Section 7
Mathematical Socionics and the Partially Alternative Algebra of Churyumov
Not thinking to amuse the haughty world…
The first application of mathematical apparatus to socionics by the talented mathematician G. R. Reinin produced an unexpected and exceptionally valuable result. Having given a mathematical proof of J. Ekman’s formula on the relationship between the number of types in a typology (N) and the number of DTs (differentiating traits) — n: n = N−1 — Reinin informed Aushra that alongside the four Jungian DTs there must exist 11 more DTs, so that their total number should be 15. Note that Reinin, having written the algebraic formulas for constructing traits derived from the basis ones, did not find semantic correspondences for the algebraic variables, confining himself to mathematical abstractions, and did not propose a substantive algorithm for defining the traits in terms of psychological types. Aushra, with all her socionic enthusiasm, immediately set about searching for these traits, which is by no means a simple matter, since it was not known in advance how this could even be done. Aushra completed this work within two weeks, finding the antisymmetric octaves corresponding to the sought traits, describing their presumed semantics, and giving them provisional preliminary names. There turned out to be exactly 15 such octaves — matching the number of traits — although there was no a priori reason to assume this; it would have been possible, for example, for one and the same trait to be supported by different groups of octaves. Aushra showed that the first case holds, which, together with her development of the semantics of the ARTs, gives grounds to name the 15 pairs of octaves she found after her — that is, to speak of the octaves of Augustinavichiute. In honour of Reinin’s initiative she named the traits she found Reinin traits. This was, of course, her authorial right, and it was fair given Reinin’s felicitous discovery; however, now, after her death, it would be fair to call these traits by the double name “Augustinavichiute–Reinin traits” (ARTs), acknowledging her invaluable contribution to developing the semantics of this construct.
However, let us make one small but nonetheless important clarification, emphasising the mathematical aspect of socionic sets. The point is that Reinin, being a talented mathematician, carried out the corresponding derivations quite correctly, not overlooking the fact that an algebraic group must necessarily contain an identity, or neutral, element, composition with which equals the non-identity element of the composition. This is easy to understand by recalling the product of any number with one. But it turned out that Reinin did not attach sufficient importance to this fact, and this manifested itself in the discussion still being about 15 rather than 16 traits — and this became fixed in the minds of socionists. Reinin, thinking about the psychic semantics of the neutral element, supposed that the meaning of the identity partition E consists of: “1. Traits that are generic for the set of objects under investigation. In other words, general traits of the substrate of the typology, i.e. relating to the entire set of investigated objects as a whole. 2. All traits not pertaining to the content of the given typology. For example, the trait of sexual dimorphism is not essential for a typology of temperaments” [Rei]. To this one may object that the identity trait must have a concrete typological meaning, rather than describing a blurred field of unspecified notions. And it certainly has no direct relation to generic traits. In this particular case, the generic trait for socionic traits — that is, for the ARTs — is the very concept of a trait, which makes it impossible to include it in the set of traits and thereby prevents recognising it as the group identity. It also seems strange to assign to the group identity all traits “not pertaining to the content of the given typology” simply because they are not essential for it, while their compositions with elements of the group of traits would be meaningless. The neutral trait is distinguished by being inherent in all elements of the group; but if all the other traits oppose the octaves of TIMs to one another, this trait opposes nothing — which, in the mathematical sense, may be regarded as a 0-opposition. Such a trait is existence: any object, including a trait, must exist before it can possess any parameters. Working with the ART table, I proceeded from the outset on the assumption that there are exactly 15 ARTs, but in trying to find their strict mathematical order I came to the conclusion that a 16th, neutral trait had to be added, without which the table could not be symmetrised. Later I learned that O. Karpenko, developing Reinin’s group-theoretic ideas, had discovered a “square-nested structure” of the trait table [2], whose cell is the Cayley table of the Klein four-group, and made the refinement that the ART group is commutative. Below I will show that she was mistaken in this, but her constructions actively used the notion of the group identity, although she did not semanticise it. The clarifications given here allow us to name the identity element of the ART group after Karpenko and Churyumov. Thus we rehabilitate the 16th trait, and it will be shown further on that in other socionic groups the neutral element will likewise correspond to a concrete psychological meaning, which is sufficient argument not to neglect this trait.
Reinin’s pioneering work substantiated the existence of DTs beyond those proposed by Jung. But it also contained information of fundamental importance for understanding the ARTs, namely that all traits derived from the four basis ones are linearly dependent. Note also that Reinin’s work opened up a vast layer of meaning relating to combinatorial typologies, to which theorists had previously paid not merely little attention but none at all.
Very interesting, and at the same time unexpected, was Reinin’s work devoted to the groups of tetrads whose semantics is determined by linearly dependent triples of ARTs. Reinin proved with mathematical rigour that there are exactly 35 such triples and, correspondingly, 35 groups of tetrads associated with them, and 140 tetrads in all. The existence of these tetrads with their specific semantics was likewise proven by Reinin using mathematical apparatus. The set of these tetrads has justly become fixed in the socionic consciousness as the “correct tetrads of Reinin.”
The apotheosis of Reinin’s mathematical applications in socionics was his work [Rei] proving the group character of the set of ARTs. Choosing the poles of the basis traits as a generalised coordinate framework, Reinin showed how, by combining them with one another, the components of the derived traits can be obtained. Such an operation may be considered a group operation, since applying it to two elements of the set produces an element of the same set and never leads outside it. By pointing out the existence of an identity element in this set, Reinin thereby proved the group character of the set of ARTs.
Reinin succeeded in showing [1] that the ARTs form an algebraic group — that is, a set on which a binary operation can be introduced such that two of its elements correspond to one element belonging to the same set; and, moreover, this set contains an element whose composition with any other corresponds to that other. This element is conventionally called neutral, zero, or identity, depending on the character of the group composition.
A group may be multiplicative or additive. In the first case multiplying any element by 1, and in the second adding 0 to any element, yields the chosen element. For his proof Reinin represented the ARTs in a discrete-coordinate form of the type X and X̄, where one pole is conventionally considered positive and the other negative. Then two bipolar traits X(x, x̄) and Y(y, ȳ) divide the set of people into 4 types: T1(x, y), T2(x, ȳ), T3(x̄, y), T4(x̄, ȳ). One can now introduce a set-theoretic composition over the traits X(x, x̄) and Y(y, ȳ) using conjunction and disjunction (©). Then X©Y = (xy ∪ x̄ȳ, xȳ ∪ x̄y) = Z, which is also a trait, coinciding with neither X nor Y, but which is their linear combination. This makes such three traits linearly dependent and does not lead to the appearance of new types. In particular, this means that the types Tn can be expressed not only by combining the poles of traits X and Y, but also by combining the poles of traits X and Z, or Y and Z — which is mathematically clear, but in concrete subject domains can lead to misunderstandings. Thus, in the set of dogs and cats, the subsets (male dogs, female dogs), (male cats, female cats), (male dogs, male cats), (female dogs, female cats) raise no doubts from a practical standpoint, whereas the sets (male dogs, female cats), (male cats, female dogs) are perceived as artificial. In socionics, for example, the club of scientists (the combination of intuition — IT, and logic — LG) can be obtained not only by the direct combination of IT and LG, but also by the combinations of IT and AR (aristocrats), or LG and AR — which, for people unfamiliar with the concept of linear dependence, causes cognitive strain associated with an apparent ambiguity.
Thus Reinin found a way to show that the set of ARTs is an algebraic group and that the group he described is commutative. Of course, it by no means follows from this that an analogous operation can be introduced on the set of TIMs or IRs and their group nature thereby discovered. But let us try.
The set of IRs is not just some arbitrary set — it is the set of relations or links between TIMs; but by virtue of the principle of duality between TIM and IR, every TIM can be regarded as a certain link between two relations, which is non-obvious and requires special substantiation. The simplest variant of such substantiation is the reduction of TIMs to geometric points, and relations to the lines that join them. Clearly, this intuitively understandable model of the system of TIMs and IRs allows the principle of duality to be applied to them, according to which one may assert that any true proposition linking these concepts will also be transformed into a true proposition if the concepts appearing in these propositions are replaced by their duals. Thus the geometric proposition “a line is determined by two points” turns into the equally true proposition “a point is determined by two lines.” Carrying the principle of duality over to TIM and IR, one can obtain, for example, two propositions: “an IR is determined by two TIMs” and its dual — “a TIM is determined by two IRs,” where the truth of the second follows from the truth of the first — which does not seem obvious, but must be recognised as such by virtue of the principle of duality.
Fig. MS.01. A Cartesian table of relations: b stands to c in relation x. Fig. MS.02. A configuration of aligned points in which the relation between b and c is determined by the line x. Fig. MS.03. The joining of functions in Model A by lines.
Let us nonetheless give a more thoroughly argued proof of the admissibility of transferring the principle of duality from the subject domain of mathematics to the subject domain of typology. Relations between the elements of a set in mathematics can be expressed in two ways: by means of a Cartesian table and by means of a diagram of aligned points.
Figure MS.01 gives an example of a Cartesian table establishing a relation between elements within a set. Tables of this kind are widely used to illustrate tournament results. Obviously, the lines bx and cx uniquely determine the point x corresponding to the relation of elements b and c. This simple configuration is easily transformed into its dual configuration, shown in Figure MS.02. This configuration is topologically equivalent to the configuration in MS.03, by means of which Aushra introduced into socionics the concept of a relation between TIMs, joining the functions in the Model A of different TIMs. In this case, the interaction of functions 1–3 and 1–2 defines the semantics of the relation of supervision and being supervised. It is thereby shown that a geometric configuration of points and lines can be transferred to the description of relations in socionics by virtue of their topological equivalence.
Correspondence between TIM and IR (Table MS.01)
| No. | TIM | TIM code | IR | IR code |
|---|---|---|---|---|
| 1. | Ki (Don Quixote) | 1111 | TZh (identity) | 1111 |
| 2. | Dyu (Dumas) | 0100 | DU (duality) | 0100 |
| 3. | Gyu (Hugo) | 1000 | AK (activation) | 1000 |
| 4. | Ro (Robespierre) | 0011 | ZE (mirror) | 0011 |
| 5. | Gm (Hamlet) | 1010 | PZ (supervisee) | 1010 |
| 6. | Go (Gorky) | 0010 | PK (supervision) | 0010 |
| 7. | Zhu (Zhukov) | 1101 | DE (business) | 1101 |
| 8. | Es (Yesenin) | 0110 | MI (mirage) | 0110 |
| 9. | Lo (London/Jack) | 1011 | KV (quasi-identity) | 1011 |
| 10. | Dr (Dreiser) | 0000 | KF (conflict) | 0000 |
| 11. | Na (Napoleon) | 1100 | SE (super-ego) | 1100 |
| 12. | Ba (Balzac) | 0111 | PP (semi-duality) | 0111 |
| 13. | Ge (Huxley) | 1110 | RD (kindred) | 1110 |
| 14. | Gb (Gabin) | 0101 | PD (semi-dual/benefit) | 0101 |
| 15. | Sht (Stirlitz) | 1001 | ZK (look-alike/mirror-pair) | 1001 |
| 16. | Do (Dostoevsky) | 0010 | KN (control) | 0010 |
(Codes as printed in the original; note the original contains an apparent typesetting duplication in rows 6 and 16.)
By virtue of the principle of duality between TIM and IR, a one-to-one correspondence can be established. This can be done in various ways, but here we use set-theoretic operations. To this end one can express a TIM through the ARTs, obtaining a 16-digit code that determines the complete trait structure of the TIM; however, for practical purposes it is quite sufficient to express the TIM through the poles of the Jungian basis (JB) — that is, through extraversion/introversion, irrationality/rationality, intuition/sensing, and thinking/feeling — replacing each pole of these polar constructs by 1 or 0 respectively and arranging them in the same order. The results are presented in Table MS.01.
If in this Table MS.01 we take any TIM and compare its TIM code with the TIM code of Don Quixote — in logic this operation is called equivalence, replacing identical digits with a one and differing digits with a zero — we obtain the IR code of the corresponding relation, in this case between Don Quixote and the chosen TIM. This is what is shown in the last column of the table. Attention should be paid to the fact that this operation is, in the arithmetical sense, neither addition nor multiplication — there is no carry into the next digit. Moreover, we are not even dealing with the digits of a binary number — this is a positional code. What is at issue is a certain composition possessing substantive subject-domain meaning, namely: first, here sign-markers (not numbers!) occupy strictly defined places corresponding to the systemic structure of the socion with its mathematically proven strict ordering, and these markers indicate the poles of the ARTs; and second, the meaning of such a comparison is that the identity of the corresponding poles is affirmed or denied — and hence the degree of difference of the generalised cognitive positions — which can be understood as the highly abstract component of interhuman relations, as a result of which one of two varieties of socionic relations is defined. In this case we are speaking of interfunctional relations (IF), as distinct from intertype relations (IR). Moreover, one should here speak of a type of everyday relations (ER), whose concrete everyday content may vary within wide limits while nonetheless preserving the corresponding socionic invariant. The chosen type of relations (IF) allows conclusions applying to all TIMs to be drawn on the basis of conclusions about the relations of all TIMs to the TIM of Don Quixote. Nor is this composition a comparison modulo 2 — above all because we are dealing not with arithmetical numbers but with structural codes, but also because the composition table would then be different.
Example of code comparison (Table MS.02)
| Don Quixote | 1111 |
| Robespierre | 0011 |
| Equivalence | ==== |
| IR code | 0011 |
The coincidence of TIM and IR codes is an indicator of their substantive isomorphism, since their semantics is determined by the same poles of the ARTs, which allows the elements of these two sets to be placed in one-to-one correspondence. From a symbolic point of view, a relation represents a difference of cognitive positions, which can be assessed as a composition of dichotomous codes — complete, or reduced to one basis or another, most simply of course to the Jungian one. The naturally adequate composition turned out to be positional comparison: if the signs in a given position coincide, this substantively means that both types have the same polarity there, which can be interpreted as a degree of approach to identity; in the opposite case, as a degree of deviation from it.
Let us consider as an example the composition of the TIM codes of Don Quixote and Robespierre: Table MS.02.
The IR code obtained as a result of the positional comparison — logical equivalence — is the code of the mirror relation (ZE). The codes of all the other relations are obtained analogously, as shown in Table MS.01. The identity of TIM codes and IR codes is a sufficient criterion for considering these constructs semantically isomorphic. This means that their semantics are determined by one and the same set of dichotomous poles.
Expressing the TIMs through the ARTs, we obtain Table MS.03. Let us consider it more closely.
This table expresses the TIMs through the ARTs. The basic table of this kind was first constructed by Aushra Augustinavichiute; however, there the arrangement of the ARTs was motivated by contingent circumstances, dependent on Aushra’s choice of the sequence of polar octaves. Later I succeeded in proving with mathematical rigour the unique ordering of the TIMs in the socion and of the ARTs in the structure of a TIM, which is what is presented in this table. TIMs can be regarded not only as types of people but, in a more general sense, as abstract qualities defining the generalised parameters of any objects and processes in our world. In that case, Table MS.03 can be regarded as a set of structural formulas of highly generalised qualities.
On the other hand, by virtue of the duality of the three basic sets of the socion, the table can be inverted and understood as an expression of the ARTs through the TIMs. This, of course, is no surprise, since it was precisely from this that Aushra constructed her original table. But Table MS.03 contains considerably more information, since it expresses not two but six relationships — TIM–ART, ART–TIM, TIM–IR, IR–TIM, ART–IR, IR–ART — and thus exhausts the set of links between these TIMs, ARTs and IRs.
These links are a heuristic hint permitting the hypothesis that not only the set of ARTs but the sets of TIMs and IRs are also algebraic groups. But such a conjecture is not enough. To turn it into a theorem, a corresponding proof must be given. To this end, let us represent each row of ARTs — that is, the structural codes of the TIMs — as conventional numbers in the binary numeral system.
Performing the set-theoretic operation of equivalence on any pair from this set, we obtain one of the codes of the same set.
Moreover, the composition of any code with the first row (Ki) equals the second code, which makes the row Ki the group identity. This is sufficient for the set of TIMs to be formally considered an algebraic group. Furthermore, such a group is commutative — that is, the result of the group composition does not change when the elements are permuted. Of course, purely mathematically this was to be expected, relying on the principle of duality; but within the subject domain of psychology it would be difficult to overcome the subconscious presupposition that a psychological quality cannot be something on which computational operations could be performed. But let us look at the whole situation from the informal — semantic — side. It is now no longer difficult to accept that the set of IRs may possess analogous properties and is consequently also an algebraic group. Still, matters are not as simple as they might appear.
(Table MS.04 — the Lyashkyavichus–Churyumov table of TIM relations; Table MS.05 — the TIM composition table (Churyumov group 1); Table MS.06 — the relation composition table (Churyumov group 2); Table MS.07 — the multiplication table for the ART group (Reinin group). These are large 16×16 Cayley tables reproduced in the original.)
Let us turn to the table of relations of Lyashkyavichus, widely known in socionics, but with Churyumov’s refinements, which gives the right to call this updated table the Lyashkyavichus–Churyumov table (Table MS.04).
Clearly, from a mathematical point of view this table can be understood as a table of compositions of TIMs, as a result of which relations appear. Incidentally, such an understanding of this table falls under the definition of an algebraic category, i.e. a set with relations defined on it. This idea was later taken up and developed by the talented Ukrainian mathematician A. Ya. Dubrov.
By virtue of the isomorphism of the sets of TIMs and IRs, the corresponding substitutions can be made, and we then obtain composition tables of IRs only (Churyumov group 1), of TIMs only (Churyumov group 2), and of ARTs only (the Reinin group). The ART group was first discovered and investigated by Reinin. However, following mathematical formalism, he assessed it as commutative, whereas in the psychologically substantive sense this group is partially commutative and partially associative, which can be designated by the umbrella term partially alternative. In this case it means that for some elements commutativity fails, and for some, associativity.
Since Table MS.05 is obtained directly from Table MS.04, the asymmetry present there is naturally preserved here as well. And this means that the corresponding group will be at least partially non-commutative. In fact, everything turned out to be even more complicated. The point is that when TIM codes are added, the resulting code, although it does not leave the set of TIM codes, does not always coincide with the TIM code in Table MS.05, which naturally requires explanation. Why does the greater part of the codes “behave correctly” while some yield an “irregular” result? The answer comes down to this group not being fully commutative. This resolves the problem with the TIMs corresponding to asymmetric relations. But it turned out that the codes of the TIMs corresponding to the dual code-pairs business–mirage (DE–MI) and kindred–semi-dual (RD–PD) also behave irregularly. Analysis of these cases showed that a regular “exchange” of codes takes place between these pairs and the TIMs corresponding to them. And this means that in these cases the composition can be corrected by introducing an additional multiplier. Such a multiplier for converting the codes of the pair DE–MI into the codes of the pair RD–PD and back turned out to be the code of Napoleon — or, equivalently, super-ego (SE). Thus the problem of the regularity of the group composition was solved, but the group had to be reclassified from commutative, recognising not only its partially non-commutative character but also the presence in it of partial alternativity, i.e. a regular shift of composition results. It also turned out that the Napoleon code solves the problem of composing the codes of asymmetric relations under permutation. Thus we have two multiplication operations in this group: right and left.
The situation is analogous with the group of relations. Thus this group too is partially non-commutative and partially alternative.
In the shaded fields the result of the composition changes depending on the order of the elements in the composition. So, for example, De©Kv = Pz, while Kv©De = Zk. And so on. In Table MS.04 (Lyashkyavichus–Churyumov), which had not previously been considered as a multiplication table of TIMs, non-commutativity was present from the very beginning. Thus, for example, Zhu©Gyu = Zk, while Gyu©Zhu = Pz.
The conclusion that both Churyumov groups are non-commutative compels the supposition that the Reinin group is also non-commutative, which is exactly what we see in the corrected multiplication (composition) table for this group in Table MS.05, obtained by replacing the TIMs in Table MS.07 with the ARTs isomorphic to them.
Let us see how the proposed algebra works.
For certain symmetric relations the relationship T1©T2 = T2©T1 = IR(1, 2) holds — that is, the composition of the codes of two TIMs gives in this case the code of the relation between them. Examples of the compositions of duals, conflictors and quasi-identicals are given in the tables.
For asymmetric relations, when the types are permuted, the result is determined by the position of who is “HE → TO HIM.” However, if the direct order yields the correct relation, then in the reverse order a corrective co-multiplier must be introduced in the form of the code of Napoleon, or SuperEgo (SE). And vice versa.
Here one may pose a question that is unprincipled from the standpoint of theory: “Why is all this needed, why all these complications? Can this affect the practical applications of socionics?” The point is that thanks to the investigations presented we arrive at an intra-socionic calculus — its elements will be shown below. And although within the framework of 16 types it does not seem necessary, when passing to multiplication tables of larger sets (32, 64, etc.) — the unfolding mechanism I presented in its proper place [UChK: ch. 6] — and when machine processing of such sets is required, the partially commutative, partially alternative algebra of Churyumov considered above will undoubtedly prove useful.
Let us now consider the problem of the trait basis and the expression of non-basis traits through it. Reinin solved this problem completely in mathematical terms, but at that time he did not know that the socion is a strictly ordered set, and this led to inaccuracies in the arrangement of TIMs and ARTs. The point is that there exists a strict order of arrangement of the ARTs, and violating it leads to inaccuracies and systematic errors. In particular, Reinin’s famous Table 20 [Rei] contains a large number of errors, connected with the difficulty of semantic interpretation of algebraic symbols. Moreover, Reinin describes the ARTs in abstract mathematical notation, which hinders the substantive identification of the ARTs.
Table MS.08 presents the difference in the numbering of the ARTs by Aushra Augustinavichiute, Reinin and Churyumov, and also shows the expression of some ARTs through others in Reinin’s numbering. To find in this table the result of composing two ARTs, it suffices to find the intersection of the row and column at whose heads they stand. Thus, the composition of EK (extraversion–introversion) and ST (statics–dynamics) gives IT (intuition–sensing) — see the intersection of the first column and the second row. Attention must be paid here to the fact that we are speaking of semantic invariants, not of abstract algebraic variables that can be chosen and assigned arbitrarily.
The structural isomorphism of the sets of TIMs, IRs and ARTs, and their belonging to one subject domain, gives preliminary grounds for considering them isomorphic in the semantic respect as well. Above all, this is expressed in the fact that behind the corresponding elements of these three sets stand the same archetypes. This will be expressed, in particular, in the fact that, for example, KI (TIM), TZh (IR) and SShch (existence, ART) have a common semantic multiplier, which should be used as a criterion in assessing their concretised values. Thus TZh and SShch share a common seme of fundamentality. To exist means to preserve something that remains identical to itself, which is a fundamental property of existence. Identity is a primary property of being, which is the most fundamental ontological entity. We find the same seme in KI — the orientation towards discovering the fundamental laws of nature. Proceeding from this heuristic principle, one must seek ways of clarifying the other insufficiently elucidated ARTs as well, since the TIMs and IRs, to a first approximation, have proved more accessible to semantic identification. Thus, for example, GYu (Hugo) and the IR corresponding to it, AK (activation), have their models and can be described from them with a sufficient degree of substantiation. At the same time, the ART corresponding to these elements, BE (carefree–farsighted), has no sufficient grounds to be considered as literally coinciding in meaning with the meaning of the words designating it. It is clear, however, that it must contain such semes as extraversion, rationality, sensing and emotionality, as well as activity. Analogous reasoning applies to all the other ARTs, including such relatively more comprehensible ones as extraversion and introversion — although it would be a mistake to simplify their content by reducing it, for example, to openness–closedness, sociability–reserve, greater energy, lesser energy, etc.
By establishing algebraic relations between the elements of the sets of TIMs, IRs and ARTs, we also gain the possibility of making formal substitutions in the corresponding formulas, thereby obtaining access to subtler socionic meanings.
In connection with the reasoning presented, an interesting article by O. Karpenko [2] comes to mind. This article confirms Reinin’s investigations and even partially duplicates them, which does not seem necessary, though it may be regarded as a certain refinement of what Reinin did earlier. However, Karpenko’s own reflections apropos of and in connection with the group character of the set of ARTs contain several ideas requiring closer analysis.
Karpenko writes: “By the number of types we have 16 different socions, the isomorphism of whose groups does not mean their identity. A peculiarity of a socion of a given type is its functioning as a whole in the manner of the IM model of that type. But then all people belonging to that socion will, in their information metabolism, have the peculiarities of the socion’s TIM, and these peculiarities will manifest at levels subtler than intertype relations, the spirit of the quadra, or belonging to the right or left ring of social progress. Their cause and the greatest breadth of their manifestations will belong to a plane higher than the one socionics works with — the higher Manas.” This is an example of an intuitive “estimate” without developed logical argumentation, with references to entities from other subject domains — for example, the Higher Manas. First, to speak idly of such entities would simply be frivolous. Second, references of this kind belong to the category of “explaining the unknown through the unknown,” for it is hard to suppose that the “broad masses” of socionists are fluent in the theological categories of Eastern spiritual teachings. Let us therefore set this argument aside as “beyond the pale.” As for the idea of the existence of 16 socions (if I am not mistaken, it was first voiced by G. Chikiris), it is undoubtedly beautiful, but of course it cannot be accepted without developed argumentation. Behind this idea stands the simple thought that any other type could be placed first in the socion instead of Don Quixote. And that is, perhaps, quite democratic. But surely it is illegitimate to demand democracy in, for example, the structure of chemical molecules. Suppose one demanded that “democratic” justice be restored in the water molecule by adding another oxygen atom, so that the formula be not H₂O but H₂O₂. One may demand it, but the result would be an entirely different substance (hydrogen peroxide). Therefore, in the structure of the socion one must proceed from its law — which, of course, must be discovered, not invented. The author of this idea of 16 socions may also be reproached for stopping at the figure 16. For if you can place any type first, then any of the remaining types can also be placed second. And third, and fourth, etc. As a result we obtain not 16 but 16!, which equals 20,922,789,888,000. And even that is not the limit! For one can now shuffle this whole set as well! ETC.!
If such a number of socions really existed, it would be interesting — but it would be an entirely different world. I, however, have succeeded in proving mathematically that the socion is a strictly ordered set, where each TIM stands in its own place and nothing there may be permuted — as in the water molecule. Nevertheless, the idea of a multiplicity of socions has some grounds. Namely, ethnoses and states have their own typological shades, and this means that one and the same type will evidently feel differently in different ethnoses — but the point here is not a change of socion but a different psychological environment. It is like the way relations between types are built — differently with different types. Thus, for example, the type Don Quixote will feel differently in Russia (Yesenin), England (Stirlitz) or Germany (Maxim). In Russia the people treat this type of person tolerantly, with sympathy and understanding. But the Russian authorities (Zhukov) stand towards this type of person in a relation of passive antagonism (SE). This means that they do not encourage them, but may make use of their discoveries. In England the authorities treat Don Quixotes from a position of active synergy (ZK) — they expect from this type of person results that could be laid at the foundation of new technologies; here science, whose TIM as a social infrastructure is ILE, is supported by the state. Whereas in Russia science is financed on a residual principle and depends on the “benevolence” of government officials, which leads to a high level of corruption in science — in particular, to the presence in it of unscientific ballast placed in “cushy positions” through “connections.” In Germany this type of person and the corresponding infrastructures occupy the position of socionic supervisors, which, in particular, ensures heightened rigour concerning the reliability of scientific results. The discussion presented shows that in developing theory one must rely not only on intuition but also employ logic. But let us continue developing the semantic side of the group structure of the socion.
Table MS.09 presents a comparison of the Jung basis (JB) and the Reinin basis (RB), which he in fact used in his book [Rei] when describing the group structure of the set of ARTs. True, it is unclear why he chose precisely that basis — unless to illustrate the equal validity of all bases. So, as the JB we take the polar traits extraversion–introversion (EK–IN), irrationality–rationality (IR–RA), logic–emotionality (LG–EM) and sensing–intuition (IT–SN). To simplify notation in the formulas we use the symbol of only one pole — without, of course, forgetting the existence of the second. In formal-algebraic terms the symbols X1, X2, X3, etc. are used here. Formally we take X5 = X1·X2, which, given the accepted correspondences, may be rewritten as X5 = EK·IR = DYu·ShT. But what will X5 equal here in substantive notation? To find out, let us turn to Table MS.09. Summing the codes of these TIMs, we obtain the code of DO (Dostoevsky). And this is of course natural, since DO is the dual of ShT, and the multiplier DYu gives the code of the dual. On the other hand, DYu and ShT stand in the relation PK–KN (being supervised–supervision), and one would expect the appearance of the supervisee code GO (Gorky). But if we swap ShT and DYu, we should now obtain the code indicating the role of ShT in this relation — and that code is the code of DO. Here we discover the non-commutativity of the TIM group. But since, purely arithmetically, a simple permutation of the codes does not change the result of the composition, a correcting element must be introduced to realise the non-commutativity. That element is the code of NA (Napoleon), since this code converts the code of DO into the code of GO and back. This is easy to understand, bearing in mind that DO/GO = SE, which corresponds to the code of NA. Performing all the remaining compositions, we obtain the substantive values of all the algebraic unknowns. Note that as a result we obtain the complete list of TIMs and, correspondingly, the complete list of ARTs. Of course, it must be borne in mind that the TIM of Don Quixote corresponds to the neutral ART, which received the name Augustinavichiute–Reinin trait named after Karpenko–Churyumov, since the two latter authors took an active part in clarifying the semantics of this trait.
Let us see how the substantive values of the variables were distributed by Reinin. First let us check whether the four elements he chose are independent. To do this, we find the result of the composition DYu·DO·LO·GO. Adding their codes in Table MS.09, we obtain RO — an element not coinciding with the neutral E (KI) — and this means the chosen elements are linearly independent. The equations given in Table MS.08 show that the RB generates ARTs and TIMs identical to the ARTs and TIMs generated by the JB.
Summing up the calculations presented, one can say that each of the three sets — TIMs, IRs and ARTs — taking their semantics into account, is a partially non-commutative and partially non-associative group of order 16, and every element of these groups is its own inverse.
The possibility of semantic interpretation of compositions in the TIM, IR and ART groups makes it possible to present the interdependent triples of ARTs in explicit form, as done in Table MS.10 (the 35 dependent ART triples).
A vivid example of the approximateness of the reasoning in Karpenko’s article is her intuitive (alas, non-logical!) assessment. She writes: “The periodic system of Shulman, which is built on the basis R (in fact, on a certain relation between the types of the socion), is nothing other than the projection of the four-dimensional cube (the very one that is the graph of the socion) onto a plane passing through its main diagonal. Thus the view of Shulman’s PSS as a separate part of socionics, not dovetailing with the other models, is incorrect. The PSS merely specifies a certain point of view on the socion and a prism in which the intertype relations are refracted.” I have analysed Shulman’s constructions in detail, and to put it very briefly, I can say that what Shulman proposed is neither periodic nor a system — simply because there is nothing there in which a periodic change of some quality would occur. When I asked him what was periodic there, he replied that he had named the sections of the configuration he had devised “periods.” And, strange as it may seem, all three words in this name — PERIODIC + SYSTEM + OF THE SOCION — are wrong. One can speak of adjacently placed symbols as a SYSTEM only with a great stretch. As for the last word, recall how it sounds in Mendeleev — the PERIODIC SYSTEM OF THE ELEMENTS. There are many elements, they are brought into a system, and this system is periodic. If Shulman had many socions and built their periodic system, that would be a different matter; but a periodic system consisting of one element, THE SOCION, is nonsense. Equally unfounded is Karpenko’s assertion that Shulman’s PSS is a projection of the socion onto some plane. In my book [UChK] I constructed such a projection, and it turned out that these things are not connected. Thus the view of Shulman’s PSS as a socionic artefact that does not dovetail with the other models is correct.
Karpenko: We may represent the defining relation a² = 1 as the relation of duality in the dyad. Then in the quadra we have two dual relations differing in character, a² = b² = 1 — that is, we note the differing character of dualisation in the right and left rings of social progress. This fully accords with the idea of the manifestation of the principles of Eros and Thanatos in socionics through the trait right–left, and also of the fundamental difference in the course of the dualisation process in the different rings (see [6–8] for details). In turn, the relation (ab)² = 1 can be regarded as the relation activation + mirror, taken as a paired second-order relation between two dual dyads.
Comment: The author does not say why the relation a² = 1 can be represented as the relation of duality in the dyad. It is simply decreed “so be it.” Why does the author assume that the relations a² = b² = 1 are different, when the letters here may take identical values too? And how does this accord — and “fully,” at that — “with the idea of the manifestation of the principles of Eros and Thanatos in socionics through the trait right–left”? What relation does this idea, taken from psychoanalysis, have to socionics? This is from the repertoire of hasty identification of constructs from different subject domains. At the very least it requires some explanation. Is it not too unambiguous to link the right ring only with love and the left only with death? And finally, in what could the fundamental difference in the dualisation processes in the different rings consist? For at the level of the linking of functions such a difference is impossible to detect in principle.
Karpenko: The dual dyad is the basic instrument for constructing the quadra; the dual relation is the fundamental relation in the socion.
Comment: This is an example of confidently spoken words that look plausible — but what really stands behind them? What does the author wish to say with the words “fundamental relation”? The deepest, the most important, that which underlies the rest? This is yet another attempt by the author to make something distinguished. I have proved that in socionics there are no distinguished TIMs, no distinguished relations, no distinguished ARTs — why then insist on distinguishedness without adducing the necessary arguments?
Karpenko: The quadra is the fundamental unit of the socion; it can be described by the Klein four-group, and its structure reflects the structure of the socion as a whole (the fractal law).
Comment: This is the most substantiated assertion, since it rests on mathematical constructions; but here again distinguishedness is at issue. Still, let us not forget that the socion is a complex crystal-like structure containing several structural elements. These are the correct octaves of Aushra (15), the correct tetrads of Reinin (140), the correct dyads of Churyumov (120), the ARTs (30), and, of course, the TIMs themselves. And in order to make any fundamental assertion, one needs a suitable criterion. Karpenko proceeded from the structure of the group — but did she really give a complete analysis of the structure of the socion? No, she merely touched it lightly.
Karpenko: The four-dimensional cube is the graph of the socion, and the PSS is its projection onto a plane.
Comment: I have spoken of this above. Here I will only note that this is an example of a logically unconfirmed judgement.
Karpenko: There exist 16 socions, whose structure is defined by fixing a certain type as the identity element of the group R.
Comment: As I showed above, this is not correct.
Karpenko: A correspondence can be established between the IM types, the Reinin traits and the intertype relations. A description of the socion through any of these systems will be complete.
Comment: I carried out this work earlier, and not only established this correspondence but also logically substantiated it by the principle of duality. But from the fact that a correspondence can be established between the socionic sets it by no means follows that they should be somehow opposed to one another, or that the socion should be described through any one of them — each of them describes its own domain, and that is quite sufficient.
Karpenko: In the observable world, the distinguished entities are the ILE socion, the Jung basis, Shulman’s PSS, and the trait rationality–irrationality.
Comment: This is not correct.
Karpenko: The nature of intertype relations at different levels is similar but different. Only when considering the isolated interaction of people of two IM types are we dealing with the classical relations of Aushra. When studying relations in a group, second-order relations must be considered on a par with them. And when considering the socion, we may encounter the fact of the folding-up of intertype relations and the most explicit manifestation of doubled second-order relations.
Comment: Second-order relations are a beautiful invention to which no socionic model corresponds.
Karpenko: In time, confirmation comes to the most astonishing hypotheses and discoveries. The suppositions made by people of the ILE type are often perceived as fantasy, a flight of thought, a hypothesis, a bare idea. But if the world is such that they carry within themselves a structure reflecting its laws, then let us be attentive to their words — in knowing themselves they know the world, and in expressing themselves they express the world.
Comment: Fine words. And yet Karpenko draws an incorrect conclusion regarding the commutativity of the Reinin group, as follows from the constructions presented above.
This was an example of a small but concrete episode from the attempts at mathematical analysis in socionics.
Now let us make an essential clarification. The socion is a set consisting of at least three subsets: the set of TIMs, the set of IRs and the set of ARTs. These three sets are equinumerous and isomorphic. That is, each of them consists of exactly 16 elements — which was by no means obvious in Aushra’s time. For while there were 16 types, there were nonetheless 15 traits — although the 16th element was present in Reinin’s algebraic derivations, it was regarded by him as formal-mathematical and was not included in the count of ARTs. For the so-called practitioners of socionics, incidentally, this was completely unimportant. Only one thing mattered to them, namely that they asserted themselves at socionics’ expense and earned money from it, simply exploiting the typological potential of the socionic core created by Aushra. And only the later investigations of O. Karpenko and S. Churyumov led to an understanding of the importance of the neutral element in the set of ARTs. As for the IRs, Aushra had 14 of them, and within her model this could not be corrected in any way. Only after the introduction of the principle of duality of IR and TIM was the question of the number of relations finally resolved. The principle of duality became the basis for establishing a one-to-one correspondence between the elements of each of these sets and the elements of every other. Moreover, it can be shown — and later it will be — that since the elements of each of these sets are semanticised in the subject domain of psychology, the corresponding elements of these sets relate to one and the same archetype, which covers their more concrete psychological meanings.
In Reinin’s book Socionics (2003), Table 20 contains an error in 22 triples connected with the determination of linear dependence — which it became possible to establish thanks to moving to the semantic level.
In the mid-1990s, a mathematics student, after acquainting himself with the basic ideas of socionics, proposed the use of mathematical symbolism for formalising socionic concepts; but apart from a rather cumbersome formalism, nothing followed from it. Elements of such a formalism in a tensor style were later partially employed in his works by A. V. Bukalov.
The Lviv mathematician Ya. A. Dubrov also took the formal path of applying mathematics to socionic constructs, defining the socion as an algebraic category — that is, a set with relations defined on it. Despite this professional attempt to adapt the apparatus of categorical algebra to socionics, it yielded not a single semantically substantive result, since from it there followed neither, say, the strict ordering of the TIMs nor the duality of TIM and IR — it did not even shed light on the group character of any of the socionic sets. What remained of this attempt was the impression that, apart from the exceptional cumbersomeness and difficult accessibility of the apparatus used — even for theorists of socionics — it contributes nothing to an essential understanding of socionics. He did, however, find it possible to improve Boltzmann’s famous equation by adding to it, besides mass and energy, information. I do not know how to interpret this from a physical point of view, but from the point of view of philosophy it is remarkable.
Until 2009 I did not know of the works of Eskisay, and he evidently did not know of mine. On seeing his results I was delighted by them, but fairly quickly recognised their formal superficiality as regards the semantic part of socionics. Eskisay succeeded in describing TIMs and IRs in terms of the theory of a complex variable, and he did so brilliantly. But he went no further than formal statements that the sets of TIMs and IRs constitute algebraic groups with all the mathematical consequences that follow. Undoubtedly this is a splendid result for mathematics: a unique example of a previously unencountered group was obtained; but no serious consequences for socionics itself flowed from these developments. Thus, the strict ordering of the socion was not proved — it was simply accepted in the form of its tabular representation as something self-evident. Moreover, the error present in the Lyashkyavichus table was not corrected! And this means that Eskisay did not succeed — or perhaps it simply did not occur to him — to establish the correct order of the TIMs in the socion. Indeed, not having derived Model A from simpler assumptions and not having constructed the socion itself, he could not have done so. In addition, for some incomprehensible reason he neglected the ARTs and was evidently unacquainted with Reinin’s works — which is, of course, entirely impermissible. Therefore, while paying due respect to Eskisay’s mathematical talent, I nonetheless give a somewhat simpler mathematical description of the socion, differing, among other things, from Reinin’s description. The socion is an aggregate of three (at least) mutually interpretable sets — namely TIMs, IRs and ARTs — constituting partially non-commutative and partially non-associative algebraic groups, with the astonishing peculiarity that every element of any of these groups is its own inverse, i.e. Xn·Xn = E, which makes these groups not only partially non-commutative but also trans-associative; and this permits, in three-term equations linking the elements of these groups, any term to be transferred from the right side to the left without introducing an inverse operation: Xi·Xj = Xk → Xi·Xk = Xj → Xj·Xk = Xi. But, of course, the admissibility or inadmissibility of commutativity for the elements entering into the composition must be taken into account.
In addition, Eskisay took the IRs as a set given in advance and entirely formally (tabularly), partially interpreting it as the result of a group operation — which is perfectly admissible in purely mathematical terms, but unacceptable in substantive socionic terms.
References
- Reinin, G. R. Socionics. St Petersburg, 2005.
- Karpenko, O. B. “The group structure of the socion; socionic bases.” SMiPL 3, 1996.
- Dubrov, Ya. A. “Conceptual and mathematical modelling in socionics.” SMiPL 5, 1999.
- Eskisay. Foundations of Socionics. Tyumen, 2008.
- Churyumov, S. I. “The partially alternative algebra of Churyumov.” Paper, 29th conference, 20 Sept. 2012.
- Chikirisova, G. V. “Socions.” SMiPL, 1996.
Translator’s notes: ТИМ → TIM (type of information metabolism); ИО → IR (intertype relations); АРП → ART (Augustinavichiute–Reinin traits); ДП → DT (differentiating traits); БЮ/БР → JB/RB (Jung basis / Reinin basis); СЭ → SE (super-ego / Napoleon code); Модель А → Model A. The large 16×16 Cayley tables (MS.03–MS.08) and the dependent-triples table (MS.10) are noted in place rather than reproduced cell-by-cell; the original page images should be consulted for the full matrices. The original contains an apparent typo duplicating code 0010 for both Gorky and Dostoevsky in Table MS.01.