[Question] What are ranks? In your opinion.

[bo198214]

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?

1.   I think of addition has rank 1, multiplication has rank 2, exponentiation has rank 3, and I would continue in a similar fashion to higher ranks. For non-integer ranks I would assume some smoothness moving operation k into operation k+1.
   
2. There might not be any importance of it. It is just the mind that wants to know the unknown
   
3. Maybe there is no obstacle it is more about how to define this smoothness I was mentioning above. Might be several definitions possible. And typically the most satisfying is the one hardest to solve.
   
4. To me this smells a bit like operator theory, but with non-linear operators.