How are permutations independent from arithmetic? (continued)

Thinking about last posts on permutations I realize there’s a slight contradiction in my sayings on this topic. In “Non permutational serialism (3)” I argued that the “reverse form is indubitably permutational because it is obtained easily  by composing S (or I) with the last element of the well ordered permutational set”. But in the previous article about “How are permutations independent from arithmetic?” I showed a way to understand permutations without even a call to the successor function. These two claims are not obviously compatible, since to get “the last element of the well ordered permutational set” you have indeed to make use of the successor function. Moreover, if we where to find a candidate permutation able to yield the problematic Inverse of S, there would surely be further arithmetic manipulations into the well ordered set of permutations in order to catch it. Nevertheless, finding an agent permutation within this set, is very different from computing each element of S. So as much as I have to amend my original statement about the independence of some permutational functions from arithmetic, I have to keep a distinct qualification for the kind of processes involved within the subset of permutational compositions that are targeted by this first declaration. The truth lies in the via media: permutational functions are not arithmetic in themselves, but finding distinguished permutations in their set, that prove able to produce other specific forms by composition, requires at present some arithmetic tasks. These tasks could nevertheless reveal only correlates to more elaborate permutational functions at some stage in the future, when we’ll be able to map more exhaustively arithmetic onto permutations.

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