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:
x * x
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.)