Coming Of Age

On the first day of my final semester in graduate school, my major professor assigned a single, seemingly straightforward semester-long research project, a solid-state physics problem to investigate, model, and simulate during my last class with him. 

He described this project as a graduation gift, or in retrospect, that’s how I came to think of it.

I couldn’t know then it’d change my view of our world the rest of my life. 

After all, I’d nearly completed graduate school. How tough could it be?

The problem appeared straightforward to describe, or so I believed. It had to do with resistance, an electrical property material demonstrates which changes over a wide range of temperature. 

The material’s resistance could be measured and its temperature controlled, changed from a red-hot molten state to a brittle, liquid nitrogen cold condition using an environmental chamber.

Measuring the material’s resistance appeared easy enough, but turned out, measuring its resistance was comparable to reinventing the wheel. Investigation revealed the material’s resistance had already been measured over the temperature range required by my professor. Reinventing the wheel is a cardinal sin in engineering circles, so I based my measurements on the work of those who came before me and moved on to understanding their measurements and modeling them using mathematics.

My resulting resistance versus temperature graph looked more like a high altitude photo of the Mississippi River than a line or parabola, snaking all over the chart with loops, bends and troublesome twists. And this badly behaved graph required explanation, a mathematical model which predicted every twist and turn.

Though it provided little comfort, investigation revealed I wasn’t alone in my dismay over this material’s unpredictable behavior. From what I read, a host of capable folks were as perplexed about these twists and turns as I was. This seemingly simple material’s resistance behavior apparently baffled the experts.

No one could model or predict its resistance as its temperature changed from hot to cold and back again.

But instead of expressing wonder or dismay, our engineering community continued to do what we’d always done. We took the wide range of temperature data and divided it into smaller, more manageable chunks.

Less temperature change meant smaller resistance variation. 

Smaller variation transformed the problem, making its changes in resistance easier to model, simulate, and predict.

More telling than segmenting the wide temperature range into smaller sections was the conjecture underpinning the mathematical models which now better approximated the resistance curve. The explanation underpinning these measurements read like threadbare conjecture rather than established theory built on understanding. Since established theory could not explain measured resistance variation over temperature, we made something up. 

Conjecture combined with mathematics governing a narrow temperature range now presented itself as a law of physics. 

Built on experimental measurement, the conjecture wasn’t compelling, but it was revealing. If we don’t understand something, we make up an explanation, cast it in mathematical expression, then present it as science.

An instructive insight revealing at times, conjecture encased in mathematical expression is the best we can do. 

When we don’t understand the intricate complexity of the world around us, we model it one small piece at a time and explain it with conjecture. In the absence of critical scrutiny, conjecture can be accepted as theory and we deceive ourselves by regarding fiction as a law of physics.

The last lesson my major professor taught me before leaving the university was based on this project and a single figure he drew on the board. It's shown at the top of this article. 

Decades later, it remains a lesson I’ll never forget. 

His figure shows the number of problems we can solve is tiny compared to those we can describe. Gray represents problems we describe but can’t solve. The innermost, smallest circle represents problems we can describe and solve. By contrast, the number of problems we can describe is infinitesimal compared to those whose complexity defy description. Red represents problems we cannot describe or solve.