The Geometrical Theory of Communication: How Information Becomes Meaningful Through Algebraic Operations

Abstract:
Shannon’s mathematical theory of communication revolutionized our understanding of information transmission while
deliberately excluding questions of meaning and effectiveness. This paper addresses Shannon’s ”Level B” (semantic) and
”Level C” (effectiveness) problems by demonstrating that meaning emerges when information acquires both structure and
direction through complementary geometric operations. We prove that all information processing reduces to the funda-
mental form Ax = b, where differentiation extracts structure by establishing boundaries (A̸ = not-A), and integration
extracts direction by establishing relationships (A = A across contexts). These operations form a universal cycle: con-
tinuous data → differentiation → integration → information→ self-reference → behavior → new data. Our central result
proves that Gaussian distributions emerge naturally as projections of high-dimensional uniform distributions, explaining
their ubiquity across nature. This geometric necessity extends to a universal computational form: at each infinitesimal
moment, causal continuity requires that state transitions follow Ax = b. We validate this framework through critical brain
dynamics, linguistic universals, and information processing timescales across biological systems. This work represents a
short version of a larger project, and provides a rigorous geometric foundation for understanding how information acquires
meaning.
Keywords: Information theory; Language Processing; Gaussian Distributions; Complex Systems; Mathematical Philos-
ophy; Meaning Emergence; Interdisciplinary Studies