Mathematics of the 19th Century: Mathematical Logic Algebra Number Theory Probability Theory

ISBN-13: 9783764364410 / Angielski / Twarda / 2001 / 308 str.

A. P. Yushkevich; A. N. Kolmogorov; Andre P. Yushkevich

Opis

This is the second revised edition of the first volume of the outstanding collection of historical studies of mathematics in the nineteenth century compiled in three volumes by A. N. Kolmogorov and A. P. Yushkevich. This second edition was carefully revised by Abe Shenitzer, York University, Ontario, Canada. The historical period covered in this book extends from the early nineteenth century up to the end of the 1930s, as neither 1801 nor 1900 are, in themselves, turning points in the history of mathematics, although each date is notable fo a remarkable event: the first for the publication of Gauss' "Disquisitiones arithmeticae", the second for Hilbert's "Mathematical Problems." Beginning in the second quarter of the nineteenth century mathematics underwent a revolution as crucial and profound in its consequences for the general world outlook as the mathematical revolution in the beginning of the modern era. The main changes included a new statement of the problem of the existence of mathematical objects, particulary in the calculus, and soon thereafter the formation of non-standard structures in geometry, arithmetic and algebra. The primary objective of the work has been to treat the evolution of mathematics in the nineteenth century as a whole; the discussion is concentrated on the essential concepts, methods, and algorithms.

Recenzje

"...The book, indispensable for historians of mathematics, can be warmly recommended to every working mathematician."

--EMS Newsletter (on the first edition)

Spis treści

One Mathematical Logic.- The Prehistory of Mathematical Logic.- Leibniz’s Symbolic Logic.- The Quantification of a Predicate.- The “Formal Logic” of A. De Morgan.- Boole’s Algebra of Logic.- Jevons’ Algebra of Logic.- Venn’s Symbolic Logic.- Schröder’s and Poretski-’s Logical Algebra.- Conclusion.- Two Algebra and Algebraic Number Theory.- 1 Survey of the Evolution of Algebra and of the Theory of Algebraic Numbers During the Period of 1800-1870.- 2 The Evolution of Algebra.- Algebraic Proofs of the Fundamental Theorem of Algebra in the 18th Century.- C.F. Gauss’ First Proof.- C.F. Gauss’ Second Proof.- The Kronecker Construction.- The Theory of Equations.- Carl Friedrich Gauss.- Solution of the Cyclotomic Equation.- Niels Henrik Abel.- Evariste Galois.- The Algebraic Work of Evariste Galois.- The First Steps in the Evolution of Group Theory.- The Evolution of Linear Algebra.- Hypercomplex Numbers.- William Rowan Hamilton.- Matrix Algebra.- The Algebras of Grassmann and Clifford.- Associative Algebras.- The Theory of Invariants.- 3 The Theory of Algebraic Numbers and the Beginnings of Commutative Algebra.- Disquisitiones Arithmeticaeof C.F. Gauss.- Investigation of the Number of Classes of Quadratic Forms.- Gaussian Integers and Their Arithmetic.- Fermat’s Last Theorem. The Discovery of E. Kummer.- Kummer’s Theory.- Difficulties. The Notion of an Integer.- The Zolotarev Theory. Integral and p-Integral Numbers.- Dedekind’s Ideal Theory.- On Dedekind’s Method. Ideals and Cuts.- Construction of Ideal Theory in Algebraic Function Fields.- L. Kronecker’s Divisor Theory.- Conclusion.- Three Problems of Number Theory.- 1 The Arithmetic Theory of Quadratic Forms.- The General Theory of Forms; Ch. Hermite.- Korkin’s and Zolotarev’s Works on the Theory of Quadratic Forms.- The Investigations of A.A. Markov.- 2 Geometry of Numbers.- Origin of the Theory.- The Work of H.J.S. Smith.- Geometry of Numbers: Hermann Minkowski.- The Works of G.F. Vorono-.- 3 Analytic Methods in Number Theory.- Lejeune-Dirichlet and the Theorem on Arithmetic Progressions.- Asymptotic Laws of Number Theory.- Chebyshev and the Theory of Distribution of Primes.- The Ideas of Bernhard Riemann.- Proof of the Asymptotic Law of Distribution of Prime Numbers.- Some Applications of Analytic Number Theory.- Arithmetic Functions and Identities. The Works of N.V. Bugaev.- 4 Transcendental Numbers.- The Works of Joseph Liouville.- Charles Hermite and the Proof of the Transcendence of the Number e; The Theorem of Ferdinand Lindemann.- Conclusion.- Four The Theory of Probability.- Laplace’s Theory of Probability.- Laplace’s Theory of Errors.- Gauss’ Contribution to the Theory of Probability.- The contributions of Poisson and Cauchy.- Social and Anthropometric Statistics.- The Russian School of the Theory of Probability. P.L. Chebyshev.- New Fields of Application of the Theory of Probability. The Rise of Mathematical Statistics.- Works of the Second Half of the 19th Century in Western Europe.- Conclusion.- Addendum.- 1. French and German Quotations.- 2. Notes.- 3. Additional Bibliography.- Bibliography (by F.A. Medvedev).- Abbreviations.- Index of Names.