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.

Godel's Flaw in the Constitution

There are stories about how when Kurt Godel was applying for citizenship he was going to tell the judge about a flaw in the constitution that would allow the United States to become a dictatorship. Einstein told him not to because then the citizenship might not be confirmed. The flaw was never publicly announced and so is the subject of speculation such as in this article, http://io9.gizmodo.com/the-mathematician-who-showed-how-the-us-could-be-made-a-1607024259 . The flaw is the thirteenth amendment.

"The Thirteenth Amendment (Amendment XIII) to the United States Constitution abolished slavery and involuntary servitude, except as punishment for a crime."

The constitution does not define what a crime is. The government defines that. Therefore the government can make a law that causes whatever group they want to be criminals. Those people would be subject to being literal slaves to the government turning the government into a dictatorship over those people. The people would have no legal way to fight back against the government because their right to vote is removed. So no right to vote means the government will not change. 

One could argue this is difficult to implement but Godel being a mathematician would not worry about implementing the solution. He would just worry if the solution was possible. Clearly by making everyone criminals the government would remove their rights to vote and most of the protection of the constitution. That would create a government that could not be voted out.