reviews

Chapman - Hall Series
Chapmari and Hall ire bringing out a new series of moderately priced mathematics books; in the main undergraduate texts. The book "Elementary Differential Equations" by R. L. E.Schwarzenberger was reviewed in MANIFOLD-6. It deals with the subject from a geometric, intuitive point of view. We review the other books so far published below.

Rings, Miodules and Linear Algebra. - B.Hartley and T.O.Hawkes. £2.25
This book is pre-eminently an undergraduate text book and is in fact derived from a second year course given at Warwick University. No effort seems to have been made to make it useful to any other class of reader.

Following a fairly lavish introduction to the theory of rings and in particular ideal domains the general theory of modules is developed, culminating in the proof of structure theorems for finitely generated modules over Principal Ideal Domains. Finally these results are applied to problems in group theory and linear algebra.

The general approach is one of rigour, clarity and care. Topics are introduced slowly and explained in great detail. Thus every student who has patience and perseverance should eventually appreciate the material. treated here. On the other hand it cannot be said that this approach is likely to generate sufficient interest and excitement without some outside stimulation. The layout and use of notation are such that any attempt to use this book for reference seems doomed to frustration. As a text for a lecture course essentially similir to that on which it is based this book is, therefore, very good, but the reviewer cannot reccomend its use in any other circumstance.
Robin Fellgett.

A First Course on Complex Functions. - G.J.O.Jameson. £2.25.
The opening paragraph promises "a rigorous coverage of those topics (and only those topics ) that, in the author's judgement, are suitable for inclusion in a first course on complex functions. Roughly speaking, these can be summarized as being the things that can be done with Cauchy's integral formula and the residue theorem." It conveys the raison d'etre of the book and also the author's prejudices.

No form of the Jordan curve theorem is assumed; Cauchy's theorem is proved in detail (and completely rigorously) for star-shaped sets only and then applied to series and singularities. A chapter is devoted to the application of the residue theorem; as well as the usual methods for avoiding singularities there are ingenious paths consisting of parallelograms. Winding numbers are developed in the final section and a complicated path is chosen for illiistration of the concept.

There are lots of excercises in the book but, unfortunately, no answers. The cross-referencing is quite good, particularly in revealing connections between various parts of the book. I only found one mistake, on page 64 where the suffices should have been upper rather than lower on the diagram. Theorems, lemmas, etc. are stated precisely and proved very economically in a rather terse style. As a whole the book is very rigorous and exact: the promise (and only the promise) is fulfilled. There is some "chat" in between the results yet something still seems to be lacking insight.
Ramesh Kapadia

A Preliminary Course in Analysis. - R. M. F. Moss and G.T.Roberts. £1.80.
"A Preliminary Course in Analysis" is intended as a first year undergraduate introduction to real analysis. Written by two lecturers on the basis of their experience at Hull, it is a very solid, thorough and friendly book. The authors have gone to a lot of trouble to introduce the basic concepts painlessly and with as many intuitive props as possible. The most noticeable characteristic is that differentiation is defined very simply and completely independently of the concept of a limit. But while this minimises the tedium of a more conventional approach it takes one away from the familiar lines of secondary school calculus. For those who want the real truth about A-level maths a direct redefinition of a limit would be more relevant and satisfactory.

The big weakness of the book, however, is that it it intellectually very unrewarding. To more advanced readers it is a novel readable introduction, but to the beginner it fails to bring out the significance of the results and above all does not seem to lead anywhere. But that said the book is still the most readable, if unexciting, introduction currently available.
John Langdon.

WFF'N PROOF: logic tutor, distributed by Science Systems, 173 Southampton Way, LONDON S.E.5. Basic kit as described £3.75, simpler version £1.25. Educational discount/bulk discount available::refund and return guarantee.

The notion that, if P and Q are propositions, P implies Q, may be expressed in a number of ways.

P --> Q

is probably the most familiar to students of mathematical logic in English schools or colleges. One failing of this system of notation is that the compound implication:

P --> Q --> R

is ambiguous and needs brackets:

(P --> Q) --> R or P --> (Q --> R)

A notation that overcomes. this ambiguity is associated with the Polish logician Lukasiewicz, and has become known as the Polish system. This places the symbol for implication "C" before the two operands: Cpq. There is thus no ambiguity in the bracketless forms: CpCqr or CCpqr.

Perversely, the most important form of the Polish notation is reverse Polish, where the operator follows the operands, a notation ideally suited it was discovered to the logic of digital computers. In logic, a compound proposition is commonly known as a WELL-FORMED FORMULA - the WFF of the title - which might be seen simply as a "legal" logical proposition, formed regularly from simpler propositions and the connectives of implication, equivalence, conjunction,alternation, and the operation of negation. PROOFs are legal sequences of WFFs, each inferred from the previous elements of the sequence by the rules of inference laid down by the svstem.

This makes mathematical logic sound very much like a game, and the manufacturers of WFF'n PROOF have marketed it precisely as this: a game of some chance in which the rules are those of mathematical logieg and only the scoring is not embedded in the logic in a natural way.
WFF'N PROOF, well- but expensively-packaged, consists of an instruction manual, a minute-timer, and 36 dice, half labelled with the connective symbols (unfortunately these are C-A-K-E, and N-for negation, which do not make for easy memorising) and the letter R, the other half with the symbols P,q,r,s, agreed as propositions and i and o.

There are 21 games, graded in difficulty, but of several main types: the early games are all variations on the idea that if a mixture of these dice are thrown, some WFFs may be created from the symbols that come to rest on the upper faces pf the dice. After only a couple of lessons, almost all of a class of Sixth-Formers could recognise WFFs and reject illegal combinations of symbols - a danger here is that it is possible to do this purely mechanically, with no thought as to any possible meanings: this is something that one must determine as desirable or not, and act as a teacher accordingly.

The most important bulk of the games are arranged on the following lines: a new rule of inference is learnt, and then proofs are constructed from the WFFs that can be constructed from the dice thrown - as a new rule is learnt, the scope of proofs broadens. In what is basically an entertaining idea. and a useful and instructive part of a serious logic course, there are two flaws. The first is the price, which even with a discount offered to schools and colleges seems exorbitant for 36 cubes, an egg-timer, some playing-mats and the rule-book (detailed though it is!). It is of interest that the guinea pigs who spent a term playing WFF'n PROOF for us guessed the price to be of the order of 30/-.

The second flaw is the handbook, which to be blunt is no damn good! It is a sober production, in tortuous American, which falls between two stools: it is far too obscure as a logic text - even for a teacher with some minimum acquaintance with the subject - and the rules as explained are too complex, and hidden too deeply among a maze of verbiage: one word is never used where three or four will suffice. If manufacturers take notice of reviewers' comments, one of the guinea-pig's ideas might be useful: the set needs two manuals - an optional logic text (of which there are many available, frequently cheaper than this whole set) and a simpler, more concise set of rules, or if this is impossible, a pamphlet summarising the rules, so that points of dispute can be rapidly checked. It took all of a 45-minute period to understand the basic PROOF game in our field-trial!

Some of the criticisms are negligible - a good teacher could very easily adapt the game to avoid these: experience suggests that as pupils progress, more than one rule of inference could be taught at a time - clearly a slavish trip through the text, game by game is very unimaginative, and one hopes in any case that a teacher embarking on this course would use it as an aid rather than his whole support for a course in logic.

It would be interesting to know why the cost of this set is so high: it is a worthwhile asset, that will repay the effort put into it, and could figure in the very simplest of logic courses: at least one school of my knowledge uses it as an O-level aid, but it is by no means below university standard. As a game for the intellectual and his friends, I'm afraid it is out - this is squarely (and rightly) a teaching aid!
John Jaworski

FOXRAB is concerned with the ecological balance between a population of R rabbits and F foxes. Foxes survive by eating rabbits, and in a time of scarcity will naturally die off. Without the interference of the foxes, the vegetarian rabbits would increase steadily: the interaction of species acts in the foxes' favour and the rabbits' disadvantage. These facts are summed up in the equations:

dR/dt = 4R - 2RF
dF/dt = RF - 3F

where the respective coefficients are of course variable, and have been given these values almost arbitrarily.

The idea of FOXRAB (open to anyone to make more specific) is that as the meeting of foxes and rabbits (which is always to the detriment of the . rabbits) is a chance happening that varies in probability with the total numbers involved, the total number of rabbits killed should be determined by dice-throws, converted to kills by consulting a table based on the respective numbers of each. Other than this, at each move of the game, the rabbits increase by 4 times their number, less any determined as killed, while the foxes increase at half the rate of the rabbit decline, less 3 times their original number, which die from hunger. A population of 3 rabbits to 2 foxes is stable in the sense that if infinitely divisible rabbits are allowed (making the model continuous and hence differentiable), dR/dt and dF/dt are both zero. In practice, 30 rabbits to 20 foxes suffices, and the first population to die out loses! We would be interested to see anyone's attempt at codifying and producing a game based on this most interesting idea.