Integers: Irreducible Guides in the Search for a Unified Theory

The notion of final theory results from a contrasting understanding of physical reality. Currently, different approaches aim to unify the four forces of nature and discuss whether a final theory may be possible. A key feature of a final theory is irreducibility, however this property has not been seriously exploited. In the paper we present an irreducible mathematical theory that describes physical systems in terms of formation processes of integer relations. The theory has integers and integer relations as the basic elements and is irreducible, because the formation processes are completely controlled by arithmetic. We suggest properties of the formation processes as irreducible guides in the search for a unified theory.