An Structural Concept of Empirical Approximation

Abstract

The purpose of this article is to give a first step towards a general account of the logical form of approximation in empirical science. To this purpose, some topological and model-theoretic notions are used. More concretely, the topological concept of a uniform structure (a special kind of a filter) is interpreted empirically and applied to a model-theoretic representation of approximation in empirical theories.

The philosophical tenet assumed is that any significant empirical theory has an irreducibly approximative character. The application of laws and theoretical structures to a given empirical domain as well as the relationship between different laws or theoretical structures is usually approximative. Perfectly exact science is science fiction. Philosophers of science have unduly neglected this essential aspect of empirical science.

Only in recent times, some efforts have been made in order to clear up empirical approximation formally. Two authors may be mentioned in this respect: the German philosophers of science E. Scheibe and G. Ludwig.

Ludwig’s work on approximation is relevant to the present article. His basic idea consists of introducing a formal notion of approximation into his theory concept. That is, the formal concept of an empirical theory is considered as incomplete unless it contains an approximative component as a mediating link between the theoretical structure and its empirical domain. This approximative component is defined as a uniform structure.

This basic idea of Ludwig’s has been adopted in the present article. But Ludwig’s concrete procedure has to be modified and reinterpreted. The reasons for this are, first, that the elements of Ludwig’s uniform structures are too amorphous and unmanageable, and, secondly, that we have decided to use not Ludwig’s general apparatus for reconstructing empirical theories, but rather the model-theoretic apparatus of the so-called structuralistic view, first propounded by J.D. Sneed. Our approximative component is a model-theoretically defined uniform structure, and its elements are model-theoretic fuzzy sets consisting of pairs of models which intuitively approximate each other by some degree.

The final purpose of this article is to show how this model-theoretic uniform structure may be adequately interpreted as empirical approximation and how it coherently fits into the Sneedian framework, which has not included any approximative component to date. This is done in Sections 4 and 5. After an intuitive discussion in Section 4, the improvement of the structuralistic theory concept by means of a uniform structure representing approximation is systematically expounded and explained in Section 5. Sections 4 and 5 are the central parts of the article.

Before that, some introductory remarks are made. Section 1 contains a very rough and short resume of the structuralistic view. In Section 2, a synoptic view of different approximation cases is offered. A distinction is made between two main kinds of empirical approximation: the approximative relationship between a theoretical structure and its domain of application (intra-theoretical approximation), and the approximative relationship between two different theories (inter-theoretical approximation). In Section 3, the program is stated: to develop a general apparatus which allows for a formal account of the two kinds of approximation mentioned above. Systematic application of the general apparatus to this double task is left to future work. Only some hints at the first part of the task (intra-theoretical approximation) are included in Section 5.

[Summary by C. Ulises Moulines]