Monistic conception of geometry

TL;DR

This paper advocates a monistic view of geometry based solely on the world function, contrasting it with traditional pluralistic approaches, and discusses potential inconsistencies in generalized Euclidean and Riemannian geometries.

Contribution

It introduces a monistic conception of geometry centered on the world function and analyzes its advantages over pluralistic approaches, highlighting issues in generalized Euclidean and Riemannian geometries.

Findings

Monistic conception simplifies geometric description.

Generalized Euclidean geometry may be inconsistent.

Riemannian geometry faces potential inconsistencies.

Abstract

One considers the monistic conception of a geometry, where there is only one fundamental quantity (world function). All other geometrical quantities a derivative quantities (functions of the world function). The monisitc conception of a geometry is compared with pluralistic conceptions of a geometry, where there are several independent fundamental geometrical quantities. A generalization of a pluralistic conception of the proper Euclidean geometry appears to be inconsistent, if the generalized geometry is inhomogeneous. In particular, the Riemannian geometry appears to be inconsistent, in general, if it is obtained as a generalization of the pluralistic conception of the Euclidean geometry.