The well known situation of language translation is I believe akin to the tension between thinking and explaining.
I am French, I can understand, write and explained myself in English yet I am a bit at a loss when required to translate some piece of English into French so I call this translating ability a third language. My guess or feeling or personal view is that the thinking is more semantic and - for weird (and sad) cultural reasons- mathematical explaining is too often required to be on a syntactic level.
