Hi George, and thanks for your comment.
If mathematics is a language, why should it have the precision it appears to do? Why should the universe be describable and comprehensible at all? If it is abstracted from the world why should we be able to develop it in non-world realms ("mathematical ideas themselves" as you put it)? Why are we able to make abstractions, and in a precise way?
I realise that in asking such questions I am in a way moving beyond your points and in some way beyond the point of the article. But I am trying to emphasise that in trying to "solve" this question, we employ assumptions that raise larger, wider questions.
This shouldn't be taken as an excuse not to raise such questions! Indeed, I was trying to inculcate a sense that there is still great mystery in the world.