Scrunching Mathematics

December 13, 2011

Book Review
Philip J. Davis


Number-Crunching: Taming Unruly Computational Problems from Mathematical Physics to Science Fiction. By Paul J. Nahin, Princeton University Press, Princeton, New Jersey, 2011, 400 pages, $29.95.


Flipping the pages of Number-Crunching, I immediately spied the name Karl Sundman (18731949). This sent me back to a minor thesis I wrote en route to my PhD. In this paper I described one small aspect of Sundman's work on the three-body problem. In his Chapter 5, which is devoted to the three-body problem, Nahin discusses special cases, difficulties of "closed-form" solutions and of computer solutions, and various other ramifications of the problem. Graphics, some of which depict aliens destroying the earth, illuminate the whole. But I'm getting ahead of myself.

Paul Nahin, a prolific and knowledgeable expository writer, is a professor emeritus of electrical engineering at the University of New Hampshire. What he offers in Number-Crunching might be described as a mix of (1) supplementary readings for courses in mathematics, physics, or electrical engineering, (2) "challenge problems" intended as a brain-jogging call to prospective professionals, and (3) a garage sale of mathematical miscellania and esoterica.

Nahin is master of a folksy, talky, jokey, self-advertising, laid-back prose style. Very often, he segues into a didactic mode---we can almost hear the enthusiastic teacher lecturing. The lessons on such pages consist of page after page of algebraic formulas, formulas from calculus, and differential equations, many of them worked out explicitly. All this material is interlarded with a plethora of diagrams, flow charts, electrical circuitry, and their related equations, episodes from the history of mathematics and physics, biographical and autographical material, anecdotes, and gossip. Science fiction---a genre that bores me to tears but that, I admit, has a large and zealous constituency of both readers and writers---comes on stage in a piece titled "What if Newton had owned a calculator." Curiosities, paradoxes, and puzzles make cameo appearances. The pages contain quite a bit of Matlab code, most of it creating plots. Nahin retells the history of early computers for the benefit of those who might believe that the Mark I and the ENIAC burst forth, in the manner of Athena, solely from the head of Alan Turing. Notes and references galore amplify the text, very useful for readers who want to dig deeper.

"The limits of computation"---Section 9.1---is itself limited in that computers are considered only as they exist in the milieu of mathematics and science. The author seems to ignore or forget that computers also exist in the social sphere. A reminder that some of the problems besetting society may not be solvable by computation would be a useful addition here.

Over the millennia, many authors have played with the idea of creating a universal catchall. My favorite, the eponymous library in the short story "The Library of Babel" of Jorge Luis Borges, includes all possible books, in the sense of all 200-page-long sequences of letters.

Nahin, with a clip from one of his magazine articles on the possibility of creating The Universal Photo Album, shows himself to be part of the fellowship of aspiring universalists. The book gets rolling with TUPA and very appropriately concludes with a diagram of the Halting Problem.

Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at philip_davis@brown.edu.



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