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On the epistemological analysis of modeling and computational error in the mathematical sciences

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Abstract

Interest in the computational aspects of modeling has been steadily growing in philosophy of science. This paper aims to advance the discussion by articulating the way in which modeling and computational errors are related and by explaining the significance of error management strategies for the rational reconstruction of scientific practice. To this end, we first characterize the role and nature of modeling error in relation to a recipe for model construction known as Euler’s recipe. We then describe a general model that allows us to assess the quality of numerical solutions in terms of measures of computational errors that are completely interpretable in terms of modeling error. Finally, we emphasize that this type of error analysis involves forms of perturbation analysis that go beyond the basic model-theoretical and statistical/probabilistic tools typically used to characterize the scientific method; this demands that we revise and complement our reconstructive toolbox in a way that can affect our normative image of science.

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Notes

  1. Charactering what the essential features of a system are is a delicate problem, and many proposals of very different natures have been made in order to address this intricate question. For the purpose of this paper, it suffices to think of them as contextually determined traits that are relevant to understanding the behavior of interest. Apart from the conceptual and logical approaches to relevance, one can understand this in terms of mathematical methods such as asymptotic analysis. For this latter approach in the philosophical literature, see, e.g., Batterman (2002a, b), Fillion (2012) and Pincock (2012).

  2. One such very similar traditional question results from a sceptical worry that lies at the very core of epistemology. In somewhat Kantian terms, it can be formulated as follows: given that the noumenal truths are not accessible, how should one determine the status of such knowledge-claims?

  3. A remark is in order. The mathematical structure is universal in the sense that it is treated as if it were. No particular constraints on its application is suggested by the theory. However, this is strictly true only insofar as we are dealing with classical (non-quantum) systems, in non-general relativistic space–time.

  4. This point is elucidated by Smith (2001, 2002) and Earman et al. (2002). See also Putnam (1991) and Stein (1995) for an illuminating discussion of this fact.

  5. The sense in which they contradict each other is that they cannot simultaneously apply to the same body, as they can characterize its dynamical properties in mutually exclusive ways.

  6. We draw this description of the recipe from Wilson (1998) and Smith (2002).

  7. This classification is an adaptation from Neumann and Goldstine (1947). As always, the difference between error and uncertainty should be borne in mind. An error is simply the difference between a value and the true value, whereas an uncertainty is an interval within which the true value is believed to lie. For more precise definitions, see, e.g., Taylor and Kuyatt (1994) and Joint Committee (2008).

  8. They are thus key for dynamical simulations.

  9. For more details concerning floating-point number systems, see for example (Corless and Fillion (2014), Appendix 1).

  10. This point is articulated more thoroughly by Batterman (2002a). See also Wilson (2006), Pincock (2012, Chap. 11) and Fillion (2012). The point is particularly important to understand the virtues of models at different scales.

  11. This approach is also central to the so-called method of modified equations (see, e.g., Corless 1994; Corless and Fillion 2014) and, in fluid mechanics, to the so-called method of manufactured equations (see, e.g., Roache 2001; Oberkampf et al. 2004).

  12. Notice that increasing the floating-point precision will not stop that from happening. Is this really a catastrophe? From the modeling point of view, no. The difficulty stems from radical scale changes, and in this context, it makes sense to consider scale as a fundamental factor in our search for solutions.

  13. A significance arithmetic is simply a system of calculation rules that takes into consideration the number of significant digits of the operands.

  14. For a historical account of backward error analysis, see Wilkinson (1971). For a recent exposition and application of this method, see Corless and Fillion (2014), whose afterword contains a brief discussion of potential limitations.

  15. It is important for the purpose of applying backward error analysis to numerical solutions of ordinary differential equations that we consider numerical solutions to be \(\fancyscript{C}^1\), i.e., continuously differentiable; otherwise, the backward error would not be globally defined on the interval of integration (see, e.g., Corless and Fillion 2014, part 4).

  16. Such an equation in \(\varvec{z}\) can be called a reverse-engineered problem. The name is suggestive because we first solve the problem numerically, and then we use the computed solution to determine what perturbed problem we have in fact solved exactly.

  17. Well-conditioning must be distinguished from the concepts of stability of a problem-solving method. There is no unique way of formalizing the notion of numerical stability, but its underlying intuitive idea is that an algorithm is numerically stable if it returns results that are about as accurate as the problem and the resources available (typically determined by choosing a system of floating-point arithmetic) allow. Thus, it is similar to the concept of conditioning, but it is a property of methods rather than problems. For rigorous definitions, see, e.g., Higham (2002) and Deuflhard and Hohmann (2003).

  18. Infinitely ill-conditioned problems are known as ill-posed problems in analysis, following Hadamard. See Earman (1986) for a rare discussion in the philosophical literature. Moreover, even if he doesn’t specifically discuss the concept of well-conditioning, Duhem (1906) has an extended discussion of “les mathématiques de l’à peu près” based on Hadamard’s work.

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Acknowledgments

First and foremost, we would like to thank Robert Batterman. We would also like to thank Erik Curiel, Bill Harper, Robert Moir, Chris Pincock, Bryan Roberts, Chris Smeenk, and two anonymous referees for their useful suggestions.

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Authors and Affiliations

  1. Department of Statistics and Actuarial Sciences, Joseph L. Rotman Institute of Philosophy, University of Western Ontario, London, ON, Canada

    Nicolas Fillion

  2. Department of Applied Mathematics, Joseph L. Rotman Institute of Philosophy, University of Western Ontario, London, ON, Canada

    Robert M. Corless

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  1. Nicolas Fillion
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  2. Robert M. Corless
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Correspondence to Nicolas Fillion.

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Fillion, N., Corless, R.M. On the epistemological analysis of modeling and computational error in the mathematical sciences. Synthese 191, 1451–1467 (2014). https://doi.org/10.1007/s11229-013-0339-4

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