I had to explain to my students that the proofs in their textbooks made no sense because they were discovered inductively in the opposite direction than they were written, and then sharpened for "elegance" in order to impress other professors so they will select the textbook. Take, for example, a typical epsilon-delta exercise from calculus. The official answer will start with "let epsilon equal blah blah bla" and the student will stop right there and scratch their head "how did they come up with that?". Well, that was the last thing they came up with.

We have met the enemy, and he is us — Pogo

(It wasn't the abstraction he objected to, but the disconnect.)

The Fates are kind.

Then Poincaré illustrated how a fact is discovered. He had described generally how scientists arrive at facts and theories but now he penetrated narrowly into his own personal experience with the mathematical functions that established his early fame.

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.

He described how a second wave of crystallization, guided by analogies to established mathematics, produced what he later named the "Theta-Fuchsian Series." 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 result at his leisure.

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. Another major discovery occurred while he was walking down a street. Others eulogized this process as the mysterious workings of genius, but Poincaré was not content with such a shallow explanation. He tried to fathom more deeply what had happened.

Mathematics, he said, isn't merely a question of applying rules, any more than science. It doesn't merely make the most combinations possible according to certain fixed laws. The combinations so obtained would he exceedingly numerous, useless and cumbersome. The true work of the inventor consists in choosing among these combinations so as to eliminate the useless ones, or rather, to avoid the trouble of making them, and the rules that must guide the choice are extremely fine and delicate. It's almost impossible to state them precisely; they must be felt rather than formulated.

Poincaré then hypothesized that this selection is made by what he called the "subliminal self," an entity that corresponds exactly with what Phadrus called preintellectual awareness. The subliminal self, Poincaré said, looks at a large number of solutions to a problem, but only the interesting ones break into the domain of consciousness. Mathematical solutions are selected by the subliminal self on the basis of "mathematical beauty," of the harmony of numbers and forms, of geometric elegance.

"This is a true aesthetic feeling which all mathematicians know," Poincaré said, "but of which the profane are so ignorant as often to be tempted to smile."

But it is this harmony, this beauty, that is at the center of it all.

Problem solving is what I have always enjoyed more than anything else. I think that "pattern recognition" is a good way of describing the process.

My motivation is to find the simple answer that everybody knows, but which doesn't yet exist: Naoto Fukasawa