I'm not a mathematician (more of a general science geek) but I ran across this question by chance, and I feel that mathematician John Nash deserves a mention.

He had important direct and indirect positive impact on society, both through his mathematical models of game theory and the role of money in society, as well as aspects of evolutionary psychology, where he personally demonstrated that an individual with severe mental illness (paranoid schizophrenic) can function and make useful contributions to society.