I stumbled, recently, on a blog post about "Egoless programming", a feature of a book called The Psychology of Computer Programming.
The points struck me as good general advice -- but also good advice for mathematicians at any level, so I thought I'd rework them here. I've made minor adjustments to the headings as appropriate.
As my good friend T. K. says, if you're not making mistakes, you're not trying hard enough. Some of them will be embarrassing (I, for example, have an error in my thesis that still makes me cringe every time I think of it). However, realising how I made that error helped me avoid making similar errors in the future.
Nobody ever gets everything 100% right - don't hold yourself to an impossible standard.
Related, when someone points out your errors -- either in marking your work as a student, or in producing papers as an academic -- don't take it personally. You are not your mathematical mistakes. Accept correction and advice gracefully, and learn from it. You'll be glad of it.
I struggled at sixth form, because I wasn't automatically the smartest person in the room any more. I struggled more at university, because I wasn't even automatically the smartest mathematician. I was threatened and outclassed. I didn't like it!
I only really got to grips with it in my final year, when I got friendly with one of the mathematical superheroes, Charlie. We revised for our Solar Theory module together, poring through the notes, him pointing out my failings in rigour, me trying to explain the physics side of things. We each learned something (I got 18/20 in the exam and ended up on the PhD program. He got 20 and pursued something else.)
I suppose this really only applies to collaboration and boils down to good manners: if you're going to completely redo something, at least do it in cahoots with your co-authors or group members.
Again, this is just good manners. Just because someone doesn't know as much as you doesn't make them subhuman. Even if it did, it wouldn't give you the right to behave like a jerk.
Even in maths, one of the least changeable fields, things are always changing. New theorems are discovered, new fields emerge, others become unfashionable.
For you, too, change is gonna come. You'll be exposed to new ideas, new ways of doing things, and it's not always obvious that the 'new' way is better than the old at first. But at least give it the chance to prove itself.
It doesn't matter if you're an eight-year old or a college professor -- if you prove something, it's proved. If you solve something, it's solved. In principle, maths is the ultimate meritocracy.
However, in practice, it doesn't quite work that way -- there's a hierarchy in maths as much as anywhere else, and sometimes you may need to convince your way up it to get your voice heard.
From the other side, if someone's engaging with you honestly, do your best to engage back and understand their arguments. You might learn something.
If you think you're right, in an ideal world you wouldn't let go until you were convinced. There comes a time, though, where you need to accept that you can't overrule everyone else, and -- if only to avoid deadlock -- you need to provisionally accept an alternative viewpoint.
It might turn out, in the end, that you were right. That's ok. Pretend you don't remember.
Maths is a solitary pursuit, right? Wrong. Absolutely wrong. It's a collaborative effort, and if you, as part of a group project, insist on working alone, you will damage the project. Mix in with the group. Be more than the sum of your parts. Pretend you're not shy. You'll enjoy the project a lot more (while making it better).
Bad: "You're an idiot." Good: "I think this would be better expressed like this because..." See the difference? The second comment is not aimed at the person, it's aimed at improving their work -- and it's explained, leaving room for discussion.
What are your tips for ensuring peace and harmony among mathematicians?
Edited 2015-04-28 to fix numbering.