On the structure of information

A well-known metaphor: a house is not a mere collection of bricks. It is, rather, a collection of bricks that has been organized in a certain structure. Organizing information into structure seems to be something our brains are good at. What if this organization is the means by which our brain comprehends, stores, and transmits information? In other words, the structure of information is all there is. "Meaning" may simply be certain types of structure of information that our brain distinguishes from others. When this distinction occurs, the brain provides us with emotional impulses, which creates this sensation of "aha, that is quite deep". If something like this is true, then the brain ought to be programmed to recognize structure in a way that is synchronized with the structure of the universe surrounding us, since those "aha" moments led us to a point where we can make predictions about nature, communicate over a large distance, etc. 

It is difficult to test out such a theory since perceived information is usually matched with existing knowledge in our subconscious... so it is difficult to isolate a describable portion of self-contained information, which would be necessary for a rigorous study of how the structure of information determines the meaning. Except perhaps in an art form, where the "meaning" is least dependent on existing knowledge, such as music.

Here is my (almost) first attempt at the study of how a meaning of a musical piece could be interpreted via the structure of the organization of its sounds. I improvised this short piece: 

Zurab's Music Channel · 1805

And then isolated various layers of its musical structure:

Each colored dot in the bottom layer corresponds to a half bar. Here is the score, for reference (bars are numbered):

The colors encode musical similarities. For instance, notice that the fourth and the fifth bars both have a C chord in an extended half note. This is marked by the fact that the seventh and the ninth dots (from left) both have the same red color. The higher layers are combinations of half-bars again categorized according to the similarity of their musical structure. Now, my hypothesis is that when listening to this musical piece our brain generates (partially subconsciously) the structure displayed in the image above. The mere possibility of, and the easiness by which the brain generates this structure gives us the illusion of "meaning".

Mathematically, the structure we are talking about here can be seen as a collection of subsets of a partially ordered set (poset) of "pieces" of given information. In the example above, this would be the poset of intervals of the musical piece. In the picture, these intervals are continuous bars. The set of bars of the same color constitutes one subset. What is extraordinary in this example is the organization of these subsets into partitions of the entire piece (one partition for each line of bars in the picture, so four altogether). Another interesting phenomenon is that bars of the same color always occur in the same line. I do not know to what extent these rules are universal to musical compositions (of a certain type?). 

As for applications of the study of the structure of information, well, if "meaning" can be reduced to "structure", then by embedding structure into artificial intelligence, we should be able to produce a machine that is more human-like. This feels scary, I know, but I hope that my theory has a sufficient amount of flaws that it will not bring us closer to the terminator judgment day anytime soon. 

Another possible application is in education: by identifying and emphasizing the structure in learning, the brain of a learner may be able to acquire the skill/knowledge more efficiently.

And perhaps, there can be applications in psychology too, where structure can be a key in helping a brain make sense of life experiences...

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Metaphysics of Human Function based on a Mathematical Structure

work in progress

I propose here a theory of human function, which I have been developing based on introspection. In this theory, human function is represented in terms of exchange of information of four agents, which I call the spirit, the mind, the soul and the body. Although these are surely familiar terms, having a variety of scientific, pseudo-scientific, religious, philosophical and other usage, I do not assume any insight derived from such usage. The essence of each of these agents will be revealed through the roles that they play in human function. Matching of this essence with any of the existing definitions of these entities is unintentional and may well be coincidental.

The four agents are organizes in the following directed graph:

We call it the Human Function Scheme (HFS for short). The arrows represent directions of information flow from one agent to another.

Postulate 1. Human function is marked by internal information processing within each of the four agents as well as exchange of information along the arrows of HFS. 

Postulate 2. When a human being is engaged in a particular activity, information flows between the agents in consistent cyclic patterns. Change of activity may change these patterns. 

Postulate 3. These cyclic patterns are made of three fundamental ones - the three cycles of HFS.

The cycles referred to in Postulate 3 are:

Whatever we claim to perceive consciously, is information processed in the Spirit. There are two arrows going into Spirit. Information flowing from Body to Spirit is the sensory ingredient of human's conscious perception. Information flowing from Soul to Spirit is the non-sensory ingredient human's conscious perception, such as thoughts, logical essence, etc. Rigorous research is needed to be able to develop a more refined distinction between these two ingredients of our conscious perception. We will hitherto distinguish them using the terms concrete perception (for Body to Spirit) and abstract perception (for Soul to Spirit). 

The arrow going out of Spirit is the channel through which our conscious impacts human function. We call information flowing along this arrow Will. The rest of the arrows have names too: subconscious perception (for Body to Mind), abstraction (for Mind to Soul), concretization (for Soul to Body).

There terms are somewhat suggestive of the role that the arrows of HFS play in human function. A better understanding of this role will be achieved when describing different types of human function in the context of HFS. 
 

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Abstract Structures in Mathematics and Music

These are notes for my online discussion with Georgian Student Parliament on Tuesday 7 December, 2021, organized by Nina Tsatsanashvili. All photos in this post are from Wikipedia. 

If you look up the word "structure" on the Google Dictionary, you will find the following definition: the arrangement of and relations between the parts or elements of something complex. When the "parts or elements" are subject to specific interpretation, we have at hand an "abstract structure". For example, consider a painting, which can be seen as an arrangement of colors. For instance, Leonardo da Vinci's Mona Lisa:

This is not an abstract structure, since its constituents are specific colors that can be found on a specific poplar wood panel that currently resides in a gallery of the Louvre Museum in Paris. In contrast, Ludwig van Beethoven's Für Elise is an abstract structure, since the sounds that make up this musical piece are dependent on the interpretation of musical performer of the piece, as well as the instrument on which the piece is performed.

In this sense, abstract mathematics is similar to composition in music: in both cases one builds structures that are inherently abstract. The analogy goes further: a musical performer can be compared to an engineer, for instance, who produces concrete structures by means of concrete interpretations of the abstract mathematical structures.

 

In fact, the analogy goes even further than that. A completed piece of work in music, that is, a musical composition, is analogous to a "mathematical theory". A mathematical theory is an exploration of one or several special types of mathematical structures and their interrelation, similarly how in a musical composition one explores one or several types of musical structures and their interrelation. In both cases, abstract structures often organize into more abstract forms. 

As an example of a form of abstract structure in mathematics, consider the following diagram:

This diagram displays two mathematical structures, given by Fig. A and Fig. B. Each of them are abstract structures in the sense that the points and the arrows in each structure are subject to interpretation. The two structures have a similar form: they are both made out of points and arrows. Such structures in mathematics are called directed graphs. A mathematical theory that explores various different types of graphs and their interrelation is called graph theory. Graph theory, however, is not only about directed graphs. Other mathematical structures are also part of the theory, such as natural numbers, for instance. They arise by counting various different phenomena dealing with graphs: for example, by counting how many trajectories connect one point of the graph with another. 

Similarly, in a single musical composition, there is one (or several) main forms of musical structure, whose different manifestations are being explored, along with their interrelations, in the composition. What is the main form of musical structure in Für Elise?

     

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