# Abstract

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.