Nicolas Fillion

This paper critically discusses a tradition in philosophy of science—here called ‘scientific demonology’—that, going back to Laplace, considers what science would be like if it were performed by a supernatural omniscient creature instead of being performed by puny humans. This tradition has been much discussed, and criticized, especially with respect to its commitment to determinism. Setting this aside, this paper argues that the proposed philosophical methodology is fundamentally deficient for other reasons having to do with the foundations of mathematics and the nature of knowledge. Even when there are determinate facts about objects representing states of dynamical systems and determinate functions from states to states, such facts are in most cases not representable, and as a result cannot be meaningfully said to be known, even by an all-knowing demon. The lesson to be drawn is that, far from improving on the classical philosophy of science methodology consisting in drawing lessons about the nature, scope, and limits of science by examining the best human science, speculations about demonic science obfuscate scientific epistemology.

This paper investigates the trustworthiness of computation implemented in simulations in engineering, with a specific focus on equation-based simulations. Whereas opacity discussions in the computer simulation literature typically center on modeling opacity, we direct our attention to the grounds engineers should have for trusting that the computer indeed found an acceptable solution to a given model. This is a particularly sensitive issue in situations in which analytical or experimental methods are impossible or impractical. After critically reviewing alternative views found in the literature, we propose a systematic approach for evaluating the accuracy of solutions obtained via computational methods, that involves quantifying residual errors and evaluating the model’s conditioning. We argue that, in equation-based simulations, ensuring that the residual is small enough and that the model is sufficiently well-conditioned provides a sufficient epistemic foundation for engineers to place their trust in the results obtained via computer simulations as much as they trust the models they use, thereby giving a conceptual basis for this much needed kind of conditional trust.

This paper investigates the trustworthiness of computation implemented in simulations in engineering, with a specific focus on equation-based simulations. Whereas opacity discussions in the computer simulation literature typically center on modeling opacity, we direct our attention to the grounds engineers should have for trusting that the computer indeed found an acceptable solution to a given model. This is a particularly sensitive issue in situations in which analytical or experimental methods are impossible or impractical. After critically reviewing alternative views found in the literature, we propose a systematic approach for evaluating the accuracy of solutions obtained via computational methods, that involves quantifying residual errors and evaluating the model’s conditioning. We argue that, in equation-based simulations, ensuring that the residual is small enough and that the model is sufficiently well-conditioned provides a sufficient epistemic foundation for engineers to place their trust in the results obtained via computer simulations as much as they trust the models they use, thereby giving a conceptual basis for this much needed kind of conditional trust.

This paper articulates an account of imperatives that sensibly supports the idea of a logic of imperative inferences. We rebuke common objections to the very possibility of such a logic, from a perspective based on recent linguistic work on the morphosyntax of imperatives. Specifically, we develop the notion that the content of an imperative sentence includes both a force operator alongside an imperational content to which the force applies. We further argue that this account of the content of imperatives constitutes a plausible and flexible framework to develop a logic of imperative by examining a number of reconstructions of this idea using semantical analogs of widespread modal semantics. After studying the performance of those approaches, we conclude that progress in imperative logic has been hindered by the failure to adopt conflict-tolerant and resource-sensitive semantics, but suggest that such considerations can be incorporated in this flexible framework. Finally, we also propose a simple account of the difference between operators applying to the content of imperatives and imperative operators, in a way that sheds light on some of the issues underlying the usual antinomies.

This paper starts from a dissatisfaction with the way logic is currently taught in philosophy departments. In such a context, we would want logic to be developed from first principles, in such a way that it can then be applied to any other domain of study. But standard “rigorous” logic courses make it depend on a substantial amount of set theory, which itself seems to require notions from logic—confusing students, if not striking them as circular. Instead, this paper proposes to start from a rejuvenated version of the Aristotelian notion of an argument schema, and to use this idea to develop a limited collection of concepts from logic that is sufficient for the introduction of the axioms of set theory. Once set theory is in place, mathematical logic can be taught as usual.

Modern mathematical sciences are hard to imagine without appeal to efficient computational algorithms. We address several conceptual problems arising from this interaction by outlining rival but complementary perspectives on mathematical tractability. More specifically, we articulate three alternative characterizations of the complexity hierarchy of mathematical problems that are themselves based on different understandings of computational constraints. These distinctions resolve the tension between epistemic contexts in which exact solutions can be found and the ones in which they cannot; however, contrary to a persistent myth, we conclude that having an exact solution is not generally more epistemologically beneficial than lacking one.

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.”

We demonstrate via several examples how the backward error viewpoint can be used in the analysis of solutions obtained by perturbation methods. We show that this viewpoint is quite general and offers several important advantages. Perhaps the most important is that backward error analysis can be used to demonstrate the validity of the solution, however obtained and by whichever method. This includes a nontrivial safeguard against slips, blunders, or bugs in the original computation. We also demonstrate its utility in deciding when to truncate an asymptotic series, improving on the well-known rule of thumb indicating truncation just prior to the smallest term. We also give an example of elimination of spurious secular terms even when genuine secularity is present in the equation. We give short expositions of several well-known perturbation methods together with computer implementations. We also give a generic backward error based method that is equivalent to iteration for regular perturbation.

The relatively recent increase in prominence of computer simulations in scientific inquiry gives us more reasons than ever before for asserting that mathematics is a wonderful tool. In fact, a practical knowledge of scientific computation has become essential for scientists working in all disciplines involving mathematics. Despite their incontestable success, it must be emphasized that the numerical methods subtending simulations provide at best approximate solutions and that they can also return very misleading results. Accordingly, epistemological sobriety demands that we clarify the circumstances under which simulations can be relied upon. With this in mind, this paper articulates a general perspective to better understand and compare the strengths and weaknesses of various error-analysis methods.

ACMES is a multidisciplinary conference series that focuses on epistemological and mathematical issues relating to computation in modern science. This volume includes a selection of papers presented at the 2015 and 2016 conferences held at Western University that provide an interdisciplinary outlook on modern applied mathematics that draws from theory and practice, and situates it in proper context. These papers come from leading mathematicians, computational scientists, and philosophers of science, and cover a broad collection of mathematical and philosophical topics, including numerical analysis and its underlying philosophy, computer algebra, reliability and uncertainty quantification, computation and complexity theory, combinatorics, error analysis, perturbation theory, experimental mathematics, scientific epistemology, and foundations of mathematics. By bringing together contributions from researchers who approach the mathematical sciences from different perspectives, the volume will further readers' understanding of the multifaceted role of mathematics in modern science, informed by the state of the art in mathematics, scientific computing, and current modeling techniques.

This paper articulates an account of imperatives that sensibly supports the idea of a logic of imperative inferences. We rebuke common objections to the very possibility of such a logic, from a perspective based on recent linguistic work on the morphosyntax of imperatives. Specifically, we develop the notion that the content of an imperative sentence includes both a force operator alongside an imperational content to which the force applies. We further argue that this account of the content of imperatives constitutes a plausible and flexible framework to develop a logic of imperative by examining a number of reconstructions of this idea using semantical analogs of widespread modal semantics. After studying the performance of those approaches, we conclude that progress in imperative logic has been hindered by the failure to adopt conflict-tolerant and resource-sensitive semantics, but suggest that such considerations can be incorporated in this flexible framework. Finally, we also propose a simple account of the difference between operators applying to the content of imperatives and imperative operators, in a way that sheds light on some of the issues underlying the usual antinomies.

Despite being one of the most dependable methods used by applied mathematicians and engineers in handling complex systems, the finite element method commits variational crimes. This paper contextualizes the concept of variational crime within a broader account of mathematical practice by explaining the tradeoff between complexity and accuracy involved in the construction of numerical methods. We articulate two standards of accuracy used to determine whether inexact solutions are good enough and show that, despite violating the justificatory principles of one, the finite element method nevertheless succeeds in obtaining its legitimacy from the other.

ABSTRACT This paper examines consequences of the computer revolution in mathematics. By comparing its repercussions with those of conceptual developments that unfolded in the nineteenth century, I argue that the key epistemological lesson to draw from the two transformative periods is that effective and successful mathematical practices in science result from integrating the computational and conceptual styles of mathematics, and not that one of the two styles of mathematical reasoning is superior. Finally, I show that the methodology deployed by applied mathematicians in modern scientific computing is a paradigmatic instance of this key lesson.

We argue that two powerful error-theoretic concepts provide a general framework that satisfactorily accounts for key aspects of the explanation of physical patterns. This method gives an objective criterion to determine which mathematical models in a class of neighboring models are just as good as the exact one. The method also emphasizes that abstraction is essential for explanation and provides a precise conceptual framework that determines whether a given abstraction is explanatorily relevant and justified. Hence, it increases our epistemological understanding of how one should go about reconstructing scientific practices by making clear that, at a fundamental level, a key aspect of mathematical modeling consists in exactly solving nearby problems.

Ethically permissible clinical trials must not expose subjects to risks that are unreasonable in relation to anticipated benefits. In the research ethics literature, this moral requirement is typically understood in one of two different ways: as requiring the existence of a state of clinical equipoise, meaning a state of honest, professional disagreement among the community of experts about the preferred treatment; or as requiring an equilibrium between individual and collective ethics. It has been maintained that this second interpretation makes it mandatory to minimize the number of patients receiving the treatment that will eventually be shown to be inferior by the trial. This requirement has led to the development of adaptive trials, i.e., trials in which treatment allocation is determined by data accumulated during interim analysis. Many statisticians argue that in some circumstances—typically with potentially high benefits, as in the much discussed ECMO trial—adaptive design is the only ethically permissible experimental design. Nevertheless, some proponents of clinical equipoise argue that adaptive trials are neither ethically required nor permissible. More specifically, they argue that clinical trials using adaptive designs fail to meet the moral requirement of clinical equipoise, since these trials presuppose an epistemic state that is incompatible with a physician’s duty of care to her subjects. This paper emphasizes that the debate is to a large extent resting on an epistemological confusion. Specifically, I argue that this response conflates two different conceptions of statistical evidence, and that recognizing this distinction elucidates an epistemological framework in which adaptive trials are both consistent with and recommended by the moral requirement of clinical equipoise.

One of the most unsettling problems in the history of philosophy examines how mathematics can be used to adequately represent the world. An influential thesis, stated by Eugene Wigner in his paper entitled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," claims that "the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve." Contrary to this view, this thesis delineates and implements a strategy to show that the applicability of mathematics is very reasonable indeed. I distinguish three forms of the problem of the applicability of mathematics, and focus on one I call the problem of uncanny accuracy: Given that the construction and manipulation of mathematical representations is pervaded by uncertainty, error, approximation, and idealization, how can their apparently uncanny accuracy be explained? I argue that this question has found no satisfactory answer because our rational reconstruction of scientific practice has not involved tools rich enough to capture the logic of mathematical modelling. Thus, I characterize a general schema of mathematical analysis of real systems, focusing on the selection of modelling assumptions, on the construction of model equations, and on the extraction of information, in order to address contextually determinate questions on some behaviour of interest. A concept of selective accuracy is developed to explain the way in which qualitative and quantitative solutions should be utilized to understand systems. The qualitative methods rely on asymptotic methods and on sensitivity analysis, whereas the quantitative methods are best understood using backward error analysis. The basic underpinning of this perspective is readily understandable across scientific fields, and it thereby provides a view of mathematical tractability readily interpretable in the broader context of mathematical modelling. In addition, this perspective is used to discuss the nature of theories, the role of scaling, and the epistemological and semantic aspects of experimentation. In conclusion, we argue for a method of local and global conceptual analysis that goes beyond the reach of the tools standardly used to capture the logic of science; on their basis, the applicability of mathematics finds itself demystified.

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.