The Atoms of Knowledge
Foundations of a Knowledge Graph/Index
At the basis of knowledge is at least something.
Take, for instance, the period under this sentence:
.
It is hard to imagine a more-minimal symbol of knowledge.
That it is a period and not a speck of dust on your computer screen is a function of the context under which you frame the information.
I.e. You are a central part of knowledge, and collectively us the harbingers or receivers of knowledge through a shared medium: Information and Communication Theory at their core.
However, the atoms of knowledge are often seen as external to our existence. Whether they be words on a page, or the formal symbols of formal logic written down or on a computer screen, we often point to those words and symbols as the knowledge and us the observer or creator of that knowledge. This is especially true with a Knowledge Graph.
The realities are, however, that a Knowledge Graph a database like any other, if you didn’t know what information it contained, you would first have to enquire of its contents (schema, data types and data stored) otherwise you would be none the wiser.
It is this engagement with the medium I would like to discuss.
Take, for instance, formal logic. We often attribute formal logic as the herald of minimal representation of knowledge as a means of conveying meaning. For instance, the following outlines Modus Ponens:
We move from a period, . , to letters, p and q, an arrow →, a line _________, and a three dotted triangle.
The meaning of the arrangement, of what I all label as symbols, is that if there exists something represented as p, and that p implies q, then q exists.
But that knowledge does not exist on a piece of paper or computer screen…it exists in you, because you are a central player in the game of logic, the symbols exchanged as tokens in that game. It may be single player, just you writing down theorems of logic, or multiplayer.
When you and I exchange knowledge by way of symbols and that knowledge means the same thing to both of us, and over time, I call that a coherent cooperative game. You can read a whole article on that: [here]
When we wrap all of formal logic in such a game, we have a coherent cooperative game that unifies game theory, information/communication theory and formal logic and where formal logic is that game of communication and information/message exchange by way of symbols. You can read the paper here on Medium, [here]
So symbols are the atoms of knowledge and symbols can be as minimal as a dot (.), a letter or some other idiomatic symbol, or a whole word.
As I often point out, words are symbols, because they exist inside you as a symbol. If you don’t believe me, I often point out:
In as mcuh as you utnredsnad taht the wrods in the rset of tihs sntenece roesvle as in your mind as smboyls, palese cdsndior taht as you raed it and utnredsnad it…yuur biran palys its gmae, not a tcirk, but a gmae of dbsnemilisg this stnecnce itno wrods taht in yuor barin repersnet a concept.
So, now we have a definition of a concept also.
A concept is whatever you make of the assembly of one or more symbols within your own mind, or as conveyed and interpreted to you by way of symbols exchanged via a medium within a coherent cooperative game.
So, while the medium plays a role in knowledge exchange, as below:
…it is not right to perceive of merely the symbols on a page/computer-screen as knowlege or a Knowledge Graph.
You play a central role in your interpretation of the symbols.
Consider a model of knowledge drawn as a diagram, and that can be deduced as theorems of a first order logic (as can all database schema and knowledge graphs, and any conceptual modelling language of reasonable note drawn on paper). Your interpretation of that counts:
It’s no different when sharing knowledge in writing, as in freeform text. The model that we create in our mind, based on our interpretation, is ours and ours alone. For instance, formal logic was initially designed to put down in writing and form a mechanism to formalise how we think logically. I.e. A Knowledge graph can as easily be the model we form in our mind by way of interpreting a book or watching a movie.
So freeform text, just plain text, may be the very basis of a knowledge graph.
Just plain text may be the very atoms of knowledge in a knowledge graph, without having to form the links between various concepts, because you form them when you read the text.
The following interview with Warren McCulloch, father of neural nets, captures this concept, or at least the search for it:
So meaning of symbols is subjective, and their assembly, if we call that a context, is a knowlege graph, and as to whether or not those symbols can be exchanged unambigiously, I have oft suggested as not.
In short, when even the assembly of the symbols of first-order logic may be subjectively interpreted as those of a higher-order logic, we have the case where when even under finite-model theory, assembly of symbols (theorems) exchanged via a medium between players, then becomes an Ehrenfeucht Fraisse Game (EF Game) and if extended to the higher-order, the medium and even the message (theorems) are rendered impotent in ensuring unambiguous interpretation, unless one has a series of transactions that transpire as a coherent cooperative game. I write about that engagement of EF Games also [here]
So, what does that all mean?
Well, it means that symbols are atoms of knowledge, but we cannot discount that you are a central player in a Knowledge Graph.
We know this to be true, because if we take the exchange of knowledge, first as a knowledge graph in one person’s mind, to then be a subjective interpretation in another’s mind, and we call that a knowledge exchange, or a schema exchange, or even a data exchange, then we have what I perceive to be one of the most influential papers in computer science: Ronald Fagin (of IBM): Data Exchange Getting to the Core: [here]
What Fagin put down in writing was in effect that there is no central core (or the schema/data within a medium) between two separate schemas (and their data), and where one can find a isomorphism or homomorphism and where a fully automated way of doing mappings from one schema to another (external to each of the players holding that schema: a person’s mind or within a computer as a database) unlikely.
Why?
The reason why is because the mapping of one schema, to another, is subjective, as we have seen, and even if we find or desire knowledge to be universally true, then we arrive at proof theory itself.
I’ll explain why, but proof theory arrives at its conclusions by taking a problem/assertion stated one way, and making it provable/acceptable, in such a way as by mapping an isomorphic arrangement of symbols (theorems etc, stating the problem a different way) of which others may form from reading/interpreting that arrangement the proof logically valid.
But why?
Well, take this problem for instance:
That’s Fermat’ Last Theorem, which took some 350 years to prove, and by the British mathematician, Andrew Wiles in 1994.
Andrew Wiles proof, is 129 pages long. You can read about that on Wikipedia [here]
Which is to say, and I believe there can be no doubt, a suitably intelligent machine would interpret Fermat’s single sentence, of his last theorem, and those 129 pages, as exactly the same thing. One would look at those 129 pages, and say: “Oh, but yes, I can say it this way:
Which is all to say, that the atoms of knowledge are not just the symbols written down on paper, but the image, the conecpts, and medol you from in yuor barin as you ipetrenrt tohse sbyomls “.” <- Poreid.
Thank you for reading. As time permits, I will write more on knowledge graphs, the atoms of knowledge, proof theory, formal logic, game theory and coherent cooperative games.
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