“The fault, dear Brutus, is not in our stars but in ourselves.”
Julius Caesar, Act I, Scene III, L. 140-141
“By the ancients man has been called the world in miniature; and certainly this name is well bestowed.”
Leonardo da Vinci
Perhaps no work better epitomizes the Renaissance than Leonardo da Vinci’s Vitruvian Man, which ironically was never meant to be displayed at all: the sketch was discovered among da Vinci’s personal notes and observations and was not published until 1810, nearly three hundred years after the great man’s death. It is based upon a concept proposed by Marcus Vitruvius Pollio, a Roman architect and engineer who lived in the first century B.C., in a ten-volume treatise entitled De architectura (On Architecture); Vitruvius, who was fascinated by proportions, suggested that the best man-made structures imitate nature by exhibiting three qualities which came to be known as Vitruvian virtues: firmitatis (it must be sound, stable, and strong), utilitatis (it must be functional, serve a purpose), and venustatis (it must be aesthetically pleasing). He also submitted that these traits characterize the ideal human form, the most captivating work of art; his hypothesis was that a perfectly proportioned man could be encompassed by both a circle and a square with his navel being the center-point of both geometrical figures, and he included dimensions supporting his contention:
… in the human body the central point is naturally the navel. For if a man be placed flat on his back, with his hands and feet extended, and a pair of compasses centered at his navel, the fingers and toes of his two hands and feet will touch the circumference of a circle described therefrom. And … if we measure the distance from the soles of the feet to the top of the head, and then apply that measure to the outstretched arms, the breadth will be found to be the same as the height, as in the case of plane surfaces which are perfectly square.
In his drawing, Leonardo proved Vitruvius was wrong; when da Vinci circumscribed his model with a circle, he discovered that in order for the navel to coincide with the circle’s focus, the man’s legs had to be spread slightly apart, lowering him (thus, raising his midpoint), and his arms had to be elevated as well as outstretched to touch the circumference. However, in this disposition, Leonardo could not circumscribe the figure with a square, as Vitruvius specified; to meet that requirement, da Vinci had to straighten the man’s legs and have him hold his arms out at shoulder level to their full span, fingertip to fingertip. In this pose, the center of the square is approximately at the level of the man’s genitalia.
Da Vinci’s sketch was not the first to test Vitruvius’ thesis, but it is the most celebrated, and it is an amazing work of art. A single man, portrayed in two superimposed postures, presents us with a figure simultaneously holding sixteen observable positions.
It seems probable from this that Vitruvius calculated his indefectible man to correspond with abstract benchmarks and never tested his hypothesis with a living model. Further, it appeared to be critically important to Vitruvius that his perfectly proportioned man be inscribed within both a circle and a square; yet squares are geometric constructs that do not exist in nature.[i] For all his protestations that nature must be the ultimate guide for the finest architects, Vitruvius’ ideals are generated by Mind rather than world – an odd contradiction for an engineer but not unique to him, for it is part of a dialectic that has engaged some of the most gifted thinkers for thousands of years: what is the relationship between particulars and universals?
Three hundred years before Vitruvius wrote De architectura, Plato was working out a theory of knowledge in his dialogues; true knowledge, he contended, must fulfill two essential requirements: 1st, it must be infallible or certain; and 2nd, it must concern what is, (it must concern that which is stable and constant). If we accept these criteria, then we must concede that true knowledge does not come to us by way of our senses, for the information we gather through them is neither infallible [ii] nor constant.[iii] But if we eliminate the material world from consideration, what is left that meets the requirements Plato set forth?
Plato attempts to answer this question by way of a dialogue between his mentor (and exemplar of philosophical virtue), Socrates, and a youthful student named Glaucon, suggesting, in the sixth book of his Republic, a division of potential sources of knowledge into two general streams, one of which originates in the Visible Realm [iv] or World of Appearances, and consists of the people, places, and things that constitute the material world and which we perceive through our senses, and another emanating from what Plato calls the Intelligible Realm or the World of Forms, which is a bit more difficult for us to fathom since we cannot directly experience it. Socrates tries to help Glaucon (and the rest of us) understand this concept better by eliciting responses to a series of questions he poses (and often answers himself) and through a variety of metaphors, such as the Simile of the Sun, the Allegory of the Cave and the Analogy of the Divided Line.
In his Divided Line analogy, Socrates explains that not all sources of information are equally valid or valuable, for they yield varying degrees of clarity and truth. In the diagram, the World of Forms is atop the World of Appearances. The idea is that the lower on the line we are, the further we are from true knowledge and what Socrates calls “the Good,” which is something beyond his ability to express in language; thus, he relies on metaphors.
However, before we can begin to apprehend the higher Intelligible Realm, we must consider the lower Visible Realm, for these two are analogous; thus, our understanding of the one will support comprehension of the other, and since we experience the material world, it is much easier to talk about.
The sun presides over the Visible Realm, for without its light, even though we have eyes, we would be unable to see; further, the clarity of our vision is proportional to the illumination emanating from the sun: we see most clearly at noon on a cloudless day and far less clearly when the sun is low in the sky or obscured by clouds. This is a relationship that is critical to the understanding of Platonic Forms.
Socrates subdivides the Visible Realm into two sections; the lowest, D, the one furthest from true knowledge, we can call “Images,” and the one directly above it, C, is the material world. If we place the sun at the top of the Visible Realm, as though it is at its highest point in the sky, the division makes a lot of sense, for the sun sends its rays into the world, shining upon people, places, and things [C] and casting images [D] of them onto lower surfaces; so we might imagine gazing upon the surface of a pond and seeing the reflection of a tree. The tree is not in the water; the water reflects the image back to our eyes. We can actually gather quite a bit of data from this reflection; however, it does not provide nearly the amount or accuracy of information as direct sight of the tree itself would. A shadow on the ground reveals even less: we would become aware that someone else is near, but we would know very little else; depending upon the position of the sun in the sky, the shadow might be a foot in length or eight feet long, and we might not even be able to tell the gender of the person casting it, let alone his age, ethnicity, hair or eye color, etc. In this section of images, we conjecture, and we engage in mythological thinking.[v] This is also where we find the visual arts: we are not seeing the apple, for instance, but an artist’s representation of the apple.
The Intelligible Realm works in a very similar way. Instead of the sun presiding over it, Socrates has the Good at the top. The Good does not emit light, for there is no physicality; however, we might say that it is the source of cognitive illumination. The lower section of the Intelligible Realm [B] corresponds to the lower section of the Visible Realm [D], insofar as it consists of reflections of the higher section. B includes geometry and mathematics, which are both infallible and constant: when we say two plus two equals four; it is as true today as it was four billion years ago, as it will be four billion years from now, regardless of whether we live in upstate New York, Beijing, or the dark side of the moon. It would be true even were there no sentient life to acknowledge it. A square is a quadrilateral with four equal sides and four right angles, as it was the moment when this multiverse came into being, just as it will be at the moment all creation passes from existence, at every point in time and place. B is also where we have hypotheses and scientific thinking, such as Newton’s laws of motion or Einstein’s realization that energy equals mass multiplied by the speed of light squared (E = mc2). This is where understanding resides. Scientific principles exist above nature but are focused downward, describing the world in which we live.
Plato, however, believed there must be something above this, a section even closer to the Good or true knowledge. This is where Platonic Forms exist, ideals and reason, pure thought that does not need to connect to the material world. In this region, we find such concepts as justice, truth, beauty, in addition to Forms which are projected into the material world: humanness (what it is to be human), redness (the universal color that includes every shade of red), roundness and squareness (as opposed to the reflections which are cast into the lower regions) – whatever we experience or can imagine has an ideal Form to which it corresponds in A and of which it is an individual expression. When the Good “shines” upon these fully conceptual entities, the “shadows” they cast become the hypothetical theories of section B, which attempt to explain the world, C, and lead to imaginative representation, section D.
Just to add to the confusion, Plato believed that the entire Perceptual Realm, all corporeal existence, was less real that the higher regions; just as we see section D as the domain of shadows and reflections that can confuse and mislead us, Plato saw the subjects of those shadows and reflections, C, in the same way, and even the population of section, B, to an extent when it is concentrated down the line in an attempt to explain the material world and contain it in theories and hypotheses. Thus, the true philosopher does not concern himself with the particulars that surround him, which will always, in Plato’s estimation, lead him away from the greater truths and entangle him in insignificant details; instead, he rejects the outer world in his pursuit of the Good. The way to recognition of the Good, the ultimate truths or universals, is through the Soul rather than the body; the answers are found within, not without, and this is the purpose of Plato’s dialectic: his protagonist, Socrates, engages his students in discussion, asks questions with the specific purpose of forcing them to examine axioms that they have had instilled in them, unchallenged, from childhood, and elicit responses free from the taint of rote learning and the mindless deference to acknowledged authorities and masters (a method which did not endear the real Socrates to the acknowledged authorities and masters of Athens, culminating in a conviction for corrupting the minds of young men and an execution).
The problem with universals, the concepts that constitute the highest region, A, is we cannot really know them. We can infer the existence of truth or justice or humanness, but we can no more wrap our minds around these abstractions than we can around concepts like God or Soul. My favorite definition of God (and probably the most famous one, though no one can say for certain who originally conceived it) is from Liber XXIV philosophorum, The Book of the Twenty-Four Philosophers, Deus est sphaera infinita cuius centrum est ubique, circumferentia nusquam (“God is an infinite sphere whose centre is everywhere and whose circumference is nowhere” [vi] The definition forces us into a logical contradiction that halts all thinking; the idea, I gather, is that if we were able to comprehend God, then it would not be God: it would be something very much smaller. (I had written in an earlier draft that it would require the mind of God to comprehend God, but then I thought that if God could comprehend Himself, He would not be God, for comprehension necessarily implies limitation. However, if God were unable to comprehend Himself, then that would be a limitation as well, so He would not be God… It is no wonder that scientists prefer the observable world.)
Vitruvius was inspired to connect man and geometry in a visionary consonance. Da Vinci’s illustration proved the hypothesis unworkable; however, we might say from a Platonically-oriented point of view that da Vinci’s findings are inconsequential insofar as it signified a fundamental misunderstanding when he brought Vitruvius’ vision down from the aether,[vii] so to speak, into the material world to test it. The failure is in a model unable to transcend his physical and fallen nature, not in the perfect geometric forms with which Vitruvius sought to circumscribe his indefectible man, nor in the inspiration to promote him to that eternal realm of Mind.
Turning and turning in the widening gyre
The falcon cannot hear the falconer;
Things fall apart; the center cannot hold [viii]
Man, as he is, is insufficient and must be if we are speaking of corporeal bodies, not to mention circles, and squares, however perfect – concretizing that which is meant to exist in a region of pure and idealized thought. “The center cannot hold,” and when it falls apart, man finds himself embracing a fantasy spun from cloud or plummeting back to earth.[ix]
However, in da Vinci’s illustration, man is preeminent; he “is the measure of all things,” [x] including geometric forms. If Vitruvius’ hypothesis is unworkable, then it is the geometry which is unworthy and must fall away, for man surpasses even it in his aesthetic ascension.
[i] As I write this, I realize there are people who will contend that this statement is not true: squares do exist in nature. They will point out that salt crystals, for instance, are more or less cube shaped, as are other crystalline formations; thus, (they will argue) I ought to have written “perfect squares do not exist in nature.” To these individuals I will say I did not inadvertently forget to include the qualifier “perfect”; rather I chose to leave it off since it is a redundancy and therefore unnecessary. A square is a quadrilateral with four equal sides and four right angles: there is no such thing as an “imperfect” square; all squares are, by definition, perfect squares, and every example of a square existing in nature is really the approximation of a square; thus, not a square. More to the point, Vitruvius, being an engineer, was not interested in circumscribing his perfectly proportioned human figure with something roughly square-shaped – that would have utterly defeated his purpose: he inscribed his ideal man within a perfect circle and a perfect square. And I should state at this point that while circular shapes exist everywhere in nature, perfect circles, such as those defined by a compass, do not occur naturally either.
[ii] We know now, better than Plato did, that our senses translate back to us only a fraction of the input besieging us every moment: there are many sounds we simply cannot hear, odors we are incapable of smelling, and our eyes, unaided by scientific equipment, detect only a fraction (the visible spectrum) of the electromagnetic fields that surround us. Further, some part of our brain that works below the level of consciousness acts as a kind of gatekeeper, sending on information that it deems useful and discarding information that it judges to be superfluous or unimportant; this activity is necessary for it prevents us from being overwhelmed by an unending flood of sensory data, but it also contributes to the fragmentary awareness we have of the world. Even our courts of law recognize that eyewitness testimony consistently proves to be unreliable; this is a conclusion completely at odds with the inherent belief we all possess that if we see something with our own eyes, then we can be absolutely certain that that is what happened (a conviction magicians, illusionists, and street hustlers have exploited for centuries). Additionally, the information we gather from our senses goes through a translation process which makes it more harmonious with our natural predispositions, so that two people can be witness to the same event and form two diametrically opposed conclusions regarding it based upon their individual psychological states at that time.
[iii] The material world is in a perpetual state of becoming; as Heraclitus (c. 500 BCE) so famously declared, Panta Rhei, “life is flux,” more commonly rendered as “the only constant is change.” This tenet of his philosophy led him to proclaim, “the way up and the way down are one and the same.” Every thing in this world is in the process of transitioning into something else; the tallest mountains and the deepest valleys are ultimately reconciled over the course of time, as are being and non-being. We say we know a person, but what we really mean by this is we are acquainted with a stage of his development: we know him as an infant, an adolescent, a youth, a man at the peak of his physical and mental abilities, or in his dotage. In truth, we probably are asserting that we are familiar with his external appearance, perhaps a couple details of his life and a few of his mannerisms – things which can change over the course of time. We cannot really say that we know the innermost thread that makes him the unique individual he is and constitutes identity through the course of his life.
[iv] In my diagram, I call this the “Perceptual Realm,” as we relate to it and learn about it with all of our senses. Socrates recognizes this as well, but for the purposes of his analogy, he limits himself to sight; thus, he refers to this empirical world as the “Visible Realm,” so I will use the two terms interchangeably through the remainder of this post.
[v] Mythological thinking occurs when an individual is unable to clearly distinguish between his psychological processes and events in the outer world, so he sees connections that do not exist anywhere other than within his mind. For instance, a person may believe that a certain object, such as a coin or rabbit’s foot, brings him luck or that another individual has cursed him, causing misfortune to follow him about. Fortune telling, discerning meaningful patterns in tea leaves or wine sediments, believing that one’s bad behavior has brought about a natural disaster or that one can cause rainfall through a ritual dance or prayer are other examples. Though we tend to assign this kind of superstitious thinking to primitive peoples, if we are honest and observant, we find that it remans quite prevalent even among our more sophisticated (and educated) modern societies.
[vi] This is also translated sometimes as “God is an intelligible sphere whose centre is everywhere and whose circumference is nowhere.” I very much like this, and it is how Joseph Campbell renders it in his book Myths to Live By; however, the Latin word infinita means “boundless, unlimited, endless,” and, of course, “infinite.”
[vii] In Greek mythology, Aether was a primordial deity, father of Gaea (earth), Uranus (sky), and Thalassa (sea); he was the personification of the pure essence that the gods breathed in contradistinction to the vulgar Aer or Chaos encircling which ordinary men breathe. Aether is also known as quintessence or fifth element in alchemy, the incorruptible substance into which the four quarrelling elements (earth, air, water, fire) are harmoniously reconciled. The quintessence or fifth element is identified with the lapis philosophorum, philosopher’s stone, the goal of the great work and is not subject to change and decay.
[viii] William Butler Yeats, “The Second Coming,” lines 1-3.
[ix] “Embracing a fantasy spun from cloud” references Ixion, the villainous king of the Lapiths from Greek mythology who betrayed Zeus’ hospitality and attempted to seduce Hera; Zeus fashioned Nephele, a cloud made to resemble the queen of the gods, and Ixion impregnated it, a union from which Centaurus, a deformed child, was born. Centaurus belonged with neither the gods nor men so lived on Pelion, a mountain at the southeastern part of Thessaly in northern Greece, where he roamed and eventually mated with the wild mares there, giving rise to the race of Ixionidae, better known as kéntauros, centaurs. “Plummeting back to earth” refers to both Icarus and Phaeton, two youths who soared into heights where they did not belong and fell to their deaths.
[x] “Of all things the measure is Man, of the things that are, that they are, and of the things that are not, that they are not,” Protagoras, near contemporary of Plato’s whose relativism Plato attacks in multiple dialogues as reducing all knowledge to subjective opinion, for if ten people hold ten different opinions concerning a single experience, and all are equally valid, the possibility of objective truth is negated.