Explanation and abstraction from a backward-error analytic perspective

Fillion, Nicolas; Moir, Robert H. C.

doi:10.1007/s13194-018-0208-6

Cited by 8 publications

(7 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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“…Other non-formal methods, not focused on code, can be employed to verify programs, such as the method of manufactured solutions, the use of benchmarks, or a posteriori verification methods such as backward error analysis (see, e.g., Fillion and Moir, 2018).…”

Section: Verification In Computational Sciencementioning

confidence: 99%

Formal Verification, Scientific Code, and the Epistemological Heterogeneity of Computational Science

Imbert¹,

Ardourel²

2022

Philos. sci.

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Various errors can affect scientific code and detecting them is a central concern within computational science. Could formal verification methods, which are now available tools, be widely adopted to guarantee the general reliability of scientific code? After discussing their benefits and drawbacks, we claim that, absent significant changes as regards features like their user-friendliness and versatility, these methods are unlikely to be adopted throughout computational science, beyond certain specific contexts for which they are well-suited. This issue exemplifies the epistemological heterogeneity of computational science: profoundly different practices can be appropriate to meet the reliability challenge that rises for scientific code.

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“…Moir 2010 for ordinary differential equations). In particular, it is a general approach which is not restricted to dynamical systems Corless 2014, 2019;Fillion and Moir 2018). But, despite its incentives, backward analysis, to our knowledge, is yet not widely used in practice.…”

Section: Backward Error Analysismentioning

confidence: 99%

“…For instance, it investigates the various kinds of numerical errors affecting computer simulations (e.g. Humphreys 2004;Lenhard 2019); proposes to interpret numerical errors as modeling errors through the perspective of backward numerical analysis (Fillion and Corless 2014;Fillion and Moir 2018); puts forward definitions of the concepts of exact solutions (Fillion and Bangu 2015;); explains why approximate numerical solutions are sometimes preferred to exact analytical solutions (Ardourel and Jebeile 2017); and discusses the available methods -e.g. perturbation methods, backward error analysis (Corless and Fillion 2019) or effective validity for inferences (Moir 2019) -used to produce satisfactory approximate solutions.…”

Section: Contents 1 Introduction 2 1 Introductionmentioning

confidence: 99%

Numerical instability and dynamical systems

Ardourel

Jebeile

2021

Euro Jnl Phil Sci

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In philosophical studies regarding mathematical models of dynamical systems, instability due to sensitive dependence on initial conditions, on the one side, and instability due to sensitive dependence on model structure, on the other, have by now been extensively discussed. Yet there is a third kind of instability, which by contrast has thus far been rather overlooked, that is also a challenge for model predictions about dynamical systems. This is the numerical instability due to the employment of numerical methods involving a discretization process, where discretization is required to solve the differential equations of dynamical systems on a computer. We argue that the criteria for numerical stability, as usually provided by numerical analysis textbooks, are insufficient, and, after mentioning the promising development of backward analysis, we discuss to what extent, in practice, numerical instability can be controlled or avoided.

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Explanation and abstraction from a backward-error analytic perspective

Cited by 8 publications

References 23 publications

Semantic layering and the success of mathematical sciences

Semantic layering and the success of mathematical sciences

Formal Verification, Scientific Code, and the Epistemological Heterogeneity of Computational Science

Numerical instability and dynamical systems

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