Ken Pledger wrote:
[[ This is not an answer,
[[to what???? -- I was asking after the motivation for Hjelsmlev making
positive hay out of traditional Sophist/Skeptic attacks on "ideal"
geometry. . .and my attribution of anti-Weierstrassian motivation was
speculative, though inspired by Hjelslev's vigorous defense of
Steiner's non-Weierstrassian way of solving the isopermetric problem.]]
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
bookand from the much earlier papers of J. Hjelmslev referenced above.
Bachmann's book "Aufbau der Geometrie aus dem Spiegelungsbegriff"
(Springer1958, 1973) explains his group-theoretic approach to the
foundations oforthodox geometry; and his "Hjelmslev Groups" paper
applies such methods toa theory apparently ascribed to Hjelmslev, in
which the number of linesjoining two points can be zero, one, or more
[[this occurs in the n.G. paper, but it would be nice to know if this
did have an influence on the ideas cited]]
.
Does anyone know of any study of Hjelmslev's works?] [END PLEDGER]
The matter I am interested in is the motivation for, and
influence of, the two papers I mentioned, on reality or actuality
geometry and natural geometry; it is not clearf whether Ken Pledger's
references apply, though I am extremely happy to have these references
[which maybe answer Sam Kuttler's questions rather than mine].
I did once pursue his papers (in English if I recall (or maybe
German) -- I was in Oxford in days when it was not so easy to xerocopy)
on Nonarchimedian geometries that came later than the
Wirklichkeit/Naturliche Geometrie papers. I had looked to them, for the
two philosophical papers I mentioned certainly support or inspire
nonArchimenidian considerations.
Also, another (partial) motivation might have been: a way of
handling the foundational/philosophical problems of the physicists' "dx"
as the physicist means it in their "in the small" constructions.
Hjelmslev seems to block the notion of reals as based on
convergence of rationals, and so therefore the full rationals, in
order to avoid the problem they raise or open out [the problem that is
the basis of Weierstrassian criticim] about the existence of limits.
(E.g., if lines have some thickness to them, then questions of the
existence of mninma/maxima can be naswered through a course perception!)
with gratitude,
rbrt tragesser
west(running)brook, connecticut usa