Dieses Buch bietet einen historisch orientierten Einstieg in die Algorithmik, also die Lehre von den Algorithmen, in Mathematik, Informatik und daruber hinaus. Besondere Merkmale und Zielsetzungen sind: Elementaritat und Anschaulichkeit, die Berucksichtigung der historischen Entwicklung, Motivation der Begriffe und Verfahren anhand konkreter, aussagekraftiger Beispiele unter Einbezug moderner Werkzeuge (Computeralgebrasysteme, Internet). Als Zusatzmedien werden computer- und internetspezifische Interaktions- und Visualisierungsmoeglichkeiten (kostenlos) zur Verfugung gestellt. Das Werk wendet sich an Studierende und Lehrende an Schulen und Hochschulen sowie an Nichtspezialisten, die an den Themen "Computer/Algorithmen/Programmierung" einschliesslich ihrer historischen und geisteswissenschaftlichen Dimension interessiert sind.

Der Begriff der Zahl ist ein vielfacher. Darauf weist uns schon die Mehrheit verschiedener Zahlworter hin, die in der Sprache des gewohnlichen Lebens auftreten und von den Grammatikern unter 5 folgenden Titeln aufgefiihrt zu werden pflegen: die Anzahlen oder Grundzahlen (numeralia cardinalia), die Ordnungszahlen (n. ordinalia), die Gattungszahlen (n. specialia), die Wiederho- lungszahlen (n. iterativa), die Vervielfaltigungszahlen (n. multi- plicativa) und die Bruchzahlen (n. partitiva). DaB die Anzahlen 10 als die ersten in dieser Reihe genannt werden, beruht ebenso wie die charakteristischen N amen, die sie sonst tragen - Grund- oder Kardinalzahlen -, nicht auf bloBer Konvention. Sie nehmen sprachlich eine bevorzugte SteHung dadurch ein, daB die samt- lichen iibrigen Zahlworter nur durch geringe Modifikationen aus 15 den Anzahlwortern hervorgehen (z. B. zwei, zweiter, zweierlei, zweifach, zweimal, zweitel). Die letzteren sind also wahrhafte Grundzahlworter. Die Sprache leitet uns hiermit auf den Gedan- ken hin, es mochten auch die korrespondierenden Beg r iff e samtlich in einem analogen Abhangigkeitsverhaltnisse stehen 20 zu denen der Anzahlen und gewisse inhaltsreichere Gedanken vor- steHen, in welchen die Anzahlen bloBe Bestandteile bilden. Die einfachste Uberlegung scheint dies zu bestatigen. So handelt es sich bei den Gattungszahlen (einerlei, zweierlei usw. ) um eine Anzahl von Verschiedenheiten innerhalb einer Gattung; bei den Wieder- 25 holungszahlen (einmal, zweimal usw. ) um die Anzahl einer Wiederholung. Bei den Vervielfaltigungs- und Bruchzahlen dient die Anzahl dazu, das Verhaltnis eines in gleiche Teile geteilten Ganzen zu einem Teile bzw.

As discrete fields of inquiry, rhetoric and mathematics have long been considered antithetical to each other. That is, if mathematics explains or describes the phenomena it studies with certainty, persuasion is not needed. This volume calls into question the view that mathematics is free of rhetoric. Through nine studies of the intersections between these two disciplines, Arguing with Numbers shows that mathematics is in fact deeply rhetorical. Using rhetoric as a lens to analyze mathematically based arguments in public policy, political and economic theory, and even literature, the essays in this volume reveal how mathematics influences the values and beliefs with which we assess the world and make decisions and how our worldviews influence the kinds of mathematical instruments we construct and accept. In addition, contributors examine how concepts of rhetoric—such as analogy and visuality—have been employed in mathematical and scientific reasoning, including in the theorems of mathematical physicists and the geometrical diagramming of natural scientists. Challenging academic orthodoxy, these scholars reject a math-equals-truth reduction in favor of a more constructivist theory of mathematics as dynamic, evolving, and powerfully persuasive. By bringing these disparate lines of inquiry into conversation with one another, Arguing with Numbers provides inspiration to students, established scholars, and anyone inside or outside rhetorical studies who might be interested in exploring the intersections between the two disciplines. In addition to the editors, the contributors to this volume are Catherine Chaput, Crystal Broch Colombini, Nathan Crick, Michael Dreher, Jeanne Fahnestock, Andrew C. Jones, Joseph Little, and Edward Schiappa.

The first critical work to attempt the mammoth undertaking of reading Badiou's Being and Event as part of a sequence has often surprising, occasionally controversial results. Looking back on its publication Badiou declared: "I had inscribed my name in the history of philosophy". Later he was brave enough to admit that this inscription needed correction. The central elements of Badiou's philosophy only make sense when Being and Event is read through the corrective prism of its sequel, Logics of Worlds, published nearly twenty years later. At the same time as presenting the only complete overview of Badiou's philosophical project, this book is also the first to draw out the central component of Badiou's ontology: indifference. Concentrating on its use across the core elements Being and Event-the void, the multiple, the set and the event-Watkin demonstrates that no account of Badiou's ontology is complete unless it accepts that Badiou's philosophy is primarily a presentation of indifferent being. Badiou and Indifferent Being provides a detailed and lively section by section reading of Badiou's foundational work. It is a seminal source text for all Badiou readers.

In Mathematics of the Transcendental, Alain Badiou painstakingly works through the pertinent aspects of category theory, demonstrating their internal logic and veracity, their derivation and distinction from set theory, and the 'thinking of being'. In doing so he sets out the basic onto-logical requirements of his greater and transcendental logics as articulated in his magnum opus, Logics of Worlds. Now available in paperback, Mathematics of the Transcendental provides Badiou's readers with a much-needed complete elaboration of his understanding and use of category theory. The book is vital to understanding the mathematical and logical basis of his theory of appearing, as elaborated in Logics of Worlds and other works, and is essential reading for his many followers.

The use of diagrams in logic and geometry has encountered resistance in recent years. For a proof to be valid in geometry, it must not rely on the graphical properties of a diagram. In logic, the teaching of proofs depends on sentenial representations, ideas formed as natural language sentences such as "If A is true and B is true...." No serious formal proof system is based on diagrams.
This book explores the reasons why structured graphics have been largely ignored in contemporary formal theories of axiomatic systems. In particular, it elucidates the systematic forces in the intellectual history of mathematics which have driven the adoption of sentential representational styles over diagrammatic ones. In this book, the effects of historical forces on the evolution of diagrammatically-based systems of inference in logic and geometry are traced from antiquity to the early twentieth-century work of David Hilbert. From this exploration emerges an understanding that the present negative attitudes towards the use of diagrams in logic and geometry owe more to implicit appeals to their history and philosophical background than to any technical incompatibility with modern theories of logical systems.

Bayesian ideas have recently been applied across such diverse fields as philosophy, statistics, economics, psychology, artificial intelligence, and legal theory. Fundamentals of Bayesian Epistemology examines epistemologists' use of Bayesian probability mathematics to represent degrees of belief. Michael G. Titelbaum provides an accessible introduction to the key concepts and principles of the Bayesian formalism, enabling the reader both to follow epistemological debates and to see broader implications Volume 1 begins by motivating the use of degrees of belief in epistemology. It then introduces, explains, and applies the five core Bayesian normative rules: Kolmogorov's three probability axioms, the Ratio Formula for conditional degrees of belief, and Conditionalization for updating attitudes over time. Finally, it discusses further normative rules (such as the Principal Principle, or indifference principles) that have been proposed to supplement or replace the core five. Volume 2 gives arguments for the five core rules introduced in Volume 1, then considers challenges to Bayesian epistemology. It begins by detailing Bayesianism's successful applications to confirmation and decision theory. Then it describes three types of arguments for Bayesian rules, based on representation theorems, Dutch Books, and accuracy measures. Finally, it takes on objections to the Bayesian approach and alternative formalisms, including the statistical approaches of frequentism and likelihoodism.

This is the first English collection of the work of Albert Lautman, a major figure in philosophy of mathematics and a key influence on Badiou and Deleuze. Albert Lautman (1908-1944) was a French philosopher of mathematics whose work played a crucial role in the history of contemporary French philosophy. His ideas have had an enormous influence on key contemporary thinkers including Gilles Deleuze and Alain Badiou, for whom he is a major touchstone in the development of their own engagements with mathematics. "Mathematics, Ideas and the Physical Real" presents the first English translation of Lautman's published works between 1933 and his death in 1944. Rather than being preoccupied with the relation of mathematics to logic or with the problems of foundation, which have dominated philosophical reflection on mathematics, Lautman undertakes to develop an understanding of the broader structure of mathematics and its evolution. The two powerful ideas that are constants throughout his work, and which have dominated subsequent developments in mathematics, are the concept of mathematical structure and the idea of the essential unity underlying the apparent multiplicity of mathematical disciplines. This collection of his major writings offers readers a much-needed insight into his influence on the development of mathematics and philosophy.

Originaltext mit ausfuhrlichen mathematischen sowie historischen Kommentaren von Eberhard Knobloch und aktualisierter UEbersetzung von Otto Hamborg "De quadratura arithmetica circuli" (1676) von Gottfried Wilhelm Leibniz ist eines der bedeutendsten Werke in der Analysis. Dieser Meilenstein der Mathematik- und Wissenschaftsgeschichte behandelt die arithmetische Kreisquadratur, also die Berechnung der Kreisflache mittels einer konvergenten, unendlichen Reihe rationaler Zahlen, Zykloide, Paraboloide, Hyperboloide, Logarithmusfunktionen usf. Die Schrift legte die Grundlagen insbesondere fur die Differential- und Integralrechnung, wie wir sie noch heute lernen und verwenden. Unter Berufung auf archimedische Strenge lehrt sie mit Hilfe der wohl definierten Begriffe "unendlich klein" und "unendlich gross" an Hand der Kurventheorie, wie mit dem Unendlichen in der Mathematik umzugehen ist. Kurven sind danach nichts anderes als Polygone mit unendlich vielen, unendlich kleinen Seiten. Die programmatischen Aussagen dieser Schrift sind grundlegend fur die Philosophie und die Grundlagen der Mathematik.

For more than two generations, W. V. Quine has contributed fundamentally to the substance, the pedagogy, and the philosophy of mathematical logic. "Selected Logic Papers," long out of print and now reissued with eight additional essays, includes much of the author's important work on mathematical logic and the philosophy of mathematics from the past sixty years.

"The book is indeed a classic. Virtually every philosopher of science now writing about probabilistic inference has been influenced by Edwards' book, and his ideas are now as alive and relevant as they were when the book first appeared. Edwards is an absolutely seminal thinker in the foundations of statistics and scientific inference." -- Elliott Sober, University of Wisconsin-Madison.

"Full of appropriate examples (especially from genetics) and historical commentary, this monograph offers a rare simultaneous treatment of both mathematical and philosophical foundations." -- American Mathematical Monthly.

This new and expanded edition of A. W. F. Edwards' classic volume on scientific inference presents his most important published articles on the subject. Edwards argues that the appropriate axiomatic basis for inductive inference is not that of probability, with its addition axiom, but that of likelihood, the concept introduced by Fisher as a measure of relative support among different hypotheses. Starting from the simplest considerations and assuming no more than a basic acquaintancewith probability theory, the author sets out to reconstruct a consistent theory of statistical inference in science. Using the likelihood approach, he explores estimation, tests of significance, randomization, experimental design, and other statistical topics. Likelihood is important reading for students and professionals in biology, mathematical sciences, and philosophy.

"This book is commended to all philosophers of science who are interested in the problems of scientific inference." -- Search.

"This book, by a well-known geneticist, will do much to publicize the generality of the likelihoodmethod as a foundation for statistical procedure. It is both smoothly written and persuasive." -- Operations Research.

"Likelihood is an important text and, in addition, is a joy to read, being a paragon of lucid and witty exposition." -- Mathematical Gazette

This book analyses the straw man fallacy and its deployment in philosophical reasoning. While commonly invoked in both academic dialogue and public discourse, it has not until now received the attention it deserves as a rhetorical device. Scott Aikin and John Casey propose that straw manning essentially consists in expressing distorted representations of one’s critical interlocutor. To this end, the straw man comprises three dialectical forms, and not only the one that is usually suggested: the straw man, the weak man and the hollow man. Moreover, they demonstrate that straw manning is unique among fallacies as it has no particular logical form in itself, because it is an instance of inappropriate meta-argument, or argument about arguments. They discuss the importance of the onlooking audience to the successful deployment of the straw man, reasoning that the existence of an audience complicates the dialectical boundaries of argument. Providing a lively, provocative and thorough analysis of the straw man fallacy, this book will appeal to postgraduates and researchers alike, working in a range of fields including fallacies, rhetoric, argumentation theory and informal logic.

Throughout his career, Keith Hossack has made outstanding contributions to the theory of knowledge, metaphysics and the philosophy of mathematics. This collection of previously unpublished papers begins with a focus on Hossack's conception of the nature of knowledge, his metaphysics of facts and his account of the relations between knowledge, agents and facts. Attention moves to Hossack's philosophy of mind and the nature of consciousness, before turning to the notion of necessity and its interaction with a priori knowledge. Hossack's views on the nature of proof, logical truth, conditionals and generality are discussed in depth. In the final chapters, questions about the identity of mathematical objects and our knowledge of them take centre stage, together with questions about the necessity and generality of mathematical and logical truths. Knowledge, Number and Reality represents some of the most vibrant discussions taking place in analytic philosophy today.

First published in 1903, Principles of Mathematics was Bertrand Russell’s first major work in print. It was this title which saw him begin his ascent towards eminence. In this groundbreaking and important work, Bertrand Russell argues that mathematics and logic are, in fact, identical and what is commonly called mathematics is simply later deductions from logical premises. Highly influential and engaging, this important work led to Russell’s dominance of analytical logic on western philosophy in the twentieth century.

Table of Contents

introduction to the 1992 edition, introduction to the second edition, preface, PART I THE INDEFINABLES OF MATHEMATICS, PART II NUMBER, PART III QUANTITY, PART IV ORDER, PART V INFINITY AND CONTINUITY, PART VI SPACE, PART VII MATTER AND MOTION, APPENDICES, index

Our words and ideas refer to objects and properties in the external world; this phenomenon is central to thought, language, communication, and science. But great works of fiction are full of names that don't seem to refer to anything! In this book Kenneth A. Taylor explores the myriad of problems that surround the phenomenon of reference. How can words in language and perturbations in our brains come to stand for external objects? Reference is essential to truth, but which is more basic: reference or truth? How can fictional characters play such an important role in imagination and literature, and how does this use of language connect with more mundane uses? Taylor develops a framework for understanding reference, and the theories that other thinkers-past and present-have developed about it. But Taylor doesn't simply tell us what others thought; the book is full of new ideas and analyses, making for a vital final contribution from a seminal philosopher.

From an infant's first grasp of quantity to Einstein's theory of relativity, the human experience of number has intrigued researchers for centuries. Numeracy and mathematics have played fundamental roles in the development of societies and civilisations, and yet there is an essential mystery to these concepts, evidenced by the fear many people still feel when confronted by apparently simple sums. Including perspectives from anthropology, education and psychology, The Nature and Development of Mathematics addresses three core questions: Is maths natural? What is the impact of our culture and environment on mathematical thinking? And how can we improve our mathematical ability? Examining the cognitive processes that we use, the origins of these skills and their cultural context, and how learning and teaching can be supported in the classroom, the book contextualises each issue within the wider field, arguing that only by taking a cross-disciplinary perspective can we fully understand what it means to be numerate, as well as how we become numerate in our modern world. This is a unique collection including contributions from a range of renowned international researchers. It will be of interest to students and researchers across cognitive psychology, cultural anthropology and educational research.

Dieses Buch gibt einen kompakten UEberblick uber die historische Entwicklung und Ideengeschichte derjenigen mathematischen Gebiete, die sich erst in der Neuzeit zu eigenstandigen Teildisziplinen entwickelt haben: Analysis, Wahrscheinlichkeitstheorie, angewandte Mathematik, Topologie und Mengenlehre. Die Darstellung verzichtet auf Vollstandigkeit und konzentriert sich stattdessen ganz bewusst auf wesentliche oder besonders interessante Aspekte: Einzelne Persoenlichkeiten und Ideen werden exemplarisch herausgegriffen und detaillierter dargestellt als andere - es entsteht jedoch insgesamt ein stimmiges, ausgewogenes und dennoch ubersichtliches Gesamtbild. Dabei wird insbesondere begreifbar, dass die historische Entwicklung der Mathematik von zahlreichen Einflussen angetrieben wurde, dass zahlreiche theoretische Resultate aus ganz praktischen Grunden gefunden wurden (und umgekehrt), und dass es zu den wenigsten mathematischen Problemen nur einen (richtigen) Loesungsweg gibt. Auch Querverbindungen zwischen den verschiedenen Disziplinen werden deutlich. Das Buch wendet sich an all jene, die eine ubersichtliche, kurze Darstellung der zentralen Momente in der Geschichte der Mathematik suchen - vor allem Professoren, (zukunftige) Lehrer und Studierende.

A mathematical sightseeing tour of the natural world from the author of THE MAGICAL MAZE Why do many flowers have five or eight petals, but very few six or seven? Why do snowflakes have sixfold symmetry? Why do tigers have stripes but leopards have spots? Mathematics is to nature as Sherlock Holmes is to evidence. Mathematics can look at a single snowflake and deduce the atomic geometry of its crystals; it can start with a violin string and uncover the existence of radio waves. And mathematics still has the power to open our eyes to new and unsuspected regularities - the secret structure of a cloud or the hidden rhythms of the weather. There are patterns in the world we are now seeing for the first time - patterns at the frontier of science, yet patterns so simple that anybody can see them once they know where to look.

The amazing story of one of the greatest math problems of all time and the reclusive genius who solved it
In the tradition of "Fermatas Enigma" and "Prime Obsession," George Szpiro brings to life the giants of mathematics who struggled to prove a theorem for a century and the mysterious man from St. Petersburg, Grigory Perelman, who fi nally accomplished the impossible. In 1904 Henri PoincarA(c) developed the PoincarA(c) Conjecture, an attempt to understand higher-dimensional space and possibly the shape of the universe. The problem was he couldnat prove it. A century later it was named a Millennium Prize problem, one of the seven hardest problems we can imagine. Now this holy grail of mathematics has been found.
Accessibly interweaving history and math, Szpiro captures the passion, frustration, and excitement of the hunt, and provides a fascinating portrait of a contemporary noble-genius.

Das Buch berichtet uber das Leben des Mathematikers Friedrich Hirzebruch (1927-2012) und seinen lebenslangen Einsatz fur die Mathematik. Er war einer der bedeutendsten Mathematiker seiner Zeit und leistete UEberragendes fur den Wiederaufbau der wissenschaftlichen Forschung in Deutschland nach dem Zweiten Weltkrieg und fur nationale und internationale Zusammenarbeit auf vielen Ebenen. Seine Forschung hatte grossen Einfluss auf die Entwicklung der modernen Mathematik. 1952-1954 arbeitete er am Institute for Advanced Study in Princeton und wurde weltberuhmt durch den Beweis eines Theorems aus der Algebraischen Geometrie und Topologie, des sogenannten Satzes von Riemann-Roch-Hirzebruch. Im Alter von 27 Jahren erhielt er den Ruf auf seine Professur an der Universitat Bonn. In seinen Vorlesungen vermittelte er wie kaum ein Zweiter den Hoerern einen Eindruck von der Schoenheit der Mathematik und dem Gluck, Mathematiker zu sein. Ab 1980 leitete Hirzebruch viele Jahre das von ihm gegrundete Max-Planck-Institut fur Mathematik in Bonn. Er war mit vielen fuhrenden Mathematikern und Wissenschaftlern der zweiten Halfte des 20. Jahrhunderts befreundet. Als Mathematiker und Wissenschaftsorganisator waren ihm auch die Beziehungen zu Israel und Polen und die Loesung der mit der deutschen Wiedervereinigung im Wissenschaftssystem entstandenen Probleme ein besonderes Anliegen. Seine Biografie ist zugleich ein Stuck Wissenschaftsgeschichte und daruber hinaus auch Zeitgeschichte, von der Kriegs- und Nachkriegszeit bis zu den politischen Veranderungen nach 1990.