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I have this odd dream that online resources like this can serve as a virtual thesis advisor for future mathematicians who are teaching themselves. Here's another question along these lines. You can do a lot of different things with your day as a mathematician. Some uses of your time are better than others. The following question certainly will vary from person to person...but cogent answers to the question may teach people to do mathematics more effectively. The question is this:

What is the highest "gain" mathematical activity?

A possible interpretation of this question:

If you had to pick one thing you do as a mathematician...and it was the only thing you were going to be allowed to do mathematically for the rest of your career...and you wanted to discover/learn the most mathematics by doing it....what would it be?

Here are some candidates:

Working out a simple but nontrivial example of a theorem or question. Trying to prove a theorem for yourself in full generality. Modifying a given problem. Looking for new problems. Doing exercises. Reading books. Trying to conceptualize (making things human-friendly). Talking to people. Answering Math.Stackexchange questions. Writing things up.

As simple as this is, I'd like to see answers posted here. This is closely related to an earlier question of mine about method. (I think the way to be a mathematician is this: work really hard until stuck, learn a little bit, repeat.)

NOTE: I am not suggesting that such an activity should exist (this question is as ridiculous as the practice in philosopy of leaving all but one fact in the universe unchanged). The point is to force us to think about what such an activity would be if it should exist. I'm inclined to think that for me working on the simplest nontrivial example is best, because it forces me to connect and compare the new concept with more familiar things and simultaneously helps organize all parts of a problem into a manageable whole. I'm sure Grothendieck would not have the same answer to this question...

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I don't think any of the current tags are relevant to the specific question being asked. Perhaps soft-question would be appropriate. – Antonio Vargas May 17 '12 at 17:29
Thanks, Antonio. I agree this is much better. – Jon Bannon May 17 '12 at 17:30

Short "answer": I don't know if I could identify one thing that I would limit myself to doing. I don't think a mathematician is going to go far if he/she only does one thing. If all you do is talking to people, but you never sit down and write out details, or if you only read books but never ask questions about what consequences this might have, then I don't think that you will get very far.

Now, you do also ask: "What is the highest "gain" mathematical activity?" This question is a bit different in that it doesn't say that you have to only do one thing. It asks for the thing that gives you the most gain. Again, I wouldn't really be able to answer that question, or: I don't think that I can identify the one thing that gives me the most gain. It really all depends. Sometimes I have made the most progress when I have taken a walk outside. Sometimes the progress comes when I finally decide to sit down and write out all the details of what I am doing. Sometimes the key that unlocks a problem is a hint that you get from a conversation with another person.

Doing mathematics isn't like digging a hole in the ground where we can say that now we are halfway to solving the problem. It is not like baking a cake where you can say that it only has 27 min. left in the oven. The biggest hurdles don't always appear where we expect them to.

All that said, I tend to find the most progress when I am actually sitting down in a quite room at a desk with pen and paper. Just letting the mind try out different approaches.

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This is actually my experience, too, Thomas. All around. It's kind of weird, actually, that things work this way. – Jon Bannon May 17 '12 at 17:34
I guess, though, the point of the question is to force us to think of what the best such activity is...I'll edit. – Jon Bannon May 17 '12 at 17:51
@JonBannon: Yes, I was kind of thinking that was really what you were going for. – Thomas May 17 '12 at 17:51

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