People usually tend to think about knowledge as a pure abstract concept. OK, it is, but what I mean is that knowledge is actually something really bound to human beings. So far, we do not know about any other form of knowledge beyond what we believe is knowledge. Well, we could argue about the concept in the animal kingdom but, let's face it, no animal so far can do Math.

So, in order to try to roughly formalize what is knowledge, in the hope to reach a point where it would be feasible for us to deal with the concept, we will start defining some sets. Actually, according to what I had just said above, the first set we have to define is the set of the Persons.

# The Set of The Persons

Let $\mathbb{P}_u$ be the set of all the persons that had ever existed and the will ever exist (sure, U stands for*Universal*). That is a huge set, isn't it?. We better define a couple of smaller ones to work with.

- Let $\mathbb{P}_0$ be the set of the persons that are currently alive. This is the set we will use the most of the time and for that reason, it will be usually referred as $\mathbb{P}$.
- Consequently, the set $\mathbb{P}_{-1}$ will represent all the person that had die since the first thinking human.
- And finally, let $\mathbb{P}_{+1}$ be the set of all the persons to come until human extinction.

From all the definitions above we can conclude that:

Fair enough. That was easy. Let's move on.

# Knowledge Items

Getting close to a knowledge model is quite a task. This is a very complex concept and the whole purpose of a model is to get a suitable representation for a given purpose... making stuff simpler, in a sense. But we have to start at some point, and for us, that point is what we had called**Knowledge items**.

In this context, a knowledge item is anything a person knows, whatever it is. Here, we get into some philosophical questions, that I cannot really answer. A lot of people had tried to find these answers in the past, and I would say that we are not yet there. Anyhow, there is a couple of things that, based on common sense, we could say about these knowledge items.

- Knowledge items only exists in the mind of a given person
- Even when a consensus exists among multiple persons on what a given knowledge item "means" we cannot assume that all those persons have the exact same internal representation of such a knowledge.

This is something that everybody knows (OK, some kids may not know that yet, and people with some diseases may had forgotten it, but, for the sake of the current discussion we can still say "everbody"), however we cannot assume that such a knowledge is "stored" by different persons in the very same way. We cannot even assume that such a knowledge, even when shared by everybody around the planet, has actually the same meaning for all persons.

This is a key point that would make all this thinking quite tricky. Actually, it is likely that, at some point, we will have to relax this idea a little bit in order to progress.

# The Set of Knowledge

So, in the same way we have defined the set of persons we can now define a set of knowledge. Let $\mathbb{K}^{p} =\{k_i\} \mid p \in \mathbb{P} $ be the set of all the knowledge items for person p. That is, what person p actually knows.From this definition we can derive the following:

- The set $\mathbb{K}^u = \bigcup\limits_{ {p_i} \in \mathbb{P}_u} \mathbb{K}^{p_i}$, represents all the knowledge that humankind will ever accumulate
- The set $\mathbb{K}^0 = \bigcup\limits_{ {p_i} \in \mathbb{P}_o} \mathbb{K}^{p_i}$, represents all the knowledge that humankind has right now

# Knowledge Representation Set

Even when any of us may represent knowledge in a different and even personal way, we, humans, manage to interchange knowledge all the day long. For the discussion below we will assume a very constrained scenario in which, two persons are locked out in a room trying to interchange knowledge. In this situation, one of the person will have to "explain" that knowledge to the other person. Yes, this is the**Locked out Room**scenario :).

That knowledge can be spoken out, can be written down or can be drawn in a piece of paper,... The same knowledge item can actually be represented in very different ways. That is how humans interchange knowledge. We write books, we attend lectures, we analyse diagrams, etc... So, here it goes our third set.

Let $\mathbb{R}^p_{k_i} = \{ r^j_i\}$ be the set of all possible representations, the person p can produce for the knowledge item $k_i$. For convenience let's note $\mathbb{R}^p$ as the set of all possible representation for all knowledge item for person p.

Now that the concept of knowledge representation has been introduced, we can define a couple of functions that will help us to formalize, a bit, the processes associated to knowledge interchange.

**Definition. Direct Knowledge Transformation**

$\forall r^j_i \mid r^j_i \in \mathbb{R}^p, p \in \mathbb{P}, \exists F^p_{i,j} \mid F^p_{i,j}:\mathbb{K}^p \rightarrow \mathbb{R}^p \mid F^p_{i,j}(k_i) = r^j_i$

In other words, the function $F^p$ represents the process of converting a given knowledge (represented by a knowledge item) into a knowledge representation. This function abstracts the process of creating a writing or a speech that represents some knowledge.

In the same way, we can define the reverse process.

**Definition. Inverse Knowledge Transformation**

$\forall r^j_i \mid r^j_i \in \mathbb{R}^p, p \in \mathbb{P}, \exists G^p_{i,j} \mid G^p_{i,j}:\mathbb{R}^p \rightarrow \mathbb{K}^p \mid G^p_{i,j}(r^j_i) = k_i$

So, the Inverse Knowledge Transformation, is a function that takes a given representation of a knowledge item (e.g. something that somebody had written) and converts it into a knowledge item. In plain words, it is the process of understanding what somebody else is explaining.

# Perfect Knowledge Transfer Function

Now we are actually in conditions to define what we had called the**Perfect Knowledge Transfer Function**. It is represented by the following expression:

or

$F^p(G^q(r^j_i)) = r^j_i \iff G^q(F^p(k_i)) = k_i$

Basically, this expressions indicates that, in order to transfer knowledge from one person to another, it first has to be translated in some kind of representation (function F) that some other person can interpret (function G) and convert it back into a new knowledge item.

The avid reader may had noted that, according to what we had discussed earlier, the expressions above are wrong. The point is that we had indicated that each person (p and q in our equations) will, in general, represent the knowledge item in a different way, and therefore the $k_i$ for p and q are actually different. This is how the expressions should actually look like.

or

$F^p(G^q(r^j_i)) = r^j_i \iff G^q(F^p(k_i)) = k^\prime_i$

That is the reason we had added the phrase **Perfect Knowledge** to this section title. Sure, we have to define what that means. The **Perfect Knowledge** condition indicates that two different persons (actually entities, let's see in a sec why), may have the exact same knowledge item within their respective knowledge sets. This basically means that $k_i = k^\prime_i$

We had said that this is not true for persons, however, there is a case where this Perfect Knowledge condition can be applied and the Knowledge Transfer Equation is actually exact. Suppose that we stop talking about persons for a second, and we start talking about computers. In this case, we can actually be sure that some knowledge (something in our mind) is properly represented in a specific way in a machine/computer, at least, to the extend of our own knowledge item representation. In this case $k_i$ can be at computer p and also at computer q, and for both computer the representation will be exactly the same.

# A Final Word

So this is all for know. As a final word, just note that, if at some point in time, humans manage to develop telepathy, neither the knowledge representation set nor the direct and inverse knowledge transformation will ever be required, and the Knowledge Transfer Equation will make no sense. And also note that telepathy between machines is actually a trivial concept (whenever they start thinking :)In the mean time, keep tuned for the next chapter!

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