[Question] What are ranks? In your opinion.

[MphLee]

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

[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.

[MphLee]

Thank you, very interesting,
I hope also other forum user can share their mind. I need these info in finalizing my research. Since I'll have to explain it, and the audience of potential audience I'd like to reach is people made of people with experience in this topic, aka the tetration forum users too, I'm really interested in every opinion.

@daniel
can you expand on your point 2)? This seems an interesting philosophical position, I'd like to have more details.

About answer 3) I agree that a civilization able to master rank 4, 5, 6... would need incredible computational power available and/or a major conceptual and theoretical advance in how to manage functions of high complexity. Anyways, do you see, Daniel, the problem of fractional or even complex ranks to pose an even bigger obstacle, of do you see both big integers and non-integer ranks as having comparable difficulties for such an advanced civilization?

[Daniel]

...
@daniel
can you expand on your point 2)? This seems an interesting philosophical position, I'd like to have more details.

OK, more stuff. I understand renormalization to be the process in physics based on moving up to the successor hyperoperator. I suspect the result of this is that the hierarchy of physics, chemistry and biology mapping at some level to the hyperoperators. I'm sure physics has at least the complexity of tetration. Assuming tetration manifest in physics, it's complexity would be that of tetration at least. But it seems like the Universe would be well served by creating a computer to produce higher order hyperoperators and their physical manifestation. That would be us! So humans and other equivalent alien beings create the higher hyperoperators. But the mind is amazing, it was able to create Gödel's theorems. In the same way, I hope that the hyperoperators as a whole can be understood.

About answer 3) I agree that a civilization able to master rank 4, 5, 6... would need incredible computational power available and/or a major conceptual and theoretical advance in how to manage functions of high complexity. Anyways, do you see, Daniel, the problem of fractional or even complex ranks to pose an even bigger obstacle, of do you see both big integers and non-integer ranks as having comparable difficulties for such an advanced civilization?

While I have given thought to non-integer and complex hyperoperators, I have nothing to report. My difficulty is I have no expectation of what would be found in such systems, no tests to validate any possible solutions. I do think the integer hyperoperators including \[x\uparrow^\infty \infty\] would need to be understood, before complex hyperoperators because a complex hyperoperator expression would likely be created from the entire set of hyperoperators.

[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.

[Gottfried]

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?

1) For me as an old teacher in statistics the term "rank" has a special connotation: "ranks" are only positive and "rank_a - rank_b" has no specific meaning other than "rank_b" might be larger, equal or smaller than "rank_a". Like the table of football teams - there is better/equal/worse and ranks from 1 to 18 (say 18 teams in a league) but no negative rank. So a generalization of hyperoperations on the base of imagination of something like "ranks" makes it somehow odd in my personal neuronal wiring... . A better concept is perhaps that of "index": there is no co-connotation of positiveness, only that of integral numbers. But here I've been "educated" already by literature of L. Euler and later mathematicians, who introduced "fractional indexes" for instance fractional bounds and/or indexes for infinite sums, showing that such constructions might have a meaningful evaluation.      
One concept that formed/modified my idea of "hyperoperations" has been that of Markus Müller in the 90ies, who introduced -as a teenager- the idea of fractional indices for hyperoperations like "1-do nothing", "2-increase / 1/2-decrease", "3-add,1/3-subtract" , "4-multiply, 1/4-divide" and so on even respecting the multiple inverses when higher indexes are considered. This is already completely different from any notion of "rank" - but somehow attractive on its own.                  

2) The "deep meaning" - hmmm. My take so far, and I've at best a glimpse of a deeper meaning, is that the hyperoperations are growing by the idea of iteration. Like multiplication is basically iteration of addition, and so on. So the more basic machine is the idea of "iteration" . In contrast, I've not not yet arrived at such an wider idea like one of a (continuous?) "field-of-hyperoperations". The basic engine of the paradigma of "iteration" might be inherently too weak to model such a contiuous space of hyperoperations - and even if we are able to proceed to fractional iteration in some way. Maybe. Maybe... :-) (I remember some answer of Qiaochu Yuan in MSE a couple of years ago [1] [2] [3], where he criticized this notion of "hyperoperations-by-iteration" in a very lucide way; the third link is the one which initially rose my interest in his answers on this subject) But my practical approaches are - >>sigh<< (perhaps) - so far based on hyperoperations with index formed by the idea of iteration.                       

3) This follows from 2). Perhaps that basic paradigma of "iteration" as the parameter in the hyperoperations-hierarchy is not perfect/precise enough. We might be able to shape it some way up towards the operation of tetration as "iterated exponentiation" indeed as a useful tool for computation of some processes in reality, but this might be holy ( :-) I mean: plagued by holes). For a short reminder: iteration of a clock-arithmetic doesn't make it an object which could ever include inversion ...  

4) Could not say more than vague inspirations, not yet ready in a form that could be operationable in any way. Let's see ....

[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.
   

[tommy1729]

I have no opinion about unicorns if you know what I mean.

If a definition is not given I will not say what it should be.

ranks are defined in other contexts ofcourse but I see no good analogue.

Sorry to be skeptical.

regards

tommy1729

[MphLee]

I have no opinion about unicorns if you know what I mean.

If a definition is not given I will not say what it should be.

I'm not sure I understand what are you saying. Are you really saying that you have never seen a definition of indexed family of hyper-operations formal enough to have an opinion on the possibility of extending the argument of the indexing function, something that has often been called "rank", from natural numbers to larger sets? I remember you discussing the possibility of the half iterate of the superfuction operator, so I'm sure you know what here is meant by non-integer rank.

In other words, it seems to me that you claim that at this moment a precise definition of rank in the context of hyperoperations (see question 1) is lacking. Am I understanding your point?

Allow me to rephrase the other 3 questions to address your skepticism:

2') Is there some reason to be interested in the problem extending the \(n\) argument in expression like \(a\uparrow^n b\) from the natural number to non-integers?
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?

[MphLee]

OK, more stuff. I understand renormalization to be the process in physics based on moving up to the successor hyperoperator. [...]

While I have given thought to non-integer and complex hyperoperators, I have nothing to report. My difficulty is I have no expectation of what would be found in such systems, no tests to validate any possible solutions. I do think the integer hyperoperators including \[x\uparrow^\infty \infty\] would need to be understood, before complex hyperoperators because a complex hyperoperator expression would likely be created from the entire set of hyperoperators.

Thank you for you contribution Daniel. This is a very interesting view. I find the topic of renormalization completely beyond my understanding but overall I have a feeling that higher operations has something to do with some mathematical physics. Also your last line deserve to be considered as an option. It is an interesting analogy that JmsNxn explored: finding non-integer ranks as limit of infinite composition of integer ranks. If that was the case we would be in computational deep troubles....since that would mean practical inaccessibility of non-integer ranks.