Number-Crunching: Taming Unruly Computational Problems from Mathematical Physics to Science Fiction

September 29, 2011

I was just about to deliver this review of Number-Crunching by Paul Nahin, the electrical engineer and prolific popular science writer, when two news stories within 24 hours caused me to opt for a dramatic rewrite.

The first was the widely reported criticism by Eric Schmidt, the executive chairman of Google, that the UK has lost its competitive technological edge because of "a drift to the humanities". The second was a statement by Stephen Spurr, the headmaster of Westminster School, that "although we retain A-levels in maths and science, we have found the (Cambridge) Pre-U is much more interesting, particularly in the humanities subjects".

Until these statements hit the headlines, the intended readership for this book was not immediately obvious. On the face of it, Number-Crunching introduces readers to the application of software (in this case, Matlab) to a set of exemplar problems in the physical sciences that challenge analytical solution techniques. In the process, Nahin covers topics including: bounds on the counting function for the Riemann zeta function; Richard Feynman and Fermat's famous Last Theorem; the solutions of cubics; infinite ladders of resistors; 2D temperature distributions leading to finite differences; the Monte Carlo method; the Fermi-Pasta-Ulam numerical experiment; eigenvalues; non-linear springs; the three-body problem leading to chaos; electrical circuits; simulation and a leapfrog problem. He concludes with some observations on science fiction.

The book is written in a highly accessible style (often against a backdrop of the author's personal history). However, members of the scientifically interested public might not be familiar enough with the necessary mathematics to absorb it all, while experienced computational scientists might find the computing aspects rather straightforward. In places, possible shortcuts in the analytical work are missed. In other places, the author's engineering background comes through. For example, after deriving analytically a sinusoidal solution for a non-linear spring problem, Nahin announces that his work is going to "show" that there really is such a solution - by numerically solving the problem!

Nahin displays an apparent US-centric view of the history of computing. He praises ENIAC at length as the first electronic computer, while completely overlooking the existence of Tommy Flowers' Colossus and the Manchester machines.

Number-Crunching is packed with copious notes and references and augmented by significant challenge problems that take the reader beyond the text and which would make good undergraduate projects. But it was at this point that the recent news stories pricked my conscience.

Nahin's aim is clearly to convey enthusiasm for the subject to a younger reader and to give a glimpse of what is technically possible, not merely to offer an exam-based textbook. He looks to convey the excitement that he and many of us had when first attracted to the physical sciences as we were growing up - the excitement at the realisation that, given a few tools, even an awkward teenager can make quantitative statements about the world. Any minor disagreements over the book's presentation should not detract from its appeal in this respect.

We can't expect ministers with degrees in English, modern history or philosophy, politics and economics to understand the need to allow time for unconstrained study beyond a crowded exam syllabus in the physical sciences. But the headmaster of Westminster also misses the point. It's not that more challenging exams in the humanities are needed. He should encourage his already excellently schooled pupils to read technical books such as Nahin's to broaden and to stimulate their interests and to broaden their expertise: many A* undergraduates struggle to apply their crammed knowledge to real-world problems. If the headmaster could achieve that, more of his pupils might be galvanised to go on to study natural sciences, engineering, mathematics or technology.

Who knows, perhaps inspired by Nahin's book, one of these pupils could even go on to become the next Eric Schmidt. For the future health of the UK economy, I certainly hope so.

Number-Crunching: Taming Unruly Computational Problems from Mathematical Physics to Science Fiction

By Paul J. Nahin. Princeton University Press. 400pp, £20.95. ISBN 9780691144252. Published 22 September 2011

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