Symmetry as a common origin of hierarchical emergence and abstraction

Many systems of interest exhibit nested emergent layers with their own rules and regularities, and our knowledge about them seems naturally organized around these levels. How do such hierarchical structures arise, and why should observers track these hierarchies so closely? This paper proposes symmetry as an answer to both questions. By combining principles from information theory, group theory, and statistical mechanics, it is shown that dynamical processes that are equivariant with respect to a symmetry group give rise to multiple emergent macroscopic levels arranged according to the algebraic structure of the symmetry. The same symmetries shape the Bayesian beliefs of an observer with only partial access to the system, yielding a matching hierarchy of “abstract” beliefs that can be updated autonomously at different levels of resolution. These ideas are illustrated in Hopfield networks, where familiar macroscopic quantities emerge from known symmetries, and in Boolean models of gene regulation, where symmetries are instead found from the dynamics. The computational advantages provided by symmetry are showcased in Ehrenfest diffusion and epidemic modeling, where abstract beliefs enable Bayesian inference over exponentially large state spaces. These results identify symmetry as a sufficient mechanism for hierarchical emergence and—at the same time—for the capacity of abstract beliefs to mirror the systems they track, making feasible inferential problems that would otherwise be intractable.