wandering the fractal hinterlands
July is an interesting time in the academic year. The undergraduates have largely left for the year, having received their results from the year, the masters students are deep into their dissertation writing, and the doctoral students are lost in the labyrinths of their projects. Committees have their last meetings for a few months, and we are spending part of our time winding things up.
The university is not quiet; various groups and conferences are visiting campus. There are important moments in the cycle of the academic year that take place over summer, such as the core days of the undergraduate admissions process in mid-August and the supplementary examination period in late August and early September. But it is quieter than during the year.
As such, it’s a good time for reflection, on the year (and years) past, and on the year to come. Amidst this reflection, an idea that has been marinating over the year bubbles to the surface.
There are many aspects of life that involve taking a varied and multidimensional space, and sorting all its points into a handful of buckets. For some of these points, the bucket will be an obvious bucket, whereas for others, the determination of which bucket becomes complicated.
This complication arises from the observation that there will be points in this multidimensional space that are very close together, that nonetheless end up in different buckets. And so each bucket will contain points where a reasonable argument, and a different sorting algorithm, would result in that point being in a different bucket.
Why does this matter? It matters because it’s the answer to a fundamental question, which is, is there a fair way of doing this sorting into buckets? We take a rule we think of as fair, and we will always note that there are points where we aren’t comfortable with the algorithm’s decision. But the issue is that every rule will come up with such points, unless we do something reckless like just using a single bucket (and so making any distinction between the points).
This is something of a slippery idea, because it means that there won’t ever be a way of sorting that everyone agrees is fair for all points. Mathematicians have become accustomed to things that cannot be done, impossibilities, and this is something that most people don’t like to admit. It’s one that I want to explore and so I’ll come back to it from time to time, but I think this is perhaps enough for the moment.
multijimbo 