Seeking independence is part of growing up. It starts at a young age when your child tells you, "I can do it myself!" and increases from there. It's an admirable characteristic in one regard as we want people to learn to stand on their own feet and not be forced to lean on others for everything. We applaud people for their independence as they figure out what it means to be self-sufficient. We even equate maturity with the level of personal independence.
Perhaps a better indicator of maturity would be in realizing our need for interdependence. It doesn't mean that we aren't capable of doing things on our own, but that we strive to work together with others in community. Instead of touting our ability to things on our own we find ourselves saying, "I need you and you need me." Strong willed people are usually only good with half of that statement while mature people recognize the value in mutual investment.