Pedagogical and scientific goals

I posit that understanding is the core value of mathematical modeling. There are (at least) two levels of understanding, the understanding of our own questions. The first aspect of understanding enhanced by modeling is making our spoken language precise with the aid of mathematics. The second aspect of understanding is providing an unambiguous structure to our ideas that the scientific community can use, that is, the development of useful theory.

I like to think of the scientific process of knowledge creation as a 3D spring, coil, or spiral, where a single loop represents a complete cycle of the scientific process (question, hypothesis, test, interpretation), and progress occurs as we repeat the process through multiple cycles, traveling down length of the coils. Mathematical modeling can help us at different phases of a single coil.

I think that making ourselves formalize our conceptual models helps us see and understand our ideas to a greater degree. Formalization helps us become ever more specific and thereby operationalize our hypotheses and thereby generate more testable predictions. Going through the formalization process helps us understand what a mathematical model is and and how mathematical models provide structure to theory. The process helps show us and convince us of how models are used in Science.