There is no need to use a number of axioms to explain the phenomenon of the "golden section" (0.618) in human binary choice. This parameter reflects the mean value of the majority of voices under conditions of uncertainty. A random distribution of choices -- which may be equiprobable (rectangular), needle-like, or, in the most common case, near-Gaussian (bell shaped) -- determines the mean value of dominant choices which differ just slightly from the "golden section". The negligible difference is about one per cent. Lefebvre's axiomatic theory stimulates the search for general quantitative regularities in human choice. The search, which is based on neurophysiological data like ours, is an alternative one. It explains certain peculiarities of random choice distributions.
and found, with a number of postulates and axioms, that
where x1, x2, x3 are variables extracted from explained psychological data. Their function Y may be equal in part to one of them.
2. Equation (2) is the main output of Lefebvre's theory for quantitative explanation of psychological data relating to human choice.
3. It is very interesting to investigate human behavior in the situation of uncertainty, when available stimuli, which are needed to choose only one act among others, are absent or not different in their intensity. If probability Pmin to do a certain single selection is less than 0.5, then instead of it one must keep in mind an alternative choice, probability P of which is
4. According to the condition of uncertainty, one can assert that the most probable value is P=0.5, and the distribution of probabilities at first approximation is a bell curve with its maximum at 0.5. One extreme case is a needle-like distribution, the so called delta distribution. Another extreme case is rectangular distribution with a just noticeable maximum at the 0.5 level. In the first case, probability P=0.5. In the second case, mean value does not exceed (1-0.5)/2 = 0.75, again in accordance with the condition of uncertainty. In all other cases 0.5 < P < 0.75. Therefore, at first approximation, its most probable value is equal to (0.5+0.75)/2 = 0.625. I underline that it is only a first approximation. The second approximation is based on most probable near Gaussian distribution (i.e., neither rectangular nor needle-like) of probabilities in the same range. I mean that the distribution of searched probabilities differs not significantly from Gaussian law with minimal probability -1.0 corresponding to -3*sd, and maximal probability +1.0 corresponding to +3*sd, where sd is the standard deviation, and asterisk means multiplication. In this case, the majority (3*sd+0.67*sd) of results divided by the range of all results of votes (6*sd) is equal to 0.612. This means that if every subject votes randomly with probability 0.5, then the searched mean value of the majority of voices is equal to 0.612. Lefebvre has found other theoretical values using formula (2). Substituting in it the values x1=x2=1-x3 and supposing that Y=x3 he has found the majority of voices to be Y=(SQRT5-1)/2=0.618. Our theoretical result differs a bit from 0.618 which is equal to the "golden section." The negligible difference is about one per cent.
5. In accordance with the data of many authors in Table 1 (Lefebvre, 1995), its real values are equal to 0.60, 0.61, 0.62, 0.66, 0.63, 0.62, 0.61, 0.63, and 0.62, with mean, 0.622. The difference is 0.625 - 0.622 = 0.003.
6. Lefebvre has found an alternative solution using formula (2). Substituting in it the values x1=x2=1-x3 and supposing that Y=x3, he has found that Y=(SQRT5-1)/2=0.618. The difference is just the same and may be a bit worse, 0.622 - 0.618 = 0.004.
7. Until now, both explanations are approximately equal in the sense of correctness of prediction, but our explanation requires fewer different postulates, axioms, and limits.
8. Nevertheless it is interesting to analyze concrete possible cases of distribution of probabilities P in the range 0.5 and 1. Earlier, after Zipf (1935), we found (Lebedev, 1983) using our own neurophysiological premises that probabilities of actualization of different memory items correspond to the harmonic row
1 : 1/2 : 1/3 : 1/4 ... 1/M, (4)
where M is the number of different items (the number of memory images) which are in active state simultaneously, like conceptual or perceptive codes in the model of Atkinson (1974).
9. The harmonic row is determined by cyclic structure of supposed neuronal codes of human memory (Lebedev, 1990).
10. It follows from (4) that in case of two memory images which are in an active state simultaneously, one of them dominates with probability 0.667. Another is in an active state with P=0.33 in correspondence with the harmonic row
1: 1/2 = 2/(1+2) : 1/(1+2) = 2/3 : 1/3.
11. By the way, because the second probability is less than 0.5, one must use, in accordance with condition (3), the formula 1 - 1/(1+2). Keeping this condition in mind, we have found the probabilities of actualization of memory images in case of 3 active memory images to be:
6/(6+3+2): 1-3/(6+3+2): 1-2/(6+3+2) = 0.56 : 0.73 : 0.82. (5)
In case of 4 active images, the row of computed probabilities is
13/25 : 19/25 : 21/25 : 22/25 = 0.52 : 0.76 : 0.84 : 0.88. (6)
12. In total, keeping in mind only these possible cases, we have found that peaks of distribution of probabilities in the range 0.5 and 1 are equal to 0.52, 0.56, 0.67, 0.76, 0.82, 0.84, 0.88. In reality, they are equal to 0.51, 0.55, 0.61, 0.64, 0.66, 0.71, 0.74, 0.76, 0.79, 0.82, 0.85, about 0.89 and 0.93 in accordance with graph 3.2.1 in the Lefebvre's book (1991, p.46). The graph shows real distributions of results of numerous votes in California during referendums in the years 1884-1990.
13. One can see that all computed peaks correspond to proper ones among observed peaks with a minimal difference of about 0.01. One of the real peaks at 0.61 corresponds to the above mentioned theoretical values of 0.625 and 0.618.
14. Lefebvre uses rows of parameters in his model such as
1/M : 1/(M-1): .. : 1/3 :1/2 : 1:2:3:4 .., (7)
and this solution may not only be philosophically based, but neurophysiologically based as we shall see.
15. A few suppositions, based on neurophysiological data related to cyclical structure of memory codes, explain a peculiarity of human choice in uncertain circumstances as well as some peculiarities of language, memory, and sensations (Lebedev, 1983; 1993).
16. Lefebvre explains the phenomenon of categorical scaling. For this purpose he derives equation (8) from formula (2)
Y= x1/(x1+x2-x1*x2), (8)
where Y = (R-1)/(M-1) is the subjective categorical estimation of a stimulus in relative units. M is equal to the upper limit of the categorical scale and R is the current categorical estimation; x1 is an absolute estimation on Stevens's magnitude scale, and x2 is a parameter which determines the level of convexity of the curve which is to be predicted.
17. Another solution is derived from our model of neuronal mechanism of subjective estimations (Zabrodin & Lebedev, 1976). If R is an estimation of a stimulus with intensity S, and d is the differential, then, in accordance with Zabrodin, one can write
dR/RZ =dS/S, (9)
where denotes "to the power of", Z is a parameter connected with neurophysiological constants like dominant frequency of alpha rhythm in human electroencephalogram and step-like difference between phases of alpha waves as well as with some conditions of measurement (Lebedev & Myshkin, 1988; Lebedev, 1993). If Z=0, then one can observe Fechner's logarithmic scale. If Z=1, then Stevens's magnitude scale exists. But when this parameter is about Z=0.5, one can see categorical scale.
18. Our solution differs from Lefebvre's formula (8), because ours explains not only categorical estimations, but direct magnitude estimations as well as just noticeable differences without suppositions, which can not be derived from experiment. Is it good or bad that "without"?
19. About the comparison of a human brain with a machine: In my opinion the brain is not like a heat machine. Highly organized in time and space, cyclic neuronal processes constitute the basis of human cognition, emotions, and will, as well as the human internal world in total.
20. There is no doubt that in the 21st century the neuronal code of the human subjective world will be discovered to be cellular, as subatomic, atomic, and molecular codes have been discovered in physics, chemistry and biology.
21. As well, new broad and simple laws of human cognition, together with a small number of easy understable and measurable but universal neuronal constants, will be discovered.
22. Together with Luce (1986), Link (1992), Geissler (1990), and other scientists of the same ilk, Lefebvre strongly stimulates with his publications the search for universal principles of brain functioning. I was witness to great interest in his scientific paper presented at the 23rd Meeting of the European Mathematical Psychology Group in Moscow in 1993.
23. Hence, despite my critical remarks, I highly esteem the paper of Lefebvre overall.
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