The Dialectical Laboratory:
Within western history, we meet certain watershed issues which have the power to turn our thoughts and practices in very different, alterative directions. Some of them have their roots in the sciences, yet ramify throughout society. Issues like this are rightly called “dialectical”.
One such conceptual divide is addressed in this lecture, which was given at St. John’s College in Santa Fe in 2006. Here the watershed is between the mechanical philosophy, which in one guise or another has been taught in our society since the time of Newton, and a holistic view, expressed in the Principle of Least Action, which sees natural systems, not as aggregations of mechanical parts, but as primarily whole.
The mechanical view is based on forces acting on particles; “explanation” works from the bottom up, and sees systems merely as aggregations of parts. By contrast, the Principle of Least Action sees the whole as primary, governed not by laws of force but by overarching tendencies of the system as a whole. Formulated in Lagrange’s equations of motion, it provides a working alternative to Newon’s laws of motion; in the form of Hamilton’s Principle, it has proved valuable in modern physics, where the notion of “particle” is elusive or untenable. For James Clerk Maxwell, it was crucial in grasping the idea of a new kind of entity, the “field”.
Each view of science has its social counterpart. The Newtonian view has given us the atomic society, of individualism, competition and war. A holistic view of systems projects a society based on cooperation and intelligent planning for common goals, with conflict resolved by reason rather than force.
Today, many are seeking a new start, based on principles which might restore some sense of wholeness to a fragmented world. The Principle of Least Action, which does just this within the natural sciences, actually belongs to a broad stream of thought, stemming from Greece and the east, and including thinkers such as Aristotle, Leibniz and Goethe.
This lecture introduces this principle in the context of the laboratory at St. John’s College, where discussion of such dialectical implications is one component of the study of the sciences within the tradition of the liberal arts.
The Dialectical Laboratory:
Lecture given 15 April, 2004
This is to be my last lecture. (I’ve already given one, quite different, last lecture in Annapolis, but since we’re running out campuses, this must definitely be my very last!) Which means, of course, that it has to be about everything. This presents a problem of organization, but I think we can handle that. It also demands a spirit of humilitywith respect to “everything,” I can of course only express profound ignorance. What I may seem to utter in the form of truth, we must understand in the spirit of myth, whose shape I believe I’ve discernedand offer for your consideration.
I do in a sense really intend to address “everything,” as I hope to lay before you the possibility of an alternative way of looking at natural science in generaland that must cover a good deal of ground! This will be a way which sees being as whole, with causality flowing from whole to partin contrast to our Newtonian tradition which frames its account of nature in terms of forces acting on bodies in separation, and wholes as mere aggregations. Plato puts this question as the contrast of the whole, the HOLON, and the all, TO PAN: true being which is one and whole, versus the all, which can never be more than an aggregation of parts. This will place before us a dialectical option with respect to the sciences, a seminar questionand this in turn leads us to wonder about the role of the seminar in relation to the laboratory, here at St. John’s.
“Why is it that in the tradition of St. John’s, there has always been a seminar table in the laboratory, in such close proximity to the laboratory bench?” Evidently the College is committed to approaching the sciences dialecticallyyet to say this may only deepen our problem. The question becomes most acute in its bearing on the modern sciences. They in effect define themselves as a bastion of objective truth, founded in unbiased empirical observation, with respect to which such questioning, which they call “metaphysical,” is irrelevant. They are happy to have us ask about them from outside, in effectbut not to open them up, to question their inner being. What might it mean, then, to bring such unreflective sciences to the seminar table?
We are speaking of something a bit more specific than simply an earnest discussion. A dialectical conversation, I propose, is one which presses the most fundamental questions to the point of crisisthat is, to the point at which we might be in serious danger of changing our minds. We see this in Plato’s dialogue, the Gorgias. Gorgias was confident of his prowess as a fully competent human beinguntil he met Socrates. At the crisis of that dialogue, Gorgias discovers that he has forgotten something which he suddenly recognizes as the most fundamental thing: he has forgotten to include justice. The sensation, it seems to me, is that of discovering that we’ve locked the keys in the car, the keys to life. The focus of dialectic, however, is not on that apparent disaster itself, but on what happens next.
We have many models of that dialectical experience: the Oresteia comes immediately to mind, andparticularly pertinent to our discussion tonightis the threefold structure of Euclid’s Elements. There, as in the Oresteia, we meet a devastating discovery and pass through a realm of confusion and darkness, to emerge into a new realm altogether, which is far richer than anything we had known before, and which addresses a new order of truthon which Plato reflects in the Theaetetus. In the Gorgias we envision in that third phase a rhetoric in the service of justice. In the Oresteia the dialectical outcome, in the third play, is at once the transformation of the Furies, and the birth of the POLISitself then a dialectical concept. In reading Euclidhaving discovered to our dismay that LOGOS in the form of ARITHMOS falls speechless in the effort to address that most profound of figures, the diagonal of the simple squarewe enter the darkness of Book V and wander in the realm of MEGETHOS, only to emerge at last empowered in a new, almost mystic way, to comprehend the beauty of the regular solidsall the way to the icosahedronin which rationality and irrationality are inseparably wed.
This three-fold way does hint of mystery:: the third cup, the poets say, is poured to Zeus SOTER, the savior. At that seminar table in the laboratory, I think the modern sciences are brought under examinationnot with the intention of doubting their ultimate validity, certainlybut in the hope that they too may prove to bear within themselvesin the spirit of that third cupan order of truth neither they nor we had suspected.
The mode of the seminar is that of the question, some timeless question which belongs to us as human, and cannot be altogether shaken off by all the turns of history. Aristotle, for example, speaks in the Physics of four causes, but these are really AITIAI, four questions which we bring to natural things, and cannot help asking. What is it made of?. What set it going? What IS it? And what is its TELOS, its purpose? So I propose that Aristotle above all, together with Plato, deserves a seat at the seminar table in our laboratories. Galileo, Newton, Faraday and Maxwell will all have roles tonight. Finally, and with a certain chill of regret, I acknowledge that Lucretius, with his sweet, dark comments On the Nature of Things, will elbow his way to the table. He will have something to say about that mix of darkness and light which constitutes our concept of science today. They may all, indeed, seem somewhat out of place at first, in the laboratory of the present daybut then, so do many of us, quite possibly for some of the same reasons!
I. THE WAY OF MODERN SCIENCE
Let us do, then, one of the simplest, yet most decisive experiments of our modern world: Galileo’s study of falling bodies, revealed by means of a potent scientific instrument, the inclined plane (Slide 1).
Here Galileo illustrates a pair of these scientific instrumentsone slow, one fastwhich in his skilled hands open the way to the modern world. The experimental question is, how are we to understand the motion of a falling body? Aristotle has seen that, being heavy, a stone has a tendency to move, downward. For him, this is a merely one example of a principle operative throughout naturefor everywhere in our experience, we meet nature as such an inner principle of motion. That might seem to Aristotle an end to the question. But Galileo, led by many clues, has seen that there is something more to thisthe startling realization that the stone falls mathematically. This is hard to see because the stone goes too fast, but the inclined plane is the perfect instrument for slowing free fall down, to study it in slow motion (Slide 2).
Here Galileo is demonstrating that by his arts he can design an inclined plane to slow the natural motion of free fall to any degree of slowness desired.The opening wedge of the modern world, we might say!
Galileo is investigating the way in which a falling body picks up speed, or acceleratesand is revealing this as a rigorously mathematical affair. He has found, as Slide 3 shows, that the body moves, like blessed god, according to strict and regular proportions between distance and time. Here, the velocity of fall increases in strict proportion to the time. This is uniform acceleration (Slide 4).
He shows that it follows that the distanceshere, HL, HM, HN and HImust increase in proportion to the duplicate ratio of the times AD, AE, AF and AG. The stone is a mathematical object! If a falling stone can do such things, all the worm must be mathematical! Descartes will draw that very conclusion, and show that an equation can be written for everything. Everything, Descartes says, is extensionsheer magnitude, I think he means, the dark matter of Euclid’s Book Vutterly measurable, but nothing more. Welcome to the modern world!
At the seminar table, Aristotle is devastated by the thought of all that must be lostcolor, appetite, passion, beauty, purposethose aspects which most characterized both nature and our own lives, all yield to MEGETHOS, the magnitude of nothingor of nothing in particular. And what about the causes? The seminar reconvenes, to read Newton’s PrincipiaThe Mathematical Principles of Natural Philosophy, and there to investigate Newton’s new answers to the question of causality, new ways of answering the question “Why?” (Slide 5).
What had once been axioms addressed to the intuitive mind have now, as if they were equivalent, become laws, dictated to Nature as the word of God. Aristotle can make nothing of this, thoughfascinatedhe does not leave the table. Between Aristotle and Newton, you see, “God-with-a-capital-G” has been discovered, very different from “O THEOS” of Aristotle, or of the Timaeus. The consequences of this slight distinction in spelling are everywhere to be seen, including, not least, in the foundations of the sciences. Matter, which Newton calls brute matter, which once as earth had that one quality of heaviness, now has strictly no property at all beyond sheer inertness and that one quantity, its massmerely how much of it there is in a given space. Agent cause becomes something Newton has introduced and calls force. Thus the First Law states that a body will never leave its state of rest, or uniform motion in a straight line, unless it is driven from that state by an impressed force.
TELOS is gone. There is no TELOS within Naturepurpose has been eradicated from modern science, and transferred instead to the mind of God. In that sense, Nature has been denatured. Newton’s book might be called the Mathematical Principles of Un-natural Philosophy. Newton does find many clues to the divine purpose, as he tries to read the mind of Godfor example, he suspects the comets are sent by God to replenish the waning spirits of the earthbut God’s purposes are no part of his strictly mathematical science. God’s presence, on the other hand, is for Newton immediate and felt everywhere, even space and time themselves are aspects of God’s sensorium.
Let us see how Newton might account for the motion on Galileo’s inclined plane. The next page of the Principia contains the clue, in Corollary I to the Laws
It contains this little diagram (Slide 7), fraught with consequence. If one force acting on a body imparts the motion which would in a given time carry a body the distance AB, and another impressed at the same moment would carry the body the distance AC, then the two acting together will carry the body from A to D. Newton’s forces are sharp impulses, we note, administered at a moment, and measured by the uniform motions which result. We conclude that, inversely, one force whose measure is AD can be regarded as compounded of two forces AB and AC, or resolved into them, at will.
Now let us apply this new wisdom to Galileo’s plane. There is a downward force of gravity, which is resolved in the way we have just seen into two components, one of which holds the ball to the plane, while the other impels it downward along the plane. This latter force is the cause of the motion down the plane: it is an external force that makes the body move (Slide 8).
The planets themselves (Slide 9) must now be moved in their courses by a close succession of such impulses (Newton says here of the force that it “acts with an impulse which is single but great”a hammer-blow!). When the body is at C, for example, it would by its sheer inertia continue to dbut impulse imparted from a distant center at S (indicated by the arrow) will impart a motion dD, parallel to CS, which, compounded with the original motion Cd, will take the body along the diagonal CD. The hammer-blows will be wielded by the hand of God, and from this beginning will emerge the system of the planets, which Newton calls the System of the World.
In the unity of Newton’s thought, science played an intimate role in a divine scheme. Newton was no mechanist; he found the new philosophy of mechanism a direct affront to religious truth, and it was in this spirit that he severely opposed Descartes’ strictly mechanical theory. According to Descartes. an algebra of direct collisions between bodies in a plenum of interactions (Slide 10) would serve to account for all the phenomena of the natural world. Indeed, as we see in his Second Book, Newton wrote the Principia, in part at least, as a direct polemic against Descartes. He shows there that if the planets obeyed Descartes’s rules, they would quickly get stuck in the plenum, and nothing would work. Yettry to keep this figure in mind. It has its beautyDescartes characterized it as “a very liquid heaven. It will make an interesting reappearance later this evening.
But the modern world has forgotten its theological origins and has left science stranded as a lesser, separate, truth which fiercely defends its right to stand alone, subject to no laws but those revealed in the laboratory. Lucretius, who has been quiet at our seminar table, now sees his chance. He once described, in elegant Roman hexameters, a universe assembled by blind chance through the endless engaging and disengaging of atoms armed with hooks and barbs. Now, he observes, modern science has turned those hooks and barbs into mathematical instruments of infinite refinement. But their workings are empty of meaning; and the universe of chance is darker, and more threatening, than even Lucretius could have imagined.
Lucretius, I fear, has caught our modern science with grim precision. Plato recognizes this image: he once painted a curiously similar picture, in his myth of those who dwell in a cavecut off, in essence, from the light of NOUS and the Good. That cave was elaborate, fascinatingonly vacant of meaning; full of splendid, sophistic workshops delightful to LOGOS, but transient, devoid of meaning or truth. A mechanical explanation couched in such terms we quite rightly see as reductive, we cannot wish our world, or ourselves, to be accounted for in such a way.
II. THERE IS A BETTER WAY!
Fortunately, there is a better wayone which meets the criteria of modern science in being mathematically exact and empirically rigorous, but which at the same time opens a path to the larger, human questions, of the sort that the seminar poses. Above all, this new way is cast in the mode of TELOS, and it invitesthough it does not pretend to answerthe question of final cause. Historically speaking, this way I’m calling “new” is new to most of us in thought, but by no means new in course of history. It is just that, for reasons historians may recount, it is a way which has never come to the fore in our Anglo-Saxon modes of thinking about the world. And it has never, so far as I know, become an accepted part of our approach to the Quadrivium at St. John’sthat is, in our conception of a rounded curriculum in the mathematical arts.
This new approach to nature was born with a certain curious problem proposed as a challenge to the academic world, at the time of Newton, by John Bernoulli. Everybody had a crack at it, but for one in particular, Leibniz, it became the key to a new vision of the entire worldthe myth I want to lay before you tonight. The name of this mythic creature is the brachystochrone. You may very likely never have heard of it, but you’ll know right away what the name means: it asks that we find a path of BRACHYSTOS CHRONOSleast time (Slide 12).
It is simply another, yet very different, way of looking at Galileo’s inclined plane. Instead of asking how time and distance are related during a trip down the plane, it asks, By what path will a body descend from point A at the top of the plane to point C at the bottom, in the shortest time? Since a straight line is well-known to be the “shortest” distance between two points, our first response is “why, the straight line, of course/” But that’s wrong.
This poses a question of strategy. It seems clear that we can gain time by falling quickly at the outset of the trip, and enjoying the advantage of this early high speed for the rest of the travel. But to what extent will it be prudent to carry this strategy (Slide 13)? Notice that this is a question of tuning, or finding the just solution. The term NOMOS, law, also means tuningPlato points out that you can’t tune anything better than right!
A mathematician will not be satisfied unless we evaluate precisely every thinkable alternative, at every point along the way (Slide 14). But how could we possibly do that?
I was surprised to find recently that the long string of reasoning in the Third Day of Galileo’s Two New Sciences culminates in his own solution to this problem. He has no difficulty using the principles of the inclined plane to show that descent on a circle will be quicker than on the plane (Slide 15), and concludes that the brachystochrone is a circle. But Galileo was wrong! Galileo lacked a method adequate to a problem of this kind.
Notice what a strange sort of problem this is: we have to find the best of all possible paths. It seems an inherently impossible task! If this rings familiar to some of the Juniors, you’re right: this problem was a paradigm for Leibniz in the concept, with which Voltaire later had so much fun, that this is the best of all possible worlds. The brachystochrone is a portal to thoughts about the cosmos.
But how, we might well ask, could it be possible to reason mathematically about all possible paths? Strangely, thinking about all possibilities is one of the things mathematicians do best! We have only to recall Euclid, leading us through the intricacies of the concept of same ratio in Book V. To show that two pairs of magnitudes are in the same ratio, the first to the second as the third to the fourth, we had to test all possible multiples. Euclid’s phrase is HOS ETUKES“any other, chance equimultiples.” With that breathtaking phrase, he traverses the impassibleand conquers the continuum. Dedekind would do the same thing much later in fancier form. But now it becomes a problem of all possible paths, which proves, again, to be mathematically tractable. The winner turns out to be the cycloid (Slide 16).
You can think of the cycloid as the path traced by a point on a rolling bicycle wheel (Slide 17). Here, point P begins at A and traces out the cycloid as the wheel rollson the ceiling, if that makes any sense to you!
This strange problem serves to launch a new mode of problem-solving, the variational method, which in turn will open the way to a new approach to the natural world. It is an elegant, far-reaching mathematical method, which it would be wonderful to study together. But tonight I can only assure you that indeed it does exist!
Let’s go back now to Galileo’s inclined plane, to let Leibniz explain a very different way of looking at this thing. Leibniz is, for starters, fundamentally a student of Aristotle. In fact, among the famous conservation laws in physicsthe conservation of momentum, the conservation of energyI would propose that the greatest might be the Conservation of Aristotle. For Leibniz looks at nature the way Aristotle does. Motion, Leibniz and Aristotle agree, is the actualization of the potential qua potential. “Actualization” means bringing the potential into being as act, the Greek for which is ENERGEIA. It is only a slight corruption to speak, then, of the ball at the top of the plane, at A, as having “energy” in potentia. In this sense, the energyin one form or the otherabides throughout the whole motion, first as potential and then as actual. Nowadays, with local motion primarily in view, we speak of local motion as kinetic. But local motion, is only one mode of ENERGEIA. Giving lectures, or listening to them, is another.
If we want to follow Aristotle in looking at a motion, not piece-by-piece but as a whole, it will be helpful to choose one motion which, in the spirit of the Poetics, has a beginning, a middle, and an end. The plot of a drama, or the course of a lifetime, will not be irrelevant. In the realm of local motion, the pendulum offers a good model of this.
This image (Slide 19) is borrowed from Galileo, who is evidently anticipating the concept. Motion begins at C with pure DYNAMIS, is realized in full swing at B, and then returns to rest at D, G, or I, depending upon the turns of the plot. We might, in the Question Period, try reading the pendulum as poetics: I propose the Odyssey, with C as Ithaca, B as the Trojan War and the adventures of the voyageand D as the island of Kalypso, farthest from home.
Newton would view this very differentlyisolated events, no poetics. To him, point D might look like home.
Knowing that total energy is quantitatively conserved is an important step, but it is not enough. A new Principle of Least Action, which weighs all possibilities, will be our guide. The names of many thinkers are associated with this principle, among them Maupertuis, Euler, Lagrange and Hamilton. It was Lagrange who gave it the form we will look at now. Going back now to Galileo’s study of the way a falling body picks up speed, or accelerates, the principle states that of all the possible ways we might think of, by which the ball might accelerate down the inclined plane, it will in fact do so in that one way for which the total action will be the least. The total what? The total ACTION. Evidently, we’ll have to define that term!
A = (T V) × t
Slide 21 is a sort of spelling lesson. We’re still thinking in terms of that global quantity, which belongs to the overall motion as suchthe total energy. We assume that it will remain constant. Action, on the other hand, looks at the way this energy changes form, from potential at the beginning of motion to kinetic as the body picks up speed. Let us call these T for kinetic energy, and V for potential. Then (T + V) belongs to the motion as a whole and is constant. Action tracks the difference between them, (T V)the degree of fulfillment, or realization of potential. This difference is of course changing all the time as the body falls, but the principle of least action says that over the whole motion, the product
(T V) × time t
summed over the trip, will be the least possible.
To accomplish this summation, we need to divide the motion into intervals of equal times, as this graph (Slide 22) is illustrating. Time is plotted along the horizontal axis, and the difference (T V) along the vertical. The motion starts at tl and ends at t2. We divide up the time into equal intervals Dt. Since the action is the product of (T V) by time, the action in each particular interval will be the value of (T V) during that interval, multiplied by the time Dt. That’s just the area of the rectangle for that interval. In symbols (using DA for the amount of action accumulated during that interval) we get the expression
DA = (T V) Dt.
Then to get the total action as the ball rolls down the whole plane to the bottom, we would add up these parts. So the whole shaded area measures the action from tl to t2.
Next, in Slide 23, the Dt’s have become as small as we please, and the shaded area measures the action exactlyin formal terms, as Newton teaches us, the summation has become an integral.
Action is a funny concept, which takes some getting used to. We’re looking at the gap between potentiality, V, and its realization, T. The principle of least energy wants to make that gap the least possiblewants to make V and T as nearly equal as possible, under the constraints of the situation. Not just at any one moment of glory, however, but on balanceover the whole of a lifetime. The whole shaded area in our graph is telling us how well we’ve done.
This reveals the mathematical clue by which the Principle of Least Action can be grasped (Slide 24). If A is to be least, the variation of A, that is DA, must be zero. Here, r is any parameter you might want to test out, to vary the motion. The Principle is stating that the right value of that parameter will be such that the corresponding variation of A will be zero. This is all we need to know, tonight, about a powerful new instrument, the calculus of variations.
That’s all there is to it! We have laid the foundation for a new approach to the natural world, which can be given elegant mathematical expression, one form of which is known as Lagrange’s equations. We will refer to Lagrange’s equations frequently tonight, though we can’t go into them in this lecture. We might, if you wish, say something more about them in the Question Period.
Over some wide span of the phenomena of nature, the Principle of Least Action appears to generate the way things moveto answer the question, “What makes things go?” Not force! Things are not inert, and don’t have to be pushed or pulled by Newtonian impulses: they move on their own to achieve this goal of balance or realization.
Leibniz noticed something interesting about this. We could take two points anywhere within the overall path, and the principle will apply as well over the smaller interval between these two points; however small the interval, the principle will still govern. In this sense, the part mirrors the wholeor better, makes its own motion in such a way that the motion of the whole will be best. This is the organic principle which leads Leibniz, in the end, to postulate the Monadwhereby each part of the cosmos mirrors the cosmos within itself. It may be the same thing Plato meant when he said that the cosmos is a living ZOON, with soul, and that we mirror the cosmos in ourselvesthat’s why our heads are round!
It reminds us that tins new way of looking at things is organic from the outset. Even the inclined plane can look organic, in this sense.
To see how this might work in practice, let’s turn to a very different scene in the dialectical laboratory. If our lecture is to be about “everything,” we could hardly do better than to choose a phenomenon which will fill the entire visible cosmos! We turn to a pair of authors who could hardly be thought of in separation: Michael Faraday and James Clerk Maxwell. Faraday laid a great many of the empirical foundations of the modern sciences of electricity and magnetism while Maxwell, following immediately upon him in time and thought, shaped a new mathematical rhetoric capable of capturing Faraday’s penetrating insights. To do that, Maxwell made creative use of the new approach embodied in Lagrange’s equations. Their two great books are, respectively, Faraday’s Experimental Researches in Electricity, and Maxwell’s Treatise on Electricity and Magnetism. However much their styles contrastthe one empirical, the other mathematicalFaraday and Maxwell are deeply bonded at a level of shared insight I would call “dialectical.” Their thoughts meet at the level of NOUS, which goes beyond the distinctions of LOGOSbeyond the distinctions of words, symbols or styles.
The first thing to know about Michael Faraday is that he had little formal education and was almost completely innocent of any knowledge of mathematics beyond simple arithmetic. This was not all innocence, however. Throughout his working life, Faraday remained extremely suspicious of any statements about science that were grounded in mathematical reasoning. In fact, since he had never read Euclid, I suspect Faraday had little sense of the concept of a scientific theory, as consisting of a chain of argument following from first assumptions. My sense is that in the researches which he conducted all his lifetime at the Royal Institution, which was literally his home, he functioned rather as a natural philosopher in the Aristotelian tradition, fitting a flow of rich and imaginative empirical studies as directly as possible into an ever-larger framework of thought about the way things are. Aristotle, we should remember, was himself an ardent empiricist, for whomas we see in his book on scientific method, the Posterior Analyticsthe way was open and immediate between empirical observation and scientific truth. Interesting, if I’m right in this, that an Aristotelian should be opening this new gateway to the modern technological world! Let’s look first at Faradayat the sorts of things he was seeing and the ways in which he would talk to himself about them; his remarks are recorded, we might say in “real time,” in the Diary he kept in his laboratory. We join him at his laboratory bench, then, one day in November of 1851. He is about to embark on a new series of experiments with the magnetic lines of force by which he has already become fascinated. He reveals them by scattering iron filings in the vicinity of a magnet. Now he wants to assure himself exactly how they behave; he will record the results as the day unfolds, directly in the Diary. His entries begin with this one:
First a simple magnet, being a needle of about this size [Slide 26], well magnetized by a horseshoe magnet of power.
He draws a quick sketch of the very needle, about 2 inches in length, in the margin of the diary.
Faraday has been a bookbinder by trade; this is very likely a bookbinder’s needle. Now he sprinkles the iron filings, and gets this result, which he records with undisguised delight:
It gave beautiful curves having perfect simplicity of form [Slide 27].
By the end of the day, Faraday has discovered that he can use gutta-percha to preserve a more complete image, capturing the perfection he has seen in these curves (Slide 28):
The next day, which is a Thursday, he carries this process further. Performing a further mathematical construction in this new mode, he breaks the needle and watches intently, to see what his curves do:
Faraday is describing the behavior of a new kind of natural entity. These are his words, in one breathless sentence (actually, this is only part of the sentence!):
... certain of those curves which before were entirely within the body of the magnet are expelled into the air, because of the sudden diminution of conducting condition at that spot by rupture and want of continuity, and of those which thus come out through the sides of the magnet part returns and is discharged at the nearest pole and part goes on and dips into the further half, the circuit being completed in space as in the unbroken magnet.
This is at once mathematics and high drama! He separates the two halves of his needleor is it now, two needles?watching the lines stretch (Slide 30):
By the time his results are published in the Philosophical Transactions, they have been copied by an engraver and reproduced in this most elegant form (Slide 31). Something of the spirit of those sketches has been lost. But something has become manifest as well: the full pattern, the Gestalt, has emerged. Seeing and knowing are close allies, as so many metaphors of insight attest. Here is a system with that wholeness of which we have been speaking!
In its character as space-filling, it might quite plausibly remind us of the Cartesian vortex we saw earlier. But now, for Faraday, it represents one whole entity, not Descartes’s aggregation of mechanical parts. Contemplating this new thing, he notices it looks like a beetle, and he sends off to Cambridge to have a proper academic name made for it. The answer comes:
The sphondylonor, as Faraday more often calls it, the sphondyloid. (I might caution you that this is not a word you are likely to hear, nor should you probably expect to use, at scientific gatherings today!)
The academic mathematicians, however, following in the footsteps of Newtonian science, have little patience for this sort of thing. They see Faraday’s supposed sphondyloid asin that terrible phrase of reductive science, nothing but a mere artifact, an appearance which can be explained scientifically by the construction of a simple diagram, of a sort we have seen before (Slide 33):
Once again, Newton’s parallelogram! Two forces, each centered at a pole of one of the magnets, act at the point O. The force from the further pole is less, by consequence of its greater distance: the ratio of the two vectors is precisely the inverse of the squares of the distances from their two polesa law of the magnetic force which is, most satisfactorily, of exactly the form of Newton’s own law of gravitation. Science is vindicatedFaraday’s supposed physical entity has been explained awayand Faraday, for all his ingenuity in experimenting, is relegated to a lower rank, as a mere discoverer. But Faraday has seen his sphondyloids tending to do things when north and south ends approach. He does not speak of a “force of attraction,” he says their sphondyloids tend to coalesce (Slide 34):
When two norths approach, their sphondyloids avoid one another, whether placed axially (Slide 35)
or in parallel (Slide 36); they’re feeling cramped:
For Faraday, nature is a realm of Such whole entities, moving because they have tendencies. His sketches, done in a moment, match their spontaneity with his own; Faraday and Nature are very close friends.
What are these sphondyloids made of? Faraday seems to answer that question when he calls them sphondyloids of power.
Faraday has met the Newtonian mathematicians in direct confrontation on an issue of reality and causality in the natural world: a true dialectical crisis. Yet though Maxwell is himself an expert mathematician, his sympathies are with Faraday in this matter of the sphondyloids. Maxwell sees that the Lagrangian equations can be used to preserve the sphondyloids while satisfying the demands of quantitative science. This creative embrace of mathematics on the one hand, and an almost Aristotelian realism on the other, is a vivid instance, once again, of the third phase which distinguishes the dialectical form.
How can Lagrange be brought to bear on the sphondyloid? Actually, they are as if made for one another. First, the Lagrangian equations of motion are founded on the concept of a physical reality that is whole. This will be the sphondyloid itself. Lagrange speaks of this whole in terms of its energy; this corresponds to Faraday’s perception of his magnetic figures as sphondyloids of power. For Lagrange, motion is initiated by energy in its form as potential, the motion unfolding from potential to kinetic in accord with the principle of least action, on which Lagrange’s equations are based. This catches the behavior of the sphondyloids, which Faraday sees in terms of tendencies to move. Most importantly, perhaps, Lagrange is silent about any material this system may be made of. Faraday and Maxwell are in agreement that they are looking at a very real seat of power which is extended in otherwise empty space yet has no mass, and consists of no bodies which might exert pushes or pulls upon one another. Thus Maxwell sees that Lagrange’s equations will serve perfectly to account for the behavior of the sphondyloid, which is all power and no mass.
There is, however, a difficulty. How can we apply this quantitative theory to the system which Faraday describes in such qualitative terms? We need to find something to measure! The sphondyloids must be re-conceived as a system whose state can be described uniquely by a set of measured parametersthe independent variables in terms of which Lagrange’s equations can be written.
Actually, it had long been known that these magnetic fields can be produced by electric currents, so two interacting sphondyloids can be very effectively represented by currents flowing in two circuits. And currents can readily be characterized by the readings of voltmeters and ammeters. So generating the sphondyloids by currents, rather than permanent magnets, will permit quantitative description of the state of two interacting sphondyloids, and bring Lagrange’s equations to bear.
Here, then (Slide 38), is the loop of wire carrying one of the currents. (For simplicity, I haven’t shown the source of the current; but imagine, if you wish, a flashlight battery connected in the loop.)
And here is the sphondyloid, or magnetic field, it produces (Slide 39):
We will have the system we need as soon as we introduce a second coil, to correspond to Faraday’s second magnet. Let us call the first coil the primary, the other, the secondary. The secondary will have a field, or sphondyloid, of its own (Slide 40):
Since the two currents are flowing in the same sense, they resemble magnets aligned alikethe two halves of Faraday’s broken needle. Faraday has shown how two such sphondyloids will tend to coalesce and that is just what happens! This is Maxwell’s figure of the combined magnetic field (Slide 41):
This pair of coils, with their currents and coalescing sphondyloids, is all we will need in order to apply Lagrange’s equations to Faraday’s problemindeed, taking this as a model instance, to any connected systemperhaps an ecology, or a living cell. In each case, we will need to find such a set of independent quantities, just sufficient to characterize the state of the system. It turns out that there will be one Lagrangian equation for each of these degrees of freedom of the system.
To get these quantities for the sake of which we’ve set this all up, we need to introduce appropriate measuring instruments. We can do that in this case by inserting galvanometers into the two circuits to measure flow of currents, and voltmeters to measure electrical pressureto speak a bit loosely. This will give us four independent quantities to characterize the state of this little system. Lagrange’s equations cover all the ways in which these quantities may interact when any one of the variables is alteredincluding, we might add, the distance between the two loops.
We are ready to set this system into motion. We begina thought experiment herewith current flowing in the primary, but in the secondary zero current, everything quiet. Now, interrupt the primary current by opening a switch. We observe the discovery which had once left Faraday baffled and entranceda miracle which would be extremely surprising if it happened between, say, two unrelated garden hoses carrying water instead of wires carrying current. At the moment of cessation of the primary current a sudden pulse of current occurs in the secondary, as shown by a reading on its meter. Interruption of current in the primary causes a sudden pressure, and a flow, at a distancein the secondary.
What has caused this phenomenon? Maxwell’s answer is that the field itself as a whole, through its own structure, is the cause. We need not look furtherneither for other causes inside the wires, nor for mechanical parts inside the field.
Behind the equations of Lagrange, it is always the Principle of Least Action that is causing the effects we see. Can it be, then, that a principle based on TELOS and final cause is activating the electrical machines and all the underlying processes of our modern world? The magnetic energies is involved, huge as they may be, are never in the wires, but always in the field.
Aristotle wishes to remind us at this point that this is just what we expect of a natural system: things have a nature if they have a principle of motion within. The magnetic field does all it does as consequence of an inner principleby its nature.
We have been telling only half of Faraday’s story tonight. There is also the electric field, with a spectacular life of its own that he has pioneered in investigating. Maxwell goes on to merge this electric field with the magnetic, as components of one single Lagrangian electromagnetic system, which thereupon takes on dual propertiesat once the kinetic energy T we have seen associated with the magnetic field, and the potential energy V stored, like that of a spring, in the electric field. We might think of this new entity as the “enhanced sphondyloid.” But the world knows it as the “electromagnetic field.” It has, then, the kinetic and the potential energies of a pendulum. Then it must swing like a pendulum!
Further, since the field is extended in space, its vibrations must propagate like ripples on the surface of a pondthe very waves Huygens taught us to recognize as the key to the nature of light Now Maxwell, having expressed Faraday’s insights in quantitative form, is able to calculate the velocity of that propagation. The outcome of Maxwell’s calculations, when combined with one crucial electrical measurement which our juniors perform as a ritual each year, showed that the ripple running through this dynamical system must be the very phenomenon we know as light.
There seems no limit to the distance to which that primary loop may be removed: as far, that is, as light can carryeven as far as a star or a galaxy, at the edge of space and time. Such is the reach, and the reality, of Faraday’s sphondyloids, as if they were holding the very cosmos together. If light is a true representative of physics as a whole, the cosmos might prove to be governed throughout by that lively search for balance between possibility and actuality which is called Least Action.
V. EVERYTHING ELSE
We have spoken of some thingsnow we must speak, in one image, of everything else. Slide 43 is an image of spinachone instance of a living system, which now must stand for all the rest.
Our time is nearly up, and we must switch to the optative mood now, to say something like “Might we only describe this system, and apply our new principles to it, had we only a bit more timeand I, considerably more knowledge!” In reality, we must settle for a mere hint.
Slide 43 depicts what biologistsin a delightful response to the poetic museterm the light-harvesting complex of spinach. A lovely phrase! It is a system virtually made out of purpose. In the very spirit of Leibniz and organism, every detail here serves one ultimate end, which is to capture the light of the Sunthat is, to receive Faraday’s sphondyloidas efficiently as possible, and to convert it, with the help of water and carbon dioxide, into living organic formultimately, we might say, to turn light into spinach; or better, to turn light into life. It thus stands, with all green leaves and photosynthetic organisms, at the very threshold of life. The inner core it serves has been doing this since the birth of life on earth, for some two and a half billion years. Without the work of such living jewels (this one is about 26 nanometers acrossGod’s own nanotechnology!) we would have nothing to eat, nor an atmosphere to breathe. Let this ikon, then, invoke the realm of living nature, purposeful, organic and whole: the cosmic ZOON of which Timaeus speaks.
It must be finely-tuned, and extremely fast, to catch the subtle motions of Faraday’s sphondyloid on the fly. In terms of Maxwell’s two coils, our Sun is the primary, and this is a Very Large Array of “secondaries.” It has been engineered to do its work by the quickest possible path; it’s the ultimate brachystochrone, the product of millennia of evolution, which must be the real calculus of variations. And it has found, as you see, the ultimate form for doing this: the icosahedronthat regular solid with the greatest number of faces, the ultimate construction in Euclid’s Elements (Slide 44). Nature has read her Euclid! Or should we say, written her own Euclid, in her own words.
The essence of Organism seems to be the nesting of forms and purposes. In each face of the harvesting system, we see a triplet of protein molecules (Slide 45):
And if we look at one of those proteins (Slide 46), we see that it positions, with precision, a set of smaller moleculeshere labeled “a” for chlorophyll a, “b” for chlorophyll b, “Lut” for the pigment lutein. The chlorophylls are the true secondaries, tuned to the Sun.
Here (Slide 47) is the heart of the chlorophyll molecule. It is an elegantly rigid molecular structure; a harp so firm it is ready to be tuned like crystal to resonate to a single tonea single vibration, a single color within the Sun’s white light.
As soon as this is augmented with a sounding board (Slide 48), it takes its tuning:
Across the bottom we add an antenna (Slide 49), linked on the one hand to the rest of the harvesting system, and on the other, to the cosmos:
And finally, the completed light-harvesting complex molecule (Slide 50):
I have indulged in this little optative adventure because it seems to bring us to Lifethe very heart of our true subject tonight. Here is the real object of physics, one entity imbued with TELOS; whole, and indeed existent, only because of its organic bond to all the rest. May we find a natural philosophy adequate to living entities such as this.
We have opened the door, at least a crack, to glimpse the possibility of a new way of seeing the world. The importance of our image of physics is greater than we might suppose, and it is not a matter for the scientists alone. To a greater extent than we normally consider, we picture our own natures in the image of our suppositions about the nature of the world around us. If we believe nature to be composed of little hard bodies interacting according to laws of force, we are quite likely to suppose that our own natures are competitive and conflictive, and that our social structures are aggregations, rather than natural wholes. The result at all levels is competition and the conflict of separate interests, with its endless corollary of human and ecological waste, and war. If on the other hand we believed that in nature everywhere the whole is primary, we might be persuaded to consider again, with Aristotle, that we are social and political by nature, rather than by negotiation and fragile compromise. We might even consider that the community of all peoples is itself naturalas Aristotle suggests when he speaks of friendshipand perhaps the only true vehicle of the salvation and happiness of the world.
This is a time in the world’s historyand a time for many of us individually, as wellwhen the lights which once illuminated our hopes are as if inexorably going out. It may be a good time, in short, at which to pause, and earnestly review our options.
It takes courage to break out of a rigid paradigm. But let us consider the possibility of a science that is restorative of nature, and that is respectful of our global human nature as well.