Hilbert's fourth problem

In mathematics, Hilbert's fourth problem in the 1900 Hilbert problems is a foundational question in geometry. In one statement derived from the original, it was to find geometries whose axioms are closest to those of Euclidean geometry if the ordering and incidence axioms are retained, the congruence axioms weakened, and the equivalent of the parallel postulate omitted. The original statement of Hilbert has been judged too vague to admit a definitive answer. Nevertheless a solution was found with the German mathematician Georg Hamel being the first who tried to solve the problem. A recognized independent solution for dimension 2 and 3 is considered to be given by Armenian mathematician Rouben V. Ambartzumian.

Hilbert discusses the existence of non-Euclidean geometry and non-Archimedean geometry, as well as the idea that a 'straight line' is defined as the shortest path between two points. He mentions how congruence of triangles is necessary for Euclid's proof that a straight line in the plane is the shortest distance between two points. He summarizes as follows:

The theorem of the straight line as the shortest distance between two points and the essentially equivalent theorem of Euclid about the sides of a triangle, play an important part not only in number theory but also in the theory of surfaces and in the calculus of variations. For this reason, and because I believe that the thorough investigation of the conditions for the validity of this theorem will throw a new light upon the idea of distance, as well as upon other elementary ideas, e. g., upon the idea of the plane, and the possibility of its definition by means of the idea of the straight line, the construction and systematic treatment of the geometries here possible seem to me desirable.

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