Wednesday, March 4, 2009

how to explain the two Euclids

On occasion, I find myself talking about axioms and the notion of a paradigm-relative axiom.

The standard example I use is Euclidean geometry... which usually elicits judgmental stares from my students. (*sigh* Public school isn’t what it used to be, eh? But I digress...)

One of the axioms (forgive the loose description) is that if you draw two straight lines diverging from a single point, those lines will continue to diverge forever.

This is one of those propositions that seems so self-evident that the glow of its (alleged) necessity is like philosophical fairy dust... magical and dependable to boot!

But alas... the view is false.

Suppose that Euclid were born in a very different world, and by “world” I mean only a very small difference. Hold fixed all of the most fundamental laws of the cosmos. Change only one thing: the size of our earth. Imagine that our earth was only 500 feet in diameter. Euclid, had he been born on such a small planet, might have formed his conceptions of spatial representation on the basis of curved lines.

(To those careful readers out there who accuse me of an implicit (well, maybe explicit) empiricism... guilty as charged.)

If such had been the case, it would be very easy to see that two straight lines that diverge do not diverge forever. They actually converge again, say, at the south pole, which happens to be very close by. In this scenario, two straight lines can contain an area.

To the careful reader who thinks that I’m cheating a little bit in my characterization of straight lines...

Just conceive of a straight line as a line on which the shortest distance between two points on that line runs through that line. That’s true of traditional Euclidean lines as well as curved lines.

But... it’s also true of the lines on the alternate Earth that is only 500 feet in diameter. Let’s go back to the Euclid of this world... His geometry would consist of geometric figures such as squares and triangles whose angle sums would vary with the size of the figure.

Flat surfaces are one thing; curved surfaces are an entirely different matter. On the curved surface of a sphere, you can have a finite and endless line, and that requires a different kind of geometry.

Here’s the crazy cool reality: Our universe is composed of curved space(s).

Well, that’s a rather imprecise but free-of-jargon way of explaining this stuff.

1 comment:

Dave said...

I have to say I'm not convinced that two lines necessarily have to converge, whether by the euclidean counterfactual world or our naturally curved universe. Perhaps your not saying the necessarily do converge. Anyway, would this mean our universe (or everything that exists) is really within a giant sphere-like container? It just seems to me that an axiom point may send out two lines that don't converge again, but I did go to a public school.