The perfection of a circle

The circle symbolizes spirituality, completeness, wholesomeness, and balance. It has come to represent totality, infinity and perfection. The circle feels deeply familiar and somehow representing life in general. However, a circle is an abstraction. Its ‘perfection’ is illusory. 

For 4000 years the value of Pi has been known. Its value has allowed us to penetrate the perfection of the circle. But Pi is irrational, both in a mathematical sense and in the normal sense. Irrational because the numbers after its decimal place do not terminate, or create any repeating pattern, they go on and on. The number is infinite, devoid of certainty, the value of Pi then is illogical, not perfect.

Yet Pi allows for practical results. This tells me that mathematics is not ultimately inherent in the world, but is a useful echo of it. A surrogate of truth. 

Ultimately math serves us well, so why does it matter? 

What is perfection? Perfection is a state of flawlessness; the highest level of excellence. Our perfection is a product of scrutiny, framed by knowledge and reason. A frame we set based on our experience of the world. Yet, this is a manufactured ideal relative to us. We create the ideal of perfection so that we may claim it; perfect beauty, perfect truth, perfect correctness, perfect knowledge. Yet, the word ‘perfect’ has an inherent flaw; it is relative to human experience. We forget that our view of the world is relative and incomplete in comparison to it all. Any truth we claim, is only a truth we can know. It is not ultimate. Beyond logical scrutiny lay infinite vistas that extend beyond categories of knowledge and sensory perception. This insight needs to be understood, it is why it matters.

The longer version:

The perfect circle

In the 14th century, Pope Benedict XI was looking to commission some paintings to adorn St. Peter’s Basilica. He sent his advisor out to find a painter of suitable skill to fulfill the job. Eventually he met Giotto, a Florentine painter, sculptor and architect who was known to be very talented. He asked Giotto for a sample of his work to bring to the Pope, Giotto agreed and immediately began to paint. Grabbing his brush, Giotto drew a perfect circle onto a sheet of paper. The advisor was confused. He asked Giotto to reconsider his sample but he insisted that it was what he wanted to submit. 

Returning to the Vatican, the advisor handed the Pope the painting. He explained how Giotto had crafted the circle without aid of a mechanical device. The Pope reflected on this for a while and then realized its significance; the circle had a great meaning; it wasn’t just a shape but a symbol alluding to transcendence. Giotto had demonstrated something profound, divine, the infinite, with simplicity and perfection. The sample represented a succinct demonstration of his mastery. This had impressed the Pope, and so Giotto was awarded the job.

The Circle 

The shape of a circle has always been inspirational. From the disc’s of the sun and moon, to water rings cascading across a still pond. It suggests cyclical behaviour, like the seasons or the movements of the planets and stars. As we gaze out into our world, the shape of our eyes are themselves circles, the windows to our soul. 

The appeal of the circle has become universal, grounded in our humanity, a fixture in psychology; a symbol of spirituality, completeness, wholesomeness, balance. It represents totality, infinity and perfection. Basically, it feels natural, deeply familiar and representative of life in general. 

However a circle is only a symbol, a shape, basically it is just an idea. We may feel the circle’s perfection, and see examples in our world but what perfection does it actually have? 

What is perfection?

Perfection is a state of flawlessness; the highest level of excellence. And for all is virtue, does perfection exist? 

Perfection is a product of scrutiny, that which is framed by knowledge and reason. A product of the faculties of our mind. Therefor, perfection is relative to the mind. It is not objectively in the world. Perfection is based on us and our consideration. Yet we examine it and make conclusions as if it were objectively real.

So, it is a manufactured ideal. Once created, it gives us a goal to achieve it, to claim it, to seemingly have the ‘ultimate’, whatever that ultimate may be; ultimate truth, ultimate correctness, ultimate control, ultimate knowledge, ultimate power.

If perfection is meant to be ultimate, it would need to exist beyond opinion, beyond the inferences of our senses. It would need to be a pure expression of fact, logical and consistent, inherent. What could possibly embody this expression?

Mathematics

Many feel Mathematics is the ultimate language of nature. In fact some believe that mathematics is an inborn feature of the universe; that it is discovered, not invented. 

Intuitively, math seems amazingly consistent with the world, and proves itself repeatedly in science. And so math is esteemed as a primary method of validation, and I would feel comfortable in believing this but there is a critical assumption we are making; is math perfect?

Kurt Godel was a twentieth century mathematician who scrutinized mathematical truth, analyzing its core axioms. An axiom is a fundamental principle of logic, a self-evident truth that requires no proof. Here is a simple example, ‘A+B+C=C+B+A’ —add the numbers together on either side of the equals sign, in any sequence, and their products will be equal. This is assumed true, therefor this is an axiom.

From this, other axioms can be derived, with the hope that we have a foundational set of axioms proving the truth of math’s systems of logic – hence showing that mathematics is consistent, and is an ultimate truth.

What Godel has done was test mathematic’s axiomatic statements by using a system of code numbers; a system that became like a surrogate mathematics, allowing him a way to prove ‘self-evident’ truths. Surprisingly, he discovered unresolvable paradoxes within certain axioms, tainting their perfection. Here is a sample of one such paradox rendered into plain language:

The following statement is false. 

The preceding statement is true.

The flawed logical, or double-bind, of these two statements is illustrative of what Godel discovered, demonstrating a fundamental problem with Math. 

However, does this prove that math and logic are not inherent in nature? Not necessarily. But we need to be careful not to mistake products of the mind for inherent characteristics of the world.

We see the world as a collection of objects that can be counted. And so our very neural network recognizes sets of objects and creates a numerical language to organize them, i.e. two donuts on a plate, two bugs on a window sill. However, the ‘twoness’ is a property of both sets of objects (donuts and bugs) but the objects do not share same ‘universal two-ness’ between them. Their two-ness is a result of our experience of them and the categories we wish to characterize them with. This action allows us to take account of our experience by making it comprehensible. 

Is there two-ness? 

Atomic science has shown that on microlevels objects do not have any real distinction of separateness. The boundary between two doughnuts is not definitive. And so the mind-bending fact of quantum theory is that on an atomic level matter does not exist, yet matter is the basis of our reality, the foundation of our truth. The same truth we find mathematics springing from. Again, we must be careful.

Physical distinctions are our experience, very real for us, but not absolute reality. And so here we see our vanity as thinking beings; we believe that how we regard the world is precisely the ‘is-ness’ of that world. 

In fact just as ‘two-ness’ is a property we ascribe to countable objects, even the identity of ‘doughnut’ again is just a word, an idea we impose towards it, to give it meaning so that we can intellectualize it. Words and language help us to conjure the concepts we need to comprehend our world. And so, we are always one mind away from the fullness of reality.

In a way, because everything is relative (to mind), we are constantly tempted to find the ‘ultimate’. This is why we are quick to declare that math, logic, faith, god, etc. is the foundation of reality. We can’t help it. We need the ultimate answer. But ultimately, it is merely reflections of the mind. It is all opinion. 

What is a circle?

A circle is a simple shape of Euclidean geometry. It is a closed curve that defines a plane in 2D space. One main mathematical property of a circle is the value of Pi(π). Here is the value including the first 15 decimal places:

π = 3.141592653589793…

The decimal places go on forever and never fall into a repeating pattern, making Pi a ‘Transcendental Number’. 

Pi represents the ratio between the circumference and diameter of a circle. Interestingly, Pi tells us that if either measurement is known precisely, the other is not. For instance, if the diameter is 1 (a precise value), the circumference will be imprecise. Oppositely, if the circumference is 1, the diameter is imprecise. Therefor the circle cannot be entirely and precisely calculated.

All in all, Pi allows us to calculate properties of circles, and in this way is very functional, but to know a circle entirely and precisely, is impossible. 

The math we feel is out there is actually a phantasm of our ideals, a projection of our thought processes. Simply evoking the value of Pi as the perfect character of a circle is not an absolute description of it. The description is a delusion. Of course, for mathematicians and scientists, any calculated result gives them the products they need to complete their study. It feels absolute to them, but whatever decimal places they stop at, is arbitrary—is this truth then?

The answer is no. 

If math is reality, and its expression of a circle is perfect, then a perfect circle must be achievable in physical reality. But a perfect is impossible to build. This is not because of any technical challenge, but rather any physical form you care to conjure necessarily requires being built from atoms. And atoms have no disposition towards circular-ness. And just like any shape, it is merely an expression of an idea. And shockingly, so are atoms.

Atom 

We believe atoms are distinct little entities, but this is not true. When you look closely at matter, you don’t find smaller bits of matter, you find energy. We have been taught that the atomic world is comprised of parts, but the consideration of these ‘parts’ is more of a convenience to understanding them. 

The atomic world is more accurately described as an interconnected, interpenetrated sea of energy, with no edges. The problem is our mind likes to divide things into associated parts so we can understand phenomena systematically. We like to intellectually dismantle ideas, like dismantling an engine so that we can know how it operates. The difference being, any part of an engine actually exists, clearly separate and well defined in its function. This is not the same in the atom. An atom is not simply the sum of parts.

How many waves make an ocean?  

The scientific method cannot always be applied flawlessly to every facet of the physical world. Atoms are the very reason for matter, but they do not ‘have’ matter. What they have goes beyond classical distinctions. Yet science requires these distinctions in order to apply its rigorous methods. So, science tries to break apart the atom, but the atom actually breaks apart science. It takes away the degree of certainty that science wants so desperately.  

It’s like trying to understand the ocean by capturing a wave in a jar. What was once a wave travelling on the ocean, becomes a wave in a jar. Completely different. Of course, the water contained is still very useful to examine, but it is not demonstrate a comprehensive view of the ocean itself. 

You don’t need to contain a wave to measure it however, it can be measured as it happens. But then this is just surface of the ocean, not its depth. 

I am getting very abstract and metaphorical here, my point is; How we decide to cut into the energy web of reality, is what we choose to identify and label for our purposes. We snip away at the atomic world to create parts like ‘electron’, ‘proton’, ‘nucleus’. But these distinctions are more for our purposes only, for comprehension, for knowledge. Of course the methods science uses to understand the atomic world clearly pays dividends, but ultimately, philosophically, we do not have a comprehensive understanding, we can’t, its beyond us and our relativity we impose. 

And so the entirety of the cosmos is not applicable to ‘human understanding’. We want to believe it is, and we learn much as we do. But it is just a belief system like any other.

Beyond the circle 

Perception encapsulates our experience, keeps it contained. Like how a  boundary of a circle denotes the inside and outside of its shape. And so we find ourselves on a boundary between mind and matter, subject and object. The mind and subjective experience is on the inside, and matter and objectivity is on the outside. Our experience of life feels this way; our thoughts are internal, and the world is external. The boundary then is perception, with the idea that the finer we attend to perception, the greater the clarity. But this is the delusion. We only reinforce the circle in this way. To see beyond the circle is to dissolve the boundary. We only relate to the boundary because we’ve been born inside of it, born with the biological tools of awareness. We are born into it and then impose our ideas unto it, reinforcing a circle of beliefs and abstractions. 

Of course, a perceived world is practical and natural. It helps us coordinate insight and involve ourselves in life, and contribute to the meaning of it, the significance of it. However to consider the relativity of perception, is to try to break through the circle, to consider a new vista. This is the effort to awaken inside the fundamental nature of reality. It is not easy. Many cultures over many thousands of years have tried to step beyond, with varied techniques and success. And whether there is success or not, the trying is important, because we can, for this short time.