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?"