Archive for the ‘Calculus’ Category

A Brief Tangent

October 24, 2007

I would like to get temporarily sidetracked. The straight road is often not always the best road, at least not when the journey matters at least as much as the destination. Basho often took sidetracks, and they often provided some of the highlights of his journeys. Wandering off track gives you the opportunity to learn a little more about the country that you are travelling through; that’s precisely the sort of sidetrack we’re about to take.

We’ve spent some time contemplating and discussing the intricacies of the infinite. We started off with a very natural abstraction, and quickly got lead into a mire of technicality and complexity. With a little work we came through unscathed and ended up with a new appreciation for the unfolding beauty and complexity that we were lead to. All of that, however, was a matter of “head in the clouds” contemplation as to what it truly meant to be infinite, and what the continuum actually was. Now that we have that understanding it is time to wander off the path and explore the countryside that such understanding begins to open up for us.


A Transfinite Landscape

July 2, 2007

Problems that involve infinity have a tendency to read a little like Zen koans. Take, for example, this problem: Suppose we have three bins (labelled “bin A”, “bin B” and “bin C”) and an infinite number of tennis balls. We start by numbering the tennis balls 1,2,3,… and so on, and put them all in bin C. Then we take the two lowest numbered balls in bin C (that’s ball 1, and ball 2 to start) and put them in bin A, and then move the lowest numbered ball in bin A from bin A to bin B (that would be ball 1 in the first round). We repeat this process, moving two balls from bin C to bin A, and one ball from bin A to bin B, an infinite number of times. The question is, how many balls are in bin A and how many balls are in bin B when we’re done? Think carefully!


Paradoxes of the Continuum, Part II

May 1, 2007

Mathematical arguments can be very persuasive. They lead inexorably toward their conclusion; barring any mistakes in the argument, to argue is to argue with the foundations of logic itself. That is why it is particularly disconcerting when a mathematical argument leads you down an unexpected path and leaves you face to face with a bewildering conclusion. Naturally you run back and retrace the path, looking, often in vain, for the wrong turn where things went off track. People often don’t deal well with challenges to their world-view. When a winding mountain path leads around a corner to present a view of a new and strange landscape, you realise that the world may be much larger, and much stranger, than you had ever imagined. When faced with such a realisation, some people flee in horror and pretend that such a place doesn’t exist; the true challenge is to accept it, and try to understand the vast new world. It is time for us to round a corner and glimpse new and strange landscapes; I invite you to follow me down, in the coming entries, and explore the strange hidden valley.