Admissible Approximations in Empirical Theories

Abstract

In some previous writings (listed in the bibliography) the authors have dealt with the concept of approximation in empirical science by using a formal approach that departs from the standard one: It does not consider the relation of approximation as a relation between statements, but as a relation between certain kinds of structures (potential models of a theory). Within this same approach, the question is raised now: Which approximations are (empirically) admissible and why? The authors try to give a general account of the conditions that make an approximation between models admissible. The discussion is illustrated by means of examples taken from planetary astronomy; for example, the question : When are we going to say that two descriptions of a planetary orbit approximate each other within an admissible degree of approximation? And what does this mean precisely?

In the first part of the article, it is shown that different aspects or’ factors must be taken account of when trying to give a complete answer to the preceding questions. Some of them are of a qualitative, some are of a quantitative kind, but all must be somehow “integrated” into a “global” approximative relationship. The explication of such factors can only partially be carried in formal terms. The main concepts that are involved in this explication are those of “(relative) importance” of different objects within the domain of the models; “very similar descriptions of objects”, and “quasi-bijective correspondence of models”. It is clear that the first two notions cannot be explicated in purely formal or’ quantitative terms. This does not mean that we cannot try to make them reasonably clear.

The second part of the article is devoted to the question of whether the notion of a maximum degree of admissible approximation makes sense and can be made precise. It is argued that each kind of application of a theory must have a supremum of admissible approximations and that this supremum is univoquely determined.

Finally, all the aspects considered in the discussion are summarized in a set of necessary conditions for approximation within an empirical theory.

[C. Ulises Moulines]