Abstract I suggest five research paradigms fundamental in the mathematical sciences that seem relevant to the hunt for explanations of genetic susceptibility to complex diseases. I discuss a few ways in which these paradigms are, or are not, currently being followed in complex disease genetics. This essay is intended to provoke discussion and not to resolve any problems. It represents a written version of remarks that I made as Discussant at a session at the 1997 National Institutes of Health Research Festival.