# Alien Encounters - Suggestions for Use

Author(s):
Gavin Hitchcock

This play is intended as an evocation of the actual historical story - a dramatic reconstruction of both the human controversy and the dialectic of ideas, and therefore draws upon primary sources for much of the dialogue. Since the evolution of ideas is inextricably bound up with the evolution of language, I have resisted the temptation to transcribe the dialogue into more contemporary forms. This means that the language, while full of charm and humour for some, may be quite challenging for others. As with science fiction, and the period costumes and sets of a Shakespeare play, this has the positive effect of encouraging temporary suspension of current modes of perception, thought and expression, and drawing the reader/listener into a world remote from our own. Some such transportation seems essential for appreciating the development of the ideas, but the language may prove too great a barrier for some people. The dilemma here is similar to that of updating Shakespeare. How much to abridge, change, or explain, must be left to individual choice, taking account of particular class situation and level. I would only counsel that care be taken not to do violence to the evolution of the ideas.

The two acts can stand independently, and the first scene of the second act is probably best abridged or omitted for classroom purposes. (It is given for completeness, and has intrinsic interest in demonstrating what students of the renowned lecturer Saunderson presumably had to sit through!) The somewhat lengthy prologues and epilogue may be omitted, used in part, or treated as background reading. Their function, vital for full appreciation of the significance of the drama, is to sketch the intellectual background against which the achievement of the mathematical pioneers is to be estimated. In any classroom presentation of the play, a narrator' may be used to give a brief introduction to each part, based upon the following items.

• As Prologue to Act 1, Blaise Pascal pronounces eloquently upon the sheer audacity of the new breed of scientists in Europe, in their seeking to lay hold of Nature and grasp her long-preserved secrets by intellect and experiment. Pascal's own abdication from the pursuit of mathematics occurred in response to an intense mystical experience of God, after which he resolved to devote himself to higher theological and spiritual priorities, as he perceived it. His theological and religious writings have become classics of both French literature and Christian apologetics. In this slightly edited extract from his Pensées, Pascal concludes by heaping scorn upon people who would presume to contemplate taking four from nothing and getting "minus four"!
• However much truth there was in Pascal's passionate warnings against the growing hubris, or intellectual pride, which culminated in the self-conceit of The Age of Enlightenment', negative numbers were not suddenly invented by some too-clever mathematician lacking in a proper humility! It happened, rather, that a new respect for Nature and a new trust in Nature's rationality had grown up, together with a new confidence in humankind's ability and mandate to explore and make sense of Nature, and even manage and exploit her resources - all of this encouraged by that very religion which Pascal espoused. It was in the course of the ensuing sanctified and sustained endeavour to uncover Nature's secrets to the glory of God and the benefit of humankind, employing the language and tools of mathematics, that a novel and refined instrument came gradually into being. The negative numbers insinuated themselves irresistibly into the consciousness of mathematicians, and justified their "reality" by their usefulness.
But the process of domesticating these `unruly' alien things and accepting them as legally accredited and law-abiding citizens of mathematical country, was a difficult and painful struggle. Act 1 highlights the "headaches" suffered by the pioneers, and illustrates the various responses to mathematical paradox and crisis prevailing in the late seventeenth century.
• In the Prologue to Act 2, the spirit of Pascal expresses a revised position on the matter, with the benefit of mature hindsight and divine insight. For, in the meantime, the mathematicians have tackled the extraordinary difficulties with characteristic courage and tenacity; and Pascal himself has learnt some lessons about the Creator's attitude to his creatures doing mathematics and science.
• Act 2 gives three glimpses into the ways these strange new numbers were introduced, and their behaviour justified, to the succeeding generations of the eighteenth century. The "law of signs" is presented, using various shades of rigour, heuristic, example and physical intuition, by Saunderson, Euler and Laplace.
• In the Epilogue, Jean d'Alembert looks back, with the optimistic and practical eye of the Enlightenment, at the long struggle to accept the negative numbers. He chides his fellow Europeans for their late emergence from the "metaphysical fog", and expresses his impatience with the philosophical quibblers who would argue interminably about the nature of negative numbers, when their uses and value in the new "age of mechanics" are unquestionable.

At the end, in addition to notes and references for the play, there is a set of questions and exercises intended to encourage reflection on the play, which are mainly at a level suited to pre-service or in-service teacher training workshops. Some of the issues arising have relevance for teachers far beyond the topic of negative numbers; among such I should mention the topic of convergence of sequences and infinite series.

### Disclaimer and acknowledgements

Where sentences, paragraphs or whole speeches are based closely on primary source material, these are appropriately referenced. In general, the words spoken by the characters must not be ascribed to the historical persons, who may not have conversed in the flesh (or even in writing) at all. The dialogue is the playwright's attempt to reconstruct the historical interaction of their ideas.

As in a previous article, "Teaching the Negatives, 1870-1970: a Medley of Models" (For the Learning of Mathematics: FLM Vol. 17, no. 1), I would like to acknowledge the inspiration of Abraham Arcavi and Maxim Bruckheimer, whose work in developing historical source-work collections for use in teacher training in Israel was described in FLM Vol. 3, no. 1. For more of the philosophy and motivation behind my own work in dramatizing the history of mathematics, see my article, "The Grand Entertainment", in FLM Vol. 12, no. 1.    Dr A Gavin Hitchcock

Gavin Hitchcock , "Alien Encounters - Suggestions for Use," Convergence (May 2011)