dialectics in mathematics

This post is part of the Virtual Conference on Mathematical Flavors, and is part of a group thinking about different cultures within mathematics, and how those relate to teaching. Our group draws its initial inspiration from writing by mathematicians that describe different camps and cultures — from problem solvers and theorists, musicians and artists, explorers, alchemists and wrestlers, to “makers of patterns”. Are each of these cultures represented in the math curriculum? Do different teachers emphasize different aspects of mathematics? Are all of these ways of thinking about math useful when thinking about teaching, or are some of them harmful? These are the sorts of questions our group is asking. (intro by Michael Pershan)

I want to talk about how we respond to polarities. Here I mean “polarity” in the philosophical sense (a pair of concepts that are apparently in conflict) rather than in a mathematical sense. When we encounter a struggle or tension between goals or ideas, we tend to create one of two things:

• dichotomy — a conclusion that the two ideas are irreconcilable and the choosing of sides, or
• synthesis — a selection of desirable features from each and the attempt to make those features coexist.

While each approach is at times appropriate, both have their downsides. Establishing a dichotomy means that one side tends to be silenced and its contributions lost. Creating a synthesis can mean that neither side is fully honored; everything is compromise.

I propose a third option, an alternative to dichotomy or synthesis: this approach is dialectic — upholding both sides fully, maintaining the two ideas in tension so that a conversation may arise between them. Etymologically, “dialectic” comes from the roots “dia” (“across”) and “logos” (“word” or “reason”), so its underlying meaning may be read as “speaking across a divide”. Dialectics can simply refer to discussion or debate between two opposing sides, but I use it to denote a state that seeks not resolution, but rather the fruitfulness of an irreducible struggle. Doing so acknowledges the worth, validity, and potency of both sides. It can therefore be used in the classroom to foster the inclusion of diverse perspectives, even in mathematics.

Our group’s discussion began with an essay by Timothy Gowers entitled “The Two Cultures of Mathematics”. In this piece, Gowers makes the claim that most mathematicians are either “problem solvers”, who prefer to attack specific open problems that they believe are important, or “theory builders”, who prefer to develop a large, coherent body of understanding. The former are interested in general theory mainly insofar as it provides ways to solve their problems; the latter are interested in specific problems mainly insofar as they spur deeper insights or new directions for theory.

This subdivision is similar to the pure/applied separation we often talk about in mathematics, though it is not quite the same thing. Even the problems Gowers mentions fall well within the “pure” category. But these two polarities (pure/applied, theory/problems) share the feature that adherents of one side tends to be a bit snobbish towards those of the other.

Pure mathematicians tend to look on applied mathematics as, at best, a dirty form of math or, at worst, not truly math at all. G. H. Hardy, in his famous essay A Mathematician’s Apology, describes pure mathematics as more enduring, more exciting, and more “real” than applied mathematics. (He does make clear that what he considers “applied” mathematics limits itself to “elementary” tools, which more-or-less means grade-school arithmetic up through introductory calculus, and so his notion of applied mathematics might no longer suffice. I’ll get back to Hardy shortly.)

Gowers claims that, in a similar way, theory-building is currently “more fashionable” than problem-solving in the math world. (Rather than drawing the analogy with pure and applied mathematics, however, he compares this snobbishness with one, observed by C. P. Snow in “The Two Cultures”, held by humanities toward the sciences.) He laments that “this is not an entirely healthy state of affairs” and spends most of his essay defending problem-solving areas of math (combinatorics, in particular) against some perceived criticisms. His argument suggests to me that both the theory-building and the problem-solving camps should be upheld without one attempting to overcome the other; that is, a healthier state can be reached by sustaining a dialectic.

How can we think about theory-building vs. problem-solving in our classes?

For one thing, many of our students are trained problem-solvers. For them, learning mathematics means developing an appropriate response to any given stimulus. If a problem statement includes this-or-that word or phrase, then I should use such-and-such a technique to find a solution. For many of us instructors, however, it is the abstraction of ideas that drew us to mathematics. What is possible in this situation? To what extent can the possibilities be quantified and categorized? If theory-building is currently en vogue in mathematical culture, then I suspect we who teach are not immune to that trend. But here comes the question of motivation: what will draw students into doing mathematics? In many cases, the answer is… a problem. The problem may be “applied” (e.g., how does a population grow over time) or “pure” (e.g., how does the size of a square increase when its side length increases?), but a concrete connection provides an open door to considering broader mathematical truths. Such problems can lead into developing theory (e.g., what properties do exponential and polynomial functions share, and what distinguishes them?).

But developing theory for its own sake has been a part of mathematics since at least Euclid; we do our students a disservice if we neglect this aspect of doing math. A theory crystallizes into a single lattice ideas that might otherwise have been perceived as disconnected. Algebra in particular provides a unifying framework for solving individual problems. On the other hand, non-constructive statements are by turns inspiring and infuriating. It is no small movement from the (typically algebraic) claim that “A solution exists! And you can find it by following these steps…" to the (typically analytic) claim that “A solution exists! And you may never find it exactly…” This theory in turn motivates a slew of new problems: if nothing else, how shall we find solutions as close to the true answer as we desire?

In any case, it is useful to abide by a constructivist view of knowledge: students will understand best the structures that they form in their own minds, whether by induction (problem-solving) or deduction (theory-building), and they should be presented with ample opportunities for both forms of construction.

[Side note: in his keynote post for this series, Michael describes an occasion where he side-steps, or deconstructs, the theory-building/problem-solving divide by encouraging math-doers to create their own questions based on a simple prompt, questions which could easily veer in any direction, including problem-solving or theory-building.]

It is not hard to find other places in mathematics where polarities exist and a choice must be made: dichotomy, synthesis, or dialectic? A few weeks ago, I made a bit of a fuss on Twitter, claiming that everything Hardy wrote about mathematical culture should be read skeptically. The context for my criticism was an oft-shared quote from “A Mathematician’s Apology”: “Beauty is the first test: there is no permanent place in the world for ugly mathematics.”

A question immediately presents itself: who decides what is beautiful? Any claim to objectivity is nearly always tied up with privilege. The answer cannot be “all mathematicians” because we all have such different tastes and preferences. Nor can the answer be “a special subset of mathematicians” because the choice of that subset will inevitably be determined by power structures within the mathematical community. But neither is the answer that all mathematics is equally beautiful. The standard of beauty may be subjective, but that does not mean it is arbitrary. We value beauty, but it is not the sole or even the primary standard by which we judge mathematics.

Hardy argues that all mathematics considered “useful” is essentially “dull” or “trivial”. He seeks to create a dichotomy between the beautiful and the practical. Perhaps he didn’t foresee the computing revolution. He couldn’t predict that number theory would be used in encryption, or that general relativity would be used for GPS, or that differential equations would be used for movie animations. Perhaps he would not consider these applications to be built on the deepest, truest parts of those theories. (To be fair, at the time he wrote, Hardy was distressed by the ways in which science had been used in the cause of warfare, and wanted to establish some distance between pure mathematics and that particular set of applications.)

From an evangelistic perspective, potential converts (our students in particular) may be drawn in from either side: the aesthetic or the practical. I personally was first attracted to geometric form and the lovely, counterintuitive properties of mathematical relations. Some of my students have tastes similar to mine, but many more will be convinced of the predictive power of mathematics before they accept its inherent attractiveness.

Beyond this, however, neither beauty nor usefulness can or should be subjugated to the other. Mathematics is grounded in both. They can hone each other, but they can also proceed independently. Progress flows from the pursuit of either. It would be a mistake, I believe, to claim that either is the true purpose of mathematics; we should support both of them in our minds and in our classrooms.

Not every tension needs to be handled this way, but examples of dialectical pairings abound: precision and approximation, confidence and confusion, individual and community. I encourage us all to consider times when it can be productive not to resolve such conflicts but instead to foster a breadth of understanding from them.