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As a freshly-minted CS doctoral student who plans on working on the more formal side of things, I've been studying up on my math in detail. While reading through a proof of Zorn's Lemma, I was struck by the complexity and creativity of it. (The proof was in Paul Halmos's Naive Set Theory.) By contrast, whenever I try to do practice proofs of anything nontrivial, I feel like I spend hours banging my head against the problem and worrying whether I've made any small mistakes that cause the whole tower to come crashing down.

My advisor is relatively math-savvy for the field, but even he doesn't do as math-intensive work as I'm planning on doing. My question is: Would a math professor be able to look at a novel problem like Zorn's Lemma and prove the thing in one sitting with no mistakes? Is that the level that will be expected of someone doing serious math work? Or is a nontrivial proof something that takes a lot of time, where small flaws in the proof is an understood part of the practice?

Sorry if this is a bad question; I suppose I'm partly just wondering how insecure I should be about my math skills.

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closed as off-topic by Cameron Williams, José Carlos Santos, Chappers, Ethan Bolker, Mostafa Ayaz Jul 15 '18 at 18:15

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "Seeking personal advice. Questions about choosing a course, academic program, career path, etc. are off-topic. Such questions should be directed to those employed by the institution in question, or other qualified individuals who know your specific circumstances." – José Carlos Santos, Mostafa Ayaz
If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ The story of Andrew Wiles' proof of Fermat's Last Theorem might be instructive. It took him a very long time, and after he presented it an error was found in the proof that took another lengthy effort to correct. You probably will not need to solve a problem that difficult, but I think it goes to show how open-ended this question really is. $\endgroup$ – David K Jul 15 '18 at 14:38
  • $\begingroup$ 'but even he doesn't do as math-intensive work as I'm planning on doing.' ? It would be interesting to know whar you are 'planning', especially as you then claim that you don't know what is humanly possible for a mathematicians. Your question is unclear $\endgroup$ – AnyAD Jul 15 '18 at 14:41
  • $\begingroup$ I'm voting to close this question as too broad and opinion based. That said, I think the way you come to prove theorems is to understand intuitively "why they are true". You practice that in math courses with exercises like "prove or disprove ...". By the time you get to research mathematics you've developed some instinct for turning "typical examples" into formal proofs. $\endgroup$ – Ethan Bolker Jul 15 '18 at 14:56
  • $\begingroup$ @AnyAD A lot of the work I'm going to be doing will be related to formal verification of some of the algorithms we use, cybersecurity-related. $\endgroup$ – Raiden Worley Jul 15 '18 at 15:07
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    $\begingroup$ Proofs in textbooks can appear miraculous because they've usually undergone years of optimization - choosing precisely the right definitions and lemmas to compress down the proof to a small size. Often the first proofs will be much messier. They are rarely if ever formed so nicely on the first try. Much head banging required first. $\endgroup$ – Jair Taylor Jul 15 '18 at 16:09
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It will all depend on your research question/s, topic, your supervisor, your ability and creativity and the amount of time you spend working on this.

You may be lucky and solve some question quickly (matter of hours), but then this will likely be one minor result. You will likely attempt many questions/problems without getting anywhere (especially if these are not proposed to you by your supervisor).

If you chose un unsolvable problem, you will obviously never get a direct result for this. You may on the other hand write up a a mostly trivial result. You want to spend minimum possible time doing these two things.

Ideally your advisor will have appropriate questions for you to work on. They will have a good idea on how much time it should take.

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