The Emergence of Units: A Unified Theory of Numbers, Atoms, and Organisms

We present a unified framework that explains the emergence of
discrete units across mathematics, physics, and biology. The theory
combines Additive Fields of Natural Numbers (AFN)—a
structural description of additive semigroups in terms of atoms, rank,
entropy, and regularity—with Potentially Finite Set Dynamics
(PFS)—a probabilistic formalism for growth and stabilization processes.
Our central thesis is that natural numbers are not abstract
Platonic forms nor human constructions, but emergent stable states
of additive dynamics. Prime numbers correspond to minimal-entropy
additive atoms, while composite numbers represent excited states with
higher structural entropy. The framework reveals that the discrete,
quantized nature of reality—from elementary particles to biological
individuals—stems from the same fundamental additive stabilization
principle. This represents a philosophical revolution (mathematics as
manifestation of world-structure rather than description) and provides
a novel mathematical foundation (numbers as process endpoints rather
than primitive objects). The theory has direct implications for the
philosophy of mathematics, theoretical physics, and biology, offering a
unified explanation for why discrete units appear across nature.