that data graphs were not yet part of a scientist's tools. Part I: William Playfair and the Origins of Graphical Display Chapter 1: Why Playfair? 9 All of the pieces were in place for the invention of statistical graphics long before Playfair was born. Why didn't anyone else invent them? Why did Playfair? Chapter 2: Who Was Playfair? 20 by Ian Spence and Howard Wainer William Playfair (1759-1823) was an inventor and ardent advocate of statistical graphics. Here we tell a bit about his life. Chapter 3: William Playfair: A Daring Worthless Fellow 24 by Ian Spence and Howard Wainer Audacity was an important personality trait for the invention of graphics because the inventor had to move counter to the Cartesian approach to science. We illustrate this quality in Playfair by describing his failed attempt to blackmail one of the richest lords of Great Britain. Chapter 4: Scaling the Heights (and Widths) 28 The message conveyed by a statistical graphic can be distorted by manipulating the aspect ratio, the ratio of a graph's width to its height. Playfair deployed this ability in a masterly way, providing a guide to future display technology. Chapter 5: A Priestley View of International Currency Exchanges 39 A recent plot of the operating hours of international currency exchanges confuses matters terribly. Why? We find that when we use a different graphical form, developed by Joseph Priestley in 1765, the structure becomes clear. We also learn how Priestley discovered the latent graphicacy in his (and our) audiences. Chapter 6: Tom's Veggies and the American Way 44 European intellectuals were not the only ones graphing data. During a visit to Paris (and prompted by letters from Benjamin Franklin), Thomas Jefferson learned of this invention and he later put it to a more practical use than the depiction of the life spans of heroes from classical antiquity. Chapter 7: The Graphical Inventions of Dubourg and Ferguson: Two Precursors to William Playfair 47 Although he developed the line chart independently, Priestley was not the first to do so. The earliest seems to be the Parisian physician Jacques Barbeau-Dubourg (1709-1779), who created a wonderful graphical scroll in 1753. Graphical representation must have been in the air, for the Scottish philosopher Adam Ferguson (1723-1816) added his version of time lines to the mix in 1780. Chapter 8: Winds across Europe: Francis Galton and the Graphic Discovery of Weather Patterns 52 In 1861, Francis Galton organized weather observatories throughout Western Europe to gather data in a standardized way. He organized these data and presented them as a series of ninety-three maps and charts, from which he confirmed the existence of the anticyclonic movement of winds around a low-pressure zone. Part II: Using Graphical Displays to Understand the Modern World Chapter 9: A Graphical Investigation of the Scourge of Vietnam 59 During the Vietnam War, average SAT scores went down for those students who were not in the military. In addition, the average ASVAB scores (the test used by the military to classify all members of the military) also declined. This Lake Wobegon-like puzzle is solved graphically. Chapter 10: Two Mind-Bending Statistical Paradoxes 63 The odd phenomenon observed with test scores during the Vietnam War is not unusual. We illustrate this seeming paradox with other instances, show how to avoid them, and then discuss an even subtler statistical pitfall that has entrapped many illustrious would-be data analysts. Chapter 11: Order in the Court 72 How one orders the elements of a graph is critical to its comprehensibility. We look at a New York Times graphic depicting the voting records of U.S. Supreme Court justices and show that reordering the graphic provides remarkable insight into the operation of the court. Chapter 12: No Order in the Court 78 We examine one piece of the evidence presented in the 1998 murder trial of State v. Gibbs and show how the defense attorneys, by misordering the data in the graph shown to the judge, miscommunicated a critical issue in their argument. Chapter 13: Like a Trout in the Milk 81 Thoreau pointed out that sometimes circumstantial evidence can be quite convincing, as when you find a trout in the milk. We examine a fascinating graph that provides compelling evidence of industrial malfeasance. Chapter 14: Scaling the Market 86 We examine the stock market and show that different kinds of scalings provide the answers to different levels of questions. One long view suggests a fascinating conjecture about the trade-offs between investing in stocks and investing in real estate. Chapter 15: Sex, Smoking, and Life Insurance: A Graphical View 90 We examine two risk factors for life insurance--sex and smoking--and uncover the implicit structure that underlies insurance premiums. Chapter 16: There They Go Again! 97 The New York Times is better than most media sources for statistical graphics, but even the Times has occasional relapses to an earlier time in which confusing displays ran rampant over its pages. We discuss some recent slips and compare them with prior practice. Chapter 17: Sex and Sports: How Quickly Are Women Gaining? 103 A simple graph of winning times in the Boston Marathon augmented by a fitted line provides compelling, but incorrect, evidence for the relative gains that women athletes have made over the past few decades. A more careful analysis provides a better notion of the changing size of the sex differences in athletic performances. Chapter 18: Clear Thinking Made Visible: Redesigning Score Reports for Students 109 Too often communications focus on what the transmitter thinks is important rather than on what the receiver is most critically interested in. The standard SAT score report that is sent to more than one million high school students annually is one such example. Here we revise this report using principles abstracted from another missive sent to selected high school students. Part III: Graphical Displays in the Twenty-first Century The three chapters of this section grew out of a continuing conversation with John W. Tukey, the renowned Princeton polymath, on the graphical tools that were likely to be helpful when data were displayed on a computer screen rather than a piece of paper. These conversations began shortly after Tukey's eighty-fourth birthday and continued for more than a year, ending the night before he died. Chapter 19: John Wilder Tukey: The Father of Twenty-first-Century Graphical Display 117 Chapter 20: Graphical Tools for the Twenty-first Century: I. Spinning and Slicing 125 Chapter 21: Graphical Tools for the Twenty-first Century: II. Nearness and Smoothing Engines 134 Chapter 22: Epilogue: A Selection of Selection Anomalies 142 Graphical displays are only as good as the data from which they are composed. In this final chapter we examine an all too frequent data flaw. The effects of nonsampling errors deserve greater attention, especially when randomization is absent. Formal statistical analysis treats only some of the uncertainties. In this chapter we describe three examples of how flawed inferences can be made from nonrandomly obtained samples and suggest a strategy to guard against flawed inferences. Conclusion 150 Dramatis Personae 151 This graphical epic has more than one hundred characters. Some play major roles, but most are cameos. To help keep straight who is who, this section contains thumbnail biographies of all the players. Notes 173 References 177 Index 185

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