I suspect that some cynics might argue that this is true precisely because we define mathematics as an internally consistent set of definitions and rules -- as a constructed system. Yet I myself am sympathetic to claims of the universality of mathematics beyond social construction. Second, mathematics seems particular effective as the language of science. Dehaene quotes Einstein, "How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?" Again, a cynic might claim that much of mathematics has been defined for the express purpose of describing our empirical observations. But that really begs the question. What are the patterns common to math and science that make this convergence convenient, even possible? Dehaene's explanation for universality and effectiveness rests in evolutionary biology -- and patterns:If the Pope is invited to give a lecture in Tokyo and attempts to convert the locals to the Christian concept of God as Trinity, I doubt that he'll convince the audience -- Trinity just can't be "proven" from first principles. But as a mathematician you can go to any place in the world and, given enough time, you can convince anyone that 3 is a prime number, or that the 3rd decimal of Pi is a 1, or that Fermat's last theorem is true.

The world and our brains together drive us to recognize the patterns that exist in the world. I am reminded of a principle that I think I first learned from Patrick Henry Winston in his text Artificial Intelligence, called... mathematical objects are universal and effective, first, because our biological brains have evolved to progressive internalize universal regularities of the external world ..., and second, because our cultural mathematical constructions have also evolved to fit the physical world. If mathematicians throughout the world converge on the same set of mathematical truths, it is because they all have a similar cerebral organization that (1) lets them categorize the world into similar objects ..., and (2) forces to find over and over again the same solutions to the same problems ....

The close compatibility of math and science marveled at by Einstein and Dehaene reminds me of another of Winston's principles,The world manifests constraints and regularities. If an agent is to exhibit intelligence, then it must exploit these constraints and regularities, no matter the nature of its physical make-up.

(If you never had the pleasure of studying AI from Winston's text, now in its third edition, then you missed the joy of his many idiosyncratic principles. They are idiosyncratic in that you;ll read them no where else, certainly not under the names he gives them. But they express truths he wants you to learn. They must be somewhat effective, if I remember some from my 1986 grad course and from teaching out of his text in the early- to mid-1990s. I am sure that most experts consider the text outdated -- the third edition came out in 1992 -- but it still has a lot to offer the AI dreamer.) So, math is more than "just" a mental construct because it expresses regularities and constraints that exist in the real world. I suppose that this leaves us with another question: do (or can) law and religion do the same, or do they necessarily lie outside the physical world? I know that some software patterns folks will point us to Christopher Alexander's arguments on the objectivity of art; perhaps our art expresses regularities and constraints that exist in the real world, too, only farther from immediate human experience. These are fun questions to ponder, but they may not tell us much about how to do better mathematics or how to make software better. For those of us who make analogies between math (or computer science) and the arts, we are probably wise to remember that math and science reflect patterns in our world, at least more directly with our immediate experience than some of our other pursuits. -----The longer two situations have been evolving in the same way, the more likely they are to continue to evolve in the same way.