Editorial

Zimbabwe's Parliamentary Elections are over, with all the excitement and trouble they brought to many. An article in this issue discusses how statistical methods are used to estimate whether elections are `free and fair' (page ). We offer sympathy to those of our contributors and subscribers who suffered the temporary closure of their classes or schools, but we hope all is back to normal, so that we can all get on with contributing to the positive development of the country and well-being of all its people- in particular through the subject we celebrate with this Magazine: mathematics!

When human affairs enter a crisis period (personal, communal or national) it can be tempting to run to the world of mathematics as a peaceful refuge, a haven of eternal, unchanging ideas and values - for surely mathematics is not subject to crises, it is exact and certain, aloof from controversies, conflicts, passions and revolutions! Isn't it? Well, - YES (read the beautiful quotation given on page in the article On Doing Mathematics by the Chairman of the UZ Maths Department), and NO (read on!).

It is true that mathematics conferences worldwide are distinctive for the almost total absence of acrimony and dispute. There are good-humoured disagreements about conjectures (would-be theorems which have yet to be proved or disproved), and occasionally some strong feelings about priority (who thought of/discovered/proved something first). But, on the whole, mathematicians are a peaceable and harmonious community, and Mathematics Departments often the happiest places in the University! (- what about schools, I wonder?) Why is this? Perhaps it's because mathematicians have a common love, and are generally joined together in a common quest, committed to a common end, and submitted to a common rule or authority - that of logical demonstration within the tight, formal, explicitly defined standards of their subject and their time.

But this is only half the story .... (please turn over for the other half)

Mathematicians are human

Read again exactly what we have just said on page 1 ... `Of their subject' - of course! We were talking only about their professional, intellectual activities, not about the infinitely richer and more complex world of their personal relationships; for mathematicians are also citizens, brothers, spouses and daughters.

`Of their time'? Yes! That's the thing that many people don't realize about mathematics - that standards of `proof', of rigour and demonstration, do change over time. Right now, there is a struggle within the mathematical community to come to terms with the growing realization that:

1) brute computer-power may be an indispensable part of some proofs (such as the four-colour map theorem - that four colours is sufficient to colour any geographical-type map in the plane); and

2) some proofs may be irreducibly long, and dependent on so many other results which themselves are proved using so many other results still, that it is no longer possible to demand checking of every step of a proof in the way that used to be considered essential to mathematics. Examples of this are the gigantic cooperative proof (sprawling over many journals and many years) of the classification of finite simple groups (completed in the early '80s), and the proof of Fermat's Last Theorem (achieved in the '90s). We trust these results, finally, because the mathematical community has endorsed them by consensus.

Dependence upon machines and human authority?! This seems such a radical change from an older view of mathematics that it can be (and has been) called a crisis (dubbed the ``death of proof''). And this is just the most recent of many major crises in the development of mathematics, each of which changed the standards of proof and the very concepts permitted! Among them are the following:

Mathematics has crises too

the crisis of the discovery of incommensurables (c. 450 BCE);
the crisis of integrity in the foundations of calculus (18th c.)
the crisis of reality of mathematical objects (18th-19th c.);
the crisis of identity of mathematics (discovery of new geometries and new algebras, early - mid 19th c.);
the crisis of intuition (mid - late 19th c.);
the crisis of infinity (over Cantor's transfinite set theory, late 19th c.);
the crisis of certainty (arising from the foundations of set theory, early 20th c.);
the crisis of decidability (from 1930's).

Here's GOOD NEWS for all readers going through personal crises, and also for Zimbabwe in its national crisis: even in the most abstract and exact of all sciences - mathematics, crises are not only inevitable, but can be wonderfully fruitful. Have you seen how tender new grass covers the earth after a bush-fire, and lush greenery issues from the decomposing organic matter of the compost heap? As pure metal emerges from the refinery fires, so sacrifice and struggle are the training grounds for all great creative achievement. ``Unless a grain of wheat falls into the ground and dies, it remains alone.....'' You can guess who said that. Here is a personal testimony from the great Cambridge historian Herbert Butterfield:

It would seem that one of the clearest and most concrete of the facts of history is the fact that men of spiritual resources may not only redeem catastrophe, but turn it into a grand creative moment ... in general the highest vision and the rarest creative achievements of the mind must come from great internal pressure, and are born of a high degree of distress. In other words, the world is not merely to be enjoyed but is an arena for moral striving ... The purpose of history is not something that lies 1000 years ahead of us - it is constantly here, always with us, forever achieving itself - the end of human history is the manufacture and education of human souls. History is the business of making personalities, even so to speak by putting them through the mill.

Famous disputes between mathematicians

I will close by pointing you to the story of one of these disputes in the following article, celebrating a great mathematician who stands as an example of how to respond nobly to personal attack, and how to transcend some of the petty divisions that rend the human family.