"Model" and "theory" are both words that physicists use frequently, and often with a different meaning than attached to it in the colloquial language. Since we on this blog write about models and theories all the time, I thought it worthwhile to clarify what I mean with that.
Disclaimer: This isn't meant to be a definition, just a clarification. As with all language related issues there is a large grayscale to a word's applicability. I am not claiming the following is a standard for usage-of-words.
I. The Real World Out There
If you are one of our frequent readers then you know that I occasionally feature the idea that all of our commenters are actually manifestations of my multiple personality disorder. Let me call that a 'theory' to describe my 'observation' of comments. It's not scientifically a particularly compelling theory. For one, some commenters have shown up in my office which means to make my theory viable, I should add some numbers to my ICD-10 diagnosis , like F22 (Delusional Disorder) or F44.1 (Dissociative fugue). Alternatively, I could simply argue there is no reality other than what the neurons in my brain produce. That's a theory as well, and since I can never falsify it, it's not something a scientist should spend much time on.
To begin with, it is therefore for practical purposes reasonable to consider the possibility there is something like a "real world out there", including my computer, you, and the CMB radiation - and it's this real world out there that we are trying to understand and describe.
For example, you could speculate that your phone always rings if you take a bath. There is reality out there: you, the bathtub, the phone. To make a theory out of your speculation - call it a 'hypothesis' because it sounds better - you need to do somthing more. To begin with your hypothesis isn't very useful because it's too unspecific. You would want e.g. to add that the phone is switched on. Also, some quantification of your framework is necessary. If you just say the phone will ring, you can shrivel in your bath until you die out of boredom, and never falsify your hypothesis because it could always ring the next minute . Instead, to make your theory credible you would want to clarify "If my phone is turned on, it will ring within 10 minutes after I sat down in a bathtub full of water".
|Now consider you do that, but the phone doesn't ring. Then there is always the temptation to add more specifications a posteriori. Like, it only happens on Wednesdays, and only if you use rosemary scented bubble bath, and only if you have large enough extra dimensions. Or so. You can add a long series of 'only ifs' to explain a negative outcome of your experiment. That's still a theory, but with each 'only if' it loses some of its generality, and its applicability becomes more and more limited, which makes it less and less interesting.|
II. Theories and their Limitations
What scientists usually mean by theory is a testable hypothesis about an important aspect of the real world out there that has predictable power, and a consistent and well-defined framework.
- 'Testable' means, there is a possibility to prove it false. Unless you have a theory of very limited applicability, you can never verify it to hold in all possible circumstances, for all experiments that will ever be done, by anyone. However, for your theory to be good for something it must at least be possible to falsify it, otherwise it's like claiming you have this invisible friend who is so smart nobody can ever prove he is really there because he's shy and just doesn't want to.
- 'Predictable power' means your theory tells you what will happen under certain circumstances stated in your theory - in physics this usually means prediction of experimental data. Experimental data can also be already available, and await to be explained by a theory. Though one better shouldn't call that a prediction, maybe a 'postdiction'. The possibility for experiments that can be done to confirm a theory differs greatly between the fields. For example, we can't repeat the evolution of species with slightly different initial conditions. Natural Selection such is somewhat weak on the side of useful predictions, but it does a good job explaining the evolution of species that archaeologists find documented. In physics we today have a lot of experimental data, like e.g. dark energy and dark matter, that yells at us theorists because it wants to be explained.
|- 'Well-defined framework' means your theory is in a comprehensible form, and not a big pile of glibber that one can't grab because the interpretation is unclear. Roughly speaking, if one can't understand and apply a theory without consulting its creator, it's not a scientific theory.|
[Figure: Not a Theory]
- 'Consistent' means, it does not have internal contradictions. E.g. if you have a theory that explains the world by taking the Bible literally, then I recommend you first make sure to remove inconsistencies.
- 'Important aspect' is very subjective. I'd think your bathtub theory for example isn't so tremendously important for the biggest part of the world, and wouldn't qualify as a 'scientific theory'. But at which stage scientists are inclined to promote a hypothesis to a scientific theory depends very much on the circumstances.
One can have a lot of theories. Theories that become accepted scientific knowledge are those that are experimentally well confirmed and have proved useful. The point about Natural Selection is not that it is a theory but that it is a very well confirmed theory with explanatory power. The point about Einstein's theory of General Relativity is not that it is a theory, but that it is a theory confirmed to very high precision with a large number of experiments. The theory of the luminiferous aether on the other hand is a theory as well, but one that was proved wrong with the experiment by Michelson and Morley . String theory, as well as other approaches to quantum gravity, are difficult cases because they would become important in energy ranges far outside the reach of experiments on earth, so their status is pending.
Theories are reliable typically in a limited range and up to a certain precision. Special Relativity for example reaches its limits when the curvature of space-time becomes important, in which case one has to use General Relativity. Your bath tub theory reaches its limits at a temperature of 100°C at which you'd get problems filling the tub, not to mention getting in.
If somebody proposes a new theory it is most often an improvement of an already existing one, either because it applies with larger generality, to better precision, or both. Still, the 'older' theory might remain useful. For example, the non-relativistic limit is accurate to very high precision for slowly moving objects and using the fully relativistic framework is often an unnecessary overkill.
|Okay, so far we have the real world out there, and we have a theory. Theories are usually very general concepts from which one constructs a specific model. The model is a simplified version of the real world out there, simplified in the sense that it deals only with a limited amount of details. For example if you want to compute how a cow drops out of a plane, you can forget about her milk-efficiency, and assume to good precision it is a ball, with a mass M and a radius R. You can also attempt to understand a political arguments by the gender of its proponent. This identification of features is a model, underlying which there is your theory that you want to test.|
In physics, we have for example the quantum field theories which underlie the Standard Model of particle physics. In this model, we identify particles with states in the Fock space, observables with expectation values, and particle properties with gauge charges that belong to specific gauge groups. Another example is the ΛCDM model in Cosmology, underlying which is Einstein's theory of General Relativity. The identification of the relevant ingredients to the model is crucial to make it useful.
Besides the identification of objects, typically your model will need to use some data as an input to make predictions for further data. In other words, it will have free parameters that the theory can not predict and that just have to be measured. Einstein's theory of Special Relativity for example has a constant that you can show to be the speed of massless particles. Then you go and measure this constant, commonly known as c, with which you can then apply your model to other cases. The more free parameters a model has, the less useful - not to mention, ugly - it is. A model that needs the same amount of parameters as there is data to fit isn't good for anything. There is always a n-th order polynomial which lies exactly on n data points.
Theories and models can come in many different forms. In physics our models use the language of mathematics, and our theories tell us how to identify mathematical quantities with 'real' physical objects. Models can also be computational, in which case you translate the real world into input of your computer code. But in other areas, mathematics or computation is not necessarily the language of choice. For example, in psychology one has the 'Existential Theory' which in a nutshell says humans are driven by four existential fears: death, freedom, isolation, and meaninglessness. Based on this theory, one can then try to understand a patient's problem, i.e. build a model to explain the real world out there. In this case, mathematics isn't the language of choice, mostly because it is too inflexible to cope with something as complex as human behaviour. Another example is Adam Smith's "Wealth of the Nations", which puts forward the 'theory' of the invisible hand. The tragedy with this specific example is that even though this theory is known to be wrong or not applicable in many cases, it is an argument still used that influences the lives of people all over the world.
The example from psychology also illuminates another feature of a good model. I am not much of a psychologist, but even to me the reduction of human behaviour to four existential fears seems to be overly simplistic. And it probably is, but what makes a useful theory is that after stripping off lots of details you have identified some relevant properties that can lead to an improved understanding, even though restrictions may apply.
However, a model doesn't necessarily have to be about describing the real world out there. To achieve a better understanding about a framework, it is often helpful to examine very simplified models even though one knows these do not describe reality. Such a model is called a 'toy-model'. 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).
IV. Theoretical Physics
A big challenge especially in theoretical physics is that theories potentially remain untestable for a long period of time, because the farther our theories depart from every day experience, the more effort we have to make to design suitable experiments. In these cases, internal consistency is often the only guide. Quantizing gravity for example is actually not the problem. You can quantize it if you want to. The problem is that the outcome is nonsensical. This way, one can drop a lot of theoretical approaches even without testing them on the real world. It is for this reason that apparent paradoxa appearing within a theory receive a lot of attention, as their investigation and solution can be the source of new insights and progress.
Besides consistency, some people also like to call upon more ethereal values like 'beauty', 'elegance' or 'naturalness' to argue for the appeal of their theory. It is a slippery slope however, as the relevance of these factors is a theory in itself, and not a scientifically well confirmed one.
You could for example have the theory that the real world is made out of tiny vibrating strings. Once you've made sure your theory is internally consistent, and added sufficient 'only ifs', the way to proceed is then to build a model, make a prediction, and line up for the Nobel Price. Alternatively, you could have the theory that the real world is made out of braids, and identify particle properties with braiding patterns. However, if your model doesn't reproduce gauge fields we commonly call photons and gluons, it doesn't seem to describe the real world out there, so it is at utmost a toy model. You can have all sorts of theories. Like you will be reborn as a Boltzmann brain. The value of such theories differs greatly depending on its usefulness.
Such, in theoretical physics you make a living with speculation, with the eternal hope that you manage to catch a glimpse of Nature's ways and experiment will confirm you. It is a difficult task, since every new theory first needs to reproduce all the achievements of the already established ones, plus it needs to lead to new insights. The requirement of consistency is one that people not working in the field typically underestimate - it greatly reduces the amount of freedom we have with our speculation. In a certain way it is as fascinating as frustrating if a theory you have disagrees with you and just doesn't do as you want it to. I am always annoyed by this. It's like I think if I made it, it's supposed to do what I want. Very possibly for this reason I am inclined to say that we don't actually invent theories, but that we discover them.
Another challenge in this procedure is the problem that different theories can under certain circumstances result in the same model. In such cases, one has to look for scenarios in which one can distinguish both theories. If there are none, one can call both theories equivalent, and the difference is one of interpretation. Though without any direct consequences as far as predictions are concerned, establishing an equivalence between two interpretations and a change in perspective can be very fruitful for further developments.
A scientific theory is a consistent and well-defined framework to test a falsifiable hypothesis about the real world out there. A theory that becomes accepted knowledge is one that has been confirmed to high accuracy, and has proved useful. Theories underlie the models that we use to describe the world. We can also investigate 'toy models' to understand our theoretical framework better, even though the scenario is not realistic. Internal consistency is a strong requirement on a scientific theory that is often underestimated.
After finishing this writing, I find that my above explanation disagrees with other's. For example Laurence Moran explains:
"A theory is a general explanation of particular phenomena that has withstood many attempts to disprove it. Because of the evidence supporting the explanation and because it hasn't been refuted, a theory will be widely accepted as provisionally correct within the science community."
As I said previously, arguing about words isn't something I like to engage in, but this would mean that there are no falsified theories, and it constrains the usage of the word 'theory' to those theories that are 'correct' descriptions of nature (to some degree) because there is already evidence supporting them. Though possibly I misunderstand, and he means to say that the science community will generally only call something a 'theory' if it lives up to certain quality standards, withstands the most obvious criticism, and the possibility exists that it describes the real world out there. (Like e.g. if your 'theory' does not have fermions, forget about it.)
Wikipedia quotes the National Academy of Sciences with:
"Some scientific explanations are so well established that no new evidence is likely to alter them. The explanation becomes a scientific theory. In everyday language a theory means a hunch or speculation. Not so in science. In science, the word theory refers to a comprehensive explanation of an important feature of nature that is supported by many facts gathered over time. Theories also allow scientists to make predictions about as yet unobserved phenomena."
Which I think goes well with what I wrote above.
 The phrase "The real world out there" is borrowed from Lee's book, who I believe borrowed from elsewhere, but I can currently neither recall the actual origin, nor where I put to book. Sorry about that.
 One finds an iteration of this sort of theory that is unfalsifiable within the experimenter's lifetime in Hollywood. It's called the 'One day I will be rich and famous' theory of the unknown actor. Surely fame is just around the corner, hang on for one more day.
 The aether theory however seems to have Zombie character and occasionally comes back to haunt us in various alterations that escape the constraints of Michelson-Morley.