The Anatomy of a Discovery in Mathematics Research

What are the activities that constitute mathematical research? Various well-known mathematicians such as Poincare, Hadamard and Poly a have given descriptions, but these tend to be general reflections on the process considered after the event. The aim of this paper is to describe, within practical limits, the thought processes in a specific piece of research as they happened. It highlights the research activities of the author over ten days, relating these to the previous development of ideas over a period of years and the developments which followed. There were flashes of insight, the coming together of previous experiences, analogies both useful and false, and intuitions having the ring of truth which proved to be embarrassingly inaccurate. After grappling with ideas which seemed complex at the time, the final product was a theory so inevitable that it seemed like a mathematical truth discovered. A year later it seemed naive, even trivial and, when presented to students, they found it straightforward, simple and obvious. But the tortuous route by which the author came to build up the theory is a story worth telling, if only because the way that it actually happened (as witnessed by notes taken at the time) was far less glamorous and logical than the memories that were subsequently recalled. In some instances memories a year later were quite different from the evidence as concretely represented by the notes. It seems that we remember the salient features of a past event and reconstruct the detail when required. In this way our recollections are far more rational than the actual processes. Even in the telling of the story it has been necessary to select material and so a certain amount of rationalisation has inevitably crept in. In doing this I have attempted to give an overall impression of the research activity and, within this programme, select certain themes that intertwine together as the work progresses. I have written of myself in the third person, as a separate observer might have done. This allows me to talk of the incorrect turns I took without (too much) embarrassment.