Thinking and learning to think - Nathan C. Schaeffer (romantic story to read .txt) 📗
- Author: Nathan C. Schaeffer
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Full knowledge implies a basis upon which it may rest. There may be sufficient ground for the firm belief which constitutes the essence of knowledge even when the truth cognized is incapable of full and complete demonstration.
It is natural for a child to believe. The statements of others are accepted as true without question, so long as the child has not been deceived by others. Hence many teachers have assumed that their chief function is to ask the reason why, so that belief in what is true may be based upon sufficient ground, and that nothing shall be accepted as true until it is proved. This was one of the erroneous views under which Pestalozzi labored. He justified the undue attention paid to mathematics in his school on the ground that he wished his pupils to believe nothing which cannot be demonstrated as clearly as two and two make four. Whereupon Père Girard replied, “In that case, if I had thirty sons I would not intrust one of them to you; for it would be impossible for you to demonstrate to him, as you can that two and two make four, that I am his father and that I have a right to his obedience.”[47]
The progress of a pupil may be hindered by too much emphasis upon the ground of knowledge. The human mind cannot make an exhaustive study of very many things. Exhaustion is a term applied by logicians to a method of proof in which “all the arguments tending to an opposite conclusion are brought forward, discussed, and proved untenable or absurd, thus leaving the original proposition established by the exclusion of every alternate.” Speaking positively, we may say that exhaustive study of a subject explores it in all its bearings and relations as well as in its nature and essence. In every subject the known is bounded by the unknown; new methods of preparation and investigation constantly reveal novelties in whole classes of objects which it was supposed had been studied exhaustively. The specialist seeks to know all that has been brought to light in his field of research, and to push out the limits of knowledge beyond the goal reached by his predecessors. The thoroughness of the specialist is not required in elementary instruction. The writer knows of a teacher who for an entire term kept a class of boys at work upon highest common factor and least common multiple on the plea that they did not thoroughly understand these subjects. No better plan of disgusting boys with arithmetic and algebra could have been devised. Thorough knowledge of these two subjects involves reasoning and demonstrations more difficult to grasp than half the theorems in Euclid. Instead of aiming at exhaustive treatment, the true teacher is satisfied with knowledge adequate for the subsequent work of the course. If the pupil has reached the stage where he can appreciate the reason why, it may be (though it is not always) wise to raise this question, and to insist on a comprehension of the proof. Very often the mind has enough to do in trying to see how; the question why then interferes with the mastery of the mechanical operations. Let any adult take up a system of arithmetic with which he is unfamiliar, say the arithmetic based on counting by fives, or by twelves, or by thirties (each of the last two, mathematically speaking, better than the arithmetic based on tens), he will soon find it is work enough at first for his intellect to perform the operations of adding, subtracting, multiplying, and dividing without reference to the philosophic explanations which exhaustive study would require at every step in the operations.
Descartes applied several of the technical terms of optics to the science of mind, and in this he has been followed by Locke, Leibnitz, and others. An object seen at a great distance or in insufficient light looks obscure; as the eye approaches, or as the dawn increases, the object, as a whole, becomes clear enough to be distinguished from other objects, although its constituent parts are still confused. Increasing light or a nearer approach finally enables us to discern the parts, and the vision of the object grows distinct. Clear vision occurs where the object, as a whole, can be recognized; distinct vision occurs when the parts of the object seen can be recognized. In like manner ideas are said to be clear as distinguished from obscure, when they are discerned in outline; they are distinct (opposed to indistinct or confused) when they are discerned in their elements or constituent parts. Distinct mental vision requires analytic and synthetic thinking.
Of many objects the mind needs only clear knowledge for ordinary purposes. One may distinguish two brothers by the total impression of each which he carries in his mind, and yet be totally unable to tell any specific marks by which he knows the one from the other. The painter, on the other hand, cannot be satisfied with this total impression; he studies the individual features until he has a distinct impression of their likenesses and differences.
Of the map of one’s own country it pays to know the States and Territorial divisions. Of one’s State, a knowledge of the counties, and of one’s county, a knowledge of the townships may be helpful. For specific vocations more minute knowledge may be desirable. Each individual mind can well afford to stop with a measure of geographical knowledge that is adequate for the duties of his vocation and the purposes of his reading of books and newspapers.
Very little of our knowledge of geography is based upon experience; most of it rests upon testimony. The eye at a glance may take in the outlines of an island of the Susquehanna river. The fact that Great Britain is an island rests upon the testimony of maps; our belief is based upon what we have always heard and read, and is further strengthened by the absence of testimony to the contrary. If the fact had ever been questioned, the mind might hold its judgment in suspense until sufficient ground was found to warrant a conclusion.
When the knowledge which a pupil has is to be deepened or made more distinct a series of well-chosen questions may beget the required thinking. For instance, let us take the case of a pupil who has reached the stage where his knowledge of the properties of the parts of speech should be made more complete. Let the teacher ask for the difference between a pencil and a part of speech, between a noun and a name, between gender and sex, between number in grammar and number in arithmetic, between person in grammar and a person like the President of the United States, between case in grammar and a case in division of fractions, between tense and time, between mode and manner, between action and a verb, between the object of an action and the object of a verb. Comparison will soon show the inaccuracy of the statement that the direct object of an action is in the accusative case; and the learner will see that case is a property of nouns, not of objects, and cannot be predicated of the object of an action, but of the word which denotes the object of the action, which word may be either in the nominative or the accusative case as the verb is either in the passive or active voice. Comparison will lead the pupil to see clearly that gender is a property of nouns, whereas sex or the absence of sex is predicated of that for which nouns stand. Comparison will serve to bring out the distinction between number in grammar as a property of nouns indicating one or more than one, and numbers in arithmetic, of which there are as many as there are units or collections of units in the universe. Thinking by comparison will lead to the detection of similarities and differences, to discrimination, combination, and generalization, and through these to more distinct and more adequate knowledge.
Questions which draw attention to likenesses and differences, to causal relations and logical sequences, stimulate analysis and comparison; the resulting judgments clarify the stream of thought and push the boundary of knowledge into the regions of the hitherto unknown.
The greatest minds when working under the influence of a false theory fail to arrive at truth. Socrates rejected the view of Anaxagoras that the sun is a fire, because we can look at a fire, but not at the sun, because plants grow by sunshine and are killed by fire, and because a stone heated in fire is not luminous, but soon cools, whereas the sun always remains equally hot and luminous. Newton did more than all other thinkers combined to make astronomy a science; his discoveries in physics and mathematics rank him among the greatest investigators the world has thus far known; yet he spent many nights trying to find the method by which the baser metals could be transmuted into silver and gold; his researches as an alchemist led to nothing, because he was working under the spell of a false theory.[48]
Faraday acknowledged that he was often compelled to give up his preconceived notions, and in some cases his failures are almost as instructive as his discoveries. It was characteristic of him to hold to his theories until he proved them either true or false, and he was ever ready to reject any hypothesis as soon as he found it inconsistent with the laws of nature. Newton was willing to suspend judgment for years upon his theory of gravitation, until more accurate measurements of the earth’s size and the moon’s distance showed his theory and calculations to be right. Socrates advised his followers to quit the study of astronomy, probably because he felt that in his time the data were not sufficient to warrant definite conclusions. Hosts of instances can be cited showing that the thinking of the strongest intellects does not issue in knowing when it is based upon or biassed by a wrong working hypothesis. And yet it must be confessed that wrong hypotheses may lead to valuable negative results, as in the case of Kepler’s investigations, each exploded theory making room for the construction of a theory more in accordance with the facts. The superiority of men of genius lies in their love of truth and fidelity to fact; in the facility with which they construct theories to account for observed phenomena; in the patience with which they test theory by fact, and in the readiness with which they reject every hypothesis as soon as it is found to be in irreconcilable conflict with well-established facts. The average life of a theory in science is said to be only ten years. The average would be lower still if all rejected theories had been put into books. The men possessed of a truly scientific spirit differ from ordinary men not only in the painstaking accuracy of their observations and in the surprising fertility with which they frame theories, but also in the habit of verifying every hypothesis until there is sufficient ground to establish its truth and to receive it as an addition to the sum total of human knowledge.
The common people are quite as ready to frame theories as the scientists and philosophers. It would be well if they were equally patient in testing their theories and in verifying their suppositions. The human mind cannot help generalizing. The moment a child uses a common noun it begins to classify. Its tendency to pull things to pieces and to put them together again are exhibitions of the mind’s tendency to treat everything by analysis and synthesis.
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