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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.

NEWTON / MAXWELL / MARX 1

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

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

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

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

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

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

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

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

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

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

Visit Newton /Maxwell / Marx 2

From Diagram to Visual Experience: More on the Reality of the Hypercube Theater

We recently launched our Hypercube Theater, a stage derived from the hypercube (the tesseract) on which four-dimensional figures can make real appearances before our eyes. In that posting, a series of small photographs and diagrams served to illustrate our line of reasoning.

 

Now we have transformed these diagrams into screen-filling images, a development which illustrates an important point. Beyond the difference in size, this marks the advance from a mere diagram to a true visual experience.

 

We consult diagrams for the information they convey, but we approach a true illustration in search of a visual experience. It’s the familiar difference between looking, and seeing. When we look at a powerful photograph, what we see is, not a picture on paper, but a mountain range. It is as if we were there: we experience, we say, a sense of presence.

 

In three dimensions, this same distinction applies, becoming the difference, let us say, between a cardboard stage-set, and a ship in a storm at sea. Even in daily life, this same distinction arises. Our eyes present to us, we realize, only optical images of the room around us; but we experience, not such images of a room, but presence in the room itself.

 

It’s in this spirit that we approach the hypercube, not as a mere geometrical figure, but as a theater – a stage on which all the entities of a four-dimensional world can make their appearances, present to our eyes.

 

The new version of our Gateway Theater posting tests the ability of screen-filling images to support this transformation. Ideally, of course, the computer screen would – like the theater stage it is becoming – fill our field of view. Even on a laptop screen, however, our visual imagination will have the power to fill that gap! Follow the link below to experiment with a first step in entering the fourth dimension!

Enter the updated Gateway to the Fourth Dimension here.

Let us know if this works for you.

Gateway To The Fourth Dimension: The Hypecube As Theater

Some time ago, we launched a new department on this website to be devoted to studies of the fourth dimension. In our first entry, we met the basic figure of 4D geometry, the hypercube, or tesseract. We watched as it emerged from the familiar cube, extending into the fourth dimension.

 

Now we return to the completed hypercube, to see how it can be better understood, and how it can be put to use: first, to view; and then, even to create other four-dimensional figures of whatever sort.

 

We begin with this model:

A model for imagining the Fourth Dimension.

The Tesseract Theatre

 

We plan to mount these studies on the Fourth Dimension page of this website under the overall title Gateway to the Fourth Dimension. Part I, The Hypercube as Theater, appears today. Take a look, and let us know what you think!

Read this new article here.

 

These studies are all contributions  to the preparation of a book currently  in progress, “At Home n the Fourth Dimension.”
Art work for “At Home in the Fourth Dimension” by Anne Farrell.
Model and photography by Eric Simpson.
All rights reserved.


 

The Rhetoric and Poetic Of Euclid’s “Elements”

Part II: Euclid’s Poetic

1. The Tragic Narrative

revised 8August 2010

In an earlier posting, I drew attention to Euclid’s rhetoric, ending with a promise to follow this with a post on his poetic. This entry is meant to fulfill that promise – but first, it will be essential to explain the sense in which the term poetic is being used. I follow Aristotle, who in his Poetics has drama primarily n view, so we’ll be reading the Elements as high drama.

The soul of the drama, Aristotle believes, is the MYTHOS, the story, and indeed, Euclid is telling a remarkable tale. It is a trilogy, in the pattern of Aeschylus’ Oresteia, in which we pass from a path of early triumph to the darkness of despair – and only finally, in the third play, discover a way forward of a brilliantly new sort.

In the first play, Agamemnon, returning triumphant from the Trojan war, is murdered by his queen Clytemnestra. Next, in the dark logic of timeless vengeance, she must in turn be murdered by her own son, Orestes. This most heinous of crimes leaves Orestes in terrifying darkness, the hands of the avenging Furies, against whose iron grip enlightened reason appears to hold no power. Only in Athens, the city of Athena, will rescue become possible from this endless logic of retribution. Finally, in the third play, at the point of crisis, the Furies are perceived to waiver.  At a word from Athena, they turn: time holds its breath, and through the subtle wit and rich wisdom of the goddess, they prove at last, persuadable. They catch some dawning image of hope, of purpose, of a good beyond the blind, literal logic of their law. From agents of death they might become nurses of life, to be celebrated as agents of abundant harvests. With this opening, a new Athenian law-court is founded, in which reason directed to the good triumphs over the Furies’ old law of consequence and necessity. Performance of Aeschylus’ play was to become a civic ritual, in which the assembled city would be reminded, through the experience f terror and its release, the foundation in a higher, human reason, of the Athenian polis.

Can we imagine that Euclid’s drama reaches to such extremes of darkness and of light?  Almost startlingly, we find that it does. This is not, indeed, the customary way in which a work of mathematics is to be read. Nor is it usual to find a relationship, at this deepest level, among the political, the mathematical, and the poetic. Yet this, I propose, is the reach of Euclid’s poetic.

2. Tragic Crisis in the Elements

At the heart of Greek mathematics lies the problem of continuity, for which Aristotle uses the word SYNECHEIA, “holding together’. The Pythagorean Theorem – whose traditional name already suggests its mystic portent –makes its appearance as Euclid’s Proposition I.47, at the culmination of the first book of the Elements. There, it marks as well the beginning of a reign of innocence which will last through the first four books: we build a succession of figures and explore their relationships with no apparent reason to doubt that our foundation is secure.  Silently but inexorably, however, this innocent theorem will be leading us into confrontation, at the close of Book IV, with a mystery hidden in the hypotenuse of that equilateral right triangle – the diagonal, that is, of a square.  No longer the simple line it had appeared, it will be revealed as a yawning chasm, seat of the darkest of mysteries – the mystery of the continuum.  Since any straight line can become the diagonal of such a square, every straight line must harbor the same abyss.

The proposition itself demonstrates as we all know, that in a right triangle, the sum of the squares on the sides equals the square on the hypotenuse. We learn this today as school children, and take it   as familiar knowledge thereafter, but it brought Greek thought to a standstill, plunging mind into the darkness of mystery: the very cave of the Furies. How could that be?

The answer lies in the proposition to which this one leads – one however which Euclid takes care to bury in silence, and leave unspoken. It lies too close to the heart of mystery.

It is easy to prove, on the basis of the Pythagorean Theorem, that “If a number measures the side of a right triangle, no number exists which will measure the diagonal. To appreciate its force, we have to pause to ask, “What is number?”  To which the answer is: Every number is a multiple of the unit. Thus if we write for example 1.414, we refer to a number which is 1,414 “thousandths”, or the 1,414th multiple of .001, chosen as unit. Every number is in this way some multiple of some unit.

Very well: by means of number in this way the rational mind, LOGOS, can know, and precisely name, every length – can it not? In the darkness of Euclid’s unspoken proposition, the answer is “No!”. The diagonal of the right triangle clearly has a length: we have just made that clear in the Pythagorean Theorem itself. But the secret proposition (imparted orally, we may be sure, to chosen students) makes it clear as well that LOGOS cannot know this length. If LOGOS cannot know this simplest of all entities, there must be no bounds to what mind cannot know.

Aristotle takes up this problem of the continuity of the straight line in his Physics, as foundational to the integrity of the cosmos itself, in all its aspects and all its levels. If the straight line fails, all else fails with it…

Here is the problem. Suppose we attempt to fill the line with all the thinkable multiples, however large, of all the thinkable units, however small. We can generate an infinity of points, corresponding to the rational numbers – yet the unspoken theorem implies immediately that we will have missed at least as many points as we filled. Indeed, our rational points actually leave the line infinitely more empty than it is full, and thus devoid of any semblance of intelligibility. We know that the diagonal of the square exists; but we know as well, by its dark corollary, that we cannot know its length, or the lengths of an infinity of infinites of other lengths within its length. LOGOS is at once the mental faculty by which we speak (and in that sense, the word means the spoken sentence – in turn, Latin for “thought”), and ratio or number, as a measured length is known by its ratio to the unit. Not being able to measure, means that mind, as LOGOS, has been struck dumb.

This is, then, the moment of darkness for LOGOS, the confrontation with the irrational, the ALOGOS. Plato sees this in the Dialogues as the threatening darkness of the entrance to ELEUSIS, the site of the mysteries, as well as the APORIA, the impassible sticking-point, in the dialectical argument itself. As Euclid and Plato both know well, it is figured in the despair of that tragic hero at the crisis of the trilogy. Literal LOGOS, counting sins and reckoning consequences in the manner of the Furies, leads the mind only to emptiness and distraction. If Euclid were to leave his project at this point, it would lie in ruins. In tragedy, this is the terrifying cave of the Furies, with whom it is impossible to reason in any higher sense. For Aeschylus, as we have seen, at this point of crisis, Athena intervenes with just such a new form of reason, a reason which looks beyond the reckoning, of LOGOS, to a vision of the beautiful and the good. What hint will Athena now whisper, in Euclid’s ear?

For Euclid, the hold of LOGOS is broken by the introduction of something new, which he calls magnitude: a way of measuring beyond the counting-logic of number; a way by which mind can embrace the wholeness of the line. For Plato in the Republic, it is the passage from the cave of LOGOS to the light of NOUS. The word mystery derives from the verb MUEIN, to be silent. NOUS is silent knowing, direct, wordless intellectual insight of a truth beyond words. This is the art of Athena, which awakened a saving hint of recognition in the wits of the Furies, and made the Athenian law-court, and with it the Athenian polis, possible. What word will Athena now whisper in Euclid’s ear, to save his Elements – and with them, it would seem, the wholeness of the intelligible cosmos?

3. ELEUSIS

Euclid’s plot has moved forward at a vigorous pace, offering to mind an unfolding sequence of intuitive objects. Now, in Euclid’s strange Book V, time stops, and the intuitive mind will be given no object on which to settle. There will be no countable or measurable objects: we will be making our way out of the realm of counting- number, which has betrayed us. Instead, we are to speak in words which have no objects: in terms of something Euclid calls magnitude (MEGETHOS) – the word Athena must have whispered n his ear! Today we call these irrational numbers, but as we see, for the Greek mind this is simply a contradiction in terms (ALOGOS LOGOS – the illogical logical!). It is a new and different abstract realm, in which mind enters upon a quest of its own. For Euclid, this quest is decisive: our goal is to restore mind’s relation to those real and most beautiful mathematical objects, which at the end of Book IV seemed to have been rendered utterly inaccessible.

The role of Book V, then, will be to open to mind a path which seemed to have been denied it: a path, beyond LOGOS, to the direct intuition of those most beautiful figures, the regular geometrical solids with which the Elements will close.

We cannot follow here the intricacies by which Euclid achieves this in Book V; even the definition of same ratio, with which the book begins, is daunting. We can say, however, however, that in Book V he accomplishes to perfection the limiting process we know today as Dedekind’s cut, Dedekind carries out in analytic terms essentially what Euclid had done two thousand years before. By carrying the measuring process to an infinite limit, it restores the power of mind to address all things – not however, as LOGOS, but after this mystic passage, as that intellectual intuition termed NOUS. We began with NOUS in the first books, but lost trust in it with what seemed the catastrophe of reason at the close of Book IV. Now, by way of the abstract concept of MEGETHOS – length, in a sense, without object – NOUS is once again accredited, and the way is open to our intellectual delight in the procession, in Book XIII, of the regular solids.

4. NOUS the Way of Intuitive Mind

There is a telling analogy here: Aethena in Aeschylus’ third play makes very certain that Athens will remember and celebrate the Furies, who have hitherto been so terrifying. Indeed, the tragedy is just that civic remembrance, and celebration. The Athenian mind, purified by the experience of tragedy, has been strengthened to the point of carrying the polity through situations which will continue to be suffused with the irrational.

Surely it is in the same spirit that Euclid makes certain to track the irrational as it appears at every turn, appearing everywhere the construction of the regular figures. To track it, he must first name it: endow the ALOGOS with LOGOS!

The powers of Book V enable him to do just that, by way of a newly empowered intuitive mind. NOUS contemplates the regular solids now in the full measure of their regularity and symmetry, but at the same time with hard-won awareness of their infusion with the darkness of the barely-speakable. They are no less beautiful – indeed, perhaps more wondrous – for being tragic figures.

5. The Dodecahedron: Noetic Being



The dodecahedron (regular solid of twelve faces) is the cumulating figure in Euclid’s sequence of construction of the regular solids. (Proposition 17 of Book XIII). Euclid is careful to include, in the case of each solid, that if the radius of the enclosing sphere is rational, then each s of the equal edges will be an irrational line.  And in each case, by way of the powers of Book V, the irrational has been specified and given what we might well think of as a mystic name. It has been the work of Book X, ascribed to Theaetetus, to work through this daunting project, and construct, in effect, a dictionary of the irrational.

Here in Euclid’s final figure, mind comes to rest on such a weave of the rational and the irrational. Here, if the radius of the enclosing sphere is rational (taken as our unit of measure) then every edge of the dodecahedron will be the irrational known as APOTOME. The name has been hard-won, as each of these rational-irrationals has been systematically constructed and blessed with its mystic name in the course of Book X. The APOTOME was christened in Prop.73 of that book.

We would be entering a very different mathematical world if we were to translate the APOTOME’s construction into the language of Descartes and modern algebra. But it may be of interest, and suggestive of the complexity of the heroic labors of Theaeteus, to know that the analytic formula for the edge of the dodecahedron looks like this:

Here, r is the radius of the inscribed sphere, and the rational unit in this case.

(Heath’s Euclid, III, p. 510)

6. A Closing Note

Recalling our initial claim, concerning Euclid’s rhetoric, we see that he has despite all odds remained true to his original plan. He appeals in the construction of this last figure, as he did in his first, that equilateral triangle, to the reader’s agreement by way of intellectual intuition, NOUS, rather than to any binding chain of LOGOS. That insight has become, under Euclid’s guidance, stronger, deeper and wiser than it was when it first looked upon the equilateral triangle. We could not have imagined then, as we now know, the overwhelming probability that the lines of the first triangle, if chosen by chance, would be unmeasurable, and inaccessible to LOGOS. Euclid’s Elements thus stand with the wisdom of Aeschylus, as witness to the power of intuitive mind.

We know well that our contemporary world has not followed that heroic path. There is more than one path to take out of the depths of ELEUSIS, whose abstract magnitude threatens to dissolve all substance. Euclid rescues substance; Descartes, embracing the abstraction of MEGETHOS, which he translates as extension, turns the world into one universal algebra, one universal “x”.

Book V thus stands at a crossroads of history, the point at which our contemporary culture left all ancient constraints behind. But that is by no means the only train of thought open to us in the modern world; Euclid’s wisdom survives today in other forms.

To explore more fully the depths of Book V and its implications for our position today would surely be work for a further blog posting. Meanwhile, as ever, comments on this one are earnestly invited.

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 Dividebut 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!