A: I am insane.

B: I am pure.

C: I am applied.

D: I am sane.

A: C is pure.

B: D is insane.

C: B is applied.

D: C is sane.

You are also 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.

Describe the four mathematicians.