Essays in Constructive Mathematics

ISBN-13: 9783030985578 / Angielski / Twarda / 2022

Harold M. Edwards

Opis

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 those—the majority—who 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.

Spis treści

Part I.- 1. A Fundamental Theorem.- 2. Topics in Algebra.- 3. Some Quadratic Problems.- 4. The Genus of an Algebraic Curve.- 5. Miscellany. Part II.- 6. Constructive Algebra.- 7. The Algorithmic Foundation of Galois's Theory.- 8. A Constructive Definition of Points on an Algebraic Curve.- 9. Abel's Theorem.

O autorze

Harold M. Edwards [1936–2020] was Professor Emeritus of Mathematics at New York University. His research interests lay in number theory, algebra, and the history and philosophy of mathematics. He authored numerous books, including Riemann’s Zeta Function (1974, 2001) and Fermat’s Last Theorem (1977), for which he received the Leroy P. Steele Prize for mathematical exposition in 1980.

David A. Cox (Contributing Author) is Professor Emeritus of Mathematics in the Department of Mathematics and Statistics of Amherst College. He received the Leroy P. Steele Prize for mathematical exposition in 2016 for his book Ideals, Varieties, and Algorithms, with John Little and Donal O’Shea.

Blurb

This collection of essays aims to promote constructive mathematics, not by defining it or formalizing it, but by practicing it. All definitions and proofs are based on finite algorithms, which pave illuminating paths to nontrivial results, primarily in algebra, number theory, and the theory of algebraic curves. The second edition adds a new set of essays that reflect and expand upon the first. 

The topics covered derive from classic works of nineteenth-century mathematics, among them Galois’s theory of algebraic equations, Gauss’s theory of binary quadratic forms, and Abel’s theorems about integrals of rational differentials on algebraic curves. Other topics include Newton's diagram, the fundamental theorem of algebra, factorization of polynomials over constructive fields, and the spectral theorem for symmetric matrices, all treated using constructive methods in the spirit of Kronecker.

In this second edition, the essays of the first edition are augmented with new essays that give deeper and more complete accounts of Galois’s theory, points on an algebraic curve, and Abel’s theorem. Readers will experience the full power of Galois’s approach to solvability by radicals, learn how to construct points on an algebraic curve using Newton’s diagram, and appreciate the amazing ideas introduced by Abel in his 1826 Paris memoir on transcendental functions.

Mathematical maturity is required of the reader, and some prior knowledge of Galois theory is helpful.  But experience with constructive mathematics is not necessary; readers should simply be willing to set aside abstract notions of infinity and explore deep mathematics via explicit constructions.