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  1. Conceptual and Computational Mathematics†.Nicolas Fillion - 2019 - Philosophia Mathematica 27 (2):199-218.
    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 (...)
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  • Mathematical Problem-Solving in Scientific Practice.Davide Rizza - 2021 - Synthese 199 (5-6):13621-13641.
    In this paper I study the activity of mathematical problem-solving in scientific practice, focussing on enquiries in mathematical social science. I identify three salient phases of mathematical problem-solving and adopt them as a reference frame to investigate aspects of applications that have not yet received extensive attention in the philosophical literature.
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  • Semantic layering and the success of mathematical sciences.Nicolas Fillion - 2021 - European Journal for Philosophy of Science 11 (3):1-25.
    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 (...)
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  • Numerical instability and dynamical systems.Vincent Ardourel & Julie Jebeile - 2021 - European Journal for Philosophy of Science 11 (2):1-21.
    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 (...)
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  • Explanation and Abstraction From a Backward-Error Analytic Perspective.Nicolas Fillion & Robert H. C. Moir - 2018 - European Journal for Philosophy of Science 8 (3):735-759.
    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 (...)
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  • Reproducibility and the Concept of Numerical Solution.Johannes Lenhard & Uwe Küster - 2019 - Minds and Machines 29 (1):19-36.
    In this paper, we show that reproducibility is a severe problem that concerns simulation models. The reproducibility problem challenges the concept of numerical solution and hence the conception of what a simulation actually does. We provide an expanded picture of simulation that makes visible those steps of simulation modeling that are numerically relevant, but often escape notice in accounts of simulation. Examining these steps and analyzing a number of pertinent examples, we argue that numerical solutions are importantly different from usual (...)
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  • On the Presumed Superiority of Analytical Solutions Over Numerical Methods.Vincent Ardourel & Julie Jebeile - 2017 - European Journal for Philosophy of Science 7 (2):201-220.
    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 by analytical solutions, this is actually often much easier with (...)
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  • Concepts of Solution and the Finite Element Method: A Philosophical Take on Variational Crimes.Nicolas Fillion & Robert M. Corless - 2021 - Philosophy and Technology 34 (1):129-148.
    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 (...)
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  • The Adventures of Climate Science in the Sweet Land of Idle Arguments.Eric Winsberg & William Mark Goodwin - 2016 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 54:9-17.
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