Groups of Central Sections of a 2ⁿ-Element Set and Bipolar Traits
Yu. P. Minaev, I. P. Datsenko, K. S. Shepel
Socionics, Mentology and Personality Psychology (SMiPL), No. 3, 2016, pp. 41–51 Section: Mathematical Methods in Socionics. UDC 159.9.075 Received by the editors 17.02.2016
Translator’s note. Translated from the Russian. “Признак” is rendered as trait; “сечение” as section; ТИМ as TIM (type of information metabolism); АРП as ART (Augustinavichiute–Reinin trait); ЮМП as JMT (Jung–Minaev trait); “простое произведение” as simple product. Tables 2 and 3 appear as images in the scan; they have been reconstructed and verified by direct computation (element-wise products and orthogonality checks all confirm the reconstruction against the legible parts of the scan).
Abstract
Sets containing 2ⁿ elements and the corresponding sets of their central sections are considered. A method of fixing central sections is proposed which makes it possible to formalise the binary operation introduced on this set and to simplify the procedure of determining whether two given central sections are orthogonal. It is shown that while for n = 2 there exists only one group of central sections, for n = 3 the number of such groups becomes 30, and for n = 4 it grows to 64,864,800.
The paper examines in detail the case of the 8-element set (n = 3), at which the effect of a multiplicity of partially overlapping groups of central sections already arises. Illustrative materials are offered which are intended to make this effect easier for readers to grasp.
The question of “voicing” central sections by means of bipolar traits is discussed.
Keywords: socionics, central sections of a set, bipolar traits, Augustinavichiute–Reinin group of traits, Jung–Minaev group of traits.
Clarifying the concepts, and a method of fixing central sections
In his book [7], G. R. Reinin writes about sections of the 16-element set that is the socion. The elements of this set are the TIMs (types of information metabolism). Particular attention is paid to the so-called central bipolar traits, each of which divides this 16-element set into two subsets equal in size (8/8).
If two given central sections of a 2ⁿ-element set produce 4 subsets equal in number of elements, such sections are called orthogonal. If a third central section is orthogonal to the first two and divides each of the 4 previously obtained subsets exactly in half, then these three sections can be said to be not merely pairwise orthogonal but also independent.
It should be noted that in Reinin’s book [7], on p. 146 the words “independent” and “orthogonal” are treated as synonyms, while on p. 147 he speaks of a triple of “mutually dependent” sections that are pairwise orthogonal. To avoid confusion, in the case of two central sections we will speak only of orthogonality or non-orthogonality in the sense defined above. And of three pairwise orthogonal central sections we will say that they are independent if they divide the original set into 8 parts of equal size. Four independent, pairwise orthogonal central sections must divide the original 2ⁿ-element set into 16 subsets equal in number of elements.
It should be clear that three independent sections can only be spoken of for n ≥ 3 (starting from the level of the 8-element set), and four — for n ≥ 4 (starting from the 16-element set). For the case n = 2 (a 4-element set) there exist only three central sections in total; they are pairwise orthogonal but mutually dependent. Mutual dependence is to be understood in the sense that from any two sections belonging to this triple, the third can be obtained by means of the binary operation introduced by G. R. Reinin on the set of sections [7, p. 147].
Suppose we have the 4-element set of letters {I, S, e, l}. Here are all three possible central sections: {I, S}/{e, l}; {I, e}/{S, l}; {I, l}/{S, e}. There is no other way to divide this set exactly in half. The central sections listed can be fixed by means of Table 1, where “1” means the element landed in the same half as the letter I, and “–1” that it landed in the other.
Table 1. Method of fixing the central sections of a 4-element set.
| I | S | e | l | ||
|---|---|---|---|---|---|
| 1. | 1 | 1 | –1 | –1 | upper case / lower case |
| 2. | 1 | –1 | 1 | –1 | vowels / consonants |
| 3. | 1 | –1 | –1 | 1 | ??? |
For the first two sections it is easy to invent bipolar traits that would “voice” them. With an adequate bipolar trait for the third section, matters are more complicated. This example is very similar to the one Reinin gave with the four temperament types [7, p. 236], and even the notation for membership in a given pole of the dichotomy has hardly changed (“1” and “–1” instead of ”+” and ”–” respectively).
As for the binary operation on the set of central sections: with this method of fixing central sections it can be defined as the element-wise multiplication of the corresponding rows consisting of “ones” and “minus ones”. It is easy to verify that any row of Table 1 can be obtained as the “product” of the other two. This is precisely what it means for the corresponding central sections to be mutually dependent.
Note that for this kind of “product” — where the elements of a new ordered array of numbers are obtained by multiplying the corresponding elements of two given ordered arrays of equal length — there exists a special term: the simple product [6, p. 54].
Observe that the simple product of two central sections turned out to be a central section (with equal numbers of “ones” and “minus ones”). This will happen only when we multiply orthogonal central sections. We can therefore introduce the notion of a scalar product, meaning the number obtained if, after performing the element-wise (simple) multiplication of the corresponding rows, we take the sum of all the elements [6, p. 53]. Clearly, with this definition the scalar product will equal zero only for orthogonal central sections.
It is important to realise that for n ≥ 3 the scalar product of two central sections will not always equal zero. The same thing can be put differently: applying the introduced binary operation to two central sections will not always yield a central section. Recall that for n = 2 all three possible central sections were pairwise orthogonal. One might think these are very simple things to understand. But for a long time socionists dealt exclusively with orthogonal central sections of 2ⁿ-element sets. Hence there is a certain psychological barrier that prevents many people from even beginning to think about this topic.
With the present article we continue a series of articles aimed at overcoming this barrier. Particular attention will be given to the “borderline” case n = 3 (the 8-element set). Unlike the case n = 4, to which the article by A. V. Osipov and G. R. Reinin “The structure of the set of central traits” [5] was devoted, the 8-element set will, it seems to us, make it easier to illustrate the qualitative leap that occurs in the transition from n = 2 to n = 3. What is at issue is the existence of pairs of central sections that are not orthogonal. And this in turn leads to the existence of not one but many groups of central sections of one and the same 8-element set. In the case n = 4 (the 16-element set) the situation with the multiplicity of groups of pairwise orthogonal central sections only becomes more acute.
Here the word group is used in the mathematical sense. So what is meant is not merely a set, but a set with a binary operation introduced on it, whose application to ordered pairs of elements of the set does not lead outside it — i.e., the result of the operation is an element of the same set. In our concrete case this means that only central sections which are all pairwise orthogonal to one another can end up in a single group. It is important to understand that for n ≥ 3, the orthogonality of one section to two others does not, generally speaking, imply the orthogonality of those two other sections to each other. In the exposition below, this point will be illustrated with concrete examples.
A group must necessarily include a neutral (or identity) element. Multiplying any element of the group by it must leave that element unchanged. The role of this neutral element in groups of central sections is played by the so-called identity section [7, p. 148], which under our adopted method of fixing will be represented as a digital code consisting entirely of ones.
Clearly, this identity section is not itself a central section. This may cause a certain psychological tension in readers far removed from mathematics. Indeed, how can an element that is not a central section belong to a group of central sections? We will only note that even the authors of article [5] — who certainly know that a group must contain a neutral element — write, when considering the best-known group in socionics, the group of Augustinavichiute–Reinin traits (ARTs), that it contains 15 traits. Clearly the “forgotten” one is the neutral element — the existence trait, which in this particular case is also called the Karpenko–Churyumov trait.
We should warn that in our article, which will mainly deal with central sections of an 8-element set (n = 3), the groups of order 8 consist of 7 central sections plus the neutral element (the identity section) common to all groups. In our tables indicating which groups a given specific section falls into, we will not enter the identity section, which, as already said, must belong to every group. But any of the 35 possible central sections of the 8-element set will belong to exactly 6 of the 30 existing groups of order 8. We will also explain in this article how to check for oneself whether the numbers just quoted (and some others) are correct.
The structure of the totality of central sections of the 8-element set
As an example of an 8-element set (n = 3), take the following collection of letters with superscripts: {I⁺, S⁺, e⁺, l⁺, I⁻, S⁻, e⁻, l⁻}. Each central section will be fixed by an ordered array consisting of four “ones” and four “minus ones”, where “1” will mean that the element of our set ended up in the same half as the element I⁺, and “–1” will be put otherwise. Unlike the situation for n = 2, we obtain not 3 central sections (see Table 1) but 35 (Table 2).
The column corresponding to the element I⁺ is filled entirely with “ones”, since it was decided to “take our bearings” from that element. If a specific section is “voiced” by a corresponding bipolar trait, then the pole named first will be the one to which our chosen element belongs. Thus, for example, the section listed under number 1 in Table 2 can be “voiced” by the bipolar trait “plus/minus”, section 10 by the trait “upper case/lower case”, and section 21 by the trait “vowels/consonants”. If we stay with the “letter-and-sign” content of the proposed set, serious problems can arise in “voicing” the remaining sections. If, however, we look at the elements of the proposed set as conventional designations of TIMs, then a noticeably larger number of central sections — though still not all 35 — can be “voiced” without much difficulty by socionic bipolar traits.
It should be noted that such designations for TIMs first appeared in article [4], of which the present article is in effect a continuation and development. The introduction of these designations is connected with the plan for structuring the exposition in these articles: we wanted first to consider the question of the central sections of a seemingly abstract letter-and-sign set, and only then to pass to a concrete socionic example. Of course, readers who follow the socionics literature will easily have guessed that these new designations are a simple combination of already familiar ones. The letters “came” from the notation used by S. I. Churyumov for the information aspects (IAs) (see, e.g., [9]). And the superscripts ”+/–” were used in the already fairly widespread abbreviated graphic designations of TIMs (see, e.g., [8]). In that notation our eight would be written as:
{▲⁺, ●⁺, ⌐⁺, □⁺, ▲⁻, ●⁻, ⌐⁻, □⁻}.
Table 2. Central sections of the 8-element set.
(Rightmost columns mark, with a ”+”, membership of the section in each of the six groups 1.1–1.6 of the cluster formed around section 1; see Table 3. The identity/neutral element, which belongs to every group, is omitted.)
| № | I⁺ | S⁺ | e⁺ | l⁺ | I⁻ | S⁻ | e⁻ | l⁻ | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.6 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | –1 | –1 | –1 | –1 | + | + | + | + | + | + |
| 2 | 1 | 1 | 1 | –1 | 1 | –1 | –1 | –1 | ||||||
| 3 | 1 | 1 | 1 | –1 | –1 | 1 | –1 | –1 | ||||||
| 4 | 1 | 1 | 1 | –1 | –1 | –1 | 1 | –1 | ||||||
| 5 | 1 | 1 | 1 | –1 | –1 | –1 | –1 | 1 | ||||||
| 6 | 1 | 1 | –1 | 1 | 1 | –1 | –1 | –1 | ||||||
| 7 | 1 | 1 | –1 | 1 | –1 | 1 | –1 | –1 | ||||||
| 8 | 1 | 1 | –1 | 1 | –1 | –1 | 1 | –1 | ||||||
| 9 | 1 | 1 | –1 | 1 | –1 | –1 | –1 | 1 | ||||||
| 10 | 1 | 1 | –1 | –1 | 1 | 1 | –1 | –1 | + | + | ||||
| 11 | 1 | 1 | –1 | –1 | 1 | –1 | 1 | –1 | + | + | ||||
| 12 | 1 | 1 | –1 | –1 | 1 | –1 | –1 | 1 | + | + | ||||
| 13 | 1 | 1 | –1 | –1 | –1 | 1 | 1 | –1 | + | + | ||||
| 14 | 1 | 1 | –1 | –1 | –1 | 1 | –1 | 1 | + | + | ||||
| 15 | 1 | 1 | –1 | –1 | –1 | –1 | 1 | 1 | + | + | ||||
| 16 | 1 | –1 | 1 | 1 | 1 | –1 | –1 | –1 | ||||||
| 17 | 1 | –1 | 1 | 1 | –1 | 1 | –1 | –1 | ||||||
| 18 | 1 | –1 | 1 | 1 | –1 | –1 | 1 | –1 | ||||||
| 19 | 1 | –1 | 1 | 1 | –1 | –1 | –1 | 1 | ||||||
| 20 | 1 | –1 | 1 | –1 | 1 | 1 | –1 | –1 | + | + | ||||
| 21 | 1 | –1 | 1 | –1 | 1 | –1 | 1 | –1 | + | + | ||||
| 22 | 1 | –1 | 1 | –1 | 1 | –1 | –1 | 1 | + | + | ||||
| 23 | 1 | –1 | 1 | –1 | –1 | 1 | 1 | –1 | + | + | ||||
| 24 | 1 | –1 | 1 | –1 | –1 | 1 | –1 | 1 | + | + | ||||
| 25 | 1 | –1 | 1 | –1 | –1 | –1 | 1 | 1 | + | + | ||||
| 26 | 1 | –1 | –1 | 1 | 1 | 1 | –1 | –1 | + | + | ||||
| 27 | 1 | –1 | –1 | 1 | 1 | –1 | 1 | –1 | + | + | ||||
| 28 | 1 | –1 | –1 | 1 | 1 | –1 | –1 | 1 | + | + | ||||
| 29 | 1 | –1 | –1 | 1 | –1 | 1 | 1 | –1 | + | + | ||||
| 30 | 1 | –1 | –1 | 1 | –1 | 1 | –1 | 1 | + | + | ||||
| 31 | 1 | –1 | –1 | 1 | –1 | –1 | 1 | 1 | + | + | ||||
| 32 | 1 | –1 | –1 | –1 | 1 | 1 | 1 | –1 | ||||||
| 33 | 1 | –1 | –1 | –1 | 1 | 1 | –1 | 1 | ||||||
| 34 | 1 | –1 | –1 | –1 | 1 | –1 | 1 | 1 | ||||||
| 35 | 1 | –1 | –1 | –1 | –1 | 1 | 1 | 1 |
From the socionic point of view, what we have here is an 8-element set uniting all the statics of the socion. It should now be clear that for the section listed under number 1 in Table 2, instead of the “letter-and-sign” bipolar trait “plus/minus” we can propose a well-known socionic one — right/left. But another would do just as well — positivists/negativists. Indeed, in socionics the TIMs {▲⁺, ●⁺, ⌐⁺, □⁺} are assigned both to the right and to the positivists, while the TIMs {▲⁻, ●⁻, ⌐⁻, □⁻} are assigned both to the left and to the negativists. Recall that when writing a bipolar trait we agreed to name first the pole corresponding to I⁺ (▲⁺, ILE).
The section under number 10 in Table 2 can now be “voiced” not as “upper case/lower case” but by means of the socionic bipolar trait extraverts/introverts. But the division into irrationals/rationals could be proposed with equal success.
As we can see, on switching to the socionic meaning of the 8-element set under consideration we were able to offer as many as two variants of bipolar traits for one and the same central section. Are these variants equivalent? They differ at least in that the bipolar traits right/left and extraverts/introverts are reflected in the abbreviated graphic designations of TIMs (the sign and the colour of the symbol), whereas determining membership in positivists/negativists or irrationals/rationals requires additional knowledge of socionics. Note that one should not link the use of the sign in the abbreviated TIM designations with the bipolar trait positivists/negativists, since this conspicuous coincidence holds only for statics, while for dynamics everything is exactly the other way round.
What socionic bipolar trait, then, could replace the trait “vowels/consonants” for section number 21? Among the ARTs there is no suitable trait. But if we turn to another group of bipolar traits — one that has been talked about relatively recently, but which in spirit is closer to Jung’s typology — two candidates can be found there. We mean the group of JMTs (Jung–Minaev traits). Two traits will suit the present case: internal/external–1 and δ/β–1. Here the digit “1” says that attention must be paid to the information aspect processed by the 1st function of the TIM. The first trait therefore divides TIMs into those whose 1st function deals with an internal IA and those whose 1st function deals with an external one. This division is reflected in the graphic symbols of the IAs, as Aušra Augustinavichiute wrote when introducing them [1, p. 14]. To use the second variant of the trait, one must know which IAs belong to the δ-values and which to the β-values.
Can the number of all central sections of the 8-element set be counted without writing out all the corresponding codes of “ones” and “minus ones”? Of course — this is a standard school combinatorics problem which can be solved in at least two ways. If we take our adopted method of fixing central sections as the starting point, it is formulated thus: in how many ways can three “ones” be placed in seven cells? Recall that one “one” always stands in first position (see Table 2), and there are four “ones” in total. The answer is therefore the number of combinations of seven things taken three at a time. If instead one realises that a single central section yields two 4-element subsets of the 8-element set, it becomes clear that the answer is the number that is half the number of combinations of eight things taken four at a time. Either approach to this combinatorial problem gives the answer 35.
The order in which the section codes are written in Table 2 is easy to understand if one mentally replaces “–1” by “0” and then looks at the modified codes as ordinary eight-digit numbers. These numbers will run in decreasing order. Anyone who wishes can carry out this procedure for themselves and verify that not a single central section of the 8-element set (n = 3) has been missed.
In the right-hand part of Table 2 there are six columns, in each of which one can count seven “pluses”. Each column corresponds to a specific 8-element group of central sections, and the “pluses” show which seven central sections belong to that group. The eighth — neutral — element (the identity section), which belongs to every group, is absent from the table.
The six groups presented in Table 2 in their “truncated” composition (without the neutral element) form a kind of cluster, uniting those groups which necessarily include the section under number 1. Each of the 18 sections orthogonal to the 1st belongs to only two groups of the cluster given as an example in Table 2. Four more clusters of 6 groups each are formed in an analogous way. The “cluster-forming” sections in those cases are the 2nd, 3rd, 4th and 5th. These sections, exactly like the 1st, belong to all six groups of “their” cluster, but do not belong to the remaining 24 groups. The section numbers for all 30 groups, distributed across the 5 clusters, are given in Table 3.
Table 3. Clusters of groups of central sections.
Cluster 1 (cluster-forming section 1):
| 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.6 | |
|---|---|---|---|---|---|---|
| a | 1 | 1 | 1 | 1 | 1 | 1 |
| b | 10 | 10 | 11 | 11 | 12 | 12 |
| c | 15 | 15 | 14 | 14 | 13 | 13 |
| d | 21 | 22 | 22 | 20 | 20 | 21 |
| e | 24 | 23 | 23 | 25 | 25 | 24 |
| f | 28 | 27 | 26 | 28 | 27 | 26 |
| g | 29 | 30 | 31 | 29 | 30 | 31 |
Cluster 2 (cluster-forming section 2):
| 2.1 | 2.2 | 2.3 | 2.4 | 2.5 | 2.6 | |
|---|---|---|---|---|---|---|
| a | 2 | 2 | 2 | 2 | 2 | 2 |
| b | 7 | 7 | 8 | 8 | 9 | 9 |
| c | 15 | 15 | 14 | 14 | 13 | 13 |
| d | 18 | 19 | 19 | 17 | 17 | 18 |
| e | 24 | 23 | 23 | 25 | 25 | 24 |
| f | 28 | 27 | 26 | 28 | 27 | 26 |
| g | 32 | 33 | 34 | 32 | 33 | 34 |
Cluster 3 (cluster-forming section 3):
| 3.1 | 3.2 | 3.3 | 3.4 | 3.5 | 3.6 | |
|---|---|---|---|---|---|---|
| a | 3 | 3 | 3 | 3 | 3 | 3 |
| b | 6 | 6 | 8 | 8 | 9 | 9 |
| c | 15 | 15 | 12 | 12 | 11 | 11 |
| d | 18 | 19 | 19 | 16 | 16 | 18 |
| e | 22 | 21 | 21 | 25 | 25 | 22 |
| f | 30 | 29 | 26 | 30 | 29 | 26 |
| g | 32 | 33 | 35 | 32 | 33 | 35 |
Cluster 4 (cluster-forming section 4):
| 4.1 | 4.2 | 4.3 | 4.4 | 4.5 | 4.6 | |
|---|---|---|---|---|---|---|
| a | 4 | 4 | 4 | 4 | 4 | 4 |
| b | 6 | 6 | 7 | 7 | 9 | 9 |
| c | 14 | 14 | 12 | 12 | 10 | 10 |
| d | 17 | 19 | 19 | 16 | 16 | 17 |
| e | 22 | 20 | 20 | 24 | 24 | 22 |
| f | 31 | 29 | 27 | 31 | 29 | 27 |
| g | 32 | 34 | 35 | 32 | 34 | 35 |
Cluster 5 (cluster-forming section 5):
| 5.1 | 5.2 | 5.3 | 5.4 | 5.5 | 5.6 | |
|---|---|---|---|---|---|---|
| a | 5 | 5 | 5 | 5 | 5 | 5 |
| b | 6 | 6 | 7 | 7 | 8 | 8 |
| c | 13 | 13 | 11 | 11 | 10 | 10 |
| d | 17 | 18 | 18 | 16 | 16 | 17 |
| e | 21 | 20 | 20 | 23 | 23 | 21 |
| f | 31 | 30 | 28 | 31 | 30 | 28 |
| g | 33 | 34 | 35 | 33 | 34 | 35 |
The structure of all the clusters is identical. Each cluster has its own “cluster-forming” element. As already said, these elements are the sections that go under numbers 1–5 in Table 2. In Table 3 these numbers ended up in the row marked with the letter a. In row b of the same table stand the smallest numbers of those sections which are orthogonal to the sections of their group whose numbers are in row a.
Row c records the numbers of the sections obtained by simple multiplication of the sections from rows a and b taken from the same group (from the same column of Table 3). Thus the sections whose numbers ended up in the same column and in rows a, b and c form triples of pairwise orthogonal but mutually dependent sections. Note that the numbers recorded in rows b and c fall in the range 6–15.
In row d of Table 3 stand the next-smallest (after c) numbers of those sections which are orthogonal to the sections of their own group whose numbers are recorded in rows a and b. Note that the sections in row d are also orthogonal to the sections c taken from the same column (from the same group). This is a consequence of the fact that the sections a, b and c belonging to one group are mutually dependent. The sections whose numbers stand in row d are not only orthogonal to the sections a, b and c of their group, but also independent of them. Therefore the sections a, b and d taken from one group can serve as a system of generating elements of their group of order 8. In the socionics literature a system of generating elements of a group is often called a basis.
The sections whose numbers are recorded in rows e, f and g are obtained by simple multiplication of section d with the sections of the same group whose numbers stand in rows a, b and c respectively.
A spatial model of a six-element cluster of groups of central sections of the eight-element set (n = 3)
As we have satisfied ourselves, in passing from the case n = 2 (4-element set) to the case n = 3 (8-element set), it is no longer true that all possible central sections are pairwise orthogonal. For one given central section, not all of the 34 remaining sections turn out to be orthogonal to it, but only 18. If we take a pair of orthogonal sections, they automatically generate one more section orthogonal to them both, which is their simple product and which forms with them a triple of mutually dependent sections.
There also exist 8 further central sections, each of which is orthogonal to any member of this triple. But this does not mean that all 8 of these sections are pairwise orthogonal. Four of them belong to one group of order 8, and the remaining four to another. These two groups have a common subgroup of order 4, consisting of the aforementioned triple of mutually dependent sections and the identity section. An example of such a pair with a common subgroup is presented in Fig. 1.
Fig. 1. Two groups of bipolar traits with a common subgroup.
Group 1.1 (solid outline): 29. β/δ–2 28. α/γ–1 24. detached/involved–2 21. internal/external–1
Group 1.2 (dashed outline): 30. logicians/ethicists 27. tacticians/strategists 23. yielding/obstinate 22. judicious/decisive
Common subgroup (shared block below both): 15. questims/declatims 10. extraverts/introverts
- right/left
- existence trait
For the two groups presented in Fig. 1 there are suitable socionic terms for “voicing” the central sections by means of the so-called bipolar traits, and moreover for each section (except the identity) there are two variants of trait. Instead of those shown in Fig. 1, the following could have been used: positivists/negativists (1); irrationals/rationals (10); democrats/aristocrats (15); δ/β–1 (21); intuitives/sensorics (22); constructivists/emotivists (23); α/γ–2 (24); carefree/farsighted (27); detached/involved–1 (28); external/internal–2 (29); merry/serious (30).
Note that the socionic terms we used for “voicing” specific central sections are not all equally well established. The JMTs in general appeared only recently, as a result of purely theoretical investigations of the groups of operators of the classical (in Augustinavichiute’s sense) intertype relations. The terms for those JMTs which do not coincide with ARTs have a demonstratively formalised character, showing their connection with the Model A structures of the TIMs. Their experimental study has not yet begun. Disputes over some of the ARTs that do not coincide with JMTs continue to this day. As for the common subgroup of order 8 of these groups of order 16, consensus on the terminology for its elements has essentially been reached.
Let us also emphasise the similarity of the scheme given in Fig. 1 to the scheme from article [2]. The scheme in that article included the bipolar traits that “voiced” 23 central sections of the socion (the 16-element set), plus the Karpenko–Churyumov trait “answering” for the identity section. The scheme was in fact titled: “Two groups of bipolar traits of TIMs (Augustinavichiute–Reinin and Jung–Minaev) with a common 8-element subgroup”. A year before that article came out, an article appeared whose title already featured the number 23 [3]. It must be said that into that article, as into article [5], a misprint also “crept” concerning the number of elements in the group of ARTs (15 was printed instead of 16). The central sections and corresponding bipolar traits belonging to that group number 15, but the group also includes the neutral element — the Karpenko–Churyumov trait (the existence trait).
Recall that in the 8-element set (n = 3) there exist 35 central sections, not the 11 that are “voiced” in Fig. 1, where only two groups (1.1 and 1.2) of the 30 available are presented. Analogously, in the 16-element set (n = 4) there are 6,435 central sections, not the 23 that are “voiced” in the scheme from article [2], where only two groups (the ARTs and the JMTs) of the 64,864,800 available are presented. Note that this last number is 6 times smaller than the one the authors of article [5] obtained owing to an unfortunate error in their calculations. More detail on this can be found in [4].
We chose to consider the case n = 3 in greater detail so that even people far removed from mathematics could independently double-check our results. The qualitative effect of the multiplicity of groups of central sections already manifests itself at this level. But even here the partially overlapping groups of central sections turned out to be too numerous for a detailed visual representation. And one cannot descend to a lower level (n = 2), since only a single group exists there. We managed to construct a spatial model not for the whole collection of 30 groups, which contains all 35 central sections, but for a single cluster of 6 groups, which contains only 19 central sections. We will now show how anyone who wishes can reproduce such a model.
In assembled form the model will resemble a flower bud with six petals. Fig. 2 gives sketches of the blanks for the model of that cluster of groups whose common element is section number 1 from Table 2. The element common to all 30 groups (the neutral one) will not be represented in the model at all (it is, after all, not a central section).
Fig. 2. Sketches of the blanks for the model of a cluster of six groups of central sections with one common element (the neutral element not counted).
Six elongated rounded “petals”, labelled 1.1, 1.2, 1.3 (top row) and 1.6, 1.5, 1.4 (bottom row). On each petal, dots mark the positions of the fastening holes, labelled with section numbers. Petal 1.1 carries, at one end, holes 29 and 28; near the other end, holes 24 and 15, and at the very end holes 21, 1, 10. Petal 1.2: holes 27, 30 at one end; 15, 23 and 10, 1, 22 at the other. Petal 1.3: holes 26, 31; 23, 14 and 22, 1, 11. Petal 1.6: 21, 1, 12 and 24, 13 at one end; 26, 31 at the other. Petal 1.5: 12, 1, 20 and 13, 25; then 27, 30. Petal 1.4: 20, 1, 11 and 25, 14; then 29, 28.
The dots on the blanks (petals) shown in the figure mark the positions of the holes for the fastenings (small bolts). We will need 19 bolts and as many nuts. One of the bolts is best taken longer than the rest. It will symbolise the “cluster-forming” section (in our case designated number 1) and pass through the corresponding holes in all six petals.
After joining the petals at the points located at the same end of the petal as point 1, an opened six-petalled flower should result (Fig. 3). To obtain the bud it remains to fasten the free ends of opposite petals (each pair at two points). It should be clear that dense paper or thin flexible plastic can be used to manufacture the proposed spatial model.
Fig. 3. View of the model before joining the opposite petals.
Line drawing: six elongated petals fanned out from a common centre like a six-armed star, with the fastening holes marked as dots — a pair of dots at the outer end of each petal and clusters of dots around the centre.
The impression may arise that the “cluster-forming sections” 1–5 are privileged in comparison with the remaining sections. But this privileged status is tied to the order in which the elements of the set were written down. Had we taken not the order adopted in Table 2 but a different one, other sections would have become “cluster-forming”. Consider, for example, this order: I⁺, S⁺, I⁻, S⁻, e⁺, l⁺, e⁻, l⁻. Then the first five rows of the table would correspond to these sections:
- {I⁺, S⁺, I⁻, S⁻}/{e⁺, l⁺, e⁻, l⁻} (old number — 10);
- {I⁺, S⁺, I⁻, e⁺}/{S⁻, l⁺, e⁻, l⁻} (old number — 2);
- {I⁺, S⁺, I⁻, l⁺}/{e⁺, S⁻, e⁻, l⁻} (old number — 6);
- {I⁺, S⁺, I⁻, e⁻}/{e⁺, l⁺, S⁻, l⁻} (old number — 11);
- {I⁺, S⁺, I⁻, l⁻}/{e⁺, l⁺, e⁻, S⁻} (old number — 12).
This means that with the new ordering of the elements of the very same set, the “cluster-forming” sections will be those which under the old ordering went under the numbers 2, 6, 10, 11, 12. And the cluster that would be “headed”, for example, by the 1st section of the new list (which in the old list went under number 10) would comprise those groups which under the old numbering were designated 1.1, 1.2, 4.5, 4.6, 5.5, 5.6. This is easy to work out by consulting Table 3. Using the same table, similar lists can be drawn up for the remaining four new clusters.
One more example of apparent privileged status
Let us return to the two groups of order 8 presented in Fig. 1. They can be regarded as groups of bipolar traits for the 8-element set of TIMs made up entirely of statics. We will use the method of visualising central sections that socionists often use for the 16-element set (see, e.g., [9]).
In these two groups (1.1 and 1.2), besides the neutral element (the existence trait, corresponding to the identity section), the shared elements are the bipolar traits “voicing” the sections that go under numbers 1, 10 and 15 in Table 2. Group 1.1 additionally contains elements 21, 24, 28, 29, and group 1.2 — elements 22, 23, 27, 30. The corresponding illustrative material is presented in Fig. 4. Here the digit “0” designates the neutral element (the identity section).
Fig. 4. A variant of the visual representation of groups 1.1 and 1.2.
Top row (shared by 1.1 and 1.2): four 2×4 cell-grids of the TIM symbols (top row of each grid: I⁺ S⁺ e⁺ l⁺; bottom row: I⁻ S⁻ e⁻ l⁻), labelled 0, 1, 10, 15. Middle row (group 1.1): four such grids labelled 21, 24, 28, 29. Bottom row (group 1.2): four grids labelled 22, 23, 27, 30. In the original, the cells belonging to the “first” pole of each section are shaded, so that each grid displays the partition pattern of its section; the patterns for 21, 24, 28, 29 look symmetric under this element ordering, while half of the patterns for group 1.2 look asymmetric.
Looking at Fig. 4, one might think that group 1.1 has a clear advantage over group 1.2, in which half of the elements correspond to asymmetric patterns (compare the two bottom rows of Fig. 4). But everything changes to exactly the opposite if in the ordered list of TIMs we swap e⁻ and l⁻, obtaining the list: {I⁺, S⁺, e⁺, l⁺, I⁻, S⁻, l⁻, e⁻}.
Note that the bipolar traits for group 1.1 were taken from the set of JMTs, and those for group 1.2 from the group of ARTs. In this article we used TIM designations in which the IA “processed” by the TIM’s 1st function is indicated explicitly, while the IA of the 2nd function had to be “computed” using the information about the sign. This gave the JMTs a certain advantage. Let us write our new list using S. I. Churyumov’s notation: {Il, Se, eI, lS, Ie, Sl, lI, eS}. In this form it is no worse — perhaps even “more symmetric” — than the original list: {Il, Se, eI, lS, Ie, Sl, eS, lI}. Indeed, in the new list intuitives alternate with sensorics, whereas at the end of the old (original) list this alternation breaks down!
If we recall that the bipolar trait intuitives/sensorics is a purely ART-type trait, it is not surprising that in Fig. 4 it corresponds to an asymmetric pattern, marked with number 22. This is because a JMT-oriented sequence of TIMs is used in that figure. Had the new (ART-oriented) sequence of TIMs been adopted instead, the pattern for intuition/sensorics would have gone under number 21 and would have looked symmetric.
We have dwelt at such length on the fact that illustrative materials can not only help one grasp certain ideas but also push one towards false conclusions, because in the socionics literature one encounters attempts to prove “mathematically” the privileged status of certain groups with the aid of illustrative material of the kind presented in Fig. 4.
Conclusion
In the present work a method of fixing central sections of 2ⁿ-element sets was applied which makes it possible to formalise the procedure of answering questions concerning the pairwise orthogonality and mutual dependence of central sections. This simplified the procedure of finding groups of pairwise orthogonal central sections. For the case of the 8-element set (n = 3), all 30 existing groups of order 8 were written out explicitly. It was shown how these 30 groups can be divided into 5 clusters of 6 groups each. A procedure for manufacturing a spatial model of a single cluster was described.
If the 2ⁿ-element set is unordered, then from the mathematical point of view no individual central section stands out in any way against the background of the remaining central sections. Nevertheless, considering the totality of central sections and introducing the bipolar operation makes it possible to single out individual groups of pairwise orthogonal central sections. Each section falls into an equal number of such groups of a given order.
Creating illustrative material for the visual representation of central sections is often bound up with a particular ordering of the 2ⁿ-element set. This can create the illusion that some section or group of sections is privileged. Accordingly, the absence of a certain symmetry in the illustrative material cannot serve as a mathematical proof that the given section or group of sections is not on an equal footing with the rest. On the other hand, mathematical equality does not guarantee substantive equality in a concrete subject domain.
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