Self reflection and the math (A Burgson moment)

 
As a kid, one of the first math problems I ever really thought about was division. Adding and subtracting were straightforward to me. Multiplication was just saying more. However, division was different—1/2. My initial thought was, how does the top number "relate" to the bottom number? In this relation was the conveying, exchanging, or limitation of the system. I learned more about division, of course, but that initial concept of the relationship in the math was still there. As I learned that the top was the numerator and the bottom was the denominator, I became even more moved by their relation. Why would we give them distinct names unless they were important to each other? In some ways, my initial concept was more complex than the actual problem itself. The system as a whole seemed complex, especially when you consider that the numerator could be larger than the denominator. How could the system as a whole be larger than itself? This concept led me to the idea of the greatest common denominator—how a system as a whole can be related to a foundational unit.

As an adult, when I look at complex systems, I think of the limiters, the rules, and the foundational unit. This concept gave me the idea of what a paradox truly is. That led me to Gödel's incompleteness theorem, which states that a system can't define itself. Why? Because you can't add anything new to the rules, the limits, or the denominator. This obviously makes sense because if you limit yourself, then there is no answer that can fit everything. This goes back to the greatest common denominator. The simple rule of scale can convert a non-linear system into a linear system. If we change the denominator, we can understand all the numerators.

The very act of limiting, of setting a denominator, is the reason why the system cannot encompass all possibilities within itself. It’s like saying, once you've chosen a denominator, the system's self-referential capacity to generate new rules or relationships is constrained. There’s no way for a system to completely "self-explain" because every attempt at self-definition would need to push beyond its own boundaries, which is not possible by the system's own rules. By finding the greatest common denominator between two systems, we understand how different scales (different numerators and denominators) can still operate linearly, meaning that they can still be related to one another when reduced to a common, irreducible unit. We can adjust the system to bring order to apparent chaos by aligning it with a consistent denominator, making the system’s behavior comprehensible or predictable.

However, what about concepts like pi? Where you can never fit the numerator into the denominator? This is like Zeno's paradox: out of infinite regression comes infinite possibilities. Our inside system will never fit our outside system because of this. Our denominator can never match the numerator. Unless we limit the system. Unless we say we have a point of irreducibility. What is that point? In a way, it’s mysterious. At what point do you give up and say, "This is good enough"? That any more analyzing, breaking down of the system will just lead into infinite regression that will have no real benefit for anything other than to say, "I did what I could."

However, this paradox only holds true when we assume infinite resolution. When we try to break down the system into infinite denominators. At some point, though, we need to set a boundary—to accept that infinite regression isn’t always meaningful. In this the point of irreducibility becomes meaningful. In real-world systems, there must be a moment when we say, "This is close enough."

This acceptance of a point of irreducibility, whether in mathematical systems like pi or in physical systems that we study, becomes a pragmatic decision to move forward. It’s like setting a resolution limit in physics: beyond a certain point, further analysis might not yield any practical benefit, only theoretical complications.

In summary, the denominator is not just a mathematical tool but a fundamental principle that governs the relationships in any system, from basic arithmetic to complex physical systems. The perceived complexity arises from our limitations in resolving or understanding the relationships between numerators and denominators. But at the core, we find that the system is ordered—it operates linearly once we account for the foundational limitations that govern it.

As we approach infinity, whether through irrational numbers like pi or Zeno’s paradox, we encounter the need for limits—we must set a boundary. That’s when the infinite regression stops being useful and when we begin to understand that the true simplicity of a system lies not in an endless search for more resolution but in the acknowledgment of irreducible limits or that there is an ultimate resolution and our infinite resolve is just stubbornness. 

I feel like this has been said before and its a bit cliche but.. The beauty and simplicity of reality is revealed when we acknowledge the limitations that define our systems, much like taking a moment to appreciate the little things as they are. When we stop trying to break things down into infinite parts and instead understand how the whole fits together within the constraints of its fundamental principles, we begin to see clarity. In other words, reality may be complex, but recognizing our inherent limitations helps us appreciate the simplicity that underlies it all.