1. The word that goes along well with "stereotypically" is hermetic.
2. The second blank should support a phrase "A for B". Poincaré forsook A for B. Poincaré forsook the messiness of human engagement for the precision of mathematical concepts.

Mathematician Poincaré lived a stereotypically (i) ________________ existence. Like many of his contemporaries, he (ii) ________________ the messiness of human engagement for the precision of mathematical concepts.

Since he favoured the precision of mathematical concepts to the messiness of human engagement, the word for the blank should be a synonym of rejected, hence forsook is the correct choice.

And, since he forsook human engagement, he must have lived a hermetic existence. The word stereotypically also supports this choice, since the stereotypical image of the mathematician is one who lives all by himself, immersed in his arcane ideas.
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