I ride my mountain bike in remote areas of North Georgia and North carolina where I have bumped into many strange charactors who do not seem thrilled I was there.
When in remote areas like this you cannot predict who you will run into. As a matter of fact a recent an FBI most wanted was captured in a town just a hop skip and jump away from some of my favorite riding places.
Eric Robert Rudolf was wanted in connection with terrorist activities and if you bump into someone like this you cannot know what they will be willing to do to keep you from saying you saw them.
I also run into very unfriendly locals from time to time that seem to think we are all city folk who are infringing upon them. Even though I am as much a country boy as any of them they do not typically like my presence. Sometimes they have guns. I even ran up on some locals out in the woods setting up to shoot bottles and cans in National Forest where it is illegal anyway.
Not to mention the remote possibility of angering Black bear and/or Cougar. Cougar are rare here but I have run up on black bear and surprised them many times but luckily they always ran from me. Once I ran up on Momma bear and three cubs, now that was a situation that could have turned ugly and at that time I did not have a peice, but I would have been very reluctant to shoot a mother bear anyway even if she charged me.
I'd likely throw my bike at her and pick up a big stick and hope for her not wanting to fight.
Anyway, I do nopt carry with the mentality that I want to shoot someone or something. I do so just to be armed if confronted with a life threatening conflict.
I do not carry on my road bike. But I live where it is not a threat to travel alone.