Question is pretty straightforward. My mathematics is sloppy, and I recognize my inaptitude in that my proofs are more or less too intuitive. My diagnosis dictates the fact that I attack a problem directly, which produces a, in principle, sloppy result.

How do I proceed in making a preliminary sketch of various increasing stages of rigor from a very general outline to the actual rigorous proof? There mus be a way to proceed, rather than waiting for the Eureka effect to strike for every single problem that I have, and produce an insufficient result.

  • $\begingroup$ There unfortunately isn't a real answer to this question. You just need to practice and listen to constructive criticisms about your proofs from your professors. $\endgroup$ – Jim Mar 17 '13 at 20:53
  • $\begingroup$ You might like to read the response by Issac. Also, search the site for books on problem solving and you'll find several nice examples dedicated to the art to teach you the different strategies. If you really work through a couple of those, your math career will be so enriched and easier! Enjoy! $\endgroup$ – Amzoti Mar 17 '13 at 20:54
  • $\begingroup$ Proofs come by exercise. Exercise a lot with problems and exercises, and you'll develop your own solving mechanism. Personally, I love book courses which have difficult exercises and problems after each chapter. I remember that in my first year the proofs I've seen in my courses didn't seem all intuitive, and some were hard to remember. After many years of experience, some (not all) proofs come easier, ideas are more clear. Practice is the key, but don't practice with things you know you can't solve. In time you'll be able to increase the difficulty. $\endgroup$ – Beni Bogosel Mar 17 '13 at 21:02
  • $\begingroup$ I'm not sure it makes sense to think of proceeding from a proof outline or proof idea to the actual rigorous proof. There may be many possible rigorous proofs of a given theorem, and your idea might potentially lead to more than one of them, or it might lead to none of them in the case that your intuition is wrong. $\endgroup$ – Trevor Wilson Mar 17 '13 at 21:05
  • $\begingroup$ I think the heart of the matter is not to translate an abitrary intuitive proof into a rigorous proof, but to train your intuition so that the transfer is more or less automatic. $\endgroup$ – timur Mar 17 '13 at 21:23

Doing mathematics is a practice with no general theory behind it. Like carpentry or engineering, transmission of the knowledge of "how to do mathematics" is something probably best done in the context of a master/apprentice relationship. In the absence of this ideal situation, watching people do mathematics (on this site, for instance) and trying your hand at similar problems under the critical eye of others is probably the best way to learn.

This isn't a direct answer to your question, but your question is unanswerable as it is posed. Just keep trying to prove things. You'll get the hang of it, hopefully.


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