Aspects of Order

There is no overall social convention about what is order; much less about what would be an ideal order. In order to have an intellectual concept, which is neutral and accessible to all, we propose to use the results of a sort on a data set to declare that the data set is in order after the sort has been conducted. We point to the data set in its state as a sorted one and say “the data set is presently in order <…>”, where we insert between the symbols < and > the sorting criteria which were used to bring the data set into this specific order.

To keep the discussion simple and general at the same time, we use as elements of the data set things-as-such which have retained some of their qualities that is, are not completely indistinguishable from each other. To be precise, we use numbers in the range 1 to 16. Learning from Nature that there is a dichotomy of two versions of the same idea, we build pairs of the things-as-such and call these a and b. The tool we demonstrate concepts of order on is then a collection of numbers in the range 1 to 16, always two together, where we call the left one ‘a’ and the right one ‘b’. One may visualize the collection as summands in an addition of the form ‘a+b=c’. To put it simply, we generate the 136 smallest additions and sort and resort them.

The starting order of the collection will be dependent on the way we have programmed the loops which generate a and b. If the collection has been generated so, that the data set is ordered as (1,1), (1,2), (1,3), (1,4) and so on, we speak of an order SQ_ab, if the elements come generated in the sequence (1,1), (1,2), (2,2), (1,3) and so on, we speak of an order SQ_ba.

The sort has assigned to each element a place in a linear sequence 1..136. We see that element (1,1) is in both of the sorts we have encountered so far on place 1; place 2 is occupied in both orders SQ_ab and SQ_ba by element (1,2).

We shall return to the conflicting assignments of places to elements (and of elements to places) by means of imposing an order that is different to the presently prevailing order, as this search for compromise between logical contradictions is the main theme of the present essay. Before doing so, we shall introduce the aspects of additions we shall be using.

One has been told at the age of 6 by Teacher, that the important thing on mastering rational thinking is that one neglects the differences among the summands and between additions if only the result of ‘a+b=c’, namely c is the same. To look into the effects of a₁ ≠ a₂ while a₁ + b₁ = c = a₂ + b₂ has the emotional connotation that a) one has not understood what additions are all about, and b) one disregards Teacher’s instructions. Massive resistance against this proposal is to be expected.

We use the describing aspects of  a, b, c=a+b, u=b-a; k=b-2a, t=2b-3a, q=a-2b, w=2a-3b, s=(17-a+b) to demonstrate alternatives among order concepts.

We have now introduced the data set we are using to demonstrate concepts of Natural Information Technologies. Reader is invited to generate his version of the data set, as looking up the numbers simplifies understanding the following discussion on order and disorder.