writings on math, logic, philosophy and art

Systems are theories or explanations of some phenomena. In the context of the brain, which we examined in the previous installment, we thought of them as mental images (means of explaining somethig to ourselves). In the context of communication they are more appropriately likened to viewpoints (means of explaining something to other people). If you examine the two concepts closely, you will see that the difference between them is only superficial - when a theory belongs just in the mind of its creator, it is is a mental image, but if it is shared by them with different people, it becomes a viewpoint that this person has, about a given state of affairs (although one person can have multiple viewpoints.)

Viewpoint pictures

If we view systems as something like viewpoints then communication between people would involve transfering those viewpoints from one person to another, or, more formally, transfering those systems from one metasystem to another (you might argue that there are other kinds of communication this is the one that we will be looking at here.) But, as we all know, this transfer does not happen directly mind-to-mind (and we will see later why it cannot ever happen directly), but through crafting representations of those viewpoints, which we will call picture, as per Wittgenstein, who originally came up with most of this stuff.

A logical system/viewpoint can be represented and thought of as a Hasse diagram (that is a set of points that are connected by arrows.) A picture of a viewpoint, then, is any object that represents the structure of such diagram.

A picture need not be a literal Hasse diagram (although using literal Hasse diagrams has certain benefits) - it may be expressed in text, video, conversation or any other media. What defines the picture of a viewpoint is just the fact that it contains the viewpoint within itself in some way (at least that is intended by its creator.)

Immediately we can split pictures in two main categories - formal and non-formal (fuzzy) pictures. A working definition: formal are the pictures created by a person who has a Hasse diagram of what they want to represent in their mind, and non-formal are the ones where they do not. Note that a formal picture might be misunderstood, i.e. the picture that is in the communicator’s mind end up different from the one which is received, however, a non-formal one is (in some sense) always misunderstood, simply because it doesn’t have a single logically sound interpretation.

Formal viewpoints

To understand the distinction better, consider this real-life example: we are sitting in a bar. I am going to get drinks for you and so I ask you what do you want and you reply the following:

Get me a beer, low alcoholic one, if there isn’t such, or if it’s very expensive, I just want a glass of water.

You notice that your response describes your wishes in a way that would allow the listener to replicate your thinking process in a way that would guarantee that they would always reach the exact same conclusion that you intended them to reach, provided that they understand the words in the same way as you do. This is what we mean when we say that it is formal.

And it is my thesis that it having these qualities implies that my response can be represented by a Hasse diagram, and vice versa.

A Hasse diagram, representing the (formal) logical picture, that is also represented by the sentence: "Get me a beer, low alcoholic one, if there isn't such, or if it's very expensive, I just want a box of water." - b is the symbol for (low-alcoholic) beer. and w for water, and negation is symbolized by ¬. The diagram is decidable (a choice can be made at all circumstance) , if we presume that the law of excluded middle - b or ¬b - holds (and that the bar serves water.)

Non-formal pictures

A non-formal response, in contrast, is one that cannot be adequally represented by a Hasse diagram:

It can be one that is fuzzy and allows room for interpretation, i.e. it doesn’t describe a Hasse diagram precisely e.g. if you ask a person what they want for drinks an they say something like “I don’t know, perhaps some white wine, or a cocktail. I fancy something fresh with some strong flavour…” Such response maybe contains some info, but it puts thrust to the receiver to fill the blanks.

A wrong Hasse diagram, representing a fuzzy picture i.e. one that is "nonsense".

Another type of non-formal response would be one which results in a invalid Hasse diagram e.g. one that contains circular logic (not sure if there are other examples.)

A wrong Hasse diagram, representing a non-formal picture i.e. one that "doesn't make sense".

Another type of diagrams that are somewhere in between those two categories are ones that are technically formal but useless: ones that require so many conditions as given and/or contain so little logic that they provide little to no value in making a decision.

A Hasse diagram, representing an empty logical picture.

Viewpoint Contexts

Does the existence of formal pictures mean that our communication can be completely formal? No, because although the viewpoint the person expresses can be specified formally, the context in which this viewpoint lives can never be. To see why, we examine our example formal picture and, in particular, it’s constituents (the points in the Hasse diagram)

Get me a beer, low-alcoholic one, if there isn’t such, or if it’s very expensive, I just want a glass of water.

As we can see, these terms are not precisely defined i.e. they are opened to interpretation, e.g. if the person who we are sending to get us drinks is significantly wealthier than us, or a heavier drinker, their idea of “cheap” and “low-alcoholic”, respectively, would not match ours. This is not an error in communicating the viewpoint itself, but an error which result from the presumptions about where this viewpoint is to be positioned, where it fits in the whole system of systems that is the brain.

As you can guess, what we call the context of a viewpoint is actually just a collection of other viewpoints that it depends on i.e. it’s viewpoints all the way down.

To formalize this context, and thus remove any chance of misunderstanding which might potentially result in me getting the wrong kind of drink, we would have to provide a description of the viewpoints that form this context e.g. provide a description of what I think is a reasonably-priced beer. This would certainly minimize the chance of error, but it will not eradicate it, because my viewpoint for a reasonably-priced beer would depend on some other, more basic viewpoints (e.g. the viewpoint of what is beer in the first place) which, in turn would depend on even more basic viewpoints and so on.

For this reason, even if your mental process for choosing what to drink is logical (let’s say) and the method I get from your description is also logical, that does not mean that they are the same. This is an essential quality of almost all communication it tends to be fuzzy i.e. what you think is not what I think.

For a detailed description of how I believe the system of viewpoints function, refer to this quote from the first part of the article (there they are called “logical systems” instead of “viewpoints”, as the context is different.)

Based on their relationship to observation, logical systems can be viewed as a tree with the most concrete and observation-driven ones located at the tree’s root, and the most abstract ones, at its branches. I think that this relationship is key to the way the brain uses these systems: presented with sensory data, it (the brain) probably invokes a most primitive system first - one which does not make any important assertions, but merely converts that sensory data into a model which is a little bit more structured. Having that model, the brain can map it with a number of more abstract systems, each of them producing an ever more structured model as a result. These models will, in turn, be fed to a couple more abstract systems and so on.

A set of observations which are a result of sensory data (at the bottom of the diagram) are mapped to a simple logical system (at the centre), which in turn is mapped to a more complex one (top).

With each “jump” from one less abstract system to another more abstract one, more and more is “known” about a given situation, but at the same time, there are more and more possible interpretations, each giving us a separate system of analysing the situation.

What is communication

Now that we have a basic understanding of the way information is transferred, let’s outline the communication process itself (of which we already have some idea.)

The process of communication comes in two parts: transferring of pictures and transfer of their contexts.

Transfering pictures between agents

The prerequisite for the occurence of communication is the existence of at least two agents.

A metasystem, with one

Transfering context between agents

And so the only way for us to be completely precise in our language is to start with the most basic terms there is and to communicate the whole tree of viewpoints that are relevant, as context to the one we are communicationg i.e. to more or less transfer to the other person the complete contents of our brain. This is theoretically possible, but it is not what we are doing.

Thoughts and communication

We tend to think of logical pictures as existing in an individual first and being transferred second, but it makes a lot of sense for things to be reversed - that is that communication enables us to be logical.


Art and aestetics

Ideologies (self perpetuating systems)

Comparison to Wittgenstein’s philosophy

The picture agrees with reality or not; it is right or wrong, true or false.

On mind-to-mind communication

Feelings are a stronger form of knowledge

Folks who are good at something, be it basketball, chess, programming or anything else cannot fully explain how do they do the thing they do, even to people who are equally good at the same thing. i.e. I cannot determine if someone is a good programmer just by talking to them.

This goes to show that huge part of what we call knowledge is not expressible in words or any other kind of language. Moreover, it is this inexpressible part that is actually the deeper, more important one, the expressible part is just the tip of the iceberg.

Even if the theory is perfect and even if you know it perfectly, you have to know how to map it in diverse set of circumstances. The theory itself cannot tell you this, this is the realm of feelings.

And (even though it may come last when you are learning a theory) the feeling is what comes first to for person who creates the theory - they have a feeling about a given phenomenon, and then they articulate some of it in a theory.

Many people say that thrusting the way we feel is irrational, they don’t realize that feelings are also knowledge, just one that we cannot express in a way that makes sense for others. Thrust your feelings.

And there is another point I can make regarding this, related to learning. Learning a given theory has nothing to do with memorizing it, learning means feeling a similar feeling than the creator of the theory did (or having the same intuition as they did, to use a more familiar word, (which means the same thing)). A theory is a work of art, like a novel or a symphony. Good theories resonate with a lot of people, just like any good art.

So science and mathematics is art - this is not so obvious when studying math at an elementary level, because the feeling/intuition is mostly already there for you (as it is with elementary art, like folklore, for example) and you just have to memorize the language/notation, but it is obvious when studying more advanced theories that require more dedication.

For example, memorizing the type signature that defines what a monad is, even if you understand it perfectly would not teach you what a monad is (although it is a good first step).

return :: a -> m a
(>>=) :: m a -> (a -> m b) -> m b

Learning what a monad is, means feeling a given feeling. And, this is why, there is no straightforward way to explain what a monad is. This is why everyone approaches it (explaining monads) differently and still no explanation is completely correct.

This is why for a layperson, reading a math textbook is often unnerving and unpleasant, in the same way as a child who never experienced romantic love, would find reading love stories (or any other book for grown-ups) unpleasant.

But once it “clicks” it suddenly makes sense - everyone knows that, but many don’t realize that this clicking has nothing to do with memorization, it is just the experience of a feeling.