The full paragraph (in fact, the very first paragraph of the essay) reads: > It is a melancholy experience for a professional mathematician to find himself wri

The full paragraph (in fact, the very first paragraph of the essay) reads: > It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds. I interpreted "men who explain" not as mathematicians that can explain their work well, but as people who try to explain mathematics in a "lay" way to cater for a large audience, whose explanation can very often become inaccurate, non-mathematical or just downright false, yet still get public credit for seeming to know the field very well, despite these inaccuracies, and even though they are not directly pushing the advancement of the field itself. It's of course a good thing to try to reconstruct your own thoughts, but I wouldn't say it's unreasonable for a mathematician to omit doing that. Could you go into more detail on the examples you mention, where doing so led to new insights?