A Knot Model in Psychology. Akio KAWAUCHI ABSTRACT

Transcription

1 A Knot Model in Psychology Akio KAWAUCHI ABSTRACT By evaluating the five-factors in Five-Factor Model of personality established by H. J. Eysenck-S. B. G. Eysenck and P. T. Costa-R. R. McCrae, we construct a knot model of a human mind in terms of 2-bridge knots so that the knot type of a mind-knot gives the personality We relate the classification of the self-releasability relations of mind-links to a certain classification of links The (normal) self-releasability relations for two mind-knots and three mind-knots are classified into 3 relations and 30 relations, respectively. Mathematical Subject Classification 2000: 57M25, 91E45, Keywords: Five-Factor Model, Personality, Human mind, Psychology, Knot, Link, Self-releasability relation 1. Introduction In this paper, we propose to visualize a mind situation by constructing a knot model of a human mind (briefly, a mind) as an application of knot theory. For several years ago (cf.[9]), the author considered that there is no contradiction by considering minds as knots whose knot types are regarded as the personalities and whose crossing changes are regarded as mind-changes, from the reason that several expressions on minds, such as a tame (straightforward) character:š (sunaona seikaku) in Japanese and a twisted character:šƒˆ (hinekureta seikaku) in Japanese 1, are represented in daily life by strings which are main research objects in knot theory. Besides, the author knew by R. Rucker s book [12] that B. Stewart and P. G. Tait 2 appears to say in their book [16] that the soul exists as a knotted vortex ring in the aether, although the aether hypothesis Šis known as a wrong story and the author 1 Furthermore, ˆ ( )(omoi no ito) meaning a string connecting human feelings, (kokoro no kotosen) meaning one s heartstring, ƒ (ningen kankei no motsure) meaning an emotional entanglement on human relations, (kokoro ni wadakamaru) meaning being rooted in one s mind. 2 P. G. Tait is known as a pioneer of knot theory. 1

2 does not clearly understand what they mean. For our purpose, the following points are some difficulty points which we must overcome to construct our model (cf. [3]): (1) (Birth-Time Mind Situation) By a genetic character inherited from Parents, we cannot always assume that the mind in birth time is untwisted. (2)(Estimation on Changes of Environments) There are many sources of mindchanges coming from changes of environments on (i) Age (ii) History (iii) Non-standard event factors. In our knot model of a mind, we count the conditions (1) and (2). Throughout this paper, we mean by knots unoriented loops embedded in the 3-space R 3, and by links, disjoint unions of finitely many knots in R 3. The types of knots and links are understood as the equivalence classes of them under the Reidemeister moves in Fig.1. For general terms of knot theory, we refer to [8]. Figure 1: Reidemeister moves Concluding this introduction, we mention here some studies on applying topology to psychology. E. C. Zeeman s work on the topology of the brain and visual perception in [20] and the topological psychology by K. Lewin [10] which is an application of topological space are respectively pointed out by R. Fenn and by J. Simon with an implication by S. Kinoshita during the international conference International Workshop on Knot Theory for Scientific Objects. L. Rudolph reported to the author his works of several applications of topology to psychology which are published in [13, 14, 15] and his works (in preparation) on a rehabilitation of K. Lewin s topological 2

3 psychology. Incidentally, the author also knew J. Valsiner s book [18] by Rudolph s report. 2. Constructing a knot model of a mind To clarify our viewpoint of a mind, we set up the following mind hypothesis: Mind Hypothesis 2.1. (1) A mind is understood as a knot, so that an untwisted mind is a trivial knot and a twisted mind is a non-trivial knot. (2) A personality is understood as the type of a knot, so that the untwisted personality is the knot type of a trivial knot and a twisted personality is the knot type of a nontrivial knot. (3) A mind change is understood as a crossing change of a knot(see Fig.2). Figure 2: A crossing change To construct a concrete knot model, we consider the basic factors of a mind by H. J. Eysenck [4, 5] and H. J. Eysenck-S. B. G. Eysenck[6, 7], which are described as follows: (1) Introversion-Extroversion. (2) Neuroticism. (3) Psychoticism. These basic factors are refined by P. T. Costa and R. R. McCrae [1] as Five-Factor Model (= Big-Five Model) described as follows: (1) Introversion-Extroversion. (2) Neuroticism. (3.1) Openness to Experience. (3.2) Agreeableness (3.3) Conscientiousness Our idea of constructing a knot model of a mind is to evaluate the degrees of the five-factors of the personality of a mind at the n-year-old as follows(see [19] for a concrete test): 3

4 Definition 2.2. (1) The introversion-extroversion degree IE n at n-year-old takes the value IE n = 1, 1,0 according to whether the mind is introverted, extroverted, or insignificant. (2) The neuroticism degree N n at n-year-old takes the value N n = 1, 0 according to whether the mind is neurotic or not. (3.1) The openness to experience degree O n at n-year-old takes the value O n = 0, 1 according to whether the mind is open to the experience or not. (3.2) The agreeableness degree A n at n-year-old takes the value A n = 1, 0 according to whether the mind is disagreeable or not. (3.3) The conscientiousness degree C n at n-year-old takes the value C n = 1, 0 according to whether the mind is unconscientious or not. We define the psychoticism degree at n-year-old from (3.1)-(3.3) as follows: (3) The psychoticism degree OAC n at n-year-old to be the product OAC n = O n A n C n = 1 or 0. We also need Father s and Mother s data IE F, N F, OAC F and IE M, N M, OAC M on their introversion-extroversion, neuroticism and psychoticism degrees at the baby s birth time, respectively. Namely, we have the following definition: Definition 2.3. (1) Father s and Mother s introversion-extroversion degrees IE F and IE M at the baby s birth time take the values IE F = 1, 1,0 and IE M = 1, 1,0 according to whether Father s mind and Mother s mind are introverted, extroverted, or insignificant, respectively. (2) Father s and Mother s neuroticism degrees N F and N M at the baby s birth time take the values N F = 1, 0 and N M = 1, 0 according to whether Father s mind and Mother s mind are neurotic or not, respectively. (3.1) Father s and Mother s openness to experience degrees O F and O M at the baby s birth time take the values O F = 0, 1 and O M = 0, 1 according to whether Father s mind and Mother s mind are open to the experience or not, respectively. (3.2) Father s and Mother s agreeableness degrees A F and A M at the baby s birth time take the values A F = 1, 0 and A M = 1, 0 according to whether Father s mind and Mother s mind are disagreeable or not, respectively. (3.3) Father s and Mother s conscientiousness degrees C F and C M at the baby s birth time take the values C M = 1, 0 and C F = 1, 0 according to whether Father s mind and Mother s mind are unconscientious or not, respectively. We define Father s and Mother s psychoticism degrees at the baby s birth time from (3.1)-(3.3) as follows: (3) Father s and Mother s psychoticism degrees OAC F and OAC M at the baby s birth time are the products OAC F = O F A F C F = 1 or 0 and OAC M = O M A M C M = 1 or 0. 4

5 We define Parents introversion-extroversion, neuroticism and psychoticism degrees IE P, N P and OAC P to be IE P = l F IE F + l M IE M, N P = m F N F + m M N M, OAC P = n F OAC F + n M OAC M for non-negative integral constants l F, l M, m F, m M, n F, n M. In our argument here, we take l F = l M = m F = m M = n F = n M = 1, but the values shuold be chosen by estimating carefully a genetic character inherited from Parents (cf. [3]). The total introversion-extroversion, neuroticism and psychoticism degrees of a mind at n-year-old are defined as follows: IE[n] = IE P + N[n] = N P + n IE i, i=1 n N i, i=1 OAC[n] = OAC P + Let a = IE[n] + N[n] and b = OAC[n]. We have n OAC i. i=1 2n 4 a n + 2 and n 2 b 0. Our knot model of a mind is the knot M(n; a, b) which is represented by a knot diagram with 2 a + 2 b crossings illustrated in Fig.3 and called the nth mind-knot. In the figure, we illustrate the case that a is negative. The picture for the case that a is positive is given by changing all the the crossings in this 2 a -tangle part into the crossings with the opposite sign. When the knot diagrams M(n 1,a, b ) and M(n; a, b) are distinct, we consider that some mind-changes are made during (n 1)- year-old and n-year-old. Thus, in our model, the total picture of a mind-knot from the birth time to the n-year-old is considered as a cylinder immersed properly in the 4-dimensional space R 3 [0, n]. If we evaluate the degrees of the five-factors finer, then a finer mind-knot model can be constructed. See 5 later for the other mind-knot models. The following proposition is a consequence of the well-known classification of 2-bridge knots in knot theory (cf. [8, 17]). Proposition 2.4. A mind-knot M(n; a, b) is untwisted if and only if a = 0 or b = 0. Two twisted mind-knots M(n; a, b), M(n ; a, b ) have the same personality if and only if (a, b ) = (a, b) or ( b, a). Proof. The knot M(n; a, b) is a 2-bridge knot whose type is calculated as follows: 1 2a + 1 2b = 2b 4ab

6 Figure 3: A mind-knot M(n; a, b) Thus, M(n; a, b) is a trivial knot if and only if 4ab + 1 = ±1 that is ab = 0. Let M(n; a, b) and M(n ; a, b ) be non-trivial knots. If they belong to the same knot type, then we have 4ab + 1 = ±(4a b + 1), so that ab = a b 0. Further, we have 2b 2b (mod 4ab+1) or 4bb 1 (mod 4ab+1). If the first case occurs, then there is an integer k such that 2(b b ) = k(4ab+1). The integer k must be even, i.e., k = 2k and we have b b k = 4abk. Suppose k 0. Then for N = max{ b, b, k } > 0, we have 3N 4N, a contradiction. Hence we have k = 0 and b = b, a = a. If the second case occurs, then we have 2b 2a (mod 4ab + 1). In fact, 2b (2b + 2a ) 4bb + 4a b = 4bb + 4ab 1 1 = 0 (mod 4ab + 1) and, since 2b is coprime with 4a b +1 = 4ab+1, we have 2b+2a 0 (mod 4ab+1). Using that M(n ; a, b ) and M(n ; b, a ) belong to the same knot type, we see from the argument of the first case that b = a and a = b. For example, M(n; 1, 1) and M(n; 1, 1) are trefoil knots which are each other s mirror images, and M(n; 1, 1) = M(n; 1, 1) is the figure-eight knot. 3. Self-releasability relations on mind-links When we consider a mind as a knot, it is an interesting problem to consider a mindlink, that is, a link of mind-knots. We consider that a mind-link is generated by a crossing change (called also a mind-change) between a pair of mind-knot components and a mind-change on a mind-knot component. We shall introduce here a concept of self-releasability relation on links of n( 2) mind-knots. Definition 3.1. Two non-splittable mind-links L and L have the similar selfreleasability relation if there is a bijection from the mind-knots A i (i = 1,2,...,n) 6

7 of L onto the mind-knots A i (i = 1,2,...,n) of L such that for every mind-knot component A i and every sublink L i of L\A i (1) the splittability between A i and L i coincides with the splittability between the corresponding A i and L i, and (2) the splittability between A i and L i up to finitely many mind-changes of A i coincides with the splittability of the corresponding A i and L i up to finitely many mind-changes of A i. In Definition 3.1, we say that a mind-knot component A i is self-releasable from L i if A i is made split from L i by finitely many mind-changes of A i, which is equivalent to saying, in terms of topology, that A i is null-homotopic in R 3 \L i. The similar selfreleasability relation is an equivalence relation and we call the equivalence class of a non-splittable mind-link under the similar self-releasability relation a self-releasability relation. To consider the classification of self-releasability relations, we consider a restrictive mind-link which satisfies the following hypothesis: Normal Relation Hypothesis(NRH). For every i, A i is self-releasable from L i if and only if A i is self-releasable from A j for every mind-knot component A j of L i. Figure 4: An anomalous self-releasability relation for three mind-knots We call a mind-link satisfying NRH a normal mind-link. The other mind-link is called an anomalous mind-link. A self-releasability relation on normal mind-links is called a normal self-releasability relation, and the other self-releasability relation an anomalous self-releasability relation. A self-releasability relation for 2 mind-knots is always normal. However, for three mind-knots, there are anomalous self-releasability relations. A typical example is the Borromean rings in Fig. 4 (We denote a split link of two mind-knots A and B by A B in the schematic diagram.) The anomaly follows from a classification of link-homotopy (see J. Milnor[11]). The classification of self-releasability relations for 2 mind-knots is given as follows: Proposition 3.3. The following cases (1)-(3) give the complete list of the selfreleasability relations for 2 mind-knots A and B. (1) Neither mind-knot component is self-releasable from the other. This relation is denoted by A B. 7

8 Figure 5: Self-releasability relations for two mind-knots (2) Both mind-knots are self-releasable from each other. This relation is denoted by A B. (3) One mind-knot component, say A is self-releasable from the other mind-knotcomponent B, but B is not self-releasable from A. This relation is denoted by A B. In this case, A is necessarily a twisted mind. Proof. Let A B be a 2-component non-split link. For example, if the linking number Link(A, B) 0, then it satisfies (1), because A is not null-homotopic in the space R 3 \B and B is not null-homotopic in the space R 3 \A. If A and B are untwisted mind-knots and the linking number Link(A, B) = 0, then A and B are respectively null-homologous in the spaces R 3 \B and R 3 \A which are homotopy equivalent to S 1, so that A and B are null-homotopic in R 3 \B and R 3 \A, respectively. Thus, this link in this case satisfies (2). If Link(A, B) = 0 and B is an untwisted mind and not null-homotopic in R 3 \A (in this case, A must be a twisted mind-knot), then A is self-releasable from B, but B is not self-releasable from A, satisfying (3). The typical examples of the cases (1)-(3) are given in Fig. 5. In (3) of Fig. 5, we note that A is a trefoil knot and B is a trivial knot and represents the element x 1 y in a group presentation (x, y xyx = yxy) of the fundamental group π 1 (R 3 \A) (see [2]). Since this group is non-abelian, we have x 1 y 1 in (x, y xyx = yxy) and we see that B is not null-homotopic in R 3 \A. 4. Classification of normal self-releasability relations for three mind-knots We give here the classification of normal self-releasability relations for three mindknots. Proposition 4.1. The list of 30 links in Fig. 6 and Fig. 7 is the complete list of normal self-releasability relations for three mind-knots. 8

9 Figure 6: Normal self-releasability relations for three mind-knots 9

10 Figure 7: Normal self-releasability relations for three mind-knots(continued) Proof. This construction is done by listing first all the distinct triangle relations with virtices A, B, C up to permutations of A, B, C and then realizing them by links. The non-splittability and normalness of the links listed here are easily checked if an argument similar to the proof of Proposition 3.3 is used. 5. Further discussions By taking a P = IE P + N P, b P = OAC P, ã = a a P, and b = b b P, we have a different knot model M(n; a P, b P, ã, b) of a mind, illustrated in Fig. 8. If b P = 0, then we have M(n; a P, b, ã, b) = M(n; 0,0,ã, b) where Proposition 2.4 can be used for the classification. However, if b P 0, then M(n; a P, b P, a, b) is on a different behavior from M(n; a, b) in general. We have also a further refined knot model by decomposing ã and b into the numbers a i = IE i + N i and b i = OAC i (i = 1,2,..., n), 10

11 Figure 8: A mind-knot M(n; a P, b P, a, b) respectively, and by considering the knot M(n; a P, b P, a 1, b 1,..., a n, b n ) illustrated in Fig. 9. In our construction, we used the full-twist tangles of the numbers a, a P, ã, a i and b, b P, b, b i. If we use the half-twist tangles of these numbers instead of the fulltwists, then our knot model changes into a link model which is a mind-knot or a link of just two untwisted mind-knots. Also, we note that there is a finer concept of the self-releasability relations for three or more mind-knots described as follows: Two non-splittable mind-links L and L have the same global self-releasability relation if there is a bijection τ from the mind-knots of L onto the mind-knots of L such that for every mind-sublink L 1 of L and every mind-sublink L 2 of L\L 1, we have (1) the splittability on (L 1, L 2 ) coincides with the splittability of (τ(l 1 ), τ(l 2 )), and (2) the splittability on (L 1, L 2 ) up to finitely many mind-changes on the components of L 1 coincides with the splittability of (τ(l 1 ), τ(l 2 )) up to finitely many mind-changes on the components of τ(l 1 ). Figure 9: A mind-knot M(n; a P, b P, a 1, b 1,..., a n, b n ) 11

12 The same global self-releasability relation is an equivalence relation. The equivalence class of a non-splittable mind-link under the same self-releasability relation is called a global self-releasability relation. Then we set the following problem, meaningful even in knot theory when we replace n mind-knots by general links of n components, which will be also discussed elsewhere. Problem 5.1. Classify the global self-releasability relations for n mind-knots. Finally, the author does not yet succeed in constructing any process model of generating a mind-link from mind-knots. References [1] P. T. Costa and R. R. McCrae, From catalog to classification: Murray s needs and the five-factor model, Journal of Personality and Social Psychology, 55(1988), [2] R. H. Crowell and R. H. Fox, Introduction to knot theory, Ginn and Co. (1963). [3] H. Enomoto and T. Kuwabara, Personality psychology (in Japanese), A textbook of the University of the Air(2004). [4] H. J. Eysenck, Dimensions of personality, Transaction Publishers(1997). [5] H. J. Eysenck, The biological basis of personality, Transaction Publishers (2006). [6] H. J. Eysenck and S. B. G. Eysenck, Personality structure and measurement, London: Routledge(1969). [7] H. J. Eysenck and S. B. G. Eysenck, Psychoticism as a dimension of personality. London: Hodder and Stoughton (1976). [8] A. Kawauchi, A survey of knot theory, Birkhäuser (1996). [9] A. Kawauchi, Physical-chemical systems and knots (in Japanese), in: Have fun with mathematics, 5(1998), 72-80, Nihon-hyoron-sha. [10] K. Lewin, Principles of topological psychology, McGraw-Hill (1936). [11] J. W. Milnor, Link groups, Ann. of Math., 59(1954), [12] R. Rucker, The fourth dimension, A guided tour of the higher universes, Houghton Mifflin Company, Boston(1984). [13] L. Rudolph, The fullness of time, Culture and Psychology, 12(2006),

13 [14] L. Rudolph, Mathematics, models and metaphors, Culture and Psychology, 12(2006), [15] L. Rudolph, Spaces of ambivalence: Qualitative mathematics in the modeling of complex fluid phenomena, Estudios de Psicologia, 27(2006), [16] B. Stewart and P. G. Tait, The unseen universe or physical speculations on a future state, MacMillan and Co., London and New York (1894). [17] H. Schubert, Knoten mit zwei Brücken, Math. Z., 65(1956), [18] J. Valsiner, The guided mind: a sociogenetic approach to personality, Harvard University Press(1998). [19] five personality traits. [20] E. C. Zeeman, The topology of the brain and visual perception, Topology of 3-manifolds and related topics, pp , Prentice-Hall(1962). Osaka City University, Sugimoto, Sumiyoshi-ku, Osaka , Japan kawauchi@sci.osaka-cu.ac.jp 13