Conceptual role semantics and theory understanding: the case of classical mechanics

Classical Mechanics (CM), since its first systematic presentation in Newton’s Principia in 1687, has undergone various mathematical rewritings and conceptual reorganizations. From a non historical point of view, CM can be formulated, and is still taught and applied today, under two main guises: using vectorial equations of motion and using variational principles. These two formulations (roughly, the Newtonian and the Hamiltonian versions) are both logically and empirically equivalent, since a chain of mathematical deductions leads from the one to the other. Therefore, traditional approaches to theories (the so-called syntactic and semantic views of theories), which aim at giving a formal account of their logical structures, consider them as alternative formulations of the very same theory. On the other hand, non logical studies of scientific change, in the Kuhnian tradition, usually focus on conceptual differences between theoretical frameworks separated by revolutionary shifts; therefore, they also consider that such changes within the same theory are purely formal and have no bearing on its physical meaning. Nevertheless, these two versions are different from the point of view of the understanding of the cognitive agents who are supposed to learn, apply and sometimes develop the theory. I initially define one’s understanding of a theory as consisting both in knowing what the theory means (this is the representational aspect of understanding: being able to represent the phenomena by means of its system of concepts) and how it works (this is the computational aspect: being able to compute the various equations given by the theory to draw predictions and explanations about the phenomena). The two versions of CM are both representationally and computationally different: they do not represent the phenomena the same way, and applying one and the other does not consist in the same cognitive operation. Indeed, they are not applied in the same cases (in the case of constrained systems, Newtonian formulation is practically intractable), they do not use the same mathematical language, and their core concepts are different. Moreover, they do not enable one to draw the same analogies with other fields of physics. I appeal to the so-called Conceptual Role Semantics (CRS) in order to give a more precise definition of one’s understanding of a theory and to account for the relation between its computational and representational aspects. My main claim is that such differences in understanding can be characterized as conceptual differences, which have a bearing on the physical meaning of the theory, and hence on its very identity, despite the logical and empirical equivalences of its formulations. According to CRS, the content of a concept, and hence the meaning of the term expressing it, depends on its role in the cognitive life of the agent; more precisely, it is defined by its inferential location within the set of the mental representations of the agent. One’s understanding of the meaning of one’s own words “is a kind of know-how: one knows how to proceed”1 . Newtonian and Hamiltonian versions of CM, though logically equivalent, do not consist in the same conceptual structure: they do not facilitate the same inferences and their concepts are not related the same way. For instance, the meaning of “force” in Newtonian Mechanics is given by Newton’s Second Law, and it is the basis of all inferences in that frame. It is defined as the cause of acceleration, operating by a local and instantaneous action on particles, whereas within Hamiltonian Mechanics, it is arrived at after a chain of mathematical deductions from Hamilton’s principle, whose core concept is “action”. I finally suggest that one’s understanding of a theory can be defined as a set of mental representations, and that the various versions of CM can be thought of as explicit and complete understandings of the same logical structure. This perspective enables us approach theorizing as the activity of scientific minds at work, rather than focusing on theories as purely logical structures