The passenger climbed the steps to the airplane. I waited at the top, in my pilot’s uniform, smiling a greeting. She looked up and blurted out “You look like a professor!”

“I am.” I don’t usually tell passengers this, but she asked. “And you’re the pilot?”

“I am.”

She started toward the cabin and stopped. “Of what?”

“Math.”

No reply. I gave her the safety briefing, walked into the cockpit, and started the engines.

For 25 years I have been both a Mathematics professor and a professional pilot. Most of my professional flying comes in the summer, with some weekends and nights during the school year. I teach flying all year. I have never missed a class because I was flying. Nor is it a hobby: I have the highest grade of pilot certification, and am held to the same standards as the full–time pilots. Most pilots flying the King Air I fly make it a career.

***

I started as an unimaginably abstract Math major. My electives were literature and history courses, not the courses associated with mathematical applications. I barely passed Freshman Physics. The low grade was partly because I didn’t find it abstract enough, but mostly because I was an irresponsible first year student. I took no other science courses. The Computer Science courses I took were also theoretical and abstract, covering few of the techniques I used at the summer programming job that their presence on my transcript made possible.

Former Harvard Dean Harry Lewis, himself a computer scientist, said that the apparent purpose of a Harvard education was to never study anything useful (2006). That was true for me: my limited experience with hands–on applications did not come from school. I studied wonderful abstract things with amazing people, but learned far more about how to do things from the campus radio station than I did in class.

Flying was a long–standing interest, in the abstract—even though it’s safe to say there is no abstraction in flying! I started learning when my post–doc salary meant I could afford it, and I quickly learned that when it came to aviation everything I knew from school was wrong. I had done and taught dozens of textbook problems in which airplanes flew due North at fixed speeds and altitudes. As a student pilot I learned that this is difficult, if not impossible. North? The compass is subject to all kinds of errors, some of which depend on whether you’re in the northern or southern hemisphere. Speed? The number on the airspeed dial, isn’t speed through the air at all, or speed over the ground. Altitude? Altitude is, at best, a fuzzy concept we gave up on decades ago. The Earth is not the perfect sphere the textbooks assume it to be, either. And then there was weather—but more about that later.

This was a different kind of reasoning from what my education taught me.

Let me explain. Many people are familiar with weather radar from television or internet reports. Watch it and decide whether the rain is coming. That’s the most abstract view: a mathematics textbook might say “the storm is 15 miles away and moving at 20 miles per hour,” setting up some practice calculations. In this context those are given, Platonic numbers, with no regard to how one comes to know them.

In truth, measuring distance and speed is a genuine problem that abstract mathematics does not address. It does not even acknowledge that, in fact, a measurement was made, or how (textbooks are full of measurements that are impossible or even silly to try to make). Measurement requires looking at the technical aspects of sending, receiving, and processing the radar signal. Where mathematics renders measurements in concrete values, in fact measurements of distance and speed are presented with some kind of plus–or–minus tolerance. That’s a big difference in approach. On the one hand, the number is known—concrete, without abstraction; on the other, the number is an interpretation of radar signals.

Larger airplanes have their own weather radar that helps pilots avoid the hazards of hail or heavy rain. But it takes skill to operate and interpret radar signals effectively.

There are four kinds of reasoning here; and representations of their practitioners can be seen as:

• a mathematician, who doesn’t need to measure (measurement is a foregone conclusion);
• an applied mathematician, who uses measurements but not instruments;
• an engineer, who knows the instruments, but can’t calibrate them; and
• a technician, who can calibrate instruments.

A mathematics professor is only the first type; a pilot must, at times, be any of the three, which are all very mathematical. In other words, my education missed a large amount of what people view as Mathematics.

Flying’s lack of abstraction changed my Mathematics teaching. I start 100– and 200–level classes with a measurement exercise: while the students fill out the little cards with name and class and major, I ask them to measure the textbook. They all use the same ruler. The numbers they come up with are close, but are never identical. “That happens with every number in this course,” I tell them. Flying means dealing with weather, and weather provides a lot of problems in applied mathematics. And so knowing how to fly—being more than a mathematician—changes teaching, and affects content.

The kinds of insights and calculations that weather analysis demands are exemplified in a research paper (Lorenz 1963) whose results on “The Butterfly Effect” and Chaos Theory have become part of popular culture. Students can read this paper from a mathematics perspective, some of its discussion as abstract as anything in the typical junior level “introduction to the major” course. Or they can read it for applications, learning computer methods for solving equations. Because I fly, I can help them read it as an engineer, predicting the range of validity of a model. And can show both Mathematics and flying students the technique of using one of the parameters in the study to predict the likelihood of thunderstorms.

Teaching can be affected in other ways, too. My flying career started with teaching students to fly in airplanes. “Math Anxiety” is a real thing, but in flying some students’ anxiety is fear for their very lives. Recognizing life or death anxiety has improved my ability to sense when a student is overwhelmed.

***

Research is an important part of an academic career—and flying influences my research, too. The influence of flying on academia appears on my curriculum vitae. I developed a course in the mathematics of navigation, which also featured readings from Antoine de Saint-Exupéry and Futurist poets, and works by artists such as Robert Delaunay. I set aside the book I developed from designing that course—it’s not the kind of thing that leads to tenure—waiting until my first tenured summer to complete it.

When I published “Understanding Mathematics for Aircraft Navigation” (2001), the book drew mixed reactions from academics. My department chair as much as spat on it; his successor praised it heartily. Both a major airline and the Australian Civil Aeronautics Authority recommended it to pilots. I don’t think this success resonates with academics, however.

I have not abandoned abstraction, and my research includes both abstract mathematics and ineffably abstract mathematics. Flying has influenced these as well. I now use applied techniques, especially statistical techniques, in my abstract geometric research, finding useful information about abstract questions by studying the data they generate in the context of measurement and signals.

That flying has changed the most abstract parts of my research surprises even me. Part of the influence comes from the study of computing, but I have also begun to explore the relationship between Calculus (especially the mathematics of the infinite) and the real–world in a kind of Kantian sense: what is the nature of space and time? My essay “The Queen of Limits” (2019), explicitly uses a common solution to a flying problem to address a foundational, abstract problem.

***

More broadly, it is easy for those who are expert at one thing to imagine themselves experts in many things. I have observed this in non-academics who enter academia, and in academic discussions of non-academic affairs. This is worse for someone with expertise in two fields. The realms of my expertise are so separate as to almost be antagonistic: pilots, mechanics, and dispatchers have little use for the knowledge of academics (although it might serve them well), and academics have little use for the knowledge of pilots, mechanics, and dispatchers (although it might serve them well).

It is tempting to overwhelm students with all this expertise, whether from inside of or outside of academia. We rush to introduce students to fancy ideas before they are in a position to grasp or use them. But there are always core concepts that are necessary to keep students safe—in the air or in the classroom.

For example, when teaching flying to beginners I focus on turning a complete circle without changing altitude. This seems simple, but in fact it uses all of the flight controls. It’s the core knowledge that every pilot at every level must master. Academic flight instruction (yes, there is such a thing) puts more emphasis on more advanced topics that professional pilots have to master; but I remind my students that crashing exactly on course still makes them dead.

The emphasis on basics is almost abstract—as if this basic maneuver is an axiom of flying—and thus mathematics influences flying. On the other hand, I am currently writing a Calculus book, and asking about basics—motivated by teaching flying—is central to its philosophy and content. The book begins “Pilots work with air; sailors work with water; knitters work with wool; potters work with clay. Those who do Calculus work with numbers and functions.”

What are the basic maneuvers, analogous to an airplane’s level turn, in Calculus? The standard textbook answer is that Calculus depends on two basic concepts, maneuvers if you will, “derivatives” and “integrals”. Through my real-world work I have come to think that these are too abstract to be useful. The first of the actual basic topics is linear approximation, which my book introduces before students study derivatives. Later, derivatives enable more precise approximations, but precision is an “advanced maneuver” that obscures the concept. Sometimes, the correct level of precision is “fly toward the blue sky.”

I understand this because I fly.

As much as I love abstract reasoning and being a Mathematics professor, it is not enough for me. I need balance between action and contemplation. The two are not incompatible. This should be well-known—after all, Wittgenstein started as an aeronautical engineer (Biletzki and Matar 2020).

A few years ago I was at a conference listening to a talk on a deep mathematical subject that very few have the background to understand when I got a text from a former Air Force fighter pilot I worked with asking a really obscure question about an airplane that very few pilots have the background to fly. I found the contrast satisfying, and my ability to discuss both a result of my capacity to fly as much as teach, to do research as much as stay aloft.

References

Biletzki, A. and Matar, A. (2020) Ludwig Wittgenstein. The Stanford Encyclopedia of Philosophy, E. Zalta (ed.), https://plato.stan- ford.edu/archives/spr2020/entries/wittgenstein/.

Lewis, H. R. (2006). Excellence Without A Soul: How a Great University Forgot Education. Public Affairs.

Lorenz, E. N. Deterministic (1963). Nonperiodic Flow. Journal of Atmospheric Sciences, vol. 20.

Wolper, J. S. (2001). Understanding Mathematics for Aircraft Navigation. McGraw–Hill.

Wolper, J. S. (2019). The Queen of Limits. Manuscript.