Commentary Magazine


The Study of Man: Sociology Learns the Language of Mathematics

It is an open question just how much longer the literate layman will be able to understand what social scientists are trying to say; some of the most important recent books in the field present an almost impenetrable barrier of mathematical symbols. In this article ABRAHAM KAPLAN describes, in straightforward English, what is happening on the other side of the barrier. 

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A troubling question for those of us committed to the widest application of intelligence in the study and solution of the problems of men is whether a general understanding of the social sciences will be possible much longer. Many significant areas of these disciplines have already been removed by the advances of the past two decades beyond the reach of anyone who does not know mathematics; and the man of letters is increasingly finding, to his dismay, that the study of mankind proper is passing from his hands to those of technicians and specialists. The aesthetic effect is admittedly bad: we have given up the belletristic “essay on man” for the barbarisms of a technical vocabulary, or at best the forbidding elegance of mathematical syntax. What have we gained in exchange?

To answer this question we must be able to get at the content of the new science. But when it is conveyed in mathematical formulas, most of us are in the position of the medieval layman confronted by clerical Latin—with this difference: mathematical language cannot be forced to give way to a vernacular. Mathematics, if it has a function at all in the sciences, has an indispensable one; and now that so much of man’s relation to man has come to be treated in mathematical terms, it is impossible to ignore or escape the new language any longer. There is no completely satisfactory way out of this dilemma. All this article can do is to grasp either horn, sometimes oversimplifying, sometimes taking the reader out of his depth; but hoping in the end to suggest to him the significance of the growing use of mathematical language in social science.

To complicate matters even further, the language has several dialects. “Mathematics” is a plural noun in substance as well as form. Geometry, algebra, statistics, and topology use distinct concepts and methods, and are applicable to characteristically different sorts of problems. The role of mathematics in the new social science cannot be discussed in a general way: as we shall see, everything depends on the kind of mathematics being used.

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I

 

The earliest and historically most influential of the mathematical sciences is geometry. Euclid’s systematization of the results of Babylonian astronomy and Egyptian surveying set up a model for scientific theory that remained effective for two thousand years. Plato found in Euclid’s geometry the guide to the logical analysis of all knowledge, and it was the Renaissance’s “rediscovery” of Plato’s insistence on the fundamentally geometric structure of reality that insured the triumph of the modern world view inaugurated by Copernicus. Scientists like Kepler accepted the Copernicarr hypothesis because it made the cosmos more mathematically elegant than Ptolemy’s cumbersome epicycles had been able to do.

The study of man—to say nothing of God!—enthusiastically availed itself of mathematical method: witness Spinoza’s Ethics, which claimed that it “demonstrated according to the geometrical manner.” But Spinoza’s Ethics failed, as demonstrative science, because the 17th century did not clearly understand the geometry it was applying with such enthusiasm. The discovery of non-Euclidean geometries two hundred years later revealed that the so-called axioms of geometry are not necessary truths, as Spinoza and his fellow rationalists had always supposed, but merely postulates: propositions put forward for the sake of subsequent inquiry. It is only by deducing their consequences and comparing these with the perceived facts that we can decide whether or not the postulates are true. Geometry is a fully developed example of a set of undefined terms, prescribed operations, and the resulting postulates and theorems which make up a postulational system; it is in this form that it is influential in some of the recent developments in the social sciences.

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Perhaps the most elaborate postulational system for dealing with the data of social and psychological science was constructed in 1940 by Clark Hull and associates of the Yale Institute of Human Relations (C. L. Hull et al., Mathematico-Deductive Theory of Rote Learning, Yale University Press, 1940). “Rote learning” is a very specialized branch of psychology that studies the learning of series of nonsense syllables; presumably, this tells us something about the act of learning in its “purest” form, with no admixture of influence from the thing learned.

The problems of the field revolve around ways of explaining the patterns of learning that are in fact discovered; why the first syllables in any series are learned first, the last soon after, and why the syllables a little past the middle of the given series take longest to memorize, and so on. There is a vast number of observations of this sort to be made, and Hull’s ideal was to set up a postulational system which would allow observed patterns of learning to be logically deduced from relatively few postulates.

The system consists of 16 undefined terms, 86 definitions, and 18 postulates. From these, 54 theorems .are deduced. The deductions can, in principle, be checked against the results of direct observation, in experimental situations or elsewhere. In many cases, as the book points out, existing evidence is as yet inadequate to determine whether the theorems hold true; in the great majority of cases, experimental evidence is in agreement with logically deduced theorems; in others, there is undoubted disagreement between deduced expectation and observed fact. Such disagreements point to basic defects in the postulate system, which, however, can be progressively improved.

The authors consider their book to be principally important as an example of the proper scientific method to be used in the study of behavior. And certainly, as a formal demonstration of the handling of definitions, postulates, and theorems, the book is unexceptionable. However, science prides itself on its proper method because of the fruitfulness of its results; and it is to the fruitfulness of this effort that we must address ourselves.

One example of the method may suggest better than general criticism the problem raised. Hull proves in one of his theorems that the greater the “inhibitory potential” at a given point in the learning of a rote series, the greatest will be the time elapsing betwen the stimulus (the presentation of one nonsense syllable in the list) and the reaction (the pronouncing of the next memorized nonsense syllable). “Inhibitory potential” is one of the undefined terms of the system; it denotes the inability of the subject, before his involvement in the learning process, to pronounce the appropriate syllable on the presentation of the appropriate stimulus. (It may be pictured as a force that wanes as the stimuli are repeated and the syllable to be uttered is learned.)

Now this theorem certainly follows logically from three postulates of the system (they involve too many special terms to be enlightening if quoted). However, on examining these postulates, the theorem is seen to be so directly implied by them that one wonders what additional knowledge has been added by formally deducing it. A certain amount must have been known about rote learning to justify the selection of those postulates in the first place. To deduce this theorem from them has added very little—if anything—to what was already known. In short: tie geometric method used by Hull, correct as it is formally, does not, for this reader, extend significantly what we already knew about rote learning from his and others work.

In the course of Hull’s book “qualitative postulates,” by which is meant the “unquanrified” ideas of thinkers like Freud and Darwin, are condemned because they have “so little deductive fertility”—because so few theorems may be deduced from them. In the narrowest logical sense of the phrase, this may be true. But fertility in the sense of yielding precisely determinable logical consequences is one thing; in the sense of yielding further insights into the subject matter—whether or not these can be presented as strict consequences of a system of postulates—it is quite another. The ideas of Darwin and Freud can hardly be condemned as lacking in fertility, even though they leave much to be desired from the standpoint of logical systematization.

This is not to deny that the postulational method can play a valuable role in science. But it is a question of the scientific context in which the method is applied. Newton and Euclid both had available to them a considerable body of fairly well-established knowledge which they could systematize, and in the process derive other results not apparent in the disconnected materials. Hull recognizes this condition, but supposes it to be already fulfilled in the area of learning theory. The results of the particular system he has constructed raise serious doubts that his supposition is true.

Science, basically, does not proceed by the trial-and-error method to which Hull, as a student of animal learning, is so much attached. It employs insight, itself subject to logical analysis, but too subtle to be caught in the coarse net of any present-day system of postulates. The geometric method in the new social science can be expected to increase in value as knowledge derived from other methods grows. But for the present, it is an elegantly written check drawn against insufficient funds.

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II

If the 17th century was the age of geometry, the 18th was that of algebra. The essential idea of algebra is to provide a symbolism in which relations between quantities can be expressed without a specification of the magnitudes of the quantities. An equation simply formulates the equality between two quantities in a way that shows how the magnitude of one depends on the magnitude of certain constituents of the other.

The characterization of mathematics as a language is nowhere more appropriate than in algebra; the notation is everything. The power of algebra consists in that it allows the symbolism to think for us. Thought enters into the formulation of the equations and the establishing of the rules of manipulation by which they are to be solved, but the rest is mechanical—so much so that more and more the manipulation is being done, literally, by machines. The postulational method characteristic of classical geometry no longer plays as important a part here. Derivation is replaced by calculation; we proceed, not from postulates of uncertain truth, but from arithmetical propositions whose truth is a matter of logic following necessarily from the definitions of the symbols occurring in them. The equations that express relations between real quantities are, of course, empirical hypotheses; but if the equations correctly formulate the function relating two quantities, then certain numerical values for one necessarily imply certain numerical values for the other. Again, as in geometry, the facts are the final test.

In this spirit, the mathematicians of the 18th century replaced Newton’s geometrizing by the methods of algebra, and the culmination was Laplace’s system of celestial mechanics. Laplace’s famous superman, given the position and momentum of every particle in the universe, could compute, it was thought, the entire course of world history, past and future. The development of quantum mechanics made this program unrealizable even in principle, just as the non-Euclidean geometries were fatal to the aspirations of the 17th-century rationalists. Nevertheless, this scientific ideal, so nearly realized in physics, has exerted enormous influence on the study of man, and much of the new social science is motivated by the attempt to take advantage of the powerful resources of algebra.

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Among the most ambitious, but also most . misguided, of such attempts is a 900-page treatise by the sociologist Stuart C. Dodd, Dimensions of Society (Macmillan, 1942). The author’s ambition, declared in the subtide, is to provide “a quantitative systematics for the social sciences.” What is misguided is the failure to realize that a system is not provided by a symbolism alone: it is necessary also to have something to say in the symbolism that is rich enough to give point to the symbolic manipulation. Dodd presents a dozen symbols of what he regards as basic sociological concepts, together with four others for arithmetical operations. They stand for such ideas as space, time, population, and characteristics (the abstract idea “characteristics,” and not the specific characteristics to be employed in sociological theory). In addition to these sixteen basic symbols, there are sixteen auxiliary symbols, compounds or special cases of the basic symbols, or supplements to them—for instance, the question mark, described as one of four “new operators,” used to denote a hypothesis, or a questioned assertion. With this notation, every situation studied by sociologists is to be defined by an expression of the form:

S= s s (T;I;L;P)s s

The capital “S” stands for the situation, and the letters within the parentheses for time, indicators of the characteristics specified, length or spatial regions, and populations, persons, or groups. The semicolon stands for an unstated form of mathematical combination, and the small “s” for various ways of particularizing the four major kinds of characterizations. Thus “T0” stands for a date, “T1“ for a rate of change, both of these being different sorts of time specifications. Instructions for the use of the notation require one hundred distinct rules for their formulation.

But this whole notational apparatus is, as Dodd recognizes, “a systematic way of expressing societal data, and not, directly, a system of the functionings of societal phenomena.” “Facts,” however, are data only for hypotheses; without the latter, they are of no scientific significance. Certainly a notational system can hardly be called a “theory,” as Dodd constantly designates it, unless it contains some statements about the facts. But Dimensions of Society contains only one social generalization: “This theory . . . generalizes societal phenomena in the statement: People, Environments, and Their Characteristics May Change. This obvious generalization becomes even more obvious if the time period in which the change is observed is prolonged.” The last sentence may save Dodd, but not his “theory,” from hopeless naivety.

Dodd’s hope that his system of “quantic classification” will “come to play a role for the social sciences comparable to the classification of the chemical atoms in Mendelyeev’s periodic table” is groundless. The periodic table, after all, told us something about the elements; more, it suggested that new elements that we knew nothing of existed, and told us what their characteristics would be when discovered. The fundamental point is that we have, in the case of Dodd, only a notation; when he speaks of the “verification” of his “theory,” he means only that it is possible to formulate societal data with his notation.

Dodd’s basic error is his failure to realize that after “Let x equal such-and-such,” the important statement is still to come. The question is how to put that statement into mathematical form.

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An answer to this question is provided in two books of a very different sort from Dodd’s, both by the biophysicist N. Rashevsky: Mathematical Theory of Human Relations (The Principia Press, 1947) and Mathematical Biology of Social Behavior (University of Chicago Press, 1951). In these two books the author does not merely talk about mathematics; he actually uses it. In the earlier one, the methods of mathematical biology are applied to certain social phenomena on the basis of formal postulates—more simply, assumptions—about these phenomena. In the later book, these assumptions are interpreted in terms of neuro-biophysical theory, and are derived as first approximations from that theory. The results, according to Rashevsky, are “numerous suggestions as to how biological measurements, made on large groups of individuals, may lead to the prediction of some social phenomena.”

As a natural scientist, Rashevsky is not seduced, like so many aspirants to a social science, by the blandishments of physics. Scientific method is the same everywhere: it is the method of logical inference from data provided and tested by experience. But the specific techniques of science are different everywhere, not only as between science and science but even as between problem and problem in the same field. The confusion of method and technique, and the resultant identification of scientific method with the techniques of physics (and primarily 19th-century physics at that) has hindered the advance of the social sciences not a little. For the problems of sociology are different from those of physics. There are no concepts in social phenomena comparable in simplicity and fruitfulness to the space, time, and mass of classical mechanics; experiments are difficult to perform, and even harder to control; measurement in sociological situations presents special problems from which physics is relatively free. Yet none of these differences, as Rashevsky points out, prevents a scientific study of man. That social phenomena are complex means only that techniques must be developed of corresponding complexity: today’s schoolboy can solve mathematical problems beyond the reach of Euclid or Archimedes. Difficulties in the way of experimentation have not prevented astronomy from attaining full maturity as a science. And the allegedly “qualitative” character of social facts is, after all, only an allegation; such facts have their quantitative aspects too. And what is perhaps more to the point, mathematics can also deal with qualitative characteristics, as we shall see.

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Rashevsky addresses himself to specific social subject matters: the formation of closed social classes, the interaction of military and economic factors in international relations, “individualistic” and “collectivistic” tendencies, patterns of social influence, and many others. But though the problems are concrete and complex, he deals with them in a deliberately abstract and oversimplified way. The problems are real enough, but their formulation is idealized, and the equations solved on the basis of quite imaginary cases. Both books constantly repeat that the treatment is intended to “illustrate” how mathematics is applicable “in principle” to social science: for actual solutions, the theory is admitted to be for the most part too crude and the data too scarce.

What this means is that Rashevsky’s results cannot be interpreted as actual accounts of social phenomena. They are, rather, ingenious elaborations of what would be the case if certain unrealistic assumptions were granted. Yet this is not in itself objectionable. As he points out in his own defense, physics made great strides by considering molecules as rigid spheres, in complete neglect of the complexity of their electronic and nuclear internal structures. But the critical question is whether Rashevsky’s simplifications are, as he claims, “a temporary expedient.” An idealization is useful only if an assumption that is approximately true yields a solution that is approximately correct; or at any rate, if we can point out the ways in which the solution must be modified to compensate for the errors in the assumptions.

It is in this respect that Rashevsky’s work is, most questionable. Whatever the merits of his idealizations from the standpoint of “illustrating,” as he says, the potentialities of mathematics, from the standpoint of the study of man they are so idealized as almost to lack all purchase on reality.

Rashevsky’s treatment of individual freedom, for example, considers it in two aspects: economic freedom and freedom of choice. The former is defined mathematically as the fraction obtained when the amount of work a man must actually do is subtracted from the maximum amount of work of which he is capable, and this sum is divided by the original maximum. A person’s economic freedom is 0 when he is engaged in hard labor to the point of daily exhaustion; it is 1 when he does not need to work at all. This definition equates increase in economic freedom with shortening of the hours of work; and an unemployed worker, provided he is kept alive on a dole, enjoys complete economic freedom. Such critical elements of economic freedom as real wages, choice of job, and differences in level of aspiration are all ignored.

Freedom of choice, the other aspect of individual freedom, is analyzed as the proportion borne by the amount of time an individual is not in contact with others who might interfere with his choices, to the time he spends alone plus time that is spent in the company of others with the same preferences. This makes freedom of choice decrease with increasing population, so that by this definition one is freer in # small village than a large city. Nothing is said about prying neighbors, or the presence or absence of a secret police, a most important determinant of “freedom of choice.” The whole matter of the “introjection” of other persons standards, as discussed for instance in Erich Fromm’s Escape from Freedom, is ignored, as are such fundamental considerations as knowledge of the choices available, or the opportunity to cultivate skills and tastes.

On current social issues Rashevsky betrays that he suffers from the same confusions and rationalizations as afflict students without the advantages of a mathematical training. To explain the Lysenko case, for example, he suggests that it is possible that the facts of genetics “may be interpreted from two different points of view,” thus naively substituting a scientific question (if there be one) for the real issue, which is the political control of science. His assumptions encourage him to attempt the conclusion, “even at the present state of our knowledge,” that after World War II peace will be most strongly desired by the Soviet Union, least by the United States, with England and Germany in between. And he confesses that he finds it “difficult to understand why the Soviet Union insists on repatriating individuals who left the Soviet Union during World War II and do not desire to return.” Mathematics is not yet capable of coping with the naivety of the mathematician himself.

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III

The 19th century saw the rise of mathematical statistics. From its origins in the treatment of games of chance, it was expanded to cope with the new problems posed by the development of insurance, and finally came to be recognized as fundamental to every science dealing with repetitive phenomena. This means the whole of science, for knowledge everywhere rests on the cumulation of data drawn from sequences of situations on whose basis predictions can be made as to the recurrence of similar situations. Mathematical statistics is the theory of the treatment of repeated—or multiple—observations in order to obtain all and only those conclusions for which such observations are evidence. This, and not merely the handling of facts stated in numerical terms, is what distinguishes statistics from other branches of quantitative mathematics.

The application of statistics to social phenomena far exceeds, in fruitfulness as well asextent, the use of mathematics of a fundamentally geometrical—i.e. postulational—or algebraic character in the social sciences. Social scientists themselves have made important contributions to mathematical statistics—which is a good indication of its usefulness to them in their work. Only two of the most recent contributions in the application of statistics to social phenomena can be dealt with here.

The first is a rather remarkable book by a Harvard linguist, G. K. Zipf, Human Behavior and the Principle of Least Effort (Addison-Wesley Press, 1949). Its basic conception is that man is fundamentally a user of tools confronted with a variety of jobs to do. Culture can be regarded as constituting a set of tools, and human behavior can be analyzed as the use of such tools in accord with the principle of minimizing the probable rate of work which must be done to perform the jobs that arise. It is this principle that Zipf calls “the law of least effort.” As a consequence of it, he claims, an enormous variety of social phenomena exhibit certain regularities of distribution, in accordance with the principle that the tools nearest to hand, easiest to manipulate, and adapted to the widest variety of purposes are those which tend to be used most frequently. These regularities often take the form, according to Zipf, of a constant rank-frequency relationship, according to which the tenth most frequently used word, for instance, is used one-tenth as often as the most frequently used one of all. This is the case, for example, with James Joyce’s Ulysses as well as with clippings from American newspapers.

A large part of Zipf’s book deals with linguistic phenomena, since he is most at home in this field, and it is there that his results seem most fully established. But an enormous range of other subjects is also treated: evolution, sex, schizophrenia, dreams, art, population, war, income, fads, and many others. Many of these topics are dealt with statistically, as likewise conforming to the law of least effort; and all are discussed with originality and insight. For example, the cities in any country, according to Zipf, tend to show the same regularity—the tenth most populous city will have one-tenth as many people as the most populous, the one-hundredth most populous city will have one-hundredth as many. Where this pattern does not prevail, we have an indication of serious potential conflict. It seems that starting about 1820 the growing divisions between Northern and Southern economies in the United States could be seen by a break in this pattern, which reached a peak of severity around 1840, and disappeared after the Civil War!

But while this breadth of topic endows the book with a distinctive fascination, it also makes the reader skeptical of the validity of the theory it puts forward. In the human sciences, the scope of a generalization is usually inversely proportional to its precision and usefulness—at any rate, in the present state of our knowledge. Zipfs law, or something like it, is well known in physics as Maupertuis principle of least action. But Zipf applies it to a much wider field of phenomena than physical flows of energy. It is understandable that action will follow some sort of least-action pattern if economy enters into its motivation and if the action is sufficiently rational. But that the law of least effort should be manifested everywhere in human conduct, as Zipf holds—indeed, “in all living process”—is difficult to believe.

That a theory is incredible is, of course, no logical objection whatever. And Zipf does not merely speculate; he presents an enormous mass of empirical evidence. The question is what it proves. It does not show, as he claims, the existence of “natural social laws,” but, at best, only certain regularities. Brute empiricism is not yet science. Unless observed regularities can be brought into logical relation with other regularities previously observed, we remain at the level of description rather than explanation; and without explanation, we cannot attach much predictive weight to description. As a collection of data that deserve the careful attention of the social scientist, Zipf’s work will have interest for some time to come. But something more precise and less universal than his principle of least effort will be required to transform that data into a body of scientific knowledge.

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The importance of clear conceptualization in giving scientific significance to observed fact is admirably expounded in the recently published fourth volume of the monumental American Soldier series (S. A. Stouffer, L. Guttman, et al., Measurement and Prediction: Studies in Social Psychology in World War II, Vol. IV, Princeton University Press, 1950). Measurement and Prediction deals with the study of attitudes. It is concerned with the development of methodological rather than substantive theory. It deals with the way in which attitudes—any attitudes—should be studied and understood, but says little about attitudes themselves. However, methodology here is not an excuse for irresponsibility in substantive assumptions, or for confusion as to the ultimate importance of a substantive theory of attitudes.

The major problem taken up is this: how can we tell whether a given set of characteristics is to be regarded as variations of some single underlying quality? Concretely, when do the responses to a questionnaire constitute expressions of a single attitude, and when do they express a number of attitudes? If there is such a single quality, how can we measure how much of it is embodied in each of the characteristics? If all the items on a questionnaire do deal with only one attitude, can we measure, in any meaningful sense, how favorable or unfavorable that attitude is, and how intensely it is held by the particular respondent?

This problem arises directly out of the practical—as well as the theoretical—side of opinion study. Consider the case of a poll designed to test the extent and intensity of anti-Semitism. Various questions are included: “Do you think the Jews have too much power?” “Do you think we should allow Jews to come into the country?” “Do you approve of what Hitler did to the Jews?” Some people give anti-Semitic answers to all the questions, some to a few, some to none. Is this because they possess varying amounts of a single quality that we may call “anti-Semitism”? Or is there really a mixture of two or more very different attitudes in the individual, present in varying proportions? If a person is against Jewish immigration, is this because he is against immigration or against Jews, and to what extent? And if there is a single quality such as anti-Semitism, what questions will best bring it out for study? It is problems such as these that the research reported on in Measurement and Prediction permits us to solve.

The approach taken stems from the work done in the past few decades by L. L. Thurstone and C. Spearman. Their problems were similar, but arose in a different field, the study of intelligence and other psychological characteristics. Their question was: do our intelligence tests determine a single quality called intelligence? Or do they actually tap a variety of factors, which though combining to produce the total intelligence score, are really quite different from each other? Thurstone and Spearman developed mathematical methods that in effect determined which items in a questionnaire were interdependent—that is, if a person answered a, he would tend to answer b, but not c. On the basis of such patterns, various factors of intelligence were discovered.

In opinion study, one inquires whether items of a questionnaire “hang together”—or, to use the technical term, whether they scale. A complex of attitudes is said to be scalable if it is possible to arrange the items in it in such a way that every person who answers “yes” to any item also answers “yes” to every item coming after it in the arrangement. In the case of anti-Semitism, we would consider the complex of anti-Semitism scalable—and therefore referring to a single factor in a person’s attitudes, rather than including a few distinct attitudes—if we could order the questions in such a way that if someone answered any question in the list “anti-Semitically,” he would answer all those following it “anti-Semitically.” Attitudes on anti-Semitism would then have the same cumulative character as a series of questions on height—if a person answers “yes” to “Are you more than six feet tall?” we know he will answer “yes” to the question “Are you more than five and a half feet tall?” and all others down the list. This type of reduction of an apparently complex field of attitudes to the simple scheme of a series of cumulative questions is of great value. In Measurement and Prediction Louis Guttman describes how to determine whether a group of attitudes does “scale”—that is, does measure a single underlying reality.

Guttman developed, for example, such a scale for manifestations of fear in battle—vomiting, trembling, etc.: soldiers who vomit when faced with combat also report trembling, and so on down the scale; while those who report trembling do not necessarily report vomiting too. On the other hand, it turned out, when he studied paratroopers, various kinds of fear of very different types had to be distinguished, for the paratroopers symptoms were not scalable in terms of a single variable.

One of the most direct applications of scaling methods is in the detection of spurious “causal” connections. We may find, for instance, that the attitude to the continuation of OPS by the government correlates closely with the attitude to farm subsidies. Scale analysis now makes it possible for us to provide an explanation for this fact by testing whether these two items do not in fact express a single attitude—say, the attitude to governmental controls.

Scale analysis permits us to handle another important problem. Suppose we find that 80 per cent of a group of soldiers tested agreed that “the British are doing as good a job as possible of fighting the war, everything considered,” while only 48 per cent disagree with the statement that “the British always try to get other countries to do their fighting for them.” How many soldiers are “favorable” toward the British? It is clear that we can get different percentages in answer to this question, depending on how we word our question. Scale analysis provides a method which yields what is called an “objective zero point”: a division between the numbers of those “favorable” and those “unfavorable” that remains constant no matter what questions we ask about the British. The method demands that, besides asking a few questions testing the attitude, we also get a measure of the “intensity” with which the respondent holds his opinion—we ask for example, whether the respondent feels strongly, not so strongly, or is relatively indifferent about the matter. With this method, it turns out that if we asked a group of entirely different questions about the British, the application of the procedure for measuring the “objective zero point” would show the same result. This limited area of attitude comes to have the same objectivity as the temperature scale, which shows the same result whether we use an alcohol or a mercury thermometer.

Measurement and Prediction also presents, for the first time, a full description of Lazars-feld’s “latent structure” analysis. This is in effect a mathematical generalization of the scaling method of Guttman, which permits us to extend the type of inquiry that scale analysis makes possible into other areas. Scale analysis and latent structure analysis together form an important contribution to the development of more reliable methods of subjecting attitudes—and similar “qualities”—to precise and meaningful measurement.

The prediction part of Measurement and Prediction does not contain any comparable theoretical developments. For the most part, prediction in the social sciences today is not above the level of “enlightened common sense,” as the authors recognize.

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IV

The distinctive development in mathematics in the last one hundred years is the rise of a discipline whose central concept is neither number nor quantity, but structure. The mathematics of structure might be said to focus on qualitative differences. Topology, as the new discipline is called, is occupied with those properties of figures—conceived as sets of points—that are independent of mere differences of quantity. Squares, circles, and rectangles, of whatever size, are topologically indistinguishable from one another, or from any simple closed figure, however irregular. On the other hand, if a “hole” is inscribed in any of these figures, it thereby becomes qualitatively different from the rest.

Topology, more than most sectors of higher mathematics, deals with questions that have direct intuitive meaning. But intuition often misleads us as to the answers, when it does not fail us altogether. For instance, how many different colors are required to color any map so that no two adjoining countries have the same color? It is known, from topological analyses, that five colors are quite enough; but no one has been able to produce a map that needs more than four, or even to prove whether there could or could not be such a map.

It is paradoxical that the field of mathematics which deals with the most familiar subject matter is the least familiar to the non-mathematician. A smattering of geometry, algebra, and statistics is very widespread; topology is virtually unknown, even among social scientists sympathetic to mathematics. To be sure, the late Kurt Lewin’s “topological psychology” has given the name much currency. But topology, for Lewin, provided only a notation. The rich content of his work bears no logical relation to the topological framework in which it is presented, as is clear from the posthumously published collection of his more important papers, Field Theory in Social Science (Harper, 1951). In these papers, talk about the “life space,” and “paths,” “barriers,” and “regions” in it, are elaborately sustained metaphors. Such figures of speech are extraordinarily suggestive, but do not in themselves constitute a strict mathematical treatment of the “life space” in a topological geometry. The actual application of topology to psychology remained for Lewin a program and a hope.

One further development must be mentioned as playing an important role in the new approaches: the rise of symbolic logic. As in the case of topology, this discipline can be traced back to the 17th century, to Leibniz’s ideas for a universal language; but not till the late 19th century did it undergo any extensive and precise development. Boolean algebra provided a mechanical method for the determination of the consequences of a given set of propositions (in another form, this is called the “calculus of propositions”). A few years later, De Morgan, Schroeder, and the founder of Pragmatism, Charles Peirce, investigated the formal properties of relations, leading to an elaborately abstract theory of relations. These results, together with work on the foundations of mathematics (like Peano’s formulation of five postulates sufficing for the whole of arithmetic), were extended and systematized shortly after the turn of the century in Russell and Whitehead’s monumental Principia Maihematica.

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Among the recent applications to the study . of man of this whole general body of ideas, one is especially celebrated; the theory of games presented by J. von Neumann and O. Morgenstern in Theory of Games and Economic Behavior (Princeton University Press, 1947). Here the focus is confined to problems of economics, but it is hoped that it will be extended to the whole range of man’s social relations.

It may seem, superficially, that von Neumann and Morgenstern, in selecting games as a way of approaching the study of social organization, fall into the trap of oversimplification. But unlike Rashevsky, von Neumann and Morgenstern do not so much introduce simplifying assumptions in order to deal artificially wtih the whole complex social order as select relatively simple aspects of that order for analysis. Only after a theory adequate to the simple problems has been developed can the more complicated problems be attacked fruitfully. To be sure, the decision-maker cannot suspend action to satisfy the scruples of a scientific conscience; but neither can the scientist pretend to an advisory competence that he does not have, in order to satisfy practical demands.

While the theory of games does not deal with the social process in its full complexity, it is not merely a peripheral aspect of that process which it studies, but its very core. The aim is “to find the mathematically complete principles which define ‘rational behavior’ for the participants in a social economy, and to derive from them the general characteristics of that behavior.” Games are analyzed because the pattern of rational behavior that they exhibit is the same as that manifested in social action, insofar as the latter does in fact involve rationality.

The theory is concerned with the question of what choice of strategy is rational when all the relevant probabilities are known and the outcome is not determined by one’s choice alone. It is in the answer to this question that the new mathematics enters. And with this kind of mathematics, the social sciences finally abandon the imitation of natural science that has dogged so much of their history.

The authors first present a method of applying a numerical scale other than the price system to a set of human preferences. A man might prefer a concert to the theater, and either to staying at home. If we assign a utility of “1” to the last alternative, we know that going to the theater must be assigned a higher number, and the concert a higher one still. But how much higher? Suppose the decision were to be made by tossing two coins, the first to settle whether to go to the theater or not, the second (if necessary) to decide between the remaining alternatives. If the utility of the theater were very little different from that of staying at home, most of the time (three-fourths, to be exact) the outcome would be an unhappy one; similarly, if the theater and concert had comparable utilities, the outcome would be usually favorable. Just how the utilities compare, therefore, could be measured by allowing the second coin to be a loaded one. When it is a matter of indifference to the individual whether he goes to the theater or else tosses the loaded coin to decide whether he hears the concert or stays home, the loading of the coin provides a numerical measure of the utilities involved.

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Once utility can be measured in a way that does not necessarily correspond to monetary units, a theory of rational behavior can be developed which takes account of other values than monetary ones. A game can be regarded as being played, not move by move, but on the basis of an over-all strategy that specifies beforehand what move is to be made in every situation that could possibly arise in the game. Then, for every pair of strategies selected—one by each player in the game—the rules of the game determine a value for the game: namely, what utility it would then have for each player. An optimal strategy is one guaranteeing a certain value for the game even with best possible play on the part of the opponent. Rational behavior, that is to say, is characterized as the selection of the strategy which minimizes the maximum loss each player can sustain.

If a game has only two players and is “zero-sum”—whatever is won by one player being lost by the other, and vice versa—then, if each player has “perfect information” about all the previous moves in the game, there always exists such an optimal strategy for each player. The outcome of a rationally played game can therefore be computed in advance; in principle, chess is as predictable as ticktacktoe.

Not every game, however, is as completely open as chess. In bridge and poker, for example, we do not know what cards are in the other players’ hands; and this is ordinarily the situation in the strategic interplay of management and labor, or in the relations among sovereign states. In such cases rationality consists in playing several strategies, each with a mathematically determined probability. Consider a type of matching pennies in which each player is permitted to place his coin as he chooses. If we were to select heads always, we should be quickly found out; even if we favored heads somewhat, our opponent (assuming it is he who wins when the coins match) could always select heads and thereby win more than half the time. But if we select heads half the time—not in strict alternation, to be sure, but at random—then, no matter what our opponent does, we cannot lose more than half the time. Rational play consists in what is actually done in the game: we toss the coin each time to determine, as it were, whether we select heads or tails. Of course, in more complex games our strategies are not always to be “mixed” in equal proportions. The fundamental theorem of the theory of games is that for every two-person zero-sum game, no matter how complex the rules or how great the range of possible strategies open to each player, there always exists some specific pattern of probabilities of strategies which constitutes rational play. It minimizes the maximum loss that each player can sustain, not in every play of the game, but in the long run. And there is a mathematical solution that tells us what this strategy is.

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Unfortunately, many games are not “zero-sum”: in the game between management and labor, utilities are created in the process of play; in war they are destroyed. It is simply not true in such cases that what one side loses the other gains, or vice versa. In such cases, the mathematics of the “zero-sum” game will not apply. Moreover, many games have more than two players: a number of manufacturers may be competing for a single market. Here the mathematics of a two-person game will not hold. The first difficulty, however, can be absorbed into the second. A non-”zero-sum” two-person game can be regarded as a three-person game, where the third person wins or loses whatever is lost or won by the other two together.

But how are we to solve games of more than two persons? Only if coalitions are formed, in effect making the game a two-person one. This is not an unrealistic assumption, since, obviously, if two players can coordinate their strategies against a third, they will enjoy a special advantage: odd-man-out is the simple but fundamental principle in such situations. For such games, however, the theory does not provide a detailed solution, for it cannot determine what is a rational division of the spoils between the members of a coalition in the way it can determine what is a rational strategy for the coalition as a whole. And here, of course, is the great difficulty in politics. The United States and Russia may be conceived of, in this theory, as playing a two-person non-”zero-sum” game with nature as the third player: only nature wins from atomic destruction, only nature loses if resources need not be diverted to military purposes. But the coalition of men against nature still leaves open how the utilities acquired in the game are to be divided between the participants. And here conflicting interests stand in the way of the joint interests that would make rational a coalition strategy.

From our present standpoint, the important outcome of the theory is to show that there exists a rigorously mathematical approach to precisely those aspects of the study of man that have seemed in the past to be least amenable to mathematical treatment—questions of conflicting or parallel interest, perfect or imperfect information, rational decision or chance effect. Mathematics is of importance for the social sciences not merely in the study of those aspects in which man is assimilable to inanimate nature, but precisely in his most human ones. On this question the theory of games leaves no room for doubt.

But a mathematical theory of games is one thing, and a mathematical theory of society another. Von Neumann and Morgenstern, it must be said, never confuse the two. Many fundamental difficulties remain, even within the limitations of the games framework. The theory of games involving many players is in a very unsatisfactory state; there is no way at present of comparing the utilities of different persons; and the whole jheory is so far a static one, unable to take account of the essential learning process which (it may be hoped) takes place in the course of the selection of real-life strategies. Yet the theory has already provided enormous stimulation to mathematical and substantive research, and much more can be expected from it. Above all, it has shown that the resources of the human mind for the exact understanding of man himself are by no means already catalogued in the existing techniques of the natural sciences.

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Thus the application of mathematics to the study of man is no longer a programmatic hope, but an accomplished fact. The books we have surveyed are by no means the only mathematical approaches to problems of social science published even within the past few years. For instance, K. Arrow’s Social Choice and Individual Values (Wiley, 1951) is a penetrating application of postulational method and the logical theory of relations to the problem of combining, in accord with democratic principles, individual preferences into a representative set of preferences for the group as a whole. Harold Lasswell and his associates report in The Language of Politics (G. W. Stewart, 1949) on the procedures and results of the application of the now widely familiar content-analysis techniques to political discourse, in order to objectify and quantify the role of ideologies and Utopias in politics. Shannon and Weaver’s Mathematical Theory of Communication (University of Illinois Press, 1950) reprints the classic papers in which Claude Shannon first developed the precise theory of information now being applied in cybernetics, linguistics, and elsewhere. There are a number of other books; and dozens of papers have appeared in the professional journals of a wide variety of disciplines, including such preeminently “qualitative” fields as social organization, psychiatry, and even literary history.

But if the new language is widely spoken, there are also wide local differences in dialects, and many individual peculiarities of usage. Yet on the scientific scene, this is in itself scarcely objectionable. New problems call for new methods, and if for a time these remain ad hoc, it is only from such a rich variety of approaches that systematic and general theories can emerge. No such theories of society or even of part of the social process have yet been developed, though, as we have seen, there have been some premature attempts. But the situation is promising. If the mathematics employed is not merely notational, if it is not merely an “illustration” of an abstract methodology, if it does not outstrip the availability of data and especially of ideas, there is reason to hope that it will eventually contribute as much to the study of man as it already has to the understanding of the rest of nature.

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