Archive for March, 2007

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.



A Fork in the Road

March 13, 2007

Alice came to a fork in the road. “Which road do I take?” she asked.
“Where do you want to go?” responded the Cheshire cat.
“I don’t know,” Alice answered.
“Then,” said the cat, “it doesn’t matter.”

— Lewis Carroll, Alice’s Adventures in Wonderland

In the later years of his life, after his journey to the interior, Basho lived in a small abandoned thatched hut near lake Biwa that he described as being “at the crossroads of unreality”*. Now, still early in our journey, we have come to our own crossroads of unreality. We are caught between dichotomies of unreal, abstract, objects. One road leads to consideration of finite collections, and properties of composition (the algebraic properties 1 through 5 from the previous entry); the other road leads to the continuum and questions of ordering and inter-relationship (properties 7 through 10 from the previous entry). The first road will lead to a new fundamental abstraction from finite collections, different from, and yet as important as, the abstraction that we call numbers; this way lies group theory and the language of symmetry that has come to underlie so much of modern mathematics and physics. The second road will lead to deep questions about the nature of reality, and, brushing past calculus along the way, lead to a new and minimalist interpretation of a continuous space through the concept of topology.

Which road do we take? As the cat said to Alice, It doesn’t matter. We are at the crossroads of unreality, and the usual rules need not apply. Which road do we take? Both.

* From the translation of Genjûan no fu by Donald Keene, in Anthology of Japanese Literature.

A Fraction of Algebra

March 5, 2007

As a mathematician there is a story I hear a lot. It tends to come up whenever I tell someone what I do for the first time, and they admit that they don’t really like, or aren’t very good at, mathematics. In almost every case, if I bother to ask (and these days I usually do), I find that the person, once upon a time, was good at and liked mathematics, but somewhere along the way they had a bad teacher, or struck a subject they couldn’t grasp at first, and fell a bit behind. From that point on their experiences of mathematics is a tale of woe: because mathematics piles layer upon layer, if you fall behind then you find yourself in a never ending game of catch-up, chasing a horizon that you never seem to reach; that can be very dispiriting and depressing. In the previous entries we have dealt with subjects (abstraction in general, and the abstraction of numbers) that most people have a natural intuitive grasp of, even if the details, once exposed, prove to be more complex than most people give them credit for. It is time to start looking at subjects that often prove to be early stumbling blocks for some people: fractions and algebra.