Saturday, December 21, 2013
The History of Heaven and Earth 04: The Music of the Spheres
When I learned about the truth of the Pythagorean theorem in freshman geometry, we “proved” it algebraically, and the illustration used by the teacher was a drawing of a random single triangle with sides a, b, and c. One might suppose Pythagoras did the same, right? I figured as much when I was originally taught, and, certainly, no one told me otherwise at that point. Well, while that is a way to learn that the Pythagorean theorem is true, it also both does a disservice to where the theorem came from and what it really means, and the way we are taught reveals the interpretive lens with which we now engage the theorem.
It is most likely that he demonstrated it with four geometric tiles, made of clay or wood, similar to the picture of this textile piece here. The sides of these triangles are not arbitrary lengths of a, b, and c. Rather, these triangles have sides that measure 3, 4, and 5 units, respectively (3 squared + 4 squared = 5 squared). The sides of the triangles end up being “squared” quite ACTUALLY, as the APPEARANCE of a square is what demonstrates the theorem. The square’s being made of clay or wood makes the meaning of GEOMETRY quite ACTUAL. Geometry literally means “EARTH MEASURE” in Greek.
Pythagoras MADE the truth appear THROUGH his demonstration. A previously hidden truth was, through the course of his demonstration, able to be OBSERVED. Hence the term “theorem.” Remember (from the last post in this series), the word “theory” is rooted in the Greek word “thea”, which means both “to see” and “god,” or “divinity.” Pythagoras did not “force” the truth out into the open, and he did not experimentally prove an invented hypothesis. What was being SEEN was associated with DIVINITY because of causation rather than force. Force is only possible to be applied from outside a thing. With the dawn of speculative thought, man had not attained to such a high level of exteriority that allowed the observation of force.
This means that the idea of the theorem itself actually lies latent or hidden in sensible reality. The mathematics, the number that Pythagoras saw as a governing principle of what and how appearance occurs was considered to lie hidden WITHIN apparent reality itself. For the Greeks, the cosmos was considered to have its own internal order. The philosophers, the pioneers of speculative thought, so to speak, were the first to presume that this order could be ascertained through observation of its natural workings by using their own powers of reason and faculties of intellect. Pythagoras then made four clay tiles to make appear what makes things appear.
The earliest philosophers, in fact, explicitly denied the role of divine revelation in the truths they were finding. Speculative thought, then, was not only a step towards exteriority but also the first step towards the reductionism that is a an inalienable and essential theoretical component of modern science. Speculative thought was, however, only one small step, because – as only one sample of their thought - the earliest philosophers also believed that human thought could mimic divine understanding.
Pure modern reductionism would refer to that as irrational nonsense (and at best, it probably sounds confusing to us), while the earliest philosophers considered it to be the very definition of reason. This reductionism that Christianity ultimately denies but that still fundamentally shapes our idea of what appears physically in the world (and how it does so) is part of why we would think of heaven as “somewhere else.”
Also, notably “order” for the ancient Greeks – the cosmological order that governs all that moves, changes, and exists – is not simply in reference to chronological time. “Order” for the ancient Greeks is not in reference to a chronological sequence of events. “Order” here is in reference to hierarchical causation. The same term the pre-Socratics used for the “prime substance,” or the original substance from which all things come – the Greek word “arche” – was also used as their word for “mayor,” as in the mayor of the city-state, the polis. The ancient Greek idea of a god or divinity was simply the causing of something to come to appear (and not to appear in the world, because they didn’t have a concept of a “world”).
The first conceptions of the “arche”, the prime material or substance from which all things appear, were elemental, meaning that it was associated with elements. The Greek word is “stoicheia” – element – the same Greek word Paul uses in reference to the “principalities and powers of this world”. Remember back to the idea of the mayor as prime mover or prime causation of the city-state being held together, hence referring to him as the “arche.”
Anyway, the first elemental conception of the “arche” in the realm of what we would now call physics was water. Next was air, and last was fire. In between there was a conception of the “arche” as being a “boundless indefinite”, but the general pattern moved from down to up (from earth to heaven, figuratively) and from participatory engagement toward exterior observation. And, in all cases, a primary reason for the choice of the prime substance or element from which all things appear was that element’s own internal capacity for change and transformation, the qualities and workings of the element itself. This hearkens back to the idea that the ancient Greeks thought of the cosmos as having its own internally hidden order revealed in the actual workings of what appears. That, in turn, points to why what Pythagoras did was a DE-MONSTRATION.
I mentioned previously that, when I was originally taught the Pythagorean theorem, it was proved algebraically. The Greeks had no algebra. Algebra balances two sides of an equation and implies the concept of a zero. The numbers and values in modern algebraic equations have no geo-metric value. The cosmological order of the ancient Greeks was not a balancing act in which the cosmos performs on a tightrope, teeter-tottering on either side of a vacuumous nothing. The ancient Greeks were not observing how the cosmos didn’t implode or explode into Nothing (nor how the cosmos appeared, as a whole, at some given point in time). They observed how one thing disappeared and another appeared (to the senses) in its place over and through time, but they spoke of it more poetically terms of life, death, and rebirth.
This meant that existence itself, the being-here of the cosmos, was assumed. For them, it did not appear out of no-where. The starting point for everything is One. What appears to our limited sensibilities as humans is mathematical ratios of a greater cosmic Unity that is bigger than ourselves. These ratios are the evidence of change, of motion, within the cosmos. Analogically, Pythagoras related this idea to what we could now refer to as a guitar string. The guitar string is fundamentally one. Where you divide the string determines the ratio of one part of the string in relation to the other after dividing it. The quality of the sound heard is determined by that ratio between the two parts of the one guitar string. These ratios determine what we now think of as musical notes!
Think about what an analogy is. An analogy is the GEOMETRIC link between two scales. The analogy between the ratios ½ and ¼ is 2. In other words, if you start with a square (a fundamental unity), and divide it, you get a rectangle whose sides are 1 and 2 units. Divide that rectangle again, and you have a rectangle whose sides are 1 and 4 units. The analogy between all three geometric shapes is the number two. It is a question of scale. The smallest rectangle in that scenario is related analogically and geo-metrically to the original unity of the square (by and through the number 2).
Notably, the term “highest” there, in reference to the highest scales of the cosmos, only partially refers to our looking up to the heavens. Again, in talking about a cosmic order, we are talking about a hierarchy of causation. The motion of the spheres is referred to as being at the highest scale, because it is most constant, most regular, and is seen to be what causes everything else “below” to happen, rather than the other way around. The observed changes from night to day and from one season to another – and all the corresponding changes observed in our immediate environment - are governed by “the music of the spheres.”
As ancient men who had no concept of the number zero, who were bound to their senses, whose everyday reality was shaped by the turnings of the heavens, and who had never before seen a photograph of the earth, this idea of hierarchical causation could have had something to do with how the apostles would have interpreted the ascension of Jesus to the right hand of the throne of God almighty. In fact, the lyrics to the Christmas carrol “O Come, O Come, Emmanuel” mention the idea:
“O come, Thou Wisdom, from on high,
and order all things far and nigh”
Now, part of the overall point I am making in talking about analogies, guitar strings, and the spheres of heaven returns to where I began this blog post. That Pythagoras DE-MONSTRATED his theorem means that the ancient philosophers, although they had taken the first steps up the stairs of the Tower of Babel, still did not have a view of everything from on high or outside. The Unity that they regarded as the truth was far bigger and beyond themselves and could only be engaged analogically. To say it literally, the cosmos, intellectually speaking, was like a big monster, and Pythagoras took a scientific step in taming it. He was still inside the mouth of the monster, but the demonstration was the opening of the mouth to let light in, to illuminate. Less literally, a demonstration was needed at an apprehendable scale “in order to” to get a glimpse of the workings of the whole of the cosmos.
The original Pythagorean theorem, in fact, has very little in common with a modern scientific hypothesis or theory, and, yet, when taken on its own terms, it is still true. Our difficulty in understanding what the Pythagorean theorem even IS, WHICH INCLUDES what it means in relation to apparent or physical reality, itself reveals our lens. In fact, when we learn the theorem in school, the actual meaning of it is either considered too difficult for us to understand or so unworthy of the time it would take to learn it that we completely disregard it. And yet, Corbusier used the overall conceptual framework behind it to make some of the best buildings of all of last century.
I opened this series of blogs asking where we’ve come from, playing off the scripture in Job in which God asks Satan where he came from. This leads to the basic quest-ion at hand, which is, “where is heaven?” As part of the foundation for the rest of the posts in this series, the second post presented the idea that “we become what we behold,” which set the wider future context of that idea for future posts about thought, art, and media and technology. The basic driving point of the series is that: a) we generally wear an interpretive lens, b) that lens lends itself to our thinking of heaven as “somewhere else”, and c) heaven is here.
I think, then, this particular blog post of this series has well made all three points (a, b, and c). Concerning point (a), I think I made it pretty obvious, in the particular context of discussing the Pythagorean theorem (as one example), that we have a lens. In fact, the difference between our lens for the Pythagorean theorem and the actual terms of the original Pythagorean theorem itself reveals our lens for reality itself. I started to address point (b) to a degree, but I will address that more as we go along in this blog series. On point (c), it is no great logical leap from “back in the day, there was no such thing as zero, and everything, quite actually, was One” to “heaven is here” (or, at the very least, that any ancient man would have thought of heaven as “here”).
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