the backgrounder mentioned a long string of mathematicians then unknown to me..that were in slight or not slight ways flag stones to the fractal.
Poincare, it turns out, was one of Mandelbrot's scientific heros, as he said in an interview with MIT Tech student journalists in an off-the-cuff early 2000s interview.
Said Mandelbrot of Poincare (to MIT students for The Tech in the early 2000s):
Henri Poincare is a hero .. He defied categories, because on the one hand he was, by quite a long stretch, the most amazing man of his time in many different areas but he never proved anything rigorously, so his community disliked him for his desire to leave difficulties to others because they enjoyed it and he didn’t.
Further Mandelbrot on Poincare:
his community disliked him for his desire to leave difficulties to others because they enjoyed it and he didn’t.
What i found interesting is the extent to which Poincare thought deeply about the processes of thinking, and the methods of science. His mathematics broke ground in probability theory, which has been used to fill in the natural gaps that tend to occur when difficult experiments are recreated. He may have lagged as a scientist because he was all the time thinking about what was going on - spanning, as may have been more common in earlier times, philosophy and science.
Because he thought about this intersection, and its application in probability, his work has him delve in the space between analytics and uncertainty - the mesh where prediction meets its match, or its better. [Was he ahead of his time? He did mathematical text analysis of Dreyfus Trial documents to deign validity and subsequently ascertain they were forged.)
He wrote:
Pure logic could never lead us to anything but tautologies; it could create nothing new; not from it alone can any science issue.
[On the philosophers ... objection :] Your science is infallible, but it can only remain so by imprisoning itself in an ivory tower and renouncing all relation with the external world. From this seclusion it must go out when it would attempt the slightest application."
We need a faculty which makes us see the end from afar, and intuition is this faculty.
In mathematics logic is called analysis and analysis means division, dissection. It can have, therefore, no tool other than the scalpel and the microscope.
... logic and intuition have each their necessary role. Each is indispensable. Logic, which alone can give certainty, is the instrument of demonstration; intuition is the instrument of invention.Related
http://www-history.mcs.st-andrews.ac.uk/Extras/Poincare_Intuition.html
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