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

Fillion, Nicolas; Corless, Robert M.

doi:10.1007/s11229-013-0339-4

Cited by 21 publications

(11 citation statements)

References 23 publications

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“…This second approach significantly differs from the first, and is much more in line with successful practices adopted by applied mathematicians, be it in computational mathematics, perturbation theory per se, or modelling more broadly construed. The importance of this second point of view has been highlighted in the philosophical literature on backward-error analysis (see, e.g., Fillion and Corless, 2014;Fillion, 2017;Fillion and Moir, 2018), but I won't go into this in this paper since it has already been covered. This discussion suggests that interesting considerations of this kind will also be useful in assessing arguments of the form ≈ → .…”

Section: Perturbative Reasoning As a Third Methodological Pillarmentioning

confidence: 99%

Semantic layering and the success of mathematical sciences

Fillion

2021

Euro Jnl Phil Sci

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What are the pillars on which the success of modern science rest? Although philosophers have much discussed what is behind science's success, this paper argues that much of the discussion is misdirected. The extant literature rightly regards the semantic and inferential tools of formal logic and probability theory as pillars of scientific rationality, in the sense that they reveal the justificatory structure of important aspects of scientific practice. As key elements of our rational reconstruction toolbox, they make a fundamental contribution to our understanding of the success of science.At the same time, any science, however exact, is dominated by approximation, error, and uncertainty, a fact that makes one wonder how science can be so successful. This paper articulates and illustrates general themes-e.g., that truth-preserving arguments often fail to preserve approximate truth-that highlight the need for additional semantic resources. Thus, our proposal is that persistent failures to unravel the reasons behind the success of science in the face of pervasive error and uncertainty should be attributed to an insufficiently rich way of rationally reconstructing scientific and mathematical knowledge. What is missing? This paper claims that there is a third formal method of reasoning that constitutes a distinct pillar on which rests the success of science, namely, perturbation theory. The paper outlines how the representational and inferential tools of perturbation theory differ from those of logic and probability theory, and how they enable us to understand the apparently elusive aspects of the success of science.However, compared to its peers, perturbative reasoning has not received the attention it deserves. As the paper explains, this partly results from the circumstances in which perturbation theory is taught, and partly from the fact that perturbation theory first appears to be a vaguely related collection of methods offering no systematic semantic insight. In an attempt to show that this first impression is wrong, this paper presents its contribution to the semantic dimension of scientific representation and inference in terms of what I call "semantic layering."

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Section: Perturbative Reasoning As a Third Methodological Pillarmentioning

confidence: 99%

Semantic layering and the success of mathematical sciences

Fillion

2021

Euro Jnl Phil Sci

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“…Understanding how they do so requires going beyond traditional questions in the epistemology of mathematics such as how we interact with mathematical entities, make reference to them, or access mathematical truths, since the epistemological issue of how we develop mathematical knowledge remains largely untouched by answers to these foundational questions. By contrast, the epistemology of applied mathematics and computational science deals directly with the issue of how logical and mathematical content is unfolded, knowledge extracted, and problems solved given our limited wherewithal, the complexity of the task, and the features of the formal tools that we use (see e.g., Wimsatt 2007, El Skaf and Imbert 2013, Fillion and Corless 2014, Lenhard and Carrier 2017. Accordingly, it involves analyzing how heuristics for mathematical problems work and what we can expect from them; how to describe the quality of approximate solutions, how to develop mathematical strategies to analyze and control computational errors and, more generally, which features influence how applied mathematicians crawl their way through complex problems.…”

Section: Applied and Computational Mathematics For Limited Social Agementioning

confidence: 99%

The Multidimensional Epistemology of Computer Simulations: Novel Issues and the Need to Avoid the Drunkard’s Search Fallacy

Imbert¹

2019

Simulation Foundations, Methods and Applications

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Computers have transformed science and help to extend the boundaries of human knowledge. However, does the validation and diffusion of results of computational inquiries and computer simulations call for a novel epistemological analysis? I discuss how the notion of novelty should be cashed out to investigate this issue meaningfully and argue that a consequentialist framework similar to the one used by Goldman to develop social epistemology can be helpful at this point. I highlight computational, mathematical, representational, and social stages on which the validity of simulation-based belief-generating processes hinges, and emphasize that their epistemic impact depends on the scientific practices that scientists adopt at these different stages. I further argue that epistemologists cannot ignore these partially novel issues and conclude that the epistemology of computational inquiries needs to go beyond that of models and scientific representations and has cognitive, social, and in the present case computational, dimensions.

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“…The equation describes an harmonic oscillator whose stiffness increases over time. Fillion and Corless (2014) insist that, although the infinite Taylor series of Airy function "converges uniformly, the floating-point computation diverges" (p. 1461). This means that a numerical computation of the Taylor series of the function -e.g.…”

Section: Generated Errors With Numerical Applicationsmentioning

confidence: 99%

“…Many functions satisfy this criterion and the Euler function can thus be applied in principle to all of them. 12 In addition, since problems of differential equations of order n can be rewritten with n differential equations of order 1, one can suppose that the generality of numerical method for greater orders can be explained the same way.…”

Section: Excessively Sophisticated Analytical Solutionsmentioning

confidence: 99%

On the presumed superiority of analytical solutions over numerical methods

Ardourel¹,

Jebeile²

2016

Euro Jnl Phil Sci

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An important task in mathematical sciences is to make quantitative predictions, which is often done via the solution of differential equations. In this paper, we investigate why, to perform this task, scientists sometimes choose to use numerical methods instead of analytical solutions. Via several examples, we argue that the choice for numerical methods can be explained by the fact that, while making quantitative predictions seems at first glance to be facilitated with analytical solutions, this is actually often much easier with numerical methods. Thus we challenge the widely presumed superiority of analytical solutions over numerical methods.

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

Cited by 21 publications

References 23 publications

Semantic layering and the success of mathematical sciences

Semantic layering and the success of mathematical sciences

The Multidimensional Epistemology of Computer Simulations: Novel Issues and the Need to Avoid the Drunkard’s Search Fallacy

On the presumed superiority of analytical solutions over numerical methods

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