In physics, your solution should convince a reasonable person. In math, you have to convince a person who's trying to make trouble. Ultimately, in physics, you're hoping to convince Nature. And I've found Nature to be pretty reasonable.
In physics, you don't have to go around making trouble for yourself - nature does it for you.
I went off to college planning to major in math or philosophy-- of course, both those ideas are really the same idea.
If you don't make mistakes, you're not working on hard enough problems. And that's a big mistake.