Boundaries: Robust Regularity

I’ve been pondering the difference between natural and artificial boundaries, thanks to reading another chapter of Carving Nature at Its Joints. Chapter 7 by Achille Varzi is titled “Boundaries, Conventions and Realism”.

The chapter begins by contrasting ‘artificial’ human-made political boundaries such as state and country lines that seem arbitrary, with ‘natural’ geographical boundaries such as a river, shoreline or mountain range. The E.U. is an interesting case whereby the landscape might not seem different, but the different languages on signs alerts you that you have crossed a line – the inhabitants on one side of the line might culturally behave quite differently from those on the other side. Here in the U.S., there might be less of a distinction. The author writes: “Most drivers feel nothing at all as they pass the border between Wyoming and Idaho, a line whose embarrassing geometric straightness says very little about its history (or says it all). Yet even here there are differences, and Idahoans are proud of their license plates just as Wyomingites are proud of theirs.”

How thin is the artificial state-line boundary? Is it infinitesimal? As thin as the planar node of a p-orbital? Or does it depend on the scale at which you’re drawing a map? Even then things get tricky. For example, how long is a coastline? The boundary seems more clear-cut when you look at a map. But as you start to zoom in, things get fuzzier with more detail. Strange, is it not? As you see more details, the actual boundary gets fuzzier. More fractal-like. With its twists and turns, the coastline starts to look longer at close range than from a bird’s eye view. Fractals beget more fractals.

In the world of chemistry, the same thing happens. As I write, I’m looking at the smooth surface of my laptop. It’s probably an aluminium alloy of some sort and feels smooth to the touch. But zoom in at the atomistic level and it is full of ridges, hills and valleys. Is Kansas as flat as a pancake? Possibly flatter. Depending on the relative scale of the protuberances. The atoms in the body of my laptop are chemically bonded to each other –they are joined together in some way. The piece of dead skin I just wiped off my keyboard? I consider it as separate from my laptop. Not chemically bonded. Was that skin flake previously chemically bonded to the (relatively smooth) skin on my hand? Likely so. Once again, things get more complicated when you zoom in closer.

What is the boundary of an atom? Is it natural or is it artificial? At first glance, you might think it’s natural. It seems depicted as such in chemistry textbooks. Your mind’s eye likely pictures colored spheres when you think of atoms. But as my G-Chem students learn in the first week of class when encountering the Rutherford model of the atom, it’s unclear where exactly the boundary is located. I draw a dashed line to indicate its fuzziness. When we get to the orbital descriptions of electrons, we see three-dimensional shapes once again. But since the orbital is defined as a probability distribution, we typically draw the orbital shapes with an arbitrary boundary – enclosing a 90% probability that the electron will be found. The probability distribution can be modeled mathematically (for hydrogen-like orbitals), but the wavefunction has a long asymptotic tail. There is no ‘hard cut-off’. The same is true of chemical bonds. The bond dissociation curve also features a long asymptotic tail.

Does this mean that all boundaries are arbitrary? Maybe there are no such things as clearly defined natural boundaries, and they’re all somewhat artificial. Perhaps we’ve descended into a post-modern, post-fact, post-truth cynicism with all this arbitrariness. But not being able to clearly define the boundary in detail doesn’t mean there isn’t a natural, albeit fuzzy, way to distinguish two separate states. We do it all the time, classifying things into one category and other things into other categories. We might scratch our heads when odd cases show up – hello platypus! But eventually an adequate space is found as we move other boundaries around.

Shorelines recede. Rivers erode a bank. Lakes dry up. Mountains rise slowly, volcanoes perhaps a little quicker. Even natural boundaries seem fluid in the long view of geological time. But what about atoms and chemical bonds? Unless the fundamental laws of physics change, an electron that’s less than 1.5 Angstroms from a hydrogen atom nucleus is likely to be “bound” to it. An electron further away, noticeably less so. One could quibble about whether the dividing line should be 1.4 or 1.6 Angstroms, but the scientists would very readily say that 0.5 Angstroms is bound and 3 Angstroms is not. (0.53 Angstroms is the Bohr radius, the most probable distance of the electron from the nucleus.) We might not agree on the exact point of division, but there are vast swaths of agreement about what constitutes the ‘normal’ state of affairs. (Similar arguments can be made about the lengths of chemical bonds.)

There’s a neat phrase to encompass this state of affairs: ‘robust regularity’. Boundaries may get fuzzier as you attempt to look closely at the details, but one doesn’t need to throw the baby out with the bath water. Heisenberg’s Uncertainty Principle may not allow us to exactly locate where an electron is, but chemists can do amazing things by moving those electrons around and all we need is a rough relatively fuzzy idea of where the electrons whiz around. Nature is regular. No two expert butchers would carve the cow along precisely the same lines, but they are likely to be close when superimposed. The boundaries are robust. Up to a point. We can’t exactly define that point in very, very close detail, but pragmatically it doesn’t matter.

After reading this chapter, I’m starting to see robust regularity everywhere. Not just in my field of chemistry. I think it’s a useful way to think about similarities and distinctions. The author writes: “Such is the magic of boundary lines: they are thin, yet powerful; they separate, and thereby unite; they are invisible yet a lot depends on them, including one’s sense of belonging to a country, a people, a place.” This reminds me of the first Harry Potter book where we are introduced to a boundary separating the wizarding and non-wizarding worlds – a pub called the Leaky Cauldron. I like the name. It hints that the boundary is leaky, and not a hard cut-off. Muggles do find their way into the magical world and vice-versa.

It’s amazing that we live in an ordered world. As a scientist, being able to discover how things work and design new technologies is dependent on that regularity. But nature is also robust, perhaps magically so. Things get fuzzy if we try to examine them too closely, escaping our senses as if a magic trick was being played. We may not know how the trick actually works but we’ve seen it enough times to make regular robust predictions. That’s how science works!