Three types of mathematical models
September 8, 2013 52 Comments
Whenever asked to label myself, I am overcome by existential dread: what am I? A mathematician? A computer scientist? A modeler? A crazy man with a blog? Each has its own connotations and describes aspects of my approach to thought, but none (except maybe the last) represents my mindset accurately. I have experienced mathematical modeling in three very different setting during my research and education: theoretical computer science, physics, and modeling in social and biological sciences. In the process, I’ve concluded that there are at least three fundamentally different kinds of modeling, and three different levels of presenting them. This is probably not exhaustive, but I have searched for some time and could not find extensions, maybe you can suggest some. Since this post is motivated by names, let’s name the three types of models as abstractions, heuristics, and insilications and the three presentations as analytic, algorithmic, and computational.
In physics, we are used to mathematical models that correspond closely to reality. All of the unknown or system dependent parameters are related to things we can measure, and the model is then used to compute dynamics, and predict the future value of these parameters. Sometimes, as in the case of statistical or quantum mechanics, these predictions are probabilistic (for different reasons in the two theories) but are expected to agree with reality after many independent measurements. I call these models that translate measurements of ’empirical reality’ into predictions about future results of similar measurements as insilications because they are a model ‘replicating’ the relevant parts of reality. We usually learn these models presented in analytic terms as a series of mathematical equations that we can solve explicitly. A standard example would be solving for the motion of a cannonball using Newtonian mechanics, or a more complicated example would be solving the spectrum of a Hydrogen atom using quantum mechanics; both are exercises I had to do at various stages of my education.
The reason I chose the in silico computer-inspired name for these sort of models is because — in non-classroom settings — they are usually solved numerically or by simulation. I call this the computational presentation. Classic examples would be a civil engineer using a piece of software to calculate the stresses on a bridge design, or NASA simulating the trajectory of a spacecraft destined for Mars. Examples with more research-level physics would be simulations and subsequent statistical inferences from the detectors at the Large Hadron Collider (see here for a biological example). These models are simulated on computers, but we understand the equations going into the simulation, how they interact with each other, and how the computer calculates the relevant parameters so well that we might as well think of them as in silico recreations of the ‘real’ world. If there are discrepancies from empirical measurements then the theory these models are built within usually has a way of quantifying and accounting for such errors. If a systematic disagreement is found between a well-implemented insilication and empirical measurements then this can be used to bring into question or even falsify the theory underlying the model. I think these are the models that most people think theorists concern themselves with.
In reality, though, most theorists outside of engineering and the hard physical sciences (and even some in them, like cosmologists) work on heuristic models. When George Box wrote that “all models are wrong, but some are useful”, I think this is the type of models he was talking about. It is standard to lie, cheat, and steal when you build these sort of models. The assumptions need not be empirically testable (or even remotely true, at times), and statistics and calculations can be used to varying degree of accuracy or rigor. Often, these models aren’t useful in spite of being false, but because they are false. A theorist builds up a collection of such models (or fables) that they can use as theoretical case studies, and a way to express their ideas. It also allows for a way to turn verbal theories into more formal ones that can be tested for basic consistency. However, the drastic contrast in basic goals of this sort of modeling is why people like Noah Smith that are more comfortable with insilications become uncomfortable with heuristics.
As with insilications, heuristics can be presented in several ways. In neo-classical economics, the standard presentation is analytic, our assumptions are represented as particular equations or axioms that might or might not be true or even empirically testable. Conclusions are then drawn from these assumptions by solving systems of equations or analyzing their qualitative dynamics. Sometime, this analysis can be done in general terms with some steps abstracted and replaceable by ‘any algorithm’ or ‘any model’, as I do in a biological setting with my recent work on evolutionary equilibria — this is an algorithmic perspective. By contrast, in fields like complex adaptive systems (or complexity economics, if we want to contrast with neo-classical), a computational perspective is used. Here, heuristic models are simulated on computers and conclusions are drawn based on the results of these experiments. To me, this is very dangerous, because unlike the analytic or algorithmic treatment of heuristics, it does not even provide a definitive link between assumptions and conclusions. Computational heuristics suffer from the curse of computing, and although they can be used for rhetorical purposes, it is not clear if the theorist studying them gains any understanding. In the words of theoretical physicist Eugene Wigner:
It is nice to know that the computer understands the problem. But I would like to understand it too.
Unlike computational insilications, computational heuristics do not provide reliable predictions, and thus their outputs are useless unless they generate understanding. By contrast, the best way to gain understanding, in my opinion, is through the third type of models — abstractions. These are the models that are most common in mathematics and theoretical computer science. They have some overlap with analytic heuristics, except are done more rigorously and not with the goal of collecting a bouquet of useful analogies or case studies, but of general statements. An abstraction is a model that is set up so that given any valid instantiation of its premises, the conclusions necessarily follow. These models are not build to illustrate a point, but as tools to analyze any theory. The classical example is Turing machines and other models of computation; if your theory has certain qualitative features then it is necessarily Turing complete and from this we can conclude — for example — that some general questions about your theory are not answerable. Abstractions are most useful as a way of classifying other models, or drawing concrete connections between heuristics or insilications in different fields. As far as I know, there is no real way to study abstractions through the computational perspective, and results are shown analytically (say in mathematical physics) or algorithmically (say in theoretical computer science).
I only concentrated on mathematical models in this post, and ignored two important types of models: mental and physical. The first is usually expressed as intuitions or verbal theories. The second is popular in biology as model organisms such as the rat or E. coli, where we can study theories about animals on which we cannot ethically (or practically) perform experiments. Did I miss any other important types of modeling? What is your preferred type and presentation of models?