CML 2: On Inspiration

In this article we will examine how human beings acquire knowledge in science, mathematics, and art. Our focus shall primarily be on the formation of ideas and hypotheses in science and mathematics, since the formal nature of these subjects tends to put the phenomena we are concerned with into particularly clear perspective. We will show that the phenomenon known as inspiration plays an essential part in acquiring knowledge in modern science and mathematics and the creative arts (such as music). We will argue that the phenomenon of inspiration cannot readily be explained by mechanistic models of nature consistent with present-day theories of physics and chemistry. As an alternative to these models, a theoretical framework for a nonmechanistic description of nature will be outlined. While providing a direct explanation of inspiration, this general framework is broad enough to include the current theories of physics as a limiting case.

Modern scientists acquire knowledge, at least in principle, by what is called the hypothetico-deductive method. Using this method, they formulate hypotheses and then test them by experimental observation. Investigators consider the hypotheses valid only insofar as they are consistent with the data obtained by observation, and they must in principle reject any hypothesis that disagrees with observation. Much analysis has been directed toward the deductive side of the hypothetico-deductive method, but the equally important process of hypothesis formation has been largely neglected. So we ask, "Where do the hypotheses come from?"

It is clear that scientists cannot use any direct, step-by-step process to derive hypothesis from raw observational data. To deal with such data at all, they must already have some working hypothesis, for otherwise the data amounts to nothing more than a bewildering array of symbols (or sights and sounds), which is no more meaningful than a table of random numbers. In this connection Albert Einstein once said, "It may be heuristically useful to keep in mind what one has observed. But on principle it is quite wrong to try grounding a theory on observable magnitudes alone. In reality the very opposite happens. It is the theory which determines what we can observe."1

Pure mathematics contains an equivalent of the hypothetico-deductive method. In this case, instead of hypotheses there are proposed systems of mathematical reasoning intended to answer specific mathematical questions. And instead of the experimental testing of a hypothesis there is the step-by-step process of verifying that a particular proof, or line of mathematical reasoning, is correct. This verification process is straightforward and could in principle be carried out by a computer. However, there is no systematic, step-by-step method of generating mathematical proofs and systems of ideas, such as group theory or the theory of Lebesque integration.

If hypotheses in science and systems of reasoning in mathematics are not generated by any systematic procedure, then what is their source? We find that they almost universally arise within the mind of the investigator by sudden inspiration. The classic example is Archimedes' discovery of the principle of specific gravity. The Greek mathematician was faced with the task of determining whether a king's crown was solid gold without drilling any holes in it. After a long period of fruitless endeavor, he received the answer to the problem by sudden inspiration while taking a bath.

Such inspirations generally occur suddenly and unexpectedly to persons who had previously made some unsuccessful conscious effort to solve the problem in question. They usually occur when one is not consciously thinking about the problem, and they often indicate an entirely new way of looking at it-a way the investigator had never even considered during his conscious efforts to find a solution. Generally, an inspiration appears as a sudden awareness of the problem's solution, accompanied by the conviction that the solution is correct and final. One perceives the solution in its entirety, though it may be quite long and complicated when written out in full.

Inspiration plays a striking and essential role in the solution of difficult problems in science and mathematics. Generally, investigators can successfully tackle only routine problems by conscious endeavor alone. Significant advances in science almost always involve sudden inspiration, as the lives of great scientists and mathematicians amply attest. A typical example is the experience of the nineteenth-century mathematician Karl Gauss. After trying unsuccessfully for years to prove a certain theorem about numbers, Gauss suddenly became aware of the solution. He described his experience as follows: "Finally, two days ago, I succeeded.... Like a sudden flash of lightning, the riddle happened to be solved. I myself cannot say what was the conducting thread which connected what I previously knew with what made my success possible."2

We can easily cite many similar examples of sudden inspiration. Here is another one, give by Henri Poincare, a famous French mathematician of the late nineteenth century. After working for some time on certain problems in the theory of functions, Poincare had occasion to go on a geological field trip, during which he set aside his mathematical work. While on the trip he received a sudden inspiration involving his researches, which he described as follows: "At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used ... were identical with those of non-Euclidean geometry."3 Later on, after some fruitless work on an apparently unrelated question, he suddenly realized, "with just the same characteristics of brevity, suddenness, and immediate certainty,"4 that this work could be combined with his previous inspiration to provide a significant advance in his research on the theory of functions. Then a third sudden inspiration provided him with the final argument he needed to complete that work.

Although inspirations generally occur after a considerable period of intense but unsuccessful effort to consciously solve a problem, this is not always the case. Here is an example from another field of endeavor. Wolfgang Mozart once described how he created his musical works: "When I feel well and in good humor, or when I am taking a drive or walking,... thoughts crowd into my mind as easily as you could wish. Whence and how do they come? I do not know and I have nothing to do with it.... Once I have a theme, another melody comes, linking itself with the first one, in accordance with the needs of the composition as a whole.... Then my soul is on fire with inspiration, if however nothing occurs to distract my attention. The work grows; I keep expanding it, conceiving it more and more clearly until I have the entire composition finished in my head, though it may be long.... It does not come to me successively, with its various parts worked out in detail, as they will be later on, but it is in its entirety that my imagination lets me hear it."5 (Italics added.)

From these instances we discover two significant features of the phenomenon of inspiration: first, its source lies beyond the subject's conscious perception; and second, it provides the subject with information unobtainable by any conscious effort. These features led Poincare and his follower Hadamard to attribute inspiration to the action of an entity that Poincare called "the subliminal self," and that he identified with the subconscious or unconscious self of the psychoanalysts. Poincare came to the following interesting conclusions involving the subliminal self: "The subliminal self is in no way inferior to the conscious self; it is not purely automatic; it is capable of discernment; it has tact, delicacy; it knows how to choose, to divine. What do I say? It knows better how to divine than the conscious self, since it succeeds where that has failed. In a word, is not the subliminal self superior to the conscious self?"6 Having raised this question, Poincare then backs away from it: "Is this affirmative answer forced upon us by the facts I have just given? I confess that for my part, I should hate to accept it."7 He then offers a mechanical explanation of how the subliminal self, viewed as an automaton, could account for the observed phenomena of inspiration.