The Re-Education of an Educated Mind

I once told a fellow graduate student at a nuclear physics summer school that, “I don’t speak math.”  He found this very funny, and me very funny.  But I absolutely meant it.  In fact, I was angry about it.  By that time, I had already met the sleep-depriving scientific discovery question I’ve dreamed of answering for the last decade.  I had been trying to solve it.  It’s why I attended the nuclear physics summer school at all.  It was considered “outside my area”, since my Ph.D. advisor and I had agreed I would declare my concentration as particle physics.  I thought gaining more knowledge would help me make progress.  But then I discovered that I don’t speak math.  I read math.  I calculate math.  I derive math.  But I don’t speak math.

In my current conception of the scientific discovery cycle the flow goes like this: question → ideation → articulation → evaluation → verification, with constant feedback between phases, and the ability to reset to an earlier phase as needed.  At the time of the nuclear physics summer school, I had the question in mind and I’d come up with three possible ideas for answers.  But my efforts completely died at “articulation”, re-phrasing my mental conceptualization of each answer as mathematical equations, because I didn’t speak math.

What do I mean by “speak” math?  And how is this different from reading and calculating?

Put it in another context.  As someone who idly studied five languages besides my native English (no, I can’t speak them all now) and who has a parent who raised me as semi-bilingual and does her professional work in at least two languages, I’ve experienced the feeling of “reading without speaking” many times before.

“Read” means I can identify things on signs that I’ve memorized or seen before.  “Read” means that I can sometimes derive related things, like word signs for the women’s toilet in a restaurant versus the signs I saw at the airport.  “Read” means I can muddle through restaurant menus, especially if there are pictures.

“Speak”, on the other hand, means I can mention to a restaurant server that the ladies’ room is out of toilet paper.  “Speak” means I can make a special meal request that’s not on the menu at all.  “Speak” means I can compose a Physicist’s Log entry about scientific discovery, even when I’m not sure how to define it, how to describe it, or how to achieve it.

“Read” means recognition, “speak” means “creation”.  While I can read math just fine, I can’t create new mathematical expressions with meaning off the top of my head, the way I can churn out sentences in a log entry.  Because I “can’t speak math”, there’s a bottleneck in my discovery cycle, right at the phase of articulation.

I’ve spent years since that summer school digging around looking for practices to help relieve the bottleneck:  Do more math! (Funny how more reading doesn’t equal better speaking.)  Try Fermi questions! (Back of the envelope calculations to answer odd questions about everyday life; but mostly just add and multiply things.)  Just practice modeling!  (Writing down just the starting equation, given any kind of physics word problem.  But this assumes you already know the physics and just need to recognize it in the problem.  What happens when nobody knows the physics yet?)

It wasn’t until I started studying cognitive psychology and scientific discovery that I came across a new option in a book called Where Mathematics Come From:  How the Embodied Mind Brings Mathematics Into Being, written by George Lakoff and Rafael Nunez, a linguist and a psychologist team who study the mind and mathematics.  Their theory is simple: all mathematics comes from lived sensory-motor experience that we then translate into the domain of mathematics via conceptual metaphor.  ALL mathematics; addition, subtraction, the concept of numbers, imaginary numbers, algebra, trigonometry, and on and on.  The final case study they do of the famous Euler equation and all the conceptual metaphors it requires is fascinating.  Most interesting in their theory is the sense that mathematics is not just derived (recognized, manipulated, objectively discovered), but that it can also be contrived (built, constructed, subjectively created).

In Lakoff and Nunez’s scheme, one could learn to speak math.  One could learn to construct mathematical expressions in the same way we construct sentences by consciously, explicitly building math expressions based on careful selection and combination of the underlying embodied metaphors (and still strictly adhering to the operational ground rules of math).  That this is based on conceptual metaphor (closely aligned to analogy and, hence, scientific discovery), and that the metaphors are based on physical experience (suited to a physics focus on the natural world), was music to my ears.

So, I may not speak math yet.  What’s more, taking Lakoff and Nunez’s approach may require a little re-education when it comes to how I think about math.  But now I know speaking math is possible.  And in the pursuit of scientific discovery, the re-education of an educated mind is a small price to pay to keep the discovery cycle alive.