An uninformed observer who came across a Mandelbrot set would presumably think that it was the product of an intelligent designer:
But in fact, the Mandelbrot set is the product of a relatively simple mathematical equation. Ratzsch addresses this in an interesting footnote:
prior to any familiarity to their mode of generation, it might be intuitively reasonable to take Mandelbrot pictures as designed. In fact, were their structure artifactual I suspect that we would so construe them. But, of course, their structure is a result of mathematical necessity, and some would argue that necessities cannot be products of agent activity and design. (p. 185)
I take it from what Ratzsch does say that he recognizes that he doesn’t have a completely satisfying answer here. What I would (tentatively) say is that the structure (despite appearances) is not complex — it can be generated via a simple mathematical process. Ratzsch wouldn’t be happy with this complexity answer — after all, as discussed in my previous post, a titanium cube is not complex, and yet Ratzsch takes it as evidence for design — but the complexity answer fits with the answer that for example Dembski would give. (Dembski’s filter wouldn’t infer design unless the pattern had specified complexity, but the Mandelbrot set is not complex. Dembski’s filter sometimes won’t infer design even when something is designed though, so from the fact that Dembski’s filter doesn’t infer design, we can’t infer that the pattern was not designed.)
I should note that, in practice, Mandelbrot sets are produced by designers — the people who wrote the computer program to produce them. What I’m interested in is if we (somehow) came across something like a Mandelbrot set in nature — say, in the pattern of a leaf. Would that pattern provide evidence for the existence of God?
To read more about these sorts of issues, see my new book, Seeking God in Science: An Atheist Defends Intelligent Design.