Chapter V: Part III: (concluded) (2)

Bacon notes two ways particularly in which words impose on the understanding--they are employed sometimes "for fantastic suppositions ... to which nothing in reality corresponds," and sometimes for actual entities, which, however, they do not sharply, correctly, and completely describe. The eighteenth century speculated at length on a state of Nature and the social contract, unaware that it was deluding itself with unrealities, and we have not yet done with such abstractions as the Rights of Man, Nature (personified), Laws of Nature (conceived as analogous to human laws), and the Vital Principle. The more common and serious danger of language, however, lies in the employment of words not clearly or firmly grasped by the speaker or writer--words which, in all probability, he has often heard and used, and which he therefore imagines to represent ideas to him, but which, closely analyzed, will be found to cover paucity of knowledge or ambiguity of thought. Cause, effect, matter, mind, force, essence, creation, occur at once as examples. Few among those who so glibly rattle them off the tongue have ever taken the trouble to inquire what they actually mean to them, or whether, indeed, they can translate them into thought at all.

Among the Idols of the Market Place we must also class the evils arising from the tendency of words to acquire, through usage and association, a reach and emotional value not inherent in their original meanings. This is what Oliver Wendell Holmes happily described as the process of polarization. "When a given symbol which represents a thought," said the Professor at the Breakfast Table, "has lain for a certain length of time in the mind it undergoes a change like that which rest in a certain position gives to iron. It becomes magnetic in its relations--it is traversed by strange forces which did not belong to it. The word, and consequently the idea it represents, is _polarized_." The larger part of our religious and no small portion of our political vocabulary consist of such polarized words--words which, on account of their acquired magnetism, unduly attract and influence the mind. We can never hope to think calmly and clearly while the very symbols of our thoughts thus possess a kind of thaumaturgic power over us, which in turn readily transfers itself to our ideas.

If, then, "words plainly force and overrule the understanding and throw all into confusion and lead men away into numberless empty controversies and idle fancies," it behooves us to watch closely the interrelations of language and thought. To put it in the vernacular, we must at all times make sure that we know what we are talking about and say what we mean. To this end the study of language itself is useful, but the habits of precise thought and expression will never be acquired by linguistic exercise alone. To use no word without a distinct idea of what it means to us as we speak or write it; to check, when necessary, the process of thought by constant redefinition of terms; to depolarize all language that has become, or threatens to become, magnetic, thus translating familiar ideas into "new, clean, unmagnetic" phraseology, these may be set down as first among the rules to which we should tolerate no exception.

We now come to the last group of Idols--those "which have immigrated into men's minds from the various dogmas of philosophies, and also from wrong laws of demonstration." These Bacon calls Idols of the Theater, "because in my judgment all the received systems are but so many stage-plays, representing worlds of their own creation after an unreal and scenic fashion." And perhaps this conceit carries further than Bacon himself intended, for it not only suggests the unsubstantial character of philosophic speculations, but also reminds us how, in the world's history, these airy fabrics have succeeded each other as on a stage, some to be hissed and some applauded, but all sooner or later to drop out of popular favor and be forgotten.

Dealing with these Idols of the Theater, or of Systems (of which there are many, "and perhaps will be yet many more"), Bacon takes the opportunity of criticising, briefly but incisively, the methods and results of ancient and mediæval philosophers. His classification of false systems is threefold: The sophistical, in which words and the finespun subtilties of logic are substituted for "the inner truth of things"; the empirical, in which elaborate dogmas are built up out of a few hasty observations and ill-conducted experiments; and the superstitious, in which philosophy is corrupted by myth and tradition. Under the first head, Bacon again instances Aristotle, whom he accuses of "fashioning the world out of categories"; under the second he glances especially at the alchemists; and under the third he refers to Pythagoras and Plato. To follow Bacon into these historic issues does not belong to our present purpose. Suffice it to notice the continued vitality of these three classes of speculative error. Bacon's judgment of Aristotle--that "he did not consult experience as he should have done, in order to the framing of his decisions and axioms; but, having first determined the question according to his will, he then resorts to experience, and, bending her into conformity with his placets, leads her about like a captive in a procession"--is at least equally applicable to thinkers like Hegel and his followers. Empiricism has by no means been eliminated from the scientific or would-be scientific world. And as for the philosophy which is corrupted by myth and tradition, the countless attempts that are still made to "reconcile" the facts of science with the data and prepossessions of theology are enough to prove that, _mutato nomine_, the methods of Pythagoras and Plato and of those who in Bacon's day sought "to found a system of natural philosophy on the first chapter of Genesis, on the book of Job, and other parts of the sacred writings," are as yet far from obsolete.

It is hardly necessary to call attention to the fact that there is a close similarity between systematic empiricism and some of the dangers brought out in connection with the Idols of the Tribe, for in each case stress must be laid on the tendency to generalize hastily, depend on scattered and inadequate data, and seek for light in the "narrowness and darkness" of insufficient knowledge. This matter is important only as showing how a common weakness may be caught up and dignified in a philosophic system and rendered more dangerous by the adventitious weight and influence which it gains thereby. Another point, not distinctly dealt with by Bacon, calls, however, for special remark. While the various Idols of the Theater, or of Systems, exercise their own peculiar and characteristic influences for evil, they all tend to the debasement of thought by reason of the authority which they gradually acquire. Associated with great names, promulgated by schools, officially expounded by disciples and commentators, they finally settle into a creed which is regarded as having oracular and dogmatic supremacy. The formula "Thus saith the Master" closes discussion. Not the fact itself, but what this or that teacher has said about the fact, comes at last to be the all-important question. In the condition of mind thus engendered there is no chance for intellectual freedom, self-reliance, growth. Lewes related an anecdote of a mediæval student "who, having detected spots in the sun, communicated his discovery to a worthy priest. 'My son,' replied the priest, 'I have read Aristotle many times, and I assure you that there is nothing of the kind mentioned by him. Go rest in peace, and be certain that the spots which you have seen are in your eyes, and not in the sun.'"[26] Such an incident forms an admirable commentary on the saying of the witty Fontenelle that Aristotle had never made a true philosopher, but he had spoiled a great many. The position assumed is simple enough: Aristotle _must_ be right, therefore whatever does not agree with the doctrines of the Stagirite must be wrong. Are your facts against him, then revise your facts. Come what may of it, you must quadrate knowledge with accepted system. Here is the theological method in a nutshell. And the theological method has only too often been the method also of the established philosophic schools.

[Footnote 26: History of Philosophy, vol. ii, pp. 95, 96.]

In our own relations with these Idols of the Theater the first and last thing to remember is that all systems are necessarily partial and provisional. "They have their day and cease to be," and at the best they only mark a gradual progress toward the truth. There can be no finality, no closing word authoritatively uttered. Our attitude toward the systems of the past and the present, toward long-accepted traditions, and dogmatically enunciated conclusions, must be an attitude of firm and steady--of respectful, it may be, but still firm and steady--independence. We must resist the tendency to passive acquiescence, and endeavor to combine with generous hospitality to all ideas the habit of not accepting anything merely because it is stated _ex cathedra_, or is backed by an influential name, or can "plead a course of long observance for its use." Perhaps to wean ourselves from this particular form of idolatry there is nothing so helpful as a wide and constant study of the history of thought. The pathway of intellectual development is strewn with outgrown dogmas and exploded systems. How fatuous, then, to accept, whole and untested, the doctrine of any master, new or old, believing that his word will give us complete and undiluted truth!

* * * * *

So much, then, we may say with Bacon "concerning the several classes of Idols and their equipage, all of which must be renounced and put away with a fixed and solemn determination, and the understanding thoroughly freed and cleansed; the entrance into the kingdom of man, founded on the sciences, being not much other than the kingdom of heaven, whereinto none may enter except as a little child." It may perhaps be urged that the result of such a survey as we have taken of the obstacles to clear thought is to leave the mind dazed and discouraged, partly because the suggestions made for the conquest of these obstacles, though easily formulated in theory are difficult and sometimes impossible in practice, and partly because the general if not expressed tendency of our analysis is (it may be said) in the direction of that Pyrrhonic skepticism which "doomed men to perpetual darkness." To the former objection I have only to reply that it is one to which all discussions of the principles and problems of conduct are necessarily open. "If to do were as easy as to know what were good to do, chapels had been churches, and poor men's cottages princes' palaces."[27] None the less, to state as lucidly as we can what were good to do under certain circumstances is properly regarded as part of the business of ethics. The other point is touched upon by Bacon himself in words which it would be impertinent to seek to better: "It will also be thought that by forbidding men to pronounce and set down principles as established until they have duly arrived through the intermediate steps at the highest generalities, I maintain a sort of suspension of the judgment, and bring it to what the Greeks call _acatalepsia_--a denial of the capacity of the mind to comprehend truth. But in reality that which I meditate and propound is not _acatalepsia_, but _eucatalepsia_; not denial of the capacity to understand, but provision for understanding truly; for I do not take away authority from the senses, but supply them with helps; I do not slight the understanding, but govern it. And better surely it is that we should know all that we need to know, and yet think our knowledge imperfect, than that we should think our knowledge perfect, and yet not know anything we need to know."

[Footnote 27: This quotation is _not_ from Bacon.]

MATHEMATICS FOR CHILDREN.

BY M. LAISANT.

Except with persons having specially favorable surroundings, I believe that the vast majority of parents have a feeling of dread at the thought of putting their children to the study of mathematics. They know that the child must learn something about it in order to pass his examinations; but with this knowledge goes an apprehension of loading his mind with those ideas which are so complicated and hard to acquire, and we put off the dreaded moment of setting him to work as late as possible.

While I believe it is wise to spare the child all useless overwork, I am persuaded also that the best way of sparing him is not to shrink from initiating him into hard work, if that can be done in a rational way.

I regard all the sciences as, at least to a certain extent, experimental, and, notwithstanding the views of those who would regard the mathematical sciences as a series of operations in pure logic, resting upon strictly ideal conceptions, I believe that we may affirm that there does not exist a mathematical idea that can enter our brain without the previous contemplation of the outer world and the facts it offers to our observation. This affirmation, the discussion of which now would carry us too far, may help to a clear idea of the way we should try to convey the first mathematical ideas to the mind of the child.

The outer world is the first thing the child should be taught to regard and concerning which he should be given as much information as possible--information which he will have no trouble in storing, we may well believe, and from this outer world the first mathematical notions should be borrowed; to these should succeed later an abstraction, which is less complicated than it seems.

Our primary teaching of arithmetic now follows in the tracks of that of grammar, as we might as well say that the teaching of grammar follows in the tracks of that of arithmetic. That is, in either case we teach the child a number of abstract and confusing definitions which he can not comprehend, imposing on him a series of rules to follow under the pretext of giving him a good practical direction, and we force him to learn and memorize these rules whether they are good for anything or not.

When the child has grown older he is given two or three short lessons a week in science, nine tenths of which, with his fleeting memory, he forgets before the next week's lessons come on. He can not relish anything that is taught him in that way, and it would be vastly better to give him no scientific ideas at all than to scatter them around in such a way, for all teachers agree that a fresh pupil is more easily dealt with and can be taught more satisfactorily and thoroughly than one who has been mistaught.

When the student has passed through it all and has established himself in life he is apt to look back upon his experiences under such teachings in no very amiable mood, and to regard such matters in the light of barriers that were set up to prevent his getting his diploma with too little work; and even if his profession is one that calls for applications of mathematics he prepares himself with sets of formulas that enable him to dispense with the imperfect instruction he has received.

When we think of giving a child a mathematical education we are apt to ask whether he has special aptitudes fitting him to receive it. Do we ask any such questions when we talk of teaching him to read and write? Oh, no! we all acknowledge that reading and writing are useful, practical, and indispensable arts, which every human being not infirm or defective should learn. Now, elementary mathematics, which represents a tolerably extended equipment, is no less useful and indispensable than the knowledge of reading and writing, and I assert further, what may seem paradoxical to many, that it can be assimilated with much less fatigue than the earliest knowledge of reading and writing, provided always that instead of proceeding in the usual way and giving lessons bristling with formulas and rules, appealing to the memory, imposing fatigue, and producing nothing but disgust, we adopt the philosophical method of conveying ideas to the child by means of objects within reach of his senses. The teaching should be wholly concrete and applied only to the contemplation of external objects and their interpretation, and the instruction should be given continually, especially during the primary period, under the form of play. Nothing is easier than this, then, in arithmetic; for instance, to use dice, beans, balls, sticks, etc., and by their aid give the child ideas of numbers.

Do we do anything of this kind? When I was taught to read and write I knew how to write the figure 2 before I had any idea of the number two. Nothing is more radically contrary to the normal working of the brain than this. The notion of numbers--up to 10, for example--should be given to the child before accustoming him to trace a single character. That is the only way of impressing the idea of number independently of the symbol or the formula which is only too ready to take the place in the mind of the object represented by it.

When a child has learned to count through the use of such objects as I have mentioned he may be taught what is called the addition table. This table can be learned by heart easily enough, but when we reach the multiplication table we come upon one of the tortures of childhood. Would it not be simpler and easier to make the children construct these tables, instead of making them learn them?

Let us first take the addition table, and suppose that we trace ten columns on suitably ruled paper, at the top of which we write the first ten numbers, for example, and then write them again at the beginning of a certain number of horizontal lines (Fig. 1). Let us suppose, too, that we have a box divided into compartments arranged like the squares in our table, into which we put heaps of balls, beans, or dice corresponding to the numbers indicated in the table. The child will take, for example, two balls from one compartment and three from another, will put them together and place his five balls in the case corresponding with the point where the lines of two and three will meet, and will thus gradually accustom himself to the idea that two added to three are equal to five, four and two to six, etc., before he knows how to write the corresponding figures. As soon as he has learned how to write them he can himself make the table with figures (Fig. 2), showing that one and one make two, one and three four, etc.

This will be all the easier for him because he will only have to write the figures in their order in the lines and the columns. This furnishes an excellent writing exercise after the children have begun to write figures, and affords besides a certain method of teaching them the addition table up to nineteen at least. I insist that all this can be done even before the child knows how to write the figures by means of an arrangement like a printer's case, and that it will be as a play, rather than a study, to the child. Hardly anything more will be required than to bring the toy to the child's notice and leave him to himself after he has been started with it, and he will get along the faster the less he is bothered.

A similar process may be adopted with the multiplication table. With a case like the other, it is only necessary to tell the child that if he wants to know how much are three times four he has only to make heaps of four things each, take three of them and put them in the box at the intersection of the line three and the column four. If he can write the figures he will write 12, instead of gathering up the twelve objects that represent the product. When he has played at this for some time he may become acquainted with all the products up to ten times ten or beyond without having to make any abnormal effort of memory.

The idea of numeration, which is usually put off till a later period, should also be given at the beginning. Children soon understand the decimal numeration and learn to write 10 for ten, and other numbers composed of one of the nine ciphers and zero. But the fact which, however, though quite essential to know, receives very little attention is that there is nothing particular about this number ten, and that systems of numeration can be devised resting on any basis that may be taken; that the principle of every system of numeration consists in taking a certain number of units and grouping them. Take, for example, a system having five as its basis. All the numbers of such a system can be represented with the figures 1, 2, 3, and 4, the symbol 10 standing in this case for five. To construct a number we have only to group the units by fives and observe the result.

To learn decimal numeration by this process we put tens of objects into little boxes, tens of little boxes into larger ones, and so on. The child can in this way acquire an exact idea of the units of successive order in any system that may be desired.

This method of teaching was developed in a remarkable way about thirty years ago by Jean Macé in a little book entitled _L'Arithmétique du Grand-Papa_--Grandpa's Arithmetic--which made some impression when it appeared, but has been substantially forgotten.

In this method I attach much importance to giving these exercises a form of play. I believe that nothing in primary instruction should savor of obligation and fatigue. It would, on the other hand, be better to try to induce the child to desire himself to go on, and it would always be well to try to give him the illusion, in all stages of instruction, that he is the discoverer of the facts we wish to impress upon his mind.

We need not stop with arithmetic, but may go on and give the child a little geometry. To accomplish this we should give him the idea of geometrical objects, and to some extent their nomenclature, and this can be done without causing fatigue. To accomplish this he should be taught to draw, however rudely. He can begin with straight lines, of which he soon learns the properties; then, when he has drawn several lines side by side, he will learn that they are parallels and will never meet. He will learn, too, after he has drawn three intersecting lines, that the figure within them is called a triangle, that the figure formed by two parallel lines meeting two other parallels is a parallelogram, and he can go on to make and learn about polygons, etc (Fig. 3). All this nomenclature will get into his head without giving abstract definitions, but in such a way that when he sees a geometrical object of definite form he will recognize it at once and give it the name that belongs to it.

In the practical matter of the measurement of areas we convey immediate comprehension as to many figures without special effort, provided we do not present the demonstration in professional style, limiting ourselves to making the pupil comprehend or feel things so clearly and definitely that it shall be equivalent, as to the satisfaction of his mind, to an absolutely rigorous demonstration. At any rate, he will be better provided for the future than by rigorous demonstrations that he does not understand. Taking the parallelogram, for example, let us suppose a figure made like Fig. 4, and we saw through it along the lines _A A'_ and _B C_. It does not need a very great effort of attention to recognize, experimentally if need be, that the two triangles _A A' D_ and _B B' C_ may be placed one upon the other and are identical. If, from the figure thus formed, we take away the right-hand triangle the parallelogram will remain; if we take away the other triangle a rectangle will be left, or a peculiar parallelogram, of which also we give the idea to the child as a figure in which the angles are formed by straight lines perpendicular to one another. Here, then, the child gains the notion of the equivalence of a parallelogram and a rectangle of the same base and height; and this notion, obtained by cutting up a piece of board or pasteboard, he will carry so seriously and firmly in his head that he will never lose it. By cutting the same parallelogram in two, along a diagonal _A C_, it may be easily shown that the two triangles can be placed exactly one upon the other, and that, consequently, they have equal areas. These lessons constitute a series of classical theorems in geometry which the child can try with his fingers and learn without even giving them the form of theorems. I might show the same as to the area of the trapeze and with many other theorems, but my purpose is only to present as many examples as will make my idea understood, without going into details.

Yet I can not leave this subject without showing how we can make a very child understand some of the geometrical theorems that have acquired a bad reputation in the world of candidates for degrees, including even such as the _pons asinorum_ of Pythagoras; the demonstration, that is, that if we construct the triangles _B_ and _C_ on the sides of a right-angled triangle, their sum will be equal to the square _A_ constructed on the hypotenuse. The usual demonstration of this theorem is not very complicated, but there is something tiresome, artificial, and hard in it. The demonstration I propose is almost intuitive, and the reasoning of it is both simple and rigorous.

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.

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.

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

n (n + 1)
S = ----------
2

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.

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.

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?"

[Footnote 28: Present Condition of Sociology in the United
States. Ira W. Howarth. Annals of the American Academy,
September, 1894.]

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."

[Footnote 29: Fairbanks. Introduction to Sociology, p. 1.]

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]

[Footnote 30: See for the following: H. H. Powers. Terminology
and the Sociological Conference, in Annals of the American
Academy, March, 1895.]

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.

[Footnote 31: See Giddings. Principles of Sociology, p. 29.]

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."

[Footnote 32: Lester F. Ward. Purpose of Sociology. American
Journal of Sociology, November, 1896.]

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.

[Footnote 33: Ward. Ibidem.]