The following is based on an excerpt from In the Mind's Eye, providing us with a wonderful example of the enduring power of visual thinking--even before it is recast into mathematical form.

With this reference to "lines of force," one is immediately brought back to Michael Faraday, the self-educated scientist of the early nineteenth century whom we met at the beginning of this chapter. A tireless worker, Faraday was responsible for a great many fundamental discoveries in chemistry and physics, although he hated these specialist terms--he preferred to call himself a "philosopher." Among many achievements, his greatest was that he originated the concept of subtle electromagnetic "lines of force"--also originating the associated concept of the nonvisible electromagnetic "field" as well. (These are the same lines as those produced by the effect of a strong magnet on iron filings spread on a piece of paper.) So sensitive was Faraday to these "lines of force" that for him they were "as real as matter." His powerful visual conception of these ideas provided the basis for James Clerk Maxwell's famous mathematical equations which, in turn, provided the foundation for modern physics by defining the relationship between light, electricity and magnetism. The ideas set forth by these men have been remarkably enduring--remaining virtually unchanged up to the present time.

Both Faraday and Maxwell are extraordinarily important in the history of modern physics, and yet, unlike Einstein, who greatly respected their work, neither is well known to the lay public. The enduring position of these scientists, as well as the nature of their contributions, is summarized in Isaac Asimov's History of Physics : "Faraday . . . perhaps the greatest electrical innovator of all, was completely innocent of mathematics, and he developed his notion of lines of force in a remarkably unsophisticated way, picturing them almost like rubber bands.” Asimov is somewhat uneasy about the foregoing description of Faraday's "unsophisticated" pictures, and he comments in a footnote to his own text: “This is not meant as a sneer at Faraday, who was certainly one of the greatest scientists of all time. His intuition was that of a first-class genius. Although his views were built up without the aid of a carefully worked out mathematical analysis, they were solid. When the mathematics was finally supplied, the essence of Faraday's notions was shown to be correct.”

It is worth noting that the ambivalence toward Faraday shown here is repeated over and over again by scientific writers, showing the difficulty of their taking seriously a scientist who is not a mathematician, no matter how original, productive or prescient this scientist may be. Asimov continues: “In the 1860's, Maxwell, a great admirer of Faraday, set about supplying the mathematical analysis of the interrelationship of electricity and magnetism in order to round out Faraday's non-mathematical treatment. . . . In 1864, Maxwell devised a set of four comparatively simple equations, known ever since as "Maxwell's equations." From these, it proved possible to deduce the nature of the interrelationships of electricity and magnetism under all possible conditions. . . . Maxwell's equations were more successful than Newton's laws. The latter were shown to be but approximations that held for low velocities and short distances. They required the modification of Einstein's broader relativistic viewpoint if they were to be made to apply with complete generality. Maxwell's equations, on the other hand, survived all the changes introduced by relativity and the quantum theory; they are as valid in the light of present knowledge as they were when they were first introduced a century ago.”

The long term consequences of Faraday's ideas recast in Maxwell's mathematical formulations have been extraordinarily broad and pervasive down to the present time. One of Maxwell's biographers points out that “there is hardly an area of modern technology and physics in which Maxwell's theory has not contributed something of importance--from electrical power generation and transmission to communication systems or the monster accelerators of modern physics. The scientific, practical, and engineering consequences of Maxwell's equations have been seminal, all-pervasive and quite impossible to list. Maxwell's theory, however, was more than a synthesis or a source of future technologies. It involved a radical change in our conception of reality, a fundamental shift in point of view--it was . . . a scientific revolution.”

The full significance of Maxwell's achievements and of Faraday's ideas, on which they are based, is little known to the nonprofessional. Consequently, it is, perhaps, worth the risk of laboring the point with another reference in order to convey the full weight of accomplishment. Richard Feynman, a Nobel Prize winning physicist and author (who came into the public eye shortly before his death through his involvement with the investigation of the space shuttle explosion), provided this assessment: “From a long view of the history of mankind--seen from, say, ten thousand years from now--there can be little doubt that the most significant event of the 19th century will be judged as Maxwell's discovery of the laws of electrodynamics. The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade.”

Although Maxwell was recognized as one of the foremost mathematicians of his time, he did not consider Faraday's total ignorance of formal mathematics a reason to take his ideas less seriously. On the contrary, he found the precision and logic of Faraday's conceptions so compelling that he termed them "mathematical." Indeed, Maxwell explicitly stated that the development of his own equations was merely a translation of Faraday's ideas into conventional mathematical form. In the preface to A Treatise on Electricity and Magnetism, his major work first published in 1873, Maxwell explains: “. . . before I began the study of electricity I resolved to read no mathematics on the subject till I had first read through Faraday's Experimental Researches in Electricity. I was aware that there was supposed to be a difference between Faraday's way of conceiving phenomena and that of the mathematicians, so that neither he nor they were satisfied with each other's language. I had also the conviction that the discrepancy did not arise from either party being wrong. . . . As I proceeded with the study of Faraday, I perceived that his method of conceiving the phenomena was also a mathematical one, though not exhibited in the conventional form of symbols. I also found that these methods were capable of being expressed in the ordinary mathematical forms, and thus compared with those of the professed mathematicians.”

Also in this preface, Maxwell provides a particularly illuminating description of Faraday's thought in comparison with that of the mathematicians of the time. An extended quotation (noted previously, in part) shows the contrast in approach and ways of thinking. Maxwell explains: “For instance, Faraday, in his mind's eye, saw lines of force traversing all space where the mathematicians saw centres of force attracting at a distance: Faraday saw a medium where they saw nothing but distance: Faraday sought the seat of the phenomena in real actions going on in the medium, they were satisfied that they had found it in a power of action at a distance impressed on the electric fluids. When I had translated what I considered to be Faraday's ideas into a mathematical form, I found that in general the results of the two methods coincided . . . but that Faraday's methods resembled those in which we begin with the whole and arrive at the parts by analysis, while the ordinary mathematical methods were founded on the principle of beginning with the parts and building up the whole by synthesis. I also found that several of the most fertile methods of research discovered by the mathematicians could be expressed much better in terms of ideas derived from Faraday than in their original form. The whole theory, for instance, of the potential, considered as a quantity which satisfies a certain partial differential equation, belongs essentially to the method which I have called that of Faraday. . . . Hence many of the mathematical discoveries of Laplace, Poisson, Green and Gauss find their proper place in this treatise, and their appropriate expressions in terms of conceptions mainly derived from Faraday.”

One does not have to be a scientist or a mathematician to see the sincere admiration Maxwell had for Faraday's ideas and his deep appreciation of the complete originality of Faraday's approach. He saw that Faraday's conception was as capable of explaining the same phenomena as was that of the professional mathematicians, but his approach involved a clearer vision of the whole and provided a "much better" way of expressing some of the "most fertile" ideas--presumably seen as more elegant as a result.

Maxwell's ready acknowledgment of his intellectual debt to Faraday is admirable. What seems more remarkable is the ease with which Maxwell absorbs and translates Faraday's relatively unfashionable ideas and the vigor with which he defends the uneducated originator of these ideas. There seemed to be an unusual correspondence in modes of thought between the two men concerning concepts that were apparently unintelligible to other scientists of their time. Their shared, unusual visual-spatial proficiency might have been a major factor in their special basis of mutual respect and understanding.

From In the Mind’s Eye, chapter 1, “Slow Words, Quick Images: An Overview.”

## Thursday, April 2, 2009

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