A two week absence from this site is pretty large, at least when its unannounced. As you can probably tell, I'm snowed under. Slowly digging through that inbox and the pile of administrative work which requires doing. You might even get a link recommendation or two from me in the next few days. But I'm not promising anything. First, I must get back to that email.
It seems that the discussion group has formed a life of its own. Here, as elsewhere, the action is below the surface.
Quite a lot about the structure of the universe (well, at least the
relationship between wishes and choices) can be inferred from folk wisdom. Here is an example.
If wishes were horses,
beggars would ride.
So, if wishes were choices,
beggars would be choosers.
But beggars can't be choosers.
Therefore, wishes can't be choices.
This is something I really love. (Warning, it's a dvi file. If you don't know what a dvi file is, it's almost certain that you won't have the software to deal with it. Sorry.) Anyway, this paper "A nonstandard proof of the Jordan curve theorem" by Vladimir Kanovei and Michael Reeken, is a beautiful piece of mathematics. The Jordan Curve theorem is a particularly easy theorem to state, and a particularly hard theorem to prove with the kind of rigour mathematicians desire and enjoy. Here's the theorem:
Every closed curve in the plane divides the plane into two regions, the inside and the outside.
That's pretty darn obvious, you might think, but it's not obvious when you realise that closed curves can be
really complicated. (Curves of high fractal dimensions, for an example.) Anyway, the paper by Kanovei and Reeken shows that it's pretty easy to prove if you move from the standard "real" plane to consider it embedded in the "nonstandard" plane of nonstandard coordinates (think
infinitesimals). Anyway, considered as a curve in the nonstandard plane, a curve is actually a polygon with infinitely many (very short) straight "sides". But for polygons, it's easy to tell whether a point is on the inside or the outside. Start with a point arbitrarily far away from the curve (which we'll call
outside), and then draw a line to the point you want to consider. Just count the number of times you cross the
polygon along the straight line to the target point. Even number of crossings? You're
outside. Odd number of crossings? You're
inside.
The remaining trick is to deal with the case in where the number of crossings is infinite. In this case, Kanovei and Reeken show that a sensible notion of "even" and "odd" can be defined for the kind of infinities applied here, and the result still applies.
It's a lovely piece of work. It's the kind of thing that I find so enjoyable in mathematics. It's sometimes easier to solve a problem by initially making it more complicated. The seemingly harder theorem is sometimes simpler to prove, because in the new space, different rules apply.
Our immediate past prime minister and the current incumbent are two very different people. There's no way, for example, that the current model would ever call anyone "pre-Copernican obscurantists." That was Keating on the current government, on the debate on cross media ownership rules. For more Keating-isms, try the Paul Keating Insults Archive.
For something a little more uplifting than that, try Keating's 1992 speech at Redfern. It ends:
We cannot imagine that the descendants of people whose genius and resilience maintained a culture here through 50 000 years or more, through cataclysmic changes to the climate and environment, and who then survived two centuries of dispossession and abuse, will be denied their place in the modern Australian nation.
We cannot imagine that.
We cannot imagine that we will fail.
And with the spirit that is here today I am confident that we won't.
I am confident that we will succeed in this decade.
That speech was given in December 1992.
