By Dr. T. D. Singh (Bhaktisvarüpa Dämodara Swami) and Sadäpüta däsa
The requirements that this process of choice must meet are strikingly illustrated by some further examples of mathematical inspiration. It is very often found that the solution to a difficult mathematical problem depends on the discovery of basic principles and underlying systems of mathematical relationships. Only when these principles and systems are understood does the problem take on a tractable form; therefore difficult problems have often remained unsolved for many years, until mathematicians gradually developed various sophisticated ideas and methods of argument that made their solution possible. However, it is interesting to note that on some occasions sudden inspiration has completely circumvented this gradual process of development. There are several instances in which famous mathematicians have, without proof, stated mathematical results that later investigators proved only after elaborate systems of underlying relationships had gradually come to light. Here are two examples.
The first example concerns the zeta-function studied by the German mathematician Bernhard Riemann. At the time of his death, Riemann left a note describing several properties of this function that pertained to the theory of prime numbers. He did not indicate the proof of these properties, and many years elapsed before other mathematicians were able to prove all but one of them. The remaining question is still unsettled, though an immense amount of labor has been devoted to it over the last seventy-five years. Of the properties of the zeta-function that have been verified, the mathematician Jacques Hadamard said, “All these complements could be brought to Riemann’s publication only by the help of facts which were completely unknown in his time; and, for one of the properties enunciated by him, it is hardly conceivable how he can have found it without using some of these general principles, no mention of which is made in his paper.”8
The work of the French mathematician Evariste Galois provides us with a case similar to Riemann’s. Galois is famous for a paper, written hurriedly in sketchy form just before his death, that completely revolutionized the subject of algebra. However, the example we are considering here concerns a theorem Galois stated, without proof, in a letter to a friend. According to Hadamard this theorem could not even be understood in terms of the mathematical knowledge of that time; it became comprehensible only years later, after the discovery of certain basic principles. Hadamard remarks “(1) that Galois must have conceived these principles in some way; (2) that they must have been unconscious in his mind, since he makes no allusion to them, though they by themselves represent a significant discovery.”9
It would appear, then, that the process of choice underlying mathematical inspiration can make use of basic principles that are very elaborate and sophisticated and that are completely unknown to the conscious mind of the person involved. Some of the developments leading to the proof of some of Riemann’s theorems are highly complex, requiring many pages (and even volumes) of highly abbreviated mathematical exposition. It is certainly hard to see how a mechanical process of trial and error, such as that described by Poincare, could exploit such principles. On the other hand, if other, simpler solutions exist that avoid the use of such elaborate developments, they have remained unknown up to the present time, despite extensive research devoted to these topics.
The process of choice underlying mathematical inspiration must also make use of selection criteria that are exceedingly subtle and hard to define. Mathematical work of high quality cannot be evaluated simply by the application of cut-and-dried rules of logic. Rather, its evaluation involves emotional sensibility and the appreciation of beauty, harmony, and other delicate aesthetic qualities. Of these criteria Poincare said, “It is almost impossible to state them precisely; they are felt rather than formulated.”10 This is also true of the criteria by which we judge artistic creations, such as musical compositions. These criteria are very real but at the same time very difficult to define precisely. Yet evidently they were fully incorporated in that mysterious process which provided Mozart with sophisticated musical compositions without any particular effort on his part and, indeed, without any knowledge of how it was all happening.
If the process underlying inspiration is not one of extensive trial and error, as Poincare suggested, but rather one that depends mainly on direct choice, then we can explain it in terms of current mechanistic ideas only by positing the existence of a very powerful algorithm (a system of computational rules) built into the neural circuitry of the brain. However, it is not at all clear that we can satisfactorily explain inspiration by reference to such an algorithm. Here we will only briefly consider this hypothesis before going on to outline an alternative theoretical basis for the understanding of inspiration.
The brain-algorithm hypothesis gives rise to the following basic questions.
(1) Origins. If mathematical, scientific, and artistic inspirations result from the workings of a neural algorithm, then how does the pattern of nerve connections embodying this algorithm arise? We know that the algorithm cannot be a simple one when we consider the complexity of automatic theorem-proving algorithms that have been produced thus far by workers in the field of artificial intelligence.11 These algorithms cannot even approach the performance of advanced human minds, and yet they are extremely elaborate. But if our hypothetical brain-algorithm is extremely complex, how did it come into being? It can hardly be accounted for by extensive random genetic mutation or recombination in a single generation, for then the problem of random choice among vast numbers of possible combinations would again arise. One would therefore have to suppose that only a few relatively probable genetic transformations separated the genotype of Mozart from those of his parents, who, though talented, did not possess comparable musical ability.
However, it is not the general experience of those who work with algorithms that a few substitutions or recombinations of symbols can drastically improve an algorithm’s performance or give it completely new capacities that would impress us as remarkable. Generally, if this were to happen with a particular algorithm, we would tend to suppose that it was a defective version of another algorithm originally designed to exhibit those capacities. This would imply that the algorithm for Mozart’s unique musical abilities existed in a hidden form in the genes of his ancestors.
This brings us to the general problem of explaining the origin of human traits. According to the theory most widely accepted today, these traits were selected on the basis of the relative reproductive advantage they conferred on their possessors or their possessors’ relatives. Most of the selection for our hypothetical hidden algorithms must have occurred in very early times, because of both the complexity of these algorithms and the fact that they are often carried in a hidden form. It is now thought that human society, during most of its existence, was on the level of hunters and gatherers, at best. It is quite hard to see how, in such societies, persons like Mozart or Gauss would ever have had the opportunity to fully exhibit their unusual abilities. But if they didn’t, then the winnowing process that is posited by evolution theory could not effectively select these abilities.
We are thus faced with a dilemma: It appears that it is as difficult to account for the origin of our hypothetical inspiration-generating algorithms as it is to account for the inspirations themselves.
(2) Subjective experience. If the phenomenon of inspiration is caused by the working of a neural algorithm, then why is it that an inspiration tends to occur as an abrupt realization of a complete solution, without the subject’s conscious awareness of intermediate steps? The examples of Riemann and Galois show that some persons have obtained results in an apparently direct way, while others were able to verify these results only through a laborious process involving many intermediate stages. Normally, we solve relatively easy problems by a conscious, step-by-step process. Why, then, should inspired scientists, mathematicians, and artists remain unaware of important intermediate steps in the process of solving difficult problems or producing intricate works of art, and then become aware of the final solution or creation only during a brief experience of realization?
Thus we can see that the phenomenon of inspiration cannot readily be explained by means of mechanistic models of nature consistent with present-day theories of physics and chemistry. In the remainder of this article we will suggest an alternative to these models.
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