Essays in Constructive Mathematics

Harold M. Edwards

Publisher:

Springer

Publication Date:

2022

Number of Pages:

336

Format:

Hardcover

Edition:

2

Price:

109.99

ISBN:

978-3030985578

Category:

Monograph

MAA Review
Table of Contents

[Reviewed by

Benjamin Linowitz

, on

12/31/2023

]

This is the second edition of Harold Edwards' Essays in Constructive Mathematics See Bonnie Shulman's review of the first edition.

In the preface to the first edition, Edwards writes that the 1880's marked a turning point in mathematics. It was then that mathematicians like Cantor, Dedekind, and Weierstrass began employing and advocating the acceptance of transfinite constructions such as the one needed to prove the Bolzano-Weierstrass theorem. These non-constructive methods were further popularized by Hilbert and soon became accepted as an important part of mainstream mathematics. On the other hand, there were mathematicians like Kronecker who were opposed to these non-constructive methods and advocated adherence to the same standards of proof as mathematicians of earlier generations like Dirichlet and Gauss. Although Kronecker did not "win the day", his approach to mathematics (in particular to algebra and number theory) is not without influence. Indeed, Edwards himself was deeply influenced by Kronecker's work and wrote his collection of essays in order to illustrate the power of this constructivist approach to mathematics.

As Shulman explains in her review of the first edition, these essays are deeply influenced by the work of mathematicians of previous generations and in particular by classical results from the 19th century. And yet this is not a work whose genre is the history of mathematics. Nor is it a work of philosophy. The essays contained in this volume are serious works of mathematics done from a constructivist perspective. Most of these essays deal with topics related to Galois theory, algebraic number theory, or the theory of algebraic curves. As such, I think that these essays would be extremely difficult reading for someone without a graduate level understanding of these topics. On the other hand, I think that most mathematicians already familiar with these topics will find Edwards' constructivist approach to the topics covered to be fascinating.

After publishing the first edition of this text in 2005 Edwards continued to reflect and write on the topics covered in his original essays. The major change in this second edition is the inclusion of four additional chapters (each chapter being a collection of essays focussed on a fixed theme). This is a very significant addition and adds more than 100 pages to the 211 page length of the first edition. The four new chapters concern Constructive Algebra, The Algorithmic Foundations of Galois Theory, A Constructive Definition of Points on an Algebraic Curve, and Abel's Theorem. The latter three topics were all discussed in the first edition, though Edwards' thinking on them evolved dramatically over time, leading to treatments that are both more nuanced and much expanded.

Edwards was in the process of writing these essays when he passed away in 2020. David Cox assisted in the subsequent preparation of the essays for publication as part of a second edition of Essays in Constructive Mathematics. Cox's contributions mainly involved formatting and fixing small gaps in proofs and are listed in a postscript.

Benjamin Linowitz (benjamin.linowitz@oberlin.edu) is an Associate Professor of Mathematics at Oberlin College.

See the publisher's website.

Tags:

Constructive Mathematics