How did non-Euclidean geometry change ideas about the foundations of geometry?

The development of non-Euclidean geometry fundamentally transformed the philosophy of mathematics by dismantling the long-held belief that Euclidean geometry was the unique, intuitive, and a priori description of physical space.12

Shift in Foundational Understanding

Prior to the 19th century, philosophers like Immanuel Kant argued that Euclidean geometry was a "synthetic a priori" truth—known through mental intuition and necessarily applicable to the external world. The discovery of consistent non-Euclidean systems proved that multiple, mutually exclusive geometries could be logically sound, which shattered the reliance on physical intuition as a basis for mathematical truth.234

The Move Toward Formalism

This realization forced a shift toward formalism, championed by mathematicians like Moritz Pasch and David Hilbert. Key changes included:45
* Axiomatic Rigor: Geometry was redefined as an abstract system of uninterpreted symbols and postulates. Truth was no longer derived from "self-evident" physical reality but from strict deductive inference.15
* Independence from Intuition: Mathematicians recognized that the validity of a system depends on internal logical consistency rather than its correspondence to sensory experience.52
* Separation of Disciplines: Geometry became bifurcated: it is either an abstract, analytic formal system or an empirical science used to model physical space, such as in general relativity.12