[Question] What are ranks? In your opinion.

[Shanghai46]

1) What do you think when you ear the term rank in the context of hyperoperations?
2) What do you think is the deep meaning and importance of the rank parameter?
3) What is, in your opinion, the main obstacle to solving the problem of non-integer ranks?
4) How do you see the mathematician of the future surpass this obstacle, if they ever manage to do it at all. In which field of mathematics do you see the key to the solution residing in?

3) for me the main obstacle is that doing so doesn't really make any intuitive sense. A bit like \(e^i\) which in the context of the first definition of exponentiation mean \(e*e*e*... \) i times, which doesn't make sense. Also, there's no real nice property we would use to easely extend fractional iterations and ranks of hyperoperations. Like associativity or commutativity.