I think that language (at least English) fails us here. The word "creative" can be used to mean both "relating to or involving the imagination or original ideas, especially in the production of an artistic work" [1] or "resulting from originality of thought, expression, etc.; imaginative" [2]. It seems that most people identify the word more strongly with the artistic sense of the word, in which creativity is a proxy for a kind of self-expression that is not bound by any rules, logic, or structure. In doing so, they seem to mistake one of the more visible manifestations of creativity with its essence, which really lies in the "originality of thought" and "imagination" of the creative person.
If we had a specific, unique, and widely used word for the "artsy" free-expression type of creativity, I think the confusion many people express when you try to convince them that mathematics is a creative endeavour would be greatly diminished.
> creativity is a proxy for a kind of self-expression that is not bound by any rules, logic, or structure
I do not find this to be the case. Art is full of structure, rules and logic. When artists "break" the rules, it usually means they been able to operate using underlying rules, and understand well the rules they break.
Yes, indeed. For instance, the creation of bebop in jazz. Of course Bird could only do this by a comprehensive understanding of the existing rules and seeing deeper.
The way I read the essay, it seems to me that math is compatible with [1]. You don't have to follow any given rules. Make up your own ideas and see what they lead to. What if square roots of negative numbers exist? What if the sum of angles of a triangle is not 180 degrees? What if a proposition can be both true and false at the same time?
If we had a specific, unique, and widely used word for the "artsy" free-expression type of creativity, I think the confusion many people express when you try to convince them that mathematics is a creative endeavour would be greatly diminished.
[1] http://oxforddictionaries.com/us/definition/american_english...
[2] http://dictionary.reference.com/browse/creative