The maths may not add up, but it's a model tool kit for the new philosophy

The Nature of Mathematical Modelling
September 24, 1999

In his introduction, Neil Gershenfeld makes great play of his attempt to capture the universality of "mathematical modelling". He states: "I hope that my presumption in reducing whole disciplines to ten or so pages apiece of essential ideas is exceeded by the value of such a compact presentation. For my inevitable sins of omission, commission and everything in between, I welcome your feedback: nmm@medi.mit.edu ."

It will shortly become clear why your reviewer finds this book something of a curate's egg and thus would approach this address with trepidation.

Most people who research, teach courses, write books or, better still, lead groups in mathematical modelling, have a different motivation from that of Gershenfeld. The Nature of Modelling with Computers describes the spirit of his approach more accurately than the chosen title, and this review inevitably contrasts these two approaches to quantitative analysis.

Let us take the traditional mathematician's modelling first. This means taking a practical situation, which could arise in any context, social or scientific, business or leisure, and endowing it with a theoretical mantle that allows quantitative assessment of what can happen in that situation.

By definition, this requires the modeller to have sufficient familiarity with the practical background for the dominant mechanisms to be identified and written down so as to make sense to a reasonable human brain, and enough understanding of the necessarily mathematical nature of the model to know whether it is feasible to make reliable numerical predictions therefrom.

This interdisciplinary activity demands many skills, the most important of which is a feel for mathematical statements of all kinds, their logical interplay and their robustness. More precisely, when the model comes as an equation or inequality, knowledge is required about its well- posedness - does it have a solution, is the solution unique, does it depend continuously on the data? These questions have puzzled talented researchers for centuries and have spawned some of the most delicate mathematical questions of all time. But the results of these investigations immediately refute the two most pervasive themes of this book: that linear models (crudely, ones whose solutions can be superimposed to generate new solutions) are "easy", and that discrete models (crudely, where the solution only exists at integers) are preferable to continuous ones (where the solution depends on a continuum of numbers). I cannot adduce the arguments for the misguidedness of these themes here but, concerning the latter point, maybe it is only fair that those who spend their lives agonising over statements such as "let n = infinity" should be rewarded with a continuous model that is infinitely safer than a discrete one.

The first 100 pages of the book do, in fact, purport to deal with these issues, but the less said about them the better. They contain profound technical inaccuracies, abuse of mathematical terminology and historical errors and may enrage experts in mathematical analysis, numerical analysis or statistical mechanics.

But these pages are not the heart of the book and they could be omitted, save for the admirable sections on statistics, cellular automata and lattice gases. These pave the way for the good things that follow in the middle 100 pages, which are quaintly headed "observational models". Together with the obligatory associated software and programming ideas, they are what make this book distinctive and valuable. They reflect the new modelling philosophy in which, rather than taking a situation and synthesising a model, you start with data, often a surfeit thereof, and then decide what you can do about the situation. (Of course intuition concerning the situation is still worth its weight in gold, but it is no longer the starting point.)

The tools switch dramatically from, say, laws of physics or biology and calculus and numerical analysis, to curve and surface fitting, wavelets, basis functions, neural nets, genetic algorithms, filtering and signal processing, and culminate in time series, linear and nonlinear. With these come descriptions of all the dedicated mathematical software packages, and the skills required for network programming (including, of course, Java).

It is in these pages that the author's ability to condense vast chunks of statistics, computation and artificial intelligence really give the reader value for money. They provide so much more than a scientific dictionary and can only inspire and comfort any researchers whose computers and/or floor space are brimful with data.

It would be wrong for most mathematicians to use this book to teach mathematical modelling. Nonetheless I will keep it on my shelf as a valuable aide-memoire for all the intricate methodologies of modern data processing, most of which are very physics-oriented.

Had this book been written by a physicist say, 30 years ago, there would have been much more emphasis on particles, black boxes and linear models than on computer science and nonlinearity.

On the other hand, the 1960s mathematician would probably have stressed mechanics and mathematical biology, and would now say more about finance, industry, medicine or materials science. That the two disciplines really do only meet intimately in areas such as field theory, relativity and geometry should perhaps have persuaded Gershenfeld to choose a title in which the word mathematics did not appear.

John R. Ockendon is lecturer, Centre for Industrial and Applied Mathematics, University of Oxford.

The Nature of Mathematical Modelling

Author - Neil Gershenfeld
ISBN - 0 521 57095 6
Publisher - Cambridge University Press
Price - £24.95
Pages - 355

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