Mathematics and Statistics in the Social Sciences

Over the years, mathematics and statistics have become increasingly important in the social sciences1 . A look at history quickly confirms this claim. At the beginning of the 20th century most theories in the social sciences were formulated in qualitative terms while quantitative methods did not play a substantial role in their formulation and establishment. Moreover, many practitioners considered mathematical methods to be inappropriate and simply unsuited to foster our understanding of the social domain. Notably, the famous Methodenstreit also concerned the role of mathematics in the social sciences. Here, mathematics was considered to be the method of the natural sciences from which the social sciences had to be separated during the period of maturation of these disciplines. All this changed by the end of the century. By then, mathematical, and especially statistical, methods were standardly used, and their value in the social sciences became relatively uncontested. The use of mathematical and statistical methods is now ubiquitous: Almost all social sciences rely on statistical methods to analyze data and form hypotheses, and almost all of them use (to a greater or lesser extent) a range of mathematical methods to help us understand the social world. Additional indication for the increasing importance of mathematical and statistical methods in the social sciences is the formation of new subdisciplines, and the establishment of specialized journals and societies. Indeed, subdisciplines such as Mathematical Psychology and Mathematical Sociology emerged, and corresponding journals such as The Journal of Mathematical Psychology (since 1964), The Journal of Mathematical Sociology (since 1976), Mathematical Social Sciences (since 1980) as well as the online journals Journal of Artificial Societies and Social Simulation (since 1998) and Mathematical Anthropology and Cultural Theory (since 2000) were established. What is more, societies such as the Society for Mathematical Psychology (since 1976) and the Mathematical Sociology Section of the American Sociological Association (since 1996) were founded. Similar developments can be observed in other countries. The mathematization of economics set in somewhat earlier (Vazquez 1995; Weintraub 2002). However, the use of mathematical methods in economics started booming only in the second half of the last century (Debreu 1991). Contemporary economics is dominated by the mathematical approach, although a certain style of doing economics became more and more under attack in the last decade or so. Recent developments in behavioral economics and experimental economics can also be understood as a reaction against the dominance (and limitations) of an overly mathematical approach to economics. There are similar debates in other social sciences. It is, however, important to stress that problems of one method (such as axiomatization or the use of set theory) can hardly be taken as a sign of bankruptcy of mathematical methods in the social sciences tout court. This chapter surveys mathematical and statistical methods used in the social sciences and discusses some of the philosophical questions they raise. It is divided into two parts. Sections 1 and 2 are devoted to mathematical methods, and Sections 3 to 7 to statistical methods. As several other chapters in this handbook provide detailed accounts of various mathematical methods, our remarks about the latter will be rather short and general. Statistical methods, on the other hand, will be discussed in-depth

Deterministic Dynamics and Chaos: Epistemology and Interdisciplinary Methodology

We analyze, from a theoretical viewpoint, the bidirectional interdisciplinary relation between mathematics and psychology, focused on the mathematical theory of deterministic dynamical systems, and in particular, on the theory of chaos. On one hand, there is the direct classic relation: the application of mathematics to psychology. On the other hand, we propose the converse relation which consists in the formulation of new abstract mathematical problems appearing from processes and structures under research of psychology. The bidirectional multidisciplinary relation from-to pure mathematics, largely holds with the "hard" sciences, typically physics and astronomy. But it is rather new, from the social and human sciences, towards pure mathematics

Wise use of mathematical models in policy

Mathematical models are used widely, in economic forecasting, in the social sciences more generally, climate research and work on infectious diseases. But what do the social sciences have to say about modelling and what can modellers learn from the social sciences

What other sciences look like

In order to have references for discussing mathematical menus in political science, I review the most common types of mathematical formulae used in physics and chemistry, as well as some mathematical advances in economics. Several issues appear relevant: variables should be well defined and measurable; the relationships between variables may be non-linear; the direction of causality should be clearly identified and not assumed on a priori grounds. On these bases, theoretically-driven equations on political matters can be validated by empirical tests and can predict observable phenomena.Natural and social sciences, econometrics, political science methods, mathematical models, regression analysis

Modeling and control in physical, life, and social sciences: Some remarks

International audienceMathematical —or mathematico-numerical— models have pervaded all branches of knowledge, and once control theoreticians have become mainly model-builders. Their mathematical skills are called upon to analyze these models, much less to build them. This confers a renewed significance to an epistemological reflection upon what has become the heart of their occupation. We offer some general remarks, and then attempt to reflect on the different epistemological status of (mathematical) models in physics ("the unreasonable effectiveness of mathematics in the natural sciences") and engineering, life sciences, and social sciences, constrained by my limited experience of the second, and very limited experience of the third (except of mathematical finance, hardly a social science)

Indonesia embraces the Data Science

The information era is the time when information is not only largely generated, but also vastly processed in order to extract and generated more information. The complex nature of modern living is represented by the various kind of data. Data can be in the forms of signals, images, texts, or manifolds resembling the horizon of observation. The task of the emerging data sciences are to extract information from the data, for people gain new insights of the complex world. The insights may came from the new way of the data representation, be it a visualizations, mapping, or other. The insights may also come from the implementation of mathematical analysis and or computational processing giving new insights of what the states of the nature represented by the data. Both ways implement the methodologies reducing the dimensionality of the data. The relations between the two functions, representation and analysis are the heart of how information in data is transformed mathematically and computationally into new information. The paper discusses some practices, along with various data coming from the social life in Indonesia to gain new insights about Indonesia in the emerging data sciences. The data sciences in Indonesia has made Indonesian Data Cartograms, Indonesian Celebrity Sentiment Mapping, Ethno-Clustering Maps, social media community detection, and a lot more to come, become possible. All of these are depicted as the exemplifications on how Data Science has become integral part of the technology bringing data closer to people.Comment: Paper presented in South East Asian Mathematical Society (SEAMS) 7th Conference, 10 pages, 7 figure

Maximality with or without binariness: transfer-type characterizations

We provide the first necessary and su±cient conditions in the literature for the existence of maximal elements in a non-binary choice framework. As an application, the characterization of the existence of maximal elements for acyclic binary relations obtained in Rodriguez-Palmero and Garcia-Lapresta, 2002, Mathematical Social Sciences 43, 55-60, is deduced as a Corollary. Further characterizations in di®erent settings given by Tian and Zhou, 1995, Journal of Mathematical Economics 24, 281- 303, follow as well. Analogous characterization problems in the k- acyclic binary cases are solved too.maximization; non-binary choice function; transfer-continuity; k-acyclicity