What is a theory, What is a model?

During my first semester I coincidentally found out that the guy who often sat next to me, one of the better students, believed the Earth was only 15,000 years old. Once on the topic, he produced stacks of colorful leaflets which featured lots of names, decorated by academic titles, claiming that scientific evidence supports the scripture. I laughed at him, initially thinking he was joking, but he turned out to be dead serious and I was clearly going to roast in hell until future eternity.

If it hadn’t been for that strange encounter, I would summarily dismiss the US debates about creationism as a bizarre cultural reaction to lack of intellectual stimulation. But seeing that indoctrination can survive a physics and math education, and knowing the amount of time one can waste using reason against belief, I have a lot of sympathy for the fight of my US colleagues.

One of the main educational efforts I have seen is to explain what the word “theory” means to scientists. We are told that a “theory” isn’t just any odd story that somebody made up and told to his 12 friends, but that scientists use the word “theory” to mean an empirically well-established framework to describe observations.

That’s nice, but unfortunately not true. Maybe that is how scientist should use the word “theory”, but language doesn’t follow definitions: Cashews aren’t nuts, avocados aren’t vegetables, black isn’t a color. And a theory sometimes isn’t a theory.

The word “theory” has a common root with “theater” and originally seems to have meant “contemplation” or generally a “way to look at something,” which is quite close to the use of the word in today’s common language. Scientists adopted the word, but not in any regular way. It’s not like we vote on what gets called a theory and what doesn’t. So I’ll not attempt to give you a definition that nobody uses in practice, but just try an explanation that I think comes close to practice.

Physicists use the word theory for a well worked-out framework to describe the real world. The theory is basically a map between a model, that is a simplified stand-in for a real-world system, and reality. In physics, models are mathematical, and the theory is the dictionary to translate mathematical structures into observable quantities.

Exactly what counts as “well worked-out” is somewhat subjective, but as I said one doesn’t start with the definition. Instead, a framework that gets adapted by a big part of the community slowly lives up to deserve the title of a “theory”. Most importantly that means that the theory has to fulfil the scientific standards of the field. If something is called a theory it basically means scientists trust its quality.

One should not confuse the theory with the model. The model is what actually describes whatever part of the world you want to study by help of your theory.

General Relativity for example is a theory. It does not in and by itself describe anything we observe. For this, we have to first make several assumptions for symmetries and matter content to then arrive at model, the metric that describes space-time, from which observables can be calculated. Quantum field theory, to use another example, is a general calculation tool. To use it to describe the real world, you first have to specify what type of particles you have and what symmetries, and what process you want to look at; this gives you for example the standard model of particle physics. Quantum mechanics is a theory that doesn’t carry the name theory. A concrete model would for example be that of the Hydrogen atom, and so on. String theory has been such a convincing framework for so many that it has risen to the status of a “theory” without there being any empirical evidence.

A model doesn't necessarily have to be about describing the real world. To get a better understanding of a theory, it is often helpful to examine very simplified models even though one knows these do not describe reality. Such models are called “toy-models”. Examples are e.g. neutrino oscillations with only two flavors (even though we know there are at least three), gravity in 2 spatial dimensions (even though we know there are at least three), and the φ⁴ theory - where we reach the limits of my language theory, because according to what I said previously it should be a φ⁴ model (it falls into the domain of quantum field theory).

Phenomenological models (the things I work with) are models explicitly constructed to describe a certain property or observation (the “phenomenon”). They often use a theory that is known not to be fundamental. One never talks about phenomenological theories because the whole point of doing phenomenology is the model that makes contact to the real world. A phenomenological model serves usually one of two purposes: It is either a preliminary description of existing data or a preliminary prediction for not-yet existing data, both with the purpose to lead the way to a fully-fledged theory.

One does not necessarily need a model together with the theory to make predictions. Some theories have consequences that are true for all models and are said to be “model-independent”. Though if one wants to test them experimentally, one has to use a concrete model again. Tests of violations of Bell’s inequality maybe be an example. Entanglement is a general property of quantum mechanics, straight from the axioms of the theory, yet to test it in a certain setting one has to specify a model again. The existence of extra-dimensions in string theory may serve as another example of a model-independent prediction.

One doesn’t have to tell this to physicists, but the value of having a model defined in the language of mathematics is that one uses calculation, logical conclusions, to arrive at numerical values for observables (typically dependent on some parameters) from the basic assumptions of the model. Ie, it’s a way to limit the risk of fooling oneself and get lost in verbal acrobatics. I recently read an interesting and occasionally amusing essay from a mathematician-turned-biologist who tries to explain his colleagues what’s the point of constructing models:

“Any mathematical model, no matter how complicated, consists of a set of assumptions, from whichj are deduced a set of conclusions. The technical machinery specific to each flavor of model is concerned with deducing the latter from the former. This deduction comes with a guarantee, which, unlike other guarantees, can never be invalidated. Provided the model is correct, if you accept its assumptions, you must as a matter of logic also accept its conclusions.”

Well said.

After I realized the guy next to me in physics class wasn’t joking about his creationist beliefs, he went to length explaining that carbon-dating is a conspiracy. I went to length making sure to henceforth place my butt safely far away from him. It is beyond me how one can study a natural science and still interpret the Bible literally. Though I have a theory about this…