It's not so much that programs are isomorphic to proofs; it's that mathematics and programming are the same activity. It's essentially a historical accident that the people who practice this activity have ended up in different groups, and that those groups use different terminology for the same things ("program" vs "proof", "type" vs "proposition", "execute" vs "simplify", "tuple" vs "conjunction", "union" vs "disjunction", "function" vs "implication", "dependent function" vs "universal quantification", "interface" vs "existential", "programming language" vs "logical system", etc.).

All of the relationships between these are the same too (e.g. a program `p` has type `T`, iff the proof `p` proves the proposition `T`; the proof `w` proves that the existence of `x` implies the proposition `y AND z` iff the program `w` is a function which generates a tuple of `(y, z)` when given an argument implementing the interface `x`; and so on).

Mathematicians and programmers do care about different things. Mathematicians are more concerned with informally proving previously unknown results; which corresponds to something like learning that a type has a value by sketching some pseudocode. Programmers care deeply about how many steps their proofs (programs) take to simplify; how easy it is to modify a proof to prove a slightly different proposition, etc. Yet there are so many good ideas on each side that are immediately applicable (e.g. dependently typed languages take rigorous logical systems and view them as programming languages; proof engineering takes software engineering ideas and applies them to building and maintaining proofs).

I'm struggling to think of an activity involving rock gardens that corresponds precisely to, say, universal quantification.