[Paper Review] Logical reloading as overcoming of crisis in geometry

Yuri A. Rylov|arXiv (Cornell University)|May 11, 2010

Advanced Theoretical and Applied Studies in Material Sciences and Geometry9 references3 citations

TL;DR

This paper proposes logical reloading—a reformulation of Euclidean geometry using different foundational concepts—as a solution to the crisis in modern geometry, where geometry is reduced to abstract logical constructions rather than geometric intuition. By shifting to a world function-based ($\sigma$-representation) formulation, the approach restores geometry's original focus on shape and spatial relationships, enabling the construction of nonaxiomatizable, multivariant geometries essential for modeling physical space-time in microphysics and cosmology.

ABSTRACT

Properties of the logical reloading in the Euclidean geometry are considered. The logical reloading is a logical operation which replaces one system of basic concepts of a conception by another system of basic concepts of the same conception. The logical reloading does not change propositions of the conception. However, generalizations of the conception are different for different systems of basic concepts. It is conditioned by the fact, that some systems of basic concepts contain not only propositions of the conception, but also some attributes of this conception description. Properties of the logical reloading are demonstrated in the example of the proper Euclidean geometry, whose generalization leads to different results for different system of basic concepts.

Motivation & Objective

• To address the crisis in modern geometry, where geometry is treated as a purely logical construction rather than a science of shape and spatial disposition.
• To demonstrate that logical reloading—replacing foundational concepts without altering geometric content—reveals different generalization potentials across representations.
• To show that the $\sigma$-representation, based solely on the world function $\sigma$, maximizes generalization capacity and enables multivariant, nonaxiomatizable geometries.
• To argue that physical space-time geometry, especially in microphysics and cosmology, requires non-Riemannian, discrete geometries beyond conventional differential geometry.
• To advocate for a return to metric geometry as a physical foundation, now empowered by the deformation principle and $\sigma$-representation.

Proposed method

• Logical reloading is formalized as a logical operation that re-expresses the same geometric conception using different basic concepts, preserving all propositions but altering generalization potential.
• The $\sigma$-representation uses the world function $\sigma(P,Q) = \frac{1}{2}(\rho^2(P,Q))$ as the sole primitive, replacing vector spaces and coordinates.
• Geometrical objects like spheres, ellipsoids, and segments are defined purely via distance relations, such as $\mathcal{T}_{[F_1F_2]} = \{ R \mid \rho(F_1,R) + \rho(F_2,R) = \rho(F_1,F_2) \}$.
• The deformation principle is applied to the Euclidean $\sigma$-representation to generate non-Euclidean, multivariant geometries that are nonaxiomatizable but physically meaningful.
• The paper uses Heron’s formula and area expressions in terms of world functions to analyze geometric invariants under logical reloading.
• Finite difference equations replace differential equations in dynamics, reflecting the potential discreteness of physical space-time geometry.

Experimental results

Research questions

• RQ1 How does logical reloading—replacing foundational concepts—alter the generalization capacity of a geometric theory?
• RQ2 Why does the $\sigma$-representation of Euclidean geometry allow for greater generalization than other representations?
• RQ3 What happens to geometric objects like cylinders when logical reloading is applied in non-Euclidean, multivariant geometries?
• RQ4 Can a geometry that is nonaxiomatizable still possess geometricity—i.e., describe shape and mutual disposition of objects—without relying on logical construction?
• RQ5 How does the deformation principle, when applied to the $\sigma$-representation, generate physically relevant, non-Riemannian geometries for microcosm and cosmology?

Key findings

• Logical reloading preserves all geometric propositions but changes generalization potential, with the $\sigma$-representation offering maximal generalization capacity.
• In the $\sigma$-representation, the Euclidean geometry is fully described by the world function $\sigma$, enabling a pure metric formulation without auxiliary structures like vector spaces.
• The segment $\mathcal{T}_{[F_1F_2]}$ defined by $\rho(F_1,R) + \rho(F_2,R) = \rho(F_1,F_2)$ is one-dimensional in Euclidean geometry but becomes a non-one-dimensional surface when the triangle inequality is violated.
• In multivariant geometries, cylinders $Cl_{PF_1F_2}$ and $Cl_{PF_1F_3}$ with $F_3$ on the same segment differ in shape, showing that the Euclidean cylinder splits under deformation.
• The equation $Cl_{PF_1F_2} = Cl_{PF_1F_3}$ in Euclidean geometry is interpreted as a degeneration, not a structural identity, highlighting the role of auxiliary structures in conventional geometry.
• Physical space-time geometry is better described by finite difference equations and non-Riemannian, multivariant geometries, which arise naturally from the $\sigma$-representation and deformation principle.