During the first half of the 20th century the Danish geometer Johannes Hjelmslev developed what he called a geometry of reality. It was presented as an alternative to the idealized Euclidean paradigm that had recently been completed by Hilbert. Hjelmslev argued that his geometry of reality was superior to the Euclidean geometry both didactically, scientifically and in practice: Didactically, because it was closer to experience and intuition, in practice because it was in accordance with the real geometrical drawing practice of the engineer, and scientifically because it was based on a smaller axiomatic basis than Hilbertian Euclidean geometry but still included the important theorems of ordinary geometry. In this paper, I shall primarily analyze the scientific aspect of Hjelmslev's new approach to geometry that gave rise to the so-called Hjelmslev (incidence) geometry or ring geometry. I den første halvdel af 1900-tallet udviklede den danske matematiker Johannes Hjelmslev en såkaldt virkelighedsgeometri. Den var et alternativ til det idealiserede euklidiske paradigme, som kort forinden var blevet perfektioneret af Hilbert. Hjelmslev hævdede at virkelighedsgeometrien var bedre end den euklidiske både didaktisk, videnskabeligt og i praksis: Didaktisk, fordi den var tættere på erfaring og intuition, i praksis, fordi den lå tættere på ingeniørens praktiske geometriske konstruktioner, og videnskabeligt, fordi den byggede på et smallere aksiomatisk fundament end Hilberts, men stadig indeholdt de vigtigste sætninger i den almindelige geometri. I denne artikel vil jeg først og fremmest diskutere de videnskabelige aspekter af Hjelmslevs nye geometri, som gav anledning til den såkaldte Hjelmslev-geometri eller ring-geometri.
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