Various
Appletons' Popular Science Monthly, October 1899
"The vulgar thus through imitation err;
As oft the learned by being singular."
These and similar congenital differences in men's intellectual constitutions might be illustrated indefinitely if it were necessary. A further remark of Bacon's must, however, be quoted, for it goes deeper in mental analysis and touches a less obvious point. "There is one principal and, as it were, radical distinction between different minds in respect of philosophy and the sciences, which is this: that some minds are stronger and apter to mark the differences of things, others to mark their resemblances. The steady and acute mind can fix its contemplations and dwell and fasten on the subtlest distinctions; the lofty and discursive mind recognizes and puts together the finest and most general resemblances." Men belonging to the former class we should call logical and critical; those belonging to the latter, imaginative and constructive. Each class tends to the excesses of its own predominant powers, and in each case excess interferes with calm reasoning and sound judgment.
To correct the personal equation it is imperative that we should study ourselves conscientiously, consider dispassionately the natural tendencies of our birth, early surroundings, education, associations, and interests, and do our utmost to conquer, or at least to make allowance for, every individual peculiarity, temperamental or acquired, likely to turn the mind aside from the straight line of thought. Such self-discipline every one must strenuously undertake on his own account if he would wish to see things as they really are. Stated in more general terms, our aim must be to rise above all kinds of provincialism and personal prejudice, and to overcome our natural proneness to rest content in our own particular point of view. Bacon quotes with approval the words of Heraclitus: "Men look for sciences in their own lesser worlds, and not in the greater or common world." We must strive to escape from our own lesser world, and to make ourselves citizens of the greater, common world. For this we need the widest and most generous culture – the culture that is to be found in books, in travel, in intercourse with men of all classes and every shade of opinion. Left to ourselves we only too sedulously cultivate our own insularity; we mingle simply with the people who agree with us, belong to our own caste, and share our own prejudices; we read only the papers of our own party, the literature of our own sect; we allow our own special interests in life to absorb our energies, color all our thoughts, and narrow our horizon. In this way the Phantoms of the Cave secure daily and yearly more despotic sway over our minds. Self-detachment, disinterestedness, the power of provisional sympathy with alien modes of thought and feeling, must be our ideal. "Let every student of Nature," says Bacon, "take this as a rule, that whatever his mind seizes and dwells on with particular satisfaction is to be held in suspicion, and that so much the more care is to be taken in dealing with such questions to keep the understanding even and clear." A hard saying, truly, yet one that must be laid well to heart.
While the Idols of the Tribe, then, are common human frailties in thought, and the Idols of the Cave the perturbations resulting from individual idiosyncrasies, there are other Idols "formed by the intercourse and association of men with each other," which Bacon calls "Idols of the Market Place, on account of the commerce and consort of men there." By reason of its manifold and necessary imperfections – its looseness, variability, ambiguity, and inadequacy – the language we are forced to employ for the embodiment and interchange of ideas plays ceaseless havoc with our thought, not only introducing confusion and misconception into discussion, but often, "like the arrows from a Tartar bow," reacting seriously upon our minds. A large part of the vocabulary to which we must perforce have recourse, even when dealing with the most abstruse and delicate subjects, is made up of words taken over from vulgar usage and pressed into higher service; they carry with them long trains of vague connotations and suggestions; the superstitions of the past are often imbedded in them; no one can ever be absolutely certain of their intellectual values. While, therefore, they may do well enough for the rough needs of daily life, they prove sadly defective when required for careful and exact reasoning. And even with that small and comparatively insignificant portion of our language which is not inherited from popular use, but fabricated by philosophers themselves, the case is not much better. Every word, no matter how cautiously employed, inevitably takes something of the tone and color of the particular mind through which it passes, and when put into circulation fluctuates in significance, meaning now a little more and now a little less.25 What wonder, then, that "the high and formal discussions of learned men" have so often begun and ended in pure logomachy, and that in discussions which are neither high nor formal and in which the disputants talk hotly and carelessly the random bandying of words is so apt to terminate in nothing beyond the darkening of counsel and the confusion of thought?
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.
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."
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?
Fig. 1.
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.
Fig. 2.
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.
Fig. 3.
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.
Fig. 4
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.