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.
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.
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.