How did David Hilbert make Euclidean geometry more rigorous through axioms?

In his seminal work The Foundations of Geometry (1899), David Hilbert transformed Euclidean geometry by shifting it away from reliance on physical intuition and diagrams, which had been identified as significant weaknesses in Euclid’s original Elements.123

Hilbert’s approach to rigor involved several key principles:

• Formalization: He argued that mathematical terms should be treated as meaningless symbols defined only by their relationships within a system of axioms. Validity was derived not from physical interpretation, but from the logical application of rules of inference.1
• Uncovering Hidden Assumptions: Hilbert identified and corrected implicit assumptions in Euclid’s work. For example, he refined Euclid’s postulates to ensure greater precision, such as explicitly stating the existence and uniqueness of a line through two points.4
• Consistency: He emphasized that the choice of axioms is arbitrary, provided the resulting system is self-consistent. This allowed for a clearer classification of geometries and provided a framework for exploring metamathematics.12

By treating geometry as a formal system, Hilbert successfully separated abstract mathematics from spatial intuition, establishing a model for the axiomatic method that influenced the development of modern mathematics.23