Hi,
Below is a short passage taken from the book
"Letters to a Young Mathematician" by Ian Stewart, which provides
us with a good insight on how mathematicians think, from which we
could model after them to improve our mathematical thinking.
According to Jacques Hadamard, author of The
Psychology of Invention in the Mathematical Field, ideas in
mathematics seem to arise in three stages.
First, it is necessary to carry out quite a lot of
conscious work on a problem, trying to understand it, exploring
ways to approach it, working through examples in the hope of
finding some useful general features. Typically, this stage bogs
down in a state of hopeless confusion, as the real difficulty of
the problem emerges.
At this point it helps to stop thinking about the problem
and do something else not related to the problem. This gives the
subconscious mind a chance to mull over the original problem and
try to sort out the confused mess that your conscious efforts have
turned it into. If your subconscious is successful, even if all it
manages is to get part way, it will "tap you on the shoulder" and
alert you to its conclusions. This is the big "aha!" moment, when
the little lightbulb over your head suddenly switches on.
Finally, there is another conscious stage of writing
everything down formally, checking the details, and organizing it
so that you can publish it and other mathematicians can read
it.
The great mathematician Henri Poincare called the first
stage "preparation,", the second "incubation followed by
illumination," and the third "verification." He laid particular
emphasis on the role of the subconscious in one famous section of
his essay Mathematical Creation:
For fifteen days I strove to prove a difficult result. I
was then very ignorant; every day I seated myself at my table,
stayed an hour or two, tried a great number of alternatives and
reached no results. One evening, contrary to my custom, I drank
black coffee and could not sleep. Ideas rose in crowds; I felt them
collide until they converged to a solution. By the next morning I
had established the approach for the proof, which
took but a few hours to write.
To end, I'd like to quote an interesting phrase from
Albert Einstein which reads as follows: Do not worry about your
difficulties in Mathematics. I can assure you mine are still
greater.
Thank you!
Cheers,
Wen Shih