The See-Saw of Logic
[Cartman]: “Kyle, all those times I called you a stupid Jew, I didn’t mean it – you’re not a Jew.”
[Kyle]: “Yes I AM, Cartman! I AM a Jew!”
[Cartman]: “No, no, don’t be so hard on yourself.”
–Exchange between Cartman and Kyle, South Park: Bigger, Longer, and Uncut
It has occurred to me that multi-variable calculus can serve as a potent metaphor for certain kinds of reasoning.
In single-variable calculus, the equations you deal with are simple: one variable is defined in terms of a series of mathematical operations applied to another. When graphed in a Cartesian system, the result is a line embedded in a plane. Direct, elegant, uncomplicated. The techniques for determining the slope of that line at any point are relatively straightforward, so that form of calculus is the one people learn first.
When you’re dealing with more than two variables, when one variable is defined in terms of operations applied to two or more, things get a little more complex. The simplest case involves three variables, and the result is often a surface embedded in a space. The methods you were taught in single-variable calculus can’t be applied to those equations.
One of the things you’re taught to do is take the original equation and rewrite it, treating all but one of the operational variables as constants each time. The standard techniques can then be applied to the modified versions of the equation. Essentially, you’re talking cross-sections through the surface in the Cartesian space along different directions, and determining the slope of the surface in a particular direction.
To measure the slope in terms of any variable, you must treat the others as constants.
There’s a certain kind of logic that operates in much the same way. We have beliefs about the world, and we compare those beliefs to incoming data from the world. We can hold those beliefs constant, or we can be uncertain as to their validity; we can accept the data as accurate and representative, or we can be uncertain as to its accuracy and representativeness. If we let the status of both the beliefs and the data vary, we cannot evaluate either.
The quoted exchange above is an excellent example of this phenomenon cast in a humorous light. For context: Cartman is a vile and bigoted child whose innate charisma barely serves to compensate for his many prejudices. He has an established set of ideas about what properties are associated with the concept of ‘Jewishness’ – various negative and unflattering ones. One of his friends, Kyle, is Jewish. Cartman frequently needles Kyle about being Jewish, much to Kyle’s annoyance, but in truth he recognizes that Kyle has a variety of laudable traits.
So: what are the variables? Kyle’s properties, and the properties of the concept.
Cartman’s idea of what Jews are like is incompatible with his observations of what Kyle is like. He must therefore hold one variable constant and set its value, while letting the other vary and determining its value in relation to the set given. The humor comes from the contrast between our expectations and what actually happens: the cliche is that the bigoted individual learns to reject his prejudices by interacting with a member of the group he prejudges, but Cartman does the opposite and instead denies that his friend is actually a member of the despised group. Instead of adapting his ideas about Jews to fit Kyle’s identification with them, he adapts his ideas about Kyle’s identification and holds fast to his prejudices.
This illustrates a very important point: The manner in which we decide which of the variables to accept as a constant given determines much about what conclusions we derive from them.
If we accept the data, we must let our evaluation of our beliefs teeter, with the data determining which way the outcome leans. If we accept the beliefs, we must let our evaluation of the data teeter, with our beliefs determining which way our thinking tips. One or the other must be held constant. How we decide which will be conserved and which will be altered depends on our evaluation of each.
If we have a well-established way of thinking, one that has served us well and endured many challenges, we are not likely to abolish it due to some contrary evidence. We are far more likely to dismiss the evidence as a fluke or error because it is incompatible with our existing theory. Logically, the conclusion could go either way: rejecting the theory or rejecting the evidence. In practice, it takes a great deal of evidence that we are unable to reject to overturn the theory, and unless a very high standard of proof is met, we reject the evidence every time.
One of the tasks of the rationalist is to always hold back from reaching definitive conclusions, to let the inclusion of this and the exclusion of that be merely provisional. Human beings aren’t good at this – when we note that something is incompatible with our understanding, we tend to reject it completely, deleting it from our awareness. It takes effort and resources to keep track of the possibilities we exclude, and it’s far easier to abolish them. But this tends to prevent us from storing up enough evidence to overturn our theories – each is evaluated individually, instead of collectively, and so a mass of data that might force us to change our minds is rejected one part at a time.
Rationalists must balance atop the see-saw, maintaining equilibrium, identifying and countering inclinations to lean one way or the other, always sensitive to changing conditions and ready to react appropriately.