Various
Appletons' Popular Science Monthly, October 1899
Suppose we take two equal squares, and, making equal lengths on the four sides of one of them, join the points so obtained as indicated in the first of the two figures (Figs. 5 and 6) so as to form four right-angled triangles, and then place four other squares in the corners of the original square. These right-angled triangles are of such sort that the sum of their sides is equal to the side of the square. This can be demonstrated, but it strikes the eyes without that. We see, too, that the interior figure is a square, and that it is constructed on the hypotenuse of the triangles in question.
Fig. 5.
Fig. 6.
It is easy to see in the other figure, which is formed after the same measures as its alternate, that the triangles 1, 2, 3, 4 can be arranged so as to occupy the positions 1', 2', 3', 4' in such way as to leave in the main square two smaller squares constructed on the sides of one of the right-angled triangles. It follows that the square A is equivalent to the sum of the squares B and C. The theorem thus becomes a kind of intuition, a thing evidently indisputable.
It is a curious fact that the origin of this demonstration is lost in the obscurity of the past; it probably goes back to thirty or forty centuries, at least, before the Christian era, and apparently to India. Bhascara, in his Bija Ganita, after tracing a figure, a simple combination of these two, says, "There you see it." I remark that such a demonstration, even if dressed with geometrical terms, assuming a character that conforms to existing ways of teaching, would be vastly superior, even in secondary schools, to the demonstrations of Legendre and others, which are much harder. The return to what was done very long ago in this case constitutes a great advance upon what we are doing now.
Fig. 7.
Having given our little one an initiation into the mysteries of arithmetic and geometry, we introduce him to algebra, a branch which passes in the majority of families as the hardest, most complicated, and most abstruse that can be imagined. I do not pretend that algebraic theories enter easily into the child's delicate brain; rather the contrary; but I declare that some ideas in algebra can be made comprehensible to children without fatigue. We can, for instance, make them understand, in the way of amusement and without great difficulty, the formula that gives the sum of the first numbers. We take a sheet of paper ruled in squares and shade the first square of the first line, then the first two squares of the second line, the first three of the third, etc. (Fig. 7). The whole number of squares shaded in this manner represents visibly the sum of the first whole numbers up to any one we may choose – to 7 in the figure. If we give this paper to the child and ask him to return it, he will very easily perceive that the figures formed by the white and the black squares are alike. The number sought for will therefore be equal to half the sum of the squares – that is, in the present example
1 + 2 + 3 + 4 + 5 + 6 + 7 = (7 X 8): 2 = 28,
we can prove by reasoning that if n be taken to represent the last number we shall have for the sum
I introduce this formula to define my thought better, but one can make the child perceive the numbers that are wanted without writing down a single character.
Somewhat similar is the method of finding the sum of the odd numbers. For this it will be enough to take our square-ruled sheet of paper and shade the first square on the left, then the three squares around it, which will form with it a square (1 + 3 = 4); continuing thus we obtain, as the figure readily shows (Fig. 8), a square formed of a series of shaded zones, representing the series of odd numbers, the examination of which will illustrate the property to the child.
Fig. 8.
In another direction it is possible to give the child algebraic ideas much beyond anything we would imagine. Suppose, for example, we want to give him a conception of addition. He easily realizes that objects – material bars, for example – can be selected so as to represent numbers by their length. He can be readily made to understand that if he has one bar three and another five inches long he can obtain the sum of these lengths, in what we might call a material way, by placing them lengthwise, one at the end of the other – an essentially practical notion and easily carried into effect. If we take a line and mark a starting point on it, calling it zero, then measure off segments on it representing the bars we have been talking about one after another, we can get the sum represented by the length of the two segments. If, instead of measuring three plus five inches I measure three plus two I reach another point. If, instead of adding two and three, I wish to take one of the bars or numbers away (3-2), or subtract, the operation will be easily performed by measuring the two in the opposite direction. The difference will be represented by the length that is left. If we try to form the quantity 3-5 in arithmetic we can not do it; but in proceeding in this method and measuring back on the bar we get to a point back of the original starting point which represents this difference – say two inches behind where we began. Here we have in the germ the whole theory of negative quantities, concerning which thousands and thousands of pages have been written. Yet we find that by carefully graduating our lines we can make it intuitive and accessible to a child who has learned that the common operations of addition and subtraction can be represented with material objects. The generation of negative and positive quantities follows quite naturally.
These examples, I think, are sufficient to show that we might considerably enlarge the field of the investigations within reach of the child. For this purpose a small amount of very simple material, which we can vary as we please, is needful. The first element of this material is paper ruled in squares, a wonderful instrument, which everybody dealing with mathematics or with science generally should have. It is of special pedagogic use in giving children their first ideas of form, size, and position, without which their early instruction is only a delusion. Add to this paper dice, buttons, beans, and match-sticks – things always easy to get – and we have all the material we need.
There is no amusement, however puerile it may appear, not even a play of words, that can not be utilized in teaching of this sort. For instance, when your child has learned his addition table, if you put him to a demonstration, assuming to prove to his comrades that six and three make eight, his curiosity will be excited, and you may be very sure that, once his attention has been given to this amusement, he will never forget that six and three make nine and not eight. To make the demonstration, we have only to group the nine match-sticks as in the figure (Fig. 9) below. We might demonstrate in a like way that half of twelve is seven by cutting the Roman numeral XII in two, leaving the upper part visible. Such pleasantries have a pedagogical value, because the paradox is precisely of a kind to attract the attention of the child, and he will always afterward be sure not to fall into the trap.
Fig. 9.
The side of this kind of instruction on which I insist most is that, given under the form of play, it is free from every sort of dogmatic character. No truth should be imposed on the child; on the contrary, he should be allowed to discover it as a fruit of his own activity. He will be thoroughly impressed with the truths which he has thus found out himself. They had better be few at first; the important thing is for him to know them completely.
The instruction should also be essentially objective and free from all abstraction. The absence of abstraction should, however, be rather apparent than real. Abstraction is indeed one of the elements that contribute most to give mathematical science a fearful air to outsiders, and yet it is most usually a simplification of matters – quite the contrary of what is generally supposed. It is, in fact, such a simplification and so necessary that we all make it as if by instinct, and the child makes it, not in mathematics only, but in all the considerations of life.
Thus, when I want to give the child his first idea of the number two I put two beans in his hand and let him contemplate them. He gets a perfect notion of the collection two. Yet, if you look at them a little closer and he himself looks at them closer he will find that the two beans, whatever else they may be, are not identical, for there exist no two objects in Nature that are not different. So when the child introduces this idea of collection into his mind in a wholly instinctive way, by identifying the things he sees, he begins to perform abstraction. This abstraction delivers him from all the complications and all the annoyances that come to him from the contemplation of real objects. By the philosophic process of abstraction it has been possible to construct all the sciences, and especially the science of magnitudes.
The ideas I have been setting forth in outline are not mine, and are, unfortunately, not recent. They may be found in somewhat different form, but substantially the same in principle, in l'Essai d'education nationale, published by Le Chalotais in 1763. The paper furnishes a programme of studies and education which, if put into execution, would, I believe, constitute a long advance over the present conditions. At a later period Condorcet was occupied with the subject. At the close of the nineteenth century the name of Jean Macé, which I have already cited, should be held among those of men who have tried to infuse sound and just views concerning the pedagogy of mathematics. Another man, from whom I have borrowed a considerable part of the examples I have cited, is Edouard Lucas, who, in his Récréations mathématiques, of which one volume was published during his lifetime and two others after his death, and in his lectures before the Conservatoire des Arts et Métiers, strove to develop views concerning the primary mathematical education of childhood – views which did not differ, except in form, from those which I have presented. —Translated for the Popular Science Monthly from the Revue Scientifique.
PRESENT POSITION OF SOCIOLOGY
By F. SPENCER BALDWIN
The present condition of sociological thought is confused, if not chaotic. It needs only a brief examination of the writings of professed sociologists to discover the want of agreement among them. There is no consensus of opinion regarding either the scope and method of the new science, so called, or its fundamental laws and principles. The name sociology stands for no definite body of systematic knowledge. It is applied to an inchoate mass of speculation, often vague and conflicting, which represents the thought of various thinkers about social phenomena.
A few years ago a student of sociology in Chicago wrote to "all the teachers of sociology in the United States, and to others known to be deeply interested in the subject and entitled to express an opinion," asking them to answer a number of pertinent questions regarding the nature and function of the "science."28 About forty replied; of these, three discreetly pleaded knowledge insufficient to entitle them to an opinion. Comparison of the views expressed in the remaining twenty-seven replies led the investigator to conclude that the science is in a more or less undefined and tentative position. So little progress toward unanimity of opinion has been made by sociologists since the date of this census that its results may be taken as typical of present conditions. Among the questions asked were these: "Do you think the study is entitled to be called a science?" "In what department does it belong?" "What is its relation to political economy, history, political science, ethics?"
The question whether sociology is entitled to be called a science is answered by "fully three fourths" of the correspondents in the affirmative. Some hedge, by affirming that it is "becoming a science." Prof. John Bascom, of Williams College, appears to have entered into the humor of the situation; he writes, "It will do no harm to call it a science if we do not abate our effort to make it one."
The opinions regarding the department in which sociology belongs are entertainingly diverse. Prof. John Dewey, of the University of Chicago, is frank enough to admit that he doesn't "feel at all sure" where it belongs. "It would seem well," he adds, "to have it a separate branch, in order to make sure that it received proper attention." This feeling of uneasiness lest the claims of sociology be slightingly treated appears to be general among the representatives of the new study. Most of the teachers of sociology are of the opinion that it ought to form a department by itself. Some would place it in the department of the social sciences, along with politics, economics, jurisprudence, and the like. Others would change the order, making all the social sciences divisions of sociology. On the other hand, Professor Giddings, of Columbia University, says: "General sociology can not be divided into special social sciences, such as economics, law, and politics, without losing its distinctive character. It should be looked on as the foundation or groundwork of these sciences, rather than as their sum or as their collective name." Scattering replies place it under psychology, moral and political science, political economy, and anthropology. One teacher thinks it belongs under the "humanities"; while two say it has no natural boundaries, and is therefore not included in any one department. Altogether the impression left by the replies to this question is that the teachers of sociology are quite at a loss to know where to put the study in the university curriculum. They appear to realize confusedly that they have on their hands a pedagogical white elephant, which defies classification.
The opinions concerning the relation of sociology to political economy, history, political science, and ethics are almost delphic in their vagueness. Says one, "History is its material, ethics its guide, political economy its interpreter, and a rational system of political science its proposed end." Says another, "Sociology is political economy in practice, history in the making, political science as an art, and ethics applied." After worrying over these oracular epigrams it is refreshing to be told by another teacher that "the relation of sociology to political economy, history, etc., is close."
It would be superfluous to cite further illustrations of the unsettled state of sociological thought. The quotations that have been made show conclusively that the accredited representatives of the new "science" are at loggerheads upon fundamental questions. This fact the sociologists themselves readily admit. The author of a recent treatise on sociology speaks of the "confusion and perplexity among its teachers, and declares that its forms are as yet varied, and perhaps would suggest a series of pseudo-sciences instead if one genuine science."29 Even Professor Giddings confesses in the preface of his Principles of Sociology that "much sociology is as yet nothing more than careful and suggestive guesswork." Professor Small, of the University of Chicago, in his Introduction to the Study of Society, speaks of sociology as an "inchoate science," and remarks that "only ignoramuses, incompetent to employ the method of any science, could claim for sociology the merit of a completed system."
Sociologists themselves, then, confess that differences of opinion exist among them. Let us look more carefully at the nature of these differences. They relate to the scope, the method, the object, and the ground-principles of the "science."
The province of sociology is defined by some very broadly, to include the whole range of the phenomena of human association. By others the scope of the study is limited to a narrower range of social phenomena. Among the latter, again, there are some who would identify sociology with the study of social origins, or the genesis of social institutions. Others would restrict sociology to a study of the history and function of the family. Still others understand by sociology merely the pathology of society, devoting themselves to the diagnosis of social diseases, as crime and pauperism.
Professor Giddings has called attention to the natural tendency on the part of each social philosopher to create a sociology in the image of his professional specialty. "To the economist," he says, "sociology is a penumbral political economy – a scientific outer darkness – for inconvenient problems and obstinate facts that will not live peaceably with well-bred formulas. To the alienist and the criminal anthropologist it is a social pathology. To the ethnologist it is that subdivision of his own science which supplements the account of racial traits by a description of social organization. To the comparative mythologist and the student of folklore it is an account of the evolution of culture."
The narrower conceptions of sociology, however, have been discarded by the best-known sociologists of the present time. There is a general tendency to adopt a broad definition of the province of sociology, to include in the field of investigation all the phenomena of social structure and growth.
But what is the relation of this general social science to the special social sciences – that is, the sciences dealing with special groups of social phenomena, as economics, politics, and jurisprudence? Is sociology anything more than a convenient collective name for the sum of all these? Touching this point opinions differ.30
At least three different conceptions of the relation of sociology to the various special social sciences may be distinguished. Sociology has been defined as (1) the "inclusive," as (2) the "co-ordinating," and as (3) the "fundamental" science of society. 1. The first conception is that of Spencer and De Greef. Spencer defines sociology as "the science of society," and defends his adoption of the term on the ground that "no other name sufficiently comprehensive existed." This implies that he conceives of sociology as an inclusive science. De Greef, the Belgian sociologist, makes the science all comprehensive; his scheme of classification "includes everything, from the husbanding of corn and wine to electioneering contests in the Institute of France."31 2. The second conception is that of Professor Small, of Chicago. He defines sociology as "the synthesis of all the particular social sciences." It does not include, it coordinates these sciences. It concerns itself with the relations which the various special groups of social phenomena hold to each other and to society as a whole, leaving to special social sciences the study of each group in minute detail. The conclusions won by these special sciences are taken by sociology and worked over into a body of correlated social principles. Sociology is, therefore, subsequent to the particular social sciences and dependent upon them. 3. The third conception is that of Professor Giddings, of Columbia University. He defines sociology as "the science of social elements and first principles." It is "not merely the sum of the social sciences; it is rather their common basis." It undertakes to analyze the general characteristics of social phenomena and to formulate the laws of social organization and evolution. Sociology furnishes a body of fundamental principles which make a common basis for the special social sciences. The latter rest on sociology, which is the antecedent and fundamental social science.
Now a little reflection will show that these three conceptions of sociology do not conflict, but harmonize. There is no real opposition between them, rightly understood. Each emphasizes correctly one phase of the relation between sociology and the special social sciences. Sociology is both an inclusive, a co-ordinating, and a fundamental science. In the first place, sociology is a general science, having as its subject-matter social phenomena of all kinds. Therefore it comprehends all the sciences dealing with special kinds of social phenomena. These particular sciences are, in the nature of things, closely related to each other. They must possess in common certain laws and principles. These it is the task of sociology to formulate; for as the inclusive social science it should exhibit the mutual relations of the included social sciences. Thus sociology becomes a co-ordinating as well as an inclusive science. Furthermore, the laws and principles of the special social sciences, which sociology, as the co-ordinating science, undertakes to formulate, are necessarily fundamental. And in this respect sociology may be regarded as the fundamental social science. The three rival conceptions of sociology must be combined in the correct view. As Mr. Arthur Fairbanks remarks in his admirable Introduction to Sociology: "Sociology may embrace all the sciences dealing with society, but it does not destroy the partial independence of any of these branches. It includes economics, politics, and the like, but, instead of supplanting them, its sphere is to lay the foundation of these particular social sciences."
It appears, then, that the disagreement among the leaders of sociological thought regarding the scope of their "science" is more apparent than real. The same may be said regarding the contention about method. The debate here is over the question whether deduction or induction is the proper method of investigation in the social sciences. One party holds that the only legitimate method is the abstract-deductive, the investigator arriving at his conclusions by reasoning a priori from certain fundamental assumptions regarding the nature of man in general. What these thinkers aim at is a subjective interpretation of social phenomena in terms of human motives, principles, and ideals. Another party maintains that the only fruitful method is the concrete-inductive, the investigator reaching his conclusions by observing the facts of social life and reasoning from them to general laws and principles. The aim here is to give an objective interpretation of society in terms of race, environment, and historical conditions. The controversy has been especially violent among the economists. The English classical school of political economy made exclusive use of the deductive method; economic laws were deduced from the fundamental postulate of human selfishness. The German historical school employed the inductive method; economic laws were inferred from a study of the concrete facts of industrial life.
This academic discussion over method is tiresome and futile. Neither method will ever drive the other from the field. The exclusive employment of either deduction or induction will yield only half results in the social sciences. The two methods effectually supplement each other and should be used together. They are not rivals, but allies. Induction without deduction is blind; deduction without induction untrustworthy. This fact is recognized by recent writers on sociology. So Professor Giddings remarks that "history without deductive illumination is chaos. Deduction without verification is undoubtedly the very light that never was on sea or land!"
The principal method in the social sciences must undoubtedly be the inductive. The nature of the subject-matter determines this. The social sciences deal with the facts of social structure and growth. The task of the investigator is the explanation of these facts. He has first, then, to observe and compare the facts. But his observation must be guided and his conclusions verified by deduction.
Concerning the purpose of sociology, as touching its method, there are two conflicting opinions. But here again the seeming disagreement is not absolutely irreconcilable. It is held by some that the purpose of the sociologist should be merely the acquisition of knowledge, without further thought of the practical use to which the results of his researches might be put. He should aim to discover and formulate the laws of social forces, not to propose ideals of social reform. Sociology is a pure science and has no utilitarian end. By others it is held that the purpose of the sociologist should be the regulation of social forces in the interest of human progress. The object of sociology is the betterment of society, the acceleration of social evolution. It is an applied science and has a practical end.
Both these views are tenable. In fact, sociology, like all sciences, has a double purpose. The primary purpose is to acquire knowledge; the secondary purpose is to apply that knowledge to the attainment of practical ends. This duality of purpose is clearly set forth by Mr. Lester F. Ward in a recent essay.32 "Sociology," he says, "has both a pure and an applied stage." It "should be studied first for the sake of information relating to the laws of human association and co-operative action, and finally for the purpose of determining in what ways and to what extent social phenomena may, with a knowledge of their laws, be modified and directed toward social ideals."
Modern society is a complex of difficult problems. And this fact furnishes a background of motive for the studies of the sociologist. Not even the veriest stickler for pure science can deny the imperative need of established knowledge of the laws of social activity. The people perish for lack of wisdom. To enlighten the public mind on vital social questions and thus to promote an intelligent direction of social conduct toward rational ends is the high function of sociology. This practical purpose, however, should be kept always secondary to the pursuit of knowledge. "The knowledge is the important thing. The action will then take care of itself."33 The discussion of the what-ought-to-be must wait on the investigation of the what-is. The neglect of this caution has been responsible for much false doctrine and foolish counsel. Sociologists have allowed their enthusiasm for ideals to blind the eye and bias the judgment. Panacea hawkers of all sorts have attempted to prescribe for social diseases, without making any study of social structure and function. Communistic quackery has masqueraded as sociological wisdom. The wild-cat sociology of the present day is a result of the over-addiction to social reform which besets students of society. It can not be too strongly emphasized that the primary object of the sociologist is the impartial investigation of facts. The man who forgets this becomes dangerous. He is liable to run amuck.
The differences of opinion as to the scope, method, and purpose of sociology have been found upon examination to be less serious than they at first sight appeared. But in regard to the fundamental principles of sociology, the confusion is hopeless. The student will search in vain in the systematic treatises on sociology for any definite body of established doctrine which he can accept as the ground-principles of the science. He finds only an unmanageable mass of conflicting theories and opinions. Each treatise contains an exposition of what the author is pleased to label the Principles of Sociology. But the "principles" are not the same in any two treatises; and by no process of analysis and synthesis can they be brought into harmony. They are fundamentally contradictory. It is impossible, I believe, to discover a single alleged ground-principle of sociology that has commanded general assent.
Some of the recent writers on sociology have devoted themselves particularly to the task of establishing one basal principle which may be applied to the interpretation of all social phenomena. At least half a dozen claims to the discovery of such a principle have been put forward. Prof. Ludwig Gumplowicz finds the elementary social fact to be conflict; Prof. Guillaume De Greef finds it to be contract; M. Gabriel Tarde contends that the fundamental principle of society is imitation; Prof. Emile Durkheim argues that it is "the coercion of the individual mind by modes of action, thought, and feeling external to itself." Professor Giddings criticises all these explanations of society, as either too special or too general, and undertakes to prove that "the original and elementary fact in society is the consciousness of kind." This is the determining principle to which all social phenomena are to be referred.34 But Professor Giddings's sociological postulate has been promptly rejected by his American colleagues, Prof. Albion W. Small and Mr. Lester F. Ward. The former speaks contemptuously of the consciousness of kind as a remote metaphysical category, and declares that the whole system of sociology based on the principle is "an impossible combination of contradictions."35 This opinion is approved by Ward, who riddles Giddings's book with criticism, and complains of the author's inability to handle principles correctly.36
It is hardly necessary to penetrate further into this debate over first principles. The most exhaustive examination of the writings of the leaders in sociological thought would fail to discover any fundamental unity of opinion. The so-called principles of the science are multiform. They represent merely the unsupported conclusions of individual thinkers. If we except the barest commonplaces, no truths have been established; no scientific laws have been agreed upon. The content of the science of sociology, as expounded in treatises bearing this name, varies with the particular bias of the writer. In fine, there are systems of sociology galore, but there is hardly a sociology.
Of the various systems of sociology that have been developed since the new "science" was first outlined by Auguste Comte, that of Herbert Spencer is undoubtedly the most coherent and self-consistent. But even the genius of Mr. Spencer has been unequal to the task of working out a body of firmly grounded principles which should furnish a basis for the convergence of opinion on social questions. He has not succeeded in giving permanent form and content to sociology. His work is disparagingly criticised by other living sociologists. Small declares that "Spencer's sociology ends precisely where sociology proper should begin," and quotes approvingly De Greef's assertion that "Mr. Spencer not only fails to show that there is a place for sociology, but his own reasoning proves more than anything else that there is no social science superior to biology."37 Ward, while commending the logical consistency of Mr. Spencer's work, pronounces him "unsystematic, nonconstructive, and nonprogressive."38