[Paper Review] More than discrete or continuous: a bird's view

Marius Buliga|arXiv (Cornell University)|Nov 19, 2010

Topological and Geometric Data Analysis12 references3 citations

TL;DR

This paper proposes that both physical reality and brain-based perception emerge from a primitive, non-geometric substrate through processes akin to differential calculus, using binocular vision as a foundational mechanism. It argues that the brain's local, intensive processing—modeled via dilation operations—can generate continuous geometric structures without assuming spatial geometry a priori, offering a unified explanation for perception and the emergence of spacetime.

ABSTRACT

I try to give mathematical evidence to the following equivalence, which is based on ideas from Plato (Timaeus): reality emerges from a more primitive, non-geometrical, reality in the same way as the brain construct (understands, simulates, transforms, encodes or decodes) the image of reality, starting from intensive properties (like a bunch of spiking signals sent by receptors in the retina), without any use of extensive (i.e. spatial or geometric) properties.

Motivation & Objective

• To investigate how continuous geometric structures in physics and neuroscience could emerge from a more fundamental, non-geometrical substrate.
• To challenge the assumption that spatial or geometric properties are necessary for perception or physical theory, proposing instead that intensive, local processes suffice.
• To establish a mathematical link between the brain's front-end visual processing and the foundations of differential geometry and calculus.
• To argue that binocular vision provides a minimal, biologically plausible mechanism for generating the algebraic structures underlying geometry and continuity.
• To reframe the discrete vs. continuous debate in physics and neuroscience as a problem of representation and computation, not ontology.

Proposed method

• Model the brain's visual processing as a syntactic, deterministic machine operating on intensive properties (e.g., spiking signals), avoiding explicit spatial representations.
• Use the concept of 'dilation gates'—derived from binocular comparison of visual inputs—as the primitive computational unit analogous to logical gates in Boolean logic.
• Formalize the brain's local, parallel processing as a system of algebraic identities that encode geometric invariants without assuming prior geometry.
• Apply this framework to the front-end visual system, where columnar organization supports local, intensive representations of structure.
• Demonstrate that the algebraic closure of dilation operations can generate the formalism of differential geometry and calculus.
• Use the metaphor of a binocular explorer jumping between points to model how a non-geometric system can reconstruct geometric structure through relational comparisons.

Experimental results

Research questions

• RQ1 Can the emergence of continuous geometry in physics and perception be explained without assuming spatial or geometric primitives from the start?
• RQ2 How does the brain construct a coherent, continuous perception of reality from discrete, intensive neural signals?
• RQ3 What is the role of binocular vision in enabling the brain to generate geometric structure from non-geometric, local inputs?
• RQ4 Can the mathematical formalism of differential calculus and differential geometry be derived from a minimal computational process based on comparison and dilation?
• RQ5 Is there a universal, undedicated mechanism in neural processing that allows the construction of geometric structure from intensive, local data?

Key findings

• The brain’s front-end visual system operates as a deterministic, syntactic transformer that encodes structure without semantics, relying solely on intensive properties and local representations.
• The columnar organization of the primate visual cortex supports a massively parallel, local representation system that avoids explicit spatial geometry.
• Dilation operations—derived from binocular comparison—serve as a primitive computational gate analogous to the NAND gate in Boolean logic, forming the basis of differential geometry.
• The algebraic closure of these dilation operations can reproduce the formalism of differential calculus and differential geometry, suggesting that geometry emerges from non-geometric computation.
• The brain’s ability to simulate continuous perception from discrete neural spiking signals is not a flaw but a feature of a deeper computational principle rooted in binocular comparison.
• The paper provides a mathematical framework in which both physical reality and perception emerge from a non-geometric substrate through a process analogous to the brain’s local, intensive processing.