Fascinating stuff!
I learned about Hjelmslev's work when I was researching for my PhD in the
late 60s. As a student of Hilbert around 1910, Hjelmslev produced a major
piece of work in deriving some central theorems of absolute geometry from
reflection principles. That's where I saw it first, in Friedrich Bachmann's
Aufbau der Geometrie aus dem Spiegelungsbegriff (AGS). Hjelmslev's work
provides one of the critical steps early in the book. I can't give many
further details because I'm summering at our place on the Klamath, 300 miles
from my books and files.
There are several slightly different ways to develop absolute geometry from
reflection principles. You can find works of Lingenberg, Karzel, and others
referenced in Bachmann. I recall that one concept, H-Ebene, is named for
the three H's--Hilbert, Hjelmslev, Hessenberg. Perhaps Diller's edition of
Hessenberg's book goes into that, or maybe it's in AGS.
I have a feeling that you've missed a major series of Hjelmslev's works.
The title "Einfuehrung (or Einleitung) in die allgemeine Kongruenzlehre"
rings a bell, perhaps published by the Danish Academy of Sciences during the
20s and 30s. The parts of his work that I looked through were all
technical, in the sense that AGS is technical, so I can't comment on the
philosophical aspects. I believe that Hjelmslev used reflection principles
as an underpinning for his not-quite-Euclidean geometry.
As soon as his work matured, there came the Depression and the War.
I remember there were some talks at Oberwolfach in the late 60s and early
70s about Hjelmslevsche Geometrie, as a generalization of (I think) finite
projective geometries. That is, the approach was combinatoric. Lines could
meet in a sort of cloud of points, rather than just a single point. You
might find some leads in Pickert's Projektive Ebenen, or Dembowski's book on
finite geometries. The person who gave at least two of those talks was from
the Florida state university at Gainesville--perhaps his name was Drake.
Also, I think a man named Zimmer from somewhere in the central U.S. did
something along those lines.
Distant memories are crystallizing. I think Bachmann wrote an expository
paper, with the name Hjelmslev in the title. It would be in the
Literaturverzeichnis of AGS.
Who might know more about Hjelmslevsche Geometrie? I believe Prof. Pickert
is still active, and perhaps Prof. Hanfried Lenz. Prof. Helmut Karzel.
Prof. Bachmann's students Peter Klopsch, Hilger Wolff, Wolfgang Pejas,
Henner Kinder, Wilhelm Klingenberg, Erich Ellers, Joachim Ahrens, Justus
Diller. (Not all these continued university careers.) Perhaps Hansjoachim
Groh. Try Prof. Karl Strambach.
Also, there are several German mathematicians formerly in the East, who
worked on logical and reflection theoretic foundations of geometry:
Wolfgang Rautenberg, Benno Klotzek, E. Quaisser come to mind.
James T. Smith
Professor of Mathematics
San Francisco State University
1600 Holloway Avenue
San Francisco, CA 94132
415-338-1368 (business/message)
415-338-1461 (fax)
smith@math.sfsu.edu
http://math.sfsu.edu/smith