Turning now to the role of mathematics in natural science, this chapter takes up a number of themes. First, the Quine/Putnam idea that the indispensability of mathematics for much of science confirms the existence of mathematical objects: from a second-philosophical point of view, this argument ignores the fine-structure of scientific justifications (in favor of superficial holism) and the idealizations and simplifications involved in the use of mathematics. If indispensability arguments are rejected, then what does the evidence tell us about the mathematical structure of the world? The KF-structure already detected is enough to support elementary arithmetic, but based on a return to the developmental studies prominent in §III.5, the Second Philosopher argues that as soon as we reach the infinitary ‘and so on’ of number theory, we've entered the realm of idealized, abstract theorizing. The chapter concludes by applying the lessons of Diaconis's statistical work on coincidence to argue that Wigner's ‘miracle’ of applied mathematics is less miraculous than it might seem.
Keywords: applied mathematics, arithmetic, coincidence, Diaconis, holism, idealization, indispensability argument, infinity, KF-structure, Putnam