The examples from the previous section illustrate how model equations are derived from modelling assumptions. However, the cases that we examined do not really reflect the complexity involved in the mathematical modelling of real systems, since the set of modelling assumptions from which we derived model equations are extremely simplified compared to what would faithfully capture real physical systems. Needless to say, when we build a model for a system of real bodies, the inaccuracy and incompleteness of the modelling assumptions could very well lead us to answer questions about the behaviour of interest incorrectly. A traditional and widespread view to remedy this sort of situation has been discussed by Batterman (2002a: p. 21):
A traditional view about modeling in the physical and applied mathematical sciences holds that one should try to find the most accurate and detailed math- ematical representation of the problem at hand. [. . . ] The aim here is to effect a kind of convergence between model and reality. One tries, that is, to arrive at a completely accurate (or ‘true’) description of the phenomenon of interest. On this view, a model is better the more details of the real phenomenon it is actually able to represent mathematically.
Using the terms employed here, the traditional view enjoins the modeller to use an accurate and complete set of modelling assumptions in order to guarantee
that the model equation derived from them correctly answers our questions about the system.
However, as Batterman (2002a: p. 22) argues, “the more details that are built into the model, the more intractable the mathematical equations repre- senting the behaviour of interest are likely to be.”That means that, even if we can somehow derive model equations from our accurate and complete set of modelling assumptions, it is likely that we will not be able to use them to make predictions and to obtain answers to our questions concerning the behaviour of interest. For that reason, one could say that those models are too true to be good.
In the case of Kepler’s two-body problem, the discrepancy between the modelling assumptions and real systems lies in the fact that real bodies are not mass-points and that there are no gravitationally isolated systems of two bodies. Moreover, there normally are other forces that act on real bodies. If we add these elements to the list of modelling assumptions, we can derive another set of differential equation that will in principle better describe the evolution of the system through time. This will lead us to model equations for a many-body problem—potentially with additional non-gravitational forces— that will present serious problems for the extraction of solutions from the model equations.25 Thus, the improvement of the accuracy of the modelling
assumptions brings about a decline of the tractability of the model.
In the case of the machines, the modelling assumptions we used are mis- guiding to the extent that real machines are not ideal. If we consider what happens when a person uses a real lever, as opposed to an ideal lever, the situation is as in figure 2.8. The lever used has a mass, a volume, a variable density, it is deformable, there is internal shear and bending in the beam, there is friction with the fulcrum, the fulcrum might wiggle, there is dissipation of energy, and there are other forces acting upon the system. If we consider in- stead a system of masses and pulleys, we find that the pulleys are not perfectly circular, that they are not perfectly uniform (so that their moments of inertia 25More specifically, there will be no closed-form solution. We will discuss in section 4.1 what are the consequences of the lack of a closed-form solution.
Figure 2.8: Real levers do not satisfy the idealizations used to study perfect levers (source unknown).
change), that there is friction and shear, that the cables stretch, that there might be wind moving the cables, that the temperature changes (so that the sizes of the pulleys and cables might change), etc. Thus, to obtain a reliable model of such systems, many factors that the analysis of ideal machines ne- glects would have to be considered. However, once again, the improvement of the accuracy of the modelling assumptions brings about a decline of the tractability of the model. In fact, we could no longer use simple geometri- cal methods applied to free-body diagrams to determine the consequences of our modelling assumptions, as we did in the previous section. As a result, the task of constructing the mathematical representation and of tracing its consequences would be significantly harder to execute.
Thus, there is a crucial dilemma between accuracy and completeness of modelling assumptions and tractability of model equations at the very core of the logic of mathematical modelling. What makes mathematical modelling difficult is that above all we must find a balance between accuracy, complete- ness, and tractability, as in figure 2.9. This being said, the fact that there are no gravitationally isolated systems of two mass-points or no real systems of perfect levers does not mean that there are no real systems that can betreated as if there were one. If it turns out to be the case that a real system can be understood by means of such a comparatively simple representation, then we have found a way to overcome the dilemma.
accuracy and completeness of the modelling assumptions
tractability of model equations
Figure 2.9: Balancing factors in mathematical modelling.
Such models, which have been successfully simplified and idealized in order to gain on the side of tractability, are calledminimal models. What constitutes a successful simplification? How can we compare the consequences of minimal and complete models, given the intractability of the latter? The idea is to use perturbation methods, asymptotic analysis, and other forms of analysis of sen- sitivity to perturbation. From the exact solutions of the simplified model, one can use such methods to add correcting factors that correspond to rectification of the assumptions. We will return in more details to this later. For now, one should understand that the question of the accuracy of models cannot in gen- eral be simply addressed by saying “add more details to have a more accurate and complete set of modelling assumptions.” There is always a cost-benefit analysis to perform, and the most important contribution of mathematics to modelling is that it provides the tools to do just that. Thus, counter-balancing the view of the role of applied mathematics as the language for formulating true representations of systems, there is the view that mathematics is “the art of finding problems we can solve,” to quote the great mathematician Hopf.26