A Mathematical Epiphany

I have just had a mathematical epiphany.

As some of you may know, for the past five years I have been studying for a degree in mathematics and statistics with the Open University. However, the first four years of this have mainly focused on the practical application (generally using statistical software) of statistical tools, techniques and proofs, without getting too hung up on the mathematics behind them. The proofs are given without detail; it’s what you do with them that counts.

This year’s course, however, is different. The title of the course is “Mathematical Statistics” and, as the name suggests, is much more about the mathematical theory behind the statistical techniques employed daily by statisticians. In other words, this year’s course is bloody tough.

And I have to confess, I’ve struggled with it somewhat so far. Calculus, my long-term bug-bear, is biting me with a vengeance this year, and it’s been hard-going sometimes to bring myself back up to speed with it.

All of that said, today I had an experience which gave me a glimpse—a snapshot taken through a keyhole, perhaps, but a glimpse nonetheless—of what it feels like to be a professional, or even a proficient amateur, mathematician.

Today I proved a result. A result, I must point out, that has been known for the best part of 70 years, and a result for which regulation undergraduate mathematics is quite sufficient. It was also a result whose answer I was given. Had I not been given this final answer, it is entirely likely that I may not have discovered it for myself at all.

Nevertheless, I did prove it myself, without reference to other texts, cribbing from the course materials, or consulting with that information whore Google. The experience sits in my memory now as though a journey through a new and unfamiliar landscape, seeking out a path towards a final destination whose location is known but to which the route is not.

I was air-lifted into this terra incognita without map or compass; merely a photograph of my destination, in order that I would recognize it when I arrived. I also commenced the journey with a familiarity not of the area itself but of the various features I was likely to encounter.

As I sought a path through this mysterious land, I was, at first, presented with a multitude of possible directions in which to strike out. However, on closer inspection, it was possible to rule out certain possibilities immediately: this path, I could see, led towards a broad, raging river, down in the valley. Unlikely, I felt, that the correct path would require me to ford such an obstacle.  This path, on the other hand, appeared to peter out at the foot of a steep, craggy cliff. Not a good option either.

By these means one can reduce the number of potential routes to explore, but perhaps not down to a single option, and in the end one must simply choose a way, perhaps at random, to explore. Eventually, one will discover whether one has chosen wisely or not. It is perhaps as likely as not, at this early stage, that one will be forced to retrace one’s steps, perhaps right back to the very start point, before setting out in a different direction.

After a while, however, one particular path seemed to become more promising. There were still occasional backtrackings—the path might, for example, run towards an area of boggy ground which could not be crossed and which therefore required circumnavigation. However, this required not a return to the start but merely a return to a suitable branching point.

By little and little, guided by experience, instinct and with a certain degree of trial and error, I picked my way through. And then it happened; as I emerged from a thicket of woodland, I realized that there was, in fact, just a single path laid out in front of me, and from my present standpoint, I could see my destination ahead. From here I was able to divine the entirety of my remaining journey: down this hillside, avoid the outcropping to the north, cross the stream by the stepping-stones, through the stone wall at the style, and on home.

Those final few steps were taken with a sense of excitement and awe the like of which I have never before experienced. Reaching my goal, I looked back at the journey taken and realized what an event of sublime beauty had just occurred.

I was not the first—not even the million-and-first—person to have walked that route; I had not required climbing equipment, specialist clothing or navigational aids with which to traverse the land. Yet I had been able, by repeated applications of induction and deduction, to find my way to a given omega from an unknown alpha. This, I realized, was the daily pleasure and privilege of the mathematician: to live and work (and play) in this landscape, explore its ways, its paths, its routes and roads, to discover new routes yet untrod by human foot; to bring familiarity to the unknown, and to map the terrain for future travellers.

It is a land in which I could happily pitch my tent and explore forever.