“Faraday’s Mathematics” is a lecture I gave at at a conference on Faraday at St. John’s College in Annapolis. Its subtitle is “On Getting Allong Without Euclid”, for Faraday had neither studied Euclid, nor taken on board the plan of formal demonstration which most of us learn from the study of geometry. In short, Faraday thought in his own way, following the lead of nature and experiment. He was in effect liberated from the presuppositions about thought and physical theory with which others in the scientific community were encumbered.
The result was that Faraday hit on a fundamentally new way of understanding the phenomena of electricity and magnetism – by way of the new concept of the “field”. Maxwell deeply respected Faraday’s way, and dedicated much of his own life to comprehending how Faraday worked, and what it was that Faraday had done. The field is a fully connected system, and fields interact, not by way of their parts, but as wholes. This was clear enough to Faraday, but it required recourse to a new sort of mathematics – Lagrangian theory – and a major reversal of conventional thinking, to articulate a formal theory in which the whole is primary, and with it a new rhetoric of explanation. This was Maxwell’s accomplishment in his Treatise on Electricity and Magnetism, a transformation I trace as a rhetorical adventure in my book Figures of Thought.
In the end, Maxwell emerged with the astonishing claim that of them all, it was the uneducated Faraday who was the real mathematician. If that could be so, what is mathematics? That’s the question pursued in this lecture, which aims to find out what Maxwell could have meant.
Maxwell was clearly in earnest, and seems to be pointing to a mathematics embodied in nature, which lies deeper than either its symbolic or its logical forms.