Theoretical Terms: Defining E and B

It is sometimes claimed by philosophers of science that the meanings of theoretical terms are implicitly fixed by the total theory $\Theta$ (i.e., theoretical laws/equations plus correspondence rules) in which these terms appear. This is then the basis for the philosophical claim that a Carnap sentence,

$\Re(\Theta) \to \Theta$.

is analytic -- i.e., true in virtue of meaning. In a sense, on this view, theoretical terms are (second-order) Skolem constants (or, equivalently, Hilbertian $\epsilon$-terms).

This claim about the semantics of theoretical terms is, however, inconsistent with the standard practice of physics, for example. In physics, one usually adopts far more local definitions of theoretical terms.

For example, in electromagnetism, the Lorentz force law plays a crucial role, but Maxwell's equations do not. So, the following formulation by Professor James Sparks at the Mathematical Institute in Oxford corresponds fairly closely to the definitions that I learnt as a physics undergraduate a long, long time ago (at the Other Place):

The force on a point charge q at rest in an electric field $\mathbf{E}$ is simply

$\mathbf{F} = q \mathbf{E}$.

We used this to define $\mathbf{E}$ in fact. (Sparks, Lecture notes on "Electromagnetism", p. 12)

Notice that Maxwell's equations are not mentioned. The meaning of

"the electric field at point $\mathbf{r}$"

is not implicitly defined in terms of Maxwell's equations. Rather it is explicitly defined using the notion of a force on a charged test particle.

When the charge is moving the force law is more complicated. From experiments one finds that if $q$ at position $\mathbf{r}$ is moving with velocity $\mathbf{u} = d\mathbf{r}/dt$ it experiences a force

$\mathbf{F} = q \mathbf{E}(\mathbf{r}) + q \mathbf{u} \wedge \mathbf{B}(\mathbf{r})$. $\text{        }$ (2.8)

Here $\mathbf{B} = \mathbf{B}(\mathbf{r})$ is a vector field, called the magnetic field, and we may similarly regard the Lorentz force $\mathbf{F}$ in (2.8) as defining $\mathbf{B}$.
(Sparks, Lecture notes on "Electromagnetism", pp 12-13)

Again, notice that Maxwell's equations are not mentioned. The meaning of

"the magnetic field at point $\mathbf{r}$"

is not implicitly defined in terms of Maxwell's equations. Rather it is explicitly defined using the notion of a force on a charged test particle.

This is not the end of the story, of course. But it casts considerable doubt on the claim that the meanings of theoretical terms are given by implicit definitions within global theories. The definitions are far more local. It is not even clear that these local definitions fit the mould of what a logician would call a genuine explicit definition. But, in any case, one does not use the whole apparatus of Maxwell's equations to define the expressions "electric field" and "magnetic field".