zu der hinterlassenen Abllamllullg VOll Abel, S. 57-81. -- 1 Die Definition der Ordnung eines algebraischen Ausdrucks, wie sie auf Seite 67 gegeben ist, ist incorrcct und nach der auf S. 10 angefiihrten zu berichtigen. Die Ordnung eines algebraischen Ausdrucks ist also nicht gleich der Anzahl der in ihm ausser den bekannten Groessen auftretenden Wurzelgroessen, sondern vielmehr, wenn man sich des Symbols V-Wie ublich zur Bezeichnung der Wurzelgroessen bedient, gleich der groessten von denjenigen Zahlen, welche angeben, wie viele solcher Wurzelzeichen sich in dem gegebenen algebraischen Ausdruck uber einander erstrecken. Dabei wird vorausgesetzt, dass, wenn ein Wurzelzeichen einen Index hat, welcher eine zusammengesetzte Zahl ist, dasselbe nach der Formel 1Jtn m -V-= VFso weit umgeformt werde, bis siimtliche Wurzelzeiehen Primzahl exponenten tragen, und dass sich keines dieser Wurzelzeichen durch Ausfuhrung der durch dasselbe angedeuteten Operation beseitigen Hisst. Kommen in einem algebraischen Ausdruck mehrere solcher auf einander oder auf algebrai. che Ausdrucke niederer Ordnung nicht reducierbarer Wurzelgroessen vor, in denen jene, die groesste Anzahl der iiber einander sich erstreekenden 'Wurzelzeichen angebenden Zahlen einander gleich sind, so giebt die Anzahl derselben den Grad des algebraischen Ausdrucks an. - Ist In die Ordnung des algebraischen Ausdrucks und bezeichnet man die einzelnen Wurzelgroessen in der Reihenfolge, wie sie numerisch berechnet werden ter mussen, um den Wert der Wurzelgroesse m Ordnung zu erhalten, mit ""m-l . . . .

Now available in a one-volume paperback, this book traces the development of the most important mathematical concepts, giving special attention to the lives and thoughts of such mathematical innovators as Pythagoras, Newton, Poincare, and Godel. Beginning with a Sumerian short story--ultimately linked to modern digital computers--the author clearly introduces concepts of binary operations; point-set topology; the nature of post-relativity geometries; optimization and decision processes; ergodic theorems; epsilon-delta arithmetization; integral equations; the beautiful "ideals" of Dedekind and Emmy Noether; and the importance of "purifying" mathematics. Organizing her material in a conceptual rather than a chronological manner, she integrates the traditional with the modern, enlivening her discussions with historical and biographical detail.

During the period of national isolation, a mathematical tradition called wasan flourished in Japan. Though virtually unknown to Europeans before the Meiji Restoration, its practitioners, the wasanka, produced some results comparable to (and sometimes in advance of) those of mathematicians of the European Enlightment. This volume, a companion to Unger's earlier translation of solutions by Aida Yasuaki (1747-1817), focuses on problems that Aida most likely used as a teacher. Unger explains the reasons for believing this, and sheds further light on the intellectual milieu in which Aida worked by discussing other books by Aida, including one in which he describes Dutch techniques of navigation.

Generality is a key value in scientific discourses and practices. Throughout history, it has received a variety of meanings and of uses. This collection of original essays aims to inquire into this diversity. Through case studies taken from the history of mathematics, physics and the life sciences, the book provides evidence of different ways of understanding the general in various contexts. It aims at showing how collectives have valued generality and how they have worked with specific types of "general" entities, procedures, and arguments. The books connects history and philosophy of mathematics and the sciences at the intersection of two of the most fruitful contemporary lines of research: historical epistemology, in which values (e.g. "objectivity", "accuracy") are studied from a historical viewpoint; and the philosophy of scientific practice, in which conceptual developments are seen as embedded in networks of social, instrumental, and textual practices. Each chapter provides a self-contained case-study, with a clear exposition of the scientific content at stake. The collection covers a wide range of scientific domains - with an emphasis on mathematics - and historical periods. It thus allows a comparative perspective which suggests a non-linear pattern for a history of generality. The introductory chapter spells out the key issues and points to the connections between the chapters.

A comprehensive collection of historical readings in the philosophy of mathematics and a selection of influential contemporary work, this much-needed introduction reveals the rich history of the subject. An Historical Introduction to the Philosophy of Mathematics: A Reader brings together an impressive collection of primary sources from ancient and modern philosophy. Arranged chronologically and featuring introductory overviews explaining technical terms, this accessible reader is easy-to-follow and unrivaled in its historical scope. With selections from key thinkers such as Plato, Aristotle, Descartes, Hume and Kant, it connects the major ideas of the ancients with contemporary thinkers. A selection of recent texts from philosophers including Quine, Putnam, Field and Maddy offering insights into the current state of the discipline clearly illustrates the development of the subject. Presenting historical background essential to understanding contemporary trends and a survey of recent work, An Historical Introduction to the Philosophy of Mathematics: A Reader is required reading for undergraduates and graduate students studying the philosophy of mathematics and an invaluable source book for working researchers.

The world around us is saturated with numbers. They are a fundamental pillar of our modern society, and accepted and used with hardly a second thought. But how did this state of affairs come to be? In this book, Leo Corry tells the story behind the idea of number from the early days of the Pythagoreans, up until the turn of the twentieth century. He presents an overview of how numbers were handled and conceived in classical Greek mathematics, in the mathematics of Islam, in European mathematics of the middle ages and the Renaissance, during the scientific revolution, all the way through to the mathematics of the 18th to the early 20th century. Focusing on both foundational debates and practical use numbers, and showing how the story of numbers is intimately linked to that of the idea of equation, this book provides a valuable insight to numbers for undergraduate students, teachers, engineers, professional mathematicians, and anyone with an interest in the history of mathematics.

The logician Kurt Goedel (1906-1978) published a paper in 1931 formulating what have come to be known as his 'incompleteness theorems', which prove, among other things, that within any formal system with resources sufficient to code arithmetic, questions exist which are neither provable nor disprovable on the basis of the axioms which define the system. These are among the most celebrated results in logic today. In this volume, leading philosophers and mathematicians assess important aspects of Goedel's work on the foundations and philosophy of mathematics. Their essays explore almost every aspect of Godel's intellectual legacy including his concepts of intuition and analyticity, the Completeness Theorem, the set-theoretic multiverse, and the state of mathematical logic today. This groundbreaking volume will be invaluable to students, historians, logicians and philosophers of mathematics who wish to understand the current thinking on these issues.

An unabridged, unaltered printing of the Second Edition (1920), with original format, all footnotes and index: The Series of Natural Numbers - Definition of Number - Finitude and Mathematical Induction - The Definition of Order - Kinds of Relations - Similarity of Relations - Rational, Real, and Complex Numbers - Infinite Cardinal Numbers - Infinite Series and Ordinals - Limits and Continuity - Limits and Continuity of Functions - Selections and the Multiplicative Axiom - The Axiom of Infinity and Logical Types - Incompatibility and the Theory of Deductions - Propositional Functions - Descriptions - Classes - Mathematics and Logic - Index

2011 Reprint of 1928 Edition. Full facsimile of the original edition, not reproduced with Optical Recognition Software. Illustrated with 31 illustrations. Florian Cajori was one of the most celebrated historians of mathematics in his day. Cajori emigrated to the United States at the age of sixteen. He received a Ph.D. at Tulane University, where he taught for a few years before settling in Berkeley. Even today his "History of Mathematical Notations" has been described as "unsurpassed." In 1918, he was appointed to a specially created chair in history of mathematics at the University of California, Berkeley. He remained in Berkeley, California until his death in 1930. "The Early Mathematical Sciences in North and South America" covers the contributions made in the field of mathematics by early practitioners in North and South America. He begins with the Mayan system of numbers, and the book contains chapters on Practical Astronomy and Surveying, Meridian Measurements of the Earth, Transit of Venus, 1761 to 1769, Comets, Almanacs, Orreries, Earliest Permanent Observation in America, Physics, Societies, Academies and Journals.

Two 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.

2013 Reprint of 1931 edition. Full facsimile of the original edition, not reproduced with Optical Recognition Software. Frank Plumpton Ramsey (1903-1930) was a British mathematician who also made significant and precocious contributions in philosophy and economics before his death at the age of 26. He was a close friend of Ludwig Wittgenstein, and was instrumental in translating Wittgenstein's "Tractatus Logico-Philosophicus" into English, and in persuading Wittgenstein to return to philosophy and to Cambridge. This volume collects Ramsey's most important papers. Contents: The foundations of mathematics.--Mathematical logic.--On a problem of formal logic.--Universals.--Note on the preceding paper.--Facts and propositions.--Truth and probability.--Further considerations.--Last papers.

Smart Moves: Developing Mathematical Reasoning with Games and Puzzles is designed to improve your sequential reasoning, explore some mathematics, and have fun along the way. The games and puzzles were created to encourage perseverance and logical thinking. The Mathematical Connections highlight key math concepts. The Game of Racetrack is the perfect introduction to vectors, Tour Puzzles lead to graph theory and Euler paths, and the mathematics behind Magic Squares is revealed.Smart Moves is a very effective way to support mathematical learning and reduce the anxiety that often accompanies the subject.Each chapter is designed to strengthen sequential reasoning, which is necessary for everyday living and problem solving. Whether you are nine or ninety, in the classroom or at home, I invite you to make a smart move and discover how much fun math can be!

Cognitive mathematics provides insights into how mathematics works inside the brain and how it is interconnected with other faculties through so-called blending and other associative processes. This handbook is the first large collection of various aspects of cognitive mathematics to be amassed into a single title, covering decades of connection between mathematics and other figurative processes as they manifest themselves in language, art, and even algorithms. It will be of use to anyone working in math cognition and education, with each section of the handbook edited by an international leader in that field.

From the author of "Zero," comes this "admirable salvo against quantitative bamboozlement by the media and the government" ("The Boston Globe")

In "Zero," Charles Seife presented readers with a thrilling account of the strangest number known to humankind. Now he shows readers how the power of skewed metrics-or "proofiness"- is being used to alter perception in both amusing and dangerous ways. Proofiness is behind such bizarre stories as a mathematical formula for the perfect butt and sprinters who can run faster than the speed of sound. But proofiness also has a dark side: bogus mathematical formulas used to undermine our democracy-subverting our justice system, fixing elections, and swaying public opinion with lies. By doing the real math, Seife elegantly and good-humoredly scrutinizes our growing obsession with metrics while exposing those who misuse them.

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.

The Discourse on the Method is a philosophical and mathematical treatise published by Ren Descartes in 1637. Its full name is Discourse on the Method of Rightly Conducting the Reason, and Searching for Truth in the Sciences. The Discourse on Method is best known as the source of the famous quotation "Je pense, donc je suis," "I think, therefore I am."This is one of the most influential works in the history of modern science

Philosophy of mathematics is moving in a new direction: away from a foundationalism in terms of formal logic and traditional ontology, and towards a broader range of approaches that are united by a focus on mathematical practice. The scientific research network PhiMSAMP (Philosophy of Mathematics: Sociological Aspects and Mathematical Practice) consisted of researchers from a variety of backgrounds and fields, brought together by their common interest in the shift of philosophy of mathematics towards mathematical practice. Hosted by the Rheinische Friedrich- Wilhelms-Universitat Bonn and funded by the Deutsche Forschungsgemeinschaft (DFG) from 2006-2010, the network organized and contributed to a number of workshops and conferences on the topic of mathematical practice. The refereed contributions in this volume represent the research results of the network and consists of contributions of the network members as well as selected paper versions of presentations at the network's mid-term conference, "Is mathematics special?" (PhiMSAMP-3) held in Vienna 2008.

Die Mathematik im mittelalterlichen Islam hatte gro en Einfluss auf die allgemeine Entwicklung des Faches. Der Autor beschreibt diese Periode der Geschichte der Mathematik und bezieht sich dabei auf die arabischsprachigen Quellen. Zu den behandelten Themen geh ren Dezimalrechnen, Geometrie, ebene und sph rische Trigonometrie, Algebra sowie die Approximation von Wurzeln von Gleichungen. Das Buch wendet sich an Mathematikhistoriker und -studenten, aber auch an alle Interessierten mit Mathematikkenntnissen der weiterf hrenden Schule.