Extracting Computable Bounds (and Algorithms) from Classical Existence Proofs: Girard Domains Enable Us to Go Beyond Local Compactness

In classical mathematics, the existence of a solution is often proven indirectly, non-constructively, without an efficient method for constructing the corresponding object. In many cases, we can extract an algorithm from a classical proof: e.g., when an object is (non-constructively) proven to be unique in a locally compact space (or when there are two such objects with a known lower bound on the distance between them). In many other practical situations, a (seemingly) natural formalization of the corresponding practical problem leads to a non-compact set. In this paper, we show that often, in such situations, we can extract efficient algorithms from classical proofs -- if we explicitly take into account (implicit) knowledge about the situation. Specifically, we show that if we consistently apply Heisenberg's operationalism idea and define objects in terms of directly measurable quantities, then we get a Girard-domain type representation in which a natural topology is, in effect, compact -- and thus, uniqueness implies computability.