The exchange of ideas between mathematics and biology was very important during the 20th century. This constructive dialogue has been helpful in passing mathematical ideas to biology and in passing biological ideas to mathematics. On the one hand, the mathematical ideas of S. Wright, R.A. Fisher, and J.B.S. Haldane in the nineteen thirties enabled the synthesis between proponents of Darwin and Mendel. On the other hand, neural networks, genetic algorithms, Lindenmayer systems, and chaotic systems are branches of applied mathematics which have directly emerged from biological examples. Recent work has included approaches that re-apply these branches of mathematics to new areas of biology. Dennis Bray has applied neural nework methods to protein interactions within cells (1995). Denis Noble has used supercomputers to model the functioning of the heart from communication between its individual cells (Noble).
The successes of this dialogue asks us to question the relationship between the two subjects. The successes of the mathematical methods in physics has lead some philosophers to suggest that the universe is fundamentally mathematical. This might lead us to assume that all of biology should be expressed as mathematics. On the other hand, some, such as biologist Ernst Mayr, that there are significant differences between mathematics and biology. Specifically, Mayr suggests that mathematics finds single answers to problems, whereas "it is quite possible that in biology the majority of phenomena and processes must be explained by a plurality of theories" (Mayr, 1997, p.68). Mayr concluded that biology has more in common with history (Note: qualify this caricature). In the middle ground are those that admit both differences and similarities between the subjects. For them, a common discrimination is that mathematics deals with universal answer whereas biology deals with unique examples. (Stewart, Sigmund, etc)
How does this dialogue relate to the mathematics education of biology students? We cannot assume that students who are good at biology will be good at mathematics. If we require biology students to be good mathematicians, we may exclude students with the potential to advance the field of biology. We are looking at two questions: