Abstraction is leaky.
Abstraction is expensive.
Abstraction is inescapable.
This is the water in which we swim.
In Chaos, James Gleick is teaching me about “universality”. If I’m understanding it correctly, he’s referring to the fact that the properties of chaotic systems all follow the same rules. This should not come as a surprise; universality is a property of systems of abstraction.
I remember my dad trying to explain water pressure to me, and I couldn’t understand the difference between pressure and flow. (I was quite young.) Until I’d had some grounding (pun intended) in electronics, I couldn’t understand that high flow required both high pressure and low resistance. A closed valve will block 20 PSI just as well as 0.
Voltage is pressure. Current is flow. Resistance is resistance. An arc and a broken pipe are the same event in different systems.
I still refuse to recognize “dynamical” as a word. Fight me.
Not much of a step is the next realization: that pressure and flow and resistance apply to all fluids, not just water. They try to reach zero PSI just as an electrical circuit tries to reach zero Volts.
And every system like this has an “equivalent electric circuit” that describes its behavior.
“Hoffman’s Iron Law“, in loudspeaker design, dictates that you can have loud bass, low bass, or a small cabinet: pick two. The other must be sacrificed. I spent months and built (I think) five prototypes before finding what I believe to be the corner of the curve: a vented chamber of about 2 cubic feet with a very efficient 10″ woofer, tuned to 38Hz. It reaches about 120dB SPL per cabinet, and each cabinet costs less than $100 to build.
Every kind of engineering has a law like this. Given three desirable properties, one must be sacrificed to satisfy the other two. In most systems, experience eventually teaches you that there’s actually only one choice. A subwoofer simply cannot be small and good. A better sound is always achieved by moving a large volume of air gently than a small volume violently.
Another example is database administration. The choice is between consistency, availability, and partition tolerance. This is called the CAP Theorem. In practice, you have to give up on 100% availability; consistency and availability aren’t optional. Failure to make the correct choice guarantees that, at some point, the administrator will have to go without sleep for a week while desperately searching for a way to recover from the disaster they could have prevented if they’d made the right choice from the beginning.
Sorry, boss. The source code for the website is gone. It’s not recoverable.
An audio engineer has the unfortunate requirement of venturing into reactive components and behavior that depends on the frequency of a signal. DC resistance is less important than frequency-dependent impedance. We learn of the Fourier transform, and those of us with competent instructors learned that if you ignore the imaginary component of the result, the needle will skip, and half your song will disappear as the listener moves around the room. Well done, grasshopper.
This is what you get for asking your math teacher “when am I ever going to use this?”
I get it. Algebra isn’t fun. It’s vegetables. It’s what you have to endure to get to the dessert: Calculus.
If you had a bad calculus teacher, you memorized a bunch of rules for how to derive and integrate. If you had a good calculus teacher, you saw that buried under all that algebra is this beautiful way of understanding the world that just flows out of your pen. Using algebra to do physics is painful. Calculus is the easy way.
It’s another of these magical ways of thinking about problems that applies to system after system. It has universality.
But I digress. We’re talking about chaos, not calculus. The latter lends itself nicely to reductionist problem solving. And reductionism is fantastic! It’s brought us sanitation, computers, the Green Revolution; it’s gotten us to the moon and back. But for some problems, it is not well-suited, or even useful.
In the field where I spend most of my time–behavioral science–we’ve long since passed the point where reductionism alone can actually solve problems. (One could argue that it never solved any problems. Not so, I’d argue: reductionism is very good at explaining the behavior of rats.) The more insightful practitioners have realized that Descartes did us the gravest dirty by convincing us that mind and body are separate. To understand human behavior, you must examine the entire stack: genes, personal history, hormones, neurotransmitters, thoughts, feelings. Your approach must be “holistic”.
Then you must face an uncomfortable fact: while reductionism has helped you understand all the pieces, and holism has helped you understand their interactions, you still can’t account for most of what’s going on. Human behavior is chaotic.
But I believe there’s an even more profound insight waiting for us: reductionism, holism, and chaos are not separate ways to look at the same problems. There is a phase change between each, and a triple point where they meet, and the behavior of science itself becomes chaotic as you approach that point.
One last insight before I go. This didn’t come from the study of any scientific field; it emerged from an experience I had, exploring the “inner space” through the use of psychedelic drugs.
Really. Just don’t even go there. You don’t have to. Read what others have written about their experiences. The risk is not worth it.
LSD asked me (among other things) about the nature of the boundary between discrete and continuous phenomena. Is it discrete or continuous?
Imagine the two sitting across a table from each other. Discrete sees only two positions: its own, and continuity across from it. Continuity sees a straight line with an infinite number of possible positions between where they’re sitting. Then the subject (me) drops way too much acid (SERIOUSLY, DON’T) and they’re allowed to start talking.
Here is the first bifurcation. We who experience the table as a table now see two positions flanking it.
Discrete goes first, pointing out that there are only two chairs, and you can only be sitting in one chair at a time. Continuous slowly crosses her legs and leans forward, silently demonstrating smooth variation. She’s bifurcated her position. Discrete starts writing a list of all the possible positions one could take relative to the chair and table, now bifurcating like mad. Continuous keeps varying from state to state smoothly, and the chaos begins flowing from their function as smoothly as she does, as clearly as he does. By now the question has devolved into a mess of non-answers with little islands of clarity, like the times where continuous takes a break to rest against the table and is, for a moment, still; or discrete wipes off a bead of sweat from a part of his face that’s in between named features.
The famous (infamous?) sexologist Kinsey made a scale to describe sexuality among men, where 0 represents men who are only attracted to AND only engage in sexual behavior with women; 6 represents men who are “gold-star” gay. There are fewer men in those two categories than you would expect.
Categories 1 and 5 are described as “incidentally” homosexual or heterosexual. Discrete leaps to his feet, pointing out that the dividing lines between 0 and 1, and between 5 and 6, completely depend on whether there was an “incident” or not. It either happened or it didn’t; there’s no in-between.
“Nonsense,” Continuous responds calmly. “How many ‘incidents’ were there? How long did each one last? How much did he enjoy himself?”
She hesitates slightly before asking, “was it consensual?” Everyone present feels a little queasy, knowing that the answer was sometimes no. Sometimes it was yes. And sometimes it was “complicated”.
After an awkward silence, the men present start placing themselves on the scale, some more certain than others. They speculate about where various dead celebrities would have put themselves, then everyone goes out for coffee.