In the previous chapter we looked at the historical roots of scientific models. We will now turn to their contemporary use and view them from a philosophy of science perspective. As a starting point for this discussion we will specify what we actually mean by models, and discuss the different classes of models that exist. In order to broaden the concept and connect it to other concepts within science, we will relate modelling to phenomena and theories, and also analyse the relation between simu- lations of models and experiments. These notions will all be tied together, and made concrete in a worked example, where we show how a model typically is constructed.
Lastly, we will try to draw conclusions about desirable properties of models, and discuss the dangers of modelling.
Since this chapter focuses on discussing general concepts that both span different types of models and their use in many different disciplines, the contents lean towards the abstract. To discuss and analyse the general ideas we have had to sacrifice the particulars. Still it is our hope that the examples contained in the chapter will illumi- nate the problems discussed and clarify the conclusions drawn. Concrete examples of different types of models are also presented in the chapter Worked Examples.
28 Structure, Relation and Use
A model is an interpretative description of a phenomenon that provides access to the phe- nomenon.1
…[Models are] physical or mental systems that reflect the essential properties of the phe- nomenon being studied.2
…a common notion is to view scientific models as representations of separate real-world phenomena.3
These suggestions point in the same direction, and we will use the following definition, which is similar to all of the above, namely: Models are descriptions, abstract or material, that reflect or represent, and hence provide access to, selected parts of reality.
We have added the clause “and hence provide access to” to exclude trivial repre- sentations that do not provide any understanding of reality. The word “bear” can be said to represent a subset of reality, namely all animals known as bears, but the word itself does not give any insight into the nature of bears nor does it provide us with any access to the properties of the animal.4On the other side of the spectrum we can imagine extremely complex, but completely pointless, representations that do not provide any meaningful access to the phenomena under study. In fact this reasoning, if taken to its logical extreme, implies that anything can be a model of anything else.
For example we could let a tea kettle be a model of an elephant (they both have mass and spatial extension, and might even have the same colour), but this would hardly give us any insight into the nature of an elephant.
At this point, it might be a good idea to stop and think about what is actually meant by the phrase “selected parts of reality”. We take it to denote a phenomenon, a concept which in itself is not unproblematic, and will be discussed at length later in this chapter.
Model Taxonomy
Models can be divided into separate classes depending on how they represent the phenomenon in question. A common division is into conceptual, iconic, analogous, symbolic, phenomenological and statistical models.
Conceptual modelsare the most basic type and serve as the foundation for more concrete and mathematical models. They manifest themselves as ideas and notions about mechanisms and entities within a system, sometimes purely mental constructs, but often expressed in everyday language. For example the greenhouse effect is
1Bailer-Jones, Daniela (2009). Scientific Models in Philosophy of Science, University of Pittsburgh Press.
2Hansson, Sven Ove (2007). Konsten att vara Vetenskaplig.
3Sellerstedt, Bo (1998). Modeller som metaforer, Studier i Kostnadsintäktsanalys, EFI, Stockholm.
4The exception here is possibly onomatopoeic words, whose sound imitate their meaning, e.g.
‘crow’.
often expressed in linguistic terms as “the gases in the atmosphere of a planet block the outward radiation from the surface of the planet and this leads to an increased average temperature”. This conceptual formulation might be useful if one wants to communicate the model, but it can also be put in mathematical terms that allow for predictions with higher accuracy. With a mathematical formulation of the greenhouse effect it is e.g. possible to calculate how many degrees the average temperature of the Earth would increase by if the concentration of carbon dioxide was doubled.
Another example of a conceptual model is the description of how stars are shaped by a balance between gravitational forces that pull matter towards the centre of gravity, and radiation pressure that pushes matter outwards, which (in most cases) leads to a stable equilibrium. This is also a general formulation that can be made quantitative by dressing it in a mathematical guise.
Iconic models are often viewed as the simplest type of models since they are direct representations of a system, only magnified, miniaturised or projected. To this class belong scale models, pictorial models and blueprints. The main reasons why iconic models are used is that they are far easier to handle than the original systems, and they save both time and money. Instead of building a full-scale version of the hull of a ship, and testing its properties, one is often content with constructing a scale model with which similar tests can be carried out, and based on the outcome, conclusions about the properties of the real ship can be drawn. In some cases the model is scaled in time and not space, such as the case of experiments with bacterial populations, with a short generation time (a couple of hours), which makes it possible to investigate phenomena on evolutionary time scales. Iconic models also allow for visualisation of certain systems, both in terms of miniatures, such as blueprints, and magnifications, such as atoms being modelled as plastic spheres that can illustrate how molecules are structured and interact.
Analogous modelsare not classified by their structure, but rather by the process of construction. The common feature is that they have been devised in analogy with a known system. Some striking examples of this are the use of hydraulic models to describe the flow of money and value between actors in economic systems, models of atoms that describe the orbits of electrons in analogy with the planets orbiting the sun, and the perennial harmonic oscillator. The latter model has been used to describe a wide array of physical systems, such as the dynamics of a simple pendulum or how molecules absorb electromagnetic radiation by storing the energy as internal oscillations (the phenomenon on which microwave ovens are based). The essence of an analogous model is that the system one tries to describe is compared with another system whose dynamics are known, and this knowledge is then transferred to the new system, which hopefully leads to novel insight. These models spring directly from the method of modelling that arose among physicists during the 19th century, where classical mechanics was used as a means to construct useful analogies. An analogous model can be both mathematical and informal. For example certain properties of light can be explained by making analogies with undulations on the surface of a fluid or in a gas, but these explanations will never be exhaustive since the two phenomena (light and ripples) are fundamentally different.
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Symbolic modelsmake use of symbols and formal systems in order to describe phenomena. A typical example is a mathematical model that utilises equations in order to represent the properties of a system. Such equations can vary both in number and kind, but the basic idea is that they describe features and components of a phe- nomenon that are considered interesting, and interactions between these variables.
There are models that only consist of a single equation, such as the Fisher model, which describes how an advantageous genotype spreads in a population as a function of time and space, but also models that are composed of hundreds of equations, such as models of genetic regulation in molecular biology. A common type of equations, especially within physics, are partial differential equations, which describe how con- tinuous quantities such as a magnetic field or wind speed changes in space and time.
Also discrete models in the form of difference equations are common. In addition to models formulated in terms of equations, there are also agent-based models, which have become popular within economics and biology. These models are defined by assigning the agents properties and rules according to which they act under certain conditions. This makes them difficult to understand from a mathematical perspective, and instead simulations are required to analyse their behaviour.
Phenomenological models are often symbolic in nature, and the two types of models are differentiated by their structural similarity. With symbolic models one tries to capture interactions between relevant, and in reality existing variables, while phenomenological models are used when the end result is prioritised and capturing the actual mechanisms within the system is viewed as less important. Such models, where only the outcome is of importance, are often viewed as “black boxes”, since the internal mechanisms are considered uninteresting and hence can be neglected.
An example of a phenomenological model is the Gompertz growth model, which is used for describing how the size of a growing population changes over time. It was originally derived as a means to estimate the expected life span of humans, and is based on the assumption that “…the average exhaustions of a man’s power to avoid death are such that at the end of equal infinitely small intervals of time, he loses equal portions of his remaining power to oppose destruction”.5 Its most common use is however to model the growth of populations, and in particular the growth of tumours.
Another phenomenological model that has been applied in cognitive science is the concept of the brain as a computer. The structural similarity between a brain and a computer is non-existent, since the brain consists of billions of neurons that process information in parallel, while in a computer information is processed by a central processing unit. Despite this, certain similarities can be seen between the composite behaviour of all neurons and the computations that take place in a computer.
Statistical modelsare a subset of symbolic models that make use of tools from probability theory. A statistical model of a system describes relations between vari- ables that are not fixed but take on values from a certain distribution. These distri- butions are characterised by a number of parameters and statistical modelling often
5Gompertz, Benjamin (1825). “On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies”. Philosophical Transactions of the Royal Society of London 115: 513–585.
boils down to estimating such parameters. With the aid of statistical models it is possible, by observing and analysing data from a phenomenon, to determine which interactions are important and what variables are relevant.
Let us briefly return to the example about rain and umbrellas that was discussed in the introduction (see p. 6). If we were to describe the situation with a statistical model we would proceed by introducing a stochastic variable X that represents the amount of precipitation that will fall during the entire day. This stochastic or random variable has a certain distribution, which is a function that assigns a probability to each level of precipitation. Initially we need to estimate this distribution by for example consulting previous meterological data. In order to avoid the rain that might
Fig. 1 The house of models is built on top of the Conceptual basement, which forms the basis for quantitative modelling. The ground floor consists of two rooms. On theleftis the Iconic playground where one can study and play with scale models of the Earth, ships, trains and molecules. On the rightis the analogous billiard room where the effects of every single shot can be predicted with perfect precision, the winner of the game is even announced in advance. The first floor has a bright and open design and fancy mobiles hang from the ceiling of the Symbolic suite. Theleft corner is considerably murkier due to the bigblack boxand the functionalist design. On theright sidein the Symbolic suite we find the Statistical corner with direct access to the Sample space. For more information about the house of models we would recommend a visit to the Iconic playground where one can study a detailed scale model of the house itself. © Lotta Bruhn based on an idea by Torbjörn Lundh
32 Structure, Relation and Use
fall one could take an umbrella, but there is always the chance of leaving it behind somewhere. We let the probability that this occurs be represented by a variableY, whose distribution depends on the degree of forgetfulness. When we are to leave the house we have to make a decision as to whether we are willing to take the bus (and pay the bus fare) or use the umbrella with the risk of losing it, were it to rain sometime during the day. If we know the distributions of the variablesX andY, the bus fare and the price of an umbrella, then with the aid of probability theory we can calculate the expected cost and choose the cheapest option.
This classification of models is hardly exhaustive and many models fall into several categories. This subdivision does however provide us with a framework with which to understand the similarities and differences between various models, and also how they relate to the systems they describe.6An artistic representation of the various classes of models that shows how they all fit under one roof is shown in Fig.1.