Saturday, 20 April 2013

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".


  1. Isn't this pretty much what Hertz tried to say in his famous "Maxwell's theory is Maxwell's system of equations"? That is, the system of equations is characterized as the common and inner significance of a plurality of modes of representation, but the definitions of (physical) representations cannot be given by those equations.

  2. Possibly, yes. I'm not sure though, as I don't know much about Hertz beyond his famous slogan. It's not clear what "modes of representation" means and explaining "representation" is the heart of the problem. It's not clear if, for Hertz, theories are linguistic, or propositional, or perhaps even mental entities.

    On the other hand, I've always thought his slogan meant that further commentary and hypotheses beyond the equations for E and B (such as certain mechanical models suggested by Maxwell himself) are distinct from the theory (i.e, the equations) proper.

  3. According to Hertz, they (theories, representations, images, symbols) are (or form) isomorphic pictures, hence eg Wittgenstein & Tractatus. But that's a long story ... I'd guess they would be propositional in that sense.

    What he means, I think, is that the inner structure of a physical theory must mirror the structure of the group of equations, but definitions of theoretical terms are more local. He says: "What is ascribed to the images for the sake of appropriateness is contained in the notations, definitions, abbreviations, and, in short, all that we can arbitrarily add or take away". And he continues: "But we cannot decide without ambiguity whether an image is appropriate or not; as to this differences of opinion may arise. One image may be more suitable for one purpose, another for another;..." ("Introduction" in The Principles of Mechanics).

  4. Ah, thanks. So it seems Hertz has some sort of ancestor of a mirror-theory like Wittgenstein's? I didn't know that. (I hadn't know also that Hertz died when he was 36.) Perhaps his view can be thought of as a kind of structuralism about representation. Such views came to be in the air shortly after the 1890s - e.g., Hilbert on geometry, Poincare more generally, and then Russell.

    The second quote suggests that his overall theory of meaning involves first this abstract structural content (given by the equations), along with "locally" associating images with particular terms, but these we can "arbitrarily take away".

  5. Yeah - Hertz is very interesting, and Wittgenstein revered him (as did his whole family, apparently). Anyhow, thanks for the discussion. I've been anonymous, cause I don't know how to make my name appear :-) ... let's see if I can do it this time.