Category Archives: Maxwell

The works and thought of James Clerk Maxwell

What is “Action”, that Nature Should be Mindful of It?

Newton/Maxwell/Marx: Spirit, Freedom and Scientific Vision

We have been tracing the course of the book, NEWTON/MAXWELL/MARX by way of a dialectical tour of three worlds of thought. We have seen Maxwell replace Newton’s “Laws of Motion” with the Principle of Least Action as the foundation of the natural world. Here, we seek the meaning of this curious phrase, Least Action.

Let’s grant that Maxwell – along with perhaps most of the mathematical physicists of our own time – is right in supposing that the Principle of Least Action governs all the motions of he physical world. How can we make sense of this truth? What is Action, and why is essential that it be Least?

First, we must begin by recognizing nature is not inert, but in some sense purposeful: every motion in the natural world (and that includes practically everything we can point to, once we take our hands off the controls!) will begin with a goal (Greek TELOS). Think, for example, of that complex process by which an acorn develops into a flourishing oak. This Motion will unfold in such a way that its goal will be achieved in the most efficient way possible. Sound like good economics? We’re asked to see every natural motion as directed to some goal, and as unfolding in such a way that waste or loss en route be the least possible under the given circumstances.

This principle can be expressed elegantly in mathematical terms, rather esoteric and belonging to the hushed domain of mathematical physics. But since it is actually in play everywhere around us, in actions going on at all times, it’s time we reclaimed it and demanded to know what it means. Let’s make a serious effort here to understand the implications that the physicists – Maxwell chief among them – have been saying.

For Maxwell, the true paradigm of physics is the laboratory of Michael Faraday, working immediately with phenomena and tuned always to hear, without complication of intervening symbols, the authentic voice of nature. The Principle of Least Action is about the world we live in.

However we may distort and engineer it, it is always nature, ever-active, with which we begin, and our projects end. We may think we begin with a tabula rasa and design with total mastery to purposes of our own, but every blade of grass, infinitely quantum-mechanical-wise, will laugh at us. It is in the fields and the mountains, the atmosphere and the oceans, and the endlessly-complex workings of our own bodies, that Nature’s economics is inexorably unfolding. High time, that we take notice of it!

We begin always with some process – the fall of a stone, from cliff’s edge to the beach below or the slow unfolding of an acorn into a flourishing oak. The principle applies in every case. Further, nature thinks always in terms of the whole process as primary: the economic outcome cannot be conceived as the summation of disparate parts, however successful each might seem in its own terms.

The unifying principle throughout any motion is always its TELOS, and it is this which in turn entails an organic view of the motion as one undivided whole process. Each phase of the motion is what it is, and does what it does, precisely as it contributes to the success of the whole. If this seems a sort of dreamland, far from practical reality, we must remind ourselves that we are merely rephrasing a strict account of what Nature always does! Things go massively awry (the seeding gets stepped on by the mailman) but these events are external constraints upon the motion: under these constraints, the Principle holds, strictly. Ask any oak tree, blade of grass, or aspen grove. Each has endured much in the course of its motion, yet each has contributed, to the extent possible, to the success of the ecology of which it is a part.

Economic achievement of the goal, we might say, is Nature’s overall fame of mind. Within this frame, exactly what is the economic principle at work? Everything moves in Nature in such a way that Action over the Motion will be least.

So, what is action? Action is the difference, over the whole motion, between two forms of energy: kinetic and potential Nature wants that difference to be minimal: that is, over the whole motion, the least potential energy possible to be expended, en route, as kinetic – i.e., as energy of motion. (One old saying is that Nature takes the easy way.) Or we might suggest: nature enters into motion gracefully.

Think of the falling stone: the stone at the edge of a high cliff has a certain potential energy with respect to the beach below. That potential is ready to be released – converted into kinetic energy, energy of motion. Thus the TELOS is given: to arrive at the beach below, with that high velocity equivalent to the total potential with which the fall began.

Our principle addresses the otherwise open question, how exactly to move en route? There is just one exact answer: the rule of uniform acceleration – steady acquisition of speed. Galileo discovered the rule; Newton thought he knew the reason for the rule. But Maxwell recognized that Newton was wrong, and we need now to get beyond this old way of thinking.

The real reason for the slow, steady acceleration is that the final motion, which is the TELOS, be acquired as late in the motion as possible, and thus that total-kinetic-energy-over-time be least.

Our principle may turn out to be of more intense interest to biologists than to physicists, as the ”kinetic energy” in this case becomes life itself. The seed bespeaks life in potentia. The ensuing show, steady conversion of potential—its gradual conversion to living form as the seedling matures – is the growth of the seeding, the biological counterpart of the metered, graceful fall of the stone.

Our principle governs the whole process of conversion: the measured investment of potential into kinetic form defines the course of maturation. Nature is frugal in that investment: the net transfer of energy-over-time is minimal; transfer in early stages of growth is avoided. Growth, like the fall of the stone, is measured, and graceful. Growth is organic in the sense that every part of the plant, at every stage of the way, is gauged by its contribution to the economic growth of the whole plant.

As it stands, our analogy to the falling stone may be misleading. It is not, of course, the case that the seed holds in itself (like loaded gun!) the potential energy of the oak; the case is far more interesting. The acorn holds in its genome the program for drawing energy from the environment in a way which will assure Least Action over the whole growth process. Once again, frugality reigns, since that energy not drawn-upon by the seedling will be available to other components of the ecology. Since the solar energy is finite, whatever is not used by one is available to the others.

We are ready now to ask in larger terms, “What sense does it make, that Nature be thus frugal in expending potential energy – minimizing its “draw” upon potential in early stages of growth, though total conversion by the end of motion be its very TELOS?

The question is a difficult one, touching on the very concept of life itself. Here, however, is my tentative suggestion. Let us consider Earth’s biosphere as a newborn project, awaiting Nature’s design. Our Earth (like, no doubt, countless other “earths” in Nature’s cosmic domain) is favored with a certain flux of energy, in the form of light from our Sun: just enough, on balance, to sustain water in liquid form, one criterion, at least, for the possibility of life. With regard to Earth, then, Nature’s overall TELOS may reasonably be characterized as the fullest possible transformation of sunlight into life. Earth also offers a rich inventory of mineral resources, which Nature will utilize to the fullest, over time, in the achievement of this goal.

Might we not think of this immense process, still of course very much ongoing, in the terms we’ve used earlier – as one great motion, transforming as fully as possible the potential energy of sunlight, into the living, kinetic energy of life? (It might be objected that the flux of solar energy is kinetic, not potential. It is so in space, en route, but is made accessible as potential by that immense solar panel, the green leaf system of the world – which by its quantum magic captures photons, uses them to split water, and thus generate the electrochemical potential on which the motion of life runs.)

That said, we may apply the logic of Least Action to life on every scale: life’s TELOS is to encapsulate our allotted solar potential energy in living form, always by way of the most frugal path possible. What is saved by the Least Action of one life-motion, is grist for the mills of others – so that overall, the solar flux is utilized as fully as possible. “As fully as possible” at this stage: but the long, slow motion of evolution continues – always, no less governed by Least Action, towards a TELOS we cannot envision, yet of which we must be organically a part, today.

For an expansion of this concept, you can read an earlier lecture:
The Dialectal Laboratory: Towards a Re-thinking of the Natural Sciences

NEXT: Karl Marx and his place in Newton/Maxwell/Marx.

NEWTON / MAXWELL / MARX 3

Maxwell and the Treatise on Electricity and Magnetism:
A DIALECTICAL WORLD CRUISE II

AT SEA

Maxwell Between Two World-Views

Many of us may know what it means to feel “at sea”: without beacons to steer by, without terra firma on which to set our feet. A dialectical passage between two world-views is like that, and James Clerk Maxwell’s life-story might be read as the log-book of just such an expedition: a lifelong search for a clear and coherent view of the physical world. Maxwell’s voyage would almost precisely fill his lifetime, but it would in the end be rewarded by his recognition of one single principle, the principle of least action, which would be key to a virtually complete inversion of the Newtonian world order from which he was escaping.

 

BEGINNINGS

In a sense, Maxwell was born into a dialectically-divided family. His father was Scottish, and Maxwell spent his early, formative years at the family home of Glenlair in rural southwestern Scotland. His mother on the other hand was English, and though she died while Maxwell was very young, her family was to have a strong influence on his career. While the English spirit would lead him eventually to Cambridge and the epicenter of an aristocratic, Newtonian concept of both science and society, the Scottish channel would lead him to a democratic view of society, and with it an appreciation of experiment and the evidence of the senses, together with a profound mistrust of the mathematical abstractions Newtonian theory.

These two themes met in abrupt confrontation when he was dispatched to Edinburgh to enter a new academy, designed to prepare students for entrance to English universities. In an encounter which must been a rude awakening, he was beaten up by his new fellow-students for his rural attire and his country ways. He stood his ground and soon became a leading student, but the encounter must have thrown light on an issue which would abide throughout his life.

 

Edinburgh University

Maxwell was clearly ready for entrance to Cambridge, for which his interest in science and his skill in mathematics surely qualified him; but he delayed for a year at Edinburgh University, and then, against the advice of family and friends, persisted in continuing there for a second year. At Edinburgh, he was encountered with excitement a truly liberal education; he loved, as he affirmed later, his professor of natural philosophy, and he became confirmed in his skepticism by the metaphysics of Kant as taught by Sir William Hamilton. Maxwell was not so much following Kant as agreeing with him: he left Edinburgh with a lifelong disbelief in the inert particles and forces upon them, on which Newtonian science rested.

After these two years he went on to Cambridge, where his skill in mathematics earned him entrance to Trinity College–the college of Newton–and high standing in the rigors of the tripos examinations. But he had brought his Edinburgh education with him, utterly abandoning Newton’s world, as we shall see when we turn to the first of his scientific papers.

 

Three Papers on Electricity and Magnetism

At this point, we find Maxwell, having obtained a fellowship at Cambridge, fully embarked on his voyage on the open seas. He is fascinated especially by the phenomena of electricity and magnetism, but he has no interest in joining the scientific community of his time elaborating Newtonian “laws of force” acting on electric “charges” or magnetic “poles”. He has met Michael Faraday, the self-proclaimed unmathematical philosopher, who has been dong brilliant experiments at the Royal Institution in London. Maxwell has, I think we can say, begun a lifelong devotion to this unassuming character, who represents the very opposite of the Cambridge/Newtonian concept of science–and almost defiantly takes up Faraday’s cause as his own. These are open seas: how to proceed?

 

Paper 1: An Analogy

Maxwell turns, not to theory, but to analogy. He shares common ground with Faraday regarding an interest in visual thinking–Faraday has presented his insights into the magnetic field by way of patterns formed by iron filings. Maxwell perceives these as the very lines of flow of a fluid. Here, then, is a gift from Faraday, a visual scientific language Maxwell can use! So is conceived the first of three papers on electromagnetism: On Faraday’s Lines of Force. This instrument of analogy, and with it, the goal of writing for the common man (for the democratic intellect, as one student of Scottish thought has put it) is to become one of the sure signs of the new world-view, towards which Maxwell has already begun steering.

 

Paper 2: A Physical Theory Maxwell has a great propensity for wit, though his friends remark on the difficulty of catching his intent. It may be a shield for a person who is living between worlds: not fully a member of the world friends suppose him to share with them. Maxwell is now in the position of having arrived at a whole view of the interrelations among the electric and magnetic phenomena: yet having no structure of theory in which to compose such a vision. I have proposed that his recourse is that of Aristophanes, in Peace, or the Birds–to stage his vision in the mode of comedy. Maxwell has been playing with the subject of mechanism (he and Karl Marx happen to have taken a course from the same teacher in London, though perhaps not at the same time!). He cannot mean that he proposes that such a mechanism actually exists, but he invents a great machine, for which he writes all the appropriate equations, which would do all the things the electromagnetic field actually does. Maxwell calls it A Physical Theory of the Electromagnetic Field, but he is not proposing that these vortices and idler-wheels actually exist. Like Aristophanes’ world of peace, it is an object for the mind, a project of pure thought. Again, this is a major step toward the goal Maxwell is seeking: in the new world, we will not take mechanisms seriously.

Maxwell’s jeu d’esprit is so successful that he can calculate from it the speed at which vibrations would be transmitted: it is very close to the speed of light! He is the point now of announcing the electromagnetic theory of light. But his discovery hangs in the air (or floats on the waves) it is no more than a beautiful play of thought.

 

Paper 3:Dynamical Theory

At last, the gods smile on Maxwell’s endeavor. He meets dynamical theory–and a new world begins to take shape. The mode of this encounter is deeply ironic, and correspondingly confusing. One of the most obdurate and imperious of Newtonian advocates is Lord Kelvin, once more modestly Maxwell’s colleague, William Thomson. He, with Maxwell’s close scientific friend P. G. Tait, have undertaken to write an ambitious, one might say proud, Treatise on Natural Philosophy. It’s intended to lay, once for all, the secure foundations of Newtonian science. An edifice of all physical science is to be built on this solid foundation, of which they’ve published only Volume 1.

At sea in uncharted waters, very strange things can happen! Kelvin and Tait, building their arguments on solid Newtonian foundations, expound a new approach to physical problems in terms of energy, rather than force: it is termed dynamical theory (they are importing it to England from the Continent, where it has been developed.) Though Kelvin resolutely insists that it is really still Newtonian, and nothing new, Maxwell sees light at the end of his tunnel (or a beacon on a new continent!) If equations can be written in terms of energy rather than force, nothing further needs to be said about forces acting upon those underlying particles, which he has always been convinced, do not exist!

 

The Treatise

The new dynamical equations are named after Pierre Lagrange, who wrote them, and Maxwell now uses them to characterize the electric and magnetic fields as regions on energy and momentum. Lagrange makes no explicit reference to ponderable mass, but that no longer matters–the equations carry all the energy that reaches Earth from the Sun. Maxwell publishes his Dynamical Theory of the Electromagnetic Field, and confidently announces his electromagnetic theory of light, based on the new equations.

Maxwell’s problem is not yet solved. Either the equations stand, as Kelvin maintains, on Newtonian theory – in which case we have only avoided the issue by not referring to some underlying particles, hardly more than a subterfuge, certainly not worthy of Maxwell, or they flow from some higher principle which Maxwell has not yet named. This is perhaps the darkest night of his voyage: he has glimpsed the new shore, but it has slipped away in the obscurity of this night.

 

The Principle of Least Action

Blessedly, Lagrange’s dynamical equations of motion can be derived from another source: indeed, this new source is their natural home, for this new origin is itself expressed in dynamical terms, i.e., in terms of the potential and kinetic energies of the system as a whole. Causality of the whole natural world is at stake here, so this “derivation” of Lagrange’s equations is no mere mathematical question! For Newton, causality flows from below to the whole: the “reason” things happen is mechanical, the whole moves as a consequence of the motions of its parts. So it was with Maxwell’s joking physical theory; he knows very well there are no such underlying parts. The new derivation of Lagrange’s equations flows from above–and with it, causality likewise flows downward, from some inclusive whole.

That inclusive whole–from which all the motions of he natural world flow–is the Principle of Least Action. The motions of the natural world arise ultimately from potential energies, such as the calories in a loaf of bread, or the BTUs in a gallon of gasoline. The conventional symbol for potential energy is V. Motions arise as potential energy is converted to kinetic energy, whose symbol is T. The difference (T–V) is called the Lagrangian, and the action (A) associated with any motion is nothing more complicated than the product of the Lagrangian and the time (t) the motion takes:

 A = (T – V) x t

 With that modest introduction, we can now state the principle on which it seems, nature runs. For any system:

The motion will be such that the action is least.

It can get complicated when systems are complex, or when relativity or quantum principles are involved, but it works, too, for systems as simple as a falling stone. Since each system is characterized first of all as a whole, it is inherently organic, and applies especially well to ecologies, which nature appears to see primarily as wholes, and organic.

Maxwell learned of this from the writings of William Rowan Hamilton of Dublin; he jokes of his “two Hamilton’s, saying their metaphisics are valuable in proportion to their physics. He means, I think, that the Kantian metaphysic espoused by Sir William Hamilton of Edinburgh was geared to the Newtonian world-view. The “new” Hamilton of Dublin is geared to a new, very different world-view in which the whole is primary as such, and not an assemblage of parts, and causality flows organically from whole to part. Wholes of course do not have to be big, the quantized hydrogen atom, a protein molecule, or the living cell, are instances.

We spoke earlier of Maxwell’s devotion to Faraday. Now we must ask, has he brought Faraday with him to this new land of Least Action? The answer, I can say confidently, is Yes.

How do we characterize a “system”? In the old, Newtonian way in which the parts were causal, it was important to describe a system in terms of those parts which constituted it and caused it to move. But now, parts are no longer causal. Our concern will be, instead, to characterize the state of a whole connected system. Interestingly, there is no one right way to do that! Any set of measurements sufficient to characterize the state of the system will serve. They don’t have to be readings of meters; Faraday’s diagrams of lines of force will serve very well to characterize a magnetic field. His intuitive interpretations of the behavior of his galvanometers serve him better than columns of numbers. Further, Maxwell’s analogy to fluid flow may serve very well to comprehend the structure of the magnetic field. Indeed, the Principle of Least Action in effect restores life to nature, which tends to move, as Faraday observed of his magnets. We have indeed arrived at a whole new world, yet one which Faraday, and Maxwell in his devotion to Faraday, already had in view.

Why has the modern world so resisted recognition of this principle, leaving it to rather esoteric studies within mathematical physics rather than teaching and embracing it generally as a far better way of understanding and caring for the natural world? Any thoughts on this will be very much appreciated.

NEWTON / MAXWELL / MARX 1

A Dialectical World Cruise – Part 1Cover Image of Thomas K. Simpson's new book: Newton, Maxwell, Marx

Good news! The Green Lion Press has now released in a single volume three of my earlier essays, collectively titled Newton/Maxwell/Marx. Many of their themes are familiar to readers of this website, but these essays are extensive, and gathered in this way, with new introductions and an overall conclusion, they reveal surprising relevance to one another. These essays speak to our troubled world today.

Does Marx, for example, have anything to do with Maxwell? Not on the surface—but at some deeper levels, which the book calls dialectical, each lifts us out of the Newtonian world in which we have lived since Newton wrote. Let us call this tour of three contrasting world-views, a dialectical world-cruise!

Edward Abbott once wrote of a realm called Flatland, whose citizens—confined to life in a table-top—had no idea how flat their world-view might be. They had never viewed themselves and their confinement from outside. Now, no less than they, we too need fresh perspectives and new insights, if we are to take the measure of own confinement and our net of unquestioned habits of thought. Newton/Maxwell/Marx navigates these unexplored waters, becoming a dialectical journey between worlds of thought, each based on its own fundamental premises concerning, as we shall see, even the nature of science itself. In turn, our concept of the nature of nature has ramifying consequences for our beliefs concerning society and human freedom.

In these essays, each port of call is represented by one of the great works of our western tradition—so these thoughts are in one sense, rather timeless, than new. But this is to be a spirited, not a scholarly investigation. We are no mere tourists, but earnest inquirers. Our purpose is not that of the objective scholar, to know about the works, but of the free mind, reading as if their authors addressed their words to us to us—as indeed, in some sense they surely did.

Reading in this mode is itself an art, and calls for skills which collectively have been known as the liberal arts, because these are the arts meant to set our minds fee. Not surprisingly, then, these three essays concern three books read at St. John’s College, in Annapolis and Santa Fe, whose curriculum is designed to capture the liberal arts in the modern world. Our essays ion emerged from this cauldron, and first appeared in the pages of the Great Ideas Today, once an annual al publication devoted to critical studies of the great books and their corollaries in our time. I express my indebtedness to John Van Doren, then executive editor, who guided these essays to their first appearance.

Our three ports of call will be, to give them their full and proper titles: Isaac Newton’s Principia Mathematica Philosophiae Naturalis (Mathematical Principles of Natural Philosophy – the philosophy of all the natural world—by no means that part we now call “physics”; James Clerk (inexplicably pronounced Clark) Maxwell’s Treatise on Electricity and Magnetism, and Karl Marx’s Capital. These works are in dialogue with one another—not literally, for the first two were far apart in time, and while Maxwell and Marx overlapped London for a time, and indeed shared an interest in lectures on mechanism, it would be hard to imagine they ever met! No: their dialogue is the more real for being conceptual—belonging to a world of ideas—and there, Newton/Maxwell/Marx will show, their ties are deep, and very real.

This set of essays, then, becomes a book for adventurous spirits, and in that sense may be a book whose time has come. People today are restless, questioning institutions which no longer make sense. Long-held assumptions are subjected to doubts reaching to the foundations of our societies and their economic systems. Even our sciences come into question, as in thrall to a limiting, encompassing world-view.

All this is of a piece with the dialectical sprit of our authors themselves: imperial in Newton’s case, gentle in Maxwell’s, boldly ironic in Marx’s – but in one style or another, each is a revolutionary, questioning the foundations of the world which surrounds them.

 A posting to follow soon will offer a brief synopsis of this Dialectical World Cruise.

Visit Newton /Maxwell / Marx 2

Maxwell’s Mathematical Rhetoric: Rethinking the “Treatise on Electricity and Magnetism”.

The Green Lion Press has just announced the publication of my study Maxwell’s Mathematical Rhetoric: Rethinking the “Treatise on Electricity and Magnetism”. Although this is by no means a new work, its implications for the most part still remain to be explored, and I am delighted to greet its appearance in this form.

Maxwell's Mathematical Rhetoric

What is meant by this curious phrase, mathematical rhetoric? To explain, it may be best to go back to the problem which first led me to undertake this project. Maxwell’s Treatise had been a candidate for the list of “great books of the western world” from the outset of the seminar program at St. John’s College in Annapolis – but it soon became apparent that no one could “crack’ this massive work. It introduced, indeed, Maxwell’s equations of the electromagnetic field, and with them, the recognition that light is an electromagnetic phenomenon. But these equations, and that theory, could much more quickly be reached by way of any modern textbook. What secrets might Maxwell’s work harbor, beyond the stark narrative those textbooks could offer? I set out to explore this question by reading the Treatise as a work of literature. By great good luck, I discovered that Maxwell had written with just just that intent: to compose a work of literature artfully shaped to convey a weave of interconnected messages. To this end, his primary instrument would be the art of rhetoric.

The basis of the art of rhetoric is the distinction between what is said, in a simple declarative sentence, and the way that thought is expressed.  A nuanced statement may convey meanings very different from the literal content of a sentence. Surprisingly, perhaps, the same is true of a mathematical equation. Its literal content is the numbers which it serves to compute; but its rhetorical content is the thoughts it suggests to the mind of the reader. Rhetoric is often used to win arguments, but Maxwell’s intention is very different. His aim is to suggest new ideas, and he shapes his equations to open our minds to new ways of viewing the natural world.

Maxwell’s Treatise has, in effect, two plots. Its first, overt role, is to provide a text in electricity and magnetism to support the addition of those subjects to the highly mathematical, severely demanding tripos examinations weeding out candidates for a degree at Cambridge. Maxwell however weaves into his work a much richer, more subtle plot, very nearly antithetical to the first. Throughout the book, this second plot increasingly shapes equations to give expression to the new and far more interesting ideas of Michael Faraday — who himself knew no mathematics whatever. Late in Part IV of the Treatise, a sharp turn of the narrative and the adoption of an altogether new rhetoric – a new form of the basic equations of physics — gives final victory to Faraday.  Thus when Maxwell’s field equations emerge in the Treatise, they belong to a breathtakingly new view of the natural world, while the conventions of the tripos exams have been left far behind.

That new rhetorical form, shaped to fit Faraday’s way of thinking as well as the idea of the space-filling field itself, is collectively known as Lagrange’s equations of motion. They speak not of forces, but of energies, and through them, explanation flows from a whole system to its parts, not from part to whole.

Whether we use Newton’s equations or Lagrange’s, the calculated results may be the same; but the contrasting form of the equations bespeaks a correspondingly transformed view of the natural world. Our very idea of causality is reversed. As we increasingly come to recognize the deep connectedness of the systems which surround us – from ecosystems to single cells, our own bodies and minds or a global economy – we desperately need the insight which Maxwell’s Treatise has so carefully crafted.

In that sense, perhaps, both Maxwell’s work and this study of its rhetorical trajectory are more timely today than ever before. We have already spoken on this website of Lagrange’s equations and their contrast to Newton’s, which I have called a truly dialectical alternative, and further studies of Maxwell’s rhetorical strategies, in direct reference to Maxwell’s Mathematical Rhetoric, are planned. “Stay tuned” — and as ever, comments are warmly encouraged.

An Ecosystem As A Configuration Space

In my most recent posting, I’ve been exploring a quite classic mathematical model of an ecosystem: the Salt Marsh ecosystem model developed at Sapelo Island and described in the fascinating 1981 volume, “The Ecology of a Salt Marsh”. For those of us who are devoted to grasping the “wholeness” of an ecosystem, the question arises whether matching such a system to a mathematical model helps in grasping this wholeness – or whether it may even detract. The concern would be that true unity is broken when a whole is described in terms of relationships among discrete parts: as if the “whole” were no more than a summation of parts – in Parmenides’ distinction, an ‘ALL” (TO PAN), exactly the wrong approach to a true “WHOLE” (TO HOLON).

An excellent guide in these matters is James Clerk Maxwell, who faced this question as he searched for equations that would characterize the electromagnetic field in its wholeness. As soon as he learned of them, he embraced Lagrange’s equations of motion, and as he formulated them, his equations derive from Lagrange’s equations, not from Newton’s. For Lagrange, the energy of the whole system is the primary quantity, while the motions of parts derive from it by way of a set of partial differential equations. Fundamentally, it is the whole which moves, the moving entity, while the motions of the parts are quite literally, derivate.

The components of such a system may be any set of measurable variables, independent of one another and sufficient in number to characterize the state of the system as a whole. Various sets of such variables may serve to characterize the same system, and each set is thought of as representing the whole and its motions by way of a configuration space. If we have such a space with the equations of its motion, we’ve caught the original system in its wholeness: not as a summation of the components we happen to measure, but in that overall function in which their relationships inhere.

Now, it seems to me that a mathematical model of an ecosystem, to the extent that it is successful, is exactly such a configuration space, capturing the wholeness of the ecosystem whose states and motions it mirrors. Specifically, the authors of the Sapelo Island Marsh Model were if effect working toward just this goal, though it may not have appeared to them in just these terms. All their research on this challenging project was directed toward discovering and measuring those connections, and the integrity of the resulting mathematical system was exactly their goal.

They had chosen to construct their model in terms of carbon sinks and flows; the measures of these quantities were sufficient to characterize the state of the system and its motions, and therefore constituted a carbon-configuration space of the marsh. A different set of measures might have been chosen, and would have constituted a second configuration space for the same system: for example, they might have constructed an energy-model, which have been equivalent and represented in other terms the same wholeness of the marsh. Carbon serves in essence as a representative of the underlying energy flows through the system.

I recognize that this discussion may raise more questions than it answers, and I would be delighted to receive responses which challenged this idea. But I think it sets us on a promising track in the search for the wholeness of an ecosystem – an effort, indeed, truly compatible with the wisdom of Parmenides!