The present paper puts a strong emphasis on similarities between mathematical sociology and non-mathematical sociology in that both attempt to explicate sociological concepts and theories as well as reasoning, although it also shows that explication by way of formalization is unsurpassed using some examples. Also some problems that mathematical sociology in Japan or otherwise are facing are discussed.

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     What is the role of the formalization of sociological thought in intellectual history? This paper attempts to discuss formalization oriented research and its present situation.
     First. we define a theory as a set of statements systematically correlated to give explanations and predictions of objects with laws. This may develop cumulatively. Second. we examine the development of substantive research by the mathematical as well as quantitative approach. Third, we point out the aporiae of scientific thought: methodological individualism, relation of systems to history, relation of value to science, and differentiation and integration of science. We refer to some attempts to break through these aporiae: structuralism, self-organizing systems, control of cultural relativism, and formalization.

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     Mathematical approaches in sociology are reviewed in terms of micro-macro perspectives. Historical survey of multi-level analysis has made clear that it started with the synthesis of macro-level variables from micro-level ones, followed by ecological correlation, contextual analysis, and mathematical formulation of micro-macro problems (e.g., social decision-making and social dilemma or unintended consequences of purposive action). It also suggests the trend from statistical-methodological concern to mathematical-substantial concern on the part of researchers. We also reviewed some mathematical models on social processes to find the following trends: 1) from macro-level formalization to micro-level formalization, 2) from little concern for micro-macro problem to explicit formalization of micro-macro relations, and 3) from behavioral model to cognitive model. These developments, which we may call the trend toward “mathematical sociology as cognitive science” transformed mathematical sociology, born as the direct application of mathematical tools to social phenomena, into the closer examination of the intrinsic mechanism of sociological processes. They also suggest that mathematical sociology as cognitive science calls for a new “language of social research”.

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     A defect which has been inherited in the theoretical and methodological discussions of “Status Inconsistency Effects” and “Mobility Effects” was indicated, was derived with help of mathematical formalization of theoretical interpretations, was shown. While the existence of the phenomena and/or their usefulness as sociological variable were suspected by some of sociological researchers, but the defect was produced because of the negligence of differentiation of endogenous and exogenous factors in the explanatory models. Contrary to the psychological model which could also be employed because the phenomena are closely related with psychological process, and in which social structural factors might be assumed as exogenous, the psychological factors such as cognitive processes, attitudinal balance, and so forth, are to be introduced as exogenous variables(i.e., parameter). These notions would lead us to a conditional interaction model shown in this article.

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     The aim of this paper is twofold. (a) generalization of Simon's analysis of Homans' small group theory, using qualitative comparative statics and Jeffries' color point method. (b) to point out that the methods used here can contribute to the further development of mathematical formalization of Homans' theory and other sociological theories.
     It is well known that Simon-Homans model is mathematical formalization of Homans' small group theory. It has been considered as a classic in the field of mathematical sociology. However, there are three major deficiencies in Simon's analysis. (1) In the case of nonlinear system, there are no analytical method to be substituted for his phase diagram approach. His phase diagram is a gross substitute for qualitative comparative statics which we use in this paper. To use the diagram, he made some assumptions(eg. ψ=0 moves to the left, when the value of the exogenous variable E reduces.). (2) His phase diagram approach is always confined to only two endogenous sociological variables. That is, no more than two differential equations are used simultaneously in Simon's analysis. (3) In the non-linear case, Simon-Homans model is not qualitative stable. But, we point out that a slight modification of Fararo's analysis of Homans' theory reveals qualitative stability, according to Jeffries' color point method. Fararo used another part of Homans' theory, which was not utilized by Simon. This finding suggests the flaw in Simon's formalization of the internal and external systems in Homans' theory.

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     This is a personal monologue about what sociologists should do for the development of sociology.
     The “crisis” in current sociology seems to be attributable to its lack of theories, which threatens the identity and cohesiveness of sociology as a discipline. However, instead of authentic theories, many pseudo-theories are presented and mistakenly accepted as theories in sociology. They are “perspectives”, “conceptual schema or definitions”, “empirical generalizations”,“‘the more.., the more’ statements”, and “statistical models such as path models” The first two types lack the true-or-false property; i.e., they cannot be falsified. The latter three types lack the explanatory power.
     Four methodological misconceptions are identified. The empiricism bias denies the value of explanation and satisfies with description. On the other hand, holistic bias tends to belittle the significance of small theories on specific phenomena. Both the foundation thesis, which asserts that the truth of theory depends on its methods or conceptual schema, and the methodological unidimensionality, which asserts that true thories are produced through only one certain method, fail to recognize that a theory should be accepted only as a temporary truth surviving empirical tests and rational discussions. Finally, the piling thesis, which asserts that a significant new knowledge can be obtained by analyzing data or by deducing statements from known statements, fails to recognize the necessity of creative imagination for a new successful explanation.
     For the development of sociological theories, we need a challenging spirit to take a risk of failure in creating and presenting imaginative ideas, and a critical mind to test them rationally by ourselves.

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     The Yasuda index Y is based on a representation of “pure” mobility that contradicts fundamental assumptions underlying the definition of mobility components. Using a matrix representation of pure mobility the index is modified so that it retains its theoretical appeal and validity. Under the proposed modification the index is a ratio of the two amounts of pure mobility - that which is extracted from the matrix of observed mobility and that which is extracted from the corresponding matrix of “perfect mobility.” For a given matrix of observed transitions both quantities are determinable by means of linear programming.

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     We tried to show how and why sampling survey outcomes over (under) estimate the real world results. We examined voting turnout rates in the Japanese Lower House election in 1980. The sample of 3000 people was selected by two stage national random sampling and the amount of the overestimation of the turnout rate was 12.4%.
     First, we verified mathematically that overestimation in sample survey is the sum of three components: sampling bias, ‘misreport effect’, nonresponse bias.
     Second, among the three we can estimate the size of the sampling bias through sampling theory. The confidence interval for sampling bias in our case was just 2.2% and the direction could be either over- or under-estimation. Thus, we estimated that 10.2% to 14.6% of the overestimation was due to the other two causes.
     Third, we assumed that there was no serious sampling bias and tried, under this assumption, to find the way to decompose the 12.4% overestimation into the other two causes, nonresponse bias and ‘misreport effect’.
     Since we got four unknown parameters with two simultaneous equations we suggested two methods. Our first method is to use statistical analysis, such as nonlinear multiple regression, in the case where we have enough data. Our second method is to introduce assumptions either a priori or from other data for two of four unknowns and solve mathematically. We tried the latter method andadopted three types of assumptions: no-misreport, like U.S., like former Japan.
     As a result we concluded that nonresponse bias was most influential and caused 7 to 13% overestimation while ‘misreport effect’ caused 0 to 6% bias.
     We also showed that we can decompose the bias into causes for each characteristic such as sex. We found that the overestimation is larger for male (15.3%) than for the female (9.9%), and the nonresponse bias is more serious for male.
     Finally, we found that the relationship between voting and response is highly positive, and that the nonresponse bias for nonvoter is much more serious.
     The method which we developed here is quite helpful to understand the character of data and can be used for other types of items such as party identification, education, and income.

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     P. V. Marsden proposed the method to analyze differential association. This method is the application of log-linear model, and used for testing several hypotheses to patterns of social association. One that he stresses is the “ingroup bias” hypothesis ; sociable intercourse is more prevalent within social groups than between different groups. The other is the “social distance bias” hypothesis ; sociable intercourse is inversely related to the social distance between groups. To investigate these hypotheses empirically, he reformalized log-linear models, and introduced new parameters in models assuming that differential association has relation to social distance. However, the derivation of these parameters is problematic. In this paper, we will examine his method, and then show the inadequacy of his derivation.

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     In human groups, everyone does not treat others in the group alike. Someone may give discriminative treatment to someone, while someone may not. As is well known, there are not a few cases where these discriminative treatments concentrate on a certain one person in the group. It is named, for example, bashing or scapegoating in groups. This paper is an attempt to explore the mechanism of the occurrence of this concentration.
     For this purpose, we present a model in which one's discriminative treatments depend on that of others in the group. This is a simplified version of Hunter's(1978) model, which explains a change in network of feelings in groups. After discussing the model, using a simulation method, we compare two type of groups. In one group, everyone(I) feels other(J) different from himself if and only if I and J treat a third person(K) in the group differently. While in another group, everyone(I) feels other(J) different from himself if and only if a third person(K) in the group treats I and J differently.
     The result of the simulation shows that in both groups, the concentration of discriminative treatments occurs, but to different persons. And in the former group, the discrimination will be dissolved if the discriminated person stops giving discriminative treatment to others in the group. On the contrary, in the later group, the discrimination will not be dissolved by the discriminated person.

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     A causal model of domestic task sharing among married couples was proposed with power, relative resources, time availability, and ideology as the main predictor variables. The model was then examined using a large American sample (N=3649) and multiple indicator model with the LISREL program. Confirmatory factor analysis led us to eliminate some indicators which have loadings on both endogenous and exogenous dimensions. The structural analysis utilizing the LISREL indicated that the theoretical model proposed here was generally valid but some new causal paths must be added. Findings showed us that the husband's relative share in housework decreases when he works full-time, earns more money, holds a traditional sex-role orientation, and gets less satisfaction from performing household tasks. His share also decreases when his wife does not work, holds a traditional sex-role orientation, and gets more satisfaction from doing housework. In addition, the husband does relatively more housework when he is less powerful in the relationship with his wife. A sensitivity analysis was conducted to check the effect of arbitrarily fixed factor loadings of single indicators and it proved that the model was quite stable among combinations of differently fixed indicator loadings on unmeasured dimensions. Although explanations based on the relative resources principle have been the most common in this topic, other factors such as the power relationship and sex-role orientations must be considered in a more plausible theoretical model of domestic task sharing within a family.

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