• Backward Error Analysis for Perturbation Methods
    In James Robert Brown, Shaoshi Chen, Robert M. Corless, Ernest Davis, Nicolas Fillion, Max Gunzburger, Benjamin C. Jantzen, Daniel Lichtblau, Yuri Matiyasevich, Robert H. C. Moir, Mark Wilson & James Woodward (eds.), Algorithms and Complexity in Mathematics, Epistemology, and Science: Proceedings of 2015 and 2016 Acmes Conferences, Springer New York. pp. 35-79. 2019.
    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 demon…Read more
  • 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 retur…Read more
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    The Content and Logic of Imperatives
    with Matthew Lynn
    Axiomathes 1-18. forthcoming.
    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 i…Read more
  •  5
    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 …Read more
  •  41
    Conceptual and Computational Mathematics
    Philosophia Mathematica. forthcoming.
    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 superi…Read more
  •  18
    Explanation and abstraction from a backward-error analytic perspective
    European Journal for Philosophy of Science 8 (3): 735-759. 2018.
    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…Read more
  •  17
    Clinical Equipoise and Adaptive Clinical Trials
    Topoi 38 (2): 457-467. 2019.
    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 b…Read more
  •  9
    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 de…Read more
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    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 de…Read more
  •  88
    (No abstract is available for this citation)