  We propose a new notion of positivity for topological field theories (TFTs),
based on S. Eilenberg's concept of completeness for semirings. We show that a
complete ground semiring, a system of fields on manifolds and a system of
action functionals on these fields determine a positive TFT. The main feature
of such a theory is a semiring-valued topologically invariant state sum that
satisfies a gluing formula. The abstract framework has been carefully designed
to cover a wide range of phenomena. For instance, we derive Polya's counting
theory in combinatorics from state sum identities in a suitable positive TFT.
Several other concrete examples are discussed, among them Novikov signatures of
fiber bundles over spacetimes and arithmetic functions in number theory. In the
future, we will employ the framework presented here in constructing a new
differential topological invariant that detects exotic smooth structures on
spheres.
