Category Archives: Principle of Least Action.

A holistic approach to mathematical physics stemming from Aristotle, passing through Leibniz and taking full form in modern variational mechanics.

THE FOURTH DIMENSIONAND THE PRINCIPLE OF LEAST ACTION: Why the Clock Says “Tick-Tock”

It has been suggested that two major themes of this website are now converging in such a way that each throws light on the other – namely, the Fourth Dimension, and the Principle of Least Action. The following essay aims to explain this connection. This may in turn throw special light on the discussion of Karl Marx, the final stop on our Dialectical World Tour – coming next!

WHAT IS “MOTION”?

Least Action – whatever our understanding of this challenging term – is certainly a way of characterizing natural motion. Natural systems, the principle asserts, move in such a way that their action shall be least. Before we concern ourselves with the mystic term action, we might do well to focus on the no less mysterious term, motion. Not only will this help us to understand the intent of the principle, I think it will lead us to see a deep link between the Principle of Least Action, and the Fourth Dimension.

What, then, is motion? We moderns tend to think of it in instrumental terms – as the means of getting from intention to accomplishment: in between planning to be something, and actually being that intended thing. In this view, motion is a sort of intermediate, between intent and accomplishment: something to be gotten over, usually as quickly as possible. “Time is money,” we say – meaning, a cost. This colorless idea time accords with the mathematical view of time as a line: our lives pass along this continuum from goal to goal, the moments between, mere means, to be passed through as expeditiously as possible. Time is money, we say, meaning cost. This uniform, colorless continuum Newton mastered by finding a way to measure motion at a point, as if the now were its dwelling-place. Taking this to the limit, he gave us the mathematics of conventional physical science: the differential calculus. Absolute, mathematical time, Newton was sure, flowed calmly through absolute space, in God’s divine sensorium.

As so often happens, the ancients had a different, more interesting point of view, suited to a richer and wiser concept of Nature itself. Calling Aristotle as our witness, we’re told that being does not lie merely at the two ends of a span of intervening motion. Rather, being inheres in the notion itself. There, in that very motion, we find ourselves in the fullness of our being. As we might expect, Aristotle has a word for this. energeia: being-at-work. And since the work is at every stage shaped to its end – its TELOS – we can call this richer, organic concept of living, EN-TELECHY.

Rest assured, this old view is by no means a threat to modern science. Leibniz, already in Newton’s time, was developing a more complex form of the calculus suitable to this richer view of motion. Instead of zeroing-in on the passing moment, it looks to the whole span of the motion, from inception to closure. It’s called the integral calculus. In its variational form, it weights every moment with respect to the goal, and hence meets Aristotle’s test of entelechy. Whether it’s a radiating atom or a busy mouse, every stage of its motion is inherently – by Nature – shaped to its goal. The motion, then, is truly whole.

Aristotle goes on to say a funny thing about time. Time, he seems to say, is the number of motion(s). To clarify, he illustrates by saying we count the number of times the horse goes around the track. More generally, in the order of being, first there is the race, and then, secondarily, we count the laps, and time arises. The whole, which is the motion, is primary; motion doesn’t happen in time. Being comes first; time is merely the count of the generations of being.

Come to think of it, that’s the way we encounter time in daily life. Our encounter with time is mediated by some motion: we count the clock which tells the time. That’s why the classic clock says, tick-tock. The swings of its balance are counted by ah escapement, going first tick, and then tock, to mark the completion of one cycle: one motion The time it tells is the count of its motions. More modern clocks, it is rue, speak in other voices, but they’re all counting motions of some sophisticated sort.

 

HOW THE FOURTH DIMENSION UNDERLIES LEAST ACTION

If we’re satisfied that a motion is essentially whole, we’re ready to turn to Least Action. All the natural world runs on the Principle of Least Action, so this is important to know about. Here is the Principle: Every natural motion, atom or mouse, unfolds in such a way that over the whole motion, total “action” will be least. Think of “action”, then, as activity-summed-over-the whole motion. Action thus refers to something Newton missed. Contra Newton, you can’t have action at a moment! More positively put, nature accomplishes the overall goal with the least possible fuss. There’s a “good reason” for that; fuss (haste) makes waste. Every activity entails heat-loss (that’s why our bodies run so hot). The horse will ultimately run at top speed, but getting-up-to-speed will be accomplished by nature as gradually as possible. In turn, once up to speed, the myriad processes throughout he body will themselves run, collectively (organically), in such a way hat the speeding horse will be expending as little energy over each cycle of the gait, as possible.

If we image a running horse by means of a three-dimensional snapshot, we’ll evidently miss Nature’s point. We need to see that motion whole: each whole cycle of the gait as one single image. Our three dimensions are not enough: we must add a fourth axis to our image. In addition to our three spatial axes, we need a time axis as well. The resulting image will then encompass in a single geometrical figure the motion as that whole which, by its very nature, it is.

Though such four-dimensional imaging can indeed present this wholeness effectively to our physical eye, a larger aim must remain: through this visual experience, to extend this same insight to the eye of the mind. We might then perceive all natural motion in this four-dimensional way – and thus, in turn, achieve a larger grasp of the wholeness of that motion of ultimate interest, life itself.

Mathematical physics has widely accepted the Principle of Least Action as its basis. Taking the term physics in its old, true meaning, as the science of all nature – the fall of a leaf, or the beat of a heart – it’s nice to know that the more modern the science, the more it attests to wholeness, and to the richness of every moment: not as transient as it may seem.

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.

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 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 2

NEWTON and His “PRINCIPIA”  
A DIALECTICAL WORLD CRUISE I

This is the continuation of a discussion of “Newton/Maxwell/Marx”, a new work of mine, from Green Lion Press. This overview has been envisioned as a “dialectical cruise”, visiting in succession the world-views of Newton, Maxwell and Marx. Here we visit the first of these “worlds”, that of Isaac Newton. Read part 1 here.

FIRST PORT OF CALL:
NEWTON and his “PRINCIPIA”

 

A FEW QUESTIONS ON ARRIVAL

The work known familiarly as Newton’s Principia is the foundation stone upon which our concept of science has been erected. Despite all the transformations by way of quantum physics and relativity, this bedrock image of objective, scientific truth remains firm. Arriving now, however, as if from outside our own world, we may feel a new sense of wonder, and presume to ask a few impertinent questions about core beliefs normally taken for granted:

 

Why, in our system of modern western science, do we suppose that the natural world is composed throughout of inert masses, with no inner impulse to move? Why are we convinced that nature is thus ruled by external forces, and that truth lies in finding mathematical laws of force?

 

In short, why do we suppose that nature is purely quantitative and, despite all appearances, deep down, essentially mechanical? Is the life we see everywhere infusing the natural world merely an illusion? Who killed nature?

 

These are dialectical questions, meaning that they go straight to the first principles of our systems of belief. Such principles normally go unquestioned, but challenging them is exactly our business here, on this dialectical world cruise! They all lead back to a fresh reading of Newton’s Principia. And as we shall be seeing in the course of this cruise, they do have interesting answers.

 

WHAT NEWTON WROTE: THE “PRINCIPIA”

What Newton actually wrote, and what the world has on the whole supposed him to have written, are two very different things, as we shall see. Let us begin, however, by taking Newton at his own word, with a thumbnail sketch of his Principia Mathematica Philosophiae Naturalis (“Mathematical Principles of Natural Philosophy”). In relation to this title itself, we might point out that Newton’s topic is by no means limited to the discipline we now call physics. Newton is prescribing for the entire natural world – the universe of objects, living or non-living, that meet our senses in the directions of the large or the small, by means of any instruments, however advanced, in any domain which assumes the role of science. The Principia is discussed in detail in the essay on Newton in Newton/Maxwell/Marx; here we give only a thumbnail sketch.

Newton builds his Principia in a geometrical mode with a clarity reminiscent of Euclid’s Elements. Like Euclid, he lays a secure foundation, now of definitions and laws of motion, from which propositions flow with the same intuitive conviction we feel as we follow Euclid’s Elements. A world is unfolding before our eyes; if the foundations are secure. The edifice must stand.

Newton thus builds an edifice of science as firm as Euclid’s, though crucially now this consists of nothing but inert masses, deflected from rest or straight lines only under the action of external forces. Bodies move according to strict, mathematical laws of motion, and the forces are defined by equally precise mathematical relations. All this unfolds in a structure of true and mathematical time and equally absolute, mathematical space. Within the Principia, Newton develops the range of all possible motions under central forces, and applies these results to describe with precision, as merely one possible case, the system of planetary motions about our sun. This beautiful result emerges as just one example of his universal method at work; he will go on, for example, in his Optics to provide an equally mathematical system of color and the visual spectrum. Where Euclid gave us the precise forms of the things of our world, Newton gives us the things themselves, though they enter strictly as quantities. Apart from inert matter whose measure is mass, there is nothing behind these mathematical forms.

This reduction to stark mathematics might well strike a modern reader as the very spirit of mathematical physics today, an account we might call mechanical. At this point, however, an important distinction arises. In fact, Newton writes in fierce opposition to mechanism.

Newton is responding to Rene Descartes, who had indeed described the world as mechanical – a plenum, each part acting on its neighbors by simple rules of contact. Once set going, the cosmos runs on its own, like a fine watch. God’s role at the Creation was as watchmaker, but since that moment, the cosmos has run, and will forever continue to run, on its own.

This exclusion of God from His cosmos is anathema to Newton, and motivates the Principia. Where Descartes had filled the heavens with ethereal mechanism, Newton sweeps the cosmos clean. And where Descartes had seen nature moving entirely on its own, Newton very deliberately cancels any such powers, leaving nature utterly inert, everywhere dependent entirely on the ongoing operations of God’s active law. Hence the introduction of law at the foundation of the Principia. The orderly motions of the planetary system, which Newton calls The System of the World, is for him a vivid testament to the wisdom and active power of God. To bring this vision to mankind is, he says in the Principia, the reason he wrote. Might we not add, it’s the reason the concept of law structures our scientific discourse today?

We see now, indeed, the answer to our question, “Who killed nature?” It was Newton! And we see, too, why he did it Newton made sure that nature would be strictly powerless, and thus fit subject for God’s continuing rule. Nature must be mathematical to admit the precision of divine rule. Force is the modality of divine command, and law enters physics as the voice of God, who speaks in the medium of mathematics. Scientists today who, in their opposition to “creationism”, may cite Newton as the founder of modern science, freed from religion, are assuredly calling the wrong witness!

 

THE “PRINCIPIA” WITHOUT GOD

Newton, then, intended his Principia as a testimonial to God’s active presence in His Creation. He thus writes as a theologian, but by the strangest of fates, has been read as a mechanician! How this happened is indeed a fascinating story, recounted in my Newton essay, but need not detain us long at this point.

Briefly, it turns out that Newton was dedicated experimenter and theorist in the realms of alchemy, and devoted much effort to detailed interpretation of scripture. It seems clear that for him the Principia itself was but one component of a far larger project. It appears that all this was regarded as an embarrassment by his executors, who took pains to sequester it from public view. In turn his denuded Principia was welcomed by a society more interested in science than in theology. A strictly mathematical world picture. Only in recent years have manuscripts been recovered, revealing the role of the Principia in a much larger, and very different, project.

Believing however that it was loyal to its mentor, the west has accepted embraced the structure of the Principia, with its assumption of nature as in itself inert, moved by forces defined by law, as if Newton had intended such a vision as the very truth of the natural world. We have conjured a Principia divested of God, a feat comparable perhaps to reading the Old Testament without mention of the Lord. We have an empty shell, a narrative with no plot, law with no lawgiver. The appearance of life, but assuredly, no role for life itself.

No one could doubt that modern science works; its success in its own terms speaks for itself, though the direction of its interests and the delimitation of its scope leaves room for important questions. Now that our dialectical inquiry has probed the foundations of our notion of modern science, which turn out to be curiously accidental, we are in a good position to ask, reasonably, whether some alternative, a different foundation for modern science, might be possible. As we shall see at our next port of call, visiting James Clerk Maxwell, the answer will be a resounding “Yes!” And nature will indeed spring to life once again, before our very eyes.

 

NATURAL PHILOSOPHY AS BEDROCK OF SOCIETY

Newton had fused natural philosophy and theology into one, truly apocalyptic vision. With that union dissolved, religion has been left to go its own way, with natural philosophy as the stark bedrock of our daily lives, our social and political associations, even our concept of freedom. We see ourselves as by nature separate and individual, while liberty becomes no more than the absence of restraint. At all levels, our associations are deliberate, held together by law in the form of agreements, to which we willingly bind ourselves for rational expectations of ultimate gain.

Our practical relationships thus rest ultimately on this understanding of the nature of nature – like Newton’s planets, we are separate bodies constrained by law, following trajectories in time and space. We group by aggregation; we are not social by nature.

In this world in which community is essentially an option, reasonable people can be heard to speculate that the brutality of war is part of our human nature. Despite all evidence, we find no place for life in the natural world: what appears as life we must accept, in scientific reality, as an artifact of complex mathematics – nothing real.

Religious convictions of course are another matter, not founded in nature but independently, in direct relation to the divine. The result, perhaps understandably, is that religious differences divide us even more fiercely than our perpetual struggle for the resources of the earth.

Surely there must be a better way – a more promising understanding of nature and natural philosophy. And indeed there is, as we shall see in our next port of call. Stay tuned!

Visit  NEWTON / MAXWELL / MARX  1

This has been the first in a series of three ports of call in a Dialectical World Cruise. The second, to James Clerk Maxwell and his “Treatise on Electricity and Magnetism”, will appear in this space soon. Stay tuned – and meanwhile, your comments will be most welcome.

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]

What Do we Mean by the Term “Elementary”?

What do we mean when we use the term ,”elementary”, in relation to a science? Does it refer to an easy introduction, as contrasted with an “advanced” treatment of the same subject? Or does it mean a solid account of the very foundations of the science? Or, for that matter, are these the same thing?

Maxwell had a tendency toward writing “elementary” texts: he wrote one on heat, and another on mechanics, both for use in classes for workingmen – a project to which he was deeply committed. Finally, at the time of his death he was at work on his “Elementary Treatise on Electricity and Magnetism, intended to serve as the Cambridge text to support a new degree in experimental natural philosophy at Cambridge University.

My sense is that Maxwell endowed each of these with earnest attention – that he regarded the “elements” not as evident, but as a topic to be approached with great care. Our decision as to what is elementary in a science has a great deal to do with our sense of the form the finished product will take – so that the most difficult issues may focus on the most elementary beginnings.

For example, Maxwell wrote his workingmen’s text in mechanics, Matter and Motion, only after he had hit on the fundamental idea, new to him, of Lagrnagian mechanics and generalized corrdinates. This would not be a mechanics in Newtonian form, in which the elements would be assumed to be hard bodies acting upon one another according to laws; rather, elements of this sort would be the least known components of the system, represented by generalized coordinates.

In this view, what we observe initially is a whole system of some sort; it is this whole which is fundamental, and truly elementary. The parts which compose it, we may never know. Our science can be complete and secure even if that question remains unresolved, or unresolvable.

This is the point of view I believe Maxwell had come to, underlying his approach to the new program at Cambridge as well. If so, must it not represent a truly revolutionary inversion of our very concept of scientific knowledge?

It fitted the primacy he – following the path of Farday – was giving to the concept of the electromagnetic field. In this view, he field would not be a secondary phenomenon, a composite or consequence of simpler “elements”, but itself both simple and whole.

If the elementary is what is primary, then in the case of the field it is the whole which is the element, from which we deduce what we can, concerning lesser components. Faraday had felt strongly that in the case of electricity, there was no “charge” lying on the surface of a charged body, but what we call a “charge” was a field, which filled the room.

Isn’t it the case that when we ask for the “explanation” of a physical system, we are asking for an account in terms of its elements? If so, then the field is itself explanatory, and we would not seek explanation in terms of the actions of some lesser parts. What will be the consequences if we extend this view to physical explanation – or explanation beyond the realm of physics — more generally?

“The Dialectical Laboratory”: A lecture on behalf of holism in the sciences

 

My lecture, the “Dialectical Laboratory ” (see the “lectures” section of this website) , was given as a sort of parting statement to the St. John’s College community in Santa Fe.  But though directed to the college, and expressed by way of references to certain of the “great books” of that tradition, its message is of far broader import.  The “dialectical” issue – meaning, a watershed of western thought – is between a science based on mechanical actions between disparate parts, and a holistic science in which wholeness is respected, and whole systems are regarded as fundamental, not as mere aggregations of parts.  

Each of these two very different scientific approaches has its rigorous theory, and either can be used to solve engineering problems.  But conceptually they are worlds apart, and I am convinced it’s crucial that we follow the way of holism, and learn, before it’s too late, to appreciate and work with systems – from the least living organism to the global environment – which are more than the sum of mechanical parts.  Science is moving in this direction, but there is now no time to lose! 

Comments on these remarks, as well as on the lecture itself, will be welcome in reponse to this posting. 

 

 

 

In Praise of Generalized Coordinates

I’ve been expressing my enthusiasm for a holistic approach to the understanding of nature — in relation to my favorite topic, the electromagnetic field, this takes the form of the Lagrangian equations for the field as a single, connected system characterized by its energy, not by forces.  It was crucial to Maxwell’s development of the equations of the field in his “Treatise on Electricity and Magnetism” that they be formulated as instances of such a connected system — i.e., in Lagrangian terms, and NOT on the basis of Newton’s laws of motion.  (The difference — very fundamental to our understanding of nature — is developed in “The Dialectical Laboratory”, in my “Lectures” menu.) Now, the question arises: “If we start in this way, from the ‘top down’, how do we ever arrive at the elements?”   The answer is, “We DON’T!” We move logically “downward” by finding the dependence of the energy of the whole system upon ANY set of measurements we want to make — provided only that it’s a complete (i.e. sufficient to determine the state of the system), with each measure “independent” of the others. We find such a set of measurements by doing experiments — and when we get them, they are called “generalized coordinates”.  The important thing is that there may be many ways we can define them, each set as good as the others: and in the whole process we never get any”real,underlying elements” — we don’t need them!  Reality is founded at the top, not the bottom, of the chain of explanation.   This is Maxwell’s new view of physical reality, founded upon the field.  It is the opposite of the notion of “mechanical explanation”, and it is the direction which our approach to nature desperately needs to take as we approach the challenges which lie before us today.  In terms of the philosophy of science, Maxwell it seems was far ahead of his time.  I propose to call this the “Maxwellian Revolution”.