In the beginning of 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 mastery to bring to the Pope, Giotto agreed and immediately set to work. Grabbing for his brush and a piece of paper, Giotto drew a perfect circle onto a sheet of paper. The advisor was confused and asked Giotto if he would reconsider his offering to the Pope. Giotto insisted that it was the sample to be given.
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 greater meaning, it wasn’t just a shape but a symbol alluding to transcendence. Giotto had demonstrated something profound, divine, and infinite, with simplicity and perfection. The most succinct demonstration of his mastery. This had impressed the Pope, and so Giotto was awarded the job.
Circles have inspired mankind across the ages. From the disc’s of the sun and moon in the sky, to the water rings that cascade across the surface of a still pond. The circle suggests cyclical behaviour, like the seasons or the movements of the planets and stars. As we gaze out into our world, our very eyes themselves are circles, the windows to our soul. The appeal of the circle has become universal, grounded in our humanity, a fixture in our psychology; a symbol of spirituality, completeness, wholeness, 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 unto itself?
Perfection is a product of scrutiny, that which is framed by knowledge and reason. Perfection is a product of the mind. A derivative of our faculties. Therefor, perfection is relative, it is not objectively in the world. Perfection is based on us and our consideration. Perfection is a manufactured ideal, that once created, gives us direction towards it, to claim it, to seemingly achieve ‘an 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.
What could possibly embody ‘perfection’?
What language is apt at expressing this?
Many feel Mathematics is the ultimate language of nature. In fact some believe that mathematics is an inborn feature of the universe; that math is discovered, not invented.
Intuitively, math does seem amazingly consistent with the world, and proves itself repeatedly in science. And so math seems rightly considered as a primary method of validation, and I would feel comfortable in believing this but there is a critical assumption we are making here; is math perfect?
Kurt Godel was a twentieth century mathematician who scrutinized mathematical truth. He critically analyzed math’s foundation; 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. As we derive more and more, the hope is that we will have a set of axioms that can prove the truth of this system of logic – hence showing that mathematics is consistent, and an ultimate truth, a perfect truth.
What Godel has done was to test the truth in mathematics by testing its axiomatic statements using a system of code numbers. His system became like a surrogate mathematics that allowed for confirmation of ‘self-evident’ truths.
Surprisingly, under his system, he discovered unresolvable paradoxes within certain axioms, tainting its 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, found in these two statements is representative of what Godel discovered oin many of the axioms. This hints towards a fundamental problem with Math. However, does this prove that math and logic is not inherent in nature? Not necessarily, but it does open an idea that we must consider: To consider math to be inherent in the world is to mistake products of the mind with the world itself. This is a mistake in my opinion.
We see the world as a collection of objects that can be counted. Our very neural network recognizes sets of objects and creates a a system of language to organize them, i.e. two donuts on a plate, two bugs on a window sill. However, the ‘two-ness’ 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, in three spatial dimensions and one temporal. The rest is mind and the categories we wish to impose. 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, this is the mind-bending fact of quantum theory: that on an atomic level matter does not exist, which is the basis of our reality.
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. Wrong.
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. Words and language help us to conjure the concepts we need to comprehend our world. We are always one mind away from the fullness of reality.
In a way, because everything is relative for us, we are constantly tempted to find the ‘ultimate’. This is why we are quick to declare that math, logic, faith, god, what have you, is the foundation of reality. We can’t help it. We need the ultimate answer. But ultimately, it is the same short-sightedness, it is merely reflections of the mind. It is all opinion.
Thank you for coming this far down the rabbit hole with me. But there is more, let’s return back to the idea of a circle:
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, beyond the strange fact that the decimal places in Pi go on endlessly after the period and never repeat, there is another paradox to be found. 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 properties of a circle cannot be entirely and precisely calculated.
All in all, Pi allows us to calculate the properties of circles, which makes it practical, 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. It feels absolute to them, but whatever decimal places they stop at, is certainly arbitrary—after how many decimal places do you finally land on the truth?
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 circle is impossible to build. This is not because of any technical challenge, but rather any physical form you care to conjure necessarily involves atomic structure. 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.
We believe atoms are distinct little entities, but this is not true. When you look closely at matter, you do not find smaller bits of matter, you find energy. We have been taught that the atomic world is comprised of parts, but these parts are more of a convenience to understanding than actuality. The atomic world is more accurately described as an interconnected, interpenetrated sea of energy, with no edges. The problem is our mind likes to dismantle things into associated parts so we can understand phenomena. It’s like dismantling an engine so that we can know how it operates. The difference being a part of an engine actually exists unto itself, clearly separate and well defined in 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 must operate with defined distinctions in order to apply it to the scientific method. 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 on an ocean, becomes a wave in a jar. Completely different. Of course, the water contained is still very useful to examine, but it does 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 the surface of the ocean, not it’s depth.
The point here is; what we decide to cut away from the energy web on micro-scales, is what we choose to identify and label for our purposes. We snip away at the atomic world to create ‘electron’, ‘proton’, ‘nucleus’. But these distinctions are more for our purpose 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, it’s beyond us.
And so the entirety of the cosmos is not applicable to ‘human understanding’. We like to believe it is, and we learn much as we do. But it is just a belief system like any other.
Speaking of belief, if you are in some doubt so far, you need only ask a scientist how truly distinct atoms and subatomic particles are. Ultimately, they will admit to a fundamental degree of uncertainty. This is because uncertainty is the only certain character of the atomic world – that sounds absurd but that is why the call it quantum strangeness. Which isn’t ‘perfect’, but in a way alludes to a new definition of perfect. The perfection that lay beyond knowing. And so in a backwards way, an electron says more about us then it does describing a ‘thing-that-orbits-a-nucleus’.
Faith and Determinism
Some place all their chips on faith, others on determinism. Either helps us close in on the ultimate truth, gives us that foundation to our lives and what we believe.
Science crafts determinism, makes it mathematically rigorous and consistent, so that it can be validated logically. Yes, science is wonderful… but ultimately its abstraction. I don’t use the word ‘ultimately’ arbitrarily here, I use it to point out something greater, an unnamable fundamental reality. Not fact, but beyond fact. Not truth and belief, but beyond truth and belief. And I am not talking about God, for like math, that is belief too (remember Godel?).
As discussed, for some people mathematics is the foundation of reality; that math is discovered, not invented. Some believe in a god in the same way, that god is the foundation and we have discovered it. And like science regards mathematical logic as it’s proof, the religious cite faith as theirs. Here are these two ideas declared and contrasted to each other:
- Mathematical theorems are true because logic is true.
- God is true because of the Bible, and the bible is the word of God.
There is circular reason in both these statements. Either can be deeply meaningful for those that wish to believe but in essence; arithmetic is fallible, and cannot be used to prove logic, as much as a Book cannot be used to prove god’s existence. This might be seen as heresy on multiple fronts. But as math, science and religion may function in people’s lives, they are all a shallow consideration. It is like someone capturing a wave in a jar and saying they have the ocean.
Is math a product of mind or reality?
Is god a product of mind or reality?
Are ideas of God and Math ‘proof’ of something more? In short, no. They are conventional thought, which means ‘generally agreed upon’, or ‘an accepted usage’. Quite flimsy if you ask me.
Beyond the circle
Perception encircles our experience, and in a way our experience is contained, like a circle. A boundary of a circle denotes the inside and outside of its shape and space, 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 an effort to awaken to 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.