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Logic Taught by Love/Chapter 17


"How beautiful are the feet of them that make Peace.'"

We have traced some of the harm that is done by prejudiced rejection of new truth; we have also glanced at the far greater harm that may be done by over hasty acceptance of it. We have now to consider the question:—What are the steps which might be taken with a view to prepare the masses to receive in an orderly manner whatever new truths about the Pulsation of Life may in future be revealed; so as to secure the greatest amount of good with the least harm? I do not propose to speak of any wholesale measures which need governmental action before they can take effect; but only to suggest possible endeavours for individuals. The present Chapter will relate to the Art of Critique, and address itself chiefly to those who have some experience of the work of teaching elementary mathematics. Any teacher of elementary mathematics might make his class work a real training in the Art of true Critique; and thereby not diminish but increase his success in his own proper subject.

Let us think for a few moments of the whole reading public as a College, which Truth, as Head-Master, is endeavouring to instruct and educate; and of men of Genius, discoverers, reformers, as assistant-masters, to each of whom is committed the task of teaching the subject of which he knows most. The comparison would be an impertinent one, if we illustrated it by reference to classes in such subjects as History or Language; because the school-teacher in such subjects is obliged to require from his pupils a kind of docility which readers are not expected to give to an author. But every mathematical teacher who deserves the name endeavours to accustom his pupils to take nothing for granted till it has been proved to their own personal satisfaction. The mathematician, therefore, is related to his class in much the same way as a writer to his readers. In my imaginary College, the various teachers are in the habit of attending each other's classes whenever they wish to do so; a practice which actually does prevail to a certain extent in some schools, and with the happiest results; for the teacher who takes his place among the pupils of a colleague, can, if he will, set the class an excellent example of courteous and intelligent questioning. But if the various teachers thought of each other's work in the same spirit as the leaders of Literature and Science do, and if the pupils thought of their teachers as the reading public do of writers and preachers, the result would be not only moral confusion, but intellectual chaos.

The chief reason why courtesy, reverence, and a certain kind of docility are needful for those who would learn, is this:—Truth is never received into the human mind without an admixture of conventions, of what may be called fictions. These fictions have to be introduced, used, and then withdrawn. It would be impossible to teach even so straightforward a subject as mathematics without the temporary use of statements which are not true to the nature of things. The history of a child who is learning mathematics, like that of human thought, is very much a record of alternate introduction of convenient fictions and subsequent analysis of their true Nature. A class, like a public, tends at times to become groovy and mechanical; to mistake the accidental for the essential; to treat necessary aids to learning as if they were actual truths; to lose sight of the relative importance of various kinds of information. A class in Botany tends to forget that classification and terminology are not so much part of the life of plants as circulation and fertilization; a class in Analytical Geometry forgets that the co-ordinates are no part of a curve. Just so, the reading public forgot, till Charles Darwin woke it up, that intermittence is no necessary part of Creative Action; although it is convenient for man, for purposes of classification, to imagine a series of intermittent acts. A student tends to such forgetfulness in proportion as he becomes mechanical in his work; the genius of a teacher is very much shown by the manner in which he contrives to arouse the interest and correct the errors of a class which is becoming too mechanical.

Theorists in education sometimes imagine that a good teacher should not allow the work of his class to become mechanical at all. A year or two of practical work in a school (especially with Examinations looming ahead) cures one of all such delusions. Education involves, not only teaching, but also training. Training implies that work shall become mechanical; teaching involves preventing mechanicalness from reaching a degree fatal to progress. We must therefore allow much of the actual work to be done in a mechanical manner, without direct consciousness of its meaning; an intelligent teacher will occasionally rouse his pupils to full consciousness of what they are doing; and if he can do so without producing confusion, he may be complimented and his class congratulated.

Let us now go into the subject more in detail. We teach laws of curves by reference to certain straight lines—tangents, co-ordinates, radii, etc. These lines bear the same kind of relation to the curves which the framework of sticks fastened into a pot bears to the climbing plant, which is the true object of the gardener's care. The plant itself is living and growing; the justification for the existence of the framework consists in the fact that it would be impossible to get the true enjoyment of the plant without its aid. The coordinates form no part of what we want to teach about; but we cannot learn without their help. They enable us to see how the curve came into being, and whither it is tending.

Suppose then that a class, while becoming skilful in working problems, seems to have forgotten that the axes are no part of the curve itself. The teacher may wake it up by saying, "You don't imagine, surely, that axes form any necessary portion of a curve. An ellipse, for instance, is the path of a body moving round a focus of attraction. Suppose a planet were endowed with the power of leaving a track in the sky, the track would be an ellipse only, unencumbered with axes. What are the axes, then? Indications of invisible forces? Not so. No line of attraction at any point of the orbit corresponds to the minor axis. The axes are human devices to enable us to measure and express the various elements of the orbit." "Well, but," exclaims perhaps some clever pupil, "if the straight lines are unreal, if they mean nothing, why were they invented, and why were we made to study them?" Such reasonable criticism is a great help to the teacher. He proceeds to picture the state of Astronomy in the days when nothing was known of the movement of the planets. He describes the first bewildered effort of the human mind to represent to itself the path of these wanderers. He shows how some man may have at last conceived the brilliant idea of projecting imaginary straight lines across the sky from one fixed constellation to another, thus forming a sort of background of measuring-rods; how the constellations, with these imaginary connecting lines, might be copied on a tablet, and the path of the planet registered thereon from day to day; and how Science might grow up by man inventing modes of measuring and registering curves which the living forces of Nature were describing in Space.

After such a lesson, the class goes on with its work with renewed interest and quickened intelligence. But how would the case be if a group of other teachers were present, who should comment on the lesson in this wise? One says "All the good books present curves with axes; you think yourself cleverer than our best writers." A second says, "It is perfectly crazy to attempt to teach people to think of curves without axes." A third says, "I really do think there is something in your view; but this much is clear: when you accepted a post in this College, you undertook to use a standard text-book, and to teach on ordinary lines. If you have theories different from those generally accepted, you were at least bound to give up your post before you began to express them; then we could have respected you. To stop here and to suggest doubts to the pupils is dishonest and scandalous; and your conduct makes me doubt the honesty of everything you say." A fourth says, "You are right, and all the books are wrong. It is a shame that children should be taught to believe fictions; away with these stupid books that have misled the world so long." A fifth is indignant that the grand old sages who created Astronomy should be accused of inventing falsehoods to serve their own ends; the sixth, on the contrary, is angry that the time of the class should be wasted on listening to, and trying to follow, the mental History of a set of savages who lived before Analytical Geometry had become a Science. And some clever young lecturer, who happens to have heard that our solar system, as a whole, is in motion, and who fancies that such knowledge is his own peculiar property, triumphantly asserts that the earth-path is not an ellipse, but an elliptical spiral; and that any statement based on the premise that the ellipse represents a planet-path must be false throughout. Now what chance would any of us have of teaching anything to a class subject to such interruptions of the normal current of thought? In a College where such disorder prevailed, would the pupils be in a frame of mind to receive instruction? Would the teachers themselves be likely to preserve the calmness necessary for the investigation of Truth? That the picture I have drawn is no exaggerated one, that those whose mission is to arouse the public to a perception of the relation between the essential and the accidental have to run the gauntlet of a style of criticism as senseless and frivolous as I have represented, no one can doubt. We are all aware of the absurdity of our present modes of receiving new truth; few, I fear, are sufficiently aware of its evil effects. Therefore it is well for us to reflect what effect it would have on the teaching of so simple a thing as Geometry, if teachers introduced into each other's classes the element which is so rife in our literature.

True Critique is one of the fine Arts; as sacred and beautiful as all true Art. There is no keener pleasure for a good teacher than genuine criticism from a pupil. If a question is asked which shows that the teacher has failed to make intelligible a fact perfectly clear to himself, even that is a source of great enjoyment. But when, as sometimes happens, a remark is made which proves that we have not gone deep enough into our own subject, which opens up new avenues of thought, and forces us to reconsider a demonstration, to re-investigate a solution; when, in fact, the relation between teacher and pupil is for a time inverted, it is then that that relation becomes fullest of pleasure and profit. (And so of course it is, in a still higher degree, between a writer and the critical reader.) But what chance would a pupil have of making intelligent criticisms, or of asking suggestive questions, if it were the fashion of the class to indulge in a carnival of absurdity wherein senseless accusations were flung about at random?

Let us return to the subject of Analytical Geometry.

The tangent to an ellipse is an imaginary straight line, representing the path which would be followed by the body tracing the ellipse, if its connection with the attracting focus were suddenly to cease. In its essence, the tangent is a sublime effort of the scientific imagination; it pictures the result of a sudden cessation of the action of gravity. In practice the tangent is a convenient line for indicating the curvature at any given point. The educational sentimentalists who object to mechanicalness, ought, if consistent, never to use a tangent in working a problem, without stopping to realize the grandeur of the idea involved. As a matter of fact, such incessant strain on the imagination and on the perception of the sublime is unhealthy and deadening. It is far better to use tangents, mechanically, as mere measuring-rods. But a good teacher will take care that no pupil goes through a year's work at Analytical Geometry without having been, once or twice, aroused to perceive the wonderful poetic conceptions represented by the lines he is using.

In our supposed disorderly College, a sentimental poet might take upon himself to reprove the Geometry teacher for allowing so awful a conception as the sudden cessation of gravity to be degraded by being talked of as a mere convenience, without due realization of its horror; a scientist holding utilitarian views might retort, that the function of the instructor being to impart truth, no such thing as the cessation of gravity ought ever to be mentioned before a class; because, as a matter of fact, no instance of any such event is on record.

Let us now pass to the subject of Arithmetic. Instead of fictitious lines, we have here to deal with fictitious statements, i.e., with tatements which, if treated as truths, are false, but which, when clearly understood to be mere convenient fictions, do actually convey truth. Such a statement, for instance, is, " Twelve pence equal one shilling." No one is ever really deceived by this particular statement; but that is because all are familiar with the actual coinage, and know that, as a matter of objective fact, "a shilling" is not identical with "twelve pennies" (in the sense in which "a dozen apples" is identical with "twelve apples"). But a similar statement, made about unfamiliar objects, or about abstractions, might be misleading, unless the teacher took care to prevent misconceptions. And even in the case of our familiar coinage, it is essential to good mental discipline that the pupils should occasionally be made to define carefully the sense of the word "equal," in the sentence "twelve pence equal one shilling," and have their attention directed to the fact that, if accepted as true, it becomes false; that it contains and conveys truth only while clearly understood to be a fictitious but well-arranged convention. If the teacher forgets to do this, he ought to be grateful to any one who reminds him of his omission.

Now let us suppose that, in our imaginary College, the pupils are unacquainted with the actual coinage, and the teacher not sufficiently awake to the meta-physical necessity for defining, for them, words the real meaning of which is present to his own mind. He simply states: "Twelve pennies are one shilling." A lecturer on physics may afterwards happen to make a statement, the obvious outcome of which is that twelve pennies are (in weight) equal to about twenty-one shillings; and a chemist might prove that silver and copper are not identical, and that no amount of silver can produce the medicinal properties or chemical reactions of a grain of copper. Each teacher will then have made a statement, judged by which his two colleagues will each have spoken falsely. Such a thing is quite conceivable as that a party-spirit should arise in the College; the pupils each taking the side of his favourite professor, and accusing the others of gross ignorance or of wilful per- version of truth. The unlucky mathematician might then do what he ought to have done, but forgot to do, at first, i.e., point out that he was using the word "equal" in a conventional sense; a sense perfectly legitimate for his purpose, though legitimate only within the scope of that purpose. But if party-spirit had already been aroused in the matter, and angry or contemptuous accusations flung about, the explanation would only add to the confusion; for the teacher would now be accused by the adverse party of paltering with truth, of using words in a double sense with intention to mislead; his very explanation would be quoted against him in triumph as a confession of guilt. Again I ask, what chance would the teachers in such a College have of discovering truth, or the pupils of learning what had been discovered?

A very prevalent form of criticasterism might be parodied as follows:—The teacher states a question thus: "The rate of exchange is 9½d. for a franc; how many francs shall I receive for so many shillings?" A colleague interrupts the lesson to ask what evidence there is to show that this is the exact rate. Another asserts that the last rate quoted was 9/875d. per franc; and a third insists that the last quotation was 9/876d. They engage in a vehement discussion of the point; but all agree that such gross ignorance of facts as the teacher betrays, proves him to be incompetent; and that, as the statement on which his procedure rests is proved false, his whole chain of reasoning falls to the ground.

I will close these mathematical illustrations by narrating an incident which actually occurred. An intelligent girl, who had been badly taught arithmetic, joined my class. I set her a sum about some damaged articles worth, originally, £3 1 5s. each, but which were to be sold in a lump at an abatement of one pound and some shillings on the price of each. She was required to find what would be received for the whole. It so happened that the lump sum amounted to so many pounds and fifteen shillings. She came to me saying that she could not get her sum right; the shillings were right, but the pounds were wrong. I worked it for her; beginning, of course, by subtracting the shillings of the abatement from the original fifteen shillings. "But I had the shillings right," she cried; "now you are getting it all wrong!" There was something very touching in her dismay at seeing the poor little bit of rightness, which she had secured, put wrong, and I felt like a ruthless destroyer of the last refuge of an innocent soul; but Arithmetic tolerates no sentimental hesitations; and I had, of course, to persist. At last, by trusting in blind faith to accurate Logic, we got the whole right, to her great surprise. I asked her to explain her own method. It appeared that she had first set down the original price as given in the book, and then looked at the answer in the end of the book, and, finding that the shillings there corresponded with what stood before her on her paper, she copied them straight into the "Answer" place, thinking that it would be useless (and, I am sure, feeling vaguely that it would be irreverent) to meddle with what the book pronounced to be right. She had therefore set to work to manipulate the pounds with a view to get them into accordance with the official standard! Of course she failed. Of course,—from our point of view; but from hers, it seemed that adhering to the book as long as you could must be the right thing to do! I was then able to explain to her a little about the right use of authoritative standards; to show her that, if used as a check on results, they may often be of use by revealing errors in our reasoning which might otherwise have passed unperceived; but that, if taken advantage of to spare ourselves the effort of working out our own problems at our own cost, they are generally found misleading. My pupil profited, I hope, by the lesson, in more ways than one. But what chance should I have had of teaching her either arithmetic, or the right use of standards, or anything else, if any other teacher had confused her mind, before I had half finished my explanation, by reproving me for teaching her to rebel against the authority of the text-book, and to work sums as she liked, without reference to the only infallible standard; and by warning her that I was only pandering to license and carelessness?

"Truth for ever on the scaffold; wrong for ever on the throne." The scaffold on which Truth is murdered, the throne on which wrong sits to rule, are built of careless, irreverent, senseless criticisms. When the account of the good and ill which we have done on earth is summed up, the heaviest item in the account against many of us will perhaps consist in the record of our idle words.

The aim of all students should be, so to pass through things temporal that finally they lose not the Eternal; so to pass through temporary aids to knowledge as not to miss perceiving the Eternal Truths. Only unpractical dreamers suppose that Truth can be grasped without adventitious and fictitious aids; all true students know that they are dependent on such aids; and their hope is so to use them as not to abuse them. We are in danger of becoming entangled in these adjuncts of Truth; of mistaking them for actual truths. Therefore The Unseen Wisdom which guides the destinies of mankind, raises up occasionally what we call "a Reformer"; whose function is to give to Humanity such a lesson as I have described the teacher of Analytical Geometry giving to a class. The Reformer reminds us that our framework of aids to investigation is not, in itself, Truth. As the mathematician teaches us to see the actual path of a planet, in contrast, on one hand, with the tangent conceived by imagining an impossible state of things from which the action of Gravity is suspended, and, on the other hand, with the Axes which are well-arranged measuring-rods, so the Reformer teaches us to look at the actual Life of Humanity, in contrast, on one side, with the Vision of an impossible Idea made simple by the omission of certain elements, and, on the other side, with that convenient mechanism by which the comprehension of our duty is facilitated for us.

How our Heaven-sent Reformers are received, we know: Truth on the scaffold; and, on the throne, whoever helps us to confuse convenient fiction with fact. The world is surely old enough to behave less like a class of ill-bred school-boys than it has as yet done; it might prepare itself to give to its future Seers a reception more worthy than it has given to Seers in the past. All we who are engaged in mathematical teaching may contribute to that end, if we take pains to make our pupils distinguish between Truth and the mere accessories of study. We may also do something to train the rising generation in a comprehension of the difference between true and false criticism; we may accustom them to combine the fearless loyalty which compels a learner to express frankly his real difficulties and doubts with the respectful courtesy which checks irreverent and thoughtless cavil.