It is always interesting to recognize a simple pattern in your own thinking. Recently, I was wondering why I am so attracted to thinking about the margins of civilization, ranging from life on the ocean (for example, my review of The Outlaw Sea) to garbage, graffiti, extreme poverty and marginal lifestyles that I would never want to live myself, like being in a motorcycle gang. Lately, for instance, I have gotten insatiably curious about the various ways one can be non-mainstream. In response to a question I asked on Quora about words that mean “non mainstream,” I got a bunch of interesting responses, which I turned into this Wordle graphic (click image for bigger view)
Then it struck me: even in my qualitative thinking, I merely follow the basic principles of mathematical modeling, my primary hands-on techie skill. This interest of mine in “non mainstream” is more than a romantic attraction to dramatic things far from everyday life. My broader, more clinical interest is simply a case of instinctively paying attention to what are known as “boundary conditions” in mathematical modeling.
To build mathematical models, you start by observing and brain-dumping everything you know about the problem, including key unknowns, onto paper. This brain-dump is basically an unstructured take on what’s going on. There’s a big word for it: phenomenology. When I do a phenomenology-dumping brainstorm, I use a mix of qualitative notes, quotes, questions, little pictures, mind maps, fragments of equations, fragments of pseudo-code, made-up graphs, and so forth.
You then sort out three types of model building blocks in the phenomenology: dynamics, constraints and boundary conditions (technically all three are varieties of constraints, but never mind that).
Dynamics refers to how things change, and the laws govern those changes. Dynamics are front and center in mathematical thought. Insights come relatively easily when you are thinking about dynamics, and sudden changes in dynamics are usually very visible. Dynamics is about things like the swinging behavior of pendulums.
Constraints are a little harder. It takes some practice and technical peripheral vision to learn to work elegantly with constraints. When constraints are created, destroyed, loosened or tightened, the changes are usually harder to notice, and the effects are often delayed or obscured. If I were to suddenly pinch the middle of the string of a swinging string-and-weight pendulum, it would start oscillating faster. But if you are paying attention only to the swinging dynamics, you may not notice that the actual noteworthy event is the introduction of a new constraint. You might start thinking, “there must be a new force that is pushing things along faster” and go hunting for that mysterious force.
This is a trivial example, but in more complex cases, you can waste a lot of time thinking unproductively about dynamics (even building whole separate dynamic models) when you should just be watching for changes in the pattern of constraints.
Inexperienced modelers are often bored by constraints because they are usually painful and dull to deal with. Unlike dynamics, which dance around in exciting ways, constraints just sit there, usually messing up the dancing. Constraints involve and tedious-to-model facts like “if the pendulum swings too widely, it will bounce off that wall.” Constraints are ugly when you first start dealing with them, but you learn to appreciate their beauty as you build more complex models.
Boundary conditions though, are the hardest of all. Most of the raw, primitive, numerical data in a mathematical modeling problem lives in the description of boundary conditions. The initial kick you might give a pendulum is an example. The fact that the rim of a vibrating drum skin cannot move is a boundary condition. When boundary conditions change, the effects can be extremely weird, and hard to sort out, if you aren’t looking at the right boundaries.
The effects can also be very beautiful. I used to play the Tabla, and once you get past the basics, advanced skills involve manipulating the boundary conditions of the two drums. That’s where much of the beauty of Tabla drumming comes from. Beginners play in dull, metronomic ways. Virtuosos create their dizzy effects by messing with the boundary conditions.
In mathematical modeling, if you want to cheat and get to an illusion of understanding, you do so most often by simplifying the boundary conditions. A circular drum is easy to analyze; a drum with a rim shaped like lake Erie is a special kind of torture that takes computer modeling to analyze.
A little tangential kick to a pendulum, which makes it swing mildly in a plane, is a simple physics homework problem. An off-tangent kick that causes the pendulum bob to jump up, making the string slacken, before bungeeing to tautness again, and starting to swing in an unpleasant conic, is an unholy mess to analyze.
But boundary conditions are where actual (as opposed to textbook) behaviors are born. And the more complex the boundary of a system, the less insight you can get out of a dynamics-and-constraints model that simplifies the boundary too much. Often, if you simplify boundary conditions too much, the behaviors that got you interested in the first place will vanish.
Dynamics, Constraints and Boundaries in Qualitative Thinking
Without realizing it, many smart people without mathematical training also gravitate towards thinking in terms of these three basic building blocks of models. In fact, it is probably likely that the non-mathematical approach is the older one, with the mathematical kind being a codified and derivative kind of thinking.
Historians are a great example. The best historians tend to have an intuitive grasp of this approach to building models using these three building blocks. Here is how you can sort these three kinds of pieces out in your own thinking. It involves asking a set of questions when you begin to think about a complicated problem.
- What are the patterns of change here? What happens when I do various things? What’s the simplest explanation here? (dynamics)
- What can I not change, where are the limits? What can break if things get extreme? (constraints)
- What are the raw numbers and facts that I need to actually do some detective work to get at, and cannot simply infer from what I already know? (boundary conditions).
Besides historians, trend analysts and fashionistas also seem to think this way. Notice something? Most of the action is in the third question. That’s why historians spend so much time organizing their facts and numbers.
This is also why mathematicians are disappointed when they look at the dynamics and constraints in models built by historians. Toynbee’s monumental work seems, to a dynamics-focused mathematical thinker, much ado about an approximate 2nd order under-damped oscillator (the cycle of Golden and Dark ages typical in history). Hegel’s historicism and “End of History” model appears to be a dull observation about an asymptotic state.
How the World Works
In a way, the big problem that interests me, which I try to think about through this blog, is simply “how does the world work?”
At this kind of scale, the hardest part of building good models is actually in wrestling with the enormous amount of “boundary conditions” data. That’s where you either get up off the armchair, or turn to Google or Amazon. Thinking about boundary conditions — organizing the facts and numbers in elegant ways — becomes an art form in its own right, and you have to work with stories, metaphors and various other crutches to get at the right set of raw data to inform your problem. Only after you’ve done that do dynamics and constraints get both tractable and interesting.
Abstractions and generalizations, if they can be built at all, live in the middle. Stories live on the periphery.
This is part of the reason I don’t like traditional mathematical models at “how the world works” scale, like System Dynamics. They ignore or oversimplify what I think is the main raw material of interest: boundary conditions. A theory of unemployment, slum growth and housing development cycles in big cities that ignores distinctions among vandalism, beggary and back-alley crime is, in my opinion, not a theory worth much. If you could explain elegantly why some cities in decline turn to crime, while others turn to vandalism or beggary, then you’d have interesting, high-leverage insights to work with.
It’s not surprising therefore, that one of the most seductive ideas in abstract thinking about history, the deceptively simple “center periphery” idea (basically, the idea that change and new historical trends emerge on the peripheries and in the interstices of “centers”) is extremely hard to analyze mathematically, since it involves a weird switcheroo between boundary conditions and center conditions. Some day, I’ll blog about center-periphery stuff. I have a huge, unprocessed phenomenology brain-dump on the subject somewhere.
So in a way, thinking about things like the words in the graphic is my way of wrapping my mind around the boundary conditions of the problem, “how does the world work?” If I just made up a theory of the mainstream world based on mainstream dynamics, it would be very impoverished. It would offer an illusion of insight and zero predictive power. A theory of the middle that completely breaks down at the boundaries and doesn’t explain the most interesting stories around us, is deeply unsatisfying.
I have proof that this approach is useful. Some of my most popular posts have come out of boundary conditions thinking. The Gervais Principle series was initially inspired by the question, “how is Office funny different from Dilbert funny?” That led me to thinking about marginal slackers inside organizations, who always live on the brink of being laid off. My post from last week, The Gollum Effect, came from pondering extreme couponers and hoarders at the edge of the mainstream.
So I operate by the vague heuristic that if I pay attention to things on the edge of the mainstream, ranging from motorcycle gangs to extreme couponers and hoarders, perhaps I can make more credible progress on big and difficult problems.
Or at least, that’s the leap of faith I make in most of my thinking.