The "geometry" Hjelmslev develops deviates significantly from
the ideal geometry of Euclid, the tradition; more directly, he sought
to create a geometry that could not be assimilated to an "analytic"
geometry, "analysis" understood in the spirit of post-Weierstrassian
arithmetic and logical rigour 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 in at least the following sense --
If we assume that a minimum exists for a range of phenomena, we
may be able to infer what this min. is (or valuable traits of it). When
the range of phjenomena are physical, we may know that the minimum has
to exist. Weierstrassian criticism required that the determinations if
minima to be mathematical, then a mathematical existence proof of the
existence of a minimum must be given. (The isoperimetric problem,
Dirichlet's problem. . .) [The clearest, simplest, discussion of
these criticisms I find in -- of all places -- Courant & Robbins, What
is Mathematics? Chapt. 7, para. 7; pp.355f).
It seems that Hjelmslev had the idea of developing a kind of
mathematics tuned to the physical-geometrical; and so untouched by
Weierstrassian criticism. And since in physical matheamtics, it is
approximations in and approximations out, going through ideal geometry
or post-Weierstrassian analysis seemed to Hjelmslev to be not only an
unnecessary detour, but a detour (a) in which one loses contact with
the phenomena one is studying, (b) a detour that burdens one with
purely formal anomalies (such as the everywhere continuous, nowhere
differentiable curve).
Otherwise, it is hard to see what Hjelmslev's intentions were.
There are a number of considerations why Hjelmslev's project failed
(there were no "takers", right?).
(a) His papers contained little more than suggestions; unlike
e.g. Brouwer he apparently never tried to execute his programme in
detail (???).
(b) At no point could one rely on logical deduction alone.
(c) Post-Weierstrassian monsters, such as the everywhere
continuous nowhere differentiable functions found applications (e.g.,
theory of Brownian motion).
(d) The logician Sol Feferman's "reduction" of almost all of the
mathematics needed for physics to primitive recursive arithmetic shows
that resources for "arithmetization" Hjelmslev could draw on would be
those of a we or strange arithmetic, a kind of approximation
arithmetic; I'm not clear whether or not some "ultrafinitist"
arithmetic would do. Perhaps a "feasible arithmetic" like that of
Parikh would do the job.
In any case, philosophically Hjelmslev is good to contemplate
because of the understanding the emergent flaws in his attempt give of
the necessity for "idealization" in mathematics, if mathematics is to
do a significant job of work.
robert tragesser
west(running)brook, connecticut usa