How did Rudolf Carnap distinguish formal space from physical space and intuitive space?

In his 1921 doctoral dissertation, Rudolf Carnap addressed the prevailing confusion surrounding theories of space by arguing that scholars were often talking past one another because they were analyzing three distinct subjects. He categorized these as follows:1

• Formal space: This refers to the abstract, axiomatic systems of geometry. In this context, concepts such as points and lines are defined implicitly by postulates and are treated as uninterpreted, purely logical structures.12
• Physical space: This is the space studied by the natural sciences. It is an empirical subject where the validity of spatial theories depends on observation, measurement, and the physical structure of the universe, such as the Riemannian geometry utilized in general relativity.12
• Intuitive space: This category pertains to the subjective or psychological experience of space, which Carnap distinguished from the formal and physical domains to clarify the fundamental differences in how space is conceptualized across various disciplines.1

By separating these three concepts, Carnap sought to resolve the conflicts between competing spatial theories of his time.1