Reviewed by
Giangiacomo Bravo
Department of Economics and Statistics, University of Torino and Collegio Carlo Alberto
Introduction to Mathematical Sociology is an handbook aimed at introducing the building blocks of algebra, probability, game theory and social network analysis to undergraduate sociology students with no mathematics beyond their high-school background. Covering such a variety of topics in a relatively short volume is clearly an ambitious and, potentially, risky business. Indeed, it is easy to get lost while dealing with definitions, concepts and examples on more subjects than what everyone could seriously hope to study in her/his lifetime. Actually, most book contents are brought to the reader's attention without being developed as needed, a fact that sophisticated scholars could find deceiving. On the other hand, this is probably the best way to motivate young (and older) sociologists to consider analytical thinking and formal models. It must be said that the large majority of sociology students hold a humanistic background and are dramatically sceptical towards any "reductionist" approach to the "complexity" of human social behaviour (yes, this is the sad world where we live in!). By keeping everything as simple as possible and including more examples than definitions and demonstrations, this book can help us to overcome the common prejudice against mathematical and computational methods and promote the use of formal models in sociology.On the other hand, someone could find deceiving that social simulation and, especially, agent-based modelling are not covered here. This is an explicit choice of the authors, who decided not to include ABMs, first, because this should be the subject of a different (and complementary) course/book and, secondly, because they feel "that one can go pretty far with mathematics alone" (p. xvi). Although this could be questioned, it is clear that including also social simulation would have resulted in a much longer and complex book, eventually difficult to digest for naive undergraduates.
This said, it is worth noting that simulation is not entirely excluded from the volume. Rather, it runs in the background. Nearly all examples and "demonstrations" are supported by online numerical simulations based on the free Wolfram's Mathematica Player. Although the only ABM example is Schelling's segregation model, the book shows that computers and simulation are important also for a mathematical approach to sociological models. Moreover, the volume offers certain important building blocks that can represent a bonus for students willing to learn simulation in the future. Indeed, it is much simpler to approach social simulation having some basics in set theory, probability, game theory and social network analysis. From this point of view, Bonacich and Lu's work brillantly introduces much of what ABM students will be requested to know in their subsequent studies.
To sum up, Introduction to Mathematical Sociology is exactly what the title states: a simple and wide-range introduction to the building blocks of formal sociology. For any comprehensive training, these basics of mathematical sociology should be completed with others introducing social simulation, statistics, and experimental research. On the other hand, it must be said that the methodological training allowed by this book will help sociology students not only to understand the true complexity of the social world, but also to analyse crucial social mechanisms, building models and creating scenarios. This can be done without assuming that reality is too complicate, scientifically intractable or, even worse, a pure social construction, as many social researchers (wrongly - in my opinion) do.
Let us close this review with a general point, which is motivated by the book. Looking at the current state of affairs in sociology, the book authors seem rather optimistic about the fact that "assimilation [to mathematics and the use of computers] is imminent and resistance is futile" (p. 213). I agree that in most sciences the trend is now towards a more intensive use of mathematics and computer models to analyse increasingly complex data and build credible representations of the universe. Nevertheless, a strong resistance against this is still strong in the social sciences. Probably, this is due to the widely shared belief of many scholars that scientific methods are not fully appropriate to understand human affairs. Resistance may be futile, as suggested here, but it presently looks institutionally strong (possibly stronger in Europe than in America) and well organized. This implies that increasing the number of classes teaching mathematical sociology is absolutely needed, especially to attract people with different attitudes towards the study of human societies. This could significantly improve the long term development of sociology and its capacity to understand an increasingly complex and interdependent world.
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