Four mathematicians have the following conversation: 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. You are also given that: a) Pure mathematicians tell the truth about their beliefs. b) Applied mathematicians lie about their beliefs. c) Sane mathematicians' beliefs are correct. d) Insane mathematicians' beliefs are incorrect. Describe the four mathematicians.

Here are the statements made by each of them: 1. Alice: I am insane. (A1) 2. Bob: I am pure. (B1) 3. Charlie: I am applied. (C1) 4. Dorothy: I am sane. (D1) 5. Alice: Charlie is pure. (A2) 6. Bob: Dorothy is insane. (B2) 7. Charlie: Bob is applied. (C2) 8. Dorothy: Charlie is sane. (D2)

We can divide the statements into two categories: self-description statements (A1, B1, C1, D1) and statements about others (A2, B2, C2, D2). Let's begin with the first category: If A1 is true (Alice is insane), then Alice's belief is incorrect, so A2 must be false, which means Charlie is not pure. If A1 is false (Alice is sane), this means Alice's belief is correct, so A2 must be true, which means Charlie is pure. If B1 is true (Bob is pure), then according to the given information, he tells the truth about his beliefs. So his statements B1 and B2 are both true. If B1 is false (Bob is not pure), then it means Bob is applied or insane. In either case, he cannot tell the truth about both his beliefs, so B2 is false, and Dorothy is not insane. Now, we have contradictory situations: if Alice is sane, Charlie cannot be pure since both A1 and A2 statements are true. But if Alice is insane, Charlie is also not pure because both A1 and A2 statements are false.

Let's look at statement C2: "Bob is applied." If C2 is true, it means Charlie is sane, making statement D2 also true. Consequently, D1 (Dorothy is sane) must be false. However, this contradicts the fact that both B1 and B2 statements are false if Bob is applied. Therefore, C2 must be false, meaning that Bob is not applied, making him pure. Now we know that Bob's statements B1 and B2 are true, which means that Bob is pure, and Dorothy is insane. With this information, we can now conclude: - Alice is insane (A1). - Bob is pure (B1). - Charlie is not pure (A2 is false because Alice is insane), but he is also not sane (D2 is false because Dorothy is insane). This means Charlie is applied. - Dorothy is insane (B2).

In conclusion, the four mathematicians can be described as follows: - Alice is insane. - Bob is pure. - Charlie is applied. - Dorothy is insane.