Problem #26

This month's problem is a logic puzzle. Four mathematicians make the following statements:

Alice: "I am insane."
Bob: "I am pure."
Charlie: "I am applied."
Dorothy: "I am sane."
Alice: "Charlie is pure."
Bob: "Dorothy is insane."
Charlie: "Bob is applied."
Dorothy: "Charlie is sane."

Describe the four mathematicians, given that:

Pure mathematicians tell the truth about their beliefs.
Applied mathematicians lie about their beliefs.
Sane mathematicians' beliefs are correct.
Insane mathematicians' beliefs are incorrect.

Source: Mathematics and Informatics Quarterly (via Tom Halverson)