Chapter 4: Mathematics: Language, Modeling, and Comparison Assisting Inference

Recently a student in a graduate mathematics class I was teaching asked me after class, “Dr. Martin, what does math study?” I asked him to elaborate on what he was asking. (Actually, I said, “What?”). The student said, “Oh you know . . . botany studies plants and geology studies rocks, but what does math study?” I realized there were about six different ways to answer the question, all being true, and I needed to be careful with my answer.

The fact that the graduate student, with many years of experience studying mathematics, and a faculty member, with many years working/teaching as a mathematician, wanted to ponder this simple-sounding question speaks to the broadness and multifaceted nature of the mathematical enterprise. One can observe that over the years, many scientists and mathematicians have exploited the connections between mathematics and science to further the development of both; physicist E. F. Winger, if not the first, then very eloquently expressed the amazing truth noted in this paper’s introductory quote from Winger’s paper entitled “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” (Winger, 1960). Three years after this paper, Winger won the Nobel Prize in physics for his work in describing the nucleus of the atom (and coincidentally using algebraic group theory in the process). His paper is well-known for having generated controversy amongst scientists and the general public (who took turns arguing both sides of the premise that his observations supported the independence of science from mathematics and religion). In this chapter, as we attempt to answer this student’s question, we will contemplate the nature of mathematics, the role mathematics can play in inference, distinguishing mathematics from/as a science, and some ideas regarding paradigm shifts. Our thoughts will develop as we examine how various tools and branches of mathematics have developed side-by-side with the progress of science; in particular, we will be exploring the distinctions that distinguish mathematics and science and yet make each a necessary companion to the other. For the purposes of this discussion on inference, science, and mathematics, we emphasize the connections between paradigm shifts and new knowledge.