[Question] What are ranks? In your opinion.

[Daniel]

The field of study of hyper-operations is relatively young. In its current width we cannot say it's older than 20 years.
As every young field, even more for fields unknown to the mainstream community of mathematicians, we have a situation in which there is not an established school of thought, a defined glossary of terms and standard definitions.

It is not a secret that I regard the problem of defining ranks and hyperoperation as the core of my research, and that I plan to bring such a needed unification of terms and formalization. Stay tuned, I'll be dropping a major update from november to late december.
I also expect, for the reasons explained in the first paragraph, that every member of this forum will have different answer to the questions of what are ranks and what are hyperoperations in general.

I hope you can help me learn your position by answering in few words to the following points.

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?

I don't need formulae, just the first thing that pops in your head. Take it as a philosophical chat.

I thank you in advance.
Regards

1) The third argument of the Ackermann function.
2) It is the number of levels at which mathematical and physical reality are interconnected. I believe our Universe has a rank parameter, although it could possibly be infinite. Our very ability to conceive an infinite rank hyper operator may be part of the reason why there could be one. Real science fiction stuff.
3) I've had to learn a surprisingly diverse number of branches of mathematics to support my research into tetration - complex dynamics, category theory, combinatorics are some. Higher rank operators may require much deeper integration. I have imagined that alien civilizations could be identified by the rank they have mastery of.
4) I think the answer to your question is the assimilation of enough required mathematics to push the entire system to a higher level of sophistication. Langlands on steroids.