12/18/2022, 09:08 AM
(10/18/2022, 07:31 AM)Gottfried Wrote: 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 ....
@Gottfried: Excuse me for the long delay. I hadn't enough time to share my comments to your great answers. Really interesting indeed!
1) You are right in some sense. Index seems more appropriate but also too vague. Rank is really cemented in my mind as something related to hyperoperations... mainly because as binary operations can be arranged into a hierarchy based on "computation hardness/complexity" a position of an operation inside such a hierarchy should be termed it's rank inside the hierarchy... but also the various levels of the hierarchy should be called ranks? This choice is supported by well established logic-set-theoretic usage.
I like a lot the linguistics and terminological problems in mathematics and the issues you bring to the discussion are really interesting.
Also the term hyperoperations is problematic because there is a completely unrelated branch of mathematics, at least 30 years old and now much more mainstream than our beloved hyperoperations, that deals with multi-valued operations and claims the term.
Also rank is used in many other places: in linear algebra it is the vector space dimension of the image of a linear map... but then also the dimension can be generalized to be non-integer.
Maybe a totally new term should have been coined for the "grade" of an hyperoperation. But how can we fix this?
2/3) What do you have in mind when you say "The basic engine of the paradigma of "iteration" might be inherently too weak [...] Perhaps that basic paradigma of "iteration" as the parameter in the hyperoperations-hierarchy is not perfect/precise enough."? I kinda feel the same way... it is a vague and mathematical unsubstantiated feeling but I'd love to know if you can expand on this.
Also Qiaochu Yuan makes some good points that any foundational work on hyperoperations needs to be pair with Devlin's critique of "multiplication as repeated addition"...
@Catullus: Thanks for the contribution.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)