When was the last time you heard someone say, "I was wrong"?
I don't hear it often. I hear "Well... you see I had made a different assumption, and if you follow my assumption, you would have ended up at my answer." To me, that sounds like "I am still right you know!!!"
In Mandarin, there is a particular verb that means "to think or believe incorrectly" (以为). It is an ordinary word that is quite commonly heard in casual conversations. Of course, a magic word doesn't change a fear of looking stupid. Native Mandarin speakers have all the same ego problems that you do. Nevertheless, I wondered, what if I change the English words I say?
What if I say, "I'm sorry, I was wrong", instead of trying to explain how my solution might be correct in a parallel universe? What if I admit that I don't know everything, learn something useful, and move on without ego?