Mental imagery

In the last section of Station 4 Maxwell commented on Neumann's mathematical concept of potential:

In the theory of Neumann, the mathematical conception called Potential, which we are unable conceive as a material substance, is supposed to be projected from one particle to another, in a manner which is quite independent of a medium, and which, as Neumann has himself pointed out, is extremely different from that of the propagation of light. In the theories of Riemann and Betti it would appear that the action is supposed to be propagated in a manner somewhat more similar to that of light.¹²⁷

According to Maxwell, Neumann failed to offer any “mental image” of his concept of potential, and this is a real deficiency in Neumann's theory.

Maxwell believed it was incumbent on Neumann to explain how his potentials can be conceived (mechanically or otherwise). Maxwell, however, did not require that Neumann come up with a consistent interpretation of his various equations. Presumably, this inability to construct a consistent mental image of the theory was one reason for Maxwell’s dissatisfaction with Neumann’s proposed theory.¹²⁸

At the very end of Station 4 Maxwell elaborated his view on mental representation:

All these theories [e.g., Neumann’s] lead to the conception of a medium in which the propagation takes place, and if we admit this medium as an hypothesis, 1 think it ought to occupy a prominent place in our investigations, and that we ought to endeavour to construct a mental representation of all the details of its action, and this has been my constant aim in this treatise.¹²⁹

These are the very last words of the Treatise. Whether this was Maxwell’s “constant aim” is not at all clear; in any event, he did not accomplish it. Perhaps he intended to get back to this issue in the second edition, which Maxwell had not completed at the time of his death.¹³⁰ Be that as it may, Maxwell regarded the continental theories as leading to the concept of a medium, despite the fact that these theories presuppose action at a distance, which does not require a medium. The demand “to bring electrical phenomena within the province of dynamics,”¹³¹ that is, turning electromagnetism into electrodynamics, convinced Maxwell that a mental image of the seat of action, the medium, was mandatory. In the case of the properties of the dielectrics the move was to form a “mechanical illustration,” an arrangement that could “serve to represent the state of a dielectric acted on by an electromotive force.”¹³²

Indeed, in Station 4 Maxwell applied the formulation of dynamics by Lagrange and Hamilton, but he sought to translate “some algebraical process” into a physical property.¹³³ Maxwell emphasized this critical point in a paper he read in 1873 at a meeting of the Cambridge Philosophical Society: “In this way our words will call up the mental image, not of certain operations of the calculus, but of certain characteristics of the motion of bodies.”¹³⁴ Evidently, the required mental image refers to a physical effect, in this case the motion of bodies, rather than some symbolic operation. At another juncture in Station 4, where Maxwell developed a theory of magnetization, he appealed to the same criterion, namely, the value of a theory lies in its capacity to facilitate mental imagery:

The scientific value of a theory ... in which we make so many assumptions, and introduced so many adjustable constants, cannot be estimated merely by its numerical agreement with certain sets of experiments. If it has any value it is because it enables us to form a mental image of what takes place in a piece of iron during magnetization.¹³⁵

Here we have an explicit requirement: the theory should facilitate mental imagery of a physical phenomenon.

We note the dual approach of verbal descriptions and mathematical symbolism. In the previous stations Maxwell translated Faraday’s verbal descriptions into symbolic form; in Station 4, however, the process is sequential, from verbal account to symbolic language, and then from mathematical symbolism to verbal form. The second stage in the sequence, namely, the retranslation, is from symbolic language to ordinary language, which facilitates the formation of a mental image. In Maxwell’s view, the second stage is essential for completing a proper argument in physics and, clearly, a mental image is not a matter of mathematical symbolism.