Approximation and Abstraction in Solid

Physical reasoning often involves approximating or abstracting the situation or the theory at hand. This paper studies the nature of approximation and abstraction as applied to the kinematic theory of rigid solid objects. Five categories of approximation are considered: 1. Geometric approximation. 2. Abstraction of a complex kinematic structure by a simpler kinematic structure. For example , the abstraction of a collection of tightly-linked objects as a single object. 3. Abstraction of a kinematic structure by a simpler theory. For example, the abstraction by a connectivity graph in connguration space. 4. Approximation of a complex kinematic structure by more complex theory. For example, the approximation of a chain by a string. 5. Approximation of a more complex theory by a kinematic theory. For example, the approximation of solid object dynamics by kinematics. We discuss how some of these types of approximation can be implemented and combined. We conclude that abstraction and approximation are open-ended styles of reasoning, rather than neatly categorizable meta-relationships. A reasoner encounters a real, physical situation and wishes to answer some question about it. She proceeds by invoking a detailed theory of the physical phenomena at hand and carefully measuring the relevant physical parameters in her situation. She now has a well-deened formal problem P which is susceptible to calculation. Unfortunately, she nds that it is computationally infeasible to compute the information she wants directly from P. In this position, one option that the reasoner may explore is to consider instead a diierent, problem A which may be a less accurate account of the real situation P, but which, she believes, is similar to P in key respects and easier to compute with. In many situations, the simpler theory A may also have the advantage of being more robust