A lot of the discussion going on above is about the fact that we do not understand our mind's working. I do not even know if mathematical thinking is language-bound. I have the impression that it is not: mathematical discussions are more tedious in foreign languages but not more difficult. An analogy is perhaps a walk on a barely visible path. Sometimes you lose it and have to search for it. Glimpsing it again you tell yourself: "Ah, here it is again" but language is completely irrelevant even if it can be used to describe it to a friend. How do other people feel about this? Perhaps we are all different and do not see not the "same" reality. (Is your color "red" the same as my color "red"). Perhaps we are making a huge mistake assuming that different people think very similarly: For a dog, reality is made of a lot of scents, for a horse it is perhaps prairies and vast avenues where it can gallop. Some of us are perhaps "dogs" and other "horses" of mathematics and since most scents and prairies have not been named, we have to communicate through a limited common denominator, even among people of the same "mathematical species".