Perception of complexity

In previous posts, the nature of reality and its complexity has been approached from the point of view of Information Theory. However, it is interesting to make this analysis from the point of view of human perception and thus obtain a more intuitive view.

Obviously, making an exhaustive analysis of reality from this perspective is complex due to the diversity of the organs of perception and the physiological and neurological aspects that develop over them. In this sense, we could explain how the information perceived is processed, depending on each of the organs of perception. Especially the auditory and visual systems, as these are more culturally relevant. Thus, in the post dedicated to color perception it has been described how the physical parameters of light are encoded by the photoreceptor cells of the retina.

However, in this post the approach will consist of analyzing in an abstract way how knowledge influences the interpretation of information, in such a way that previous experience can lead the analysis in a certain direction. This behavior establishes a priori assumptions or conditions that limit the analysis of information in all its extension and that, as a consequence, prevent to obtain certain answers or solutions. Overcoming these obstacles, despite the conditioning posed by previous experience, is what is known as lateral thinking.

To begin with, let’s consider the case of series math puzzles in which a sequence of numbers, characters, or graphics is presented, asking how the sequence continues. For example, given the sequence “IIIIIIIVVV”, we are asked to determine which the next character is. If the Roman culture had not developed, it could be said that the next character is “V”, or also that the sequence has been made by little scribblers. But this is not the case, so the brain begins to engineer determining that the characters can be Roman and that the sequence is that of the numbers “1,2,3,…”.  Consequently, the next character must be “I”.

In this way, it can be seen how the knowledge acquired conditions the interpretation of the information perceived by the senses. But from this example another conclusion can be drawn, consisting of the ordering of information as a sign of intelligence. To expose this idea in a formal way let’s consider a numerical sequence, for example the Fibonacci series “0,1,2,3,5,8,…”. Similarly to the previous case, the following number should be 13, so that the general term can be expressed as fn=fn-1+fn-2. However, we can define another discrete mathematical function that takes the values “0,1,2,3,5,8” for n = 0,1,2,3,4,5, but differs for the rest of the values of n belonging to the natural numbers, as shown in the following figure. In fact, with this criterion it is possible to define an infinite number of functions that meet this criterion.

The question, therefore, is: What is so special about the Fibonacci series in relation to the set of functions that meet the condition defined above?

Here we can make the argument already used in the case of the Roman number series. So that mathematical training leads to identifying the series of numbers as belonging to the Fibonacci series. But this poses a contradiction, since any of the functions that meet the same criterion could have been identified. To clear up this contradiction, Algorithmic Information Theory (AIT) should be used again.

Firstly, it should be stressed that culturally the game of riddles implicitly involves following logical rules and that, therefore, the answer is free from arbitrariness. Thus, in the case of number series the game consists of determining a rule that justifies the result. If we now try to identify a simple mathematical series that determines the sequence “0,1,2,3,5,8,…” we see that the expression fn=fn-1+fn-2 fulfills these requirements. In fact, it is possible that this is the simplest within this type of expressions. The rest are either complex, arbitrary or simple expressions that follow different rules from the implicit rules of the puzzle.

From the AIT point of view, the solution that contains the minimum information and can therefore be expressed most readily will be the most likely response that the brain will give in identifying a pattern determined by a stimulus. In the example above, the description of the predictable solution will be the one composed of:

  • A Turing machine.
  • The information to code the calculus rules.
  • The information to code the analytical expression of the simplest solution. In the example shown it corresponds to the expression of the Fibonacci series.

Obviously, there are solutions of similar or even less complexity, such as the one performed by a Turing machine that periodically generates the sequence “0,1,2,3,5,8”. But in most cases the solutions will have a more complex description, so that, according to the AIT, in most cases their most compact description will be the sequence itself, which cannot be compressed or expressed analytically.

For example, it is easy to check that the function:

generates for integer values of n the sequence “0,1,1,2,3,5,8,0,-62,-279,…”, so it could be said that the quantities following the proposed series are “…,0,-62,-279,… Obviously, the complexity of this sequence is higher than that of the Fibonacci series, as a result of the complexity of the description of the function and the operations to be performed.

Similarly, we can try to define other algorithms that generate the proposed sequence, which will grow in complexity. This shows the possibility of interpreting the information from different points of view that go beyond the obvious solutions, which are conditioned by previous experiences.

If, in addition to all the above, it is considered that, according to Landauer’s principle, information complexity is associated with greater energy consumption, the resolution of complex problems not only requires a greater computational effort, but also a greater energy effort.

This may explain the feeling of satisfaction produced when a certain problem is solved, and the tendency to engage in relaxing activities that are characterized by simplicity or monotony. Conversely, the lack of response to a problem produces frustration and restlessness.

This is in contrast to the idea that is generally held about intelligence. Thus, the ability to solve problems such as the ones described above is considered a sign of intelligence. But on the contrary, the search for more complex interpretations does not seem to have this status. Something similar occurs with the concept of entropy, which is generally interpreted as disorder or chaos and yet from the point of view of information it is a measure of the amount of information.

Another aspect that should be highlighted is the fact that the cognitive process is supported by the processing of information and, therefore, subject to the rules of mathematical logic, whose nature is irrefutable. This nuance is important, since emphasis is generally placed on the physical and biological mechanisms that support the cognitive processes, which may eventually be assigned a spiritual or esoteric nature.

Therefore, it can be concluded that the cognitive process is subject to the nature and structure of information processing and that from the formal point of view of the Theory of Computability it corresponds to a Turing machine. In such a way that nature has created a processing structure based on the physics of emerging reality – classical reality -, materialized in a neural network, which interprets the information coded by the perception senses, according to the algorithmic established by previous experience. As a consequence, the system performs two fundamental functions, as shown in the figure:

  • Interact with the environment, producing a response to the input stimuli.
  • Enhance the ability to interpret, acquiring new skills -algorithmic- as a result of the learning capacity provided by the neural network. 

But the truth is that the input stimuli are conditioned by the sensory organs, which constitute a first filter of information and therefore they condition the perception of reality. The question that can be raised is: What impact does this filtering have on the perception of reality?

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