Essays in Constructive Mathematics

"... The exposition is not only clear, it is friendly, philosophical, and considerate even to the most naive or inexperienced reader. And it proves that the philosophical orientation of an author really can make a big difference. The mathematical content is intensely classical. ... Edwards makes it warmly accessible to any interested reader. And he is breaking fresh ground, in his rigorously constructive or constructivist presentation. So the book will interest anyone trying to learn these major, central topics in classical algebra and algebraic number theory. Also, anyone interested in constructivism, for or against. And even anyone who can be intrigued and drawn in by a masterly exposition of beautiful mathematics." Reuben Hersh

This book aims to promote constructive mathematics, not by defining it or formalizing it, but by practicing it, by basing all definitions and proofs on finite algorithms. The topics covered derive from classic works of nineteenth century mathematics---among them Galois' theory of algebraic equations, Gauss's theory of binary quadratic forms and Abel's theorem about integrals of rational differentials on algebraic curves. It is not surprising that the first two topics can be treated constructively---although the constructive treatments shed a surprising amount of light on them---but the last topic, involving integrals and differentials as it does, might seem to call for infinite processes. In this case too, however, finite algorithms suffice to define the genus of an algebraic curve, to prove that birationally equivalent curves have the same genus, and to prove the Riemann-Roch theorem. The main algorithm in this case is Newton's polygon, which is given a full treatment. Other topics covered include the fundamental theorem of algebra, the factorization of polynomials over an algebraic number field, and the spectral theorem for symmetric matrices.

Harold M. Edwards is Emeritus Professor of Mathematics at NewYork University. His previous books are Advanced Calculus (1969, 1980, 1993), Riemann's Zeta Function (1974, 2001), Fermat's Last Theorem (1977), Galois Theory (1984), Divisor Theory (1990) and Linear Algebra (1995). Readers of his Advanced Calculus will know that his preference for constructive mathematics is not new.

Auteur

Harold M. Edwards is Emeritus Professor of Mathematics at New York University. His previous books are Advanced Calculus (1969, 1980, 1993), Riemann's Zeta Function (1974, 2001), Fermat's Last Theorem (1977), Galois Theory (1984), Divisor Theory (1990) and Linear Algebra (1995). Readers of his Advanced Calculus will know that his preference for constructive mathematics is not new. In 1980 he was awarded the Steele Prize for mathematical exposition for the Riemann and Fermat books.

Résumé
He [Kronecker] was, in fact, attempting to describe and to initiate a new branch of mathematics, which would contain both number theory and alge­ braic geometry as special cases.Andre Weil [62] This book is about mathematics, not the history or philosophy of mathemat­ ics. Still, history and philosophy were prominent among my motives for writing it, and historical and philosophical issues will be major factors in determining whether it wins acceptance. Most mathematicians prefer constructive methods. Given two proofs of the same statement, one constructive and the other not, most will prefer the constructive proof. The real philosophical disagreement over the role of con­ structions in mathematics is between thosethe majoritywho believe that to exclude from mathematics all statements that cannot be proved construc­ tively would omit far too much, and those of us who believe, on the contrary, that the most interesting parts of mathematics can be dealt with construc­ tively, and that the greater rigor and precision of mathematics done in that way adds immensely to its value.

Contenu
A Fundamental Theorem.- Topics in Algebra.- Some Quadratic Problems.- The Genus of an Algebraic Curve.- Miscellany.