Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfangen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen fur die historische wie auch die disziplingeschichtliche Forschung zur Verfugung, die jeweils im historischen Kontext betrachtet werden mussen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

An erster Stelle bringt dieses Doppelheft der Mater i a I i e n einen Abschnitt aus dem in Vorbereitung befindlichen Aufsatze von A. Ga 11 e uber die geodatischen Arbeiten von Gauss; da es sich um einen Gegenstand handelt, der uber die Geodasie hinaus greift und ein selbstandiges Interesse besitzt, schien es angebracht, diesen Abschnitt .dem eigentlichen Aufsatze vorauszuschicken. Das Zahlenrechnen greift in die ganze Tatigkeit von Gauss in der reinen wie in der angewandten Mathematik ein, und darum war dem Verfasser die gutige Unterstutzung der Mitarbeiter an der Herausgabe der Werke von Gauss, sowie einiger anderer Gelehrten sehr willkommen; er mochte auch an dieser Stelle M. Brendel, F. Klein, L. Kruger, A. Loewy, P. Maennchen, L. Schlesinger, P. Stackel und G. Witt fur verschiedene Bemerkungen und Hin weise seinen verbindlichen Dank zum Ausdruck bringen. An zweiter Stelle erscheint der Aufsatz von P. Stackel uber Gauss als Geometer, der sich den Aufsatzen von Bach mann uber die zahlentheoretischen und von Schlesinger uber die funktionentheoretischen Arbeiten von Gauss anreiht. Der Ver fasser ist F. Klein und L. Schlesinger fur das grosse Interesse verpflichtet, dass sie seiner Arbeit wahrend ihrer Entstehung ent gegenbrachten, nicht minder auch fur eine Reihe wertvoller Be merkungen, die sie wahrend der Korrektur beigesteuert haben. An der Korrektur beteiligten sich ferner F. Engel, A. Galle und A."

While we are commonly told that the distinctive method of mathematics is rigorous proof, and that the special topic of mathematics is abstract structure, there has been no agreement among mathematicians, logicians, or philosophers as to just what either of these assertions means. John P. Burgess clarifies the nature of mathematical rigor and of mathematical structure, and above all of the relation between the two, taking into account some of the latest developments in mathematics, including the rise of experimental mathematics on the one hand and computerized formal proofs on the other hand. The main theses of Rigor and Structure are that the features of mathematical practice that a large group of philosophers of mathematics, the structuralists, have attributed to the peculiar nature of mathematical objects are better explained in a different way, as artefacts of the manner in which the ancient ideal of rigor is realized in modern mathematics. Notably, the mathematician must be very careful in deriving new results from the previous literature, but may remain largely indifferent to just how the results in the previous literature were obtained from first principles. Indeed, the working mathematician may remain largely indifferent to just what the first principles are supposed to be, and whether they are set-theoretic or category-theoretic or something else. Along the way to these conclusions, a great many historical developments in mathematics, philosophy, and logic are surveyed. Yet very little in the way of background knowledge on the part of the reader is presupposed.

Algebraic Art explores the invention of a peculiarly Victorian account of the nature and value of aesthetic form, and it traces that account to a surprising source: mathematics. The nineteenth century was a moment of extraordinary mathematical innovation, witnessing the development of non-Euclidean geometry, the revaluation of symbolic algebra, and the importation of mathematical language into philosophy. All these innovations sprang from a reconception of mathematics as a formal rather than a referential practice-as a means for describing relationships rather than quantities. For Victorian mathematicians, the value of a claim lay not in its capacity to describe the world but its internal coherence. This concern with formal structure produced a striking convergence between mathematics and aesthetics: geometers wrote fables, logicians reconceived symbolism, and physicists described reality as consisting of beautiful patterns. Artists, meanwhile, drawing upon the cultural prestige of mathematics, conceived their work as a 'science' of form, whether as lines in a painting, twinned characters in a novel, or wavelike stress patterns in a poem. Avant-garde photographs and paintings, fantastical novels like Flatland and Lewis Carroll's children's books, and experimental poetry by Swinburne, Rossetti, and Patmore created worlds governed by a rigorous internal logic even as they were pointedly unconcerned with reference or realist protocols. Algebraic Art shows that works we tend to regard as outliers to mainstream Victorian culture were expressions of a mathematical formalism that was central to Victorian knowledge production and that continues to shape our understanding of the significance of form.

Pythagorean thought, from the civilisation of Ancient Greece, is still prevalent in religion, poetry, philosophy, music, architecture and the classical sciences today. This fascinating and insightful collection of essays by experts in their fields explores the Pythagorean tradition, drawing out connections in form, number and geometry as well as light, colour, music and poetry. The contributors include Robert Lawlor, Keith Critchlow, Kathleen Raine, Anne Macaulay and Arthur Zajonc. Previously published as Rediscovering Sacred Science.

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.

The teaching of mathematics proceeds from simple calculations to complex conceptualisations. As numerical figures and symbols and shapes morph towards complex abstractions, there seems to be a `natural selection' in society between a few who experience sheer joy from the subject and phobia for the majority. Today, the world places great value on mastering mathematics as a basis for integration into the world of work in a global epoch of rapid technological change. Thus, understandably, most nations obsess about their ability to impart and absorb mathematical knowledge. Global comparative studies on that issue are taken as grounds for national pride and self-perceptions of intelligence. Sheer horror greets poor outcomes; the empirical and the substantive merge into a confounding vortex of misconceptions. Finger-pointing and hyperbole ensue, and politics enters the fray in its most shameful and destructive forms. Post-1994, South Africa has had its own share of self-flagellation. There has been much research on the reasons behind the country's poor comparative performance. While there have been some improvements in the recent period, those are barely enough. This book on the pedagogy of mathematics reasserts some of the findings of previous studies. Those relate to: the impact of a racist system that perversely reckoned that keeping mathematical knowledge from the oppressed would prove their supposed inferiority; the relevance of the language of teaching; changing school curricula, and the questions of how to speed up movement from universal access to better outcomes. The authors go beyond that to pose the simple but telling question: why, at all, do we teach mathematics, and what is its actual utility to life? As this book clearly reveals, teaching mathematics through dialogue that is linked to a concrete social environment is fundamental to speeding up the improvements South Africa has started to experience. So are the joint efforts of government, the unions, and private partners to improve the situation.

In line with the emerging field of philosophy of mathematical practice, this book pushes the philosophy of mathematics away from questions about the reality and truth of mathematical entities and statements and toward a focus on what mathematicians actually do--and how that evolves and changes over time. How do new mathematical entities come to be? What internal, natural, cognitive, and social constraints shape mathematical cultures? How do mathematical signs form and reform their meanings? How can we model the cognitive processes at play in mathematical evolution? And how does mathematics tie together ideas, reality, and applications? Roi Wagner uniquely combines philosophical, historical, and cognitive studies to paint a fully rounded image of mathematics not as an absolute ideal but as a human endeavor that takes shape in specific social and institutional contexts. The book builds on ancient, medieval, and modern case studies to confront philosophical reconstructions and cutting-edge cognitive theories. It focuses on the contingent semiotic and interpretive dimensions of mathematical practice, rather than on mathematics' claim to universal or fundamental truths, in order to explore not only what mathematics is, but also what it could be. Along the way, Wagner challenges conventional views that mathematical signs represent fixed, ideal entities; that mathematical cognition is a rigid transfer of inferences between formal domains; and that mathematics' exceptional consensus is due to the subject's underlying reality. The result is a revisionist account of mathematical philosophy that will interest mathematicians, philosophers, and historians of science alike.

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.

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.

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

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.

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.