Hilbert's 1902 motion-based foundations of plane geometry: the open questions of axiom redundancy, the spatial analogue, elliptic geometry, and the replacement of Axiom III

Hilbert's 1902 memoir 'Ueber die Grundlagen der Geometrie' (Mathematische Annalen 56, 381-422) founds plane geometry on the group of motions instead of the congruence axioms of his 1899 book: the plane is a system of points in one-to-one correspondence with the finite points of the number plane or with a subsystem of it, every point having a neighborhood of plane points; motions are its sense-preserving self-homeomorphisms; and three axioms — the group property (I), infinitude of every true circle (II), and a closedness condition on limits of motions (III) — force the geometry to be Euclidean or hyperbolic (Bolyai-Lobachevsky), with no differentiability assumption. Hilbert explicitly left four investigations open: which components of the system are dispensable; the analogous axiom system for space; the modifications needed to include elliptic geometry; and, as he asked in a footnote to Axiom III, with which further requirements the property he had used as a separate axiom in his 1901 Göttingen lecture — that any two points can never be brought into arbitrary proximity by a motion — could replace Axiom III. Later literature resolved the spatial case (de Kerékjártó 1928; Montgomery-Zippin 1940) and the elliptic case (Lubben 1928; Fischer 1933; Podehl-Reidemeister 1934; Bachmann 1936), and produced weakened or alternative axiom systems (Moore 1919; Cairns 1933; Montgomery-Zippin 1940). No resolution of the independence analysis of Hilbert's exact 1902 system, or of the footnote's replacement question, is known; the problem remains open.

Background

In the tradition of the Riemann-Helmholtz-Lie space problem, Euclidean and non-Euclidean geometry is to be characterized by the properties of its group of rigid motions alone. Riemann and Helmholtz had phrased this program in terms of freely movable rigid bodies; Lie gave it a precise group-theoretic form but had to assume that the motion group is differentiable. Hilbert's memoir 'Ueber die Grundlagen der Geometrie' (Mathematische Annalen 56, 1902, 381-422 — the detailed version of a note presented to the Göttingen Academy of Sciences on 8 November 1901, and not to be confused with his 1899 Festschrift (commemorative volume) 'Grundlagen der Geometrie') carries out the program for the plane with no differentiability assumption, using instead Cantor's point-set theory and the Jordan curve theorem. A Jordan curve is a simple closed plane curve; the Jordan curve theorem states that such a curve separates the plane into exactly two regions, an interior and an exterior. The memoir was later reprinted essentially unchanged as Anhang IV (Appendix IV) of the editions of the book 'Grundlagen der Geometrie' (7th edition 1930).

The primitive notion is the motion. The number plane is the ordinary Euclidean plane with a fixed rectangular coordinate system, that is, the set of real coordinate pairs $\mathbb{R}^2$ ; its finite points are all of its points (as opposed to points at infinity). The plane of the memoir is defined as a system of points that can be put in one-to-one correspondence with the finite points of the number plane or with a certain subsystem of it, in such a way that every point has a neighborhood: there are Jordan domains in the number plane containing its image all of whose points likewise represent points of the plane. A motion is a single-valued, invertible, continuous transformation of the plane into itself that preserves the sense of traversal of Jordan curves (an orientation-preserving homeomorphism). A rotation about a point $M$ is a motion that leaves $M$ fixed, and the true circle (wahrer Kreis) about $M$ through a point $P$ is the orbit of $P$ under all rotations about $M$ ; it is called 'true' because, unlike a circle defined by a distance notion, its existence and its properties (for example, being a Jordan curve with infinitely many points) must be derived from the axioms. The axiom system is: Axiom I — the motions form a group (identity, composites, inverses are motions); Axiom II — every true circle contains infinitely many points; Axiom III — if there are motions carrying point triples in arbitrary proximity of the triple $A$ , $B$ , $C$ into arbitrary proximity of the triple $A'$ , $B'$ , $C'$ , then there exists a motion carrying $A$ , $B$ , $C$ exactly into $A'$ , $B'$ , $C'$ (a closedness or completeness condition on the group; the points of a triple need not be distinct from one another). From these, Hilbert derives the structure of the true circles, introduces straight lines, proves the congruence theorems, and concludes that the plane's geometry is the Euclidean or the Bolyai-Lobachevskian (hyperbolic) one, its motions forming the corresponding metric motion group. The construction is a direct ancestor of Hilbert's fifth problem (1900) — whether a topological group that is locally homeomorphic to Euclidean space must admit differentiable coordinates — which made precise, for topological groups in general, how far continuity assumptions alone force group structure.

Within this framework Hilbert explicitly left four investigations open. (i) Redundancy: 'Gewiss sind auch in dem von mir aufgestellten System noch einzelne Bestandtheile entbehrlich' (certainly some components of the system are dispensable), but he refrained from investigating which. (ii) Space: he treats only the plane and believes that an analogous axiom system for space can be set up, without carrying it out. (iii) Elliptic geometry: the narrow definition of the plane (a one-to-one image of the finite points of the number plane or of a subsystem of it, with orientation-preserving motions) excludes elliptic geometry from the outset; he notes that the modifications of the proof needed under the wider concept of a plane are not difficult to recognize, but does not develop them. (iv) Axiom III replacement: in a footnote he recalls that in his oral lecture at the 1901 anniversary session (Jubelfeier) of the Göttingen Academy of Sciences he had used the property 'Irgend zwei Punkte können durch Bewegung niemals in beliebige Nähe zu einander gerathen' (any two points can never be brought into arbitrary proximity by a motion) as a separate axiom, and asks ('Es wäre zu untersuchen') which requirements, together with this property, could replace Axiom III. An adjacent footnote adds that it suffices to assume Axiom III for sufficiently small neighborhoods. The later literature that took up the memoir's program became known as the Lie-Riemann-Helmholtz-Hilbert problem of the foundations of geometry; a parallel tradition (Bachmann's reflection geometry) rebuilds the same geometries from the group generated by point and line reflections.

Problem Statement

Complete the axiom-analysis program that Hilbert explicitly left open for his 1902 motion-based axiom system for plane geometry. The system consists of: the definition of the plane as a system of points that can be put in one-to-one correspondence with the finite points of the number plane $\mathbb{R}^2$ or with a certain subsystem of it, in such a way that every point has a neighborhood (a Jordan domain in the number plane containing its image, all of whose points also represent points of the plane); the definition of a motion as an invertible continuous transformation of the plane into itself preserving the sense of traversal of Jordan curves; Axiom I (the motions form a group); Axiom II (every true circle, that is, the orbit of a point under the rotations about a fixed center, contains infinitely many points); and Axiom III (if there are motions carrying point triples in arbitrary proximity of the triple $A$ , $B$ , $C$ into arbitrary proximity of the triple $A'$ , $B'$ , $C'$ , then there exists a motion carrying $A$ , $B$ , $C$ exactly into $A'$ , $B'$ , $C'$ ). The program has four jointly defined parts: (a) determine which components of this system are dispensable, by an independence analysis of the definitional components (the correspondence with the number plane, including its subsystem and neighborhood clauses, and the orientation-preserving character of motions) and of Axioms I, II, III; (b) formulate the analogous motion-based axiom system for three-dimensional space and prove the corresponding characterization of its motion group; (c) determine the modifications by which the 1902 construction covers elliptic plane geometry; (d) answer Hilbert's footnote question: find the requirements with which the property used as an axiom in his 1901 Göttingen lecture — that any two points can never be brought into arbitrary proximity by a motion — together replace Axiom III, that is, give companion requirements such that, in the presence of the remaining 1902 assumptions, the 1901 property together with them implies Axiom III, so that Axiom III can be dispensed with; the companion requirements must be stated in the 1902 plane and motion vocabulary, and any companion that by itself already implies Axiom III in the presence of the remaining 1902 assumptions (in particular Axiom III itself or any restatement of it), or that presupposes the target conclusion — for example a conditional whose antecedent is the combined 1901-plus-remaining setup — does not answer the question, which asks what the 1901 property must be combined with. Parts (b) and (c) have already been carried out in the literature in different frameworks (de Kerékjártó 1928 and Montgomery-Zippin 1940 for space; Lubben 1928, Fischer 1933, Podehl-Reidemeister 1934, and Bachmann 1936 for the elliptic and spherical cases), while part (a) for the exact 1902 system and part (d) have no known resolution; part (a) must be carried out on the exact 1902 definitions, whereas for part (d) a complete answer must make explicit which framework (the exact 1902 definitions or a stated modernization) is being analyzed, since the content of 'replacing Axiom III' depends on this.

References

D. Hilbert, 'Ueber die Grundlagen der Geometrie', Mathematische Annalen 56 (1902), 381-422. DOI: 10.1007/BF01444168, https://doi.org/10.1007/BF01444168
D. Hilbert, 'Grundlagen der Geometrie', 7th edition, series Wissenschaft und Hypothese vol. 7, B. G. Teubner, Leipzig, 1930 (Anhang IV is the reprint of the 1902 memoir). zbMATH (JFM) record 56.0481.01
R. L. Moore, 'On the Lie-Riemann-Helmholtz-Hilbert Problem of the Foundations of Geometry', American Journal of Mathematics 41 (1919), 299-319. DOI: 10.2307/2370289, https://doi.org/10.2307/2370289
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