He's saying design is trial-and-error, but it helps to have an approximately right model to start from. It seems to me that mathematics could lead us directly to the right model without mistakes... or is the truth that the same trial-and-error still occurs, but just in more abstract and general terms? Does that seem right to the mathematicians here?

... the nature of design, as they discovered, is so intensely cognitive, happening inside the mind at mind speed, the researchers could conceive of no useful tools to help in that process!

This is exactly my experience. The closest is diagrams with pen-and-paper, but it doesn't help that much. The more difficult part is not solving the problem, but showing that you've solved it i.e. a proof, for all possible cases. Finding a proof of a solution is as much work (if not more) than finding the solution in the first place.

It is by logic we prove, it is by intuition that we invent - Henri Poincaré

The article is talking about consciously formulated conjectures, not about instantaneous insight, discussed here: p.62 of Intuition in science and mathematics: http://books.google.com/books?id=qqGWlEwWj5UC&pg=PA62...