A short while ago, noted skeptic Michael Shermer wrote an interesting article about Steven Hawking's new book "The Grand Design" (another brief review from the Washington Post is here). In Shermer's analysis, Hawking's philosophy of model-dependent realism sounds a lot like radical constructivisim. Skeptical of the extent to which this philosophy should be pushed, Schermer asserts that Hawking's model-dependence has its limits, and that science provides an unprecedented means of overcomming the relativism that model-dependent realism and constructivism seem to lead to.
Coincidentally, there are a number of articles in the most recent issue of Constructivist Foundations (volume 6, number 1) that discuss the same issues that Shermer brings up in his assessment of Hawking's model-dependent realism.
What Shermer and the authors in Constructivist Foundations wrote about, and what seems to trouble people most about model-dependence and radical constructivisim vis-a-vis science, is whether or not science tells us anything about "reality." Shermer answers affirmatively:
"Yes, even though there is no Archimedean point outside of our brains, I believe there is a real reality, and that we can come close to knowing it through the lens of science — despite the indelible imperfection of our brains, our models, and our theories."
"...a scientific theory is regarded as a model, constructed to address certain questions that we want to ask, and then imposed on natural phenomena. If the model is successful, fine – but this is then better seen as due to the capabilities of the constructors (scientists) [rather than being closer to reality]."
What does this have to do with math? The debate between scientific realists and relativists maps partially (but perhaps not exactly?) onto the debates between neo-platonists and constructivist/intuitionists and formalists in the philosophy of mathematics (see a few quotes here). Also, after reading a few articles by radical constructivists, I am surprised how many of them are mathematicians and/or mathematics educators (a notable example is Ernst von Glasserfeld, another is Andreas Quale, quoted above).
Few doubt the constructed nature of scientific theories. As Shermer states, the issue is how far you go with constructivist arguments (is there an ultimate reality, or is it turtles all the way down?), and whether ultimately you are a realist, or a relativist. Perhaps asserting the existence of reality requires a Kierkegaardian leap, not of faith, but of rationality - and also marks the reasonable limits of skepticism.