Degrees of Freedom: An Essay on Competitions between Micro and Macro in Mechanics

and not concrete entities. I shall call it physical dependence. The correlative, and more basic, notion is that of physical independence: it is in fact the notion of degree offreedom. The idea is this: X is a physically independent quantity whenever it is among those quantities whose magnitudes shape the state of a system to which they belong, and it is physically dependent when it is given shape to by other quantities. For example, momentum shapes kinetic energy, whereas kinetic energy does not shape momentum. The concept of shaping is primitive in our treatment, and is governed by the following axioms, which entail that shaping is an unmediated and irreflexive activity. A I When quantity X shapes the state of a system a, it is false that X is given shape to by any other quantity. A II When quantity Y of a system a is given shape to by a quantity X, it is false that X is given shape to by any other quantity. In the physical sciences the concept of independent quantity, which is a variant on the concept of degree of freedom, is formulated typically in the epistemological mode: X is an independent quantity whenever X is among quantities whose magnitudes must be specified in order to specify completely