This paper utilizes Scott domains (continuous lattices) to provide a mathematical model for the use of idealized and approximately true data in the testing of scientific theories. Key episodes from the history of science can be understood in terms of this model as attempts to demonstrate that theories are monotonic, that is, yield better predictions when fed better or more realistic data. However, as we show, monotonicity and truth of theories are independent notions. A formal description is given of the (...) pragmatic virtues of theories which are monotonic. We also introduce the stronger concept of continuity and show how it relates to the finite nature of scientific computations. Finally, we show that the space of theories also has the structure of a Scott domain. This result provides an analysis of how one theory can be said to approximate another. (shrink)

In this paper, I will develop a nontrivial interpretation of Feyerabend's concept of a hidden anomalous fact. Feyerabend's claim is that some anomalous facts will remain hidden in the absence of alternatives to the theories to be tested. The case of Brownian motion is given by Feyerabend to support this claim. The essential scientific difficulty in this case was the justification of correct and relevant descriptions of Brownian motion. These descriptions could not be simply determined from the available observational data. (...) An examination, however, of this case shows that no alternative theory is or historically was thought to be necessary in order to justify descriptions of Brownian motion that "directly" refute thermodynamics. While Feyerabend's appraisal of this case therefore is incorrect, a sense is developed in which successful alternatives lend inductive support to the correctness of refuting experimental descriptions. Crucial though to the explanation of this support is the notion of arguments that show the possibilities for improving experimental fit. (shrink)

The use of idealizations and approximations in scientific explanations poses a problem for traditional philosophical theories of confirmation since, strictly speaking, these sorts of statements are false. Furthermore, in several central cases in the history of science, theoretical predictions seen as confirmatory are not, in any usual sense, even approximately true. As a means of eliminating the puzzling nature of these cases, two theses are proposed. First, explanations consist of idealized deductive-nomological sketches plus what are called modal auxiliaries, i.e., arguments (...) showing that if the idealizations used in the initial conditions are improved, then there will be an improvement in the prediction. Second, a theory is confirmed if it can be shown that its idealized sketches can be improved; similarly, a theory is disconfirmed if its idealized sketches cannot be improved. Several examples are given to illustrate both confirmation and disconfirmation achieved by means of the modal auxiliary. These cases are compared with Scriven's bridge example. (shrink)

Using the Schwarzschild calculation of the Relativistic bending of starlight near the sun as an illustration, it is shown that the relationship between theory and data requires a hierarchy of structures of different logical type. An essential feature of this hierarchy is the use of idealizations and approximate truths. On the basis of a counterfactual analysis of these concepts, it is shown that confirmation is possible even though statistical measures of goodness of fit are not satisfied. The consequences of this (...) view of confirmation and hierarchical structure for scientific realism are then considered. (shrink)

The problem for the scientist created by using idealizations is to determine whether failures to achieve experimental fit are attributable to experimental error, falsity of theory, or of idealization. Even in the rare case when experimental fit within experimental error is achieved, the scientist must determine whether this is so because of a true theory and fortuitously canceling idealizations, or due to a fortuitous combination of false theory and false idealizations. For the engineer, the problem seems rather different. Experiment for (...) the engineer reveals the closeness of predictive fit that can be achieved by theory and idealization for a particular case. If the closeness of fit is good enough for some practical purpose, the job is done. If not, or there are reasons to consider variation, then the engineer needs to know how well the experimentally determined closeness of fit will extrapolate to new cases. This paper focuses on engineering measures of closeness of fit and the projectibility of those measures to new cases. (shrink)

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)

Maxwell claimed that the electrostatic inverse square law could be deduced from Cavendish's spherical condenser experiment. This is true only if the accuracy claims made by Cavendish and Maxwell are ignored, for both used the inverse square law as a premise in their analyses of experimental accuracy. By so doing, they assumed the very law the accuracy of which the Cavendish experiment was supposed to test. This paper attempts to make rational sense of this apparently circular procedure and to relate (...) it to some variants of traditional problems concerning old and new evidence. (shrink)

Grunbaum has argued that the Lorentz-Fitzgerald contraction hypothesis is not ad hoc since the Kennedy-Thorndike experiment can be used to provide a test that is significantly different from that provided by the Michelson-Morley experiment. In the first part of the paper, I show that the differences claimed by Grunbaum to hold between these two experiments are not sufficient for establishing independent testability. A dilemma is developed: either the Kennedy-Thorndike experiment, because of experimental realities, cannot test the uncontracted Fresnel aether theory, (...) or if experimental difficulties are ignored, the Kennedy-Thorndike experiment degenerates into a version of the Michelson-Morley experiment. The second part of the paper is a feasibility study of the prospects for defining experimental types according to aims of measurement and determination. This approach is applied to the contraction hypothesis, where it is suggested that the usual analysis of independent testability be modified. (shrink)

It’s uncontroversial that notions of idealization and approximation are central to understanding computer simulations and their rationale. So, for example, one common form of computer simulation is to abandon a realistic approach that is computationally non-tractable for a more idealized but computationally tractable approach. Many simulations of systems of interacting members can be understood this way. In such simulations, realistic descriptions of individual members are replaced with less realistic descriptions which have the virtue of making interactions computationally tractable. Such simulations (...) can be supplemented with empirically determined correction factors which render the output produced by means of the idealizations more in accord, one hopes, with what the more realistic approach would have produced had it been computationally tractable. Another way to utilize computers is to replace an idealized but analytically tractable account of some phenomenon with a less idealized account which has no closed form or analytical solutions but where the computer can be used generate approximations to the desired solutions. (shrink)

One of the things I attempted to do in my paper on independent testability was to illustrate convincingly the very real difficulties of distinguishing between the accidental and essential features of a scientific experiment. The importance of this distinction is that independent testability presumably requires differences that are essential and telling and not merely accidental reflections of existing experimental technique or of the procedural preferences of the experimenter. In the case of the Michelson-Morley and Kennedy-Thorndike experiments, I showed, following a (...) suggestion of M. G. Evans, that Grünbaum's attempt to distinguish these two experiments does not work. I also showed that certain obvious and natural variations of Grünbaum's approach do not adequately distinguish these experiments. So, for example, it will not do simply to claim that MM is characterized by equal arms and KT by unequal arms, since equal arms are required for MM only in the absence of adequate photographic registration methods. But if these methods are not available then KT is not possible. Conversely, if these methods are available, then while both MM and KT are experimentally possible, MM need not be restricted to equal arms. (shrink)