  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.
