The allegory includes a group of people who have spent their entire lives imprisoned in a dark cave. Shadows play along the walls of this cave, and they watche these shadows, study them, believing them to be the fundamental observable manifestation of the reality they live in. One day, though, one of them is freed. In the world outside, he sees a camp fire, with figures moving around the fire. It is these forms gathered around the fire which have been casting the shadows. They are a deeper reality which he was never able to see until he was freed.
The cave, according to Plato, represents our daily life, and the shadows are all of our earthly experiences. The purpose of the allegory is to suggest that the things we experience in our lives are but shadows of a deeper and truer world. He refers to this world as the world of pure form. Our experiences are but imperfect shadows cast by these “true” forms. For example, he might say that every apple you’ve ever tasted or even seen was an imperfect manifestation of the one true, perfect apple that exists in the world of pure form.
This idea continued to influence western thought by way of the Neoplatonists, members of a loosely associated group of mystery religions that took the idea of Platonic Forms as a central tenet. In these mystery schools, and also for later branches of western esoterica, such as Renaissance and Enlightenment Hermeticism, and the nineteenth century Gnostic revival, the idea of Platonic forms found a ready partner in Pythagoreanism. Pythagoras taught that number and geometry were the ultimate reality. For the the Neoplatonists, and possibly for Plato himself, whom many believe to have been influenced by Pythagorean philosophy, the realm of platonic ideals was mathematical and geometric in nature; it’s reasonable to say that for many of them, mathematics was the realm of ideal forms in a veral literal sense: that numbers, equations and proofs had a reality of their own that is independent of and in many ways truer than our own.
The concept of sacred geometry sprang from this Pythagorean/Platonic way of thinking. A common tool of western mystics for over two thousand years (or perhaps much longer), sacred geometry studies various geometric forms believing they conceal great secrets of a spiritual nature. For example, the Pythagoreans held the pentagram to be a powerful and sacred symbol of the creative powers that give birth to the universe. They studied its mathematical properties, believing that a deeper understanding of the many complex relationships and ratios hiding throughout the shape would allow a deeper understanding of the world around them and how it arose from those hidden first causes.
Though it may be myth, there is an old story which gives a sense of the extent to which the ancient Pythagoreans invested their mathematical heaven with an imposing aura of power and sanctity. One of the groups of geometric figures they held especially sacred are the three dimensional polygons that we know of today as Platonic Solids. A platonic solid is a convex three dimensional figure that has identical faces which are all regular polygons. There are five such figures, the most familiar of which is an ordinary cube. They’re named after Plato, who wrote about them in his Timaeus, but the first mathematician known to have described all five was Teaetetus, a contemporary and friend of Plato’s. One hundred years before Plato and Teatetus wrote freely about these shapes, Hippasus, a mathematician and supposed member of the Pythagorean sect wrote a description of how to construct one of the five - a dodecahedron, which is a shape with twelve pentagonal sides. The dodecahedron was particularly important to the Pythagoreans; it’s pentagonal faces were associated with the pentagram, which they believed to be a force of creation, and the number twelve, the number of faces on the dodecahedron, was associated with heaven and the stars. According to the myth, Hippasus was killed by his Pythagorean brothers for revealing the mysteries of this holy object.
While it is certainly true that most people in the modern age would find the idea of executing someone for publishing a geometry proof preposterous, the philosophy behind the use of sacred geometry has remained virtually unchanged for thousands of years. The study of mathematics and geometry - typically classical mathematics and geometry as understood prior to Newton and Leibniz - is only part of the process. The mathematics they study is not understood as a literal description of something in the “real world,” but a set of relationships that describes a type or form. This type can be the true form behind physical objects, and also behind philosophical or ethical concepts, such as justice, or peace. To explore the nature of these forms and how they relate to perceived reality, the student relies on free associations that are guided primarily by intuition and need not conform to strict rational techniques; although generally a student will discard results of this intuitive process that don’t “make sense” when complete.
We’ll talk more about this process later. It has strengths and weaknesses, and while it can provide many great insights, most who use it also misuse it in my opinion. In the mean time, it might have occurred to you that there is a second, particularly modern way of exploring the nature of reality that has a profound link to the Pythagorean idea that math is the ultimate reality, and the Platonic idea that these mathematical constructs are forms which define how reality as we perceive it manifests. This same philosophy is one of the basic assumptions of science; one which drives how we use mathematical concepts and equations to explain physical processes.
First, though, a question. Many of the ancient Greek philosophers were intimidatingly brilliant. Their understanding of the basic laws of logic were very similar to our own. While they lacked a calculus, they had powerful algebraic and geometric tools at their disposal, including the geometry of conic sections which has come to play such an integral role in modern physics. They were keen observers of the world around them, and they had already settled on the fundamental idea that that world was governed by mathematical principles. So why have we been so much more successful at producing repeatable mathematical principles that consistently describe how the world behaves? It seems that, 2300 years ago, the Greeks already had all of the tools at their disposal to produce many of the discoveries that we have made in the last 400 years.
The most common explanation for this discrepancy is the scientific method, a rigorous approach to experimentation which seeks to dispense with dogma and unnecessary assumptions and allow observation to be the sole arbiter of the truth of any assertion. Certainly, this has played a role in our success. However, while the burden of past assumptions would clearly have slowed the progress of any ancient Greek equivalent to modern science, there’s no reason to think it would have prohibited it. In fact, while they possessed no precise analog to our scientific method, they were not strangers to the idea of discarding unnecessary dogma when it contradicted reason or observation.
However, there is a second difference between the ancient Greek approach and the modern scientific one. Remember that, for Pythagorean and Platonic philosophers, our sensory experience is an imperfect copy of a pure world defined strictly by the laws of mathematics and reason. For the Greeks, math described that which IS, and what we experience of that is imperfect.
That is important for two reasons. First, it would have slowed the process of experimentation. Because the ultimate truth was composed of pure reason which could only be perceived by the intellect, and our senses experienced only imperfect copies of this world of pure reason, experimental evidence contradicting a theory would have been mistrusted. The second, and less recognized, limitation is this: modern science has experienced a fundamental, if largely un-analyzed, shift in how those ideals themselves are seen.
Consider Newton’s Apple. Newton had been working on his calculus and his theories of mechanics, and was struggling to come up with some theory to tie everything together. He was sitting under an apple tree pondering how this might be done when an apple fell from the tree and smacked him on the head. He had an epiphany: the one concept which he could use to tie the others together was gravity.
To the Greeks, math described the perfect apple and the perfect human, of which this particular Newton and this particular apple were as reflections on the wall of a cave. To Newton, however, it explained why the apple hit him on the head.
Let’s take that a step farther by generalizing. To the ancient Greeks, math and reason described pure, perfect forms and we experienced reflections. This is a dualistic understanding of the world, albeit a soft, graduated dualism of a less and more real world. To modern science, math and reason do not describe forms, they describe interactions.
The Platonic-like bond between math and physics has been so successful that it seems they must have gotten something right. They haven’t just proposed a link between mathematics and reality, they seem to have discovered what that link is and how it operates. If they’re right, though, the effect on any metaphysics built around the concept of platonic ideals - such as one that draws upon a technique like sacred geometry - is both drastic and profound.
The core assumption of such a metaphysics is this - the world described by reason and geometry, the world of “pure forms,” is a deeper and more real world than the world of our daily experience. However, according to science, these aren’t exactly two different worlds. The duality is removed. Rather, reality as we experience it confirms precisely to the dictates of reason, which determines how things in this world interact with one another, and the forms we perceive are as they are as a result of these interactions.
To put it another way, when we talk about “reality” we aren’t delusioned prisoners trapped in a cave, metaphorically speaking. Rather, we simply have things upside down. When we look out at the world we see “things” that are real, and we see them interacting with one another. What science tells us, independent of any particular theory that may be right or wrong, is that what we see as “interaction” between things that are real is, in fact, the true reality, far deeper and more real than the “things” which we believe to be interacting.
This may seem like a strange thing to say, but it’s been said before in a spiritual context. There is an ancient Zen koan (a puzzle used as a meditation aid) that talks about a master and a student sitting and listening to a rain shower (emphasis is mine).
The teacher turns to the student and asks “What is the sound outside?”
“That is the sound of rain striking the roof,” the student responds.
“Sentient beings are upside down. They lose themselves in the pursuit of things.”
The masters response confuses the student, and after a moment’s thought he asks “Master, what should I do?”
The master responds. “I am the sound of the rain.”
