## Friday, September 16, 2005

Models of reality

I think Godel has been somewhat misunderstood. What he actually proved was that within a logical system with a given set of axioms and rules, some true statements can be made which cannot be proved to be true from within the system.

This does not mean that nothing can be known—quite the contrary. Many things within the system can be known to be true—in some systems an infinite number of them. And we can know that some things are true even if they are unproveable from within the system. This requires information from outside the system, of course, but that isn’t so terribly rare. All that the theorem says is that there exist some meaningful statements which you cannot prove.

It is very tempting to try to expand this finding into other fields, such as politics. The theorem isn’t strictly applicable, but the humility it engenders is something political theorists desperately need. One major reason you can’t use Godel in political theory is because none of the political theories come within shouting distance of describing human behavior, and the predictions of their models are so badly wrong to begin with that it does not make sense to try to define what you mean by true statements within the system.

It may make some things clearer if I explain what scientists do.

The world is quite complicated, but when you look at isolated bits of it you see that its motions follow relatively simple patterns. You can enumerate all the patterns, if you have the time, but we found early on that the patterns fell into categories which we could describe mathematically.

Post-modernist criticisms to the contrary, mathematics itself has no politics or cultural bias. Considered as language (which seems to be the post-modern favorite approach) mathematics is essentially pure syntax, and will give you whatever degree of precision you need for description or prediction—provided your model is correct.

So how do we know how correct our models are? The scientist’s fundamental job is to understand the models and the limits of the models he uses. The best known example is Newtonian mechanics, which works like a champ provided you don’t get too small (you get quantum mechanical effects) or work around too great a gravity (which distorts space and time).

Or if your model is incomplete.

The most famous example of an incomplete model is the falling rock you learn about in elementary physics. For the equations you learn (v=a*T, d=1/2 a*T*T) to be precisely true, there can be no air resistance, the acceleration can’t change as a function of time, measurement of the position mustn’t perturb the system, space-time must have a Euclidean geometry, and so on. For a rock falling short distances, air resistance is a small effect, the acceleration doesn’t change noticeably, space-time is nearly Euclidean, and you can use light to measure the positions without worries. But for a small glider of the same mass, air pressures are as important as gravity—you cannot use the same simple model to describe the airplane’s motion; it isn’t complete enough.

A model has a “domain of validity:” the conditions under which the model usefully describes reality. You do not understand the model completely until you understand its domain of validity.

You can model the interactions of molecules bouncing off each other as a set of equations with a set of initial conditions, but unfortunately there isn’t a simple procedure for producing the exact equations that describe the solution. You must use approximations. To make matters worse, just writing out the position of every molecule becomes unuseably tedious. If you are interested in “bulk” properties (such as pressure), you don’t really care that much about individual positions, and you may use a different model: the ideal gas law or variants of it. It looks like a large jump from the model with billions of bouncing particles to PV=nRT, but we can justify and derive the new model from the old with understood simplifications and statistical mechanics. Let me emphasize this: the new simpler model for gases is justified in terms of the detailed model for individual molecular collisions. It isn’t an ad-hoc add-on anymore; the models for the few and for the many are connected.

You can model the supply and demand for hamburgers: demand and supply having mostly seasonal variation. You can then use the model to predict how much money you’ll need to spend on cows and bread and pickles, and what price to charge for your burgers. You understand that there is some error in the model, due to the uncertainties in weather which can drive up feed prices, and hence beef prices. So you allow for that: “McBurgers will need \$450 million next year, but we might need as much as \$35 million more if the weather in Patagonia is bad, so keep an eye on the weather during the year.” The model is good enough that you can commit millions of dollars.

Of course, if someone pretends to find a finger in one of your burgers, you’ll need a lot less beef, and your competitors might find that the price of beef is lower this year. The model doesn’t cover that contingency. With more experience you might be able to estimate the rate of those kinds of losses as well, and make them part of a more comprehensive model. (It would make business courses livelier.)

When you use a model you have to check that it is appropriate for the circumstances. To borrow from a familiar joke:

Three men--a mathematician, a physicist, and an engineer--are brought to a magical football field, to one goal line. At the other goal line is a beautiful woman. The referee tells them: “You can roll a die. Every time you roll an even number you get to move half the remaining distance down the field.”

The mathematician says “This is dumb, it’d take an infinite number of rolls to get there. I quit.” And he leaves.

The physicist tries rolling the die a few dozen times, verifies that the rule is correct, and he quits too.

The engineer just keeps on rolling the die and moving down the field.

The referee asked him “Why are you still playing? Don’t you know you’ll never reach the other goal?”

The engineer replied: “I’ll get close enough.”

The engineer noticed a limit to the validity of the “half the remaining distance” model: the human body has a thickness, and the model of movement makes no sense when that thickness is smaller than the distance to the target. The model is fine when you’re 20 yards away, but not when you’re 2 inches away.

Another famous example of an incomplete model is the argument that this is, or was originally, the best of all possible worlds; for how could God make anything but the best? The assumption is that we can compare worlds using a simple better-than/worse-than relation. This set of all possible worlds and the better-than/worse-than relation would be an example of what mathematicians call a “well-ordered set.” But mathematicians know of many sets which aren’t “well-ordered” (vector spaces, for example), and there is no obvious reason to believe that you can always say that one world is better planned than another. Maybe a lot of possibilities are equally well-planned. I can’t design a better world, but that’s no surprise.

How do you model risk in loans? The old joke went: “If you owe the bank a hundred thousand dollars, you have a problem. If you owe the bank a hundred million dollars, the bank has a problem.” The scale of the loan shifts the direction of the risk model.

You can have useless models, and models that are wrong: that have no domain of validity. You can, if it amuses you, model the electrical conductivity of copper by a mesh of rubber bands, but you won’t get any useful predictions or even descriptions from it. Electron motion in copper isn’t anything like that model, even near absolute zero. The model is simply wrong—it has no domain of validity.

You can model a river as a solid block, which doesn’t make much sense for most work but is an OK approximation if you are studying what happens in very high speed impacts or looking at landscape radar reflections. The river as a solid block model has a limited domain of validity.

Political economics provides many incomplete models.

It is obvious that someone who provides great benefit to society ought to be rewarded commensurately with the benefit. It is quite proper for him to ask that the reward benefit his children in turn. But if one person or family amasses too huge a fraction of a society’s wealth, they’re are apt to use it to distort the economy to benefit themselves unjustly. And this needn’t even be intentional. So justice in reward can lead to systematic unfairness.

On the other hand, you must not try to reward everyone equally—we all know what a disaster that produces, and what hideous injustice. Or if you try to temper a reward by heaping conditions on it, that reduces the reward—sometimes to the point of uselessness.

Neither justice nor “equity” stand alone: they conflict. The best ways I’ve seen for trying to satisfy both are ad-hoc collections of unsatisfactory laws about monopolies and taxation and education. (Presumably these collections have to be dynamically modified over time, but so far the only experiments have been in the direction of adding new laws.)

Godel’s work tells me that even if I have a complete mathematical model (which I obviously don’t in political economics), there will be things that are true within that model that I cannot prove from within that model. In practice this may not matter—getting “close enough” may be enough for useful knowledge. In practice, a model can be good enough to stake your life on. Or your eternity on.

I’ll not take up a discussion of the sources of knowledge at this time, except to note that simple observation tells us that the senses are not our only way of knowing.

With that rather extended preface out of the way, I want to look at the post-xxx isms. This isn’t always easy. (And I remain to be convinced that the study of philosophy reduces to the study of language.)

I have a lot of trouble making sense of what seems Rorschach writing by the disciples of Derrida et al. The Sokal and Social Text incident strongly suggests that this is not a failing on my part. So to understand what they meant I have to rely on what I hear, which I admit is biased towards sampling his noisier disciples. Those I hear about are firmly wedded to an oppressor/oppressed model of human relations. Certainly that’s all I ever find them talking about. (The actual philosophy department has a bit more variety to it.) Do I need to point out how terribly limited this model’s domain of validity is? I have to conclude that either these writers have never tried to compare their models to the world around them, or else that they have had such miserable lives that normal human relations are a mystery to them.

In a less political example, consider the philosophy professor Karen Barad, who the Physics Department brought in for a panel discussion about ethics in science, in conjunction with sponsoring the play Copenhagen about Heisenberg and Bohr. I was (as usual) unable to attend, so I searched around the net for samples of her work, which often had to do with science. In one paper she attempted to show that observations of the physical world should be “privileged” texts. There’s nothing objectionable about that, but the fact that it is necessary to argue for this suggests that the philosophical model that understands reality in terms of “texts” is, to put it charitably, not ready for prime time.

She was trying to argue for her own “Agential Realism,” to replace/supplement some other theory; but I’m not going to try to analyze her system. Contemplate one of her article titles: "Performing Culture / Performing Nature: Using the Piezoelectric Crystal of Ultrasound Technologies as a Transducer Between Science Studies and Queer Theories." Could Sokal do better?

Perhaps there exist practitioners of post-modernism/post-structuralism/neo-Marxism/etc that are doing substantive work that actually makes sense, but I am not familiar with them. I will stipulate that they do exist, provided you will allow me to also stipulate that what is popularly taken as post-modernism uses of models of human relations and human knowledge that have very limited or no domain of validity. And yes, I am aware that there are sometimes bitter differences between post-modernism, post-colonialism, post-structuralism, and so on. And when someone uses the phrase “post-modernism,” they may be referring to quite a range of applications. But whether in sociology or philosophy, it does not escape the charge of relativism. The basis of knowledge is not “discourse,” we don’t look to any human/political “hegemony” to define all meaning, and there is no reason why a traditionally “subordinate” factor should automatically be privileged.

There is a world of difference between the revolutionary statement “Blessed are the poor” and the ultimately meaningless (even nihilist) assertion that the subordinate side of every binary must be privileged. And so when I hear of churches informed with a post-modern viewpoint, I feel a little "cognitive dissonance," and suspect that something is wrong.