Introduction to Mathematical Sociology PDF
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INTRODUCTION TO MATHEMATICAL SOCIOLOGY INTRODUCTION TO MATHEMATICAL SOCIOLOGY JAMES S. COLEMAN Johns Hopkins University The Free Press of Glencoe Collier-Macmillan Limited, London Copyright © 1964 by the Free Press of Glencoe A DIVISION OF THE MACMILLAN COMPANY Printed in the United States of America All rights in this book are reserved. No part of this book may be used or reproduced in any manner whatsoever without written permission except in the case of brief quotations embodied in critical articles and reviews. For information, address: The Free Press of Glencoe A DIVISION OF THE MACMILLAN COMPANY The Crowell-Collier Publishing Company 60 Fifth Avenue, New York, N.Y. 10011 Collier-Macmillan Canada, Ltd., Toronto, Ontario Library of Congress Catalog Card Number: 64-13241 TO Paul F. Lazarsfeld PREFACE In the development of any science, two things are crucial: systematic empirical study and systematic conceptual elaboration. Each of these requires its special tools. Experimental techniques, systematic observation in natural settings, and the whole apparatus of research methodology is necessary if empirical study is to carry the science beyond the distillation of personal experience. Similarly, if conceptual elaboration is to progress beyond the proverbs of the ancients, special tools are necessary. The most remarkable of these is mathe matics. Mathematics provides a battery of languages which, when carefully fitted to a set of ideas, can lend those ideas great power. The mind falters when faced with a complex system or a long chain of deductions. The crutch that mathematics provides to everyday reasoning becomes essential as sociology moves toward the analysis of complex‘systems and predictions based on extended chains of deductions. This book aims to provide mathematical tools for conceptual elaboration in sociology. Such tools are not easily come by in an area of behavior as complex as that which sociology covers, and consequently this book only makes a beginning in this direction. But its principal aim is exactly that—to begin the development of a mathematical language which is equally at home with the empirical results of social research and the ideas of social theory. Chapter 1 provides a survey of uses of mathematics in sociology. This chapter attempts to indicate both the variety of styles of mathematics presently used in sociology and the variety of purposes to which the mathematics is put. VII viii Preface This chapter is not encyclopedic, but gives examples representative of the different styles and purposes employed. Chapter 2 examines in some detail the problems of measurement that confront systematic sociology. Are the criteria for fundamental measurement as developed in physical science essential to the development of mathematical sociology, and if so, how can they be met? Strategies for surmounting or bypassing the problem of measurement are discussed. This provides the context for the development in Chapters 3—13 of one such strategy which I believe will be particularly productive. In Chapters 3-13, a mathematical language for the study of social and psychological processes is introduced. When sociology is more fully developed perhaps a book on social theory can be written in two sections, labelled “Dynamics” and “Statics,” or “Processes” and “Structure.” In such a book, Chapters 3—13 of this book would fall squarely in the center of the “Processes” section. The work presented in Chapters 3-5 was first developed in 1955-56, as a mathematical framework within which both the results of cross-sectional surveys and panels of successive waves of observations could be stated. Such a framework seemed to me then, and seems to me now, essential if survey research and other qualitative observations are to be relevant to social theory. Every science requires a language in which both its empirical results and its theoretical propositions can be stated. In the early stages of the science— the stages through which sociology has been passing—ordinary statements in natural language, abounding in terms like “more” and “less,” can suffice. However, when these stages are passed, when more discriminatory power is required of the theory and more precision of the data, then such looseness can no longer suffice. It is this stage at which sociology presently finds itself, and it is for traversing this stage that the mathematical models of Chapters 3-5 are introduced. Formally, the processes may be described as continuous time stochastic processes. The mathematics used in this section is rather simple, being confined in most cases to algebra and elementary calculus. Chapters 6-9 are elaborations of this language and application of it to varying forms of problems or data. For example, Chapter 6 shows its applica tion to problems of multivariate analysis with qualitative data, while Chapter 7 deals with the problem of shifting between individual and group levels of aggregation. Chapters 10 and 11 are elaborations of the basic mathematics of these processes to systems involving a simple repetitive action, and thus many states. The processes into which these models lead are all related to a Poisson process; and models of medical epidemics bear a close resemblance to some of these processes. Chapters 12 and 13 continue the elaboration of these processes by allowing the relation between an individual’s state and the response he makes to be a probabilistic one. Such an elaboration becomes especially valuable for Preface IX handling data from multi-wave panels, which almost never conform to the simple processes described in Chapters 3-5. The next three chapters, 14, 15, and 16, are a minor excursion into prob lems of structure in sociology. Chapter 14 treats the problems of structural measurement by the use of processes imposed upon the structure. Such processes by their outcomes provide measures of the structure which derive directly from its functioning under these processes. Chapter 15 examines a particular structural problem, that of relations between spatially separate groups of varying size, and gives an approach to its solution. Chapter 16 uses a very simple situation involving the interaction between members of different subgroups in a community to show the logical implications of certain varia tions in group structure. Chapter 17 is a kind of amalgam of the structure and process models. It takes a problem—non-homogeneous structure—which is poorly handled by all diffusion models and presents two partial solutions. Finally, Chapter 18 suggests tactics and strategies which I believe will be profitable in the use of mathematics in sociology. Most of these strategies are exemplified in the earlier chapters, and Chapter 18 makes explicit the tactics and strategies which led to the development of these chapters. Two other chapters were to have been included in this book, one entitled “Reaction to Events: Individual and Social,” and the other “Decay of Attention.” However, the data for the first of these were too insubstantial, and the methods of analysis of the data for the second could not be developed in time for this publication. In addition, the developments of Chapter 13 have led in extremely profitable directions, resulting in a monograph which elaborates and applies this model (Coleman, 1964). I hope that this book will help stimulate such work in mathematical sociology as to make it quickly outmoded. It is truly an introduction to mathematical sociology, and I trust that this introduction will be quickly followed by systematic growth, particularly in the area of mathematics for social processes. The mathematics used in different chapters of this book is at quite different levels. Chapters 1 and 2, for example, require only a good grounding in algebra, while Chapters 10-13 and 17 make substantial use of calculus. Throughout the volume I have avoided the use of difficult mathematics and mathematics likely to be unfamiliar to the general reader. This is in part because of my own greater facility with simpler mathematics, and in part because I think it is important that we keep to as simple a mathematics as possible. If we allow the complexity of the mathematics to reach an insoluble stage (as can easily happen) before the social phenomena it represents becomes complicated enough to be substantively useful, then our mathematics is obviously of little value. It will become evident to the reader that further developments in mathe matical sociology will soon require techniques that go beyond those used here. X Preface Thus the knowledge of advanced probability theory, advanced calculus, and other areas of mathematics will soon become necessary skills for those who will innovate in certain branches of mathematical sociology. My endeavors in these areas have been greatly aided by many people. My greatest debt is to Paul Lazarsfeld, who first engaged my interest in mathematical sociology and who posed many of the problems to which the chapters of this book are directed. Indeed, his stimulation has been so import ant to my work in this area that I have dedicated this book to him. For the very conception of this book I am indebted to Jeremiah Kaplan, who proposed nearly ten years ago that I begin such an undertaking, before either he or I had any idea of the ultimate shape it would finally assume. I owe particular thanks to several other persons: William McPhee and Lee Wiggins have long been engaged in related enterprises, and my work owes much to their stimulation. To Richard Savage I am grateful for guiding me toward continuous-time stochastic processes as the clothing for data on social and psychological processes. I regard this as perhaps the greatest innovation in this book, and Savage was in part responsible for it. To many persons, especially Duncan Luce, Herbert Menzel, and Arthur Stinchcombe, I am grateful for reading and commenting on drafts of various chapters. Several graduate students, in particular Louis Goldberg and Seymour Spilerman, checked equations and solutions of examples. Virginia Bailey carried out the arduous task of typing numerous drafts of the manu script. My wife, Lucille Coleman, survived the more arduous task of living with the discontent which an unfinished book creates. Work on this book was carried out at the University of Chicago and Johns Hopkins University, to which I am grateful for the free time that allowed such activity. Grants from the Ford Foundation and the National Science Foundation helped make this free time productive. A special warmth, however, I reserve for the Center for Advanced Study in the Behavioral Sciences, where this book was begun. Had I not had the extended periods of contemplation that the Center provided, the ideas which form the central core of the book might not have emerged. James S. Coleman Baltimore, Maryland March 1964
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