The Rhetoric and Poetic Of Euclid’s “Elements”

Part II: Euclid’s Poetic

1. The Tragic Narrative

revised 8August 2010

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

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

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

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

2. Tragic Crisis in the Elements

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

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

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

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

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

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

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

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

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

3. ELEUSIS

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

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

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

4. NOUS the Way of Intuitive Mind

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

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

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

5. The Dodecahedron: Noetic Being



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

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

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

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

(Heath’s Euclid, III, p. 510)

6. A Closing Note

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

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

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

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

On the Concept of a Dialectical Divide

Commenting on my lecture, “The Dialectical Laboratory”, Tony Hardy has raised a number of interesting issues. They take us to one overriding question: “What do we mean b the term dialectical? A dialectical question, I believe, splits our world-view down the middle: virtually all of our perceptions, and our purposes as well, are placed at risk.  A dialectical question, therefore, is not one which can be resolved by negotiation among familiar options. We stand before a new and altogether different court of review.

The principal model for this is the Socratic dialogue, which places a respondent’s life, and that life’s central goals under devastating review.  Worst, we might say – that review is not that of an external judge, but a hitherto unrecognized standard within. The orator Gorgias, acclaimed political expert of Athens of his day, is a prime target of Socrates’ dialectical art. His very life crumbles before our eyes as he recognizes that it has lacked one transforming concept, that of justice. He holds great powers, but he has used them to serve no good end.  This is a tragic fall, mirrored in the fall of Oedipus. The classic term for a life, or a world-view, based on false pretention is HYBRIS (pride). To live on the wrong side of a dialectical divide is an invitation to disaster.

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

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

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

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

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

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

Organism vs. Mechanism: Science at the Lagrangian Divide

The Lagrangian equations are a powerful set of differential expressions describing the motion of a complex system.  With one equation for each component of the system, they would seem to offer a powerful expression of the relation of part to whole.

They are, however, seriously ambivalent: they can be read in either of two opposite ways. They present, then, a stark problem for the art of interpretation, the highest branch of rhetoric, as it comes from Augustine to Bacon and Newton.  The same statement becomes a watershed; it may belong to one world, or its opposite – but not both.  Each is a containing frame, within which we picture, and live, our lives

Read in one way – the way most common today – they are seen as derived from Newton’s laws of motion, and thus adding nothing fundamentally new. From this perspective, they merely rephrase Newton in terms of the concept of energy, a mathematical convenience in certain circumstances but making no fundamental change in our understanding of the natural world. In this interpretation, they express what we today call mechanism, which sees the motion of any system as the mere aggregation of the motions of its individual parts. Causality flows upward; motions of the parts explain the motion of the whole.

Seen from the other side of the Lagrangian watershed, however, the same equations express a world of a totally different sort. Here, the same equations are derived from the Principle of Least Action – a concept which readers may recognize as one of the recurring themes of this website.  The system itself as a whole, described in terms of potential and kinetic energy, becomes the primary reality and the source of the motions of the parts. Causality arises from the  interplay of these energies, and flows in the reverse direction, from whole to part.

Within the world of mechanism – the first interpretation – there is no place for goalor purpose. These are concepts considered far too vague to meet the standard of objectivity, the signature of modern science.

Remarkably, however, Least Action reconciles purpose with quantitative objectivity. By means of the mathematical technique of variation, which considers all possible paths, this principle seeks the optimum path by which potential energy may, over he whole course of any natural motion, be transformed to kinetic. In this interpretation of Lagrange, then, our world-view is transformed. Science itself, while remaining strictly objective and quantitative, becomes at the same time goal-oriented – all at once!

More than this, however, science on the Least Action side of the Lagrangian divide becomes, at last, fundamentally organic. This arises from a further, crucial feature of Least Action: if a system as a whole moves in such a way as to minimize action,so also will, within the bounds of external constraints, every part of that system. The goal which belongs primarily to the whole, is pervasive: it is shared by every part.

It was important in stating this principle to add “within given constraints”, because a rigid part of a man-made machine has few options. By contrast, the myriad components of a leaf, or of a cell or enzyme within the system of a leaf, navigate among unimaginable options toward the common goal of turning sunlight into life, over the season of the leaf, the life of the tree, or the evolution of photosynthesis on earth.

It is this community of purpose, nested and shared, which renders a system trulyorganic – a living being, something fundamentally beyond any bio-molecular mechanism, however intricate.

It is hardly necessary to add that it is this sense of nested purpose and shared membership in natural communities which has been so lacking during the long reign of mechanism. Our so strongly-held worldview has diverted us from that other option, which has nonetheless long formed a strong alternative flow of thought and practice in science, mathematics, politics and the arts. Now in many ways, not least the earth’s biosphere itself, the demand is upon us to recognize that we do have an option of immense importance. Viewing this whole scene now, we might say, from the Lagrangian ridge-line itself, with both worldviews clearly in view, our task is truly dialectical: leaving none of the insights of the past behind, we are in a position to move forward into a new, far richer and wiser world.

That new world-view, which has appeared here as a richer interpretation of Lagrange’s equations, is the ongoing theme of this website – always with an eye to Maxwell’s turn to Lagrange as mathematical vehicle for the launch of his concept of the electromagnetic field, paradigm, if ever there was one, of that whole system of which we have been speaking.

[A brief introcution to the Principle of Least Action is given in my lecture, “The Dialectical Laboratory” .

It is important to add that in this thumbnail sketch, many nuances of the application of Least Action have been left without mention]

“There’s no Space in Euclid!” or Euclid’s Rhetoric

“There’s no ‘space’ in Euclid!”
I remember vividly the moment when a talented young tutor at St. John’s College in Annapolis, Maryland, came careening down the stairway our old library, unable to contain this startling realization. As a new tutor at the “Great Books College”, he had likely been assigned to teach a class in the subject he knew least, and was as a result making his first encounter with Euclid’s Elements in the company of a class of first-year students – an experience as fresh and surprising to him as  to his student fellow-readers.

Euclid writes in a style we no longer expect of mathematicians: simple, confident, almost buoyant. In the scale of rhetorical possibilities, Euclid’s is the style Aristotle calls “simple”. His figures stand upright and firm, on their own: they have no dependence on a “space” in which to be.

Reading the classics in this unscholarly way, as if they had been written for us, is an experience filled with such surprises; as myself a student at the College at the moment of this proclamation, I recognized the experience: the same revelation had struck me not long before. Euclid’s idea of geometry is not what we moderns have been led to expect!

One aspect of this dis-conformity between Euclid’s world and our modern expectations is this troubling absence in Euclid’s mind of any notion of “space”. Euclidean figures, all the way to the great regular solids in which the thirteen books of his Elements culminate, are organic webs of relationships, each standing whole before the mind’s eye, increasingly vivid  as the plot  thickens. Our relationship with them is direct and immediate: they are not “in” anything!

Similarly, Euclid never proves anything, in our modern sense. His style is strikingly at odds with the ways of today’s formalism, which tends to bind the mind in chains of consequence, rather than liberating it. Euclid leads us to contemplate the figure which evolves as the demonstration (epidexis, a showing) unfolds. We construct the figures as we go along, and take well-deserved satisfaction, as they unfold, in our workmanship.

At each step in the course of a demonstration we are tacitly invited to lend assent. There is, on the other hand, nothing as it might seem, loose or casual about all this. Euclid relies at each step on a very real power of intellectual intuition – on our human ability to discern truth when we see it. Euclid as author is, at the same time, our teacher. Under his guidance we develop confidence and soon find ourselves taking delight in the exercise of new-found power of geometrical insight. Skeptics today will accuse us of self-deception, and Euclid, of naivete. But we and Euclid may stand if we wish by our own agreement, when we confirm a geometrical truth.

An illustration from Euclid: Book 1-PropositionsA striking example of Euclid’s method at work would not be far to seek: the first words of his Elements make a strong demand on our visual intuition. We are asked to construct an equilateral triangle:  (please excuse the informality of these images derived from my aging copy of Euclid!)

The procedure is very simple.  We begin with the straight line AB:

And on it at point A draw a circle ACB of radius AB:

(Point C is here no more than a label, not yet specified, to identify the circle in question.)

Similarly, at point B, we construct a second circle. ACE of the same radius:

(“C” is still functioning as a label, not yet located.)

But now, Euclid begins the final stage of the construction, with no hint of apology or explanation, by giving the mysterious “point C” a specific location, and a crucial function: “from the point C, at which the circles  cut one another ….”.

We stop to catch our breath!  Point C has now been specified, without ceremony or justification.  How do we know that it exists – that the circles do indeed intersect?  Euclid’s answer is simple: we know it, because we see it, in our mind’s eye – and of course we never really doubted.

We proceed to draw the sides, complete the figure and carry our new triangle with us, a secure foundation on which the great structure of the Elements will rest.

This is to be Euclid’s style throughout: even first principles are not legislated, but offered for our agreement. They are things asked of us or postulated, as questions (AITIAE) or proposals – and the rhetoric of the Elements will be consistent throughout.

Similarly, when we pause at the close of a long stretch of reasoning, to review the steps we have just passed through, this is not a matter of mere logical bookkeeping.  It is, rather, clearing the way to that moment of commanding insight, in which we say, in the spirit we now think of as that of Gestalt, “Aha!”I see!” And indeed, we do.

This is the rhetoric of Euclid, which so shapes our path that the we are led to see.  What department of mind is this, which Euclid is invoking?  I’m sure he would be in easy agreement with Plato, that while the logical mind grinds away at syllogisms, another, higher department of mind sees truth, and says “yes” to an argument not because it is bound by chains of syllogism to do so, but because it can view truth directly, and know it for what it is.  Plato calls this higher, defining power of mind NOUS, and as his dialogue Theaetetus makes clear, mathematics, practised in this mode, is crucial preparation for an approach to the highest things.

I have been drawing attention to Euclid’s rhetoric, but just as the rhetoric of Plato’s dialogues is essentially philosophic – skillfully leading the respondent to a question which is philosophic because it leads logic to an impasse and thus invites a higher end– so Euclid’s rhetoric leads as well to an end beyond the familiar realm of figure. I propose we should identify this further mode, having to do with the matters of plot and character, as Euclid’s poetic. It will be the topic of a separate posting, to follow soon on the heels of this one.

——

Footnote Concerning Other Geometries

Followers of this website may find it surprising that on the one hand I praise Euclid for his clarity about a three-dimensional world – while on the other, I announce a new expedition on this very website into a world of four dimensions. I even claim that we will be experiencing an intuitive sense of relationships in a four-dimensional world. What sort of contradiction is this?

My own proposition is this: Euclid invokes the power of geometrical intuition, but he does not set bounds to it. We have the ability to keep track of the agreements we make in signing-on to sets of postulate belonging to worlds quite different from Euclid’s: we may very well learn to see in our minds’ eyes rooms with four directions at each corner, or left shoes turning readily into right – powers of the visual imagination we have not yet learned to use. This website will post images from this project as unfolds. Stay tuned, and feel free to share any comments you may have

A Project For Viewing Objects In The Fourth Dimension

A new menu item has just gone up on this website, at which images will be displayed of objects in the fourth dimension.  Since it is widely agreed that we are by nature incapable of seeing four-dimensional objects, something evidently needs to be said at the outset to justify this new approach.

The Case Against

The case against seeing 4D objects is easy to make, and does sound convincing. We human beings by our very nature belong to a three-dimensional world; we have neither retinas nor brains capable of seeing objects in higher dimensions. We can see projections of such objects as the  4D hypercube, for example, into our 3D world–but we have no power to see the thing  itself. We are like Abbott’s Flatlanders (Edwin Abbott, Flatland (1884), confined in their case to a 2D world, and unable to conceive a world beyond it.  We laugh at them, but Abbott’s initial point is that we are the Flatlands–confined in our case to a world of three dimensions, and unable to envision a world beyond.

But Abbott makes a further point, and his real story is one of courage and release.  The turning-point of Flatland is the breathtaking escape of his hero, A-Square, who is swept out of Flatland and indeed does view, to his amazement, a world beyond his own, as well as his own from a new vantage point–outside. Surely Abbott’s real point is to challenge us, stuck in three dimensions, to break out of our own confinement. That is the experiment we will be undertaking on this website.

The difficulty is not mathematical. In our drawings, we place the viewer’s eye at a definite, fixed position in a four-dimensional coordinate frame, and define simple objects within it. The objects lie before the eye, in relations which can be calculated and depicted. The huge question remains, however: what will such an eye actually see? Not much, we might think–for our limited, 2D retinas would seem to have no power to capture 4D images.

Response to These Objections

But here’s a problem: by the same argument, the same 2D retinas must be inadequate to see the very 3D world in which we live!  Mathematically, it is true, we’ve never actually seen our own, familiar world: we would have to be outside it, to actually view it. But that does not stop us from knowing it intimately, and seeing it in another sense.

Evidently, real vision is not simply a mathematical question: the eye is not a camera. Rather, it is a powerful extension of the brain, actively searching and interpreting, constructing a meaningful and coherent understanding of the world we live in, and love. We “see” objects growing smaller as they recede into the distance; but from infancy, our interpretive visual system has learned the tricks of 3D visual intuition: we know automatically that the objects remain unaltered.  And if this is the case, there would seem to be no obstacle to carrying out our project, exploring the possibilities of a visual experience of four-dimensional space. Maybe our visual intuition is capable of learning new interpretive tricks!

The New Proposal

We propose therefore to look directly at the mathematically defined objects within it, with the aim of building a new structure of visual intuitions appropriate to the fourth dimension. It’s hardly necessary to stress the importance such a capability might have, given the striking ability of the visual cortex to “see”–graphically or otherwise–the relationships among groups of interrelated factors. The ability to visualize complex functions in a four-dimensional coordinate frame might be enough to convince mathematicians and scientists of the practical value of such an augmented power of visual intuition.

For an initial example of the method at work go to my webpage on the fourth dimension. Where you can post any comments which occur to you.

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!

Can An Ecosystem Model Help Us Think About Wholeness?

Readers of this website will be aware of my preoccupation with the question of “wholeness”. The more I observe the world’s current struggle to find its way through complex economic structures or global systems, the more convinced I become of the degree to which our deep-rooted commitment to individualism is betraying us. Individualism is both an ethic, which we are determined to impart to the world, and a habit of thought. This is not the moment to follow that line of thought further; it has been the subject of other postings, and it will be of more in the future.

My concern at the moment is to offer a new approach to this issue. On a visit to the Key School in Annapolis recently, on the shores of the Chesapeake, I was struck by the widespread awareness there that the Bay is sick: 27% of true health was the figure I was hearing. That led me to wonder about the concept of “health” of an ecosystem, and how it might be grasped. With the aid of the computer, I knew, the human mind is today able to reason about problems hitherto too complex to analyze. Could I find a computer model of an ecosystem?

By good luck, I’ve found not only such an ecosystem model, but a revealing account of a team project by which it was achieved. Teams of experienced scientists agreed to set aside their normal researches into separate compartments of the ecosystem, and direct their efforts  instead to a different kind of learning: to the common goal of constructing a coherent computer model which would capture the intricate interrelationships of these many components of one single system.

The system to which fortune had led me was a salt marsh at Sapelo Island on the coast of Georgia. The Book, edited by L. R.Pomeroy and R.G.  Wiegert, is “The Ecology of a Salt Marsh” (New York, 1981). Its innocent title fails to suggest the very special interest of the project it narrates. Quite elegantly, the book pulls together a fascinating account of the scientists’ experience in disciplining their work to this goal.

An aesthetic of wholeness is invoked at the outset, with lines from  Sydney Lanier’s poem, “The Marshes of Glynn”. We learn much about this new sort of scientific endeavor when the book closes with a section on the aesthetic of the marsh, and a final quotation from that same poem.

Though a layman in matters of biology, I’ve since been making an effort to follow the turns of this inquiry. I won’t say more how, beyond the remark that the effort proved successful only after the scientists had learned of a fundamental error they had been making, and accepted correction from the computer.

People whose judgment I very much respect have expressed their doubts as to the whether such a computer model is an appropriate means for approaching wholeness, or whether at this point I’m confusing true wholeness with a mere assemblage of parts by complicated aggregation. (My thoughts go back to Plato’s “Parmenides”, and the paradigm there of Hesiod’s wagon: I agree that the “wagon” is something quite other than an assemblage of its parts!)
In these terms, is a working computer model helping us to grasp the wholeness of a system, or betraying us into confusing true wholeness with a merely clever example of aggregation? In the case of a living ecosystem, in which the wholeness is manifestly organic, is the computer misleading us, tempting us to confuse organism with a complex structure of inherently inorganic parts?

My case for the computer as a welcome aid in advancing toward a  grasp of true wholeness must be made in future remarks which I plan to post soon.