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.
