Mathematical sociology, often known as the sociology of mathematics[1], is an interdisciplinary topic of study that looks at the connections between math and society as well as how mathematics is used in sociological research.[/3]
As a result, the definition of mathematical sociology might vary based on the authors and the type of study being done. This gives rise to debate on whether mathematical sociology is a branch of sociology, a point where the two fields converge, or a distinct field in and of itself.*[4]
There are gray areas and a need for more research to strengthen mathematical sociology’s academic quality because of this dynamic, ongoing academic development, which occasionally leaves the field hazy and lacking in consistency.(5)(6)
History of Mathematical Sociology
Large social networks, where the nodes are individuals and the linkages are acquaintanceships, can be characterized using a relational and probabilistic approach, first devised by Nicolas Rashevsky in the early 1940s[7][8] and then by Anatol Rapoport and others in the late 1940s.
In the late 1940s, formulas were developed that linked global network properties like connection to local factors like closure of contacts (if A is linked to both B and C, then there is a greater than chance probability that B and C are linked to each other).In [9]
Additionally, camaraderie is a positive tie, but what about bad ties like hostility between people? Graph theory, the mathematical study of abstract representations of networks of points and connections, was used to address this issue.
The fundamental theorem of this theory was created by mathematician Frank Harary’s formalization of this study. As an example, the psychological principle “my friend’s enemy is my enemy” illustrates how a network of interrelated positive and negative ties can be balanced. It states that in this case, the network is composed of two sub-networks, each of which has positive ties among its nodes and only negative ties between nodes in separate sub-networks.10]
This suggests the notion of a social structure that divides into two cliques. However, there is a unique situation when there is only one sub-network, which can happen in very small networks. Relative strengths of ties are present in another model. It is possible to see “acquaintanceship” as a “weak” bond and “friendship” as a strong one.
These two discoveries contain mathematical models that are relevant to the structural analysis. There were other early, significant advances in mathematical sociology that dealt with processes.
For example, Herbert A. Simon created a model consisting of a deterministic system of differential equations in 1952 to generate a mathematical formalization of a published theory[11] of social groupings. Theorems concerning the dynamics and implied equilibrium states of any group were obtained by a formal analysis of the system.
In the 1940s and 1950s, a number of new interdisciplinary scientific innovations, including information theory, game theory, cybernetics, and the development of mathematical models in the social and behavioral sciences, emerged. This period also saw the rise of mathematical models in the social sciences.In [12]
Approaches
Mathematics in sociology
Mathematical sociology, which focuses on mathematics in sociological research, employs mathematics to build social theories. The goal of mathematical sociology is to translate sociological theory into mathematical language. Increased clarity and the capacity to apply mathematics to infer implications of a theory that cannot be reached intuitively are two advantages of this technique.
The preferred method in mathematical sociology is best described as “constructing a mathematical model.” This entails formulating certain assumptions regarding a social phenomenon, formalizing those assumptions mathematically, and offering an empirical explanation of the concepts. It also entails determining the model’s features and contrasting them with pertinent actual data. The most well-known contribution made by this subfield to social network analysis and sociology.
Society and mathematics
Mathematical sociology, also known as the sociology of mathematics, is an academic field that complements sociology of knowledge and sociology of science by examining the relationship between society and mathematical knowledge and the social roots and social effects of mathematics.
In [14]In [15] This reflexivity on the history and application of mathematics in sociology aims to comprehend the relationship between mathematical facts and social constructs as well as the potential biases that arise from the use of mathematics in attempts to comprehend social phenomena.In 16
Further developments
Sociologist James S. Coleman wrote a critical explanatory analysis of Rashevsky’s social behavior theories in 1954.18] The dilemma of how to relate such theoretical models to sociological data—which frequently takes the form of surveys with results stated as percentages of respondents believing or acting—is raised by both Rashevsky’s models and Simon’s model.
This proposes using a method widely recognized in the mathematics of stochastic processes—namely, deriving the equations from assumptions about the likelihood of each individual changing state in a brief period of time.
In his 1964 book Introduction to Mathematical Sociology, Coleman incorporated this concept by demonstrating how stochastic processes in social networks could be examined.
Coleman used mathematical concepts from economics, such as general equilibrium theory, in earlier works to argue that broad social theory should start with the idea of purposeful action and use rational choice models to approximate such activity for analytical reasons (Coleman, 1990).
This argument is comparable to the opinions put forth by other sociologists who have attempted to apply rational choice theory to sociological study, despite the fact that these attempts have encountered philosophical and substantive objections.20]
Simultaneously, the previously mentioned structural analysis was expanded to include social networks that rely on formalized social relationships, particularly kinship. This instance of the connection between mathematics and sociology utilized group theory in abstract algebra.21] Thus, attention turned to a data-analytical version.
Experiments are used in several sociology research programs to examine social interaction mechanisms. This kind of program was started by Joseph Berger and his associates, and its main idea is the application of the theoretical notion of the “expectation state” to the development of theoretical models to explain interpersonal processes, such as those that relate the external status in society to the differential influence in local group decision-making.
A large portion of this theoretical work is associated with the construction of mathematical models, particularly following the adoption in the late 1970s of a graph theoretic representation of social information processing, as Berger (2000) outlines in reflecting on the evolution of his research program. In 1962, he and his associates provided an explanation of model building by referencing the model builder’s objective, which may be the explanation of a concept.
Present research
Although it is still a minor topic within the discipline, mathematical sociology has been successful in inspiring other subfields that share its objective of formally modeling social life. The most prominent of them is social network analysis, which is one of the sociological subfields that is expanding the fastest in the twenty-first century.29]
The emergence of computational sociology, which uses computer simulations, artificial intelligence, and sophisticated statistical techniques to broaden the mathematical toolkit, is the other significant advancement in the field. Utilizing the enormous new data sets on social activity produced by online social contact, the latter subfield likewise does so.
A crucial measure of mathematical sociology’s importance is the fact that popular journals in the discipline.
Contributions to The Journal of Mathematical Sociology (JMS) demonstrate more recent developments in the field of mathematical sociology.
A number of trends come to light, including the advancement of formal theories that explain experimental data pertaining to small group processes, the persistence of structural balance as a central mathematical and theoretical concept, the integration of theory-oriented mathematical models with cutting-edge quantitative methodology techniques, the application of computer simulations to the study of social complexity issues, interest in micro-macro linkage and the issue of emergence, and the ever-growing body of research on social networks.
Because of this, ancient subjects like balance and network models are still relevant today. The formal approaches used still include many of the accepted and widely used mathematical techniques: variations in equations.
Research programs
Leading mathematical sociologists and formal theorists made groundbreaking contributions that paved the way for many significant decades of advancements in mathematical sociology, including formal theory. This offers an additional method of noting new developments while maintaining continuity with earlier work by employing the concept of a “research program,” which is a logical sequence of theoretical and empirical investigations predicated on a core idea or methodology.
There are numerous such programs; the following is merely a succinct synopsis of the most notable examples of this concept, with a focus on the pioneering leadership in each program and its subsequent evolution over many years.
(1) James S. Coleman and the Rational Choice Theory:The Foundation book brought together easily understood examples of how rational choice theory may be applied to the study of sociological subjects including social capital, authority, trust, and norms (especially how they emerged).
The book demonstrated how rational choice theory could serve as a useful foundation for advancing sociological explanation from the micro to the macro levels. A key aspect of the book is how mathematical concepts are used to extend the rational choice model to include interpersonal sentiment relations as outcome modifiers. This is done in a way that makes the original, more self-oriented theory a special case in the generalized theory, as noted in a subsequent theory analysis.(31 )
(2) Harrison C. White and Formal Structuralism: Harrison White has been at the forefront of applying mathematical and empirical foundations to social structural analysis in the decades since his initial contributions.
One such contribution was the 1970 publication of Chains of Opportunity: System Models of Mobility in Organizations, which outlined and applied a vacancy chain model for mobility within and between organizations to data. His other, highly influential work includes the operational concepts of structural equivalency and blockmodel, which are used to derive analytical results from a corpus of social relational data. Together with François Lorraine, Ronald Breiger, and Scott Boorman—three of his former students—these concepts and techniques were created.
(3) Joseph Berger and the expectation states theory: Expectation States Theory expanded into a great number of specialized research programs on specialized issues under the organizational and conceptual guidance of Berger. Each of these programs was approached using the overarching idea of expectation states.
Along with producing their own work, he and colleague and frequent collaborator Morris Zelditch Jr. founded a doctoral program at Stanford University that resulted in a massive body of work by prominent alumni such as Murray Webster, David Wagner, and Hamit Fisek. Mathematical graph theory was developed in collaboration with mathematician Robert Z. Norman to model and analyze social information processing in self-other interactions
. Additionally, Berger and Zelditch expanded work in mathematical modeling and formal theorizing.
(4) Formalization in Theoretical Sociology and Thomas J. Fararo: This sociologist has made significant contributions to the field by increasing the interaction between mathematical reasoning and sociological theory.(41) He arranged a symposium where formal theorists presented papers that were published in 2000, and social theorists attended.42]
His own theoretical research program, through collaborations with students and colleagues, addressed subjects like subjective images of stratification (with former student Kenji Kosaka), tripartite structural analysis (with colleague Patrick Doreian)[44], macrostructural theory and E-state structuralism (both with former student John Skvoretz), and computational sociology (with colleague Norman P. Hummon).45][46]- He elaborates on his theoretical sociological approach in two of his works.(47)48]