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Mathematicians usually focus on the products of their creativity: theorems (equations and inequalities, existence-uniqueness results), algorithms, modeling, and conjectures.

A different question: What are the major aspects of the mathematical creative process?

After substantial reflection, let me offer a Socratic perspective to sharpen this question.

  1. Knowledge. What is the right balance of depth versus breadth for a mathematician? How does one continue learning? How does one learn the art of making connections to disparate areas of mathematics?

  2. Collaboration. When is the lack of collaboration hurtful in the creative process? Too much collaboration? Have electronic means of communication caught up (or even surpassed) the value of face-to-face collaboration?

  3. Computers. This modern tool has served several purposes: perform rote calculations, inspired the development of algorithms, resurrected dormant areas (fractal geometry), visualization, and allowed automated symbolic exploration.

  4. Self Exploration. A toxic mindset for the researcher is to feel that the whole landscape of an area must be digested before any exploration can be conducted. In contrast, Jacobi writes “... [young mathematicians should be pitched] into the icy water to learn to swim and drown by themselves. Many students put off attempting anything on their own account till they have mastered everything related to their problem that has been done by others. The result is that but few acquire the knack of independent work.”

  5. Rest. This includes getting an appropriate night's sleep, power naps, exercise, etc. Where does having a "calm mind" --- the "voice in the head" is quiet --- fit into the creative process? Jacques Hadamard's “The Psychology of Invention in the Mathematical Field” is essentially a book about the value of rest. What habits are helpful?

  6. State of Mind. What is the role of aesthetics in the creative process? Is the ideal mental state a balance between being too skilled (leads to boredom) and too challenged (leads to anxiety)?

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This is of course a fascinating and important topic, but it's much too subjective for MO. Voting to close. –  Qiaochu Yuan Aug 18 '12 at 21:14
Indeed; this sort of question is not the right fit for MO. Perhaps Quora would be a better place for it? –  Ben Webster Aug 18 '12 at 22:08
Among many reasons this post is not a good fit here (it is a great post for other discussion fora) is that it asks hella questions all at once. I wish you luck finding a good place to have this discussion, and when you do I hope that you post a link to it here in the comments (unless other MOers feel that that would be in bad taste). –  Theo Johnson-Freyd Aug 21 '12 at 17:34
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closed as not constructive by Qiaochu Yuan, Ben Webster Aug 18 '12 at 22:07

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