continuous functions. This is still justifiable if one thinks of this as substituting idealized concepts for the particulate concepts of matter. The ultimate justification comes from measurement. Measurement of gas pressure, fluid velocity, the density of materials, or field strength, represents an averaging over microscopic differences. Such smoothed out models have continuity built in through the process of idealization. Such quantities, accordingly, can be represented in the small by smooth, continuous, generally analytic, functions.34 Thus, physical assumptions, once used to justify the treatment of mathematical functions, now serve to justify the selection of a proper subset of mathematical functions. In analyzing functioning physics one can treat the language as basic and accord mathematical expressions a functional, rather than a foundational, role. They may have more than a functional role. Many famous formulas were inspired guesses whose physical significance was originally misconstrued or underestimated. Yet, basic formulas must have a functional role in the normal working of physics.
2.4 The Objectivity of Physics
Many defenses of the objectivity of physics rely on ontological assumptions, e. g., the real existence of objects independent of our knowledge of them. The thrust of the preceding considerations is to put the issue of objectivity in the context of the language and practice of physics. Does this entail an abandonment of objectivity? Instead of a simple “Yes” or “No” I will reply with distinctions and qualifications. The first distinction is between internal and external questions about physical claims. By ‘internal’ I mean internal to the normal practice of physics, rather than analyses of physics as an object of study. The distinction is fuzzy. One cannot practice physics without some interpretation. Yet it supplies a useful point of departure.
For the internal question we begin with some very simple claims involving the type of quantitative concepts we have been considering:
This body has a mass of 8 kg.(8)
This body has a temperature of 13° C(9)
This body has a net charge of 11 Coulombs(10)
In the normal practice of physics such claims are routinely accepted as true. Without the routine acceptance of an indefinitely large number of such claims as true the normal practice of physics would be impossible. Few would contest this. Complexities arise when such claims are put in a larger context. The two most basic contexts here are the context of language and the context of theories. They have opposed orderings with regard to the relation between truth and ontology. Roughly in language truth seeps down from the surface. In theories it seeps up from the foundations.
Contrast the previous statements with the type of ordinary language claims considered earlier:
This shirt is yellow(11)
Being colored is a real property of objects(12)
Objects are the primary existents(13)
The acceptance of (11) as true entails a functional acceptance of (12), i.e., acceptance of it as a normal presupposition of discourse rather than an ontological claim. Statement
34 In contemporary terms, C (M), the set of smooth, real-valued functions on a manifold, M, having infinitely many continuous derivatives is a commutative algebra over the real numbers on which one may define a vector field. See Baez and Munian, chap. 3.