5. Clans
Step Five of Learn to Count in Twelve Easy Steps
Reminder Step 4:
Ordering the data set on all pairs of aspects brings forth 72 variants of realisations of the order principle based on {<|=|>}. Some of the sorting orders are identical, some contradictory. The contradictions are visible on {place|amount|frequency|order}. The task is to consolidate the contradictions.
Step 5:
Identities:
We group those sorting orders together, which assign identical sequential numbers to each addition. To do so, we create Vector V (please see 5.num ). Vector V is 72*72 long, as for formal reasons we allow comparisons of a sort with itself, too. Each of the cells of Vector V contains the result if (SQαβ=SQγδ, .t.,.f.).
Sequencing:
The sequence of elements in Vector V depends on the sequence of steps that drive the comparison loops. Producing Vector V is driven by the sequence of the first-level arguments {a,b,c,k,u,t,q,s,w}. Changing the sequence of comparisons will result in a differing representation of the results.
Clans:
We call a group of sorts that sort in an identical fashion a ‘clan’. Each clan has a chieftain, who gives the clan its name. As the members of the clan are indistinguishable with respect to the places they assign to the elements, the members of the clan can carry a common name. The first of the members we encounter in the course of the comparison is the chieftain, who lends his name to all the members of the clan. We point out that in the variant of Vector V present in 5.num as the result of the specific permutation of first-level arguments {a...w}, we find 20 different clans. This is reminiscent of one of the findings of applied science, where 20 basic building blocks of organic material are pointed to by a surjection from 64 logical cases. The accuracy of the 72 as such distinguishable realisations of order is lost in translation; thereafter only 20 factually distinguishable groups remain. The overview of the clans’ strength is in table Klan of 5.num.
Teeth:
The sorting aspects finely comb the data set differently. Aspects a and b, both have 16 gradations. Aspect c=a+b possesses 31 different subdivisions. Yet, obviously, SQab = SQac, that is V[ab,ac]=.t., where if b increases, a+b increases, while a remains the same. The differing number of teeth of the aspects will come into play when investigating, in which sequence the ordering was made (described in step ten).
Ties:
The denticulation of the aspects is visible in the context of ties. A tie with respect to aspect a can be pointed out in a deictic fashion. Under sorting orders ab, ac, ak, ...., aw,the elements (1,1), (1,2), (1,3) ... (1,16) are in a tie with respect to aspect a, as these elements all share the first place equally. Generally, the place of an individual element within a tie can be predictable, as it depends on the history of having been sorted previously on a differing aspect. Step 5 shall only introduce the concept of ties and prepare the discussion (in Step 10) of changes in a linear sequence having consequences in more dimensional readings of the assembly.
Cartography:
We offer a planar graphical representation of Vector V. The rows and the columns of Table 3 – see 5.num – are SQab to SQws, in this order. This presentation allows for the diagonal to be tautologically .t.. For ease of understanding, we have coloured mod(4) the clans differently, as seen in.
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