Monday, April 17, 2006

Is this philosophy of mathematics, or is this bullshit?

Lately, I've been curious as to why the number 2 is so important. The number 2 crops up everywhere, so pervasive that I’ve never noticed it until lately.

Concave, convex.
Positive, negative.
Left, right.
Yes, no.
Sky, earth.

The importance of two turns out to be very fundamental- it is precisely because two is one more than one. Symbolically, 2 = 1 + 1.

Say we have a "dimension". For now, we will restrict the geometric properties of this dimension to being a straight line. We are on an arbitrary point along this one dimension, and we can only travel along this dimension. Think of a railcar running on its tracks.

Strangely enough, there is only one station on this rail line. It is Piccadilly Circus, far away in the distance. The railcar travels in one direction, towards Piccadilly Circus.

So far, we have only worked with one's. There is one dimension, and along this dimension is a preferred direction. And now, to invoke "two".

If the railcar was made to reverse its direction of travel, such that it moved further and further away from Piccadilly Circus, then it would be travelling in the opposite direction. Obviously, this direction is not the same as the direction that leads to Piccadilly Circus.

And thus there are now two directions: to Piccadilly Circus, and away from Piccadilly Circus. Relating back to the earlier equation 2 = 1 + 1, we can see that the two comes from going along the dimension's 'preferred direction', and in the reverse direction.

Thus the "two" is constructed.

Of course, this dimension's geometric restrictions can be lifted, and the dimension can then be made to fit between any two extremes.

North and South.
On and off.
Stupid and clever.
Civilised and barbaric.
Bright and dim.
Light and dark.

As one can see, the dimension can be fitted not only to geometric extremes like North and South, but also to conceptual poles such as civility.

Edit 20 April 2006:

Here is a better derivation of 'two' that does not depend on vectors nor time derivatives of position.

On that railway, there lies one railcar. The presence of this one railcar seperates the railway into two distinct regions. So yeah, one dimension, one obstruction, two regions. Simple.