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This on the workings of the conscious and the subconscious in problem solving...

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


"The future is already here -- it's just not very evenly distributed" William Gibson
by ChrisCook (cojockathotmaildotcom) on Wed Jan 2nd, 2008 at 12:14:03 PM EST
[ Parent ]
Yes, it's remarkable how important beauty can be to those in maths and physics - there are a number of examples in Kaku's "Parallel Worlds".

Maybe it's because I'm a Londoner - that I moved to Nice.
by Ted Welch (tedwelch-at-mac-dot-com) on Wed Jan 2nd, 2008 at 12:44:04 PM EST
[ Parent ]
Why is it remarkable? Isn't aesthetic pleasure important to engineers, architects, sculptors, painters, photographers, filmmakers, writers...?

We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Wed Jan 2nd, 2008 at 12:47:53 PM EST
[ Parent ]

Because it is so different from the general image the general public has of the nature of science and maths (I don't recall it being mentioned during my education :-)), and, has been noted here, even many scientists do not like to acknowledge the creative/intuitive nature of their work.

Maybe it's because I'm a Londoner - that I moved to Nice.
by Ted Welch (tedwelch-at-mac-dot-com) on Wed Jan 2nd, 2008 at 12:56:20 PM EST
[ Parent ]
As Sam has mentioned perviously, we come from a college background where drinking in the pursuit of theorems was a semi-serious  strategy. I just find this whole part of the discussion culturally weird.
by Colman (colman at eurotrib.com) on Wed Jan 2nd, 2008 at 12:58:49 PM EST
[ Parent ]
It is remarkable that the general public has the wrong idea about the nature of science and maths?

We have met the enemy, and he is us — Pogo
by Migeru (migeru at eurotrib dot com) on Wed Jan 2nd, 2008 at 12:59:33 PM EST
[ Parent ]
A big factor is that a vast majority of people in the written press have arts degrees, and are activly disinterested in science. Unless you are reading more specific science press, the quality of science reporting is poor.

On top of this you have a vast amount of celebrity reporting that compresses the space that's available for other reporting. The vast increase in the number of TV channels also allows people to avoid any exposure to science.

why is Science that important to the average person when Brittneys got no underwear on?

Any idiot can face a crisis - it's day to day living that wears you out.

by ceebs (ceebs (at) eurotrib (dot) com) on Wed Jan 2nd, 2008 at 01:15:25 PM EST
[ Parent ]
And elegance as a criteria for suggesting truth in mathematics is a widely known heuristic, surely? Not that it always works, but people are extra suspicious of inelegant hacked together proofs.
by Colman (colman at eurotrib.com) on Wed Jan 2nd, 2008 at 01:00:09 PM EST
[ Parent ]
This book, edited by Graham Farmello - subtitled 'Great Equations of Modern Science' goes into great detail on this subject.

You can't be me, I'm taken
by Sven Triloqvist on Wed Jan 2nd, 2008 at 01:08:50 PM EST
[ Parent ]

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