A Finite Memory Argument for an Axiomatic Conception of Scientific Theories

Abstract

This article concerns the split between syntactic and semantic approaches to scientific theories. It aims at showing that an axiomatic representation of a scientific theory is a precondition of comprehending if the models of contain infinite entities. This result is established on the basis of the proposition that the human mind—which is finitely bounded for all we know—is not capable of directly grasping infinite entities. In view of this cognitive limitation, an indirect and finite representation of possibly infinite components of the models of a scientific theory proves to be indispensable. Sets of axioms and sets of axiom schemes provide such a representation. These considerations will be cast into an argument for an axiomatic conception of scientific theories. The article concludes with a case study of the ideal gas model.

Acknowledgements

This work was supported by the Alexander von Humboldt Foundation. I am also grateful to three unnamed referees of this journal for very helpful comments on an earlier draft of this article.

Notes

1 For a related, yet different line of criticism of the semantic view, see Halvorson (2012, 2013) and Lutz (2014). They attack the semanticist's claim that the representation of a scientific theory by model-theoretic structures can be understood independently of a vocabulary interpreted by such structures.

2 See also Lutz (2012) for further evidence that the received view in philosophy of science was not confined to first-order theories.

3 The presence of the framework of model theory is most obvious in the case of Suppes (1960), French and Ladyman (1999), Suppe (2000), and Da Costa and French (2003). The notion of embedding being crucial to van Frassen's semantic approach implies a commitment to the model-theoretic framework. Semanticists sometimes prefer to speak of configurated state-spaces rather than model-theoretic models (Suppe 2000). However, such state-spaces contain infinite entities to the same extent as model-theoretic models of a scientific theory do. Hence, the central argument of the present article applies to state-spaces inasmuch as it does to model-theoretic models.

4 I omit the details of this deduction, because, first, they are not relevant to the finite memory argument of the present article, and, second, they can be found in standard undergraduate textbooks of physics (see e.g. Feynman, Leighton, and Sands 2011, ch. 39). The ideal gas model continues to be used in statistical physics (see e.g. Reif 2008, ch. 5). For the purpose of the present article it suffices to deal with the non-statistical use of this model. Note that the axioms of classical mechanics are not explicitly stated as a premise in the derivation of the ideal gas law, because satisfaction of these axioms by an ideal gas is, in the present account, part of the meaning of such a gas. This will become more obvious soon.

5 , , and are further set-theoretic predicates which are assumed to be predefined. , for example, says that is a model of classical particle mechanics. stands for classical collision mechanics and for elastic classical collision mechanics.