Notice how it goes from "we proved Fields-medal level stuff" to "promising leads" when we go to the world of biology. Still awesome, but speaks to the limitations of pure thinking in a more applied subject. No automatic cancer cures today.
"Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?"
It's a fight! I envision myself as Jacob wrestling the angel.