# duality

## Table of Contents

## 1. is it One or is it Two?

Is it me, or is it you? Is it false, or is it true? Many things can be explained by a bimodal or binary system. Everything, from boolean logic to art, any system whose goal is to make a prediction.

My hypothesis is that whenever someone believes something, the opposite belief is equally valid. In order to demonstrate, let me introduce:

### 1.1. The Great Filter

No, not the one about aliens dying or something. I'm talking about the everyday systems that you use in order to make predictions about the world. For example, this one, that wants to explain everything.

#### 1.1.1. The problem of explaining everything

Because this is indeed a system that explains everything, I must also demonstrate why it might be impossible to explain everything. I'll give the argument in a short set of syllogisms.

- If something must explain everything, it must also explain itself.
- If it explains itself, it means it is circular by definition, and therefore not objective.
- If it does not explain itself, then it does not explain everything, and it is therefore incomplete.

So either the system we're describing is circular, in which case it is arbitrary and subjective, or it is incomplete. So there's no way to construct any system with both qualities. Right?

#### 1.1.2. The problem with the problem with explaining everything

Notice that our above set of syllogisms is extrapolative. This means that it makes predictions about what is possible about a system. Due to this, the system results in some duality: either a valid system is incomplete or subjective. And there's another problem too: the system applies to itself.

If either systems are subjective or incomplete, then the system we are using to describe subjective and incomplete systems is also subjective. Which presents a problem: we have no reason to believe that this framework is worth believing in at all!

#### 1.1.3. The problem with the problem with the problem with explaining everything

But the fact that the system applied to itself breaks it is just a confirmation that our predictions are accurate! We predicted that things that explain themselves must be circular, and we were correct!

#### 1.1.4. etc…

The claim is that we can do this forever, although even this statement can go on forever. What's interesting is that it seems like there's an inherent link between duality and infinite recursion; infinite self reference. But the real question is: is this statement about frameworks true, or is it false? Well, according to the principle of duality, both of them can be true.

### 1.2. The Filter is based on what you choose to believe

So what you think is true or false is just what you choose to be true or false. Of course, even this statement is an infinite contradiction and confirmation, but what I am trying to communicate is that what you believe filters what's true and what's false, and as a result, leads to different prescriptions, or different actions for the same situation. In this way, frameworks act like filters. They shield us from the infinite opposite side of what we currently believe. Does this "filter" I am describing exist or not?

### 1.3. Of course it Doesn't!

The mindmap explains everything, without a filter.

## 2. The problem and solution with duality

The theory of duality is self deconstructing and self constructing in the same way via self reference. If the opposite stance is valid, that means that not believing in duality is valid, but that also is a data point that confirms our hypothesis.

We're also describing duality using a dual framework here, which is another pretty interesting thought.

## 3. Logic Explains Duality

Logic is a self affirming structure, and that might give you the clue that it is also self destructing. Nevertheless, one of the axioms of logic is:

\begin{align*} p \neq \neg p. \end{align*}This statement filters for binary, or as I would call it, dual mode frameworks, and gets around the principle of explosion. We have an intuitive understanding of truth and falsehood, and we can use those general terms whenever there is a mutually exclusive divide. In short, you can view the logical framework as an abstraction of all other dual frameworks. I propose that you can do analysis on all dual frameworks in much the same way group theory does analysis on groups. Mathematics in general has the same recursive binary structure to it, because it is based on a couple of axioms and utilizes logic as an extrapolation mechanism. By , everything that utilizes mathematics is also an inherently dualistic structure.

## 4. Programming Explains Duality

Of course, there is logic in programming, but that is kind of boring. What I am going to explain here is a recursive, binary structure known as the binary tree. It seems like you can model a lot of things in this way as well; John Conway's surreal numbers are a manifestation of this phenomenon.

class BinaryTreeNode: def __init__(self, value): self.left = None self.right = None self.value = value def insert(self, value): if value < self.value: if self.left is None: self.left = BinaryTreeNode(value) else: self.left.insert(value) else: if self.right is None: self.right = BinaryTreeNode(value) else: self.right.insert(value) def print_node(self, level=1): print(f"level {level}: {self.value}") if self.left is not None: self.left.print_node(level + 1) if self.right is not None: self.right.print_node(level + 1) root = BinaryTreeNode(5) root.insert(3) root.insert(10) root.print_node()

level 1: 5 level 2: 3 level 2: 10

Currently, all that this binary tree has is an insert method, but that is all that is needed in order to see the recursion in the structure. Each node "height" is self similar, and it works of a dual-mode sorting algorithm. That is, smaller values go on the left side, and bigger values go on the right side. It is the binary tree model in my view that unites the concept of duality with the concept of recursion. Duality is self similar at every abstraction level, and the duality is crucial for subdividing processing power in order to break a big task into small tasks, which is needed for recursion to be finite.

## 5. Why duality, and not Any other Modality?

This is a good question, and one that I've still yet to answer completely. However, I would still like to try my hand at this, because there are things that make the number two specially suited for the task of subdividing.

### 5.1. Two is a Natural Number

From a biological perspective, we're probably more used to dealing with whole numbers. We did not even come up with the concept of any others until much later, and negative numbers, and even zero, were a construct invented much later as well. Yes, there are an infinite number of natural numbers, but at least it's a filter we can use.

### 5.2. Two is Prime

Of course, there are an infinite number of other numbers that are prime, but this is yet another filter that can be used. Any number that is not prime can be represented by a smaller factor of that number. For example, 4-ality can be represented by a longer chain of dualities.

What's interesting is that one is a factor of everything. This represents the "null filter", or "anti filter", which doesn't filter any data and simply represents it all as one thing. Very interesting.

### 5.3. Two is small and not One

The number two is also the smallest natural number that is not one. This means it is the simplest way to subdivide any particular object. This makes it more elegant compared to some other modalities.