George Cantor first worked out the basic ideas about infinity in 1890. His theories were criticized at first, but now are accepted as one of the foundations of modern mathematics. Among other things, Cantor first showed that there are different types of infinity and that a given set has exactly as many positive numbers as there are positive even numbers (while conventional wisdom would suggest that there are twice as many positive numbers than positive even numbers).
This brilliant mathematician battled depression throughout much of his life, and died in 1918 in an institution. His work was initially received with harsh criticism, before finding acceptance in the mathematical community. Though some of his findings about infinity were certainly mind-boggling, his mental state likely had more to do with the criticism he received for his work than the concept of infinity itself!