“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 effectliberated 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.
The series of three segments constituting the article, “The Anglo Revolution in New Mexico” was published in 1977, but it seems likely that they will raise questions just as pertinent today.I’ve described the circumstances of the articles in an Introduction to them on the “Articles” page of this website, where the articles themselves will appear.The first, on the Santa Rita copper mine, has already been posted; the other two are scheduled to appear shortly.
They refer to a clash of cultures which has taken many forms, overt or otherwise, over the years.But contrasts of cultures need not take the form of conflict: each has much to learn from the others, and the possibility is real that out of their interaction will arise, dialectically, something far better than either could be alone.
That was my hope when this series was written, and far more, it remains my hope today.My own current involvement with the “Cosmic Serpent” project, referred to in earlier postings, is one vehicle for that conviction: it brings together indigenous and western approaches to the natural world.These begin in sharp contrast, but each has much to learn from theother – and the global environment cannot wait forever for us to straighten this out!
So it seems to me.Comments will be welcome to this posting, but more, to the articles themselves.I’ll be waiting toi hear!
When I wrote yesterday about the “deep roots” of Western science, I intended to point to a possible relation this opens up between the domain of “science” and Indigenous views of the natural world. If we follow that line of development which leads from Aristotle through Leibniz to the holistic mathematical physics based on the Principle of Least Action, we find ourselves in a position much closer to that of Native American thinkers than we might have expected.Modern science in its mechanical mode cuts off “science” from any sense of wholeness or, especially, of purpose. It wants to reduce all quality to quantity, all motion to the operation of laws which bind matter apart from any sense of goal or meaning, and sees “nature” exclusively as an object from which we stand apart as mere observers. None of these limitations apply to the physics in the holistic mode. Least Action applies to whole systems, and sees systems moving directionally toward the optimization of a quantity which applies to the system as a whole. Although this goal may be no more than the optimization of a mathematical quantity, it opens the way to thinking of systems such as organisms or ecologies as moving as wholes toward ends — a line of thought of which the modern world is in desperate need.One more link in this line of thought: the modern computer is bridging the gap ;between “quantitative” and “qualitative” thinking. What goes in as number typically comes out on the computer screen as a graphical image readily grasped by the intuitive mind and conducive to interpretation in terms of purposes and goals. We can see how systems are moving, and where they “are going”. Nothing stands in the way of reading these in terms of purposes, and that is what we do on a daily basis — think for example of evidences of the consequences of global warming emerging from complex computer modeling. Thinking in this way in terms of whole systems, understanding their motions in terms of a mathematics of optimization, and bridging the gap between quality and quantity — all this is yielding an approach to science at once new and old — in a continuous thread leading from Aristotle into the age of the modern computer. If we follow that path and think of modern science in terms like these, then it seems to me the gap between a holistic science and Indigenous relations to the natural world is not as deep as it had seemed. Set aside mechanistic thinking, embrace the sense of nature as a whole of which we ourselves are part, admit goal as a category amenable to science — and then the old gap between Indigenous, or simply hunan views of the world, and those of “western science”, begins to dissolve. Thus the Cosmic Serpent project, designed to consider this relationship, begins to look much more promising than it otherwise might have.