Category Archives: Cosmic Serpent Project

A project to relate indigenous views of nature with those of modern science.

“Faraday’s Mathematics”

“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.

 

 

 

 

 

“Reason”, Old and New

Somewhere in the course of our western history, something fundamental has been lost: we have lost track of the wholeness of the psyche, and its membership; in a world which was whole and in which it might feel at home.   

Where did this happen?  The psyche was whole in Athens – its membership in the family, the polis and the cosmos were so presupposed that there were perhaps no words to express the separation and fragmentation so vivid to us today.   I don’t think there was a word for “objective” or “subjective”, nor was there a mind which might be thought of as a blank tablet, upon which an outside world might write. In society there was work, but no word for “job”, with the radical alienation that term implies.  I’m not suggesting life was in any sense idyllic – only that for better or for worse, the psyche was intact, and seated in the world. 

I’ll leave it to others to explain how this has come about, but somehow we now find ourselves equipped with a mind which is well-furnished with knowledge, indeed, but all too easily likened to a calculative engine, with a memory bank stored with data from an “outside” world.  We understand the mind better and better – but only as a marvelously equipped machine.  

What is missing would seem to be that faculty once called “intellectual intuition” – the power to see directly and immediately, without the intervention of words, truths which are timeless and fundamental.  That old intellect — for which the Greeks did have a word: NOUS — was inherentlyi drawn to beauty, which it deeply loved. 

I don’t see this as an exercise in nostalgia: there are ways open to us today by which we can recover this power, which is perhaps rather hidden than lost.  Other cultures have preserved it in ways we haven’t, and we have much to learn from them.  To a large extent it is our conception of “modern science” which denies the evidence of intuitive reason, and reduces the concept of “reason” to accurate symbolic calculation.  But there is another way within modern science, equally mathematical and rigorous, but founded in a concept of wholeness, and looking to the whole rather than the parts as the ground of “explanation”.  I have spoken about this way – the “Pinciple of Least Action” — in my lecture, “The Dialectical Laboratory”, elsewhere on this website.   

 

Nothing prevents, I believe, our mending this split between that classic concept of intuitive reason, seated in the world and knowing and loving truth directly — and the concept current today of reason as a calculative engine making what it can of an “outside” world.   We need only retrace our steps and pick up that thread of truth wherever we dropped it.  Not easy to do, of course, but worth every effort!

 

Any suggestions as to how to begin? 

 

        

 

 

 

Away!

I will be away from this site for some time, during which I’ll be attending the “Cosmic Serpent” project I’ve mentioned in my earlier blogs.

On my return, I’ll definitely have a report to post on the outcomes of that gathering-see you then!

Why Aristotle? Why Now?

 Here’s a brief posting, not unrelated to the previous two. I spoke in the first of a “tap root” running back from what we think of as “modern science” to sources in an ancient past. Such a tap root is not just a connection to the past, something of interest to academic historians, but potentially a powerful source of nourishment today.  This may seem a strange claim to make for Aristotle in relation to modern science, but I do put it forward in earnest. Aristotle generally gets a bad rap from those who tell the story of modern science, but to a large extent it’s latter-day Aristotelians (such as Galileo’s Simplicio), not Aristotle himself, who are the targets of such criticism.  It is well-known, and widely acknowledged, that Aristotle was a serious empiricist, conducting dissections and drawing such generalizations as he could perceive. But what was his account of scientific method, that we might give it serious attention today?  I’m writing this from memory, so my references for the moment must be inexact.  But in crude summary, here is the account which culminates in his “Posterior Analytics”.  

 

He has said elsewhere that the objects of true knowledge which Plato calls the “forms” are “nowhere”, not in the sense that they do not exist, but that they do not exist in separation.  The forms are everywhere in the observable world. We meet them when mind grasps something as whole and true. He says somewhere that scientific inquiry, as we gather data, is like an army in retreat: first one soldier takes a stand, then another, then more – and soon, the whole column stands fast. That standing fast is the mind grasping something true: “seeing something” whole, as we say, or achieving an intellectual intuition.  Such an intuition is not the additive sum of the component data. Between such an empirical summation (which Plato calls the “all”), and the grasp of a truth, (a “whole”), lies the difference between data-processing and great science. 

 

 We are so concerned today to emphasize the “objectivity” of true science, that we fail to acknowledge the role of mind – a function which grasps something the data do not themselves present. In that sense, great science, serious science, cannot be reduced to objectivity.  It cannot fly in the face of the data, but it cannot be reduced to those data, either.  We live at a time when it is becoming increasingly urgent that we rise to the challenge of recognizing whole systems as such. An ecology is something more than the sum of any quantity of data.  In biology, this whole beyond the parts is termed an “organism”; perhaps Aristotle would be reminding us today that we are in danger of failing to recognize life itself when it lies before us in our laboratories or in the seas.   

Indigenous Views of Nature and the Deep Roots of Western Science

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.   

The Deep Roots of “Western Science”

I’m very excited to have been invited to participate in the Cosmic Serpent project, which will be exploring the relationships – likenesses and differences – between Indigenous views of Nature, and the world-view of “western science”.

My first thought about this is that what we are accustomed to calling “western science” is not one well-defined, monolithic structure, but rather a growing and changing, organic body including strongly contrasting strands and a deep tap root which reaches far back in history to ancient Greece and beyond.

It is this richness and diversity of our present notion of “science”, together with its vigorous signs of growth, which make the Cosmic Serpent conversation something far more than a confrontation of two distinct ideas. Whether there’s the same degree of diversity and growth within Indigenous approaches to Nature is something I don’t pretend to know, but the coming conversation may reveal.

I feel impelled to say something more about that deep “tap root” of modern science, as it lies close to my heart and has been the subject of much that I’ve thought about and written. (I wrote about one aspect of it in the lecture “The Dialectical Laboratory”, elsewhere on this website.) For me, as we look backwards from our present stance toward a distant past, it is Leibniz who’s the key. Between Leibniz and Newton lay a split far more important than the question of prioty in laying the foundations of the calculus usually referred to. In ways not always recognized, Newton was looking to Christian scripture, especially the Old Testament, when he placed the notion of “law” at the foundation of his Principia. Leibniz, by contrast, was looking to Aristotle and saw intelligible principle – not “?law” – as the basis of our approach to Nature. Two of Leibniz’ crucial terms: energy – potential and actual – come straight from Aristotle’s Physics, and remain to this day beacons of an alternative path in physics. Not forces between particles, the dominant concept of the mechanical view of Nature – but motions of whole systems guided by principles rightly thought of as holistic – set this other course. It becomes formulated mathematically as the law of least action, which evolves in turn into the equations of Lagrange and Hamilton, and in general into the Variational approach to natural motions. It is an approach inherently compatible with the notion of TELOS, or goal. In a broader arena, it is at home, for example, with Gestalt theory in psychology and the theory – at once of art and science – which Goethe sets over against that of Newton in his Farbenlehre, the Theory of Color.

For the practicing physicist, the Newtonian and the Lagrangian methods may seem convenient alternatives to be called upon as occasion demands. But in truth they reach very deep into alternative conceptions of the natural world and its ways. As I explore in Figures of Thought, it was not for convenience but out of deep conviction that Maxwell chose the Lagrangian approach in his own development of the equations of the electromagnetic field in his Treatise on Electricity and Magnetism. That this is an issue for human thought in general, and not a problem whithin mathematical physics alone, is shown beautifully by the fact that Maxwell chose the Lagrangian method as the way to express within mathematics the insights of Michael Faraday, who knew, and wanted to know, no mathamatics.

I have to acknowledge that there’s a manifest contradiction in what I’ve just written: I spoke at the outset of one “tap root” of science, but this whole discussion has been of two: one Newton’s, and the other that of Leibniz. I’m convinced these two lead back, by way of Alexandria, to one lying still deeper – but that must be the subject of another “blog”!