Every mathematical invention is at first only an hypothesis that must be demonstrated, i.e., must be brought under previously established general principles: prior to the decisive moment of rational verification it is only a thing imagined. "In a conversation concerning the place of imagination in scientific work," says Liebig, "a great French mathematician expressed the opinion to me that the greater part of mathematical truth is acquired not through deduction, but through the imagination. He might have said 'all the mathematical truths,' without being wrong." We know that Pascal discovered the thirty-second proposition of Euclid all by himself. It is true that it has been concluded, wrongly perhaps, that he had also discovered all the earlier ones, the order followed by the Greek geometrician not being necessary, and not excluding other arrangements. However it be, reasoning alone was not enough for that discovery. "Many people," says Naville, "of whom I am one, might have thought hard all their lives without finding out the thirty-two propositions of Euclid." This fact alone shows clearly the difference between invention and demonstration, imagination and reason.
In the sciences dealing with facts, all the best-established experimental truths have passed through a conjectural stage. History permits no doubt on this point. What makes it appear otherwise is the fact that for centuries there has gradually come to be formed a body of solid belief, making a whole, stored away in classic treatises from which we learn from childhood, and in which they seem to be arranged of themselves. We are not told of the series of checks and failures through which[113] they have passed. Innumerable are the inventions that remained for a long time in a state of conjecture, matters of pure imagination, because various circumstances did not permit them to take shape, to be demonstrated and verified. Thus, in the thirteenth century, Roger Bacon had a very clear idea of a construction on rails similar to our railroads; of optical instruments that would permit, as does the telescope, to see very far, and to discover the invisible. It is even claimed that he must have foreseen the phenomena of interferences, the demonstration of which had to be awaited ten centuries.
On the other hand, there are guesses that have met success without much delay, but in which the imaginative phase—that of the invention preceding all demonstration—is easy to locate. We know that Tycho-Brahé, lacking inventive genius but rich in capacity for exact observation, met Kepler, an adventurous spirit: together, the two made a complete scientist. We have seen how Kepler, guided by a preconceived notion of the "harmony of the spheres," after many trials and corrections, ended by discovering his laws. Copernicus recognized expressly that his theory was suggested to him by an hypothesis of Pythagoras—that of a revolution of the earth about a central fire, assumed to be in a fixed position. Newton imagined his hypothesis of gravitation from the year 1666 on, then abandoned it, the result of his calculations disagreeing with observation; finally he took it up again after a lapse of a few years, having obtained from Paris the new measure of the terrestrial meridian that permitted him to prove his guess. In relating his discoveries, Lavoisier is lavish in expressions that leave no doubt as to their originally conjectural character. "He suspects that the air of the atmosphere is not a simple thing, but is composed of two very different substances." "He presumes that the permanent alkalies (potash, soda) and the earths (lime, magnesia) should not be considered simple substances." And he adds: "What I present here is at the most no more than a mere conjecture." We have mentioned above the case of Darwin. Besides, the history of scientific discoveries is full of facts of this sort.
The passage from the imaginative to the rational phase may be slow or sudden. "For eight months," says Kepler, "I have seen a first glimmer; for three months, daylight; for the last week I see the sunlight of the most wonderful contemplation." On the other hand, Haüy drops a bit of crystallized calcium spar, and, looking at one of the broken prisms, cries out, "All is found!" and immediately verifies his quick intuition in regard to the true nature of crystallization. We have already indicated[114] the psychological reasons for these differences.
Underneath all the reasoning, inductions, deductions, calculations, demonstrations, methods, and logical apparatus of every sort, there is something animating them that is not understood, that is the work of that complex operation—the constructive imagination.
To conclude: The hypothesis is a creation of the mind, invested with a provisional reality that may, after verification, become permanent. False hypotheses are characterized as imaginary, by which designation is meant that they have not become freed from the first state. But for psychology they are different neither in their origin nor in their nature from those scientific hypotheses that, subjected to the power of reason or of experiment, have come out victorious. Besides, in addition to abortive hypotheses, there are dethroned ones. What theory was more clinging, more fascinating in its applications, than that of phlogiston? Kant[115] praised it as one of the greatest discoveries of the eighteenth century. The development of the sciences is replete with these downfalls. They are psychological regressions: the invention, considered for a time as adequate to reality, decays, returns to the imaginative phase whence it seems to have emerged, and remains pure imagination.
IV
Imagination is not absent from the third stage of scientific research, in demonstration and experimentation, but here we must be brief, (1) because it passes to a minor place, yielding its rank to other modes of investigation, and (2) because this study would have to become doubly employed with the practical and mechanical imagination, which will occupy our attention later. The imagination is here only an auxiliary, a useful instrument, serving:
(1) In the sciences of reasoning, to discover ingenious methods of demonstration, stratagems for avoiding or overcoming difficulties.
(2) In the experimental sciences for inventing methods of research or of control—whence its analogy, above mentioned, to the practical imagination. Furthermore, the reciprocal influence of these two forms of imagination is a matter of common observation: a scientific discovery permits the invention of new instruments; the invention of new instruments makes possible experiments that are increasingly more complicated and delicate.