Why I like hash table and neural nets

Most mathematicians do no like hash tables. They prefer trees. All kinds of trees. The prefer trees because theorems can be proven on the properties of trees. Hash tables are not amenable to mathematical analysis. That makes hash tables uninteresting to mathematicians. I like them because of that because they are an example of what happens at the edge of computability.

In 1931 Kurt Godel proved an incompleteness theorem. The theorem is that any system complicated enough to represent integers has true statements that that cannot be proven to be true. I believe that hash tables are a construct that has true statements that cannot be proven to be true. Statements that would be fundamental to showing the effectiveness of hash table are not provable. They are cool because they are clearly defined and effective yet remain hidden from analysis using the foundation of mathematics, First Order Logic. We use them and they work great, are relatively simple and yet not analyzable. That is very cool. Neural nets are the same way.