7. Standard Chains     

Running Battle

At any moment, we may decide that we declare sorting order αβ to be relevant. The relevant order to be existing, we may effect a sort on αβ, resulting in the data set being in order αβ. The order αβ is then a fact; we say “αβ  is the case”. The order αβ we have declared to be relevant is recognisable on the fact that the subscripts αβ appear in describing sentences relating to the data set. Those potential orders that are not relevant will not appear as subscripts in the sentences describing the data set. (In this sense, we follow Wittgenstein’s dictum that one does not talk about things that are irrelevant.) The possible orders that are not relevant, are a contrast-background to it, these constitute that what is not the case. Relative to one specific sequencing of the elements, there will be 19 other sequencings that are in deviation to this specific one, which is relevant in the moment. The concept of an overall quasi-constant of system stability describing the total amount of displacement with respect to misallocation may be helpful. In all cases of having selected a sorting order, an overall coefficient of misplacement is calculable. There is a quite stable proportion of what is the case to what is not the case.

Political Solution

The compromise solution between the requirements of

αβ ‘on p₁, a₁ should stay’

and of

γδ ‘on p₁, a₂ should stay’

(and of their pairs ‘a₁’s place is p₁,’ etc.) is accessible to the human brain by our ability to look into the future and predict what will happen. Human culture is based on our ability to learn, to have a memory and to have inner pictures predicting what will happen. The cultural inventions of the past and of the future are well established in our thinking. One may then point to a piece of logic and say, “I call this Future.” There is a cultural solution to dealing with contradicting claims regarding places and amounts, a diplomatic compromise by pushing unresolved issues off into the future. Maybe these contradictions will cease to be relevant and the conflict will somehow resolve itself. That kind of future, where this contradiction will become critical for the stability of the system, will maybe not happen at all, so why worry?

Multitude of Arrival Times

The table as it stands (in 4.num) is a frozen moment in time. The spectator moves as he chooses any αβ and says, “This is now relevant; it is relative to this that I calculate the deviations; this is the case now.” Using the chains we can say in how many steps of how many strings the reordering into γδ will be achieved. If there are chains of differing lengths in this reorder, the state of reorder will be achieved in a protracted fashion, with as many results as there are differing lengths among the chains that produce the reorder. As the chains have different lengths, the complete reorder will not be observable after one specific number of steps. The mental picture of partly arrived, partly not arrived yet may choose among ideas on how to explain a logical state being achieved peu-à-peu, during a linear sequence which we may well experience as time. Either one will see the elements as having arrived at their correct destination and there staying put, or one will assume that the shorter chains run more slowly or more often. The concept of the smallest common multiple may come into play.

Standardizing

During the transmission of the genetic information, Nature works by using triplets-based units. It is the sequence of the triplets that translates a one-dimensional realization of the order (the sequence of the triplets of the DNA) into a three or more dimensional object that is the living organism. Table T contains variants of reorderings where the series of place changes happen in three elements changing place (see 7.num).

The standard chain connects three amounts with three places within a standard reorder; the standard reorder consists of 45 standard chains, the remaining logical statement is 6+11=17. This is the, unique, central element and also the average of every standard chain.

Grammatical Rules

We expand the scope of the investigations of the Tractatus by allowing sentences of the form, ‘Z can be the case’ and ‘Y will be the case’ to be valid. We propose to call that place of a standard chain which is closest to the central element its X-corner, the second closest to be called the Y-corner and the farthest the Z-corner of a triangle drawn on a plane of which the axes are S_αβ and S_γδ, the prefix S_ meaning that the orders come from among the standard reorders.

In other words, we allow for the past and the future to be connected with the present within the moment, in the same way that the places among the elements of a standard chain are connected. We know what can be the case because we have encountered it before – or have imagined it by using rules that have been proven to be valid rules of grammar, that is, we have relied on experience. Of that what can be the case we do not know whether it will be the case again. We distinguish among the possible that what is certain. Thereby, we allow logical sentences to be more true than others: this is what will be the case. Concurrently, we allow logical sentences to be less true than others: these detail what can be the case, without the added certitude that they will be the case again. presently, we know it to be different to that what will be the case. Common to the past and the future is that they are not the case; they are distinct by our ability to describe the difference between what we are sure about and what we know. (There is a difference between knowing that one knows a telephone number and the content of the attention being the correct number. The closed, or “packaged” contents of the memory and the contents after ekphoresis are logically different. The grammar of *.zip files and of *.doc files is different.

The grammar of the logical language allows compromises among statements relating to what is where. Against the gain of flexibility stand some local costs within the tautology of the language. Roughly two thirds of Sachverhalte are in any moment not the case in the strict sense, and yet we can still talk about them, because one of the missing thirds were the case just an instant ago, and most of the remaining third will evidently be the case in an instant again. The compromise between logical contradictions consists of pushing presently unsolvable conflicts into the not-now, experienced as future, slicing them up and not talking about them.

Minkowski

The numbers in the table support Minkowski’s model (please see 7.graph). If one thinks the time slice to be one Sachverhalt thick, one will use that corner of the triplet that is presently the case. Maybe it is practical to think the Minkowski-moment to be one standard Zusammenhang, that is three standard Sachverhalte, thick. Chains longer than three steps cross the plane of ‘now.’ The predictability of that which will come, and where it will come can be read off the properties of the chains that co-exist with the standard chains.

Islands of Stability

Biology can only exist within niches of parameters’ restrictions. For the information transfer in genetics to function, the environment has to be extremely well regulated and ordered. To serve as a conceptual tool for theoretical genetics, we have to use some idealised properties of the accounting tool. We idealise the discussion here into cases where reorders αβ →γδ and γδ→ αβ are utilised as descriptions of one and the same organism. The logical interdependences between place, amount, quality, order, history can only take place embedded in a very stable and predictable logical background. Watch repair can not be done during earthquakes. The matches between  the linear and the spatial descriptions of one and the same organism, e.g. the development of embryos, only can take place in a perfect environment. The combinatorics behind the biochemical processes of pointing out a complicated spatial arrangement by a linear sequence and later pointing back into one-dimensional description works with a degree of precision that allows for the back and forth to function in actual life, this combinatorics can only work under very detailed rules, under very specific circumstances. We have to look for the ideal case, where everything functions best – however improbable such a scenario may appear - and see how the tautology is maintained between a linear sequence and a spatial arrangement.