Category Archives: Lagrange

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.

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.

Organism vs. Mechanism: Science at the Lagrangian Divide

The Lagrangian equations are a powerful set of differential expressions describing the motion of a complex system.  With one equation for each component of the system, they would seem to offer a powerful expression of the relation of part to whole.

They are, however, seriously ambivalent: they can be read in either of two opposite ways. They present, then, a stark problem for the art of interpretation, the highest branch of rhetoric, as it comes from Augustine to Bacon and Newton.  The same statement becomes a watershed; it may belong to one world, or its opposite – but not both.  Each is a containing frame, within which we picture, and live, our lives

Read in one way – the way most common today – they are seen as derived from Newton’s laws of motion, and thus adding nothing fundamentally new. From this perspective, they merely rephrase Newton in terms of the concept of energy, a mathematical convenience in certain circumstances but making no fundamental change in our understanding of the natural world. In this interpretation, they express what we today call mechanism, which sees the motion of any system as the mere aggregation of the motions of its individual parts. Causality flows upward; motions of the parts explain the motion of the whole.

Seen from the other side of the Lagrangian watershed, however, the same equations express a world of a totally different sort. Here, the same equations are derived from the Principle of Least Action – a concept which readers may recognize as one of the recurring themes of this website.  The system itself as a whole, described in terms of potential and kinetic energy, becomes the primary reality and the source of the motions of the parts. Causality arises from the  interplay of these energies, and flows in the reverse direction, from whole to part.

Within the world of mechanism – the first interpretation – there is no place for goalor purpose. These are concepts considered far too vague to meet the standard of objectivity, the signature of modern science.

Remarkably, however, Least Action reconciles purpose with quantitative objectivity. By means of the mathematical technique of variation, which considers all possible paths, this principle seeks the optimum path by which potential energy may, over he whole course of any natural motion, be transformed to kinetic. In this interpretation of Lagrange, then, our world-view is transformed. Science itself, while remaining strictly objective and quantitative, becomes at the same time goal-oriented – all at once!

More than this, however, science on the Least Action side of the Lagrangian divide becomes, at last, fundamentally organic. This arises from a further, crucial feature of Least Action: if a system as a whole moves in such a way as to minimize action,so also will, within the bounds of external constraints, every part of that system. The goal which belongs primarily to the whole, is pervasive: it is shared by every part.

It was important in stating this principle to add “within given constraints”, because a rigid part of a man-made machine has few options. By contrast, the myriad components of a leaf, or of a cell or enzyme within the system of a leaf, navigate among unimaginable options toward the common goal of turning sunlight into life, over the season of the leaf, the life of the tree, or the evolution of photosynthesis on earth.

It is this community of purpose, nested and shared, which renders a system trulyorganic – a living being, something fundamentally beyond any bio-molecular mechanism, however intricate.

It is hardly necessary to add that it is this sense of nested purpose and shared membership in natural communities which has been so lacking during the long reign of mechanism. Our so strongly-held worldview has diverted us from that other option, which has nonetheless long formed a strong alternative flow of thought and practice in science, mathematics, politics and the arts. Now in many ways, not least the earth’s biosphere itself, the demand is upon us to recognize that we do have an option of immense importance. Viewing this whole scene now, we might say, from the Lagrangian ridge-line itself, with both worldviews clearly in view, our task is truly dialectical: leaving none of the insights of the past behind, we are in a position to move forward into a new, far richer and wiser world.

That new world-view, which has appeared here as a richer interpretation of Lagrange’s equations, is the ongoing theme of this website – always with an eye to Maxwell’s turn to Lagrange as mathematical vehicle for the launch of his concept of the electromagnetic field, paradigm, if ever there was one, of that whole system of which we have been speaking.

[A brief introcution to the Principle of Least Action is given in my lecture, “The Dialectical Laboratory” .

It is important to add that in this thumbnail sketch, many nuances of the application of Least Action have been left without mention]

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!