reviews

MATHEMATICAL LOGIC WITH SPECIAL REFERENCE TO THE NATURAL NUMBERS
by S.W.P.Steen. CUP 19-72, £15-00. Reviewed by Tim Poston

Mathematical publishing is a curious business. In other areas one feels that publishers work from a concept of the market for their books: their ideas of it are inevitably more inaccurate but they do exist. Other factors entirely seem to operate in producing, for instance, this book.

Who exactly (libraries apart) is expected to spend £15 on a thoroughly unexciting grind through the setting up of the natural numbers from pseudo-primitive notions of counting? Not undergraduates, at a price like that. Not research students either; a logician will already know enough to have more specialized needs than this massively expanded undergraduate course could fulfil. The expert would look at it, perhops, to see the approach, but why buy it? So who is it for? Nothing in the book or the dust jacket says anything about an intended readership, but on reading the preface it becomes clear that it is for the author. For years he had itched to do the natural numbers right and at last on retirement had the time. Hence this massive tome, mixing a pedestrian setting-up of all the relevant machinery with dubious philosophy and psychology.

(For instance, 'Thoughts primarily are languageless.' (p.4); perhaps true, but sufficiently many experts would contest it to require that it be argued for. Bold assertion is not enough.)

Steen in mystified as to why the Greeks didn't develop set theory etc., on the grounds that they had the necessary tools. He does not notice that they did not have the necessary problems: the 19th century developed the calculus to the point that they knew they were uncertain of Truth - for instances the parallel postulate was neither True nor False - and they had to go deeper. Just so, with slave labour plentiful, Hero's steam engine did not produce an industrial revolution.

Clearly, satisfying oneself about the natural numbers is as good a retirement hobby as growing roses. but exactly why did Cambridge publish it?

At £15, this book in a white elephant. At £1.50, I would still spend the cash on something else. 1 find it worth, fairly precisely, the effort of writing this review. Fo:r which I have got it.

ELEMENTS OF FORTRAN STYLE: TECHNIQUES FOR EFFECTIVE PROGRAMMING
Charles Kreitzberg, Ben Shneiderman; Harcourt Joyanovich.
COMPUTERS AND COMPUTATION
Readings from 'Scientific American'. intr, by Robert Fenichel, Joseph Weizenbaum;;W.Freemaln $4.95. Reviewed by John Jaworski

Every beginning computer programmer at some stage, comes across two mathematically equivalent statements in his computer language and is told that one of them is 'preferable' as it is more 'stylish'. The same programmer will later be dismayed to discover that this 'style' in no more than a loose bunch of prejudices and that he will not find a formal collection of techniques but will be expected to accumulate them as and when he can.

The first book is, as far as I am aware, a first published attempt to treat this question of style as an important and separate subject. Dr. A.C.Day, of University College, in a mimeographed set of notes on 'Non-numerical Techniques',justifies this approach by the existence of even one programmer who did not know these 'well-known tricks',

Kreitzberg and Schneiderman is a competent book. It identifies five areas where style in important - in reducing computational time, in reducing storage requirements, in improving accuracy, in documenting programs and in the design of programs. In each of these areas it has something to say that will be found useful.

It is liberally sprinkled with examples, and every chapter ends with a check-list. The examples purport to be photographic copies of computer listings, which is a pity as a number of them would not survive the simplest of syntax checks had they ever been run on a real computer. (e.g. P.77 closing parenthesis missing;etc.)

Nevertheless, it is a long time since I came across a computer book that I would unhesitatingly recommend, and this is indeed one such, even to users of languages other than FORTRAN.

The collection of articles on computing is taken from the pages of S.A. over the period 1950-1971. Most of the articles one remembers from that period appear here - indeed, I cannot recall anything left out. Each section has an introduction, bibliographies and author profiles are collected at the end of the book, and an index is provided.

The articles themselves are straight reprints of the originals (errors as well - e.g P71, missing brackets in FORTRAN statement). It is well to read these articles with their dates in mind: No. 13 refers to Hilbert's Tenth Problem (see elsewhere in this issue for new developments).

MARTIN GARDNER'S SIXTH BOOK OF MATHEMATICAL GAMES
Freeman $9.95.
Twenty-five chapters this time, updated from Scientific American columns, ranging from the helix to 'extraterrestial communication' calling in on graph theory, Pythagoras, bouncing-ball computers and the cycloid in passing. Have fun!