The important role of mathematical representations in scientific thinking has received little attention from cognitive scientists. This study argues that neglect of this issue is unwarranted, given existing cognitive theories and laws, together with promising results from the cognitive historical analysis of several important scientists. In particular, while the mathematical wizardry of James Clerk Maxwell differed dramatically from the experimental approaches favored by Michael Faraday, Maxwell himself recognized Faraday as “in reality a mathematician of a very high order,” and his (...) own work as in some respects a re‐representation of Faraday’s field theory in analytic terms. The implications of the similarities and differences between the two figures open new perspectives on the cognitive role of mathematics as a learned mode of representation in science. (shrink)

Faraday demonstrated electromagnetic induction in 1831 using an iron ring wound with two wire coils; on interrupting battery current in one coil, momentary currents arose in the other. Between Faraday's ring and the induction coil, coiled instruments developed via meandering paths. This paper explores the opening phase of that work in the late 1830s, as the iron core, primary wire coil, and secondary wire coil were researched and differentiated. ‘Working knowledge’ gained with materials and phenomena was crucial to innovations. To (...) understand these material-based interactions, I experimented with hand-wound coils, along with examining historical texts, drawings, and artefacts. My experience recovered the historical dead-end of two-wire coils and ensuing work with long-coiled single conductors initiated by Faraday and Henry. The shock and spark heightened in these coils provided feedback to the many instrumental configurations tested by Page, Callan, Sturgeon, Bachhoffner, and others. The continuous conductor differentiated into two segments soldered together: a thick short wire carrying battery current and a long thin wire for elevating shocks . The joined wires eventually separated, yet their transitional connection documents belief that the induced effects depend on continuity. These coiled instruments, with their intertwined histories, show experimental work and understandings in the process of developing. Seeing the nonlinear paths by which these instruments developed deepens our understanding of historical experiences, and of how people learn. (shrink)