Some comments on information theory

In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices.

Question: What is the equal sign, is it limited to two dimensions?
Question: What does a tesseract do?
A tesseract or other more than three-dimensional objects are interesting because we can easily describe them mathematically, but we cannot imagine them. Although perhaps not entirely.

I remember the dream about tesseract. When I was 14 years old, I read the book "Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the 10th Dimension" by Michio Kaku. It was also then that I made a decision that in the future I would study theoretical physics, which I then did.

After reading the book, I remember having an unusual dream. I dreamed of tesseracts and other multi-dimensional objects. When I woke up I had a vague memory, but I know that in my dream I did see a multitude of dimensions. It was one of those dreams that ended with a feeling of a hard fall, as if the soul was returning to the body again.
Infinity, on the other hand, that makes a difference.
What about countability or uncountability? Does it make a difference? In terms of dimensions, I think that may be the case. For example, think about fractal dimensions and elaborate on that thought.
I decided to share with you my thoughts on the discussed topics. Perhaps they will clarify a bit the concept of entropy and the arrow of time. In case anything is not entirely clear, as I mentioned, I love being asked and I will try to answer every question. The questions make me think, crystallize my imagination, and this is very important to me. Let me start with matter again.

Intuitively through matter there is understood the whole of the existing physical objects, cognizable by the senses. In other words matter is constituted by everything that exists in time and space, an objective reality independent of consciousness. In philosophical terms, it may be a being that extends into space and time, although the location of material objects need not be clearly defined.

Plato believed that nature is the unity of material, ideal and spiritual elements. The world was built with the already existing plastic. The Universe was created by the Demiurge of undefined, shapeless matter. In this way Plato introduced a new concept of matter, which he considered to be an abstract component of bodies.

Aristotle, on the other hand, distinguished form and matter in each object. Matter does not exist independently, it is merely the substratum of phenomena and transformations, amorphous, shapeless material from which they arise as a result of shaping it by the material form proper to the given class of beings and for the individual being. This view, so called hylomorphism, was also shared by Thomas Aquinas.

The name of the view is derived from the combination of Greek words ὕλη hyle - matter and μορφή morphe - shape. Hylomorphism is a trend in the theory of being (metaphysics) and the philosophy of nature concerning the ontological construction and operation of natural bodies. According to this theory, every being is constituted by the first matter and the substantial form. Accepting assumptions of hylomorphism may help to explain substantial changes - the transition of one substance to another substance, i.e. the loss of one substance while the other substances are formed.

However, what is the ageing of matter and what is ageing in general? Could we talk about ageing if, according to so many concepts, we are dealing only with the transition from one form to another? People understand ageing in relation to material objects as a process of changing the properties of a given material with time. Therefore, ageing seems to constitute only a certain element deeply rooted in human consciousness.

Similarly, in a wide sense, random changes occurring in stars or random epigenetic aberrations in biological organisms, also material objects are subject to some random changes that translate into a classically understood ageing process.

One of the theories that takes into account the concept of the direction of time and thus the issue of ageing is thermodynamics. According to classical thermodynamics time-reversal symmetry does not occur.

The second law of thermodynamics, which I mentioned in one of the previous posts, states that there exists a specified function that in an isolated system never decreases. This function is called entropy. It is the existence of entropy that has to explain why the broken glass is not coming back together again.

Entropy is the thermodynamic function of the state, defining the direction of the course of spontaneous processes in an isolated thermodynamic system. Entropy is a measure of the degree of disorder of the system and the dispersion of energy. According to the second principle of thermodynamics, if the thermodynamic system passes from one equilibrium state to another equilibrium state, without the involvement of external factors (i.e. spontaneously), then its entropy is always increasing.

The perfect body, called the perfect crystal, has zero entropy at 0 K, since its state can only be realized in one way (each molecule performs zero oscillations and takes up the smallest energy).

Entropy describes the evolution of matter in the Universe in quite satisfactory way (despite all the haze surrounding entropy), but this function is particularly interesting for biological systems. Entropy is called the thermodynamic arrow of time, therefore, it is important to introduce the general concept of the arrow of time.

The arrow of time claims that time is always going from the past to the future, and never the opposite, that is unidirectional, asymmetrical and irreversible. Time can be divided into past, present and future. Synonyms of time arrows include, but are not limited to, the time direction, time asymmetry, time anisotropy, time irreversibility and unidirectional time. This feature differentiates time from one-dimensional space.

Due to the reflection in time, two types of physical processes are distinguished:

1. reversible processes (symmetrical with respect to time)
2. irreversible processes (unsymmetrical with respect to time)

However, reversible processes are the approximations and the idealistic concepts. A reversible process is a process in which it is possible to return all parts that make up a given system to the previous state. In this sense, the reversible processes would be phenomena occurring in the microscopic scale described by motion equations known from dynamics that are symmetric with respect to the time parameter. Examples of such reversible processes would be phenomena such as the motion of the pendulum, as well as the movement of the Earth around the Sun (Although it is not entirely true that the Earth rotates around the Sun. The Earth and the Sun rotate around the Solar System’s centre of mass.).

The idealism of this concept, however, is that it has not been taken into account that with each rotation of the Sun, the Earth may be perceived as different from that of the previous rotation. Consequently, one can refer to the previously cited philosophical conception of perdurantism and endurantism, or only recognize that the physical parameters change with each subsequent turn.

The same problem applies to the movement of the pendulum. Assuming that outside of this motion no other processes take place, one can assume that this is a cyclic process.

However, it is known that its material structure changes as well as the parameters of the environment in which this movement occurs, or any parameters referring to the external world.

Physical laws describing the behaviour of microscopic elements of the system do not exclude the course of phenomena in the opposite direction, which would correspond to the description of changes similar to the reversal of time, but statistical laws indicate that such states are almost unlikely, meaning that they cannot occur as a state of development of the system. The probability of inverse phenomena is infinitesimally small, because in irreversible processes the energy dissipation occurs in the form of thermal motions of many molecules. The reverse process could only be achieved by re-focusing the dispersed energy, which would require synchronization of the trillions of the molecules.

With regard to the ageing processes, the biological arrow of time and the psychological arrow of time may be particularly important. The psychological arrow of time ought to be also considered as cognitive.

The biological arrow of time illustrates, among others, the evolution of living organisms over time; from simple unicellular organisms to complex, highly organized multicellular organisms.

Entropy of the system (living organism + environment) increases over time according to the second principle of thermodynamics and is termed the thermodynamic asymmetry of the world.

In the first half of the twentieth century, there was a view that living organisms violate the second principle of thermodynamics as regards entropy. The basis for clarifying the viability of living organisms that are thermodynamically open systems to increase the degree of organization and reversal of entropy changes is negentropy, the term introduced by Erwin Schrödinger in his book under title “What Is Life?”. Negative entropy (negentropy) is responsible for keeping living organisms far away from the environment and avoiding progress towards chaos. In the state of thermodynamic equilibrium with the environment comes to the death of the living organism, because balance determines the impossibility of the flow of energy into the living organism, which must constantly complement, but the distribution is no longer a state of thermodynamic balance. In nature, there are side-by-side processes in the direction of chaos, as well as the processes of the opposite – spontaneous growth and self-organization. These two types of processes (growth and decay, life and death) are inextricably linked.

Negentropy is interpreted as the maximum energy that the organism is able to transform into a work called the free enthalpy or Gibbs function, which is a very questionable concept, since this thermodynamic potential is different from zero only when taking into account the chemical potential, i.e. the flow of particles from the system to the system, as can also be offset - in other words, it is at most equal to zero.

In the book published in 1982 under title Principles of Biochemistry, American biochemist Albert Lehninger argues that the production of order in cells is accompanied by an increase in disorder in the environment that compensates and even surpasses the increase of order in the cells. Living organisms maintain their internal order by taking free energy from the environment in the form of food or light, and bring about the equivalent amount of energy in the form of heat energy along with the accompanying entropy.

The phenomenon of ageing of organisms can also be interpreted as an increase in entropy with respect to certain factors, and its decline relative to others. Most of the cells are already specialized, so their development is targeted, however, the amount of spontaneous ageing processes increases with the lifetime of the organism. In addition, as Lehninger said, even if entropy is locally decreasing in the body elements, it increases with respect to the whole ecosystem and to the Universe.

For living organisms, it is also worth considering the psychological arrow of time. Some portions of thinking are understood in the light of functioning of human memory. The human memory recollects and stores past information but cannot remember the future. The past events are recorded in the mind but not the future ones. This occurs because of the time’s directionality. The constraints imposed on the memory forbids us to take a look into the future. This is the arrow of time, which permits memory to act in this particular way only. In psychology, memory is said to be remained in the chaotic form so far as it is unused but becomes orderly when it is put to use. This particular process is related with the nature of knowledge, so more will be said while considering the epistemological dimension of the arrow of time.

The problem of the time arrow is to be tackled not merely at the physical level but essentially at some epistemological (or philosophical) level. Moreover, different arrows of time function in a different and paradoxical way, e.g., thermodynamic arrow functions from order to chaos, whereas, biological arrow functions from chaos to order. Therefore, epistemological presuppositions of the arrow of time ought to be considered.

1. Time's arrow basically depends on dualism. Dualism between past and present, right and left, up and down etc. Unless dualism is presupposed, no arrow can be realized. This dualism exists only at the apparent level. There must be beginning and end.

2. Knowledge of arrow is possible because of change. Arrow always moves from the previous point to the subsequent point. The knowledge of any system is extracted when it is moving. The knowledge of change depends upon the knowledge of a dual. In the absence of dualism no change is perceived.

3. Change is a forklike behaviour which is irreversible. Arrow exists in forklike or branching phase. It is from the roots to the branches and not the other way round that the information of the world is gathered.

4. Irreversible change makes any system to expand with time. Knowledge of any system is possible in the expanding mode.

5. Arrow always exists in an open system.

6. Small input will give high output.

The arrow of time illuminates that knowledge is always fallible. Always it is subject to change and revision. According to Popper, knowledge always grows through falsifiability. But here falsifiability is of the nature of irreversibility. Patterns of knowledge are recognized in the dynamic state.

It is possible to define a psychological arrow of time also for inanimate matter.

In the next longer post, the consciousness of inanimate objects as well as living organisms and their relation to the processes of ageing will be considered.
What about countability or uncountability? Does it make a difference? In terms of dimensions, I think that may be the case. For example, think about fractal dimensions and elaborate on that thought.
Theoretically, I can create projection of 4, 5, 6 or 32423 (etc...) dimensional object in 3 dimensions. However, what if I do a projection of infinity? Will it be constantly expanding object and projection of this object in 3 dimensions or will it go into something else entirely?

If it is an ever-expanding-dimensions object (with +1 dimension), and I draw its projection in 3 dimensions, then I will never really have infinity because I will always see the projection of the 99991 dimensional object in 3 dimensions, then projection of 99992 dimensional object in 3 dimensions, then 99993 dimensional obj in 3D, then 99999...9 obj in 3D and so on, just another object with extra dimension and its projection.

However, what if an "object" (or what is it? an object or something else?) which is an infinite dimensional "object" I project into 3 dimensions will be something completely different? I mean if it will be not simply an object with an extra dimension with its projection, but something else, then what is it? This is an interesting question...
Luks said:
I mean if it will be not simply an object with an extra dimension with its projection, but something else, then what is it?

What if, stated mathematically, "The product of the journey is greater than the sum of the destination."
Many times we went out and did jobs but did not get paid, the boss would say "just think of all the experience you're getting."
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What if, stated mathematically, "The product of the journey is greater than the sum of the destination."
Many times we went out and did jobs but did not get paid, the boss would say "just think of all the experience you're getting."
If it is an equation that might need some modifications. Perhaps...

The product of all journeys combined is only equal to the ultimate destination when gravity says they are.
I would like to ask you to develop this thought. What is real sacrifice? How do you understand it? Does it have to do with historical facts for you or just a concept? We can talk about it on the level of history or physics. Or maybe metaphysics?
I’m working on it. I foolishly thought it was self explanatory.
What if, stated mathematically, "The product of the journey is greater than the sum of the destination."
Many times we went out and did jobs but did not get paid, the boss would say "just think of all the experience you're getting."
Yeah, that's okay. :-) I will come back to the dimensions etc. Even purely theoretically, by computing (multiplying) these matrices, by a matrix representing the 3 dimensions. We probably won't get too far this way. But another thing occurred to me. I used to think of encapsulation, which is rather "top-down", from densities representing Knowledge to lower material levels. But why not in other cases, also from bottom to top as well.

The three dimensions relate to three directions that are perpendicular to each other. Following this regularity, I should say that the 4th dimension is a dimension - a vector perpendicular to the other three vectors perpendicular to each other. But how to contain it?

So I remembered that after all, 4 Density where there are 4 Dimensions (as the C's say) has no right-left discrimination and reality is selective. So in 4th Dimensional Reality we possible might have a reality split into two segments where the 4th dimension is actually perpendicular to the other 3 dimensions. Except in this case, these 3 dimensions represent one group. When we are in 4 dimensions, we may have, in a sense, two principal dimensions, arranged perpendicular to each other, as if two segments. Where the latter dimension represents our 3 dimensions "packed".

One dimension in which there is no right or left - the parent segment and the second dimension, which encapsulates the other 3 dimensions, i.e. these we know well, where we distinguish right-left directions, we can look up and down. So we would have a 1st dimension (4th dimension), and a 2nd dimension composed of the other 3 dimensions, well known to us and experienced in this density.

If we assume that this subgroup of these 3 dimensions, which is visible in 4 dimensions/4 densitity as the 2nd dimension (one of the two dimensions) is an encapsulation of these "our" 3 dimensions, then only after decapsulating them, the entity of 4th density can see into a 3 dimensional reality (which is also a 3rd density reality.)

If such decapsulation is not done then the 4 dimensional/4 density being remains in two dimensions, two segments of reality experience, where these 3 dimensions are selective and visible only when focused on them and in one way or another decapsulates them and "unfolds" them.

In the 4 Density (and 4 Dimensions also). The primary (first) dimension represents the spiritual reality, the "additional" (second) dimension represents the physical reality which, after decapsulation, constitutes the 3 dimensions of physical reality.
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