I don't think the author is advocating letting all the students come up with the "solutions" to concepts like the Pythagorean Theorem; that is, he's not saying the students should be given a blank slate and left to their own devices in order to discover mathematical concepts. Rather, it seems to me that he is simply offering a pedagogy that is opposite of that which is currently being used, where students are taught that mathematics, like painting or literature, can be considered a form of "art," or a very creative subject. The author contrasts the way we teach the art of painting and the way we teach the art of mathematics. In painting class, teachers do not simply say, "Go nuts, and do whatever." In these types of art classes, the instructors explain the ideas of contrasting colors, what colors are objectively good together, how to use a brush and the mediums, and then allow creativity, because the students are given tools and are engaged. Mathematics should be taught using this same approach. Allow the students to delve into the reasons for equations, explain the beauty of mathematics, and allow the students to think critically and come up with new ideas. I, for one, wouldn't imagine this would take too much time, and approaching mathematics instruction from the perspective that it is a beautiful art would be extremely beneficial.
Well, I'd say that in the case of the Pythagorean Theorem, we should actually explain it. There's like 100+ different proofs, so there are options to actually explore the methodology of discovering something like this. From a personal example, this year in math class I really wanted to understand math deeply, so I had to ask for a lot of explanations before class and stuff. Like the change of base formula with logarithms. I had no idea why this worked, but when I found the proof in my book and on Wikipedia, I was like "Dang, that's pretty cool." I had fun getting deep into the abstract parts of other mathematical tools, too. That's what had an impact on me, and I've never liked math as much as I do now. As for looking at new ideas, another personal example, if you will. On my test, instead of writing "ln", I wrote Log base e. I understood that more, and I looked at it differently, but I got 3 points off because of it. Maybe I'm bitter, but looking at a formula a different way or finding new ways of doing things shouldn't be discouraged, rather encouraged.