5
$\begingroup$

Caveat: I fear that people will criticize me for asking this potentially inappropriate question here, but I guess that the community here is quite unique in the ability of potentially answering my question (if you don't have an answer, then probably there is no), and that there is some (little) chance of getting some good answers to the question - and not asking this question here or banning it right away would reduce the chance of getting a good answer to 0.

The question is: If one tries to prove something, are there some tricks for getting a creative idea? Ok, I now what you think, yes, there is no algorithm to finding a creative idea, otherwise the idea wouldn't be creative. However, there are some general "tricks": if for minutes one stares at ones sheet of papers with no new ideas, just moving in the same thought cycles, it certainly helps to go and talk to a colleague, because somehow talking awakes the creative ability of the brain (and, additionally, together with a colleague one can mutually pick up an idea of the other and think it a bit further). Also, forgetting the problem for a moment and go and attend talks (even if they are about another topic) or even just rest helps. Do you have any other general "tricks" for getting creative ideas for solving mathematical problems?

To make the question a bit more concrete, do you know of any tricks for finding or looking for a good lemma (or several lemmas)? I have the feeling that often the most creativity in proving a theorem lies in finding the right lemma (not even the proof of it, but just the statement). I noticed that whenever I see a proof about which I afterwards say "wow, that's genius, I don't even rudimentally see how one could have come up with it", the crucial point was a lemma (or several lemmas). This also seems to me to be one difference between doing research and doing like homework problems: in homework assignments the proofs usually require only one or two main ideas, and if it requires a lemma, this lemma often is stated in the task as a subtask - while in research, one doesn't even know how much one has to "go down", how many levels of lemmas one has to show.

$\endgroup$
4
  • 1
    $\begingroup$ I'm afraid your question, while interesting, is almost impossible to answer, because even people who have an insightful idea are generally unable to explain, ex post facto, how it came to them. The closest thing to an answer is probably Pólya's book How to solve it. $\endgroup$
    – Gro-Tsen
    May 30, 2019 at 22:19
  • 1
    $\begingroup$ Sure, this question is probably impossible to definitively answer, but various "failures to definitively answer" might be interesting. "Interesting" is my criterion for ... almost anything... even if not quite in line with policy. $\endgroup$ May 30, 2019 at 23:00
  • 2
    $\begingroup$ I don't think this question is appropriate here. I also am not sure if you really understand what you are talking about. That "the crucial point was a lemma" piece is rather odd to me; this is about a proof-writing, about packaging an idea in a form that is user-friendly, rather than about the invention/discovery. I feel that it is not very often that a mathematical argument you see in writing is faithfully representing the process of discovery. $\endgroup$ May 30, 2019 at 23:45
  • 2
    $\begingroup$ You might be interested in the Tricki. gowers.wordpress.com/2010/09/24/is-the-tricki-dead $\endgroup$ May 31, 2019 at 0:38

2 Answers 2

4
$\begingroup$

I am not addressing your request for "tricks." One can hardly do better than Poincaré's description of creativity in mathematics:

Poincaré, Henri, translated by Francis Maitland. Science and method. Courier Corporation, 2003.

Preface by Bertrand Russell: "The writing of professional philosophers on such subjects has too often the deadness of external description. Poincaré's writing, on the contrary, ..., has the freshness of actual experience, of vivid, intimate contact with what he is describing."

More recently, one can find literature on what constitutes mathematical creativity, often in the math-education literature. Two examples:

Ervynck, Gontran. "Mathematical creativity." In Advanced mathematical thinking, pp. 42-53. Springer, Dordrecht, 2002. Springer link.

"We hypothesize that the context for creativity is set by a preparatory stage in which mathematical procedures become interiorized through action before they can be the objects of mathematical thought."

 

Sriraman, Bharath. "The characteristics of mathematical creativity." Mathematics Educator 14, no. 1 (2004): 19-34. Journal link.

"The results indicate that, in general, the mathematicians’ creative processes followed the four-stage Gestalt model of preparation-incubation-illumination-verification."

$\endgroup$
4
$\begingroup$

Here's a simple but extremely effective technique to generate creativity:

1) Work hard on a problem until you reach an impasse.

2) Take an extended break from the problem (it could be a week or up to a few months, it's your call).

3) Repeat

For creativty, Take Breaks - Article

And a piece of historical evidence to consider:

When Isaac Newton was forced to return home from school due to the Black Plague, he began to generate vast swaths of groundbreaking ideas including laying the foundation for classical mechanics and calculus. His productivity lasted for decades.

$\endgroup$