A note on a variational problem in biology and sociology

The author's suggestion that the behavior of an individual or of a society may be described not by maximizing the satisfaction function but by maximizing the integral of it over a period of time (Rashevsky,Mathematical Biology of Social Behavior, rev. ed., p. 233, Chicago: The University of Chicago Press) has been considered applicable to other biological functions in a recent paper by R. Rosen (Bull. Math. Biophysics,24, 279–290, 1962). It is pointed out that the possibility of introducing such variational principles into biology or sociology requires that the corresponding functions depend explicitly and in a nonlinear manner on the rate of change of some of the variables. An example of a satisfaction function is given that describes the behavior of an individual who derives more satisfaction from the process of approaching a goal than from the actual goal itself.     

On mass behavior

In imitative behavior, as studied previously by N. Rashevsky (Mathematical Biology of Sociol Behavior, Chapter XIII, The University of Chicago Press, 1950), the reason for the majority of a society to accept a particular behavior is based on purely voluntary action (band-wagon effect). In the present paper effects of coercion of the majority by a small minority group which poses the means for coercion, are studied. More general types of equations are thus obtained and threshold effects found, which bear a resemblance to some such effects studied previously.     

Some remarks on the mathematical bio-sociology of the relativity of injustice

The author's theory of the adoption of certain types of behavior patterns (Rashevsky, N., 1957, “Contributions to the Theory Initiative Behavior”.Bull. Maths. Biophysics,19, 91–119; 1968,Looking at History through Mathematics, Cambridge, Massachusetts: M.I.T. Press) consisting of elementary behaviors for each of which there is an opposite one and the two are mutually exclusive, is applied to describe the changes in the general type of behavior of a society. The elementary acts of which the whole problem consists may be either overt activities or beliefs or opinions. The general behavior patternsadopted by the society are considered as the “proper” or “just” ones. Any deviation from it in either one or more of the component elementary behaviors is considered as “unjust” and is subject to some punitive action. The total number of possible mutually exclusive behavior patterns is very large but finite. Within this very large range of possible patterns, we find that this notion of justice is relative, because changes from any behavior pattern to any other may occur. It is further shown that the amount of punishment for the deviation from the accepted pattern in order to be effective as well as efficient must be applied in different ways to different individuals even for the same transgression.     

Contributions to the theory of imitative behavior: The number of political parties as determined by biological and social factors

The theory of imitative behavior as applied tow mutually exclusive behavior patterns (N. Rashevsky,Mathematical Biology of Social Behavior, Rev. Ed., 1959; The University of Chicago Press) leads to the possibility of any numberw of different behavior patterns existing in a social group. Mutually inhibitory effects suppress the effectiveness of behavior of groups that are very small numerically. The manner in which the different biological and social parameters that enter into the theory of imitative behavior determine the number of different effective behaviors is discussed. The results are applied to the problem of what determines the number of political parties in different countries. This number is expected to increase with increasing spread of the distribution curves for the tendencies towards different behaviors, with decreasing imitation factors, and with increasing instability of psychophysical judgments of the average individuals.     

Studies in the mathematical theory of excitation

The general linear two-factor nerve-excitation theory of the type of Rashevsky and Hill is discussed and normal forms are derived. It is shown that in some cases these equations are not reducible to the Rashevsky form. Most notable is the case in which the solutions are damped periodic functions. It is shown that in this case one or more—in some cases infinitely many—discharges are predictable, following the application of a constant stimulusS. The number of discharges increases withS, but the frequency is a constant, characteristic of the fiber and independent ofS.     

Organismic sets and biological epimorphism

It is shown that the principle of biological epimorphism (Rashevsky,Mathematical Principles in Biology and Their Applications, Springfield, Ill.: Charles Thomas, 1960) is contained in the theory of organismic sets (Bull. Math. Biophysics,29, 139–152, 1967) if an additional postulate not directly connected to mappings is made.     

Mathematical theory of biological periodicities: Formulation of then-body case

In a series of papers, L. Danziger and G. Elmergreen (Bull. Math. Biophysics,16, 15–21, 1954;18, 1–13, 1956;19, 9–18, 1957) showed that a non-linear biochemical interaction between the anterior pituary gland and the thyroid gland may result under certain conditions in sustained periodical oscillations of the rates of production and of the blood level of the thyrotropic and of the thyroid hormone. They treated the systems, however, as a homogeneous one. N. Rashevsky (Some Medical Aspects of Mathematical Biology, Springfield, Illinois: Charles C. Thomas, Publisher, 1965;Bull. Math. Biophysics,29, 395–401, 1967) generalized the above results by taking into account the histological structures of the two glands as well as the diffusion coefficients and permeabilities of cells involved. The present paper is the first step toward the theory of interaction of any numbern of glands or, more generally,n components. The differential equations which govern the behavior of such a system represent a system of2n ²+n non-linear first order ordinary equations and involve a total of 7 n ²+3n parameters of partly histological, partly biochemical nature. The requirements of the existence of sustained oscillations demand 4n ²+2n+2 inequalities between those 7n ²+3n parameters.     

Comments on the approximate solution of the diffusion equation: II

On the basis of a previous general formulation (Bull. Math. Biophysics,15, 21–29, 1953a) a discussion is given of the error in the approximation method of N. Rashevsky. This error, inherent in the method when the metabolic rate is different at each point in the cell, is discussed in detail and numerical values are presented for two particular cases: the rate proportional to the concentration and the rate a prescribed function of the spatial coordinates. It is shown that the formulation for the first case also applies to several other cases, that the error is negligible provided the rate is sufficiently small, and that the error is fairly sensitive to the cell size. If the rate depends upon the coordinatesalone a small rate is not sufficient to insure a negligible error. The relations between the exact method, the standard approximate method, an earlier approximate method (Physics,7 260, 1936), and a more recent refinement (Bull. Math. Biophysics,10, 201, 1948) of the standard method are discussed. This work was performed while the author was a research participant, Oak Ridge Institute of Nuclear Studies, assigned to the Mathematics Panel, Oak Ridge National Laboratory.     

A contribution to the mathematical biology of automobile driving: II. Passing as a case of psychophysical discrimination

The decision to pass or not to pass in view of an oncoming car is considered as a case of comparative judgment in which it is to be decided whether the time it will take to pass safely is greater or less than the time it will take to collide with the oncoming car. H. D. Landahl's well-known theory of psychophysical discrimination is used, and it is assumed that the “distracting stimuli” considered previously (Rashevsky, 1959,Bull. Math. Biophysics,21, 375–85) tend to increase the standard deviation of Landahl's fluctuation function. Effects of the “distracting stimuli” on the threshold of the neuroelements in Landahl's circuit are also considered. On this basis an expression is derived which gives the probability of a collision accident in passing as a function of the “distracting stimuli.”     

Social behavior of captive,group-living Orangutans

The behavior of orang-utans (Pongo pygmaeus)was observed in two captive groups (one adult group, one juvenile group). Activity profiles,animal interactions, and compartmental spatial use for both adult-group and juvenile-group individuals were recorded over a 9-month period. Behavioral repertoires for both groups included large amounts of social activity. Equivalent amounts of social activity were found for each group. The social behavior of juvenile animals involved more active behavior such as play. The social behavior of the adult animals was more subtle, involving social monitoring and allogrooming. These results indicate that orangutans, at least when group-living in captivity, exhibit the potential to display social behavior which is apparently of greater frequency and complexity than that which has been observed in the wild. These findings suggest that the solitary behavior of wild orangutans is not a necessary characteristic of orangutan behavior. Under different environmental conditions orangutans appear to readily adapt socially, and, like other nonhuman primates,they have the capacity to exhibit complex and subtle social behavior. This report is based on part of a senior thesis submitted by Sara D. Edwards     

A neural model for conditioned avoidance learning

A mathematical model for learning of a conditioned avoidance behavior is presented. An identification of the net excitation of a neural model (Rashevsky, N., 1960.Mathematical Biophysics. Vol. II. New York: Dover Publications, Inc.) with the instantaneous probability of response is introduced and its usefulness in discussing block-trial learning performances in the conditioned avoidance situation is outlined for normal and brain-operated animals, using experimental data collected by the author. Later, the model is applied to consecutive trial learning and connection is made with the approach of H. D. Landahl (1964. “An Avoidance Learning Situation. A Neural Net Model.”Bull. Math. Biophysics,26, 83–89; and 1965, “A Neural Net Model for Escape Learning.”Bull. Math. Biophysics,27, Special Edition, 317–328) wherein lie further data with which the model can be compared.     

Studies in mathematical biosociology of imitative behavior: I. Effects of income distribution

In connection with previous studies (Rashevsky,Mathematical Biology of Social Behavior, chap. xii), a situation is investigated in which the two mutually exclusive possible behaviors of a society consist of the desire to keep the present socioeconomical situation and the desire to change it inany way. The psychophysiological tendency ϕ towards either of the behaviors is considered to be proportional to the difference between the actual incomei of the individual and his needsi′. Assuming that the distribution functionN ₁(i′) of the needs is a given characteristic of the population, it is shown that the distribution functionN(ϕ) of ϕ in the society can be derived fromN ₁(i′) and from the distributionN ₂(i) of the incomesi. A particular case is worked out as an example. Conditions of stability of a socioeconomic structure are studied in their dependence on the income distribution.     

Sporadic sustained nonperiodic oscillations in the activities of organismic sets

In combining the author's theories of organismic sets (Rashevsky,Bull. Math. Biophysics,31, 159–198, 1969a) and Robert Rosen's theory of (M, R)-systems (Bull. Math. Biophysics,20, 245–265, 1958), a conclusion is reached that the number of either normal or pathological phenomena in organismic sets may occur. Those phenomena are characterized by occurring spontaneously once in a while but are not exactly periodic. Some epilepsies are an example of such pathological phenomena in the brain.     

Probability distributions associated with distinct hits on targets

Some probability distributions connected with distinct hits on targets, using two different firing schemes, are developed. It is assumed that any shot has a probabilityp, not necessarily unity, of hitting the target at which it was aimed. The development uses a well-known expression for the probability that exactlyt ofN possible events occur simultaneously. Some of the formulae developed here include as special cases the probabilities derived separately and by more complicated arguments in papers by N. Rashevsky. (Bull. Math. Biophysics,17, 45–50, 1955) and A. Rapoport (Bull. Math. Biophysics,13, 133–38, 1951).     

A contribution to the mathematical biology of social behavior: Comparison of non-violent resistance and of riots by oppressed groups

A previous paper (Bulletin of Mathematical Biophysics,30, 501–518, 1968) discussed the mathematical biosociology of riots by suppressed groups. In this paper the effect of non-violent disobedience as a method of prevention of oppression is discussed from a biosociological point of view. It is found that in general, other conditions being equal, a non-violent resistance is more effective than a violent riot. In a large number of cases however, depending on the choice of the biosociological parameter, the above conclusion may not hold. The actual outcome depends not only on the attitude of the oppressed group, but also on the attitude of the oppressor group. When the choice between several attitudes is allowed in both groups, the situation depends on 224 parameters. With certain choices of those, it is possible that only violent revolts will lead to abolition of oppression.     

Studies in imitative behavior: A generalization of the rashevsky model; Its mathematical properties

This paper is based on N. Rashevsky's theory of imitative behavior, the underlying idea being that performance of one reaction by a given individual produces an increased stimulation (or tendency) toward the same reaction in other individuals. For simplicity, consideration is limited to cases in which each individual may choose only between two (or two main categories of) reactions, denoted byA andB in the following. However, upon suggestion from Dr. Rashevsky and certainly in better agreement with actual facts, the strength of imitative interaction is assumed to vary from individual to individual. More precisely, if Ψ_i denotes the additional excitation caused by imitation in theith individual,PA_i the probability for performance of reactionA, andPB_i the probability for performance of reactionB by theith individual, we postulate that where the constants α _ik ^′ and β _ik ^′ (coefficients of imitative interaction) measure the amount of imitative influence exerted by thekth individual upon theith,N being the total number of individuals in the population. The term — α_i Ψ_i accounts for the spontaneous decay of excitation, and the quantities α _ik ^′ and α _ik ^′ are assumed to benon-negative. The expressions forPA_i andPB_i are obtained from H. D. Landahl's theory of conflicting stimuli; they depend non-linearly on the values Ψ_i. It is implicit in this formulation that the theory can only be applied if the frequency of contacts between individuals is not too small. Some further shortcomings and limitations of the model are outlined, and the discussion includes suggestions for reinterpretation and improvement of the theory. If all the quantities α _ik ^′ and α _ik ^′ have the same value, sayA, we return to the case treated by Rashevsky (and Landau, 1950); these authors, however, replace the sums in the equation above by integrals, which automatically restricts the validity of their results to very large values ofN. Their work may therefore be characterized by the assumption of uniform interaction in large populations. Our equations, on the other hand, are applicable even to very small groups, and therein lies one of their main advantages. In this paper the mathematical properties of the non-linear system of equations above are studied with particular reference to the existence and stability of steady states [dΨ_i/dt ≡ 0;, i = 1 , 2, . . . N]. A sufficient condition for the existence of only one stable steady state is derived. It may be formulated roughly by stating that all the coefficients of interaction should be sufficiently small. It that is not the case, there may exist a greater number of stationary states. In particular, two of them (called “extremal”) have the following properties: they arestable and such that the average number of individuals in the group performing one or the other reaction is the largest (or smallest) possible as compared with the other steady states. Hence the situation is qualitatively similar to that found by Rashevsky and Landau.Quantitatively, however, important differences may arise, depending on the nature of the matrix specifying the interaction. A stable state may be approached through damped oscillations, but this effect is important only if the damping is sufficiently small for the oscillations to become practically observable. Little information could be obtained on this point, due to mathematical difficulties. As mentioned above, the most interesting applications of this theory will be with respect to small populations or to populations partitioned into subgroups with varying amounts of imitative interaction within as well as between groups.     

Outline of a unified approach to physics,biology and sociology

The theory of organismic sets, introduced by N. Rashevsky (Bulletin of Mathematical Biophysics,29, 139–152, 1967;30, 163–174, 1968), is developed further. As has been pointed out, a society is a set of individuals plus the products of their activities, which result in their interactions. A multicellular organism is a set of cells plus the products of their activities, while a unicellular organism is a set of genes plus the products of their activities. It is now pointed out that a physical system is a set of elementary particles plus the product of their activities, such as transitions from one energy level to another. Therefore physical, biological and sociological phenomena can be considered from a unified set-theoretical point of view. The notion of a “world set” is introduced. It consists of the union of physical and of organismic sets. In physical sets the formation of different structure is governed preponderantly by analytical functions, which are special type of relations. In organismic sets, which represent biological organisms and societies, the formation of various structures is governed preponderantly by requirements that some relations, which are not functions, be satisfied. This is called the postulate of relational forces. Inasmuch as every function is a relation (F-relation) but not every relation is a function (Q-relation), it has been shown previously (Rashevsky,Bulletin of Mathematical Biophysics,29, 643–648, 1967) that the physical forces are only a special kind of relational force and that, therefore, the postulate of relational forces applies equally to physics, biology and sociology. By developing the earlier theory of organismic sets, we deduce the following conclusions: 1) A cell in which the genes are completely specialized, as is implied by the “one gene—one enzyme” principle, cannot be formed spontaneously. 2) By introducing the notion of organismic sets of different orders so that the elements of an organismic set of ordern are themselves organismic sets of order (n−1), we prove that in multicellular organisms no cell can be specialized completely; it performs, in addition to its special functions, also a number of others performed by other cells. 3) A differentiated multicellular organism cannot form spontaneously. It can only develop from simpler, less differentiated organisms. The same holds about societies. Highly specialized contemporary societies cannot appear spontaneously; they gradually develop from primitive, non-specialized societies. 4) In a multicellular organism a specialization of a cell is practically irreversible. 5) Every organismic set of ordern>1, that is, a multicellular organism as well as a society, is mortal. Civilizations die, and others may come in their place. 6) Barring special inhibitory conditions, all organisms multiply. 7) In cells there must exist specially-regulatory genes besides the so-called structural genes. 8) In basically identically-built organisms, but which are built from different material (proteins), a substitution of a part of one organism for the homologous part of another impairs the normal functioning (protein specificity of different species). 9) Even unicellular organisms show sexual differentiation and polarization. 10) Symbiotic and parasitic phenomena are included in the theory of organismic sets. Finally some general speculations are made in regard to the possibility of discovering laws of physics by pure mathematical reasoning, something in which Einstein has expressed explicit faith. From the above theory, such a thing appears to be possible. Also the idea of Poincaré, that the laws of physics as we perceive them are largely due to our psychobiological structure, is discussed.     

Topological biology: A note on Rashevsky's transformation T.

In the bio-topological transformation between graphs denoted by (T ⁽¹⁾ X) N. Rashevsky (Bull. Math. Biophysics,18, 173–88, 1956) considers the number of fundamental sets which (a) have only one specialized point as source (and no other sources), (b) have no points in common (are “disjoined”); he proves that this number is an invariant of the transformation. In this note we show that Rashevsky's Theorem can be extended as follows:The number of fundamental sets of the first category is an invariant of the transformation. We must, however, count the subsidiary points of the transformed graph as specialized points. We recall that fundamental sets of the first category are those whose sources consist of specialized points only (Trucco,Bull. Math. Biophysics,18, 65–85, 1956). But in this modified version of the Theorem the fundamental sets may have more than one source and need not be disjoined.     

Some consequences of Henri Poincaré's hypothesis about the biological nature of the three-dimensionality of our space

Henri Poincaré (Derniere Pansées, Paris, Flamerion, 1920) makes the interesting suggestion that our space is three-dimensional because ourvoluntary movements are those of quasi-rigid bodies in three dimensional space. Inasmuch as according to the theory of organismic sets (Rashevsky,Bulletin of Mathematical Biophysics,31, 159–198, 1969) organisms are conceivable, perhaps in some remote parts of the universe, for which the primary voluntary changes may be not spatial movements but changes of other physical qualities, it is pointed out that the acceptance of Poincaré's hypothesis will require an invariance of the physical laws in an abstractn-hyperspace with respect to the choice ofm<n coordinates as the basic frame of reference.     

Ethnicity,race and class in the United States

Orlando Patterson, Ethnic Chauvinism: The Reactionary Impulse, New York: Stein and Day, 1977, 347 pp., $15.00.

William Julius Wilson, The Declining Significance of Race: Blacks and Changing American Institutions, Chicago and London: The University of Chicago Press, 1978, xxi + 204 pp. £8.85.