Chapter 9: Life in Mathematics: Evolutionary Perspectives on Subject Matter

Edgardo Cheb-Terrab is an applied mathematician who specializes in developing algebraic algorithms for Maple, a mathematical software package for symbolic computation. His algorithms solve classes of problems, including differential equations and special functions. In 1999, Cheb-Terrab used his software to investigate solutions of Abel equations—a class of first-order non-linear differential equations that was first described by the Norwegian mathematician Niels Abel in the 1820s. By the end of the 20th century, the mathematics research literature contained solutions to nearly 40 types of Abel equations; each of these types was thought to require its own method of solution. Cheb-Terrab showed that all these types are special cases of an 8-parameter hyper-class. By devising a computer algorithm for solving all equations of this hyper-class, he expanded the range of solvable Abel equations far beyond what was thought possible. These results could not have been derived without a computer.

Even though one may expect such innovation to be greeted enthusiastically, Cheb-Terrab’s work has met with suspicion and even dismissal by mainstream algebraists. “For many of them, it was heresy; surely a computer cannot solve problems that were impenetrable to such great mathematicians as Abel and Liouville” (E.S. Cheb-Terrab, personal communication, November 2, 2008). Ironically, Cheb-Terrab’s censure by the research community came at a time when thousands of mathematicians, engineers, and physicists were already using his algorithms, and verifying the solutions obtained, in a variety of applications. It took four years, and a lengthy review process, to publish the findings (Cheb-Terrab & Roche, 2003).

Cheb-Terrab has since developed innovative computer-based solutions to other long-standing problems in algebra. Yet he regularly confronts obstacles to acceptance. As Cheb-Terrab (personal communication, December 13, 2008) noted, “[m]athematicians have received with discomfort almost every algorithm that I developed which seemed to challenge ‘established truths’; but, in fact, these established truths were never anything more than incomplete truths holding back progress in their fields.”

The issue of what constitutes acceptable mathematics innovation points to a prevalent orthodoxy among mathematicians around the question “What is mathematics?” Even though today’s digital technologies enable new mathematical understandings, many mathematicians are unwilling to accept computerized solutions as “real mathematics.” For them, the only true mathematics is that which manifests in the time-honored mechanisms of formal proof. As Cheb-Terrab’s example illustrates, these mathematicians, in their strict conformity to traditional modes of mathematical knowledge production, may be stunting the evolution of mathematics and contributing to stagnation of mathematical research.

Likewise, we believe that part of the blame for the current stagnation of mathematics education and pedagogy can be attributed to a shared orthodoxy among educators around the question “What is mathematics?” From an evolutionary perspective, we understand the term orthodoxy as referring to rigid adherence to a particular worldview, and refusal to acknowledge and participate in the evolution of consciousness. Mathematics pedagogues often perceive mathematics as a treasured, monolithic, even sacred body of knowledge, which must be preserved and passed on to future generations. This perception of mathematics often leads to a model of instruction that centers on transmission of stable knowledge.

This chapter explores the range of worldviews that respond to the question “What is mathematics?” We will use the evolutionary frameworks of complexity science and integral philosophy to analyze the stages through which conceptions of mathematics have evolved to date, and where they are likely to evolve next. We examine barriers to the evolution of teachers’ views on mathematics, and some approaches from the authors’ experience to overcome these barriers. We conclude with a discussion of the implications that an evolutionary view of mathematics holds for pedagogy, and, in particular, the need for educators to balance stable and emergent dimensions of mathematics in their instruction.