Elsevier

Historia Mathematica

Volume 54, February 2021, Pages 95-116
Historia Mathematica

Hjelmslev's geometry of reality

https://doi.org/10.1016/j.hm.2020.08.003Get rights and content

Abstract

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.

Section snippets

Johannes Hjelmslev (born Petersen) (1873-1950)

When the principal character of this paper was born in 1873 in the district (herred) Hjelmslev in Jutland, Denmark, he was baptized Johannes Trolle Petersen. After completing high school in 1890 he moved to Copenhagen to study mathematics at the university. When he graduated in 1894, he began making a living as a teacher in a high school and as a tutor at the university. Parallel with his teaching duties he worked on a doctoral dissertation on infinitesimal methods in descriptive geometry that

Inspiration from his peers

Hjelmslev published his first paper on his geometry of reality2 in 1913. However, one can trace

The empirical foundation of Hjelmslev's geometry of reality

In order to clarify what Hjelmslev meant when he declared that school geometry should deal with things that belong to the practice of life I shall now explain how he introduced the basic objects of geometry. In general he required that

the definition of the basic concepts: plane, straight line, right angle etc. must be formulated in such a way that they specify a procedure for the manufacture of the thing, a procedure that allows us to check if the thing has the specified property [Hjelmslev,

Geometric constructions or experiments

The same year Hjelmslev published his first programmatic paper on his new geometry of reality he published a textbook entitled Geometric Experiments. According to the preface

This book deals with geometric constructions in the widest sense. It does not deal with approximations but with exact constructions. At the same time, it deals with the practical execution seeking to form the construction with a view to the simplest realization. And through a natural union of these purposes it tries to

The relation between geometry of reality and Euclidean theoretical geometry: fixing

Hjelmslev began his first programmatic paper on the geometry of reality [Hjelmslev, 1913a] with a historical survey of the early development of geometry. In particular, he emphasized that Euclid had partially idealized and axiomatized the previous Egyptian material geometry. In doing so, he had gone beyond experience by introducing ideal points without extension, and lines and planes without width. Euclid and the textbook authors who followed him still relied partially on intuition and

(Rejected) axioms

As we saw above, Hjelmslev in his schoolbooks did not deduce his theorems from axioms. He preferred to build on experience and intuition. However, true to his dictum “one shall not begin with abstraction but end with it” [Hjelmslev, 1913a, 50], in his scientific papers, he recognized the need for clearing up the axiomatic foundations of his new geometry.

For the time being our problem is principally the same as in the old geometry. We set up a system of axioms. However, we require that this

Differential geometry

Differential geometry gave rise to special considerations concerning the relation between the geometry of reality and arithmetic geometry. In arithmetic geometry, concepts like tangents and osculating circles are defined through differentiation, i.e. through limiting processes. However, in the geometry of reality, there is no continuity axioms and so the limiting procedures make no sense. Differential geometry had interested Hjelmslev since he wrote his doctoral thesis: Basic principles for the

The flexibility of the axioms

In his early programmatic papers [Hjelmslev, 1913a and 1916b] as well as in his textbooks [Hjelmslev, 1916a and 1918] Hjelmslev did not try to formulate a complete set of axioms for his geometry of reality. He often claimed that his geometry of reality was based on one empirical axiom, namely the existence of a square grid in the plane (e.g. [Hjelmslev, 1913a, 54]. In his first paper in German on the geometry of reality, he first repeated this formulation [Hjelmslev, 1916e, 39]) but then

Infinitesimals

In [Hjelmslev, 1916c] and in his Hamburg lectures, Hjelmslev gave an interesting model of a geometry where two different lines can have a line segment in common, so that the uniqueness axiom does not hold. This was a non-Archimedean geometry build up as the usual analytic (arithmetic) model of geometry, except one must replace the real numbers by the dual numbers, that is numbers of the form a+εb,(a,b)R where ε is a non-zero symbol (an infinitesimal) such that ε2=0. In this geometry, two lines

Geometry of great-points

The concept of a great-point (Grosspunkt) came to play an important role in Hjelmslev geometry [Benz, 1990, 249-50]. Hjelmslev introduced them in a preliminary form called a coarse point (grober Punkt) in [Hjelmslev, 1923, 28] in order to illustrate the main ideas behind his geometry of reality:

A coarse point is a domain in the drawing plane such that every pair of points that lie inside this domain have a distance smaller than 1 cm. The coarse point can be fixed by any of its fine points [

Conclusion

The geometry of reality that Hjelmslev developed in the 1910s was primarily inspired by his engagement with descriptive geometry, practical geometric drawing and teaching of elementary geometry to schoolchildren. However, he went much further than other reformers of school geometry developing his approach into a rather well rounded alternative scientific system. “[The materialization of geometry] may at first sight seem to serve exclusively utilitarian purposes. In reality it seeks its first

Acknowledgements

I wish to thank Toke Lindegaard Knudsen with whom I studied many of Hjelmslev's papers and Reinhard Siegmund-Schultze whose very informed comments and references to relevant literature have improved the paper considerably.

Jesper Lützen is professor of history of mathematics at the Department of Mathematical Sciences, University of Copenhagen. He has published books on the history of the theory of distributions, Joseph Liouville and Hertz's mechanics. Recently he has done research on the history of descriptive geometry, Hjelmslev's contributions in particular, as well as the history of impossibility theorems.

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    Jesper Lützen is professor of history of mathematics at the Department of Mathematical Sciences, University of Copenhagen. He has published books on the history of the theory of distributions, Joseph Liouville and Hertz's mechanics. Recently he has done research on the history of descriptive geometry, Hjelmslev's contributions in particular, as well as the history of impossibility theorems.

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