I know that angles can't be trisected. But suppose that I know for sure that unless I manage to find a way to trisect angles in the next ten years, the human race will be destroyed. What should I do? Surely I should try to trisect, hoping that there are mistakes in the impossibility proofs. And surely here at least is a place where I can do what I should. So it is possible to try to do what one knows to be impossible.