Why are there four letters in the genetic alphabet?

We list, without thinking, the four base types that make up DNA as adenine, guanine, cytosine and thymine. But why are there four? This question is now all the more relevant as organic chemists have synthesized new base pairs that can be incorporated into nucleic acids. Here, I argue that there are theoretical, experimental and computational reasons to believe that having four base types is a frozen relic from the RNA world, when RNA was genetic as well as enzymatic material.