*Properties of Places and of Objects*
*We have seen in this chatroom that there is recognisance of the need to come up with something dramatically new and innovative in the field of information theory, but also, that having been educated in a specific fashion, it is not easy to think new thoughts.* *So, if you kindly allow, I shall wrap the new concept in such ideas and words which are not yet filled up and reserved with abstract meanings. This is a technique which was used by Swift, as he discussed he relationships between rulers and ruled, and by Lewis Carroll, when discussing some logical syllogisms, to mention but two of the classical writers. Some say that Aesop’s and La Fontaine’s fables are also among the didactic fairy tales.* 1. *What are the objects we abstract from: what the Sumers did* *We are in Sumer, listening to the discussions among the king(s), princes and wealthy courtiers. They decide that they will invent now the positional algorithm, meaning that where a symbol is placed has as much influence over the meaning of the resulting logical statement as which symbol it is. (Simplified example: if A means 1, B means 2, C means 3, the invention of the day is that ABA means: 121 and not: 1 furthermore 2 furthermore 1. The example is simplified: they did not use the decimal system. Look up Wikipedia for a more exact explanation of the principle.)* *What are the objects that the scientists have agreed on to use as things that can have places? Not every one of them had camels, not every one of them had date trees, not every one of them had bushels of wheat. What each of them had was a harem with at least – say – 60 ladies in each of the harems.* *Now they had a set of objects the individual properties of which could be dismissed; what can be agreed on is then: only the number and therefore the position of the objects: herewith delivering a solid epistemological basis for: so many identical objects to the left (or right) of it, so many places to count. This results in the property of the i-th lady to have the property of lady number i. * *It is not the job of the narrator to speculate about the experiences of the reader with a large number of ladies, but in those long bygone days there could have been agreement among the wealthy and speculative-minded Sumerians, that this method saves a lot of discussions on who is the prima donna and why. * *The de-individuation of the individual objects comes as a side-effect of assigning places as individuating properties to objects. Lady X is that lady who comes the 4th night hence and that week where ladies ABACBAC follow each other is different to that week where ladies BCAACBC offer their charms.* *To be able to actually use the positional assignment based counting, it is necessary to go through 3 steps of abstraction:* *a) **De-individuate the objects by assigning one absolute ranking to them across all harems (women nr. 1: women represented by symbol A, women nr. 2: women represented by symbol B, etc.);* *b) **Individuate the places by enumerating them 1, 2, 3, …. This they were able to do, because they have discovered the rules connecting the 28 nights of the moon and some particularities of women and the 12 months and the year: that is, they were able to enumerate in a temporal sequence, which they then transferred to a linear (geometric) sequence;* *c) **Individuate the permutation based on a sample with replacement, which is the method what we use till today to arrive at a picture of a number. (We draw any of the symbols A,B,C,… and put it on place 1, then we draw again from the same universe, obviously having replaced the element we had drawn before, so it is again available. This 2nd element we put on place 2. Then again we draw an element, again doing so as if there was an endless supply of symbols, thinking ourselves to have replaced the sample drawn. This fallacy of our imagination will entertain us much when discussing the genetic information stricture.)* 1. *What we can improve on the Sumers: what they had no way of doing* *Had the Sumers been of such gentle and wise disposition as we are, they had done the following (also, they would have needed paper, pencil and computers):* 1. *We establish the maximal number of describing aspects of the objects* *We of course know that of a limited number of different objects, only a limited number of distinct logical sentences can be said (after a while, one will start repeating oneself. The maximal number of distinct descriptions of a set containing n objects – as can be read off OEIS/A242615 – is the number of partitions of n, raised to the power of the logarithm of the number of partitions of n. For all practical purposes, one will establish this upper limit by calculating n!, building ln(n!) and creating sqr(ln(n!)). This is the number of independent describing dimensions and agrees for n<136 quite exactly to ln(p(n)), where p(n) = number of partitions of n.)* *One can visualise this upper limit as the vertical depth of the logical sentence describing the set, where the vertical depth is understood to mean the number of sub-sentences nested within the main sentence, which sub-sentences have the form: of among which i are concurrently included in groups of cardinality k, etc. * *The horizontal width refers to the number of same-level mutually exclusive subgroups that are built by imposing one of the – roughly – sqr(ln(n!)) mutually independent describing aspects.* 1. *We place each element into one of the horizontal groups* *To return to the easily imaginable objects the Sumers have abstracted from, we make a catalogue of the objects according to some properties of the objects. These properties could be, e.g. sweetness of breath, likeness of the face to the Moon, pearl-comparable shine of the teeth, fire in the eyes, silkiness of the skin, circumferences a,b,c (for instance: wrist, ankles and neck), circumferences d,e,f (find your own), etc. (The reader is recommended to study the works of great poets in his or her own cultural tradition, e.g. Song of Songs, etc., for suitable classifying aspects). The categories (=gradations) within these horizontal groups are mutually exclusive. There can not be more gradations per aspect as there are objects.* 1. *We connect the elements across vertical groups* *Each of the ladies is now characterised – and therefore individuated – by the assembly of symbols which are not mutually exclusive. For instance, Lady X may be of excellent sweetness of breath, moderate likeness of the roundness of the face to the Moon, poor shine of the teeth and good fire in the eyes, etc. * *The differing enumerations of the categories within the horizontal and the differing enumerations of the aspects among the vertical describing dimensions are but mirroring effects, giving different appearances to the same underlying position in an sqr(ln(n!))-dimensional space. The important characteristics to pay attention to is the width of the horizontal group (how many other elements share that symbol).* 1. *We watch the patterns of re-appearances* *If we think the distribution of the category widths to be roughly close to the general rule: many is frequent, few makes infrequent, we will be prepared to find logical archetypes which arise from being in a broad category as a matter of probability in aspect A while being in a moderately broad category as a concurrent matter of probability in aspect B, etc. etc. * 1. *Advantages of using a more complex assignment method than the Sumers* *By this method we have maintained the individuality of the objects and have not flattened out their immanent differences. Learning is, as we all know, based on deepening potential associations. In order for the dog to learn that whistle means food coming, there must exist the potentiality of a connection between whistle noise and smell experience. * *One can’t imagine any kind of intelligence, be it artificial or natural-instinctive, without assuming that associations can exist. Therefore, logical objects need to have some innate, immanent, intrinsic relationship among each other.* *The Sumer method – which is brilliantly reproduced by the Shannon algorithm – makes objects uniform. Uniform objects can’t have different associations among each other.* *In order to be able to understand learning, one has to go back and find out, where our forefathers have – lacking the tools to do otherwise – had to help themselves by simplifying the complexities which we cannot evade addressing.*
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