Cheiromancer said:
It's been a while since I read Immanuel Kant, but I think he said that Euclidean geometry was hard-wired into the way we perceive reality. It is a priori.
it's been a long time since I've read Kant too, but to put it bluntly: Kant was obviously wrong on that point.
We can define other objects to be "straight lines" that have non-euclidean properties, but they won't seem straight. They may express the true nature of the universe, but they won't conform with our perception of reality.
Which is Lovecraft's thesis; the true nature of the world is unimaginable; we are not wired to perceive it.
trouble with the whole non-euclidean argument is mutlifold though.
For example: If you measure the distance between two points on a city map by measuring the lengths of the streets you are going to drive along to get there you are using a non-euclidean metric. In euclidean geometry the distance between two points is "as the crow flys". If you measure it any other way, the metric is non-euclidean.
More to the point: if while you are driving down the road you visualize in your head the path to the place you are traveling to in terms of the intersections you are going to turn at then you are visualizing in non-euclidean terms.
This is just a simple example, but it clearly indicates to me that we're not hard-wired to think in euclidean terms. If we were, navigating city streets would be hopelessly confusing. flying airplanes or sailing ships on long voyages would be dangerous. these are all examples where you use a two dimenstional non-euclidean metric in order to navigate.
If Kant had said we were hardwired to think in 3-dimensions then I might be inclined to agree with him. visualizing in higher dimension is hard. But unfortunately, that's not what Kant said (edit: then again, it may well have been what he
meant to say, after all, modern non-euclidean geometry was invented some 100 years after Kant died, so he couldn't have forseen us analysing his argument in these terms).
As an aside: Kant also said a number of things regarding the physical make up of the universe that are in direct contradiction to the general theory of relativity. specifically, Kant claimed time to be an absolute. einstien showed its not.
If Lovecraft had said the nature of the world was unimaginable for some other reason I might be inclined to agree with him, but to claim it has anything to do with non-euclidean geometry is, to me, silly.
[**grrr** are these message boards always this friggin flaky? it takes forever to get a submission through half the time. or is it just a bad weekend for the server?]