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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.)

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