I think there is a difference between being left alone and being lonely.
Sometimes, I want to be left alone as I have to have my own space at times. I will get a bit more of this now as I now live on my own, so will be able to put more time into my studies and hobbies than I did previous (or thats the theory anyway). However, I don't think anyone ever wants to be lonely as it is a horrible feeling. To me, it feels like rejection. Being lonely is an profound emotional state of mind whereas being left alone is not, that's just when either you or other people (or both) choose to keep your or their distance. You don't have to be alone to feel lonely, you can be in a room full of people and still feel this way. In the past, I experienced this because I was surrounded by people who I didn't care about (or some that I did) and they didn't care about me either. Another occasion is where I have been in a relationship and he didn't care enough about my feelings to even ask how I was yet expected me to give a toss about him. That makes you feel lonely, even though there is someone there, and it's horrible. Whereas, to me, being left alone isn't horrible, it's being made to feel I'm not worth hanging out with or outcasted that's bad, but then that's different again and another issue entirely.
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The only person in the world that can truly make you happy is yourself.