Archive for the ‘Topology’ Category

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.



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.