4. Still Life with Contradictions
Step Four of Learn to Count in Twelve Easy Steps
Reminder Step 3:
We have shown that a sort on the data set – with any of the aspects first, and a different aspect as the second sorting argument – assigns a place to an addition; different sorts may assign different places.
Step 4:
Data:
Please use Table 4.num here. The 72 columns of type SQ are each a sorting order. For ease of use, we suppress columns {c,k,u,t,q,s,w}.
Sequential place of elements:
The interactive graph called ‘sequence’ in 4.graph shows the order of the elements after a sort. The user selects the two aspects (αβ) on which the collection shall be ordered. On pushing the button “Kirajzol” (draw), 3 illustrations appear. First, we show uniformly the sequence of a,b in the sorting order αβ. Then, we show the sequence of value α, lastly that of value beta, in the sequential order αβ. We use the term ‘place’ when referring to a number between 1 and 136 that shows, where in the sequence a specific element comes to stand as the result of a sorting.
Places:
For didactic reasons, please visit 3.graph first. There, we show the places of 136 additions on the plane with X-axis SQab, Y-axis SQba. Please note that both axes are equally spaced, linear sequences. The place they assign to an addition is logically sound and geometrically precise. It is clear that sorting the data set twice assigns one place for each element on a plane, of which the axes are the two sorting orders.
One may discuss whether SQαβ is the X-axis or rather the Y-axis, therefore whether SQγδ is the Y-axis or rather the X-axis. We note that the two results are equally legitimate, logically.
Generalising this approach leads to the interactive graphic in 4.graph. Please select first the sorting order αβ, then the sorting order γδ. The distribution of the 136 additions on the plane defined by X: SQαβ, Y: SQγδ becomes visible on clicking on button ‘Kirajzol’.
Contradictions:
We see that SQαβ may or may not differ to SQγδ. The contradictions and identities are manifold. The logical dilemma involves not only (a,b) and their connotations (c…w) with respect to the places they occupy or should occupy. The dilemma is rather whether an order, which is in deviation to the defining ’correct’ order, is still within boundaries of acceptance. The idea of order is the same, whether order αβor order γδ is realized. The idea of limits of acceptability is the same, whether the test of ‘is this deviation still ok’ is conducted on {place | amount | number | order}. We see that orders αβ and γδ may well have one or more elements that do not change place, therefore the investigation of ‘are these two realizations of the order identical or different?’ gives rise to a decision tree that is multi-faceted, of discrete, minimal steps.
What is not the case:
Having established that the collection is presently in order αβ (short-speak for ‘in the realization/actualization αβof the order’) we say that this is the case. That αβ is the case implicates that element i is on place p and so forth. There is a complete match between amounts and places.
The logical assertion that ‘under order αβ element i is on place p ‘ is an elementar fact, logically a tautology, because it cannot be otherwise. The content of the statement cannot be discussed. The truth of the statement is however dependent on whether this is actually the case. If the order αβ does not exist, it may well not be the case that ‘element i is on place p‘.
We can decide, which of the 72 sorting orders we designate to be the case. The other sorting orders are then not the case. What is not the case, deviates from that what is the case in the respects of amount, place, number, order. We may say that “’a+b=c’ that is, the overall order if succession prevails, but we may also hear ‘it may well be that a+b=c that is, the overall order of succession prevails, but definitely {not here | not with these a,b | not such many times | not under this order}.
Like the concept of order has a transcendent existence beyond all its realisations – the idea of the sort as such is the same, irrespective of which sort the set is presently in -, the concept of being otherwise has also a transcendent existence beyond all its realisations.
What we see is a conflict between the effects (realizations/actualizations) of two logical principles: {+} and {<|=|>}. The sequential number of an element generates a partition on 136, so {<|=|>} translates into {+}.
The implicated contradictions visible in Table 4.num surround that what is the case. That what is the case is but a small part of all possibilities. The relation of that what is not the case to that what is the case is what Wittgenstein has declined to discuss.
Consolidation:
Making use of the technological advances of the last decades, we can tabulate, classify and quantify that what is in deviation to what is the case. We note that the connotations of the words “is the case” confer a faint touch of moral-epistemological supremacy to that what is pointed out as being the case as contrasted to its background, what is evidently not the case. This artefact of the neurology is practical to serve as a cultural understanding, namely that if we speak about something, we by default refer to the foreground and not to the background; there is however no moral or epistemological supremacy of any of the SQ-named columns of Table 4.num above the other 71 columns.
In the steps that follow we shall investigate, how to consolidate that what appears – relative to the order αβ – to be incorrectly placed, amounted, frequented or ordered with that what is the case (on the right place, the exact amount, the expected frequency, the valid order).
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