TITLE: What is about J. v. Neumann's "Continuous geometry"?
QUESTION [13 upvotes]: I am curious about von Neumann's "Continuous geometry", but found no recent text or survey on it. Does anyone know the book and would be so nice to share their impression, and if/how the concept of such geometries fits into contemporary tries to generalize geometry?

REPLY [5 votes]: In plane projective geometry, there are "subspaces" of dimension 0 (points) dimension 1 (lines) and dimension 2 (the whole space).  We may also add dimension -1 (the empty set).  Axioms may be given concerning incidence among these various subspaces.  There is a "duality" that switches points and lines; the whole space and the empty set; reverses inclusion.
In $n$-dimensional projective geometry, there are "subspaces" of dimension $-1,0,1,\dots,n$.  Axioms may be given concerning incidence again.  And duality.
The idea of continuous geometry is to allow subspaces of dimensions in more general totally ordered sets.  The main example is $[0,1]$.  Still there are axioms concerning incidence.  And duality.
As noted in the other answers, it did not turn out to be an important idea.