Category Archives: wholeness

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 1

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

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

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

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

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

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

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

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

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

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

Visit Newton /Maxwell / Marx 2

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

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

Maxwell's Mathematical Rhetoric

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

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

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

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

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

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

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.

The Two Minds of Charles Darwin

I’ve wanted for some time to write this note, but have hesitated because there are so many others who know Darwin far better than I. Nonetheless, I have a certain conviction I’d like to share.

Two minds seem to be at work as Darwin surveys the natural world and its evolution. One sees natural selection in terms of confrontations between individuals or species in the search for limited resources. We all know that scenario, which in most of our discussions has become the very paradigm of Darwinian selection.

But Darwin has unmistakably another line of thought, which grasps the utter complexity of the selection process: not as a competition between individuals, but as a system whose complexity defies analysis. If we were to make an improvement in a breed in order to increase its chances of survival, we would not, he remarks, know what to do. In another passage, he remarks on the flourishing of a certain flower in one particular English village. What advantage does this plant have here, which it lacks elsewhere? The answer, he has decided, is the absence of dogs. (Dogs, he reasons, eat cats; cats eat mice; mice eat seeds.) I’ve forgotten why there are no dogs, it might be some village regulation. Whatever it is, there lies the strength of the flower: not in its own design alone, but in the structure of that ecosystem, which has at least for a time stabilized in a pattern collective survival –a pattern, we might say simply, of collective health.

This I believe is an overriding principle, which we have tended since Darwin’s time to miss. That principle, almost systematically ruled out of all facets of our thinking – even our very ideas of medicine or science itself, is the overriding concept of organism, the recognition that we live, flourish and evolve as a whole – not as a sum of individual parts. Only in recent years have we begun to study ecosystems, of all sorts and levels, as wholes. As a society, we’re far behind the demands pressing upon us in catching Darwin’s other, and I believe higher, insight.

The stereotype in describing the components of living systems, to ever-higher levels of resolution, is mechanism. Wrong! We will never understand living organisms as summations of mechanisms. A living system is a different concept altogether from a machine, and study of it calls for different strategies, and different conceptual tools.

Much new work is being done now in the spirit of this new understanding. I’ve found exciting studies of ecosystems to which I want to call attention in an upcoming blog posting. Indeed, it’s not a new thought on this blogsite, which has traced the idea of organism back to its rich source in the writings of Aristotle, and fast-forward through western history to Leibniz, Euler, Lagrange, Maxwell, Hamilton, Feynman and modern physics. But in the din of our celebration of Newton, isolation and competition, we haven’t heard, or perhaps have deliberately rejected, these other voices. We’ve caught only the lesser of the two voices of Charles Darwin.

Cancer and Ecosystems

Peter Gann was a member of our Aristotle discussion group at Pemaqud Point in Maine this summer.  In response to a question I had raised in the wake of our discussions, Peter has written a letter which I find so interesting that, with his permission, I’m reproducing it here a a sort of “guest blog”.  Dr. Gann is Professor and Director of Reearch in the Department of Pathology of the University of Illinois in Chicago.

Dear Tom,

Your question about cancer and ecosystems naturally leads to Virchow! It was he who recognized cancer (and other diseases)as disorders within the community of cells that make up an organ or an organ system. I find this to be a very useful analogy.

The healthy function of the organ requires that each differentiated cell carry out its designated role while remaining in its designated space. How this unfolds during organ development is fascinating and deeply mysterious, but it seems to involve special “tunes” – primitive ones – played out within the genome as well as lots of direct chemical communication between nearby cells.

At some point, once the organ has developed, these signals must change so that such rapid growth and morphogenesis can stop and a more “mature” ecosystem of stable, collaborating cells can emerge.

Cancer cells overcome the signals that maintain this stable ecosystem, and, even appear to hijack some of the genetic programs that are used to control normal development.

This is not too far from how the Ailanthus tree in our backyard (which Wendy identified this summer) threatens our local ecosystem by hyperproliferation, exploitation of local energy sources, and evasion of organisms that would otherwise control its spread. Left undeterred, the Ailanthus could be viewed as a pathological force that would eventually destroy the native Midwestern woodland that we consider to be healthy.

I suppose one could look at all invasive exotic species through the same analogical lens. [But then, thinking of that awful tree in the backyard, maybe this is just demonizing the enemy before going to war!]

The response of an ecosystem to this type of imbalance raises very interesting questions and it would not surprise me to learn that there are numerous examples of stressed ecosystems righting themselves, through adaptation, since the invasive force can be seen as a stimulus to natural selection, just as a change in climate would be. It would take a serious ecologist to deal with that question.

I believe I do recall that some of the early thinkers in the field of ecology (as well as some of the post-Darwin evolutionary biologists) were very interested in the analogy between cell communities and ecosystems. It would be interesting to know what Virchow thought of Darwin.

All the best,

Peter

The Aristotelian Pathway to the Modern World and Beyond

I’m just back from a week of seminars in Maine: an overview of Aristotle’s world-view, based on a sequence of selected readings.  Although I’ve long been curious about Aristotle’s thinking, and written about this to some extent on this website, I’ve never before caught the full coherence and impact of his world-view. I’ll leave details to future posts to this blog, but here’s an overview of a few highlights.

Tradition has misleadingly titled many of Aristotle’s works. His “Physics” is not limited to what we today call “physics”, but actually addresses the foundations of the entire natural world, of all things that move, from stones to living creatures, including ultimately ourselves. Aristotle’s “Physics”, then, lays the foundation for his other works, and in the “Metaphysics”, of the cosmos itself. We ourselves he will say, are rational by nature.

What is “nature”?  An inner principle of motion, Aristotle says; things move not because they are pushed or pulled, but through inner tendencies. This is by no means nonsense. Within what we call “physics”, think for example of the second law of thermodynamics, which asserts, in more formal terms, that heat “tends” to flow downhill. Within our own lives, think of fear or love, and our innate desire to know. Thus in Aristotle’s inclusive world-view, there’s no occasion for the infamous split which today appears to divide our sciences from the humanities.

Such unification need not threaten the integrity of the sciences. Remarkably, within this encompassing perspective Aristotle lays a secure foundation for a fully valid alternative approach to modern science. Key is his concept of “energy” (the word, energeia, is his!); motion consists in the unfolding of energy from potential to kinetic form. Importantly, energy belongs primarily to whole systems, so wholeness and living, organic unity are foundational in Aristotelian science.

In the 17th century Leibniz, who knew his Aristotle, put this into mathematical form. He introduced, in open opposition to Newton, a version of the calculus which served to open alternative path into not just modern physics, but modern thought more generally.

As a result, we can discern two very different, parallel pathways through the history of western thought – one leading to Newton, Descartes, and a world of force, competition and mechanism; the other, prefigured by Aristotle, leading to wholeness, cooperation, friendship and life.

The path through Newton, Locke and Hobbes is very familiar to us; it has led t the world we know today, a world of strife, competition, and ever-escalating warfare. That other thread, which runs from Leibniz, Euler, Lagrange, Hamilton, Faraday, Maxwell and Einstein, bespeaks unity and intelligent cooperation. Within physics, this appears especially in the concept of the field; but more generally, it looks to a society of intelligent cooperation in the solution of our common human problems. It is easy to see, I believe, which is better suited to address the problems of warfare and environmental catastrophe which beset human society today.

Nobody, of course, is offering us this choice of roads into the future.  But we have independent minds, and it would be good to know that there is a difference in principle even if we see no way at present to pursue it in practice. I propose to write more about this in upcoming postings – and it will be good to know what others think of this Aristotelian way I’m convinced I’m seeing.

An Ecosystem As A Configuration Space

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

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

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

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

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

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