Vladimir A. Lefebvre (1995) The Anthropic Principle in Psychology and Human Choice. Psycoloquy: 6(29) Human Choice (1)

School of Social Sciences

University of California, Irvine

Irvine, California

VALEFEBV@UCI.EDU

We introduce a model of a subject facing a choice of an alternative out of a set. The model ties together three aspects of human activity: behavioral, mental, and neural-computational. Parameters of this model cannot be estimated experimentally. Thus, a problem arises of determining them by means of theoretical considerations. Similar problems appeared in cosmology as well: the values of the fundamental constants necessary for constructing models of the Universe cannot be determined empirically. One possible solution is to use the "anthropic principle," that is, an abstract statement which allows narrowing the number of combinations of values. We show that a similar methodological gambit can be used in psychology. We formulate an abstract statement and find the parameters of the model with its help. Here we establish the relation of this model to other theories of choice and to experimental psychology. Then we demonstrate that our model is formally isomorphic to the process of gradual minimization of work lost by a heat engine system. The existence of such an isomorphism supports a hypothesis that mental phenomena are related to macro-characteristics of an ensemble of neuron states. KEYWORDS: choice; computation; decision theory; ethical cognition; mathematical psychology; model building; parameter estimation; probability; rationality.

1. In 1937, Paul Dirac, one of the founders of quantum mechanics, noticed surprising coincidences between numbers characterizing the micro-world and the Universe as a whole. Twenty years later, Robert X. Dicke showed that this coincidence could be explained by assuming that the conditions necessary for the appearance of an observer similar to a human being correspond to a certain period in the evolution of the Universe (Barrow & Tipler, 1986). Continuing this line of research Brandon Carter suggested using a principle, which he called "anthropic," for choosing constants in cosmological models. Its essence is expressed in the statement:

...what we can expect to observe must be restricted by the conditions necessary for our presence as observers. (Carter, 1974, p.291)

2. This declaration "forbade" constructing models of Universe in which an observer similar to a human being could not appear. Since the existence of human beings depends on the existence of organic molecules, only those relations between fundamental constants that allow organic molecules to appear in the Universe are acceptable. Therefore, the anthropic principle can be regarded as a special method of reasoning, on which to base our selection of cosmological models (Barrow & Tipler, 1986).

3. According to this methodological scheme we formulate a statement which cannot be empirically falsified but which allows us to generate hypotheses concerning possible values of parameters in mathematical models. In this way we can progress from models of isolated phenomena to more general theoretical constructions.

4. We consider a concrete model of human activity related to choice. This model ties together three aspects of this activity: behavioral, mental and neural-computational. We could not construct such a model, however, if we followed a contemporary paradigm which requires that its parameters must be estimated experimentally. Let us consider this problem in more detail. We seek a function of variables x1, x2, and x3 by assuming that this function, X1=f(x1,x2,x3), is linear in each of its arguments in the sense that if the values of any two are fixed, X1 is linear in the third. It follows from this assumption that X1 can be represented as a polynomial

f(x1,x2,x3)=a0+a1x1+a2x2+a3x3+a4x1x2+a5x1x3+a6x2x3+a7x1x2x3. (1.1)

5. The right side of (1.1) contains eight parameters a0, a2,..., a7 whose values are unknown. Suppose that we have established experimentally that /a0/<0.001 with probability 0.99. For predicting behavior this information might be sufficient. But the properties of f(x1,x2,x3), which must be known for establishing connections between the different aspects of the model, may depend on the answer to a cardinal question: whether a0 is equal to 0 or not? Experimental data, in principle, cannot answer this question. Similar problems appear concerning other parameters as well.

6. We will demonstrate later how the idea similar to the anthropic principle in cosmology allowed us to find the exact values for the parameters a0, a1,...,a7 in a particular psychological model and how knowledge of an analytical form of a function allowed us to suggest a concrete hypothesis about the nature of computational processes in the human brain.

7. Our point of departure is the following statement, which we will call the Principle of Freedom:

We have free will and under certain circumstances also freedom of choice.

8. We will try to show that on this base a theory can be constructed from which falsifiable models of psychological phenomena related to choice can be deduced.

9. To give the terms "free will," "freedom of choice," and "certain circumstances" an operational meaning, imagine an abstract subject who is to choose one of six alternatives marked 1, 2, ..., 6 in the following way. He must pick from a continuum of dies with different centers of gravity, so that there is a one-to-one correspondence between the set of dies and the set of all possible probability distributions of choosing an alternative from the six. If there are no obstacles, the subject (i) plans to use a certain die; (ii) picks it from the continuum, and (iii) casts it, thereby choosing one of the six alternatives. Note that the subject can make a completely determined choice, namely, he can plan to use only those dies which always show the same face up.

10. Imagine now that a state of the external world that may or may not interfere with the process of actually taking from the continuum the die the subject planned to use. For each plan there are such states of the world.

11. The statement asserting the subject's freedom of will means that he can plan to use any of the dies from the continuum independently from the state of the world.

12. The statement that under certain circumstances the subject has freedom of choice means that there are states of the world that do not interfere with the process of using the die the subject planned to use. In this way we distinguish between "free will" and "freedom of choice."

13. Complete absence of freedom of choice means that some state of the world exactly predetermines the die to be taken. In this case the subject's choice does not depend on his intention.

14. We will call a choice realistic if the subject plans to use only those dice that can be taken under a given state of the world. A subject who makes only realistic choices will be called a Realist.

15. This scheme reflects two different tendencies in psychology. Miller, Galanter and Pribram (1960) emphasized the plan as a central element of human activity. The distinction between a readiness (or setting) to act, on the one hand, and the actual act, on the other, evolved from gestalt-psychology and was developed by Uznadze (1968) and his school (Prangishvili, 1967). In our scheme, the subject with a die in his hands represents the state of readiness to act, and casting a die the act itself.

16. A subject facing a choice from a set of n alternatives (n>=2) will be represented by the function

a=G(b,c), a and c are elements of D, b is an element of S. (2.1)

17. The set D={(p1,p2,...,pn)}, where pi>=0, i=1,2,...,n, and p1+p2+...+pn=1, is the set of all probability distributions on a given set of n alternatives, and S is the set of all possible states of the world. Variable a represents the probability distribution characterizing the state of the subject's readiness to choose an alternative. Variable b represents the world influence on a particular choice. Variable c represents the probability distribution which the subject plans to use for choosing an alternative.

18. The statement that the subject has a freedom of will corresponds to the assumption that variables b and c are independent.

19. The statement that under certain circumstances the subject has freedom of choice corresponds to the assumption that there exist an S1 subset of S such that for every b element of S1, the following identity holds

G(b,x)==x, (2.2)

where x is any distribution from D and "==" means identical.

20. The existence of states of the world that predetermine the subject's choice behavior implies that there exists an S2 subset of S such that for every b element of S2 there is exactly one y(b) element of D that satisfies the identity

G(b,x)==y(b), (2.3)

where x is any distribution in D.

21. The Realist is represented by the Equation

G(b,x)=x, (2.4)

where x is element of D and b is element of S.

22. Let us put each element b which is element of S into correspondence with the set D[b] using the following rule: if with a given b Equation (2.4) does not have a solution, then D[b] is an empty set; otherwise, D[b] consists of all the elements x which are elements of D for which Equation (2.4) holds. Therefore we introduce a single-valued function g(b):

D[b]= g(b). (2.5)

23. Not all sets D[b] are empty, because when b is element of S1, D[b]=D (follows from (2.2)), and when b is element of S2, D[b] is a one-element set {y(b)} (follows from (2.3)).

24. Let us emphasize that on the right-hand side of (2.5), there is only one variable b corresponding to the state of the world. By comparing (2.5) with (2.1) we see that in the case when a choice is realistic, that is, when a=c, we can exclude variable c from consideration; its values are subjective phenomena and as such, non-observable. Imagine that we want to predict a Realist decision if we know the function g(b) and having observed the value of variable b. If D[b] is empty set, this means that with a given b realistic choice is impossible, and that the Realist refuses to choose. If D[b] contains only one element, this means that the subject will use one particular distribution, hence Equation (2.5) allows us to make a precise prognosis. If D[b]=D, this means that the subject has freedom of choice and can decide to use any of the distributions in D. Therefore, the state in which the subject has the ability to make a free choice cannot in principle correspond to any kind of concrete probability distribution. In this case observation of the world does not allow us to make a more precise prognosis. If D[b] consists of more than one element and D-D[b] is not an empty set, we can predict that the Realist will not use distributions belonging to D-D[b]. We cannot say, however, which distribution from D[b] he will use; in this case we say that the subject has partial freedom of choice.

25. We use the ideas described above to construct a concrete model of a subject making binary choices (Lefebvre, 1992a) and a model of choice among any number of alternatives (Lefebvre, 1994a). In the binary choice model, we consider a subject choosing between a positive and a negative pole. The subject is represented by the function

X1=f(x1,x2,x3), (3.1)

where X1,x1,x2,x3 are elements of [0,1]. The value of X1 is interpreted as the readiness to choose a positive pole with probability X1, and the value of x3 as the subject's plan or intention to choose a positive pole with probability x3. Variables x1 and x2 represent the world influence on the subject. Thus, variable b in Equation (2.1) above corresponds to the pair (x1,x2) in Equation (3.1).

26. We assume that in every choice situation the world influences the subject by two kinds of impulses. An impulse of the first kind stimulates the subject to choose a positive pole, one of the second kind a negative pole. The value of x1 is the frequency of the impulses toward a positive pole (at the time of the subject makes his choice). The value of x2 is the mean frequency of the impulses toward a positive pole, sent to the subject in similar previous situations (including the present one). Thus, the value of x2 reflects the subject's earlier experience, in other words, the subject's "knowledge" about a class of similar experiences.

27. Next we introduce

POSTULATE OF SIMPLICITY:

Function (3.1) is linear in each of its three arguments.

As mentioned in the Introduction, it follows from this assumption that f(x1,x2,x3) can be represented as in a (1.1), i.e. containing eight unknown parameters. We can find these parameters by formulating certain limitations for function (3.1). Those limitations will correspond to the requirements posed to the subject who possesses free will and the ability for free choice. They are given by Equations (2.2) and (2.3). First we formulate these limitations as axioms; then we will discuss their meaning.

28. Equation (2.2) corresponds to

For any x3 element of [0,1], f(0,0,x3)==x3. (3.2)

The set S1, in this case, contains at least one element b=(0,0).

29. Expression (2.3) corresponds to the following two axioms:

For any x3 element of [0,1], f(0,1,x3)==0. (3.3)

For any x2,x3 elements of [0,1], f(1,x2,x3)==1. (3.4)

The set S2 in this case consists of at least an element (0,1) and a set of elements (1,x2), where x2 takes on values from [0,1].

30. The restrictions corresponding to Equations (3.2), (3.3), and (3.4) stem from moral philosophy, which has a rich history of logical manipulation involving the categories of good and evil. In that field freedom of choice is usually linked with responsibility: a person is responsible for his actions if at the moment of its performance he was free to perform it and conversely (Losskii, 1927). Thus expression (3.2) corresponds to the statement: If the state of the world inclines the subject to perform a negative action (x1=0), and the subject knows this (x2=0), then the subject is responsible for the choice. Expression (3.3) states: If the state of the world inclines the subject to perform a negative action (x1=0), but the subject believes that the world always presses toward good (x2=1), then the subject yields to the world's pressure (X1=0). Finally, expression (3.4) corresponds to the statement: A human being is not a source of evil; temptation can come only from an external agent. Hence if the world urges the subject to perform a positive action (x1=1), only the positive action can be performed (X1=1). Such recourse to moral philosophy may create the impression that I suppose a human being to be an innately moral being. Karl Popper wrote in his note about my book (Lefebvre, 1992a):

No psychologist will like this. Everyone will say this means thinking of man as an essentially moral being. (Popper, 1992)

31. Here is my answer: The problem of morality in a religious or any other conventional sense is not important for constructing a model. Relying on moral philosophy is a technique which helps to find the values of initial conditions.

32. Using the Postulate of Simplicity, Axioms I, II, and III we can write a system of equations for parameters a0, a1, ..., a7:

f(0,0,x3)==x3, f(0,1,x3)==0, (3.5) f(1,x2,x3)==1.

33. After solving this system we find that a0=0, a1=1, a2=0, a3=1, a4=0, a5=-1, a6=-1, a7=1. Substituting these values into (1.1) we obtain:

X1=x1+(1-x1-x2+x1x2)x3. (3.6)

34. Heretofore, we have found a specific function from the class of functions given by Equation (1.1) without experimental estimation of the parameters a0, a2, ..., a7 but using an abstract principle of freedom, instead (Lefebvre, 1992a).

35. To represent the Realist, (X1=x3), expression (3.6) can be written as

x1/(x1+x2-x1x2), if x1+x2>0X1 = { (3.7)

any number from [0,1], if x1+x2=0.

36. Equation (3.7) corresponds to expression (2.5), where b is an element of the set of all pairs (x1,x2). If x1+x2>0, then the set D[(x1,x2)] consists of a single distribution (X1,1-X1) given by Equation (3.7), and if x1+x2=0, then D[(0,0)] coincides with the set of all the distributions.

37. Recently Schreider (1994) showed that in another interpretation, Equation (3.6) can be deduced from pure probabilistic considerations. We can also deduce this equation from an earlier version of our model (Lefebvre, 1977a; 1980; 1982). Based on this version a special method of simulating human decision making has been developed (Shankin, 1994). Experimental aspects related to the early version of the model were discussed by Adams-Webber (1987) and Zajonc (1987), and its logical and epistemological aspects were analyzed by Rapoport (1982; 1990), Townsend (1983; 1990), Levitin (1987), McClain (1987), Wheeler (1987; 1990), Batchelder (1987), and Kauffman (1990).

38. Further we will show that the functions (3.6) and (3.7) can be directly related to observable psychological phenomena (Lefebvre, 1990, 1992a, 1994a).

39. The BTL model is a distinctive starting point in the psychology of choice. It helps to compare various probabilistic models of choice behavior. In the case of binary choice, the BTL model is based on the assumption that the probabilities of choosing alternatives belonging to a certain set are proportional to their utilities. Thus, if utilities of alternatives A and B are respectively v1 and v2, then the probability that A will be chosen from the set {A,B} is given by

p(A,B)=v1/(v1+v2). (4.1)

40. Imagine now that the Realist, represented by function (3.7), is making a choice between A and B. Without loss of generality, we can assume v1>=v2. Let A be the positive pole, and B the negative pole.

41. We assume that the world's pressure toward the positive pole is given by

x1=(v1-v2)/v1. (4.2)

We will explain the reasons for choosing this expression at the end of section 8.

42. If the subject has had no prior experience in similar situations, then in accordance with definition of x2, x1=x2. Substituting the values of x1 and x2, given in (4.2), into (3.7) we find the relation between the probability X1 that the subject is ready to choose alternative A and utilities v1 and v2:

v1/(v1+v2), if v1>v2X1 = { (4.3)

any number from [0,1], if v1=v2.

43. We see that on the basis of assumption (4.2) our model coincides with the BTL model for the case of v1>v2. We can even say that the BTL model is deduced from ours. However, for the case v1=v2 the model predictions are different. According to the BTL model each alternative will be chosen with probability 1/2, whereas our model implies that in this case the subject has freedom of choice: he can select any probability of choosing alternative A (Lefebvre, 1994a).

44. Almost forty years ago Stevens and Galanter (1957) found that the results of magnitude estimation of stimuli intensities are related non-linearly to their categorical estimation. In the first case, subjects estimate intensities comparing them with a unit of measurement. In the second case, subjects attribute the intensity of each stimulus to a certain level. Such levels can be represented by a set of numbers, where 1 corresponds to the weakest stimulus, and k to the strongest. If we plot magnitude estimations against categorical estimations, the graph will be an upward convex curve rather than a straight line. Further, the convexity is greater when the mean values of stimuli are shifted toward the weakest stimulus. The interrelation between these two methods of estimation was debated for a long time (Stevens & Galanter, 1957; Galanter & Messick, 1961; Parducci, 1965; Marks, 1968; Haubensak, 1992; Parducci, 1992). However, no persuasive theoretical explanation has been offered.

45. Lefebvre (1992a) attempted to explain this non-linear relation with the help of the model described in this paper. A subject is represented by equation

X1=x1/(x1+x2-x1x2), (5.1)

where x1+x2>0 (cf. (3.7)). Suppose the subject is able "to answer a question" about his readiness to choose a positive pole. Let him do this by marking a point X1 on the interval [0,1]. In this way we extend our model by assuming that the final result of the subject's activity, in addition to the actual choice, can also be the subject's "report" about the value X1. We interpret such a report as a categorization.

46. Consider first a case of bipolar categorization. Let magnitude estimations of stimuli found in earlier experiments be between Smax and Smin. The subject's task in the given experiment is to categorize each stimulus as "strong" or "weak." Suppose the attractiveness of alternative "strong" is v1=Smax-Smin, while that of "weak" is v2=Smax-S, where S is the intensity of the presented stimulus: the more intensive is the stimulus, the less attractive is the alternative "weak." Interpreting the alternative "strong" as the positive pole and using Equation (4.2) we obtain

x1=(S-Smin)/(Smax-Smin). (5.2)

47. We will call this value a normalized magnitude estimation. In accordance with the definition in section III, x2 takes on the mean value of normalized magnitude estimations of stimuli preceding a given one (including itself). The value of X1 is the subject's readiness to choose "strong" with probability X1.

48. When a scale with a large number of categories is used, the subject marks the value of X1 on the scale. For well-randomized long sequences of stimuli presentation, we can assume x2 to be constant in a given experiment. In this case expression (5.1) can be regarded as an equation of a hyperbola with variables X1 and x1 and a constant parameter x2. Larger convexity of the hyperbola corresponds to smaller x2. Such hyperbolas appear to fit published experimental results on categorization (sf. Lefebvre, 1992a). It is also noteworthy that values of variables x1 and x2 can be estimated from experimental data. Thus, the model contains no free parameters.

MEASURABLE INTENSITY

49. For many centuries the golden section [g=(SQRT5-1)/2=0.618...] was regarded as especially attractive (cf., for example, Ghyka, 1946). Attempts to refute or confirm this hypothesis began with Fechner (1876); but the question of whether this phenomenon really exists remained unanswered (cf. reviews in Valentine, 1962; Plug, 1980; Lefebvre, 1992b). A new wave of interest in the golden section was generated by the works of Adams-Webber and Benjafield, who found that subjects evaluate their acquaintances positively in bipolar constructs with a frequency close to 0.62. They hypothesized that the theoretical value of this frequency is identical with the golden section (Adams-Webber & Benjafield, 1973; Benjafield & Adams-Webber, 1976; Adams-Webber, 1990).

50. Analyzing other subsequent experiments we found that the frequencies of choice of the positive pole lie in the interval 0.60-0.64 (see review in Lefebvre, 1992b) and suggested that the golden section is not limited to evaluations of other people but appears in binary choice or categorical estimation of stimuli without measurable qualities (Lefebvre, 1985; 1990). We also proposed a model similar to the one described here from which an explanation of the phenomenon is deduced (Lefebvre, 1985).

51. Suppose a subject represented by Equation (3.6) faces a choice between two alternatives without objective measure in the context of the given situation. Let the subject relate the first alternative to a positive pole, the second to a negative pole, and let the attractiveness of these alternatives be v1=1 and v2=x3. The latter means that the greater the probability with which the subject is planning to choose the first alternative, the more attractive becomes for him the second alternative. Let also the attractiveness of the positive alternative be always greater than that of negative one, that is, v2=x3<1. In view of (4.2), we have x1=(v1-v2)/v1=1-x3. Next suppose the subject had never faced a similar choice in the past. In view of the definition of x2 (see section 3) we must set x1=x2. By substituting the values x1=x2=1-x3 into Equation (3.6), we obtain

X1=1-x3+x3^{3}, where x3^{3} means "x3 to the power of 3." (6.1)

52. Suppose that the subject is a Realist, that is, X1=x3. Then Equation (6.1) is transformed into a cubic equation

x3^{3}-2x3+1=0. (6.2)

53. Positive roots of this equation are x3=1 and x3=(SQRT5-1)/2. The first one does not satisfy the condition x3<1. Therefore, the subject chooses a positive pole with the probability equal to the golden section value.

54. Consider an experiment conducted by Kunst-Wilson and Zajonc (1980), in which the ideas presented here played no part. The stimuli were 20 irregular octagons. During the first stage of the experiment, ten of these octagons were presented one by one with a very short expositions, 1 millisecond, five times each. During the second stage, octagons were presented in pairs: one which was presented earlier and one new one. The subjects were not told that one of the octagons was among those seen earlier; their task was to indicate which octagon they liked more. The octagon seen earlier was preferred with frequency 0.60. Table 1 shows the frequencies of preference for objects seen earlier in a number of similar experiments:

Table 1

Experimenter Frequency

Kunst-Wilson et al.(1980) 0.60 Seamon et al.(1983) 0.61 Mandler et al.(1987) 0.62 Bonanno et al.(1986) 0.66, 0.63, 0.62, 0.61, 0.63, 0.62

55. As can be seen, the frequencies cluster around 0.62. We found no indications in the literature that this fact was noticed.

56. Our model suggests an explanation of this phenomenon. The first stage of the experiments determined the polarization of the objects: the object seen earlier became the positive pole; the new one the negative. The stimuli had no measurable properties which might determine the preference between the objects. Consequently the pressure of the world was determined by the subject's intention, which led to a preference for the positive pole with probability close to the golden section ratio.

57. In 1908 a psychiatrist from Prague, H. Loewy, discovered a new psychological phenomenon (see Reznik, 1969). One of his patients complained that she had lost all her inner feelings. Her statements, however, contrasted with the objective observations made by Loewy: her tone, look, and style of behavior suggested that she had deep feelings making her suffer. Loewy was confronted with a paradox: the patient suffers because she is convinced that she does not.

58. According to Mayer-Gross,

The paradoxical occurrence of such complaints in patients who obviously suffer from the alleged death of their feelings has aroused great interest in all writers on the subject. (Mayer-Gross, 1935, p.108)

59. Here are typical patients' reports:

"My emotions are gone, nothing affects me." "All the feeling is gone, no remorse, no passion. I have only the feeling of being alive, that my heart beats..." "My feeling is dead, I have no interest for my husband and the baby." "Nothing impresses me, ... I have lost all my love for the children, they seem miles away." "I am unable to have any emotions, everything is detached from me." "The trees in the garden seem unreal, I get no feeling, no thrill, no joy ... ." "I cannot see the beauty of flowers -- the feeling does not get past my eyes." (Mayer-Gross, 1935, pp. 107-108)

60. Sometimes the patients themselves find that the meaning of their statements contradicts their form:

"One of my patients remarked on her own surprise at this paradox, viz. that she should weep in the very act of complaining about loss of feeling." (Mayer-Gross, 1935, p.108)

61. One of the attempts to construct an explanatory scheme for this phenomenon was made by Reznik (1969). He assumed that a person in a normal state has the ability to sense his own feelings; that is, in addition to the feelings as such, there is also a secondary ability to "feel" the process of feeling, although this ability is not consciously registered. In the case Loewy observed, the patient had lost this secondary ability, so that she did not sense her own feelings and believed that they were missing.

62. Although this phenomenon has been known to psychiatrists for a long time, it did not attract attention of psychologists studying the cognitive realm of human beings. Perhaps the inner mechanism of "awareness" of one's own feelings is akin to the mechanism of awareness of one's own visual perception (Gazzaniga, 1970; Griffin, 1976). In that case, the loss of the ability to see one's own feeling is close to the phenomenon of paradoxical vision or blindsight, the essence of which is that subjects with damage to the visual cortex can move around and not collide with objects but at the same time be sure that they don't see these objects (Zihl, 1980; 1981; Zihl & von Cramon, 1985; Paillard et al., 1983; Perenin & Jeannerod, 1978).

63. Another phenomenon which might be close to those already described is the "feeling of knowing": sometimes subjects report that they have a feeling that they will be able to retrieve unrecalled information under specific conditions (Hart, 1965; see also Metcalfe & Shimamura, 1994).

64. On the other hand, although all those phenomena are usually described in similar terms, such as awareness, metacognition, and attention (Metcalfe & Shimamura, 1994), we cannot rule out the possibility that by their neural physiological nature they belong to different functional systems in the human brain (Gazzaniga, 1995).

65. We will show further that the theoretical representation of the subject described in the previous sections allows us to develop a model of the subject capable of becoming aware of his own feelings (Lefebvre, 1980; 1992a,b).

66. We consider a human being possessing a specialized functional processor generating an "image of the self," which, in turn, generates a second "image of the self." The first "image of the self" is related to the subject's direct feelings; the second order "image of the self" allows the subject to "see" himself feeling. The generation of the first and second order of the "image of the self" goes on automatically; it is not related to any voluntary constructive activity (Lefebvre, 1967; 1977b). Under certain disruptive conditions, however, a secondary image is not generated. This gives rise to the paradox discovered by Loewy: a person has feelings but does not sense himself having them. A connection between this scheme of the process of automatic awareness and the model of the subject developed in the previous sections is given by the following

67. STATEMENT: The function X1=x1+(1-x1-x2+x1x2)x3 of three variables x1, x2, and x3 can be represented as composition

X1=F(x1,F(x2,x3)), (7.1)

of one function, F(x,y), of two variables x and y, where

F(x,y)=1-y+xy. (7.2)

This representation is unique. (The proof is given in Lefebvre, 1992a).

68. We consider the function

X2=F(x2,x3) (7.3)

as a formal analogue for the subject's image of the self. In this interpretation, the subject's self-description and the objective description by an external observer are presented by the same function F(x,y). We thus obtain a model of a subject "aware" of the self. In the case in which the subject is a Realist (X1=x3), Equations (7.1) and (7.3) transform into Equations (7.4) and (7.5) respectively:

X1=F(x1,F(x2,X1)), (7.4)

X2=F(x2,X1). (7.5)

69. It follows from these Equations that, if x1+x2>0, then X1=x1/(x1+x2-x1x2),

(7.6)X2=x2/(x1+x2-x1x2),

and if x1+x2=0, then

X2=1-X1, (7.7)

where X1 is an arbitrary number from [0,1].

70. Variables X1 and X2 can be represented as identities

X1==F(x1,X2), X2==F(x2,X1). (7.8)

71. In order to simplify the analysis, we represent X1 as a composition of F1=F(x1,u) and u=X2, and X2 as a composition of F2=F(x2,v) and v=X1 by using a standard notation for function composition:

X1==F1oX2, X2==F2oX1. (7.9)

72. It follows from (7.9) that X1 can be put into correspondence with the sequence of its representations as compositions of F1, F2, and Xj:

X[1]==F1 o X2, X[2]==F1 o F2 o X1, (7.10) . . . . . . . X[n]==F1 o F2 o F1 o ... o Fi o Xj,

_where i,j=1,2; j=i.

73. A subject corresponding to composition X[1] has an image of the self X2, but this image does not have an image of the self. A subject corresponding to composition X[2] has an image of the self F2oX1, and this image, in turn, has an image of the self X1. We identify an image of the self, F2oX1, with the subject's inner feelings, and an image's image of the self; X1, with the "sense" of one's own feelings (using Reznik's term). Loewy's paradox appears when there is a disruption such that a subject in the state X[1] cannot proceed to the state X[2]. The subject has feelings connected with his image of the self, but his image of the self does not have feelings, and the subject is incapable of "seeing" the self with the feelings.

74. In the general case, we put an act of automatic awareness into correspondence with the transformation X[k]->X[k+1] (Lefebvre, 1967; 1977b). Then composition X[n] will correspond to a state of the subject appearing as a result of (n-1) acts of awareness starting with X[1].

75. When the subject is in the state X[n], every image of the self (in the hierarchy of images) is presented together with the influence of the world. Therefore, in addition to the hierarchy of self-images, the subject has a hierarchy of world's images. These images are connected with the subject's feelings of the world. So, the subject has feelings of the two types and both are included into multiple awareness: some are connected with self-reference, and the others with the reference to the external world (Lefebvre, 1967; 1977b).

76. Every composition from (7.10) can be regarded as a formal operator mapping a set of pairs of values (x1,x2) onto a set of pairs of vectors (A,a), (See Fig. 1).

---------------------------------------------------------------

1 2 3 n-1 n .................. ............. : : : : : : : A = : X1 : X2 : X1 : ... : Xi : Xj : .................. .............X[n] == F1 o F2 o F1 o ... o Fi o Xj :

................................... : : : : : : : a = : x1 : x2 : x1 : ... : xi : xj : .................. .............

Figure 1. Connection between composition X[n] and vectors of feelings A and a. The upper lane corresponds to vector A which is a theoretical analogue of the feelings linked to self-reference. The lower lane corresponds to vector a which is a theoretical analogue of the feelings linked to the reference of the external world. ---------------------------------------------------------------

77. The upper lane corresponds to vector A=(X1,X2,X1,...) whose periodical components are the values of functions F1, F2, and Xj (j=1,2) occurring in composition. This vector is a theoretical analogue of the subject's feelings (and their multiple reflexion) linked to self-reference. The lower lane corresponds to vector a=(x1,x2,x1,...) whose components also constitute a periodical sequence. This vector is an analogue of the subject's feelings (and their multiple reflexion) linked with reference to the external world (Lefebvre, 1992b).

78. In the next section we will try to move further and show that the model that we have constructed allows us to make some assumptions about computational brain work.

79. One of the hypotheses concerning the general principles of the functioning of real brain neural networks is that the ensemble of neuron states is a similar statistical ensemble of physical particles obeying the laws of statistical physics and thermodynamics (Cowan, 1967). The models of some types of such networks already exist (Hopfield, 1982; Kirkpatrick et al., 1983; Hinton & Sejnowski, 1983; Ackley et al., 1985; Churchland & Sejnowski, 1992).

80. In the framework of this "thermodynamic" representation, mental phenomena can be connected with macro-characteristics of an ensemble of neuron states, such as "heat," "temperature," and "entropy." If we accept this hypothesis, we may expect that formal schemes of mental processes reflect thermodynamic correlations between different macro-characteristics of neuron ensembles. In the previous section we constructed a formal model of one particular mental phenomenon, the awareness of one's own feelings. No specific information taken from statistical physics and thermodynamics was used to do this. Therefore, if we show that our model can be easily and clearly translated into the language of thermodynamics, this will serve as an argument in favor of plausibility of both our model and the thermodynamic hypothesis as a whole. Further we will make such a translation by demonstrating that the process of multiple awareness described in the previous section is isomorphic to the process of gradual minimization of work lost by a heat engine system.

81. Let us consider an abstract heat machine consisting of a sequence of heat engines and two tapes, on which the engines print their working parameters (see Fig. 2).

-----------------------------------------------------------------

................................ ............................. : : : : : : : : M1 : M2 : M3 : ... : M{m-1} : Mm : ................................ .............................

................................ ............................. : W1 : W2 : W3 : : W{m-1} : Wm : : : : : : : : : Q1 Q2 : Q2 Q3 : Q3 Q4 : ... : Q{m-1} Qm : Qm Q{m+1} : : -> -> : -> -> : -> -> : : -> -> : -> -> : ................................ ............................. T1 T2 T3 T4 T{m-1} Tm T{m+1} ................................ ............................. : : : : : : : : R1 : R2 : R3 : ... : R{m-1} : Rm : ................................ .............................

Figure 2. A heat machine. Reservoirs correspond to vertical lines. Each engine prints its efficiency Rm=(Qm-Q{m+1})/Qm on the bottom tape, and the values Mm=Rm/[(T1-T2)/T1] on the upper tape. -----------------------------------------------------------------

82. In this sequence each succeeding engine performs work compensating for the loss of available work by the preceding engine by receiving from a reservoir the amount of heat equal to that yielded to this reservoir by the preceding engine. The temperatures of reservoirs make a decreasing geometrical progression:

T1/T2=T2/T3= ..., (8.1)

where T1>T2. Engine m performs work

Wm=Qm-Q{m+1}, where {m+1} is a subscript. (8.2)

83. We do not require an engine to be necessarily reversible, so

Wm<=Um, where "<=" means "lesser than or equal to," (8.3)

and where

Tm-T{m+1} T1-T2Um = Qm --------- = Qm ----- (8.4)

Tm T1

is the work which would be performed by a reversible engine if it were located between reservoirs m and (m+1). The loss of available work by engine m is equal to

Q{m+1} QmdWm = Um - Wm = T2(------- - --) = T2dH, (8.5)

T2 T1

where dH is the change of entropy of the whole system caused by engine m performing work Wm. Engine (m+1) receives from a reservoir (m+1) heat Q{m+1} and performs work dWm. Each engine measures the work it performs and registers it in two ways: as a portion of the heat received from the reservoir, Rm, and as a portion of the work which would be performed by a reversible engine under the same conditions, Mm. The engine prints on the lower tape the value

Rm=Wm/Qm=(Qm-Q{m+1})/Qm, (8.6)

that is, the efficiency of engine m, and on the upper tape the value

Wm RmMm = -------------- = ----------, (8.7)

Qm[(T1-T2)/T1] (T1-T2)/T1

that is, the ratio of the efficiency of engine m to the efficiency of reversible engine.

84. We consider these printed quantities to be theoretical analogues for subjective processes. We proved that the "text" printed by such a heat machine is formally equivalent to the "text" generated by a composition X[n] (Lefebvre, 1994b). The two following statements are true concerning this sequence of heat engines.

85. STATEMENT 1. Sequences Rm and Mm are periodic and

R1 if m is oddRm = {

R2 if m is even , (8.8) M1 if m is oddMm = {

M2 if m is even ,

where

R1=(Q1-Q2)/Q1, R2=[Q2-(T2/T1)Q1]/Q2, (8.9)

M1=R1/[(T1-T2)/T1], M2=R2/[(T1-T2)/T1]. (8.10)

86. STATEMENT 2.

M1=R1/(R1+R2-R1R2),

(8.11)M2=R2/(R1+R2-R1R2).

We can see that equations (8.11) are equivalent to equations (7.6).

87. Consider an arbitrary heat machine and establish the correspondence

R1=x1, R2=x2. (8.12)

88. By using (8.9) we find that

Q1/Q2=1-x1, T2/T1=(1-x1)(1-x2). (8.13)

---------------------------------------------------------------

................................ .................... : : : : : : : : X1 : X2 : X1 : ... : Xi : Xj : ................................ ....................

................................ .................... : : : : : : : : -> -> : -> -> : -> -> : ... : -> -> : -> -> : : : : : : : : ................................ ....................

................................ .................... : : : : : : : : x1 : x2 : x1 : ... : xi : xj : ................................ ....................

Figure 3. A heat machine which prints out vectors of feelings on tapes. ---------------------------------------------------------------

89. Let us choose an arbitrary pair (x1,x2), where neither x1 nor x2 is equal to 1 and x1+x2>0, and two arbitrary pairs (Q1, Q2) and (T1, T2) which satisfy equations (8.13). Construct now a finite machine in which those four values are used (Fig. 3). In accordance with (7.6), (8.11), and (8.12), composition X[n] in Fig. 1 and heat machine in Fig. 3, with given values of x1 and x2, generate the same pair of vectors of feelings A and a. Let us choose now x1=0 and x2=0. In this case T1=T2, which follows from (8.13). Therefore, the subject's state in which he has the ability to make free choice corresponds to the state of heat equilibrium in which engines cannot perform work.

90. By establishing one-to-one correspondence between a sequence of functions constituting the composition on Fig. 1 and a sequence of heat engines constituting the heat machine on Fig. 3, we settle the isomorphism between the analytical and the thermodynamic models.

91. Let us return to Equation (4.2). Its meaning becomes clear from the thermodynamic model. Let utility v1 correspond to the heat Q1 received by engine 1 from reservoir 1 and utility v2 to the heat Q2 yielded by engine 1 to the reservoir 2. We showed in this section that x1 corresponds to the efficiency R1 of engine 1 (see (8.9) and (8.12)). Equation (4.2) reflects this parallel.

92. We will conclude by adding to the formulation of the Anthropic Principle in cosmology the formulation of the Principle of Freedom:

What we can expect to observe must be restricted by conditions necessary for our presence as observers. We have free will and under some circumstances also freedom of choice.

93. The first sentence helps us construct concrete models of Universe; the second concrete models of the human being. However, the combined statement is more than a sum of its two parts. We can now say

What we expect to observe must be restricted by conditions under which we at least sometimes have a free choice.

94. This means that we are not only observers but also to a certain degree creators of the reality we observe. This is the content of the connections between psychology and cosmology on the level of abstract principles.

I am sincerely grateful to Anatol Rapoport for numerous constructive suggestions and for his invaluable help in expressing in English the content of this paper. I am also grateful to Victorina Lefebvre without whose constant help this work would never be completed.

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