### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Nineteenth-century developments in electricity and magnetism
- 3. Faraday’s experimental and theoretical practices
- 4. Maxwell and the mathematical representation of fields
- 5. The cognitive sources of Maxwell’s representations of fields
- 6. Conclusions
- Acknowledgments
- References

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.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Nineteenth-century developments in electricity and magnetism
- 3. Faraday’s experimental and theoretical practices
- 4. Maxwell and the mathematical representation of fields
- 5. The cognitive sources of Maxwell’s representations of fields
- 6. Conclusions
- Acknowledgments
- References

Open at random any page from Michael Faraday’s *Experimental researches in electricity* (1839–1855) and compare it to any page chosen at random from James Clerk Maxwell’s (1873/1891)*Treatise on electricity and magnetism*. In the case of the Faraday page, you will not find a mathematical equation—there are none in any of his publications. Maxwell, in contrast, will have mathematical equations, usually more than one, on almost any page. Clearly, this is some sort of “style” difference; Maxwell used mathematics, and Faraday did not. But, as I will argue below, the gross difference is not what is most important here. When Maxwell (1873a, p. 360) described Faraday as “in reality a mathematician of a very high order,” he meant it quite literally. There are many references to Faraday in his writings on electricity and magnetism, and Maxwell’s *Treatise* is, in part, a defense of Faraday’s style as being actually *the same as* Maxwell’s, even given the different use of mathematics.

The use of mathematics is widespread in science today, and especially so in physics. Ironically, there have been almost no studies of the cognitive role of mathematics in scientific thinking, in spite of many existing studies on the cognitive nature of science. By examining the similarities and differences between Faraday and Maxwell, I argue that we can see what must be involved in a cognitive understanding of the role of mathematics as a representational medium for scientific concepts and as a set of skilled practices used by scientists in the development of scientific concepts. In particular, I will argue for a *central* role of mathematical representations as aspects of the scientific practices of Faraday and Maxwell, rather than as an added-on “module” otherwise independent of physical conceptions.

Using historical figures for such analysis is unusual among cognitive scientists, although the overall approach is not new (see, e.g., Gooding, 1990; Nersessian, 1995; and my earlier analyses of Faraday, e.g., Ippolito & Tweney, 1995; Tweney, 1985, 2006; Tweney, Mears, & Spitzmüller, 2005). In an extensive review of cognitive studies of science, Klahr and Simon (1999) argued that historical studies were especially strong in the possibility of uncovering new phenomena that could be subject to cognitive analysis; the present study offers one such possibility. This reverses the order of what might seem a more intuitive approach to cognitive scientists, that of using cognitive laws and theories as interpretive frameworks for real-world science (Tweney, 2001). For example, in earlier laboratory work on the heuristics of confirmation and disconfirmation, we found that a small number of subjects appeared to use a “confirm early, disconfirm late” strategy, in which disconfirmatory data was ignored early in a hypothesis development process, but actively sought after a certain amount of confirmatory data had been obtained (Mynatt, Doherty, & Tweney, 1978). Such heuristics were later shown to be very prominent in Faraday’s research (e.g., Tweney, 1985), and they have been seen in subsequent studies of scientific thinking as well (e.g., Dunbar & Fugelsang, 2005).

A number of historical–cognitive studies of scientific thinking have centered on the manipulative practices of science, especially those deployed in experimental research, and on the representations used by scientists, often with a focus on visualization. For example, Gorman (1997) explored the experimental approach taken by Alexander Graham Bell in the invention of the telephone. Basing his approach on notebooks and patent drawings, Gorman used a problem-space approach to map the successive steps by which Bell’s electro-mechanical analogy of the human ear was refined across a series of models until the goal, transmission of sounds via an electric wire between distant speakers, was achieved. Similarly, Tweney et al. (2005; Tweney, 2006) replicated aspects of Faraday’s research on thin films of gold, to uncover the manipulative laboratory skills used by Faraday in the discovery of metallic colloids. Note that both studies, by Gorman and by Tweney et al., focused upon external representations (diary entries, drawings, replicated practices, etc.) to reconstruct scientific practices. The focus upon practices means that historical–cognitive studies often resemble ethnographic studies of science (Tweney, 2004). In fact, ethnographic approaches are explicitly used in the work of Nersessian and her colleagues, which seeks to understand the practices and material representations used in a bioengineering lab (e.g., Nersessian, 2005).

In all of these examples, the theories and concepts of cognitive science have served as “mapping tools” for the understanding of science. The present study seeks to understand the sophisticated use of mathematics in science using a cognitive–historical approach. By centering on a case in which the use of advanced analytic methods was still new, we can take advantage of the explicit rationale and description given by the original proponents as aids in understanding what otherwise would be obscure. The case study is, in effect, a “lens” through which previously unstudied aspects of scientific thinking are visible. Thus, I argue that new aspects of the nature of mathematical–physical skill are revealed by the cognitive–historical approach. These otherwise “invisible” skills (because they are taken for granted and are part of the tacit knowledge of contemporary scientists) play a central role in modern physics, chemistry, and biology—even in some aspects of psychology.

The representational characteristics of Faraday’s and Maxwell’s field concepts can help to establish a cognitive account of the development of the concepts, particularly since there have been extensive biographical and historical studies of the transition from Faraday’s ideas to Maxwell’s—I hope to add a significant cognitive dimension to these studies.^{1} Following a brief overview of the relevant historical context, I present the argument in three parts. First, I characterize Faraday’s practices, and the extent to which they were mathematical, as Maxwell claimed. Second, I characterize Maxwell’s mathematical practices and contrast Faraday’s practices with those of Maxwell. Finally, I discuss the cognitive nature of Maxwell’s skilled representations and present an account of their development.

### 2. Nineteenth-century developments in electricity and magnetism

- Top of page
- Abstract
- 1. Introduction
- 2. Nineteenth-century developments in electricity and magnetism
- 3. Faraday’s experimental and theoretical practices
- 4. Maxwell and the mathematical representation of fields
- 5. The cognitive sources of Maxwell’s representations of fields
- 6. Conclusions
- Acknowledgments
- References

Up to the early 19th century, the phenomena of electricity and those of magnetism were generally seen as separate topics. While electric charges attracted or repelled each other, just as magnetic poles did, and both involved “inverse square” laws of motion, as did gravity, all three were seen as independent domains. Things became more complicated in 1820, however. In that year, Oersted discovered that a current-carrying wire influenced a magnetic compass needle held near the wire. The discovery suggested an underlying unity among electricity and magnetism and immediately inspired research by others (Caneva, 2005). Michael Faraday (1791–1867) quickly became aware of Oersted’s finding and developed a series of experiments to determine the direction of the attractive forces associated with currents. Faraday showed that the influence of the current-carrying wire on a magnet involved *transverse* forces (i.e., forces oriented in a circular fashion in a plane perpendicular to the wire), rather than simple attractions or repulsions. Using this knowledge, in 1821, he was able to make a magnetized needle revolve around a current-carrying wire. The demonstration devices made by Faraday to show these phenomena are sometimes called the first electric motors; further, Faraday’s results suggested that electricity and magnetism could not be treated as independent domains—theory had to account for their interactions as well (Cantor et al., 1991; Gooding, 1990).

Further, if currents could produce magnetic forces, could the reverse be true? Could magnetic forces produce an electric current? Faraday discovered the way to do this in 1831 by showing that electric currents were produced in conductors near an electromagnet being switched on or off (Faraday, 1832). Currents were also produced by a bar magnet in motion near a conductor. That is, while Oersted found that steady electric currents produced a continuous magnetic force, Faraday found that *changing* magnetic forces could induce a current in a conductor.

Faraday spent decades developing his theoretical ideas about the underlying nature of these phenomena (Faraday, 1839–1855; see Fisher, 2001 for a usefully annotated guide to a “first reading”). In effect, Faraday’s is the first true field theory in modern physics (Nersessian, 1984). The theory was centered on the notion of lines of force, which Faraday regarded as physically real, invisible, and space filling; these lines of force became the fundamental reality of the physical world for Faraday (see, e.g., Faraday, 1852a). His experimental work showed that all matter was subject to their influence, that the lines of magnetic force existed only as closed loops (unlike electric lines of force), that their effect could be quantified, and that there was an analogous relationship between lines of magnetic force and the lines of electric force in the vicinity of a static charge (Gooding, 1981; Tweney, 1992a, 1992b).

In the end, Faraday left a large body of empirical results that inspired a number of attempts, beginning with those of William Thomson (later known as Lord Kelvin), to place Faraday’s results within a mathematical framework. However, Faraday’s theoretical ideas enjoyed little success among his peers, most of whom did not accept his ideas about lines of force. James Clerk Maxwell (1831–1879) was an exception in this regard; he was the most important figure to accept Faraday’s theoretical ideas. His contribution was to “mathematicize” Faraday’s theory, showing that Faraday’s field concepts were consistent with the energy-based approaches then rapidly becoming central in physics, and to extend the theory into the form now known as “Maxwell’s Equations,” four vector expressions. These constitute a brilliant summary of classical electromagnetic phenomena, even as they carry strong implications for the theory that underlies these phenomena. The most stunning aspect of Maxwell’s equations is that they imply that light is a propagating electromagnetic wave, a claim experimentally verified by Hertz in 1888. Only after this finding was Maxwell’s account accepted by a majority of mathematical physicists.

Throughout his work on electricity and magnetism, Maxwell gave frequent acknowledgement to Faraday as the originator of the field ideas he sought to develop, even at times claiming that his own approach was Faraday’s. To understand what he meant, we must look more closely at the practices of the two scientists. In particular, what was it about Maxwell’s mathematical development of field concepts that he saw as similar to Faraday’s method? To answer this question, I first briefly summarize some aspects of Faraday’s experiment and theory, then develop a cognitive approach to understanding Maxwell’s use of mathematics.

### 3. Faraday’s experimental and theoretical practices

- Top of page
- Abstract
- 1. Introduction
- 2. Nineteenth-century developments in electricity and magnetism
- 3. Faraday’s experimental and theoretical practices
- 4. Maxwell and the mathematical representation of fields
- 5. The cognitive sources of Maxwell’s representations of fields
- 6. Conclusions
- Acknowledgments
- References

In the 1820s, as noted, Faraday began his efforts to understand the relation between electricity and magnetism, and it is during this period that his unique approach to experimentation, and the first beginnings of his field theory, emerged. Faraday relied upon successive experiments that moved from initially vague attempts to make phenomena regular and visible, through a series of increasingly clear demonstrations of the forces involved, and ultimately to demonstration devices of great persuasive power showing the nature of the forces (Gooding, 1990, 2006). He kept careful, very extensive notebooks that were used as a continually active resource. These permit close tracking of the course of his experimentation (Tweney, 1991). Faraday’s experiments were generally exploratory, rather than formal hypothetico-deductive tests (Steinle, 2002). Especially in his diaries, but even in his published papers, there is a dialectical movement in which experiments are developed in succession; “For him, the most valuable experimental results do not acquire their importance from quantitative considerations but from their capacity to *lead the mind* to ever more searching and comprehensive images under which to view the powers that are in play” (Fisher, 1992, p. 167, emphasis in original). Faraday did not first elaborate a theory and then seek confirming experiments. The “proof” of theory came only after a long series of researches, in which his physical ideas were developed by successively refined experiments, while alternative conceptions were tested and swept away by inconsistent results. To read Faraday is to join in something like a mystery story, a narrative in which truth is gradually revealed from many clues, and in which theory emerges slowly over long periods.

Faraday’s style of experimentation can be seen in an important series of researches he conducted on the nature of acoustical vibrations, particularly those on the surface of water (“crispations,” seen when, for example, a wine glass is struck and made to “ring”). In the course of these experiments, Faraday developed an intuitive model of the vibrations, which were of course too rapid to be directly observed. His mental model started with the small movements of visible particles on the water, abstracting these to the infinitely small particles of the water itself, a kind of “intuitive calculus,” that, in effect, “stopped time” (Tweney, 1992a). By tracking his successive experiments, it was possible to follow the path by which his mental model of the vibrations was refined and confirmed (Ippolito & Tweney, 1995).

The example makes clear the way in which Faraday’s ideas were “mathematical,” even as they lacked the formal symbolism of mathematics. In effect, he used a dynamic geometry in which visualizations of processes changing in time were used generatively to advance his theoretical claims (see also Gooding, 1992, 2006). Thus, his representations of the magnetic field relied upon patterns formed by iron filings sprinkled on a card near a magnet, but these were generalized to the notion of lines of force, and were ultimately abstracted to a “purely” geometric representation of interlocking rings showing the relation between electric and magnetic lines of force (Fig. 1).

In every case, the theoretical ideas were grounded in experimental practices, that is, in his manipulations of physical phenomena.

Faraday was uncomfortable with what was then a “Newtonian” orthodoxy concerning the action of forces. For the Newtonians, forces acted “at a distance”; that is, like gravity, electric and magnetic attractions, and repulsions were conceived as properties of bodies—masses, magnets, and charges. Nothing could be said about the intervening space between two attracting or repulsing bodies, but the forces could be quantified, represented by powerful mathematical tools, and shown to be consistent across broad domains of physics. The great success of Newtonian mechanics meant that most physicists up to the late 1870s regarded action-at-a-distance formulations as the best way to proceed in the understanding of electricity and magnetism. Faraday, however, refused to acknowledge that forces could act at a distance, and this is ultimately why he developed the idea of lines of force, not as abstractions but as real physical entities. This move allowed him to conceptualize forces as acting contiguously, rather than at a distance. The idea was an important motivator for the decades of experimentation that he devoted to establishing the physical reality of lines of force. Space, for Faraday, was not an empty void, within which material objects acted upon one another via mysterious forces across the nothingness, but rather was a plenum, in which the real but nonmaterial lines of force stood between material objects and constituted the medium of interaction.

In a letter to Maxwell (13 November, 1857, Letter 3357, in James, 2008; see also James, 1998), Faraday wondered whether the results of mathematical approaches to physics could be “Translat[ed] … out of their hieroglyphics, that we might also work upon them by experiment.” It was Maxwell’s achievement to carry out this translation, and to show that the kinds of mathematical representations used affected the physical conceptions represented, and hence the ease with which they could be experimented upon.

### 4. Maxwell and the mathematical representation of fields

- Top of page
- Abstract
- 1. Introduction
- 2. Nineteenth-century developments in electricity and magnetism
- 3. Faraday’s experimental and theoretical practices
- 4. Maxwell and the mathematical representation of fields
- 5. The cognitive sources of Maxwell’s representations of fields
- 6. Conclusions
- Acknowledgments
- References

Faraday and Maxwell shared many presuppositions and central concepts about the nature of fields. They followed a similar path in developing geometric representations of the electromagnetic field. Maxwell was well versed in the Newtonian worldview, and he was well-equipped to understand and advance the mathematical formulations of Newtonian ideas prominent in both Continental physics and in “north British” and “Cambridge” experimental physics (Anderson, 2001; James, 1998). Like Faraday, he ultimately rejected action at a distance in favor of contiguous action, but he did this in a fashion that attempted to reconcile the two. When Maxwell began to think about electricity and magnetism, he first dealt head-on with the experimental results of Faraday’s approach, results that caused a considerable problem for the Newtonian approach, in part because Faraday’s lines of force were *curved*, rather than straight-line forces (like gravitation). While Faraday’s practical results could not be ignored, his theoretical claim about the reality of lines of force was rejected by nearly everyone—except Maxwell.

Maxwell’s “translation” (or, better, his re-representation) of Faraday’s concepts into mathematics required an extended argument, in the course of which Maxwell changed the way in which mathematical expressions could be used as representations of physical phenomena. This was not a simple move; in fact, Maxwell developed his ideas across many years, first in a series of three studies (Maxwell, 1855–1856, 1861–1862, 1864), then in the 1873 *Treatise*. The development has been traced in detail by, for example, Nersessian (2008, esp. Chapter 2). The change was momentous because it changed the way mathematics was used in physics.

We tend to think of mathematics as useful in science in one of two ways: (a) as a numerical and computational tool, or else (b) as a derivational and proof mechanism, Maxwell was among the first to argue for an additional role. He was insistent that there is also a *representational* function. For example, in presenting the energy concepts of Lagrange’s approach to mechanical systems, he stated: “Our aim … is to cultivate our dynamical ideas. We therefore … retranslate the results [of the mathematicians] into the language of dynamics, so that our words may call up the mental image, not of some algebraical process, but of some property of moving bodies” (Maxwell, 1873/1891, vol. 2, p. 200).^{2} In effect, Maxwell is asking that mathematical representations be representations of physical realities.

The specific referent of the above quote was the Lagrangian approach to the dynamics of systems of moving particles. For example, the gravitational force between two bodies can be represented using an equation in which force is a function of the mass of the bodies:

where *m*_{1} and *m*_{2} are the masses, *r* the distance between the bodies, and *G* the universal gravitational constant. If we know the numerical values of the masses, the distance between them, and the value of the constant (all in suitable units of measurement), we can calculate the force of attraction. In the 18th century, Lagrange showed that the same dynamic laws can also be represented using formulations in which the relevant equations specify only the energy associated with each infinitesimal point of the space *between* the two bodies. The specific equations of motion of any dynamic system can then be derived. For most of Maxwell’s predecessors, the Lagrangian formulation was taken to be more or less a computational simplification, not a change in the underlying conceptualization. While it was a re-representation of Newton’s ideas, the only specific physical meaning involved the forces that would be experienced by a particle placed at that location. In contrast, Maxwell emphasized that the Lagrangian formulation pointed to a different, “field-like” representation of the physical ideas underlying the mathematical expression.

For Maxwell, the use of mathematics in physics was closely tied to the representational power of mathematical expression. As there were a variety of such expressions available for any one set of phenomena (as in the example above of Lagrange’s equations), the scientific challenge was to choose one which corresponded to the right physical ideas. Further, across *different* phenomena, it was known that similar mathematical expressions could be used: “In many parts of physical science, equations of the same form are applicable to phenomena which are certainly of quite different natures, as for instance, electric induction through dielectrics, conduction through conductors, and magnetic induction” (Maxwell, 1873/1891, vol. 1, p. 70). For Maxwell, analogies across domains were of central importance (Hesse, 1973; Nersessian, 1984, 2008), and he explored them intensively. Thus, he first used a fluid-flow analogy to conceptualize electric- and magnetic-field effects (Maxwell, 1855–1856), then changed this to a mechanical model using vortex wheels separated by idler wheels (Maxwell, 1861–1862), later adapting this to a mechanical model in which the vortex wheels were elastic (Maxwell, 1864). Finally, in the *Treatise* (1873), Maxwell developed an abstract representation in which the electromagnetic field was manifested as a continuum of stresses and strains in a material medium—an ether. In the *Treatise*, he dropped the explicitly mechanical analogies (which, even in the earliest study were not, he insisted, to be taken as *more* than simply analogies). Instead, the electromagnetic field was conceived as a whole, a space-filling continuum of electric and magnetic energy. The ether was, he insisted, a *hypothesis*. While its exact nature was unknown, “We ought to endeavor to construct a mental representation of all the details of its action, and this has been my constant aim in this treatise” (Maxwell, 1873a, vol. 2, p. 493).

Much of the *Treatise* was an argument for this representational approach. His goal was to use the mathematics in a way that reflected his underlying “physical intuitions” about fields. The “intuitions” were ones largely shared between Faraday and Maxwell. For example, consider how Maxwell used the relation between electric potential and electric intensity to change the conventional action at a distance formulation to a field formulation (see Simpson, 2005, pp. 7–12, which simplifies Maxwell’s treatment, and which I follow here). If we take an electrically charged surface and wish to find the total energy at the surface and at a point distant from the surface, we can do this knowing only the electric intensity per unit charge at the surface and the potential at the surface:

where Ψ is the electric potential, *Ε*_{n} is the electric intensity perpendicular to the surface, and d*S* is the differential element of the surface. In this case, we have to carry out a double integration (i.e., “add up”) around all the infinitesimal (differential) elements, d*S*, of the surface. But there is a mathematical identity that relates this double integration to a triple integration. Known as Green’s theorem, it states an identity:

where dζ is the differential element of volume. That is, we get the same result if we “add up” all the infinitesimal elements of volume instead of surface. While the two expressions are mathematically equivalent, Maxwell argued that the two rest upon different underlying conceptions when applied to physics. Whereas the double integral form assumes an action at a distance view, in which it is the surface that acts upon the distant point, the triple integral form states the relationship between each point of the entire region outside the surface. The triple integral form is thus representative of the field view—provided that the “translation” calls up “the mental image, not of some algebraical process, but of some property of moving bodies” (Maxwell, 1873/1891, vol. 2, p. 200).

The *Treatise* culminates in a summary representation of the properties of electric and magnetic fields and of the relation between them. Now known as “Maxwell’s equations” (shown in modern notation in Fig. 2), these describe the relationship between two field quantities (**E**, the electric field intensity, and **B**, the magnetic field intensity) and one current term, **J**. All three are vector quantities, that is, they specify a magnitude and a direction. A detailed examination is beyond the scope of this study,^{3} but there are two aspects of these equations that are of cognitive relevance to my argument, namely, the character of the divergence operator (“Div **E**” or ∇ · **E**, read “Del dot E” for the electric field) and of the curl operator (“Curl **E**” or ∇ × **E**, read “Del cross E”). Each summarizes a characteristic of the entire vector field in a given region. The divergence of a field is a number that expresses the rate of change of the field with respect to position (the Div operator itself expands into a partial differential expression, to be applied to a vector). Thus, consider a system consisting of a faucet and a sink (the example is based on Skilling, 1942). If we think of the water flowing out of the faucet and leaving at the same rate via the drain, then the total divergence of this field is zero—as much water is flowing out as in, and so the net change across the field is zero, Div = 0. If we place a stopper in the drain, then more water will be flowing in than leaving, and the divergence will therefore be nonzero.

The curl operator captures a different aspect of a field’s properties; it is a vector that expresses the “curliness” of the field. Thus, consider the field represented by the water swirling out of the drain. Because the water is swirling, it is moving at a faster velocity near the center of the whirlpool than near the edge. Immersing a small paddle wheel in the water between the edge and the center would cause the wheel to spin. In contrast, if we consider water flowing smoothly in a river, say, and we immerse our paddlewheel, there will be no movement of the wheel, as both sides will be equally pressured to move. The curl of the river field is zero, and that of the whirlpool is nonzero. You could “feel” the difference, if you immersed your hand in the river or the whirlpool. Note that this is not an expression for the curvature of the field and hence is not necessarily a visual concept. The curl would also be nonzero if, say, the water in the river flowed more slowly near the bank, compared to near the center of the flow. While all the flow vectors would be straight, their magnitude would be smaller near the bank. The immersed paddle wheel would turn, and curl would be nonzero.

When applied to electric and magnetic fields, Maxwell’s equations thus specify the divergence of each, considered individually, and the curl of each, which is dependent on the *other* field. Note that the divergence of the electric field depends upon the presence of an electric charge. The divergence of the magnetic field is always zero, however. This is another way of saying that there are always as many magnetic lines of force entering a given volume as leaving it. And that, in turn, expresses the empirical fact that there are no “magnetic monopoles”—anywhere a north pole occurs, there is a south pole. In Faraday’s terms, lines of magnetic force are always closed loops. As for the curl, note that the curl of the electric field is dependent upon the rate of change of the magnetic field, whereas the curl of the magnetic field is dependent upon the rate of change of the electric field and the current density. That is, the two fields are interdependent, if either one is changing. Put crudely, if a changing electric field is nearby, there will be a tension in the area that produces a magnetic field—“the paddlewheel will turn.” This is Oersted’s discovery of electromagnetism. If there is a changing magnetic field, then there will be an electric field in motion (a current) in a nearby conductor. This is Faraday’s discovery of electromagnetic induction. Put crudely, a changing magnetic field will make an “electric paddlewheel” turn. Thus, whereas Oersted and Faraday had shown experimentally that electric fields and magnetic fields were interdependent, the exact nature of the interdependence, and their mathematical representation, was summarized completely by Maxwell’s equations.

### 5. The cognitive sources of Maxwell’s representations of fields

- Top of page
- Abstract
- 1. Introduction
- 2. Nineteenth-century developments in electricity and magnetism
- 3. Faraday’s experimental and theoretical practices
- 4. Maxwell and the mathematical representation of fields
- 5. The cognitive sources of Maxwell’s representations of fields
- 6. Conclusions
- Acknowledgments
- References

So far, my account has been thin on the explicit cognitive underpinnings of the representational ideas at play in the work of Faraday and Maxwell, but these can now be developed. First, note that the representation of fields in terms of the vector operators’ divergence and curl is closely related to a particular kind of nonvisual sensory and motor representation. In effect, the divergence and curl specify “feels” and “imagined doings,” as well as visual representations. For both operators, I have asked the reader to imagine water flowing in a sink or a river, and to carry out thought experiments to see this (see Clement, 2009). Likewise, Maxwell asked his readers to relate the “physical intuition” of vector fields to the mathematical representation.

Faraday did something similar in representing field properties, using a kind of dynamic geometry grounded in experimental manipulations. For example, to conceptualize magnetic field forces, Faraday imagined a tiny loop of conducting wire carried around in a magnetic field and considered the electric currents generated in the wire as it crossed over the magnetic lines of force—rather like our paddlewheel. In fact, he actually did such experiments with very small loops of wire connected to a galvanometer. When moved through a magnetic field, small currents were generated that depended upon the field strength and the velocity of the moving wire (e.g., Faraday, 1852a, 1852b). In this way, he was able to map the space of the lines of force around a magnet and to determine many of the quantitative relationships later captured by Maxwell’s equations. (In just the same way, we could use a physical paddlewheel to map and to measure the forces in a river or a sink.) Maxwell defended Faraday’s conception as truly mathematical: “Every question relating to the forces acting on magnets or on currents, or by the induction of currents, may be solved by consideration of Faraday’s lines of force. … Faraday defined with mathematical precision the whole theory of electro-magnetism” (Maxwell, 1873b, p. 320). For Maxwell, Faraday’s mathematics was geometrical and tied closely to physical conceptions—all that was lacking was the technical symbolism of the usual mathematical approaches.

This suggests that sensory motor representations—“feels”—can be used to great effect in physics. Clement (1988, 2008), for example, has analyzed video tapes showing a scientifically trained subject making manual twisting motions while attempting to represent the sources of tension in a spring—these cannot be “seen” but they can be “felt.” For Maxwell, the translation of Faraday’s loop of wire into the mathematics of vector fields was of central concern, and Maxwell and Faraday were, on this level, saying nearly the same thing—for both, the field was a kinesthetically “felt” reality. The mental models were similar, except that Maxwell invoked a re-representation of the model in explicit mathematical terms.

For both Faraday and Maxwell, a reader of their works cannot understand the claims made unless in possession of specific skills that are demanded as prerequisites. In the case of Faraday, these are relatively straightforward for nonphysicists, as everything is presented in terms of the experimental path by which the claims were supported. A careful reader needs little background in physics to follow most of Faraday’s arguments because the experiments and conclusions are so carefully described. The appropriate “feels” emerge as one imagines the experiments and the extension to the invisible realm of lines of force.

Maxwell, in contrast, can be notoriously hard to follow because he presupposes a high degree of a certain kind of mathematical skill. In fact, his work was initially difficult even for most physicists, because the particular style of mathematical argument used by Maxwell was not universal—it was “Cambridge mathematics,” the mathematical physics that arose at Cambridge University in the 19th century. Andrew Warwick (2003) has examined in detail the history of mathematical physics education at Cambridge and has argued that this impacted both the formulation and the reception of Maxwell’s ideas. As Warwick has shown, there was intense competition among Cambridge students to be a “Wrangler” upon graduation; that is, to pass with high marks a “Tripos” examination in which difficult problems in mathematical physics were posed and students had a limited time in which to solve them. By 1854 (the year Maxwell placed second, i.e., as “Second Wrangler”), the problems had become incredibly hard and this in turn required students to work closely with tutors, not so much to learn the relevant physics and mathematics, but to acquire through intensive practice the unique problem-solving skills required to tackle the Tripos questions. These skills depended upon a particular kind of physical knowledge, as well as extensive knowledge of mathematics. The questions themselves frequently required analogies from one domain of physics to another and often required the student to re-represent one physical system using relations from another. For example, a problem involving gravitational potential might require an approach used in analyzing problems of heat conduction. The problems demanded that students possess “physical intuition,” especially in the realm of Newtonian dynamics, as well as skill in handling multiple representations. Nersessian (1999, 2002) and others (e.g., Siegel, 1991) have discussed Maxwell’s extensive use of physical analogies in his work on electromagnetic fields. The source of this aspect of his style is to be found partly in his educational experiences at Cambridge, which, as Warwick has shown, prioritized the use of analogy across physical domains and emphasized a mathematical physics, rather than pure mathematics or experimental physics.^{4}

Maxwell often wrote as if his audience possessed the skills of a Cambridge Wrangler, and this accounts in part for the slow acceptance of his ideas (Hunt, 1991). Thus, he remains difficult to read today, even for sophisticated readers, because he took for granted a kind of mathematical skill that is uncommon today. His works can appear maddeningly elliptical, even to physicists. Conceptually, Maxwell’s equations (even with modern methods of explaining them) remain a difficult hurdle for undergraduate physics majors, and this is due partly to the need first to become comfortable with the vector analysis (curls and divs and all) and partly because the conception of a field, with its reliance on “felt” mathematical representations is itself an acquired aspect of physics expertise.

How are such skills developed? Recent cognitive theory holds some important clues. In particular, we can see Maxwell’s skill as dependent upon a particular feature of expert memory identified first by Ericsson and Kintsch (1995), what they call long-term working memory. Thus, they showed that expert readers (but not inexpert readers) can keep the thread of a book’s argument “in mind” long after the contents of ordinary (short-term) working memory have been replaced by new information. In effect, long-term memory remains accessible for immediate call-up as needed; what experts seem to have developed, then, is a specific set of retrieval structures that make this possible. The relevant skill is more than simply possession of a set of retrieval cues. The expert retrieval structures also imply a kind of “prospective” focus, an anticipatory element that flags what *might* be relevant in the near or far distant future. Such structures are domain specific and develop only after extensive deliberate practice. In expert reading, the effect serves to keep the larger gist of text available across long stretches of text. Similarly, what Maxwell and his fellow Wranglers could do was to call up relevant skills and physical intuitions from long-term memory and use these to keep the thread of a mathematical argument in mind.

Skill of this sort is dependent upon extensive deliberative practice (Ericsson, 2006). In the case of Maxwell, this was partly a result of the training received at Cambridge and partly on his prior and subsequent work in mathematics and physics. It is commonplace among physics teachers today to insist on the need for students to practice repetitively some of the fundamental operations and representations used in physics. For this reason, all physics textbooks contain problems, usually of graded difficulty and sometimes with answers given. Still, it is a frequent complaint that too few of these are actually solved for the student to have real mastery (e.g., Arons, 1997). In the transcript of a lecture to students, given outside the regular lecture series of the course, Richard Feynman discussed some of the problems students encountered in learning physics, and he emphasized the role of repetitive drill in some of the basic operations such as differentiation: “What we have to do is to learn to differentiate like how we know how much is 3 and 5, or how much is 5 times 7. … You’ll find you need to do this all the time. … Therefore differentiation is like the arithmetic you had to learn before you could learn algebra” (Feynman, 1961/2006, p. 19).

It is thus possible to understand how Maxwell was able to use his physical intuitions about fields within the framework of his skilled knowledge of physical and mathematical representations. While his intuitions about fields were close to those of Faraday, his skilled mathematical tools allowed a new and different way to represent them. Faraday lacked the specific mathematical skills employed by Maxwell, although his intuitions corresponded closely, as did his “intuitive geometry.” What Faraday did possess was a set of experimental skills that were successively refined and practiced over decades of experimentation.

Both sets of skills, Faraday’s and Maxwell’s, were dependent upon external representations. That is, both relied upon external material objects to serve as “cognitive artifacts,” as representational media for ideas, media that could be seen as agentive within the course of thought. These were coils and wires, say, for Faraday, and paper and pen for Maxwell. It is interesting to note that Warwick (2003) noted the importance of paper and pen for the establishment of the Wrangler system at Cambridge. Prior to the 1830s, students generally used a chalk and slate to take notes or work problems. The shift to the permanent representation on paper extended long-term working memory and the scope and difficulty of the problems that could be solved. In the course of learning to do “Cambridge mathematics,” these representations played an important transitionary role in the development of the appropriate retrieval structures.

### 6. Conclusions

- Top of page
- Abstract
- 1. Introduction
- 2. Nineteenth-century developments in electricity and magnetism
- 3. Faraday’s experimental and theoretical practices
- 4. Maxwell and the mathematical representation of fields
- 5. The cognitive sources of Maxwell’s representations of fields
- 6. Conclusions
- Acknowledgments
- References

In the end, Maxwell’s equations came to have “authority” for physicists (Kurz-Milcke, 2004). That is, as specific representations of the properties of electromagnetic fields, the equations themselves became the tools of particular problem-solving efforts, while the understanding of the physical phenomena they represent and the abilities to manipulate the equations in specific applications became part of the expert knowledge of physicists and engineers. There is a social–psychological dimension to this development, of course. What counts as scientific knowledge is the outcome of a “complex process of transformative negotiation … that simultaneously formalizes the essential character of the discovery and confers upon it the stamp of objectivity as, by implication, an aspect of the physical world that was there waiting to be ‘discovered’” (Caneva, 2005, p. 176). Maxwell’s equations came to allow a set of possible “doings” for the physicist, a way of organizing an intentional effort to deal with the specifics of a problem situation. Just as Clement’s scientist used manual gestures to represent the sources of tension in a spring, the mathematical physicist can use Maxwell’s equations to conceptualize the tensions, energy relationships, and forces associated with an electromagnetic field. The equations therefore embed two kinds of expert knowledge, one kind associated with knowledge of the fields as physical entities and another associated with the skills, mathematical and otherwise, needed to work with the equations.

By concentrating on the nature of the representations (and hence focusing upon the structure of cognition, see Tweney, 2001), I do not mean to slight the importance of a process level of analysis, processes that have been the focus of much effort in the past, beginning with the problem-space analyses of Herb Simon and his students (e.g., Langley, Simon, Bradshaw, & Zytkow, 1987). In the terms of the present study, Maxwell’s equations provide a representation of the states of a problem space and the mathematical–physical skills needed to manipulate them become the operators that alter one state into another. Thus, it should be possible to use think-aloud protocols to explore the way in which physicists actually work with the equations. Similar problem-space analyses have been carried out for Faraday’s experimentation (e.g., Tweney & Hoffner, 1987), and they could be carried out for some aspects of Maxwell’s argument in the *Treatise* as well.

Still, problem-space analysis cannot account for all aspects of scientific practices; in particular, they do not easily allow for the analysis of the way representations are generated in the first place (Kurz-Milcke, 2004). Recent work in the cognitive–historical analysis of science has made this especially clear. Consider, for example, David Gooding’s (2005) model of the use of visualization in science, the “Pattern–Structure–Process” model. According to Gooding, many uses of visual representations in science depend on a recursive use of pattern-finding inferences, followed by inferences to structures, followed by inferences to processes that produce those structures. For example, paleontologists may recognize patterns in fossilized remains that lead to inferences about two-dimensional structures, which in turn lead to inferences about the geological processes that reduced a three-dimensional object to a two-dimensional fossil. The resulting three-dimensional representation then serves as another pattern in a renewed cycle of pattern to structure to process (e.g., inferences about the physiology of the fossilized organism). Gooding (1992, 2006) has applied this analysis to the case of Faraday’s electromagnetic field model as well, showing how Faraday’s recognition of patterns in electromagnetic rotation led to his conceptualizations of the three-dimensional field and its subsequent elaboration into his final field theory. As in the present analysis, Gooding characterized Faraday’s use of visualization as mathematical in much the same way that Maxwell regarded Faraday as an “intuitive mathematician.” Still, as I have argued, Faraday’s dynamic geometry is more than simply visual, important as the visual aspects may have been.

John Clement (2008, 2009) has developed a model of scientific thinking in which there is an emphasis on recursive cycles of “Generation–Evaluation–Modification.” Using detailed think-aloud and video-taped protocols of scientists and students solving problems in physics, Clement has shown how processes of analogy, imagery, and modeling are used in repeated cycles of representation and testing. Like Gooding (2005), Clement argues for a central role of imagery, including kinesthetic imagery as well as visualization. Clement’s “depictive gestures” of a scientist solving a spring problem (noted earlier) resemble the kinds of representations that I have identified in connection with the vector operators of divergence and curl.

Nancy Nersessian (2008) has provided detailed analyses of Maxwell’s development of his field theory, and her account is directly relevant to the present argument. By focusing on the model-based reasoning of Maxwell, she has explored the way analogy was used to develop his theory, anchoring her account within a general theory of model-based reasoning. Of most relevance in the present context, she has noted how the process followed by Maxwell gradually eliminated domain-specific aspects of the phenomena, such that “A new kind of representational structure emerged—one that abstracted the notion of ‘mechanical’ from the analysis, creating a representation of a nonmechanical, dynamic system” (Nersessian, 2008, p. 60).

I am suggesting that Maxwell’s “new kind of representational structure” is mathematical, but in a special sense. Rather than just the distillation of logical proof processes or quantitative relationships, Maxwell’s equations also include the visual *and the nonvisual* aspects of the scientist’s expert knowledge of the phenomena, and they allow this expertise to be used as a new source of patterns, structures, and processes, in the service of new models of the physical reality they embed. Lakoff and Núñez (2000) have recently argued that mathematical expressions are embodied metaphors, closely related to basic cognition and basic linguistic relationships. While there is little in their book about the use of mathematics in physics as such, they do mention that the periodic functions used to represent wave motion use complex numbers (i.e., numbers that include the square roots of negative numbers, so-called imaginary numbers). Such functions exist in the minds of physicists, they claim, not in the real world: “Human beings use the mathematics to calculate the physical regularities. The mathematics is in the mind of the physicists, not in the universe” (Lakoff & Núñez, 2000, p. 345). But this understates the case, as if the calculational function of mathematics was all that mattered in physics. On my account, as Maxwell argued, following Faraday, there is far more to mathematical physics than this. Instead of a *merely* calculational resource, the mathematical representations of physics constitute a central mode of cognition inseparable from the doing of physics itself, and central as well to the development of new models of physical reality. Put differently, if the purpose of mathematical representation is to capture a model of physical reality, then a cognitive account cannot ignore that fact, anymore than a problem-solving account can ignore the goal of the problem.

In the present study, I have used a cognitive–historical approach to identify aspects of mathematical representation in physics. For cognitive science, it is an exciting prospect to realize that, for the last century-plus, physicists have been using an invented mode of representation that has carried science to an extremely high level. This mode goes beyond the more familiar modes of visual and verbal representation and points to a mode of expert representation that invokes entirely new ways to think about thinking. I have suggested several cognitive models that are potentially applicable, including long-term working memory and the use of kinesthetic imagery. Exploration of these using think-aloud protocols and simulation techniques, along with ethnographic approaches based on “in vivo” science, is clearly possible, and it will extend the range of our knowledge of the “invisible” cognitive processes involved in the use of mathematics in science.