  Theoretical and computational frameworks of modern science are dominated by
binary structures. This binary bias, seen in the ubiquity of pair-wise networks
and formal operations of two arguments in mathematical models, limits our
capacity to faithfully capture irreducible polyadic interactions in
higher-order systems. A paradigmatic example of a higher-order interaction is
the Borromean link of three interlocking rings. In this paper we propose a
mathematical framework via hypergraphs and hypermatrix algebras that allows to
formalize such forms of higher-order bonding and connectivity in a parsimonious
way. Our framework builds on and extends current techniques in higher-order
networks -- still mostly rooted in binary structures such as adjacency matrices
-- and incorporates recent developments in higher-arity structures to
articulate the compositional behavior of adjacency hypermatrices. Irreducible
higher-order interactions turn out to be a widespread occurrence across natural
sciences and socio-cultural knowledge representation. We demonstrate this by
reviewing recent results in computer science, physics, chemistry, biology,
ecology, social science, and cultural analysis through the conceptual lens of
irreducible higher-order interactions. We further speculate that the general
phenomenon of emergence in complex systems may be characterized by
spatio-temporal discrepancies of interaction arity.
