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Publisher:

American Mathematical Society

Publication Date:

2003

Number of Pages:

326

Format:

Paperback

Price:

45.00

ISBN:

978-0-8218-2672-0

Category:

Anthology

[Reviewed by , on ]

Kenneth A. Ross

07/19/2004

Why would anyone want to read a drab green book with reprints of one hundred-year-old articles written by long-dead guys? I wondered, but the more I looked at this collection, the more I liked it.

First some editorial comments. It would have helped a lot to have the dates of the papers listed in the Table of Contents. Also, an index would have been very useful. Finally, the following sentence on page 162 is puzzling: "The last article in this section is Wilson's [15] review of the first of these books." I am unable to find any artlcle by Wilson in this book.

This book is a natural follow-up to the special January 2000 issue of the *Bulletin of the American Mathematical Society* (BAMS), commemorating the "the development of mathematics in the last century." Two of the articles in this book also appear in that issue of BAMS. One of these is a fascinating summary of the history of 19th century mathematics by James Pierpont (BAMS 1904). Pierpont, an American who spent most of his career at Yale, was an early fine expositor. The section on "Functions of Real Variables" lists Lebesgue "among the many not yet mentioned who have made important contributions." This is impressive considering that Lebesgue published his thesis in 1904.

One of Pierpont's articles outlines the battle of the quintic equation problem. Key players were, in order, Lagrange, Ruffini, Abel, and Galois, spanning the years 1770-1832. Pierpont refers to an improved proof by Kronecker in 1879 and offers his own improved exposition in 1896.

Florian Cajori's article succintly outlines the theory of infinite series as developed up to 1892. Most of the standard results (in Knopp's classical "Infinite Sequences and Series" and in Karl Stromberg's "An Introduction to Classical Analysis," for example) were known by 1892.

Charlotte Angus Scott, one of 17 Americans to attend the ICM in Paris in August 1900, provided a useful report for the U.S. audience; it was published in November 1900. She includes summaries of the main addresses, by M. Cantor and Volterra, as well as presentations in the sections. A morning was devoted to a resolution that "Esperanto" should be adopted as the "vehicle for all scientific communication." Scott didn't hesitate to extensively criticize the "usually shockingly bad" presentation of the papers. Finally, it is interesting to note her comments: "The communications in Sections V and VI, while not necesssarily the most valuable mathematically, were yet of the most general interest, and lend themselves best to any general report." She goes on to describe Hilbert's now famous report on problems for the future. The reading of the paper was followed by a "rather desultory discussion." The last article in the book is a translation of Hilbert's famous article. It was published in 1902.

Chapter 4 is devoted to aspects of Poincaré's spectacular career. One is a review of his 1892 book on celestial mechanics which was published in the same year. A second is his lecture at the 1897 ICM in Zurich on the relationship between analysis and mathematical physics. This excellent essay can be viewed as a sales pitch to physicists (why math is important) and also a sales pitch to mathematicians (why problems from physics are important). Each of these subjects enhances the other. It's interesting to note how the mainstream of pure mathematics in the mid-20th century ignored this basic message.

A related article (BAMS 1906), based on Poincaré's September 1904 address, is a nice short article on the present and future of mathematical physics. It has some interesting early comments about relativity including the following brief definition. "The principle of relativity, according to which the laws of physical phenomena must be the same for a stationary observer as for one carried along in a uniform motion of translation, so that we have no means, and can have none, of determining whether or not we are being carried along in such a motion." (page 187) This article also appeared in the January 2000 BAMS.

Pierpont was a good expositor regarding the mathematics of his day and earlier times. Klein seems to be the ultimate proponent of the importance of communicating mathematics to a general audience and for the importance of a balance in teaching between rigor and intuition (used in the nontechnical sense). The first point is made in his 1894 address (BAMS 1895) about Riemann, and his other points are made in his 1895 address published in BAMS 1896. Moreover, his article on Riemann makes it clear that Riemann relied as much on intuition as on rigor. What I learned from these two addresses is that there doesn't seem to be much new in the world, but we seem to need to reinvent these ideas. The other article by Klein, published in BAMS 1893, makes the case for the modern treatment of some aspects of geometry using group theory. This summarizes his "program" that he started in 1872.

The last chapter is devoted to David Hilbert. One article is Poincaré's long, very thorough review (BAMS 1903) of Hilbert's "Foundations of Geometry," published in 1899. Poincaré had some serious criticisms of the book, some of which Hilbert later dealt with. Poincaré comments that Hilbert's axiomatic development "may seem artificial and puerile; and it is needless to point out how disastrous it would be in teaching and how hurtful to mental development; how deadening it would be for investigators, whose originality it would nip in the bud. But, as used by Professor Hilbert, it explains and justifies itself, if one remembers the end pursued." The other article in this chapter is Hilbert's famous lecture mentioned above.

The editor, Donald Saari, did a fine job of selecting articles that show how the finest mathematicians of the time thought about mathematics and education.

Kenneth A. Ross (ross@math.uoregon.edu) taught at the University of Oregon from 1965 to 2000. He was President of the MAA during 1995-1996. Before that he served as AMS Associate Secretary, MAA Secretary, and MAA Associate Secretary. His research area of interest was commutative harmonic analysis, especially where it has a probabilistic flavor. His most recent work has been on Markov chains and random walks on finite groups and other algebraic systems. He is the author of the book Elementary analysis: the theory of calculus (1980, now in 14th printing), co-author of Discrete Mathematics (with Charles Wright, 2003, fifth edition), and, as Ken Ross, the author of A Mathematician at the Ballpark: Odds and Probabilities for Baseball Fans (2004).

- D. G. Saari -- Introduction
- D. G. Saari -- Spirit of the time
- J. Pierpont -- The history of mathematics in the nineteenth century
- F. Cajori -- Evolution of criteria of convergence
- C. A. Scott -- The International Congress of Mathematicians in Paris
- M. G. Darboux -- A survey of the development of geometric methods
- D. G. Saari -- Fifth degree polynomials
- J. Pierpont -- Lagrange's place in the theory of substitutions
- J. Pierpont -- On the Ruffini-Abelian theorem
- J. Pierpont -- Early history of Galois' theory of equations
- D. G. Saari -- Henri Poincaré
- E. W. Brown -- Poincaré's mécanique céleste
- H. Poincaré -- The relations of analysis and mathematical physics
- H. Poincaré -- The present and the future of mathematical physics
- D. G. Saari -- Felix Klein
- F. Klein -- A comparative review of recent researches in geometry
- F. Klein -- The arithmetizing of mathematics
- F. Klein -- Riemann and his significance for the development of modern mathematics
- D. G. Saari -- David Hilbert
- H. Poincaré -- Review of Hilbert's "Foundations of geometry"
- D. Hilbert -- Mathematical problems

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