What is the philosophical significance, if any, of the fact that we can often find interpretations of arithmetic in other mathematical realms?
I got so excited that I just bought your book "Lectures on the Philosophy of Mathematics". I have two Brazilian books on mathematical philosophy, but they don't have as many topics as yours, besides, I found your way of narrating much more friendly and exciting.
Could the properties of the natural numbers be the properties shared among all the interpretations? I just think that there does not have to be one correct interpretation. The plurality is conceptually intriguing. Also, the idea of interpreting a theory in a background theory sounds like an embedding, and there are generally many ways to embed an object into an ambient space.