Re: [HM] Hjelmslev-naturliche Geometrie
Ken Pledger (Ken.Pledger@vuw.ac.nz)
Wed, 9 Jun 1999 10:12:10 +1200
At 5.03 am +1200 9-6-99, Robert Tragesser wrote:
> The Danish geometer J.Hjelmslev has published two mathematically
>curious but philosophically fascinating papers,
> Die Geometrie der Wirklichkeit, Acta mathematica 40
> Die natu"rliche Geometrie, Einladung des Mathematischen Seminars
>der Hamburgerischen Universitaten, 1923.
> Can anyone offer any historical clarity about these works?
>
>.... The properties of objects in his
>geometry are more or less those that drawings and material,
>perceived-things-as-perceived things, do present. We have such
>propositions as: a circle and a tangent to it coincide for an arc of
>the circle (even if one considers just the edges of the lines as the
>lines); there exist triangles that violate the triangle inequality
>(triangles in which the sum of two sides is equal to the third side). A
>version of the Pythagorean theorem holds -- the sum of the squares of
>the legs of a rgt triangle are "equal" to the square of the hyp.,
>meaning that the errors are nicely damped.
> It is clear what part of the motivation is.-- Hjelmslev is
>vigorously anti-Weierstrassian in the sense that he thinks that there is
>a mathematic linked to the physico-geometrical realm of our perception
>and observation....
This is not an answer, but perhaps enlarges the question about
Hjelmslev. A few years after publishing the above works, he began a series
of papers "Einleitung in die Allgemeine Kongruenzlehre," Danske Vid.
Selsk., mat.-fys. Medd., 8, Nr.11 (1929); 10, Nr.1 (1929); 19, Nr.12
(1942); 22, Nr.6, Nr.13 (1945); 25, Nr.10 (1949). This reference comes
from a paper by Friedrich Bachmann: "Hjelmslev Groups," Colloquio
Internazionale sulle Teorie Combinatorie, (Atti dei Convegni Lincei 17),
Tomo I, 469-479; Accademia Nazionale dei Lincei, 1976. This paper is in
English. On p.470 Bachmann says that he is generalizing from his own book
and from the much earlier papers of J. Hjelmslev referenced above.
Bachmann's book "Aufbau der Geometrie aus dem Spiegelungsbegriff" (Springer
1958, 1973) explains his group-theoretic approach to the foundations of
orthodox geometry; and his "Hjelmslev Groups" paper applies such methods to
a theory apparently ascribed to Hjelmslev, in which the number of lines
joining two points can be zero, one, or more.
Does anyone know of any study of Hjelmslev's works?
Ken Pledger.