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Essays in Constructive Mathematics

Harold M. Edwards
Publication Date: 
Number of Pages: 
[Reviewed by
Bonnie Shulman
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As the Secret Master of MAA Reviews warned me when he asked me to review this book, it is NOT a book about the history/philosophy of mathematics, but rather a very serious book of mathematics. However, as the author makes clear in his preface, "history and philosophy were prominent among my motives for writing it" (p. ix). These essays are a walk down the road Kronecker opened, but most mainstream mathematicians bypassed in the late 19th and early 20th century. In a final essay, Edwards attempts to right the record and correct the legend, as popularized by E. T. Bell, of Kronecker as a "vicious" critic of Cantor and unbeliever in the existence of π (pp. 201-204). Harold Edwards succeeds admirably in his goal to show "substantial mathematics can be done constructively" (p. x) and that the "greater rigor and precision of mathematics done in that way adds immensely to its value" (p. ix).

A colleague of mine picked this book up off my desk with that familiar smirk I have come to expect from most mathematicians when they see or hear the words "constructive mathematics" (unless they have had experience with a healthy dose of computational mathematics). But his expression changed to surprise and even interest as he read off the topics covered — all of them from classical algebra and algebraic number theory. I, however, was not surprised to see splitting fields, binary quadratic forms, Newton's polygon and algebraic curves in a book by Harold Edwards, as I was already familiar with his interest in 19th century mathematics and his constructivist proclivities from his earlier books, especially Galois Theory (1984) and Advanced Calculus (1993). Last winter (2004) I taught a senior seminar on the History of the Proof, where we traced the evolution of Abel's proof of the insolvability of the quintic. We used Peter Pesic's book Abel's Proof for the history and philosophy and supplemented it with excerpts from other texts including Edwards' Galois Theory for more mathematical content. My (undergraduate) students found Edwards' writing telegraphic at best, and we spent the better part of two weeks decoding six pages of text and two problems.

Although admittedly part of the problem was my students' lack of mathematical maturity, I had to agree with them that the author's explanations were often cryptic, and the point of some of the exercises obscure, unless one was already aware of the historical and mathematical issues at stake. While I applaud Edwards' commitment to remain as faithful as possible to the original sources, I wish he'd given us more of a road map, indicating the lay of the land from his modern mathematician's viewpoint. Similarly, the book under review, while a strikingly original contribution to mathematics, is surely not one I would recommend for the "most naïve or inexperienced reader," as touted by Reuben Hersh (himself a master of lucid expository prose) on the back cover. However, even if the exposition is at times too spare for my own tastes, the mathematics is accessible to those with advanced undergraduate or graduate level courses in algebra and/or number theory, and will handsomely repay the effort to follow through the arguments with pencil in hand.

To whet your appetite, here is a sample of the flavor of the delicious mix of history of mathematics and new (constructive) mathematics Edwards uses to season his essays. Chapter 3, "Some Quadratic Problems" begins with a brief description of Greek (Archimedes, Pythagoras and Plato) approximations to √3, √2, and other irrational square roots. Questions about how these remarkably accurate approximations were arrived at, lead Edwards to (a generalization of) a formula recorded by a 7th century Indian mathematician Brahmagupta. He then conjectures that all of these approximations can be generated by a sequence of integer solutions of the form dn2 = 2sn 2 ± 1. With this historical development as motivation, he introduces in a natural way the notions of hypernumbers (y + x√A, with x and y integers, is a hypernumber for A) and modules of hypernumbers. At this point we are ready to appreciate how he skillfully wields these concepts to give constructive proofs of many results in number theory. And for dessert: a stunning treatment of Gauss' theory of binary quadratic forms "which describes how to use multiplication of modules to find an explicit composition of two given binary quadratic forms, provided they pertain to the same square-free integer" (p. 111).

Without a doubt the mathematics in this book is rigorous and serious, but the author is not humorless. Edwards' wry wit erupts unexpectedly here and there to enliven the sometimes dry prose I complained of earlier. In the final chapter, "Miscellany," he includes an essay where he reviews his own book, Linear Algebra (1995). He achieves a balanced tone that is at once self-mocking and self-justifying, contrasting his approach to teaching linear algebra with that taken by Sheldon Axler. Slyly co-opting the title of Axler's book — Linear Algebra Done Right (1996) — Edwards explains why his way is superior. With a wink he acknowledges that the reputation of mathematics as a repository of matters of fact over matters of opinion has had the unfortunate consequence of attracting more than our fair share of dogmatic thinkers — those who assert there is only one right way and it is my way!

Axler claims that the right way to do linear algebra is to avoid those pesky determinants, rejecting them as old-fashioned cumbersome formulas which are hard to motivate. Edwards agrees that calculating determinants can be intimidating, but as they are the heart and soul of linear algebra — solutions to systems of linear equations — we must find a way to help students become comfortable dealing with them, not sweep them under the rug of modern style mathematics. His solution is to "deal with the subject in an algorithmic way that I have found through teaching makes sense to students and gives them the tools they need to solve problems in linear algebra" (p. 191). He proceeds to define determinants without the formula in a clever way that I will not spoil for you, and leave to the reader as an enticement to read essay 5.3, "Overview of 'Linear Algebra.'"

The essay that follows this one is a beautiful constructive proof of the Spectral Theorem for symmetric matrices (with integer entries), which originally appeared in the last chapter of Linear Algebra, where it languished ("as far as I know no one has ever read it") until now where it appears "once again, with a few simplifications and improvements" (p. 196). This is such good mathematics, we must be grateful that the author indulged himself (indeed, who among us could resist the opportunity?) and rescued his own work from obscurity.

One of the most interesting features of the book is the statement of theorems. As good constructive theorems should, they all take the form: "Given . . . . , construct . . . ." or "Given . . . . , determine whether . . . ." For instance:

THEOREM: Given two n x n matrices of rational numbers, determine whether they are similar. (p. 193)

This may be discomfiting to some readers at first (they certainly can translate them into a more traditional format) but I think this drives home the point that we really are doing mathematics in a different, constructive way — as Edwards would no doubt say, the right way. (And, of course, such statements have a long mathematical pedigree, being found as far back as Euclid's Elements, Book I, Proposition 1...)

One other nice feature of this book is the bibliography which notes the sections where each reference appears. It should appeal to mathematicians and historians of mathematics alike. I am curious to see how it is received by the constructivist mathematical community.

Bonnie Shulman is associate professor of mathematics at Bates College in Lewiston, ME. Coming of age in the 1960s, she has always been attracted to theories and viewpoints that challenge the status quo. Trained as a mathematical physicist, her current intellectual passions include history and philosophy of mathematics and applications of game theory to studying social conflict and cooperation. She teaches t'ai chi and also practices yoga and meditation.

Preface * Synopsis * PART 1: A Fundamental Theorem * General Arithmetic * A Fundamental Theorem * Roots Field (Simple Algebraic Extensions) * Factorization of Polynomials with Integer Coefficients * A Factorization Algorithm * Validation of the Factorization Algorithm * About the Factorization Algorithm * Proof of the Fundamental Theorem * Minimal Splitting Polynomials * PART 2: Topics in Algebra * Galois' Fundamental Theorem * Algebraic Quantities * Adjunctions and the Factorization of Polynomials * Symmetric Polynomials and the Splitting Field of x^n + c_1x^{n-1} + ... + c_n * A Fundamental Theorem of Divisor Theory * PART 3: Some Quadratic Problems * Hypernumbers * Modules * The Class Semigroup * Multiplication of Modules and Module Classes * Is A a Square Mod p? * Gauss's Composition of Forms * The Construction of Compositions * PART 4: The Genus of an Algebraic Curve * Abel's Memoir * Euler's Addition Formula * An Algebraic Definition of the Genus * Newton's Polygon * Determination of the Genus * Holomorphic Differentials * The Riemann-Roch Theorem * The Genus is a Birational Invariant * PART 5: Miscellany * On the So-Called Fundamental Theorem of Algebra * Proof by Contradiction and the Sylow Theorems * Overview of 'Linear Algebra' * The Spectral Theorem * Kronecker as One of E.T. Bell's 'Men of Mathematics' * References