## Euclid's _Elements_ and Meatspace

You stand in opposition to gravity, and walk in a direction perpendicular to it. This defines a right angle, and is all the justification we need for many of the *elements* of geometry.

But what if we took a fresh look at geometry from the vantage-point afforded to us by information technology and * infinite* imagination? What if we attempted to colonize Hilbert Space?

The intersection of two volumes in four-space defines a plane:

w x y One infinite volume in four-space, and

x y z A second infinite volume in four-space

x y define an infinite plane

The intersection of two planes defines a line:

x y One infinite plane

y z A second infinite plane

y The resulting infinite line

And the intersection of two lines defines a point:

y

x * x

y

Now, given the possibility that we can extend these definitions into higher dimensions still, we have a very interesting new view of geometry, set theory, and quite a few other subjects in mathematics and possibly logic as well. Rather than beginning with the Null set, we can begin with the Universal set! (Once we've built ourselves a scaffold, of course.)