Formal proof typically refers to a branch of mathematics / philosophy concerned with the symbolic manipulation of formal systems in order to prove axioms about

Formal proof typically refers to a branch of mathematics / philosophy concerned with the symbolic manipulation of formal systems in order to prove axioms about representations of real world objects. For example, a system like Coq (https://coq.inria.fr/ https://coq.inria.fr/) is concerned with formal proofs, but is not really a foundation for most of science.