Reading Dr. Tomabechi's Theory #2: The Relationship Between Cognitive Abstraction and Solipsism

This is a continuation of my reading of Dr. Tomabechi's Latent Potential Theory (NDU paper).
The paper can be viewed on Dr. Tomabechi's blog site.

In the previous article, I deciphered the meaning of "T (Cognitive Time Horizon)" in the Ego control equation.

I considered the structure in which cognition (= grasping the world) is optimized for the future by extending T into the future.

After that, while proceeding with reading the paper using AI, I was able to gain insights into the relationship between emptiness and altruism. I would like to write an article about that, but as a preliminary step, it is necessary to establish Theorem 2 and Theorem 3. That is why I decided to write this article.

Theorem 1 showed the Ego control equation that the optimal trajectory of cognition follows, while Theorem 2 shows that the trajectory of shared cognition with others (social cognition) similarly converges to a shared stable region (Shared-TCZ), and Theorem 3 shows that the level of abstraction is incorporated into the control equation, and the shared stable region is expressed in two aspects: a lower-order shared region and a higher-order integrated shared region.

In this logic, altruism is shown not merely as something superficial, but as an optimality in evolution, and I personally perceive that the foundation for understanding this is the relationship between cognition and the level of abstraction.

"Level of abstraction" is an important concept in Dr. Tomabechi's coaching, but I did not understand it very well. By deciphering Theorem 2 and Theorem 3 this time, I was able to deepen my understanding in my own way.

What I grasped is that "cognition inherently possesses a level of abstraction". It is not that a specific cognition possesses a level of abstraction, but that every cognition possesses a level of abstraction.

Theorem 2 formula

Now, let's look at Theorem 2.

First, starting with the extended control equation.

Σ_j γ_ij S_ij(x_i, x_j) is the term newly added in Theorem 2.
The following is an overview of the items by AI.

Σ_j

This means "for all other subjects j (all j where j≠i), add up the following values." In other words, subject i is simultaneously considering its relationship with all subjects other than itself.

γ_ij (coupling coefficient/strength of social bonding)

This is a coefficient that determines how much "weight" subject j has on the cognitive trajectory of subject i.

The larger γ_ij is, the more it means that subject j is "strongly cared about or influenced by" subject i.

S_ij(x_i, x_j) (distance of cognitive state between subjects)

This is a function that measures how far apart the cognitive state x_i of subject i and the cognitive state x_j of subject j are.

The greater the distance (the more different the two individuals' cognitive states are), the larger S becomes. The closer the distance is to 0 (the more similar the two individuals' cognitive states are), the closer S becomes to 0.

The meaning of the equation as a whole

So, what does the control equation of Theorem 2 actually mean when these elements are combined?

Subject i is attempting to simultaneously minimize the following two things:

1. Their own cognitive discomfort/instability (V_i)

2. The distance of cognitive states from others (Σ_j γ_ij S_ij)

In other words, subject i is pursuing both "being stable themselves" and "being consistent with others" at the same time.

Based on this control equation, the Shared Total Comfort Zone (Shared TCZ, TCZ_η^shared) is defined.

For a shared TCZ to be established, both of the following must be satisfied:

1. All subjects are within their own TCZ_i (personal stability)

2. The degree of consistency between subjects is η or higher (social consistency)

Can one grasp the cognition of others?

An important question arises here.

In the control equation of Theorem 2, the cognitive state x_j of a subject j, which is different from subject i, is described; the question is whether subject i can actually grasp that.

In other words, it is the question of whether one can grasp cognitive states other than one's own.

The Ego control equation in Theorem 1 was a kind of solipsistic structure.

The subject's cognitive state x indicates their current location in the set of possible worlds W, representing a structure where cognition equals the world (grasping).

Even if another person (subject j) exists here, they should be contained within the subject's cognition.

However, the control equation of Theorem 2 describes the cognitive state (x_j) of a subject other than subject i. It is described from a third-party perspective, so to speak, but is such a thing possible within the solipsistic cognitive structure of Theorem 1?

Theorem 3, in which abstraction is introduced

The key to solving this problem lies in the abstraction introduced in Theorem 3.

The difference from Theorem 2 is the addition of the final term η_i A(x_i(t)).

A(x_i(t)) (Abstraction Potential)

A is a function that measures 'how low an abstraction level' the cognitive state x of subject i remains at.

In other words, it is a numerical representation of the 'distance to reaching a higher level of abstraction'.

The larger A(x_i(t)) is, the more the cognitive state remains at a low level of abstraction. And when the highest level of abstraction (the Least Upper Bound, or LUB, which encompasses everything) is reached, A = 0.

η_i (Abstraction Balance Parameter)

η_i is a coefficient that adjusts 'how much of a cost' subject i pays to increase their level of abstraction.

What is important here is that the term A(x_i(t)) does not function alone, but is combined with the coefficient η_i.

In other words, it indicates the strength of the subject's motivation to 'understand more people' or 'perceive a wider world.' It is, so to speak, the degree of desire for world understanding.

When η_i is large, it means that 'world understanding' is extremely important to that subject. In that case, in the arg min control equation, the necessity to move toward a higher level of abstraction to make A(x_i(t)) smaller becomes stronger.

Conversely, when η_i is small, 'world understanding' is not that important to the subject. In that case, even if they remain at a lower level of abstraction, no significant penalty occurs in the arg min control equation.

The three simultaneous optimizations in Theorem 3

Looking at the control equation of Theorem 3, subject i is 'minimizing three different goals simultaneously'.

First goal: V_i(x_i(t), t) (Personal Stability)
Same as Theorem 1. Minimizing one's own discomfort or instability.

Second goal: Σ_j γ_ij S_ij(x_i, x_j) (Social Consistency)
Added in Theorem 2. Minimizing the distance of cognitive states with others.

Third goal: η_i A(x_i(t)) (Abstraction increase based on the desire for world understanding)
Newly added in Theorem 3. Depending on the subject's individuality, η_i, it seeks to move toward a higher level of abstraction.
This seeks to 'perceive the world more broadly according to one's own strength of desire for world understanding'.

All cognition possesses a level of abstraction

Now, thanks to the introduction of abstraction in Theorem 3, we can solve the previous question: 'Can subject i grasp the cognitive state (X_j) of another?'

Briefly stated, it means that 'a subject with a certain level of abstraction or higher possesses a cognitive structure that allows them to grasp themselves as objectively as they do others.'

Compared to the general image of cognition, it looks like the diagram below.

While the general image of cognition involves using oneself (i) as a viewpoint to illuminate the rest of the world (including others j), the image with abstraction involves illuminating the world, including both oneself and others, from a viewpoint above oneself.

This structure explains how the cognitive state of others can be grasped in Theorem 2. Even if not explicitly stated in Theorem 2, it means that the cognitive viewpoint is in a position that can include both oneself and others.

Furthermore, it is thought that this abstraction is also latent in the Ego control equation of Theorem 1. This is because even if i = oneself, grasping the trajectory of one's own possible worlds itself suggests that the cognitive viewpoint is located somewhere apart from the self.

And although I described the control equation of Theorem 1 as 'solipsistic,' assuming this structure means that is not the case. The very fact that one recognizes the 'self' as an object of cognition indicates that the 'self' is one of the constituent elements of cognition, and that constituent elements other than the 'self' exist.

In true solipsism, since one finds nothing outside of oneself, one should conversely be unable to find oneself either. This is because in the structure of 'self = cognitive viewpoint,' the self is not included in the field of cognitive vision.

The Parallel Hierarchy of Social Cognition and Personal Cognition

Considering that abstraction is an essential element of cognition in this way, we can also deepen our understanding of social cognition (cognition that converges on a shared TCZ).

When thinking from the starting point of the individual, social cognition is difficult to visualize. Because the belief that 'cognition = individual' is so strong, cognition that transcends the individual feels as though it lacks substance.

However, when considering cognition at a lower level of abstraction than the individual, the conviction that it exists increases.

Could it be that even the individual cognition that feels absolute is constructed using the functions of the eyes, nerves, and brain that make up the body as elements? Thinking of it this way, individual cognition itself can be regarded as a high level of abstraction in cognition.

While 'cognition of bodily organs' is difficult to visualize, it could be said to be information grasped by each organ in its own way.

In any case, starting from personal cognition, these feel like mere sensations and far removed from what we call cognition, but perhaps the same gap exists between personal cognition and social cognition.

Understanding the Self through the Ego Control Equation

As a preliminary step to discussing emptiness and altruism, I have looked at the abstraction in the Ego control equation in Theorem 2 and Theorem 3.

What I felt while writing this article is the difficulty of understanding mathematical formulas. I am reading and interpreting them while using AI, but I cannot understand them all at once, and I find myself going back and forth repeatedly.

This difficulty can be expressed as the S_ij(x_i, x_j) of the control equation. When I am i and the author of the paper, Dr. Tomabechi, is j, my level of understanding of the paper's mathematics is precisely the cognitive gap between the two, S_ij(x_i, x_j).

Within me, the weight of this gap γ_ij was not small, so I can understand that I wrote this article to minimize it.

Furthermore, the fact that I can think about my own cognition and actions from a perspective detached from myself can be interpreted, as we have seen so far, as being because my cognition also possesses a certain degree of abstraction.

In the paper, abstraction is not merely a parameter, but is defined as the degree of hierarchy in an inclusion partially ordered set that has emptiness as its top element.

I would like to write in a separate article about how altruism is derived from this emptiness as the limit of abstraction.