In a long section where Pirsig summarizes Henri Poincare's thoughts on scientific and mathematical discoveries, he focuses on the fact that observing facts and applying rules and making and testing hypotheses are simply unable by themselves to supply progress. The sheer number of potentially observable facts and possible mathematical combinations and reasonably believable hypotheses is prohibitive. Poincare spoke of a 'subliminal self' that chooses among the facts, combinations and hypotheses on the basis of an emotional reaction to beauty, harmony and elegance. "This is the true esthetic feeling which all mathematicians know, " said Poincare. Pirsig uses some examples from Poincare's own brilliant mathematical career to illustrate the process:
1) "For fifteen days, he said, he strove to prove that there couldn't be any such functions. Every day he seated himself at his work-table, stayed an hour or two, tried a great number of combinations and reached no results.
Then one evening, contrary to his custom, he drank black coffee and couldn't sleep. Ideas arose in crowds. He felt them collide until pairs interlocked, so to speak, making a stable combination.
The next morning he had only to write out the results. A wave of crystallization had taken place."
2) "He left Caen, where he was living, to go on a geologic excursion. The changes of travel made him forget mathematics. He was about to enter a bus, and at the moment when he put his foot on the step, the idea came to him, without anything in his former thoughts having paved the way for it, that the transformations he had used to define the Fuchsian functions were identical with those of non-Euclidian geometry. He didn't verify the idea, he said, he just went on with a conversation on the bus; but he felt a perfect certainty. Later he verified the results at his leisure."
3) "A later discovery occurred while he was walking by a seaside bluff. It came to him with just the same characteristics of brevity, suddenness and immediate certainty."
Robert M. Pirsig
Zen and the Art of Motorcycle Maintenance
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