An Iteration Between Theory and Practice

Meehl's Philosophical Psychology, Lecture 4, Part 1

This post digs into Lecture 4 of Paul Meehl’s course “Philosophical Psychology.” You can watch the video here. Here’s the full table of contents of my blogging through the class.

I cringe every time someone drops George Box’s overused aphorism “All models are wrong, but some are useful.” This quip gets thrown around to defend the worst sorts of scientific and engineering practices. But the thing is, when I read Box in context, I agree with everything he writes. Today, let me write a defense of Box, and describe what he actually meant. It turns out that Box was making the same point about modeling, feedback, and refinement of theories that Meehl picks up in Lecture 4.

Every scientific theory is indeed technically false. A theory is a complex of statements, and if any one of them is false, their conjunction is necessarily false. You’ll always be able to find one statement in any theory that’s not precisely true. If you only use a 3-significant-digit point estimate of the mass of some quantity in some formal derivation chain, your conclusion is technically false according to the logician. But such logical pedantry is annoying. Some of these calculations let us do stuff, like build things or run experiments. They might be logically wrong, but they are more than good enough for our purposes.

Meehl says that scientific theories that are good enough have some relative accordance with truth. Different theories have different degrees of truth. The approximation of truth is what Popper called verisimilitude. I’ll use this term even though I’m not particularly concerned with what “truth” is. As Meehl points out, there’s a difference between Instrumentalism and Realism. If you’re an Instrumentalist, you validate theories based on their utility for prediction and control. If you’re a Realist, it’s the theories themselves that you care about. 

Since I’m an engineer, this blog series takes a decidedly instrumentalist interpretation of Meehl. For the engineer, verisimilitude is your estimate of how widely—dare I say—useful a theory is for practical ends. We want to do things with scientific theories. We want to make predictions, cause things to happen, or build things. 

What do we do when the predictions don’t quite line up with what the theory says? The theory is now not just literally false, but practically false too. Do we throw everything away and start from scratch? That’s impractical for the instrumentalist and aesthetically displeasing for the realist. It seems expedient to look for ways to patch the theory up rather than throwing the whole thing in the trash can.

To patch a theory, we have to go to the statements and look at which ones to change. The nice thing about scientific theories is we know in advance that some of the statements are literally false. We know because we deliberately added them to the theory with full knowledge they were false. Deliberate false assumptions in a theory are called idealizations. 

Meehl gives two examples of idealizations that illuminate their value. Scientific theories include literally false statements that simplify derivation chains or yield simple rules. I’ll call these idealizations of first principles. Theories also include literally false statements about the particulars of entities in the derivation chains because we only have collected a finite amount of information about the universe. I’m going to call these idealizations of boundary conditions. Let me now use Meehl’s examples to show how both can be adjusted to patch up false theories to make them more truthy. And in doing so, we’ll see how the dynamic feedback between theory building and experiment doesn’t admit a clean, logical set of rules. I’ll describe idealizations of first principles today and idealizations of boundary conditions in the next post.

The Ideal Gas Law

Meehl and Box have the same favorite example of a model that is wrong yet useful: the ideal gas law we all learn in high school. The development of this law nicely illustrates idealizations and the iterative feedback loop of theory building.

If you have a gas in a chamber and you compress it with a piston, the pressure the gas exerts on the piston is related to the volume of the gas, the temperature of the gas, and the amount of the gas. You get the famous formula:

P is pressure, V is volume, T is temperature, n is the number of moles of gas, and R is a constant. This law was initially derived from experiments where two variables were manipulated and the others held constant. But in the 1850s, physicists figured out how to derive this law from basic kinetic interactions of trillions and trillions of individual gas particles. This derivation connected microscopic classical mechanics to macroscopic thermodynamics: small tiny particles bouncing against each other would manifest themselves in properties that we measure as heat or pressure.

The kinetic theory relied on several idealizations. Two central ones were

• Particles are “point masses” so small that you can neglect their size. 

• There are no attractive forces between the particles.

Both of these were known to be literally false, even in the primitive molecular theories of the day. The particles, though very very tiny, were unlikely to have literally zero volume. And they clearly had mass, so they must attract each other gravitationally. These assumptions helped simplify the calculations. As you wrote out the math, you’d see a term that would depend on the radius of the particle and consider it too small to influence the downstream calculations. With these simplified equations, you grind out some calculations and, lo and behold, find PV = nRT.

The ideal gas law fits the data well over a wide range of Ps, Vs, and Ts. But it breaks down at small volumes and low temperatures. Now, if physicists had just applied the hammer of Popperian modus tollens, they’d have to throw the whole theory away at this point. The new experiments had falsified the theory. But that seems silly. If a theory is good over vast ranges of Ps, Vs, and Ts, maybe we can figure out a way to patch it in the regions where it’s not so good. We don’t consider the theory dead. We don’t even consider the theory mostly dead. We just consider it injured and in need of a band-aid.

Adjusting the kinetic theory calculations to account for the incorrect idealizations gave the needed fix. Van der Waals showed that by allowing the molecules to occupy a non-negligible volume and removing some pressure due to pairwise molecular attraction, you could get a correction that would match the data:

Here, Van der Waals added two new constants, “a” and “b.” When a and b are much smaller than V, this formula is more or less PV=nRT again. It is only at small volumes and temperatures where the a and b play a role.

The constants a and b were not universal in the way “R” was. They were properties of the associated gas and had to be fit on a case-by-case basis. But Van der Waals’ formula, with the additional free parameters, gave the right predictions in a wider region than its uncorrected version.

Adjusting the known false statements in the idealizations of first principles to be more truth-like gave better predictions over a wider range of outcomes. This wider applicability came only with the expense of uglier formulas and more difficult derivation chains. Experiments that had technically falsified the ideal gas law ended up corroborating the kinetic theory of gas. By changing what they knew was false, physicists ended up with a better prediction, and that’s a damned strange coincidence.