The Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae

ISBN-13: 9783642058028 / Angielski / Miękka / 2010 / 578 str.

Catherine Goldstein;Norbert Schappacher;Joachim Schwermer

Opis

Since its publication, C.F. Gauss's Disquisitiones Arithmeticae (1801) has acquired an almost mythical reputation, standing as an ideal of exposition in notation, problems and methods; as a model of organisation and theory building; and as a source of mathematical inspiration. Eighteen authors - mathematicians, historians, philosophers - have collaborated in this volume to assess the impact of the Disquisitiones, in the two centuries since its publication.

Recenzje

From the reviews:

"A book that traces the profound effect Gauss's masterpiece has had on mathematics over the past two centuries. ... The shaping of arithmetic is a major accomplishment, one which will stand as an important reference work on the history of number theory for many years. ... The editors and authors deserve our thanks for their efforts." (Victor J. Katz, Mathematical Reviews, Issue 2008 h)

"It's a big book, with eighteen authors and almost six hundred pages, and it mixes the work of well-established scholars with that of recent Ph.D.'s. ... This volume deserves a wide audience, both among the mathematically able and among historians of nineteenth-century science." (Thomas Archibald, Institute for Science and International Security, Vol. 102 (2), June, 2011)

Spis treści

I. A Book’s History. – C. Goldstein, N. Schappacher. II. Algebraic Equations, Quadratic Forms, Higher Congruences: Key Mathematical Techniques of the Disquistiones. - O. Neumann: The Disquisitiones Arithmeticae and the Theory of Equations.- H.M. Edwards: Composition of Binary Quadratic Forms and the Foundations of Mathematics.- D. Fenster, J. Schwermer: Composition of Quadratic Forms: An Algebraic Perspective.- G. Frei: Gauss’s Unpublished Section Eight: On the Way to Function Fields over a Finite Field.- III. The German Reception of the Disquisitiones Arithmeticae: Institutions and Ideas. – H. Pieper: A Network of Scientific Philanthropy: Humboldt’s Relations with Number Theorists.- J. Ferreirós: The Rise of Pure Mathematics as Arithmetic after Gauss.- IV. Complex Numbers and Complex Functions in Arithmetic.- R. Bölling: From Reciprocity Laws to Ideal Numbers: An (Un)Known 1844 Manuscript by E.E. Kummer.- C. Houzel: Elliptic Functions and Arithmetic. V. Numbers as Model Objects of Mathematics.- J. Boniface: The Concept of Number from Gauss to Kronecker.- B. Petri, N. Schappacher: On Arithmetization. VI. Number Theory in France in the Second Half of the Nineteenth Century.- C. Goldstein: Hermitian Forms of Reading the Disquisitiones Arithmeticae.- A.-M. Décaillot: Number Theory at the Association francaise pour l’avancement des sciences.- VII. Spotlighting Some Later Reactions.- A. Brigaglia: An Overview on Italian Arithmeitc after the Disquistiones Arithmeticae. P. Piazza: Zolotarev’s Theory of Algebraic Numbers.- D. Fenster: Gauss Goes West: The Reception of the Disquistiones Arithmeticae in the USA. VIII. Gauss’s Theorem in the Long Run: Three Case Studies.- J. Schwermer: Reduction Theory of Quadratic Forms: Toward Räumliche Anschauung in Minkowski’s Early Work.- S. J. Patterson: Gauss Sums.- F. Lemmermeyer: The Principal Genus Theorem.- List of Illustrations.- Index.- Author’s Addresses.

O autorze

Catherine Goldstein is Directrice de recherches du CNRS and works at the Institut de mathématiques de Jussieu (Paris, France). She is the author of "Un théorème de Fermat et ses lecteurs" (1995) and a coeditor  of "Mathematical Europe: History, Myth, Identity"(1996). Her research aims at developing a social history of mathematical practices and results, combining close readings and a network analysis of texts. Her current projects include the study of mathematical sciences through World War I and of experimentation in XVII th-century number theory.

Norbert Schappacher is professor of mathematics at Université Louis Pasteur, Strasbourg.His mathematical interests relate to the arithmetic of elliptic curves.But his current research projects lie in the history of mathematics. Specifically, he focuses on the intertwinement of philosophical and political categories with major junctures in the development of mathematical disciplines in the XIX\up and XX\up centuries. Examples include number theory and algebraic geometry, but also medical statistics.

 Joachim Schwermer is professor of mathematics at University of Vienna. In addition, he serves as scientific director at the Erwin-Schroedinger International Institute for Mathematical Physics, Vienna. His research interests lie in number theory and algebra, in particular, in questions arising in arithmetic algebraic geometry and the theory of automorphic forms. He takes a keen interest in the mathematical sciences in the XIX\up and XX\up centuries in their historical context.