Thinking and learning to think

by Nathan Christ Schaeffer

Images.

Higher forms of thinking.

Technical terms are very helpful in dealing with that which cannot be imaged or visualized. When Francis Galton began his inquiries into the power possessed by different minds to conceive the breakfast table, to recall vividly the various dishes and the way in which they are placed upon the table, many men of scientific habits of thought declared that there is no such human faculty. On the other hand, the educational reformer whose early training did not make him familiar with the thought-processes of higher mathematics may honestly declare that he cannot conceive an abstract number, and, as a matter of course, he can have no adequate conception of the value of the higher forms of thinking in symbols. Dr. W. T. Harris has well said that the mind can think ideas which cannot be pictorially conceived or made to stand before the mind in thought-images. In thinking this class of ideas, technical terms are indispensable as instruments of thought.

Symbols classified.

Suggestive symbols.

The value of technical terms as instruments of thought is seen in a still clearer light if we try to classify the various uses of the signs and symbols which are employed as aids in thinking. Many of these have no office beyond that of suggesting the things or ideas for which they stand. To this class belong the marks which suggest to the tramp a cross dog or a good meal. As soon as he has seen them, they could be erased; the train of thought which they started in his mind can go on without them. Of a similar character are the devices by which the merchant marks the buying and the selling prices of goods, the red and blue lights used on railways and ocean steamers, the secret signs and signals employed by the signal corps of an army, and the steps, grips, signs, countersigns, and passwords employed by secret societies as a means of identification. Very many of the artificial devices used in systems of mnemonics have no higher function than that of suggesting what otherwise might be forgotten.

Symbols as substitutes.

Very different are the signs and symbols which mathematics employs as substitutes for the quantities to be considered. In adding a column in the ledger or in a statistical table the mind thinks the figures without reference to the concrete objects which they denote. In the solution of a problem in algebra the unknown quantities are represented by symbols like x and y, the known quantities by the first letters of the alphabet or by numerical expressions; the relations between the quantities are indicated by equations; there is no thought of the quantities themselves while the mind is engaged in manipulating the symbols according to well-defined rules of operation, and only when the result is to be interpreted do the quantities reappear in the field of consciousness. The substitute symbol is a device for temporarily dropping an idea until it is needed for interpretation; the suggestive symbol is a means of bringing an idea or thought into the domain of consciousness. The latter furnishes or recalls material for the mind to act upon; the former lightens the burden which the mind would otherwise have to carry. The arithmetical solution of an age question in which the mind constantly carries the thought of A’s age and his wife’s age as compared with the algebraic solution of the same question in which A and his wife, as well as their ages, sink temporarily out of sight, shows the value of substitute signs and symbols in mathematical thinking, and explains why algebraic methods are so far superior to the clumsy and involved methods of arithmetical analysis.

Expressive symbols.

Different from either of these is the class of symbols used in expressing ideas. This class includes not only the words of written and spoken language, but also the natural signs of gesture language and the conventional signs of manual language taught to deaf mutes. The language is full of faded metaphors indicating the office of common words. They are said to express meaning, to convey thought, to embody ideas, to enshrine content. They may be likened to window-panes through which one sees what is back of them. Sometimes the window-panes, like spectacles when first worn, attract more attention from the person looking than the objects seen through them,—a parallel to what occurs when the articulate speech, or its rhetorical adornment, attracts more attention than the thought expressed. But if that which is seen through the window-pane is on the order of a Santa Claus loaded with toys and Christmas-gifts, then no notice is taken of the medium through which the object is seen. Hence the very best teaching—that which rivets attention upon the thought conveyed—always fails to teach the spelling of words incidentally. Furthermore, the instruction which frequently stops to draw attention to the grammar of the sentences, the spelling of the words or their mode of utterance, interferes with the formation of logical habits of thinking and divests the words of their function as expressive signs. When the word itself becomes an object of thought the mind is not thinking by means of that word. It has been well said that we may fail to apprehend the meaning of what a person is saying because the tone of his voice arrests our attention through its resemblance to that of some one else in whom we feel an interest; that so far as signs thus attract notice on their own account, they fail to fulfil their function as a means of attending to something other than themselves. For this very pertinent observation credit is due to Mr. G. F. Stout, who (“Mind,” lxii. page 18) has very clearly drawn the distinction between the three classes of signs or symbols used as helps in thinking. He says,—