The work known familiarly as Newton’s Principia is the foundation stone upon which our concept of science has been erected. Despite all the transformations by way of quantum physics and relativity, this bedrock image of objective, scientific truth remains firm. Arriving now, however, as if from outside our own world, we may feel a new sense of wonder, and presume to ask a few impertinent questions about core beliefs normally taken for granted.
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?
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.
“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.
A 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
It's great to be able to announce the arrival of a new entry to the Articles department of this website. One of a series of studies I wrote over the years for the Encyclopaedia Britannica's Great Ideas Today, it's titled The Abode of the Modern Muse: The Science Museum. It can be reached by going to Articles on the menu bar; there, choose Great Ideas Today,; and finally, within Great Ideas Today, select the article itself.
I took the opportunity of this assignment to reflect on a long tradition beginning with the MUSEION, the grove sacred to the Muses of ancient Greece, and leading, I claim, in a way important to us today, to the role and concurrent responsibility of the modern science museum. Along the way the essay makes major stops, first at Alexandria, where it treats the celebrated "Library" as more truly an academy, and thus just such a meeting-ground of human minds; and finally, at our own Smithsonian Institution, regarded from its inception as a centerpiece of the scientific spirit of our nation.
One crucial role at the outset of this story is that of Aristotle, who affirmed, very much in his manner as thoughtful observer, that the human community is in essence one, and that a fundamental goal, alike of ethics and of politics, must be to realize this truth in practice. The tradition seems secure that Philip of Macedon, to free his son from the distractions of the court at Pela, hired Arisotle as tutor of Alexander, and sent the two of them off to the hills of Macedonia to focus on education. The curriculum may have been cut short by Alexander's early ascent to the throne, but it seems clear that Aristotle's advice concerning the unity of the human community was foremost in Alexander's mind when he made the founding of Alexandria in Egypt one of his first, and most successful projects.
There were to be many more Alexandrias as Alexander carried his campaign of munification across the Middle East. Readers may have encountered a recent exhibit of Ai Khanoun, an Alexandria discovered to everyone's complete surprise under the sands of northern Afghanistan; my guide at the East Wing of the National Gallery in Washington reported there are believed to be perhaps a dozen more to be unearthed, if our own present wars might cease. But the Egyptian center was surely the best. It began, indeed, as a "library", whose mission was to collect books from the entire Mediterranean basin; copies were made at a publishing house (apparently the building close to the harbor destroyed by the legendary fire). The copies were sent back to the sources, while the originals were stored securely at Alexandria.
The books, however, were gathered to be studied, not simply to be stored, and in this sense Alexandria is better thought of as paradigm of the universitas, than as library, fundamental as the books themselves must be. As university, Alexandria was conceived to be a new center of human learning for the entire Mediterranean world. It succeeded in that role to a remarkable extent, and we today are its beneficiaries in ways of which we aren't always aware. This was indeed a science museum, as the works of Ptolemy and Euclid, to cite just two examples, attest. Euclid's Elements is a synoptic work, a gathering of contributions from probably widespread sources. What is most exciting in that work is Euclid's own: his brilliant grasp of a profound unity arising out of these contributions. It is a true Alexandrian moment when Euclid perceives in this mathematics the pattern of the tragic trilogy: for those tragic texts were being gathered and assembled in their own unities by that single community of thinkers. It had not occurred to anyone-least of all to Aristotle!-that the human mind need be or could be, compartmentalized into separate academic domains as we have done today. Academic labors could indeed be divided, but the human mind, as gathered at Alexandria, remained focused on the whole.
This understanding, the article claims, remained intact in the early days of our republic: it is not by chance that our corporate seal, reproduced on the dollar bill (as well as on the seal of my own college, St. John's) depicts an Egyptian pyramid and an insightful eye. Nor that the leader of the procession dedicating the new Smithsonian Institution was reportedly wearing George Washington's Masonic apron. When Smithson's benefaction was accepted as a gift to this nation, the concept of the liberal arts and the unity of learning was still very much alive, and the institution founded in his name was meant as a center of new learning very much in the Alexandrian tradition. We tend to forget this, but other science museums, here and abroad, today wear that same mantle, whether we are always aware of it or not. Most unfortunately, we forget that is not just science, conceived as domain of human endeavor separate from others, but rather science as an integral component of that spectrum of all human thought, collectively the best we can do in understanding and guiding our precarious life on this planet today.
The essay closes with a severe criticism of the abandonment by the Smithsonian, under heavy industry pressure, of a project in conjunction with an exhibit of the Enola Gay. The exhibit had been thoughtfully designed to help the public review in a social and ethical context, the decision to launch our two atomic bombs. Some readers of the essay in the past have disagreed with this judgment on my part, and in this matter, as in all others, I would welcome readers' comments.
Now more than ever we as a world community need to gather our collective wits by any means possible. Science stands at the center of many of our pressing concerns, and the science museum may still be one of the best institutions we can turn to, as the grove of our modern muse.