Archive for the ‘General Mathematics’ Category

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 I

March 27, 2007

Infinity is a slippery concept. Most people tend to find their metaphorical gaze just slides off it, leaving it as something that can only ever be glimpsed, blurry and unfocused, out of the corner of their eye. The problem is that, for the most part, infinity is defined negatively; that is, rather than saying what infinity is, we say what it is not. This, in turn, is due to the nature of the abstraction that leads to the concept of infinity in the first place.

The ideas of succession and repetition are fairly fundamental, and are apparent in nature in myriad ways. For example, the cycle of day and night repeats, leading to a succession of different days. Every such series of successive events is, in our experience, bounded — it only extends so far; up to the present moment. Of course such a series of events can extend back to our earliest memories. Via the collective memory of a society, passed down through written or oral records, it can even extend back to well before we were born. Thus, looking back into the past, we come to be aware of series of successive events of vastly varying, though always bounded, length. We can then, at least by suitable juxtaposition of a negation, form the concept of a sequence of succession that does not have a bound. And thus arises the concept of infinity. Is the concept coherent? Does succession without bound make any sense? With this conception of infinity it is hard to say, for we have only really said it is a thing without a bound. We have said what property infinity does not have, but we have said little about what properties it does have.


The Slow Road

December 12, 2006

natsukusa ya
tsuwamonodomo ga
yume no ato

The summer grasses:
The high bravery of men-at-arms,
The vestiges of dream.

— Matsuo Basho, on visiting Hiraizumi, once home to the great Fujiwara clan whose splendid castles had been reduced to overgrown grass mounds.*

A good haiku not only arrests our attention, it also demands reflection and contemplation of deeper themes. In Basho’s Oku no Hosomichi, The Narrow Road to the Interior, the haiku often serve as a point of pause amidst the travelogue, asking the reader to slow down and take in all that is being said. The slow road to understanding is often the easiest way to get there. At the same time the travelogue itself provides context for the haiku. Without that context, both from the travelogue, and from our own experiences of the world upon which the haiku asks us to reflect, the poem becomes shallow: you can appreciate the sounds and the structure, but the deeper meaning — the real essence of the haiku — is lost.

Mathematics bears surprising similarities. A well crafted theorem or proof demands reflection and contemplation of its deep and wide ranging implications. As with the haiku, however, this depth is something that can only be provided by context. A traditional approach to advanced mathematics, and indeed the approach you will find in most textbooks, is the axiomatic approach: you lay down the rules you wish to play by, assuming the bare minimum of required knowledge, and rapidly build a path straight up the mountainside. This is certainly an efficient way to get to great heights, but the view from the top is often not rewarding unless you have spent time wandering through the landscape you now look out upon. Simply put, you lack the context to truly appreciate the elegant and deep insights that the theorems have to offer; like the haiku it becomes shallow.


On Abstraction

November 10, 2006

Let’s begin with a short practical experiment. Pick up a pen, or whatever similar sized object is handy, hold it a short distance above the ground, and drop it. The result — that the pen falls to the ground — is not a surprising one. The point of the experiment was not to note the result, however, but rather to note our lack of surprise at it. We expect the pen to fall to the ground; our expectation is based not on knowledge of the future however, but on abstraction from past experience. Chambers Dictionary defines “abstract”, the verb, to mean “to generalize about something from particular instances”, and it is precisely via this action that we come to expect the pen to fall to the ground. By synthesis of many previous instances of objects falling when we drop them, we have generalized the rule that things will always fall when we drop them*. We make this abstraction so instinctively, and take it so completely for granted, that it is worth dwelling on it for a moment so we can see how remarkable it actually is.