Mule stubbornness, in a traditional sense, is a myth. By traditional sense, I mean that they are unreasonable obstinate and are so for no reason. Mules can be "stubborn", but only in situations when freezing is best.
There are two reasons while mules behave like they do: Instinct and deep thought. To understand the instinct reason for mule stubbornness, you have to look at the donkey.
The wild donkey lives in the deserts of Africa in mountainous areas. When scared, they can't just run blindly. They would run themselves right off a cliff. So what do they do? They think. They look at the predator and think, "Would I run, stay, or fight?"
So, when a mule is confronted with a scary blue tarp or creek crossing, they use the jackass lodged deep inside their brains and think. "Should I put my foot on the tarp? Will it hurt me? Why not just go around the stupid thing?" Once you convince the mule it is a good idea or the mule convinces himself, he will go on with no problems.
Second reason for mule stubbornness is deep thought. This is a lot like instinct, only it happens randomly at various times of the day (and usually right when you don't want it to happen). "What should I do next? What is good about option A? What is bad? What about option B?"
You see, mules are only stubborn because they are smart. Mules know that running into or away from everything isn't going to work as well as thinking it through. Many people will tell you "Trust your mule." Mules are sometimes smarter than people and are highly sensible.