In this paper, following the claims made by various mathematicians, I try to construct a theory of levels of abstraction. I first try to clarify the basic components of the abstract method as it developed in the first quarter of the 20th century. I then submit an explication of the notion of levels of abstraction. In the final section, I briefly explore some of main philosophical consequences of the theory.

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)

(This insularity was further promoted by the guileless duplicity of scholars in other fields, who were all too prepared to bequeath "the problem of ...

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.

Thomas S. Kuhn's classic book is now available with a new index.

Philosophical concerns rarely force their way into the average mathematician’s workday. But, in extreme circumstances, fundamental questions can arise as to the legitimacy of a certain manner of proceeding, say, as to whether a particular object should be granted ontological status, or whether a certain conclusion is epistemologically warranted. There are then two distinct views as to the role that philosophy should play in such a situation.

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)

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)

In this paper, we explore the extent to which issues of simulation model validation take on novel characteristics when the models in question become particularly complex. Our central claim is that complex simulation models in general, and global models of climate in particular, face a form of confirmation holism. This holism, moreover, makes analytic understanding of complex models of climate either extremely difficult or even impossible. We argue that this supports a position we call convergence skepticism: the belief that the (...) existence of a plurality of different models making a plurality of different forecasts of future climate is likely to be a persistent feature of global climate science. (shrink)

Introduction -- Sanctioning models : theories and their scope -- Methodology for a virtual world -- A tale of two methods -- When theories shake hands -- Models of climate : values and uncertainties -- Reliability without truth -- Conclusion.

Computer simulations are an exciting tool that plays important roles in many scientific disciplines. This has attracted the attention of a number of philosophers of science. The main tenor in this literature is that computer simulations not only constitute interesting and powerful new science, but that they also raise a host of new philosophical issues. The protagonists in this debate claim no less than that simulations call into question our philosophical understanding of scientific ontology, the epistemology and semantics of models (...) and theories, and the relation between experimentation and theorising, and submit that simulations demand a fundamentally new philosophy of science in many respects. The aim of this paper is to critically evaluate these claims. Our conclusion will be sober. We argue that these claims are overblown and that simulations, far from demanding a new metaphysics, epistemology, semantics and methodology, raise few if any new philosophical problems. The philosophical problems that do come up in connection with simulations are not specific to simulations and most of them are variants of problems that have been discussed in other contexts before. (shrink)

Computer simulations are an exciting tool that plays important roles in many scientific disciplines. This has attracted the attention of a number of philosophers of science. The main tenor in this literature is that computer simulations not only constitute interesting and powerful new science , but that they also raise a host of new philosophical issues. The protagonists in this debate claim no less than that simulations call into question our philosophical understanding of scientific ontology, the epistemology and semantics of (...) models and theories, and the relation between experimentation and theorising, and submit that simulations demand a fundamentally new philosophy of science in many respects. The aim of this paper is to critically evaluate these claims. Our conclusion will be sober. We argue that these claims are overblown and that simulations, far from demanding a new metaphysics, epistemology, semantics and methodology, raise few if any new philosophical problems. The philosophical problems that do come up in connection with simulations are not specific to simulations and most of them are variants of problems that have been discussed in other contexts before. (shrink)

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)

There was a methodological revolution in the mathematics of the nineteenth century, and philosophers have, for the most part, failed to notice.2 My objective in this chapter is to convince you of this, and further to convince you of the following points. The philosophy of mathematics has been informed by an inaccurately narrow picture of the emergence of rigour and logical foundations in the nineteenth century. This blinkered vision encourages a picture of philosophical and logical foundations as essentially disengaged from (...) ongoing mathematical practice. Frege is a telling example: we have misunderstood much of what Frege was trying to do, and missed the intended significance of much of what he wrote, because our received stories underestimate the complexity of nineteenth-century mathematics and mislocate Frege’s work within that context. Given Frege’s perceived status as a paradigmatic analytic philosopher, this mislocation translates into an unduly narrow vision of the relation between mathematics and philosophy. This chapter surveys one part of a larger project that takes Frege as a benchmark to fix some of the broader interest and philosophical significance of nineteenth-century developments. To keep this contribution to a manageable.. (shrink)

Empirical agreement is often used as an important criterion when assessing the validity of scientific models. However, it is by no means a sufficient criterion as a model can be so adjusted as to fit available data even though it is based on hypotheses whose plausibility is known to be questionable. Our aim in this paper is to investigate into the uses of empirical agreement within the process of model validation.

Philosophical concerns rarely force their way into the average mathematician’s workday. But, in extreme circumstances, fundamental questions can arise as to the legitimacy of a certain manner of proceeding, say, as to whether a particular object should be granted ontological status, or whether a certain conclusion is epistemologically warranted. There are then two distinct views as to the role that philosophy should play in such a situation. On the first view, the mathematician is called upon to turn to the counsel (...) of philosophers, in much the same way as a nation considering an action of dubious international legality is called upon to turn to the United Nations for guidance. After due consideration of appropriate regulations and guidelines (and, possibly, debate between representatives of different philosophical factions), the philosophers render a decision, by which the dutiful mathematician abides. Quine was famously critical of such dreams of a ‘first philosophy.’ At the opposite extreme, our hypothetical mathematician answers only to the subject’s internal concerns, blithely or brashly indifferent to philosophical approval. What is at stake to our mathematician friend is whether the questionable practice provides a proper mathematical solution to the problem at hand, or an appropriate mathematical understanding; or, in pragmatic terms, whether it will make it past a journal referee. In short, mathematics is as mathematics does, and the philosopher’s task is simply to make sense of the subject as it evolves and certify practices that are already in place. In his textbook on the philosophy of mathematics (Shapiro, 2000), Stewart Shapiro characterizes this attitude as ‘philosophy last, if at all.’. (shrink)

This paper identifies one way that a mathematical proof can be more explanatory than another proof. This is by invoking a more abstract kind of entity than the topic of the theorem. These abstract mathematical explanations are identified via an investigation of a canonical instance of modern mathematics: the Galois theory proof that there is no general solution in radicals for fifth-degree polynomial equations. I claim that abstract explanations are best seen as describing a special sort of dependence relation between (...) distinct mathematical domains. This case study highlights the importance of the conceptual, as opposed to computational, turn of much of modern mathematics, as recently emphasized by Tappenden and Avigad. The approach adopted here is contrasted with alternative proposals by Steiner and Kitcher. (shrink)