Ignorance Is Bliss

Even a blind squirrel finds a nut every once in a while.

What’s the difference between an engineer and a mathematician? An engineer believes equations approximate the world. A mathematician believes the world approximates equations.

The temptation for me is to side with the mathematician. Granted, one of his favorite pastimes is generating equations that — while interesting — are for all apparent purposes useless. But it is often the case that somewhere down the line an applied scientist will discover that a brute physical fact (one he has been grappling with for a number of years) is perfectly described by this same relationship, published as little more than a thought exercise by a quirky statistician.

A sequence formulated by Leonardo of Pisa is a well known example. It’s an utterly simple series that he applied to solve what might be called (by nerdy grad students) a fun math problem. As many of you know, Fibonacci Sequences continue to pop up in unexpected places from software programming to pinecones, which seems to suggest that the world does indeed approximate equations.

Pythagoras was so convinced of the supremacy of numbers that he considered them divine. The fact that we can imagine a perfect right angle but can’t actually build one led him to the conclusion that the physical world is just a distortion of a higher, mathematical and spiritual reality. (Plato anyone?)

A modern Pythagorean might say something like the following.

’’When you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot…your knowledge is of a meager and unsatisfactory kind…’’

–William Thomson Kelvin, 19th Century English Physicist

As I said, my initial inclination is to agree, but there have been times when my experience leads me in the other direction. My semester-long senior design lab was spent constructing a communications system. The end result wasn’t that exciting: a waveform on one oscilloscope that looked pretty much like the waveform on a second oscilloscope. But though I knew just about everything there was to know about the process (I’d built it), if you’d asked me how it worked I probably would’ve replied, only half-jokingly, “Magic.”

My sense was that the equations I had been using were actually serving to hide a larger, more complicated reality that I wasn’t intentionally taking into account. Not that this was a bad thing — without those formalizations, not much could have been accomplished — but something hinted that some non-number stuff was going on. (Though this is purely anecdotal, I’ve had conversations with a number of other electrical engineers who have had similar inklings, one of which sparked this post.)

This “non-number stuff” is evident when we consider the basis of the Scientific Method: the belief that the way things have behaved is the way they will continue to behave. Without accepting that dogmatically (since it can’t be proven), we are pretty helpless. Why? Because the tools we’re given to interface with the natural world are equations. And without consistency, an equal sign doesn’t even make sense, or at least it doesn’t make sense for any length of time.

What’s especially strange is that this presumption at the heart of our approach to science isn’t always technically true. For example, we’ve learned that the electron will behave in different ways when put in the exact same conditions. So, even if every factor in a system is known, quantum mechanics can’t tell us what outcome to expect; it can only provide us with a probability that a certain outcome will occur. The Scientific Method has allowed us to realize incredible technological advances, even though (maybe even because?) it hides this randomness through an assumption that is practically useful, despite being technologically unsound.

A few months ago I posted an essay about some very un-human aspects of modern engineering. Let me say that one of its very human features is a requirement to accommodate a stubborn reality, i.e., to yield to an other. The creation is unforgiving; everything must be considered for a system to function successfully. And yet, our foundational scientific approach treats nature as if she behaves in a way that makes sense to us in spite of the fact that we’ve proven otherwise. How then can we explain the success we’ve had operating from that premise?

I don’t think blind squirrels are part of the answer, and there is something to be said for the mathematician’s position—some phenomena line up pretty darn well with simple equations. But maybe Mr. Kelvin got it exactly wrong; at least in some instances, quantifying behavior is only possible when we know something imperfectly.

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