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. (shrink)

Computational methods such as computer simulations, Monte Carlo methods, and agent-based modeling have become the dominant techniques in many areas of science. Extending Ourselves contains the first systematic philosophical account of these new methods, and how they require a different approach to scientific method. Paul Humphreys draws a parallel between the ways in which such computational methods have enhanced our abilities to mathematically model the world, and the more familiar ways in which scientific instruments have expanded our access to the (...) empirical world. This expansion forms the basis for a new kind of empiricism, better suited to the needs of science than the older anthropocentric forms of empiricism. Human abilities are no longer the ultimate standard of epistemological correctness.Humphreys also includes arguments for the primacy of properties rather than objects, the need to consider technological constraints when appraising scientific methods, and a detailed account of how the path from computational template to scientific application is constructed. This last feature allows us to hold a form of selective realism in which anti-realist arguments based on formal reconstructions of theories can be avoided. One important consequence of the rise of computational methods is that the traditional organization of the sciences is being replaced by an organization founded on computational templates.Extending Ourselves will be of interest to philosophers of science, epistemologists, and to anyone interested in the role played by computers in modern science. (shrink)

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. (shrink)

Should philosophers of science be paying attention to developments in "nanoscience"? Undoubtedly, it is too early to tell for sure. The goal of this paper is to take a preliminary look. In particular, I look at the use of computational models in the study of nano-sized solid-state materials. What I find is that there are features of these models that appear on their face to be at odds with some basic philosophical intuitions about the relationships between different theories and between (...) theories and their models. My conclusion is that developments in nanoscience are not an unlikely place for novel insights in the philosophy of science to emerge. (shrink)

Reasons are given to justify the claim that computer simulations and computational science constitute a distinctively new set of scientific methods and that these methods introduce new issues in the philosophy of science. These issues are both epistemological and methodological in kind.

By stressing the act rather than the relation of approximation, I argue that the magnitude of the error introduced should not be used as the sole criterion for judging the worth of the approximation. Magnitude is a necessary but not sufficient condition for such a judgement. Controllability, the absence of cancelling errors, and the approximation's justification are also important criteria to consider when praising or blaming an approximation. Boltzmann's discussion of the types of approximations used in the kinetic theory of (...) gases at the turn of the century illustrates the use of these criteria. (shrink)

Much of the current thought concerning mathematical ontology in volves in some way the Quine/Putnam indispensability argument. The indispensability approach needs to be more thoroughly specified, however, before substantive progress can be made in assessing it. To this end I examine in some detail the ways in which pure mathematics is applied to physical theory; such considerations give rise to three specific issues with which the indispensability approach must come to grips.

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. (shrink)

Mark Wilson explores our strategies for understanding the world. We frequently cannot reason about nature in the straightforward manner we anticipate, but must use alternative thought processes that reach useful answers in opaque and roundabout ways; and philosophy must find better descriptive tools to reflect this.

Following historical developments, this article traces two basic motives for employing sets within a physical setting and discusses whether they truly pose a problem for ‘mathematical naturalism’.

It's uncontroversial that notions of idealization and approximation are central to understanding computer simulations and their rationale. What's not so clear is what exactly these notions come to. Two distinct forms of approximation will be distinguished and their features contrasted with those of idealizations. These distinctions will be refined and closely tied to computer simulations by means of Scott-Strachey denotational programming semantics. The use of this sort of semantics also provides a convenient format for argumentation in favor of several theses (...) I shall propose concerning the role computer implemented approximations and idealizations play in fixing what the acceptance of an underlying scientific theory is or should be. (shrink)

Much of the current thought concerning mathematical ontology in volves in some way the Quine/Putnam indispensability argument. The indispensability approach needs to be more thoroughly specified, however, before substantive progress can be made in assessing it. To this end I examine in some detail the ways in which pure mathematics is applied to physical theory; such considerations give rise to three specific issues with which the indispensability approach must come to grips.

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. (shrink)

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 (...) numerical methods. Thus we challenge the widely presumed superiority of analytical solutions over numerical methods. (shrink)