Book: Grundlagen der Geometrie

Overview
David Hilbert’s Grundlagen der Geometrie (1899) reshaped Euclidean geometry by presenting it as a rigorously axiomatized theory. Replacing Euclid’s informal diagrams and implicit assumptions, Hilbert isolates a small set of primitive terms, points, lines, and planes, and a carefully organized system of axioms from which all geometric theorems are derived. The work targets the structure of Euclidean geometry in two and three dimensions and pioneers a modern approach: meanings of the primitives are fixed only through the axioms, so any system of objects satisfying them counts as a model of geometry.

Primitive notions and method
Points, lines, and planes are left undefined, with relations such as incidence, betweenness, and congruence specified axiomatically. Definitions and theorems then proceed purely from these axioms using logical inference. This stance frees geometry from dependence on intuition or particular constructions, making explicit what counts as assumption and what counts as proof. The approach also clarifies the scope of geometric truth: it concerns all models of the axioms, not just the familiar space of intuition.

The axiom groups
Hilbert organizes the system into groups. Incidence axioms fix how points, lines, and planes determine one another (for instance, two points determine a unique line; three noncollinear points determine a plane). Order axioms, centered on betweenness and Pasch’s axiom, govern the behavior of points on lines and the interaction of lines with triangles, rectifying gaps in Euclid’s reasoning about intersection and separation. Congruence axioms codify equality of segments and angles, guaranteeing triangle congruence (such as SAS) without appealing to motions. The parallel axiom is adopted in a precise form equivalent to Euclid’s fifth postulate. Continuity axioms provide an analytic backbone, including an Archimedean principle and a completeness condition for lines, preventing pathological, non-Archimedean behaviors and ensuring that lengths can be coordinated with a complete number system.

Consistency, independence, and completeness
A central achievement is the logical articulation of three metatheoretic aims. Consistency is addressed by interpreting the primitives as sets of real triples and the relations as algebraic predicates, thereby reducing geometrical consistency to that of real analysis. Independence is studied through alternative models: by constructing structures that satisfy all axioms but one, Hilbert shows that no axiom is derivable from the others (notably, the parallel axiom cannot be proved from incidence, order, and congruence). Completeness is treated semantically rather than proof-theoretically: the axioms are strengthened so that the intended Euclidean model cannot be properly extended without violating them. In later editions Hilbert isolates this as a separate completeness axiom, yielding categoricity, up to isomorphism, any model of the full system is (real) Euclidean geometry.

Coordination with arithmetic
Hilbert demonstrates how coordinates can be introduced from the axioms, connecting geometry to algebra. By selecting a line as a number axis and defining addition and multiplication geometrically, one obtains a field structure; the continuity axioms ensure this field behaves like the real numbers. Conversely, analytic models over the reals satisfy the axioms, closing a circle that both clarifies the content of geometry and secures its relative consistency.

Impact and legacy
The book refounds geometry and inaugurates the axiomatic method that shaped 20th‑century mathematics, influencing model theory, formal systems, and later axiomatizations by Tarski and Birkhoff. It identifies precisely where Euclid’s arguments required supplementation, elevates Pasch’s insight on order to a cornerstone, and shows how geometric truth emerges from transparent assumptions. The resulting picture is a geometry that is conceptually clean, logically disciplined, and firmly connected to analysis, yet flexible enough to be realized by any structure satisfying the axioms.