Philosophy of mathematics Books

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

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

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Book SynopsisWritings by early mathematicians feature language and notations that are quite different from what we're familiar with today. Sourcebooks on the history of mathematics provide some guidance, but what has been lacking is a guide tailored to the needs of readers approaching these writings for the first time. This book intends to fill this gap.Trade ReviewOne of Choice's Outstanding Academic Titles for 2010 "Anyone interested in the history of mathematics should start here, especially those who teach history of mathematics courses. The text is refreshing, relevant, and surprisingly interesting. A great read!"--Choice "[This book] is well written, readable, and straightforward... It should be read by anyone who is using original source material to study the history of mathematics."--David Ebert, Mathematics Teacher "This is an extraordinary book for anyone interested in the history of mathematics. The author notes in the preface that reading historical mathematics can be fascinating, challenging, enriching, and endlessly rewarding. He then proceeds to illustrate how to analyze and get the most out of original source material."--Jim Tattersall, MAA Reviews "What Wardhaugh does exceptionally well is to break the ice for readers interested in the subject. He does this largely by training readers to ask insightful questions when they read a historical text."--Sol Lederman, Wild About Math "How to Read Historical Mathematics is filled with worthwhile advice to historians of mathematics and potential historians of mathematics. Wardhaugh's book should be readily available and kept with your personal reference books. It should also be in your school library."--Donald Cook, Mathematical Review "[A] splendid introduction to what to look for and to think about when reading historical source material in mathematics... This volume provides much food for thought in relatively few pages, yet in a pleasantly relaxed manner."--Leon Harkleroad, Zentralblatt MATH "How to Read Historical Mathematics is more than a useful aid to students being introduced to the field: it is a practical field guide to a whole new way of doing the history of mathematics. I warmly recommend it."--Amir Alexander, British Journal for the History of Science "Although Wardhaugh's examples will likely appeal mainly to those already interested in the history of mathematics, his commentary is broadly applicable to all of history of science and indeed to all students of history generally. There are occasional mentions of technological tools unknown to earlier generations of historians, but for the most part the discussion is generic enough that one expects How to Read Historical Mathematics to remain relevant even in a future where JSTOR and Google Books may no longer have the place they hold now."--David Lindsay Roberts, ISIS "Each item is preceded by a brief sketch of its author and context. The entertainment for the reader rests not only with the mathematical content but also in the evolution of expository style and often inventive presentation."--E. J. Barbeau, Mathematical Reviews Clippings "The book is a small jewel, the book to give to the student who is interested in pursuing history of mathematics. The author is apparently a talented historian."--UMAP JournalTable of ContentsPreface vii Chapter 1: What Does It Say? 1 Chapter 2: How Was It Written? 21 Chapter 3: Paper and Ink 49 Chapter 4: Readers 73 Chapter 5: What to Read, and Why 92 Bibliography 111 Index 115

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Book SynopsisExamines how advances in philosophy were led by scientific discoveries - the more humankind understood about the physical world, the more curious we became. Drawing on work by Descartes, Galileo, Hume, Kant, Leibniz, and Newton, this book helps readers understand science through the lens of philosophy.Trade Review"The translation has long been out of print, so this recent publication, with a very fine introduction by Frank Wilczek, is to be highly valued... Weyl's Philosophy of Mathematics and Natural Science should be on every mathematician's or physicist's bookshelf... What a pleasure, what a privilege, to read and contemplate Hermann Weyl's monumental achievements."--Jeremy Butterfield, Physics Today "[W]e remain ever grateful that Hermann Weyl, compromising his conscience to the extent that he did, left behind this unrivaled treasure of insights into the murkiest epistemological depths of mathematics and theoretical physics."--Thomas Ryckman, Metascience

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Book SynopsisAn anthology that gathers together nearly one hundred selections from the past 500 years of popular math writing. Ranging from the late fifteenth to the late twentieth century, and drawing from books, newspapers, magazines, and websites, it includes recreational, classroom, and work mathematics; mathematical histories and biographies; and, more.Trade Review"One of the pleasures of this book is reading the texts in the language of the day... The collection as a whole provides the general reader with a history of mathematics, biographical and otherwise, through popular writing. Because the writing was aimed at general readers of its time, it is usually accessible to the average mathematical reader of our time. The book would be an excellent reference for teachers of mathematics and for those researching the history of the dissemination of mathematical ideas."--Carol Dorf, American Scientist "[F]or the enthusiast for the history of popular maths writing this is a must-have book."--Brian Clegg, Popular Science "In A Wealth of Numbers, we have the end product of what must have been a lot of challenging research... This book works well for random browsing as well as for sustained reading; purely recreational essays and puzzle problems are well-mixed with more serious topics such as an article explaining Cantor's diagonalization proof and 'Cubic equations for the practical man.' There's something in here for everyone, and it's a great contribution to the mathematics literature to have it all in one place."--Mark Bollman, MAA Reviews "Wardhaugh provides an exciting addition to mathematics anthologies... The physical format is very reader-friendly, with especially good line spacing and margins. The book is valuable for all libraries supporting undergraduate and graduate study, as well as many public libraries. Faculty should consider this as a source of comprehensible readings for aspiring mathematics majors. Individuals interested in math history will want a copy for their personal libraries."--Choice "The Wardhaugh book is a welcome addition to anthologies that have preceded it... Although written for the general reader who is interested in mathematics, the collection is apropos for those who are more mathematically oriented as well... [T]his well-thought-out, eclectic collection will provide hours of enjoyable reading."--Jim Tattersall, CSHPM "Fascinating to browse, a delight to read, and informative... Get this book! It is as much fun to read as it is to share with others, especially students who can gain from doses of past mathematical realities."--Jerry Johnson, Mathematics Teacher "This book permits the reader to pick it up whenever he or she has a few minutes (or longer) to spare, and find a section to fit the available free time and mood. It will provide the reader, novice and expert alike, many hours of learning filled with surprise, pleasure, amazement, and sometimes laughter."--Godfried Toussaint, Zentralblatt MATH "A Wealth of Numbers explores the often overlooked history of popular mathematics in an easy to read and captivating manner. I recommend the book, not only as an excellent research text in this area of mathematics, but as an interesting and entertaining read."--Steve Humble, Mathematics Today MagazineTable of Contents*FrontMatter, pg. i*Contents, pg. v*Preface, pg. xiii*1. "Sports and Pastimes, Done by Number": Mathematical Tricks, Mathematical Games, pg. 1*2. "Much Necessary for All States of Men": From Arithmetic to Algebra, pg. 32*3. "A Goodly Struggle": Problems, Puzzles, and Challenges, pg. 62*4. "Drawyng, Measuring and Proporcion": Geometry and Trigonometry, pg. 84*5. Maps, Monsters, and Riddles: The Worlds of Mathematical Popularization, pg. 108*6. "To Ease and Expedite the Work": Mathematical Instruments and How to Use Them, pg. 152*7. "How Fine a Mind": Mathematicians Past, pg. 176*8. "By Plain and Practical Rules": Mathematics at Work, pg. 216*9. "The Speedier Expedition of Their Learning": Thoughts on Teaching and Learning Mathematics, pg. 245*10. "So Fundamentally Useful a Science": Reflections on Mathematics and Its Place in the World, pg. 290*11. The Mathematicians Who Never Were: Fiction and Humor, pg. 326*Index, pg. 367

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Book SynopsisRecalls the last words of the great Greek mathematician Archimedes before he was slain by a Roman soldier - "Don't disturb my circles" - words that seem to refer to two radically different concerns: that of the practical person living in the concrete world of reality, and that of the theoretician lost in a world of abstraction.Trade Review"Editors Doxiadis and Mazur have compiled a collection of 15 essays that look at the many possible roles narrative can play in mathematics, which is usually considered far removed from storytelling... Circles Disturbed will be of special value to collections in history of mathematics, philosophy of mathematics, and mathematical pedagogy."--Choice "Circles Disturbed presents a cohesive narrative whose strength lies in helping each side to understand the other. It should encourage scientists to grasp the logic behind storytelling and literary critics to sense the allure of mathematics."--Mel Bayley, British Society for the History of Mathematics Bulletin "Well thought and well written and with a careful balance between erudition and down-to-earthness all through it, Circles Disturbed is a highly recommended reading for mathematicians and students of mathematics, as well as for anyone who wishes to better understand what it is to do mathematics and why they are done the way they are done."--Capi Corrales Rodriganez, European Mathematical Society "Circles Disturbed will spark interest in younger readers in the commonalities among these three disciplines as well as engage other readers. Further, readers with greater background in one or more topics can see the intra- and the intersections rather naturally and inquisitively. The diverse perspectives represented by the various authors are quite refreshing."--Farshid Safi, Mathematics TeacherTable of ContentsIntroduction vii Chapter 1: From Voyagers to Martyrs: Toward a Storied History of Mathematics 1 By AMIR ALEXANDER Chapter 2 Structure of Crystal, Bucket of Dust 52 By PETER GALISON Chapter 3: Deductive Narrative and the Epistemological Function of Belief in Mathematics: On Bombelli and Imaginary Numbers 79 By FEDERICA LANAVE Chapater 4: Hilbert on Theology and Its Discontents: The Origin Myth of Modern Mathematics 105 By COLIN MCLARTY Chapter 5: Do Androids Prove Theorems in Their Sleep? 130 By MICHAEL HARRIS Chapter 6: Visions, Dreams, and Mathematics 183 By BARRY MAZUR Chapter 7: Vividness in Mathematics and Narrative 211 By TIMOTHY GOWERS Chapter 8: Mathematics and Narrative: Why Are Stories and Proofs Interesting? 232 By BERNARD TEISSIER Chapter 9: Narrative and the Rationality of Mathematical Practice 244 By DAVID CORFIELD Chapter 10: A Streetcar Named (among Other Things) Proof: From Storytelling to Geometry, via Poetry and Rhetoric 281 By APOSTOLOS DOXIADIS Chapter 11: Mathematics and Narrative: An Aristotelian Perspective 389 By G .E .R . LLOYD Chapter 12: Adventures of the Diagonal: Non-Euclidean Mathematics and Narrative 407 By ARADY PLOTNITSKY Chapter 13: Formal Models in Narrative Analysis 447 By DAVID HERMAN Chapter 14: Mathematics and Narrative: A Narratological Perspective 481 By URI MARGOL N Chapter 15: Tales of Contingency, Contingencies of Telling: Toward an Algorithm of Narrative Subjectivity 508 By JAN CHRISTOPH MEISTER Contributors 541 Index 545

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Book SynopsisWhat is algebra? For some, it is an abstract language of x's and y's. For mathematics majors and professional mathematicians, it is a world of axiomatically defined constructs like groups, rings, and fields. This title considers how these two seemingly different types of algebra evolved and how they relate.Trade Review"An excellent book; its accurate historical and pedagogical purpose offers an accessible read for historians and mathematicians."--Raffaele Pisano, Metascience "Well written and engaging with a wealth of useful material and a substantial bibliography for further reading, this book is a valuable resource for anyone with a serious interest in the history of algebra. With Taming the Unknown, Victor Katz and Karen Parshall have created a comprehensive synthesis of recent research on the subject, accessible to mathematicians, historians of mathematics and anyone involved in the teaching of algebra."--Adrian Rice, BSHM Bulletin "The authors have ... pitched their writing perfectly for their intended audience. The broad outline of the story is expressed in clear prose, combined with a judicious use of that other 'native tongue' of the college mathematics graduate, symbolic algebra... There is an extensive bibliography presenting the more detailed historical research that has been carried out... You could base a really nice third-year course on this book."--John Hannah, AestimatioTable of ContentsAcknowledgments xi 1 Prelude: What Is Algebra? 1 Why This Book? 3 Setting and Examining the Historical Parameters 4 The Task at Hand 10 2 Egypt and Mesopotamia 12 Proportions in Egypt 12 Geometrical Algebra in Mesopotamia 17 3 The Ancient Greek World 33 Geometrical Algebra in Euclid's Elements and Data 34 Geometrical Algebra in Apollonius's Conics 48 Archimedes and the Solution of a Cubic Equation 53 4 Later Alexandrian Developments 58 Diophantine Preliminaries 60 A Sampling from the Arithmetica: The First Three Greek Books 63 A Sampling from the Arithmetica: The Arabic Books 68 A Sampling from the Arithmetica: The Remaining Greek Books 73 The Reception and Transmission of the Arithmetica 77 5 Algebraic Thought in Ancient and Medieval China 81 Proportions and Linear Equations 82 Polynomial Equations 90 Indeterminate Analysis 98 The Chinese Remainder Problem 100 6 Algebraic Thought in Medieval India 105 Proportions and Linear Equations 107 Quadratic Equations 109 Indeterminate Equations 118 Linear Congruences and the Pulverizer 119 The Pell Equation 122 Sums of Series 126 7 Algebraic Thought in Medieval Islam 132 Quadratic Equations 137 Indeterminate Equations 153 The Algebra of Polynomials 158 The Solution of Cubic Equations 165 8 Transmission, Transplantation, and Diffusion in the Latin West 174 The Transplantation of Algebraic Thought in the Thirteenth Century 178 The Diffusion of Algebraic Thought on the Italian Peninsula and Its Environs from the Thirteenth Through the Fifteenth Centuries 190 The Diffusion of Algebraic Thought and the Development of Algebraic Notation outside of Italy 204 9 The Growth of Algebraic Thought in Sixteenth-Century Europe 214 Solutions of General Cubics and Quartics 215 Toward Algebra as a General Problem-Solving Technique 227 10 From Analytic Geometry to the Fundamental Theorem of Algebra 247 Thomas Harriot and the Structure of Equations 248 Pierre de Fermat and the Introduction to Plane and Solid Loci 253 Albert Girard and the Fundamental Theorem of Algebra 258 Rene Descartes and The Geometry 261 Johann Hudde and Jan de Witt, Two Commentators on The Geometry 271 Isaac Newton and the Arithmetica universalis 275 Colin Maclaurin's Treatise of Algebra 280 Leonhard Euler and the Fundamental Theorem of Algebra 283 11 Finding the Roots of Algebraic Equations 289 The Eighteenth-Century Quest to Solve Higher-Order Equations Algebraically 290 The Theory of Permutations 300 Determining Solvable Equations 303 The Work of Galois and Its Reception 310 The Many Roots of Group Theory 317 The Abstract Notion of a Group 328 12 Understanding Polynomial Equations in n Unknowns 335 Solving Systems of Linear Equations in n Unknowns 336 Linearly Transforming Homogeneous Polynomials in n Unknowns: Three Contexts 345 The Evolution of a Theory of Matrices and Linear Transformations 356 The Evolution of a Theory of Invariants 366 13 Understanding the Properties of "Numbers" 381 New Kinds of "Complex" Numbers 382 New Arithmetics for New "Complex" Numbers 388 What Is Algebra?: The British Debate 399 An "Algebra" of Vectors 408 A Theory of Algebras, Plural 415 14 The Emergence of Modern Algebra 427 Realizing New Algebraic Structures Axiomatically 430 The Structural Approach to Algebra 438 References 449 Index 477

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Book SynopsisSet in 1998 with flashbacks to classical Greece, this title investigates the confrontation between opposing views of mathematics and reality, and explores ideas from both early and cutting-edge mathematics.Trade Review"Who would have guessed that a murder-treasure mystery lay hidden behind a geometric formula familiar to every high-schooler? Weaving a wealth of mathematical scholarship into a compellingly plotted novel, Sangalli recounts a fascinating tale of ancient arson and modern sleuthing, as Pythagoras of Samos (forever linked to the triangular theorem bearing his name) perishes amid brutal intrigues sweeping an early Greek colony, yet leaves behind a tantalizing legacy of numerical reasoning and paranormal mysticism... To be sure, it is the author's own fertile imagination that generates the characters who form this resolute band and then scripts the adventures they encounter in their unlikely international quest... [R]eaders will learn a great deal about real mathematics and its history as they join Pythagoras' modern epigones in pondering the meaning of geometrical patterns, the surprising randomness in numbers, and the logic of mathematical proofs... [T]his engaging narrative will persuade many readers that mathematics offers far more excitement than they had previously supposed."--Bryce Christensen, Booklist "[The book] comes together [around] the tantalizing possibility that Pythagoras, who forbade his followers to write down any of his sayings, may just have left something tangible after all. Sangalli builds his story on this, using clues from ancient texts, bits of mathematical lore and interesting arcana, like the puzzle that couldn't be patented because it had no solution. For a total escape, this novel is perfect."--Margaret Cannon, Globe and Mail "Pythagoras' Revenge: A Mathematical Mystery is more than just a novel. It is also an introduction to several big ideas in mathematics, from infinite series to unsolvable puzzles... [T]his romp through ancient and modern mathematics is entertaining in patches, and certainly a cut above standard holiday reading. Despite occasional plot hiccups, its gripping story will likely hold readers to the end."--Physics World "Initially Pythagoras' Revenge was intended to discuss the tyranny of numbers in modern societies in the same style as Sangalli's previous book. But, as if by magic, it became instead a work of fiction... What remains after the end of this page-turner is Sangalli's impressive capacity to communicate mathematics. Let us take this book as a reminder to capitalize on the full potential of scientific storytelling."--Javier Fresan, Notices of the AMS "This is an entertaining read, and although the plot is implausible at times it succeeds in conveying a variety of mathematical and philosophical ideas in a simple and light-hearted way... Pythagoras' Revenge is a gripping novel that offers a refreshing way to learn about mathematics."--Sarah Shepherd, iSquared "Human beings are story making animals, and this book shows that there is an opportunity to make use of this approach in the field. A fascinating attempt."--Brian Clegg, Popular Science "Read this book if you like mathematics and spend some time ruminating over the larger philosophical questions that are implicit in modern math. Such questions go directly to the heart of modern scientific culture."--William Byers, European Legacy "If you like conspiracy adventure, and can dismiss the shallow characters and clunky sub-plots, it's a fun read as you get the history, philosophy, and theories on randomness and math, and of a figure who famously said, 'All is Number.'"--Phil Semler, San Francisco Book ReviewTable of ContentsPreface ix List of Main Characters xi Prologue xiii PART I: A TIME CAPSULE? Chapter 1. The Fifteen Puzzle 3 Chapter 2. The Impossible Manuscript 10 Chapter 3. Game Over 19 Chapter 4. A Trip to London 25 Chapter 5. A Letter from the Past 32 Chapter 6. Found and Lost 38 Chapter 7. A Death in the Family 46 PART II: AN EXTRAORDINARILY GIFTED MAN Chapter 8. The Mission 53 Chapter 9. Norton Thorp 63 Chapter 10. Random Numbers 69 Chapter 11. Randomness Everywhere 76 Chapter 12. Vanished 82 PART III: A SECT OF NEO-PYTHAGOREANS Chapter 13. The Mandate 85 Chapter 14. The Beacon 87 Chapter 15. The Team 98 Chapter 16. The Hunt 106 Chapter 17. The Symbol of the Serpent 115 Chapter 18. A Professional Job 122 Chapter 19. With a Little Help from Your Sister 126 PART IV: PYTHAGORAS' MISSION Chapter 20. All Roads Lead to Rome 139 Chapter 21. Kidnapped 152 Chapter 22. The Last Piece of the Puzzle 158 Epilogue 169 Appendix 1: Jule's Solution 171 Appendix 2: Infi nitely Many Primes 173 Appendix 3: Random Sequences 175 Appendix 4: A Simple Visual Proof of the Pythagorean Theorem 177 Appendix 5: Perfect and Figured Numbers 178 Notes, Credits, and Bibliographical Sources 181

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Book SynopsisHenri Poincar (1854-1912) was not just one of the most inventive, versatile, and productive mathematicians of all time - he was also a leading physicist who almost won a Nobel Prize for physics. This book explores all the fields that Poincar touched, the debates sparked by his investigations, and how his discoveries still contribute to society.Trade ReviewOne of Choice's Outstanding Academic Titles for 2013 "[M]asterly ... Gray encapsulates Poincare's multiple dimensions; his intellectual biography is both a tour de force and a triumph of readability."--George Szpiro, Nature "Gray shows us the full dazzling sweep of what Poincare accomplished, including the work on dynamical systems and chaos that only came into its own in recent years. A tour de force, Gray's masterful treatment will long remain an invaluable resource for all who want to understand Poincare, so embedded within his times and yet so far ahead of them."--Peter Pesic, Science "[A] comprehensive but uncluttered guide to Poincare's extensive oeuvres."--Madeline Muntersbjorn, Times Higher Education "Full of the mathematical, physical and metaphysical ideas of a man who was not only a dispassionate observer of the world around us, but of our way of understanding it."--Mark Ronan, Standpoint Magazine (U.K.) "[A] comprehensive assessment of Poincare's work and its importance, essential for anyone interested in Poincare's scholarship or the history of mathematics."--Laura Tarwater Scharp, Sacramento Book Review "Comprehensive."--Science News "A fundamental study of the scientific work of one of the greatest mathematicians and mathematical physicists of the three decades straddling the 19th and 20th centuries... Chapters are organized topically, not chronologically. Each illuminates in depth one or other of Poincare's works but all are set in context both historical and temathic such that each can serve as an introduction into the many subjects to which Poincare made a contribution."--Alexander Bogomolny, CTK Insights "Poincare's work is fully alive in science today. This biography is one of the first thorough introductions to his work, it should get the attention of mathematicians, natural scientists and philosophers."--Ferdinand Verhulst, European Legacy "Gray, a mathematics historian and scholar on the life and work of Henry Poincare, has, with the support of a Leverhulme Research Fellowship, produced this comprehensive and definitive 'scientific biography.' Gray offers abundant rich information on Poincare's ideas and scientific process, the evolution and maturity of his mathematics including missteps, the dexterity of his reasoning, and the influences that shaped his thought."--Choice "I recommend [this] book highly."--Robert E. O'Malley, Jr., SIAM Review "Jeremy Gray's book on Poincare's mathematics, physics, and philosophy is an important contribution to the literature and a huge step towards a full biography of this pioneer of modern science."--Reinhard Siegmund-Schultze, Zentralblatt MATH "Gray's book is a comprehensive scientific biography of Poincare. It embraces the broad scope of Poincare's work, from his philosophical speculations to his popular writing, and gives a thorough overview of his extensive mathematical researches."--Peter Lynch, Irish Mathematical Society Bulletin "[T]he author does not simply give platitudes when writing about Poincare's ideas: mathematicians will enjoy reading about his discoveries concerning the three-body problem, the theory of functions, topology, number theory, Lie theory, algebraic geometry, and probability. This scientific biography is the first to comprehensively cover all of Poincare's main contributions to mathematics, philosophy, and physics."--Alan S. McRae, Mathemematical Reviews Clippings "Jeremy Gray has done a marvelous job of exposition and of binding together the many different cognitive, social and biographical strands into the coherent whole of a challenging, but highly rewarding, 'scientific biography'."--Klaus Hentschel, British Journal for the History of Science "A good intellectual biography of an artist should help the reader see how a particular worldview shapes the pursuit of art. Gray's book does that most admirably."--Daniel S. Alexander, H-France Review "Henry Poincare is likely to remain the standard by which scientific biographies, at least those that concern physicists and mathematicians, are judged for some time."--Christopher Cumo, Canadian Journal of History "I warmly recommend the book to anyone with an interest in the development of modern mathematics. It will surely be the definitive scientific biography of Poincare for the foreseeable future."--John Stillwell, Notices of the AMS "Gray describes Poincare's scientific epoch in a beautiful way. Due attention is paid to the mathematical and further scientific aspects of his life, and the intellectual complexity of his achievements, both in their range and their depth, are amply discussed. Gray displays a mastery of his material that is rare even among historians of mathematics and science, and his biography is richly rewarding, engrossing, and informative. He deserves our congratulations."--H. W. Broer, Journal of the British Society for the History of Mathematics "Gray succeeds admirably in presenting both the conceptual and the historical context necessary to appreciate Poincare's contributions. Gray's masterful biography may well serve as a standard example for future endeavors of this kind."--Tilman Sauer, Isis "The obvious virtue of this book is its comprehensiveness. The deeper virtue is to connect Poincare's views of all the parts of his work and to encourage more of that. Gray gives us Poincare's view of Science as a whole."--Colin McLarty, Mathematical Intelligencer "The book is an endless source of interesting insights by Poincare... I would recommend the book for mathematicians, mathematics educators, and philosophers in higher education who want a rich understanding of Poincare, his work, and his times."--Mary L. Garner, Mathematics TeacherTable of ContentsList of Figures ix Preface xi Introduction 1 * Views of Poincare 3 * Poincare's Way of Thinking 6 1 The Essayist 27 * Poincare and the Three Body Problem 27 * Poincare's Popular Essays 34 * Paris Celebrates the New Century 59 * Science, Hypothesis, Value 67 * Poincare and Projective Geometry 76 * Poincare's Popular Writings on Physics 100 * The Future of Mathematics 112 * Poincare among the Logicians 123 * Poincare's Defenses of Science 144 2 Poincare's Career 153 * Childhood, Schooling 153 * The Ecole Polytechnique 157 * The Ecole des Mines 158 * Academic Life 160 * The Dreyfus Affair 165 * National Spokesman 169 * Contemporary Technology 177 * International Representative 187 * The Nobel Prize 192 *"1911", "1912" 200 * Remembering Poincare 202 3 The Prize Competition of 1880 207 * The Competition 207 * Fuchs, Schwarz, Klein, and Automorphic Functions 224 * Uniformization, 1882 to 1907 247 4 The Three Body Problem 253 * Flows on Surfaces 253 * Stability Questions 265 * Poincare's Essay and Its Supplements 266 *Les Methodes Nouvelles de la Mecanique Celeste 281 * Poincare Returns 291 5 Cosmogony 300 * Rotating Fluid Masses 300 6 Physics 318 * Theories of Electricity before Poincare: Maxwell 318 * Poincare's Electricite et Optique, 1890 329 * Larmor and Lorentz: The Electron and the Ether 338 * Poincare on Hertz and Lorentz 346 * St. Louis, 1904 356 * The Dynamics of the Electron 361 * Poincare and Einstein 367 * Early Quantum Theory 378 7 Theory of Functions and Mathematical Physics 382 * Function Theory of a Single Variable 382 * Function Theory of Several Variables 391 * Poincare's Approach to Potential Theory 402 * The Six Lectures in Gottingen, 1909 416 8 Topology 427 * Topology before Poincare 427 * Poincare's Work, 1895 to 1905 432 9 Interventions in Pure Mathematics 467 * Number Theory 467 * Lie Theory 489 * Algebraic Geometry 498 10 Poincare as a Professional Physicist 509 * Thermodynamics 513 * Probability 518 11 Poincare and the Philosophy of Science 525 * Poincare: Idealist, Skeptic, or Structural Realist? 525 12 Appendixes 543 * Elliptic and Abelian Functions 543 * Maxwell's Equations 545 * Glossary 548 References 553 * Articles and Books by Poincare 554 * Other Authors 564 Name Index 585 Subject Index 589

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

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Book SynopsisJohn Napier (1550-1617) is celebrated today as the man who invented logarithms--an enormous intellectual achievement that would soon lead to the development of their mechanical equivalent in the slide rule: the two would serve humanity as the principal means of calculation until the mid-1970s. Yet, despite Napier's pioneering efforts, his life andTrade Review"John Napier fills a gap concerning an important, and often ignored, chapter of mathematical history."--George Szpiro, Nature "In this engaging book, we learn more about Napier the mathematician, the religious zealot, the person."--Devorah Bennu, The Guardian, Grrl Scientist "Edinburgh born John Napier, the inventor of logarithms, is in danger of fading into the shadows of the scientific landscape. In the new book John Napier: Life, Logarithms, and Legacy, Julian Havil does a marvelous job of bringing Napier back into the spotlight."--Stephanie Blanda, American Mathematical Society blog "I'm sure after reading this entertaining and enjoyable book, Napier will climb some rungs on your ladder of famous mathematicians."--A. Bultheel, European Mathematical Society "Havil ... gives a rich history of Napier's involvement in the Protestant reformation, his introduction of logarithms, and his legacy."--Choice "With this book, the author continues his impressive series of illuminating, accessible monographs on the history of mathematics."--Bart J. I. Van Kerkhove, Mathematical Review "This book fills a clear gap in published work on Napier and is likely to be the standard point of departure for those interested in his life and work for some years to come."--Mark McCartney, London Mathematical Society Newsletter "It is clearly a very interesting book."--Ernesto Nungesser, Irish Math Society Bulletin "Havil's attention to detail is without equal in the opinion of this reviewer."--John A. Adam, ScotiaTable of ContentsAcknowledgments xv Introduction 1 Chapter One Life and Lineage 8 Chapter Two Revelation and Recognition 35 Chapter Three A New Tool for Calculation 62 Chapter Four Constructing the Canon 96 Chapter Five Analogue and Digital Computers 131 Chapter Six Logistics: The Art of Computing Well 155 Chapter Seven Legacy 179 Epilogue 207 Appendix A Napier's Works 209 Appendix B The Scottish Science Hall of Fame 210 Appendix C Scotland and Conflict 211 Appendix D Scotland and Reformation 216 Appendix E A Stroll Down Memory Lane 220 Appendix F Methods of Multiplying 229 Appendix G Amending Napier's Kinematic Model 232 Appendix H Napier's Inequalities 233 Appendix I Hos Ego Versiculos Feci 236 Appendix J The Rule of Three 238 Appendix K Mercator's Map 250 Appendix L The Swiss Claimant 264 References 270 Index 275

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Book SynopsisTrade Review"An equal to its companion volume, The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook this scholarly effort fills a noticeable void... Any individual who enjoys mathematics will learn a great amount about mathematical history in a context that is often not discussed or covered."--Choice "[A] very deep and detailed dive into the mathematics of the medieval era."--Charles Ashbacher, MAA ReviewsTable of Contents*Frontmatter, pg. i*Contents, pg. v*Preface, pg. xi*Permissions, pg. xiii*General Introduction, pg. 1*Chapter 1. The Latin Mathematics of Medieval Europe, pg. 4*Chapter 2. Mathematics in Hebrew in Medieval Europe, pg. 224*Chapter 3. Mathematics in the Islamic World in Medieval Spain and North Africa, pg. 381*Appendices, pg. 549*Editors and Contributors, pg. 567*Index, pg. 571

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Book SynopsisTrigonometry has always been an underappreciated branch of mathematics. It has a reputation as a dry and difficult subject, a glorified form of geometry complicated by tedious computation. In this book, Eli Maor draws on his remarkable talents as a guide to the world of numbers to dispel that view. Rejecting the usual arid descriptions of sine, cosTrade Review"Maor's presentation of the historical development of the concepts and results deepens one's appreciation of them, and his discussion of the personalities involved and their politics and religions puts a human face on the subject. His exposition of mathematical arguments is thorough and remarkably easy to understand. There is a lot of material here that teachers can use to keep their students awake and interested. In short, Trigonometric Delights should be required reading for everyone who teaches trigonometry and can be highly recommended for anyone who uses it."--George H. Swift, American Mathematics Monthly "[Maor] writes enthusiastically and engagingly... Delightful reading from cover to cover. Trigonometric Delights is a welcome addition."--Sean Bradley, MAA Online "Maor clearly has a great love of trigonometry, formulas and all, and his enthusiasm shines through... If you always wanted to know where trigonometry came from, and what it's good for, you'll find plenty here to enlighten you."--Ian Stewart, New Scientist "This book will appeal to a general audience interested in the history of mathematics. I highly recommend [it] to teachers who would like to ground their lessons in the sort of mathematical investigations that were undertaken throughout history."--Richard S. Kitchen, Mathematics TeacherTable of ContentsPreface xi Prologue: Ahmes the Scribe, 1650 B.C. 3 Recreational Mathematics in Ancient Egypt 11 1.Angles 15 2.Chords 20 Plimpton 322: The Earliest Trigonometric Table? 30 3.Six Functions Come of Age 35 Johann Muller, alias Regiomontanus 41 4.Trigonometry Becomes Analytic 50 Francois Viete 56 5.Measuring Heaven and Earth 63 Abraham De Moivre 80 6.Two Theorems from Geometry 87 7.Epicycloids and Hypocycloids 95 Maria Agnesi and Her "Witch" 108 8.Variations on a Theme by Gauss 112 9.Had Zeno Only Known This! 117 10.(sin x)/x 129 11.A Remarkable Formula 139 Jules Lissajous and His Figures 145 12.tan x 150 13.A Mapmaker's Paradise 165 14.sin x = 2: Imaginary Trigonometry 181 Edmund Landau: The Master Rigorist 192 15. Fourier's Theorem 198 Appendixes 211 1.Let's Revive an Old Idea 213 2.Barrow's Integration of sec o 218 3.Some Trigonometric Gems 220 4.Some Special Values of sin alpha 222 Bibliography 225 Credits for Illustrations 229 Index 231

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Book SynopsisOn October 23, 1852, Professor Augustus De Morgan wrote a letter to a colleague, unaware that he was launching one of the most famous mathematical conundrums in history - one that would confound thousands of puzzlers for more than a century. This book tells the amazing story of how the "map problem" was solved.Trade Review"The simplicity of the four-color conjecture is deceptive. Just how deceptive is made clear by Robin Wilson's delightful history of the quest to resolve it... Four Colors Suffice is strewn with good anecdotes, and the author ... proves himself skillful at making the mathematics accessible."--Jim Holt, New York Review of Books "Wilson's lucid history weaves together lively anecdotes, biographical sketches, and a non-technical account of the mathematics."--Science "Earlier books ... relate some of the relevant history in their introductions, but they are primarily technical. In contrast, Four Colors Suffice is a blend of history anecdotes and mathematics. Mathematical arguments are presented in a clear, colloquial style, which flows gracefully."--Daniel S. Silver, American Scientist "Robin Wilson appeals to the mathematical novice with an unassuming lucidity. It's thrilling to see great mathematicians fall for seductively simple proofs, then stumble on equally simple counter-examples. Or swallow their pride."--Jascha Hoffman, The Boston Globe "A thoroughly accessible history of attempts to prove the four-color theorem. Wilson defines the problem and explains some of the methods used by those trying to solve it. His descriptions of the contributions made by dozens of dedicated, and often eccentric, mathematicians give a fascinating insight into how mathematics moves forward, and how approaches have changed over the past 50 years... It's comforting to know that however indispensable computers become, there will always be a place for the delightfully eccentric mathematical mind. Let's hope that Robin Wilson continues to write about them."--Elizabeth Sourbut, New Scientist "An attractive and well-written account of the solution of the Four Color Problem... It tells in simple terms an exciting story. It ... give[s] the reader a view into the world of mathematicians, their ideas and methods, discussions, competitions, and ways of collaboration. As such it is warmly recommended."--Bjarne Toft, Notices of the American Mathematical Society "Recreational mathematicians will find Wilson's history of the conjecture an approachable mix of its technical and human aspects... Wilson explains all with exemplary clarity and an accent on the eccentricities of the characters."--Booklist "Wilson gives a clear account of the proof ... enlivened by historical tales."--Alastair Rae, Physics World "Wilson provides a lively narrative and good, easy-to-read arguments showing not only some of the victories but the defeats as well... Even those with only a mild interest in coloring problems or graphs or topology will have fun reading this book... [It is] entertaining, erudite and loaded with anecdotes."--G.L. Alexanderson, MAA OnlineTable of ContentsForeword by Ian Stewart xi Preface to the Revised Color Edition xiii Preface to the Original Edition xv 1The Four-Color Problem 1 What Is the Four-Color Problem? | Why Is It Interesting? | Is It Important? | What Is Meant by "Solving" It? | Who Posed It, and How Was It Solved? | Painting by Numbers | Two Examples 2The Problem Is Posed 12 De Morgan Writes a Letter | Hotspur and the Athenaeum | Mobius and the Five Princes | Confusion Reigns 3Euler's Famous Formula 28 Euler Writes a Letter | From Polyhedra to Maps | Only Five Neighbors | A Counting Formula 4Cayley Revives the Problem ... 45 Cayley's Query | Knocking Down Dominoes | Minimal Criminals | The Six-Color Theorem 5... and Kempe Solves It 55 Sylvester's New Journal | Kempe's Paper | Kempe Chains | Some Variations | Back to Baltimore 6A Chapter of Accidents 71 A Challenge for the Bishop | A Visit to Scotland | Cycling around Polyhedra | A Voyage around the World | Wee Planetoids 7A Bombshell from Durham 86 Heawood's Map | A Salvage Operation | Coloring Empires | Maps on Bagels | Picking Up the Pieces 8Crossing the Atlantic 105 Two Fundamental Ideas | Finding Unavoidable Sets | Finding Reducible Configurations | Coloring Diamonds | How Many Ways? 9A New Dawn Breaks 124 Bagels and Traffic Cops | Heinrich Heesch | Wolfgang Haken | Enter the Computer | Coloring Horseshoes 10Success! 139 A Heesch-Haken Partnership? | Kenneth Appel | Getting Down to Business | The Final Onslaught | A Race against Time | Aftermath 11Is It a Proof? 157 Cool Reaction | What Is a Proof Today? | Meanwhile ... | A New Proof | Into the Next Millennium | The Future Chronology of Events 171 Notes and References 175 Glossary 187 Picture Credits 193 Index 195

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Book SynopsisThe mathematics of ancient Egypt was fundamentally different from our math today. Contrary to what people might think, it wasn't a primitive forerunner of modern mathematics. This title provides an introduction to the intuitive and often-surprising art of ancient Egyptian math.Trade Review"Count Like an Egyptian would make an excellent addition to math classrooms at many different levels. Reimer includes problems in the text and solutions in the back of the book, so the reader can practice techniques and get a feel for exactly how the system works as they go through the book. The mathematics is basic enough to be helpful for children learning fractions or multiplication for the first time, but it's also different enough from the methods most of us know that adults will get a lot out of it as well."--Evelyn Lamb, Scientific American "History lovers will gain much more than just insight into the Egyptian mind-set. The author interleaves mathematical exposition with short essays on Egyptian history, culture, geography, mythology--all, like the rest of the book, beautifully illustrated... For a lively and inquiring mind the book has a good deal to offer. It is well written, lavishly illustrated, and just awfully interesting. The book is a pleasure to hold, to browse, and to read."--Alexander Bogomolny, Cut the Knot "You get the feeling that David Reimer must be a pretty entertaining teacher. An associate professor of mathematics at the College of New Jersey, he has taken on the task of explaining ancient math systems by having you use them. And though it's not easy, he manages to lead you, step by step, through a hieroglyphic based calculation of how many 10-pesu loaves of bread you can make from seven hekat of grain."--Nancy Szokan, Washington Post "An interesting combination of history, ancient literature and mythology, arithmetic puzzles and mathematics, and lavishly illustrated with numerous colour diagrams, this engaging book is unusual, thought-provoking and just plain fun to read."--Devorah Bennu, GrrlScientist, The Guardian "Count Like an Egyptian is a beautifully illustrated and well-written book... Reimer's overriding goal is to demonstrate that Egyptian fraction arithmetic is fascinating, versatile, and well suited for whatever calls fractions into existence... By working through the material Reimer patiently and gently presents, the reader will have a more thorough understanding and appreciation of how Egyptian scribes made the calculations needed to administer an empire bent on building pyramids and granaries, surveying flooded riverside property, digging irrigation basins, and rationing or exchanging bread and beer supplies amongst its gangs of workers... This book should find a home in libraries used by middle school and high school mathematics teachers. It also provides a good resource for mathematics education professors and their students on the college level as they explore historical beginnings of mathematical ideas, make cultural comparisons, and develop interdisciplinary connections."--Calvin Jongsma, MAA Reviews "An interesting combination of history, ancient literature and mythology, arithmetic puzzles and mathematics, and lavishly illustrated with numerous colour diagrams, this engaging book is unusual, thought-provoking and just plain fun to read."--GrrrlScientist "This amusing popular introduction to an uncommon subject is a mental adventure that sheds new light on the thought processes of a lost civilization and will appeal both to those who enjoy mathematical puzzles and to Egyptophiles."--Edward K. Werner, Library Journal "In general I really like this book and believe it is, if not necessarily a must for all Egyptophiles, then definitely one to put on the wish list as an interesting addition to your bookshelf... It is fun way of working through complicated and yet practical mathematics which makes the Rhind Papyrus come alive and gives an insight into the logical brain of ancient Egyptian scribes."--Charlotte Booth, charlottesegypt.com "Reimer succeeds very well in transferring his enthusiasm tor the Egyptian system to the reader. The reactions from his students who were used tor a try-out are claimed to be positive. But even if you do not want to graduate as an Egyptian scribe, you may be charmed by the witty Egyptian system and you will be delighted by the colourful illustrations and Reimer's entertaining account of it all."--A. Bultheel, European Mathematical Society "Count Like an Egyptian takes the reader step-by-step through the ancient Egyptian methods, which are surprisingly different from our own, and yet, in the capable hands of author David Reimer, surprisingly understandable. This lovely book has fun illustrations to demonstrate the various operations, basic geometry, and other tasks faced by the scribes... This book is a pleasure to read and makes Egyptian math a pleasure to learn."--Gretchen Wagner, San Francisco Book Review "The book is intended to be used as a teaching tool and includes practice examples for the student. It would be difficult to imagine a work that more effectively covers this aspect of the ancient civilization."--JPP, Ancient Egypt "David Reimer succeeds in keeping the mathematics in Count Like an Egyptian clever and light, raising this book into a rare category: a coffee table book that is serious and fun."--Robert Schaefer, New York Journal of Books "This volume is ideal for anyone, and I truly mean anyone, young or old, mathematician, student or teacher, who wants to learn how the ancient Egyptians did mathematics... This book has all the Egyptian mathematics a general mathematician, teacher or student could ever want to learn. In particular it would be a perfect resource for a schoolteacher, elementary through lower division college. The material is presented in a direct and accessible manner."--Amy Shell-Gellasch, CSHPM Bulletin "Overall this is a didactic and well written book, with many important illustrations, with some incursions in the mathematics of other ancient cultures."--European Mathematical Society "With Reimer's guidance, motivating stories, and lighthearted remarks, readers can become facile with Egyptian algorithms and the insights they reveal... Valuable for all readers looking for a guided of an alternative to traditional school arithmetic and the torpor that algorithmic training causes."--Choice "[T]his book is a worthwhile read for anyone interested in seeing exactly how ancient Egyptians dealt with mathematics. It will help put our present algorithms into perspective as simply one of many possible algorithms one could use to perform arithmetic operations."--Victor J. Katz, Mathematical Reviews Clippings "[Reimer] ... set himself to understand and explain the ancient methods, and the result is an approachable, thorough and lavishly-produced book."--Owen Toller, Mathematical Gazette "Count like an Egyptian is a beautifully glossy and colourful book; the presentation of hieroglyphs is particularly well done, and fully interated into the surrounding text... This book has given me a new perspective on day-to-day arithmetic."--Christopher Hollings, Mathematics Today "This is a wonderful book, very well written, filled with illustrations on every page, witty, addressing anyone interested in grade school arithmetic."--Victor V. Pambuccian, Zentralblatt MATH "Count Like an Egyptian is important for anyone interested in alternative algorithms... If you want to roll up your sleeves and learn some new mathematics, this is the book for you."--Michael Manganello, Mathematics Teacher "An engaging and beautifully illustrated book that deals with the basics of ancient Egyptian mathematics, set in the wider context of other ancient mathematical systems."--Corinna Rossi, Aestimatio "A great approach and a dedicated effort. One hopes the book will reflect that persistence and it does... This is a book that comes recommended, for anyone who wants to know where our current basis of mathematics comes from through to those with an interest in maths and history."--Gordon Clarke, Gazette of the Australian Mathematical SocietyTable of ContentsPreface vii Introduction ix Computation Tables xi 1 Numbers 1 2 Fractions 13 3 Operations 22 4 Simplification 55 5 Techniques and Strategies 80 6 Miscellany 121 7 Base-Based Mathematics 144 8 Judgment Day 182 Practice Solutions 209 Index 235

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Book SynopsisDemonstrating the different roles that logic plays in the disciplines of computer science, mathematics, and philosophy, this title covers select topics from three different areas of logic: proof theory, computability theory, and nonclassical logic. It presents relevance logic with applications.Trade Review"Overall, this is a well-written text with challenging exercises, proofs of important theorems, and a modern integrated approach."--Choice "The book can serve as material for a course that teaches the role of logic in several disciplines. It can also be used as a supplementary text for a logic course that emphasizes the more traditional topics of logic but wishes to include a few special topics. Moreover, it can be a valuable resource for researchers and academics."--Roman Murawski, Zentralblatt MATH "It's always interesting to find a text that reimagines, and offers a novel approach to, a fairly standard subject. This book does that for logic... There is a lot of interesting and well-presented material found here that cannot be easily found elsewhere in a book at this level."--Mark Hunacek, Mathematical Association of America blog "An instructor of a logic course offered by a mathematics department who is interested in some experimentation will undoubtedly find this book quite rewarding... Even an instructor who is not planning to teach a course along these lines, but who is interested in the subject, will want to look at this text; there is a lot of interesting and well-presented material found here that cannot be easily found elsewhere in a book at this level."--Mark Hunacek, MAA blogTable of ContentsPreface ix Acknowledgments xiii PART 1. Proof Theory 1 Donald Loveland 1Propositional Logic 3 1.1 Propositional Logic Semantics 5 1.2 Syntax: Deductive Logics 13 1.3 The Resolution Formal Logic 14 1.4 Handling Arbitrary Propositional Wffs 26 2Predicate Logic 31 2.1 First-Order Semantics 32 2.2 Resolution for the Predicate Calculus 40 2.2.1 Substitution 41 2.2.2 The Formal System for Predicate Logic 45 2.2.3 Handling Arbitrary Predicate Wffs 54 3An Application: Linear Resolution and Prolog 61 3.1 OSL-Resolution 62 3.2 Horn Logic 69 3.3 Input Resolution and Prolog 77 Appendix A: The Induction Principle 81 Appendix B: First-Order Valuation 82 Appendix C: A Commentary on Prolog 84 References 91 PART 2. Computability Theory 93 Richard E. Hodel 4Overview of Computability 95 4.1 Decision Problems and Algorithms 95 4.2 Three Informal Concepts 107 5A Machine Model of Computability 123 5.1 RegisterMachines and RM-Computable Functions 123 5.2 Operations with RM-Computable Functions; Church-Turing Thesis; LRM-Computable Functions 136 5.3 RM-Decidable and RM-Semi-Decidable Relations; the Halting Problem 144 5.4 Unsolvability of Hilbert's Decision Problem and Thue'sWord Problem 154 6A Mathematical Model of Computability 165 6.1 Recursive Functions and the Church-Turing Thesis 165 6.2 Recursive Relations and RE Relations 175 6.3 Primitive Recursive Functions and Relations; Coding 187 6.4 Kleene Computation Relation Tn(e, a1, ... , an, c) 197 6.5 Partial Recursive Functions; Enumeration Theorems 203 6.6 Computability and the Incompleteness Theorem 216 List of Symbols 219 References 220 PART 3. Philosophical Logic 221 S. G. Sterrett 7Non-Classical Logics 223 7.1 Alternatives to Classical Logic vs. Extensions of Classical Logic 223 7.2 From Classical Logic to Relevance Logic 228 7.2.1 The (So-Called) "Paradoxes of Implication" 228 7.2.2 Material Implication and Truth Functional Connectives 234 7.2.3 Implication and Relevance 238 7.2.4 Revisiting Classical Propositional Calculus: What to Save,What to Change, What to Add? 240 8Natural Deduction: Classical and Non-Classical 243 8.1 Fitch's Natural Deduction System for Classical Propositional Logic 243 8.2 Revisiting Fitch's Rules of Natural Deduction to Better Formalize the Notion of Entailment-Necessity 251 8.3 Revisiting Fitch's Rules of Natural Deduction to Better Formalize the Notion of Entailment-Relevance 253 8.4 The Rules of System FE (Fitch-Style Formulation ofthe Logic of Entailment) 261 8.5 The Connective "Or," Material Implication,and the Disjunctive Syllogism 281 9Semantics for Relevance Logic: A Useful Four-Valued Logic 288 9.1 Interpretations, Valuations, and Many Valued Logics 288 9.2 Contexts in Which This Four-Valued Logic Is Useful 290 9.3 The Artificial Reasoner's (Computer's) "State of Knowledge" 291 9.4 Negation in This Four-Valued Logic 295 9.5 Lattices: A Brief Tutorial 297 9.6 Finite Approximation Lattices and Scott's Thesis 302 9.7 Applying Scott's Thesis to Negation, Conjunction, and Disjunction 304 9.8 The Logical Lattice L4 307 9.9 Intuitive Descriptions of the Four-Valued Logic Semantics 309 9.10 Inferences and Valid Entailments 312 10Some Concluding Remarks on the Logic of Entailment 315 References 316 Index 319

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Book SynopsisThe history of mathematics is filled with major breakthroughs resulting from solutions to recreational problems. Problems of interest to gamblers led to the modern theory of probability, for example, and surreal numbers were inspired by the game of Go. Yet even with such groundbreaking findings and a wealth of popular-level books exploring puzzlesTrade ReviewOne of Choice's Outstanding Academic Titles for 2016 "Beineke and Rosenhouse have compiled and edited a fantastic collection of essays dealing with popular mathematics... Anybody who enjoys reading about recreation mathematics should definitely explore these writings."--ChoiceTable of ContentsForeword by Raymond Smullyan vii Preface and Acknowledgments x PART I VIGNETTES 1 Should You Be Happy? 3 Peter Winkler 2 One-Move Puzzles with Mathematical Content 11 Anany Levitin 3 Minimalist Approaches to Figurative Maze Design 29 Robert Bosch, Tim Chartier, and Michael Rowan 4 Some ABCs of Graphs and Games 43 Jennifer Beineke and Lowell Beineke PART II PROBLEMS INSPIRED BY CLASSIC PUZZLES 5 Solving the Tower of Hanoi with Random Moves 65 Max A. Alekseyev and Toby Berger 6 Groups Associated to Tetraflexagons 81 Julie Beier and Carolyn Yackel 7 Parallel Weighings of Coins 95 Tanya Khovanova 8 Analysis of Crossword Puzzle Difficulty Using a Random Graph Process 105 John K. McSweeney 9 From the Outside In: Solving Generalizations of the Slothouber-Graatsma-Conway Puzzle 127 Derek Smith PART III PLAYING CARDS 10 Gallia Est Omnis Divisa in Partes Quattuor 139 Neil Calkin and Colm Mulcahy 11 Heartless Poker 149 Dominic Lanphier and Laura Taalman 12 An Introduction to Gilbreath Numbers 163 Robert W. Vallin PART IV GAMES 13 Tic-tac-toe on Affine Planes 175 Maureen T. Carroll and Steven T. Dougherty 14 Error Detection and Correction Using SET 199 Gary Gordon and Elizabeth McMahon 15 Connection Games and Sperner's Lemma 213 David Molnar PART V FIBONACCI NUMBERS 16 The Cookie Monster Problem 231 Leigh Marie Braswell and Tanya Khovanova 17 Representing Numbers Using Fibonacci Variants 245 Stephen K. Lucas About the Editors 261 About the Contributors 263 Index 269

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Book SynopsisContrary to popular belief--and despite the expulsion, emigration, or death of many German mathematicians--substantial mathematics was produced in Germany during 1933-1945. In this landmark social history of the mathematics community in Nazi Germany, Sanford Segal examines how the Nazi years affected the personal and academic lives of those GermanTrade Review"The strength of the book lies in its many individual stories and case histories... [It] offer[s] disturbing and important accounts of the life of science and scientists under the Nazis."--The Economist "The remarkable feature of this book is that in spite of the temptation, the story-telling never succumbs to simplistic descriptions of events or people. The analysis avoids the sentimentality and moral superiority that so often accompany descriptions of the Nazi years... Perhaps this is why Mathematicians under the Nazis is so compelling... This is a perceptive analysis of an important era and well worth reading."--John H. Ewing, Mathematical Reviews "A fascinating, well-researched and richly footnoted account of what occurred within a scientific discipline during the Nazi period."--George G. Szpiro, The Jerusalem ReportTable of Contents*Frontmatter, pg. i*Contents, pg. ix*PREFACE, pg. xi*ACKNOWLEDGMENTS, pg. xix*ABBREVIATIONS, pg. xxi*CHAPTER ONE. Why Mathematics?, pg. 1*CHAPTER TWO. The Crisis in Mathematics, pg. 14*CHAPTER THREE. The German Academic Crisis, pg. 42*CHAPTER FOUR. Three Mathematical Case Studies, pg. 85*CHAPTER FIVE. Academic Mathematical Life, pg. 168*CHAPTER SIX. Mathematical Institutions, pg. 229*CHAPTER SEVEN. Ludwig Bieberbach and "Deutsche Mathematik", pg. 334*CHAPTER EIGHT. Germans and Jews, pg. 419*APPENDIX, pg. 493*BIBLIOGRAPHY, pg. 509*INDEX, pg. 523

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

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Book SynopsisThis is a book for those who enjoy thinking about how and why Nature can be described using mathematical tools. Approximating Perfection considers the background behind mechanics as well as the mathematical ideas that play key roles in mechanical applications. Concentrating on the models of applied mechanics, the book engages the reader in the typeTrade Review"A well-written general-interest introduction to classical mechanics."--ChoiceTable of ContentsPreface vii Chapter 1. The Tools of Calculus 1 1.1 Is Mathematical Proof Necessary? 2 1.2 Abstraction, Understanding, Infinity 6 1.3 Irrational Numbers 8 1.4 What Is a Limit? 11 1.5 Series 15 1.6 Function Continuity 19 1.7 How to Measure Length 21 1.8 Antiderivatives 33 1.9 Definite Integral 35 1.10 The Length of a Curve 42 1.11 Multidimensional Integrals 44 1.12 Approximate Integration 47 1.13 On the Notion of a Function 52 1.14 Differential Equations 53 1.15 Optimization 59 1.16 Petroleum Exploration and Recovery 61 1.17 Complex Variables 63 1.18 Moving On 65 Chapter 2. The Mechanics of Continua 67 2.1 Why Do Ships Float? 67 2.2 The Main Notions of Classical Mechanics 71 2.3 Forces, Vectors, and Objectivity 74 2.4 More on Forces; Statics 76 2.5 Hooke's Law 80 2.6 Bending of a Beam 84 2.7 Stress Tensor 94 2.8 Principal Axes and Invariants of the Stress Tensor 100 2.9 On the Continuum Model and Limit Passages 102 2.10 Equilibrium Equations 104 2.11 The Strain Tensor 108 2.12 Generalized Hooke's Law 113 2.13 Constitutive Laws 114 2.14 Boundary Value Problems 115 2.15 Setup of Boundary Value Problems of Elasticity 118 2.16 Existence and Uniqueness of Solution 120 2.17 Energy; Minimal Principle for a Spring 126 2.18 Energy in Linear Elasticity 128 2.19 Dynamic Problems of Elasticity 132 2.20 Oscillations of a String 134 2.21 Lagrangian and Eulerian Descriptions of Continuum Media 137 2.22 The Equations of Hydrodynamics 140 2.23 D'Alembert-Euler Equation of Continuity 142 2.24 Some Other Models of Hydrodynamics 144 2.25 Equilibrium of an Ideal Incompressible Liquid 145 2.26 Force on an Obstacle 148 Chapter 3. Elements of the Strength of Materials 151 3.1 What Are the Problems of the Strength of Materials? 151 3.2 Hooke's Law Revisited 152 3.3 Objectiveness of Quantities in Mechanics Revisited 157 3.4 Plane Elasticity 159 3.5 Saint-Venant's Principle 161 3.6 Stress Concentration 163 3.7 Linearity vs. Nonlinearity 165 3.8 Dislocations, Plasticity, Creep, and Fatigue 166 3.9 Heat Transfer 172 3.10 Thermoelasticity 175 3.11 Thermal Expansion 177 3.12 A Few Words on the History of Thermodynamics 178 3.13 Thermodynamics of an Ideal Gas 180 3.14 Thermodynamics of a Linearly Elastic Rod 182 3.15 Stability 186 3.16 Static Stability of a Straight Beam 188 3.17 Dynamical Tools for Studying Stability 193 3.18 Additional Remarks on Stability 195 3.19 Leak Prevention 198 Chapter 4. Some Questions of Modeling in the Natural Sciences 201 4.1 Modeling and Simulation 201 4.2 Computerization and Modeling 203 4.3 Numerical Methods and Modeling in Mechanics 206 4.4 Complexity in the Real World 208 4.5 The Role of the Cosine in Everyday Measurements 209 4.6 Accuracy and Precision 211 4.7 How Trees Stand Up against the Wind 213 4.8 Why King Kong Cannot Be as Terrible as in the Movies 216 Afterword 219 Recommended Reading 221 Index 223

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Book SynopsisThe interest earned on a bank account, the arrangement of seeds in a sunflower, and the shape of the Gateway Arch in St. Louis are all intimately connected with the mysterious number e. In this informal and engaging history, Eli Maor portrays the curious characters and the elegant mathematics that lie behind the number. Designed for a reader with oTrade ReviewHonorable Mention for the 1994 Award for Best Professional/Scholarly Book in Mathematics, Association of American Publishers "This is a gently paced, elegantly composed book, and it will bring its readers much pleasure... Maor has written an excellent book that should be in every public and school library."--Ian Stewart, New Scientist "Maor wonderfully tells the story of e. The chronological history allows excursions into the lives of people involved with the development of this fascinating number. Maor hangs his story on a string of people stretching from Archimedes to David Hilbert. And by presenting mathematics in terms of the humans who produced it, he places the subject where it belongs--squarely in the centre of the humanities."--Jerry P. King, Nature "Maor has succeeded in writing a short, readable mathematical story. He has interspersed a variety of anecdotes, excursions, and essays to lighten the flow... [The book] is like the voyages of Columbus as told by the first mate."--Peter Borwein, Science "Maor attempts to give the irrational number e its rightful standing alongside pi as a fundamental constant in science and nature; he succeeds very well... Maor writes so that both mathematical newcomers and long-time professionals alike can thoroughly enjoy his book, learn something new, and witness the ubiquity of mathematical ideas in Western culture."--Choice "It can be recommended to readers who want to learn about mathematics and its history, who want to be inspired and who want to understand important mathematical ideas more deeply."--EMS Newsletter "[A] very interesting story about the history of e, logarithms, and related matters, especially the history of calculus... [A] useful complement to a course in calculus and analysis, shedding light on some fundamental topics."--Mehdi Hassani, MAA ReviewsTable of ContentsPreface1John Napier, 161432Recognition113Financial Matters234To the Limit, If It Exists285Forefathers of the Calculus406Prelude to Breakthrough497Squaring the Hyperbola588The Birth of a New Science709The Great Controversy8310e[superscript x]: The Function That Equals its Own Derivative9811e[superscript theta]: Spira Mirabilis11412(e[superscript x] + e[superscript -x])/2: The Hanging Chain14013e[superscript ix]: "The Most Famous of All Formulas"15314e[superscript x + iy]: The Imaginary Becomes Real16415But What Kind of Number Is It?183App. 1. Some Additional Remarks on Napier's Logarithms195App. 2. The Existence of lim (1 + 1/n)[superscript n] as n [approaches] [infinity]197App. 3. A Heuristic Derivation of the Fundamental Theorem of Calculus200App. 4. The Inverse Relation between lim (b[superscript h] - 1)/h = 1 and lim (1 + h)[superscript 1/h] = b as h [approaches] 0202App. 5. An Alternative Definition of the Logarithmic Function203App. 6. Two Properties of the Logarithmic Spiral205App. 7. Interpretation of the Parameter [phi] in the Hyperbolic Functions208App. 8. e to One Hundred Decimal Places211Bibliography213Index217

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Book SynopsisTrade ReviewWinner of the 2006 Book Award, Mathematics/Computer Science category, Alpha Sigma Nu, and the Association of Jesuit Colleges and Universities "A definitive work, very carefully written, Complexities will inspire a wide range of women mathematicians and scientists for a long period of time... By far this is the most important study of women in mathematics that even a giant amongst men mathematicians will find himself reading with sheer pleasure."--Current Engineering Practice "[T]he variation in [the book's] content and writing styles ... is exactly its strength--it is both an excellent reference for a professor wishing to provide a student with a few inspiring gems and a comprehensive overall picture of the life of women in mathematics. Its lessons are gleaned from the trials and tribulations of a specific group, but the advice is universal."--Lisa DeKeukelaere, MAA Online "The collection documents the complex nature of the conditions women have faced while pursuing their careers in mathematics. It shows the pleasure women had in discovering new mathematics, and energy to do a good job!"--Silke Gobel, Zentralblatt "As a female mathematics student, I found that reading this book increased my appreciation for the courage and determination of the women who entered mathematics before me, while also building my personal confidence in the prospect of finding a rewarding and fulfilling life in the mathematical community."--Gwen Spencer, Math Horizons

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Book SynopsisTrade Review"[Stillwell] writes clearly and engagingly... [Elements of Mathematics] can appeal to various constituencies at different levels of mathematical sophistication."--Mark Hunacek, MAA Reviews "A great exploration of elementary mathematics, its limitations, how infinity complicates things, and how various branches of mathematics fit together."--Antonio Cangiano, Math-Blog "Stillwell is ... One of the better current mathematical authors: he writes clearly and engagingly, and makes more of an effort than most to provide historical detail and a sense of how various mathematical ideas tie in with one another... The features we have learned to expect from Stillwell (including, but not limited to, excellent writing) are present in [Elements of Mathematics] as well."--MAA Reviews "An accessible read... Stillwell breaks down the basics, providing both historical and practical perspectives from arithmetic to infinity."--Gemma Tarlach, Discover "[A] sophisticated treatment of topics usually described as elementary."--John Allen Paulos "[Elements of Mathematics] is quite a tour de force, organized by areas of mathematics--arithmetic, computation, algebra, geometry, calculus, and so on--and in each area Stillwell manages to distill down the big ideas and the connections with other areas. He is a master expositor, and the text manages to be engaging and accessible without watering down the mathematics. I definitely learned new things from the book!"--Brent Yorgey, Math Less Traveled blog "From a lifetime of teaching, Stillwell has distilled some nice examples from the entire gamut of elementary mathematics."--Mathematical Reviews Clippings "[A] wonderful book... I think that [Elements of Mathematics] will itself become a modern classic and a reference work for anyone trying to learn basic topics in any of the major fields of mathematics."--Victor Katz, Bulletin of the American Mathematical Society "Elements of Mathematicsis a fine ... overview of the field of mathematics... The writing is clear, succinct, organized, and the diagrams [and] illustrations excellent... While some of the discussion is introductory or elementary, it always leads to deeper, more challenging ideas... [T]his will make a fine basic addition to most mathematicians' bookshelves."--Math Tango "Stillwell uses his broad and impressive command of mathematics to transport a reader through each topic and to a higher level of understanding and questioning."--Convergence "[A] wonderful book ... I think that [Elements of Mathematics] will itself become a modern classic and a reference work for anyone trying to learn basic topics in any of the major fields of mathematics."--Victor Katz, Bulletin of the American Mathematical Society "[Elements of Mathematics] is a book that everybody should read. You will be the better for it."--Reuben Hersh, American Mathematical MonthlyTable of Contents*Frontmatter, pg. i*Contents, pg. vii*Preface, pg. xi*1. Elementary Topics, pg. 1*2. Arithmetic, pg. 35*3. Computation, pg. 73*4. Algebra, pg. 106*5. Geometry, pg. 148*6. Calculus, pg. 193*7. Combinatorics, pg. 243*8. Probability, pg. 279*9. Logic, pg. 298*10. Some Advanced Mathematics, pg. 336*Bibliography, pg. 395*Index, pg. 405

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