Paul Zorn on Communicating Mathematics III: What Is Mathematics?

Paul Zorn, St. Olaf College

Part I: Introduction

Part II: Mathematics in Literature

Part III: What Is Mathematics?

OK, let’s get more serious and mainstream about mathematical communication as such, in service of mathematics itself.

What, first of all, is this thing called mathematics that we’re hoping to communicate? Many people would mention patterns. For others, mathematics is just what mathematicians do. In any case, defining mathematics precisely, and non-recursively, is famously difficult.

Perhaps it’s a moving target.

The late William Thurston, who received a Steele Prize in 2012, had interesting things to say about this in a 1994 paper, “On Proof and Progress in Mathematics”, in the Bulletin of the American Mathematical Society. “Could the difficulty of giving a good direct definition of mathematics be an essential one, indicating that mathematics has an essential recursive quality?”

Continuing along these lines, Thurston proposes seeing mathematics as a sort of convex hull, or span, of simple axioms. Mathematics, he proposes, is the smallest subject satisfying three properties:

  1. Mathematics includes the natural numbers and plane and solid geometry.
  2. Mathematics is that which mathematicians study.
  3. Mathematicians are those humans who advance human understanding of mathematics.

“As mathematics advances,” Thurston observes, “we incorporate it into our thinking. As our thinking becomes more sophisticated, we generate new mathematical concepts and new mathematical structures: the subject matter of mathematics changes to reflect how we think.”

I like the expansiveness of Thurston’s definitions, both of mathematics and of mathematicians. It implies, at least to me, that mathematical communication is not just descriptive of something else called mathematics, but rather a valid form of mathematics in its own right. When we help ourselves and others to understand mathematics, through teaching, study, writing, and communication in all its forms, we are doing, not just observing, real mathematics.

If mathematics itself is not static but growing, perhaps even recursively, what it means to know and understand mathematics is open to question, too. Thurston points to difficulties with a simplified definition-theorem-proof model, in which mathematicians start with accepted axioms and definitions and try by standard transformations to prove or disprove important assertions about such systems.

A better model of mathematics and mathematical exposition, Thurston asserts, must account for more variety in how real people learn and understand the subject. Valid but variant views of the derivative, for instance, could involve infinitesimals, rates, epsilons and deltas, symbolic, microscopic views, geometric interpretations, a Lagrangian section of the cotangent bundle, and more. Effective mathematical exposition needs to take due account of this variety and richness.

To quote Thurston once more:

We mathematicians need to put far greater effort into communicating mathematical ideas. To accomplish this, we need to pay much more attention to communicating not just our definitions, theorems, and proofs, but also our ways of thinking. We need to appreciate the value of different ways of thinking about the same mathematical structure. We need to focus far more energy on understanding and explaining the basic mental infrastructure of mathematics—with consequently less energy on the most recent results.

This entails developing mathematical language that is effective for the radical purpose of conveying ideas to people who don’t already know them.

Terence Tao, writing in his blog, makes a related point, roughly dividing mathematical education into three stages:

The pre-rigorous stage, in which mathematics is taught in an informal, intuitive manner, based on examples, fuzzy notions, and hand-waving. (For instance, calculus is usually first introduced in terms of slopes, areas, rates of change, and so forth.) . . .

The rigorous stage, in which one is now taught that in order to do maths “properly”, one needs to work and think in a much more precise and formal manner (e.g. re-doing calculus by using epsilons and deltas all over the place). The emphasis is now primarily on theory; and one is expected to be able to comfortably manipulate abstract mathematical objects without focusing too much on what such objects actually “mean”.

The post-rigorous stage, in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready to revisit and refine one’s pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory. . . . The emphasis is now on applications, intuition, and the “big picture”.

I find this analysis convincing. It implies to me that really excellent mathematical communication, including with students, must take account of and sometimes span all of these stages of mathematical development. That’s a difficult task, but it’s crucial to the best mathematical communication.

Part IV: Mathematical Language: Learning from Barbie
Part V: Rule Books and Tour Guides: Two Live Questions
Part VI: Lessons from History
Part VII: Valuing Communication

What is Math Comm

MAA Mathematical Communication ( is a developing collection of resources for engaging students in writing and speaking about mathematics. The site originated in the MIT Department of Mathematics and was expanded through support from an NSF grant.