## A mathematical model of human interaction

Perhaps it's better I didn't post this earlier.

My motivation:

Some time ago I wondered how valid is socionics so I decided to make a model of human interaction my own. The model is a work in progress and although it was inspired by socionics, it doesn't deal with it. These are my current condensed thoughts on it

On human interaction

Let's take two human beings and start observing them. Naturally there is interaction between them. What I want to do is to make an abstract model that describes this interaction.

Ok, for the purposes of the observation, let's isolate them from anything else as to prevent if from interfering with the observations.
To achieve this we place the subjects in a closed system.
In that closed system rules that govern closed systems apply. Summarized, energy cannot be created out of thin air, any reaction causes an equal reaction and others.
This means that in our closed system a change in person "A" would necessarily induce an equal change in person "B". Let's call this change "information exchange".

Now we have two cases:
• when a change in person A induces a change in person B
• and when a change in person B induces a change in person A.
Let's describe this as:
• a flow of information from person A to person B
• and a flow of information from person B to person A
Let's now describe these two cases as
• f: A → B – meaning that information flows from person A to person B
• and f: B → A – meaning that information flows from person B to person A
We now define "x" as a change of person A over time
• ΔA / Δt = x
And "y" as a change of person B over time
• ΔB / Δt = y

Now we can write
• f: A → B = { y = f (x) } – meaning we can express a change in person B as a consequence of a change in person A
• and f: B → A = { x = f (y) }} – meaning we can express a change in person A as a consequence of a change in person B
What I have done now is described changes in our subjects through a mathematical formula.
For the time being what is the exact change in the person A or B is irrelevant, for now I'm just observing the properties of the change itself.

Let's now continue to observe these functions over time.

In moment "Δt" we have a change in person A, or a flow of information from person A. In our closed system this has nowhere to go other then to our person B where it triggers an equal change.
So in moment "Δt" we have a flow of information from person A to person B.

This can be written as:
• a change occurred in person A → ΔA / Δt = x ≠ 0
• a change occurred in person B → ΔB / Δt = y ≠ 0
• a change in person A causes a change in person B and they are equal → x = y
depending on the "x", or the change in person A, we determine "y", or the change in person B.

But being that we are observing people, who all by themselves are a system, this is not necessary true.

Let's review what we know.

So far A and B, despite being called people, don't have to be anything more then dots in space. If they are going to represent people then they have to be treated as such. Being that I want to stay in mathematics from now on A and B represent complex systems.

In light of this, a change in person A still triggers an equal change in person B, as the rules that govern closed systems still apply, you can't you create something out of nothing nor can you make energy disappear, but now the change in person A is not on point, but on a complex system. This means that perhaps the entire system changes, or perhaps it's just a small segment or something like that. Also a same change could be realized in a multitude of ways, perhaps through the entire system changing uniformly or perhaps through one small part changing significantly. The same goes for the complex system B.

It now becomes evident that we cannot universally speak of the same change in both people, as they are both complex systems so there is a probability that they could be different. Actually, if I remember correctly the chance of them being the same is actually 0%, that is, the chance of finding two complex systems that are exactly same, that is, the chances that when randomly throwing a dart you will hit the same point twice.

But that is not my concern at the moment, I just needed to show that they don't have to be absolutely equal. And if A and B are not absolutely equal that necessarily means that the changes on them don't have to be absolutely identical. We could have two systems that have nothing in common so there is no chance of them having the same changes.

So now we cannot speak of the same change but of a proportional change in the other person. Corrected this is
• ((ΔA / Δt) ~ (ΔB / Δt)) → x ~ y
* the "~" sign means proportional

To explain it in a more physical manner, consider it like heat exchange, the amount of "heat" radiated from object A is the amount of "heat" absorbed by the object B. But, as I have mentioned, objects A and B are people and are systems themselves so there is an interaction between the two systems. For all we know some of that "heat" radiated from the first object has been used to break molecular bonds or make them in the second object or something like that. So even though the "heat" radiated from first object will be the "heat" absorbed by the second, the temperature drop on the first object does not have to equal the temperature rise on the second, or the change in color doesn't have to be the same, or the change in size doesn't have to be the same and so on.

Now, here we have to limit ourselves to what we are interested. Here out "heat" is the information transmitted, but we don't know how much of it is wasted and how much has been beneficial to us.

So now we have
• f(ΔA / Δt) ~ g(ΔB / Δt) → x(t) ~ y(t)

that is, we have a formula to describe how this information exchange beneficially affected the people we are observing.

Now, beign that this is a closed system, no change in person A means no change in person B. That means that for Δt = 0 we have f(0) = g(0). Now it becomes truly simple as we just observe interactions between x(t) and y(t). Some hypothetical cases:
1. x(t) = t^(1/2)
y(t) = t

2. x(t) = t^2
y(t) = t

3. x(t) = t
y(t) = sin(t)

4. x(t) = (1 - cos(t))sin(t)
y(t) = (1 + cos(t))cos(t)

5. and so on

Ok, let's take the case of determining one's self, that is, the process of self exploration using this method. In this case we have a model closed system, the person A is person B. It's the closest one can get to a fully closed system.

Ok, this means that
• f(ΔA / Δt) ~ g(ΔA / Δt) → x(t) ~ x'(t)
I've marked the second functions as x'(t) so it would not seem that it's the same as x(t) but in essence they are both functions of the same person despite being mathematically separate phenomenon.

So basically we have an equation
• x(t) ~ x'(t)
and it is quite apparent that unlike the
• x(t) ~ y(t)
where one can determine the value of "x" in any moment "t" using the value of "y", in the first case one cannot.
In the first case, x(t) ~ x'(t), the variable "x" is dependent only on itself, that is it's dependent on it's self in terms of being dependent on it's initial state and the type of formula. In human terms on the persons state of mind when they entered the process of self exploration and the type of process they are using to find themselves.
Usually, as time progresses, the value of the function converges to a solution of the equation "x(t) = x'(t)". If there are any solutions that is. In human terms, there usually is an "end" to your self exploration, enlightenment as you may say. Basically you converge to another state.

But this most certainly does not have to be the case, but I don't concern myself with that just yet.

This is how it unfolds in real life.

We start observing ourselves and we observe in discrete time intervals.

In moment "t = 0" we observe the object. But because we are the object by making the measurement we have changed as the information flowed from us to us.
In moment "t = 1" we observe the object. Again, because we are the object by making the measurement we have changed.
In moment "t = 2" we observe the object. We change again.
In moment "t = 3" we observe the object. We change again.
.......
In moment "t = n" we observe the object. We change again.
.......
In moment "t = ∞" we observe the object. We either do not change again or we have entered a periodic regime of change or we are in a chaotical regime of change.

What happens is that in recursive functions over time attractors appear. An attractor is something that attracts all the values of the recursion, which mathematically we behave as when we are self exploring. These can be a single point, a periodic attractor or a chaotic attractor.

So by observing ourselves we diverge from where we were to a new location. If we end up where we started then we are lucky. Point being through self observation and exploration we go to new places and it should not be used to find stuff about us as it transforms us, that is, we shouldn't use self exploration to for example search deep inside us and find who we *truly* are as the result, what we find, is not the person that started the process.

Now let's look at another case. Let's take the original situation with two people. By interacting with other people we change and through interacting with us other people change.

Now the formulas go as following:
• f(ΔA) ~ g(ΔA, ΔB) → x(t) ~ y(t, m)
So basically we have an equation
• x(t) ~ y(t, m)
this case is the same as the one from before but now we have convergence that is decided by the factor "m" or the change of the other person in time.
Basically over time our relationship with that particular person will either converge to a stable state, enter a periodic regime, be chaotic or diverge.

This is how it unfolds in real life.

We start interacting with the individual in discrete time intervals.

In moment "t = 0" we interact. As information flows from us we change the other person, and when information flows from them they change us.
In moment "t = 1" we interact. As information flows from us we change the other person, and when information flows from them they change us.
In moment "t = 2" we interact. As information flows from us we change the other person, and when information flows from them they change us.
In moment "t = 3" we interact. As information flows from us we change the other person, and when information flows from them they change us.
.......
In moment "t = n" we interact. As information flows from us we change the other person, and when information flows from them they change us.
.......
In moment "t = ∞" we interact. The interaction either does not change either of us or we have entered a periodic regime of mutual change or we are in a chaotical regime of mutual change or there is no more relationship.

*~~~~~~~~~~~~*

Here are my conclusions:

Relationships between people depend on the information itself and the flow of information between individuals.

Conclusions regarding socionics:

Self exploration should be shunned immediately as means to determine ones type.

Only relationships, or looking at the the flow of information and the information that flows, can be used to accurately type people. And sometimes now even then.

What people say their type is bares absolutely no relevance on what their type is. It bares relevance only if it was a direct conclusion of observing relationships with other people.