  We reinterpret various properties of Noetherian local rings via the existence
of some $n$-ary numerical function satisfying certain uniform bounds. We
provide such characterizations for seminormality, weak normality, generalized
Cohen-Macaulayness, and $F$-purity, among others. Our proofs that such
numerical functions exist are nonconstructive and rely on the transference of
the property in question from a local ring to its ultrapower or catapower.
