A lecturer on wireless telegraphy will use in the course of the hour two or three more or less contradictory conceptions of electricity. If afterward you call his attention to the inconsistency and ask him which is right and which is wrong, you will not get a very satisfactory answer. He does not know and obviously does not care. You insist upon his telling you which theory he personally believes in. He really had not thought of "believing" in any of them. If he uses white chalk on the blackboard in preference to red, it is not because he denies the existence of red chalk and its occasional usefulness. So, too, the astronomer will speak of the sun's rising and in the next breath of the earth's turning toward the sun, quite innocent of his inconsistency. The botanist alludes to a certain flower as a poppy and again as Eschscholtzia. He means the same thing but is using different languages; in the first case English, in the second case I don't know what.
It is eminently desirable that people should have faith in science, but in order to do that they must have the same sort of faith in it that the scientist has. Otherwise they will regard it as a lot of ingenious fancies which are proved false by each succeeding generation. Science is moulting just now and looks queer. The public ought to understand clearly that the process means growth and not disease. There is another reason now for the popularization of the scientific mode of thought. It is beginning to be applied where entirely different conceptions have so far prevailed—to art, ethics, religion, sociology, and the like. This is already arousing a great commotion and will cause more before the process is complete. It will, for example, involve the rewriting and to a large extent the reinvestigation of history. Poincaré has hinted at this in a passage which seems to me of very great significance:
Carlyle has somewhere said something like this: "Nothing but facts are of importance. John Lackland passed by here. Here is something that is admirable. Here is a reality for which I would give all the theories in the world." Carlyle was a fellow countryman of Bacon, but Bacon would not have said that. That is the language of the historian. The physicist would say rather: "John Lackland passed by here. That makes no difference to me for he never will pass this way again."—"Science and Hypothesis", p. 102.
The aim of science is prevision, and I believe that this will eventually be recognized as the true aim of all knowledge. The historian, or let me say rather the antiquarian, for the historian may have the scientific temperament, values facts for their rarity. The scientist values facts for their commonness. A unique fact, if there be such, would have no possible interest to him. The antiquarian goes about looking for things, facts, or furniture, which have been of importance in the past. The scientist is looking only for things that will be of importance in the future.
According to Poincaré, the proper choice of facts is the first duty of the scientist. He must be able to pick out the significant and reject all the rest. "Invention consists in avoiding the constructing of useless combinations and in constructing the useful combinations which are in infinite minority. To invent is to discern, to choose." It is most desirable to bring together elements far distant from one another. Such unions are mostly sterile, but when this is not the case, they are the most fruitful of all. The successful scientist does not, like a shopper, look over one by one all available samples and pick out what he wants. Life is too short. The unsuitable ideas do not even present themselves to his mind. It is as if he were an examiner of second resort who only concerns himself with the candidates who have passed the first test. This preliminary sifting and sorting process is done largely by the unconscious mind, as Poincaré shows by telling how he came to make his first mathematical discoveries:
For a fortnight I labored to demonstrate that there could exist no function analogous to those that I have since called the fuchsian functions.[3] I was then very ignorant. Every day I seated myself at my work table and spent an hour or two there, trying a great many combinations, but I arrived at no result. One night when, contrary to my custom, I had taken black coffee and I could not sleep, ideas surged up in crowds. I felt them as they struck against one another until two of them stuck together, so to speak, to form a stable combination. By morning I had established the existence of a class of fuchsian functions, those which are derived from the hyper-geometric series. I had merely to put the results in shape, which only took a few hours.—"Science et Méthode", p. 52.
After working out the deductions from this discovery, he went on a geological excursion of the School of Mines. The distractions of travel took his mind from his mathematical labor. But at Constance, just as he was stepping into an omnibus for some excursion, the idea occurred to him, without any connection with his previous thoughts, that his fuchsian functions were identical in their transformations with those of the non-Euclidian geometry. He took his seat in the omnibus and continued his conversation, feeling absolutely certain of his discovery, which he worked out at his leisure on his return to his home at Caen.
He next devoted himself to the study of arithmetical questions, without reaching any results of importance and without suspecting that this subject could have the slightest connection with his earlier researches. Disgusted at his lack of success, he went to pass some days at the seashore, where he was occupied with other things. One day as he was walking on the cliff, the thought came to him, brief, sudden, and certain as usual, that he had been employing the same transformations in his arithmetical and geometrical work.
He thereupon went back to Caen and undertook the systematic application of his theory. But he was stopped by an insurmountable obstacle, and while in this perplexity he was called away to his military service at Mont-Valérien, where he had no time for mathematics. One day while walking on the street, the solution of the difficulty appeared to him in a flash. He did not try to think it out at the time, but after his release from the army, he completed his memoir without trouble.
These fascinating glimpses into the soul of a mathematician will remind the reader of many other instances of such subconscious assistance on record and doubtless of personal experiences as well. We think of Alfred Russel Wallace at Ternate, his brain inflamed with tropical fever, seized with the sudden inspiration of the theory of natural selection, the key to the biological problems which had perplexed him for so many months. How fortunate that his clerical opponents did not know of this and so could not dismiss evolution as the dream of a diseased imagination. But as James says in his "Varieties of Religious Experience", we have no right to discountenance unwelcome theories as feverish fancies, since for all we know 102° may be a more favorable temperature for truth to germinate and sprout in than the ordinary bloodheat of 98°.