An Invitation to Higher Arity Science

TL;DR

This paper advocates for expanding scientific thinking beyond binary interactions to include higher arity concepts, especially ternary structures, which could revolutionize algebra, logic, and network models across disciplines.

Contribution

It introduces the concept of higher arity science, explores the mathematical and scientific implications of ternary interactions, and highlights open problems and future research directions.

Findings

Higher arity ideas are prevalent in various research areas.

Ternary structures reveal new algebraic and logical phenomena.

Open problems suggest the need for developing ternary mathematics.

Abstract

Analytical thinking is dominated by binary ideas. From pair-wise interactions, to algebraic operations, to compositions of processes, to network models, binary structures are deeply ingrained in the fabric of most current scientific paradigms. In this article we introduce arity as the generic conceptualization of the order of an interaction between a discrete collection of entities and argue that there is a rich universe of higher arity ideas beyond binarity waiting to be explored. To illustrate this we discuss several higher order phenomena appearing in a wide range of research areas, paying special attention to instances of ternary interactions. From the point of view of formal sciences and mathematics, higher arity thinking opens up new paradigms of algebra, symbolic calculus and logic. In particular, we delve into the special case of ternary structures, as that itself reveals ample surprises: new notions of associativity (or lack thereof) in ternary operations of cubic matrices, ternary isomorphisms and ternary relations, the integration problem of 3-Lie algebras, and generalizations of adjacency in 3-uniform hypergraphs. All these are open problems that strongly suggest the need to develop new ternary mathematics. Finally, we comment on potential future research directions and remark on the transdisciplinary nature of higher arity science.