Data structures are similar to variables. Algorithms to functions. Objects combine both data and algorithms. Is there a mathematical object / concept that combines variables and functions?
I think you are looking for type theory. It combines mathematical objects and proofs - proofs are constructive, meaning that a proof is an algorithm, consisting of symbolic manipulations.
I am a bit vague on the details, it takes some time to give a proper introduction and it is not my field.
So, in short, type theory and constructive mathematics is what you are looking for. In constructive mathematics, a proposition of the form "there exists an X ..." must also provide an algorithm to construct said X, not just proving its existence.