Perceptual axioms are irreconcilable with Euclidean geometry

Area of Science:

• Neuroscience
• Cognitive Science
• Philosophy of Mathematics

Background:

• Axioms are foundational truths in logic and mathematics, but Gödel's incompleteness theorems demonstrated inherent limitations in axiomatic systems.
• Previous research on axiomatic systems did not account for the brain's mechanisms for generating and mediating logic.
• Optical illusions present a unique opportunity to study the relationship between perception and formal systems.

Purpose of the Study:

• To investigate the potential irreconcilability between visual perceptual axioms and mathematical descriptions, specifically Euclidean geometry.
• To explore whether perceptual axioms can be adjusted to conform to cognitive (mathematical) systems.
• To provide evidence for the distinct and independent nature of visual perception and mathematical cognition.

Main Methods:

• Participants were presented with optical illusions, creating a discrepancy between visual perception and Euclidean geometric descriptions.
• Participants attempted to adjust their perceptual geometric axioms to match the Euclidean description.
• The degree of mismatch between perceptual and mathematical descriptions was measured.

Main Results:

• A significant and irreconcilable difference was observed between visual perception and Euclidean geometric descriptions of optical illusions.
• Participants were unable to make their perceptual axioms fully conform to the Euclidean description, even with awareness of the discrepancy.
• The degree of mismatch varied individually, indicating personalized perceptual axioms.

Conclusions:

• Perceptual axioms, representing perceptually true statements, cannot be accurately described by mathematical systems like Euclidean geometry.
• The logic of the visual perceptual system is fundamentally irreconcilable with the cognitive (mathematical) system.
• This irreconcilability suggests that different brain realities exist and are not necessarily updated by cognitive knowledge, challenging the notion of a single objective brain reality.