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

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.

THE PRINCIPLE OF LEAST ACTION

First Principle of the Natural World

It’s with a certain sense of awe that I introduce a new page on this website, to be devoted to the Principle of Least Action. I’ve written about this principle, on this website (see the article here) and elsewhere, in various ways and contexts, but it appears now for the first time as the centerpiece around which writings on this theme will be gathered.

The principle itself, though simple, requires careful statement, before we do that, however, we must pause to take the measure of situation. Even as leading physicists and mathematicians have embraced Least Action as the single moving principle of the entire natural world, most others, even at the highest levels of education and professionalism in fields other than mathematics or the physical sciences, regard it as of no interest to themselves, or more likely, have never heard of it all.

The issue is acute: our old conceptions speak of mechanism, with even the most subtle of natural bodies composed ultimately of inert parts moved by impressed forces, according to equations knowable only to experts. What a different picture Least Action paints! Wholeness is, in truth primary, with causality flowing from whole to part, not from part to whole. Nature is everywhere self-moving, and throughout, life is real. In short, the era of Newton is behind us, and once again, nature lives! We cannot know yet, what the consequences of embracing this truth might be: but the time to begin exploring this question is surely now, before we have altogether destroyed this planet–the living system of which we are all organic parts, and on which our lives depend. It’s the mission of this web page to explore the concept of least action, and some of the many ways it may affect our institutions and or lives. Such a trajectory of thought, which I would call dialectical, has been the theme of the ongoing blog commentary on  Newton/Maxwwell/Marx. Between Newton and Least Action, we may be living in the acute stress-field of a dialectical advance of human understanding.

As a firm mathematical foundation for further discussions of Least Action, here is an elegant sequence of steps leading from Newton to Least Action, following closely the argument given by Cornelius Lanczos in his Variational Principles of Mechanics. I have distinguished seven steps in this argument, adding a few notes by way of commentary.

Note that this does not propose to prove the truth of Least Action (though some rugged Newtonians such as Kelvin might take it in this sense!). Rather it demonstrates the formal equivalence of Newton’s laws and the principle of least action. Lanczos points out that the argument is reversible: can be traversed in either direction. The two equations are formally equivalent, but speak of completely different worlds.

We should note that they are not of equal explanatory power. Least Action serves as foundation for General Relativity and Quantum Mechanics, areas in which Newton is powerless.

NEWTON / MAXWELL / MARX 1

A Dialectical World Cruise - Part 1

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.

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.

On the Concept of a Dialectical Divide

Commenting on my lecture, “The Dialectical Laboratory”, Tony Hardy has raised a number of interesting issues. They take us to one overriding question: “What do we mean b the term dialectical? A dialectical question, I believe, splits our world-view down the middle: virtually all of our perceptions, and our purposes as well, are placed at risk.  A dialectical question, therefore, is not one which can be resolved by negotiation among familiar options. We stand before a new and altogether different court of review. The principal model for this is the Socratic dialogue, which places a respondent’s life, and that life’s central goals under devastating review.  Worst, we might say – that review is not that of an external judge, but a hitherto unrecognized standard within. The orator Gorgias, acclaimed political expert of Athens of his day, is a prime target of Socrates’ dialectical art. His very life crumbles before our eyes as he recognizes that it has lacked one transforming concept, that of justice. He holds great powers, but he has used them to serve no good end.  This is a tragic fall, mirrored in the fall of Oedipus. The classic term for a life, or a world-view, based on false pretention is HYBRIS (pride). To live on the wrong side of a dialectical divide is an invitation to disaster.

Sight is the universal metaphor for this inner vision which can judge truth. Socrates images a dialectical emergence as the passage into sunlight, from the false lights and shadows of a cave, into the full light of the sun. Oedipus takes his own eyesight in rejection of the false vision which had guided his life.

In the spirit of the same metaphor, we rightly speak of the perspective we gain when as readers we witness a transformation of world-view, as fascinated readers of the Socratic dialogues, or terrified spectators of Oedipus’ downfall, in the theater.  We can weigh and discuss a dialectical world-change as if it were a mater for formal consideration, as I have done in a recent web posting on what I’ve characterized as the Lagrangian Divide – but we should not lose track of the stakes at issue. Dialectical issues cannot be resolved by reasonable adjustment or adjudication by a court located within either system.  From the point of view we all occupy as dedicated members of a present world system, exit from that system looks like apostasy, or a tragic fall.

Although the alternative we described as “Lagrangian” between organism and mechanism looks like a problem within natural philosophy, I believe our concept of natural philosophy sets the stage for our view of life and our social institutions.  Although it is the pride of our modern science to believe that its very success depends on its freedom from “metaphysics”, this illusion strongly suggests that of Oedipus or Gorgias. To elaborate this thought would be matter for a much longer discussion, but if I were to assume the mantle of Teiresius, the blind seer who counsels Oedipus, I think it might be enough to utter the keyword competition. The concept of lives, nations and a world, based on competition, strife and isolation, rather than (to put it simply) love and community, may well be killing our planet and leading our so-called “international community” to self-destruct.

It used to be fun to imagine a “visitor from Mars” taking a distant look at our human scene; he would regularly be thought to find us insane.  Ironists such as Swift, Aristophanes and Shakespeare have found ways to make that same judgment. If investing our lives in scenarios in flat contradiction to our own best sense of values is insanity, we can see how they might all be right.

There is a better way; we do have a choice – though in a thousand ways we forbid ourselves to think about it.

In one way or another, consideration of this option seems to be the ongoing concern of this website!