The Curve Is the Aerial Line

March 25, 2013 3 Comments

Our eyes capture a photon by the photoisomerization of retinal. The reaction is shown below.

retinal2028129

Hence, there is the digital image on our retina. Since our brain get visual information only from the retina, our first image is the raster image. It must consist of pixels, which is the regular polygon.

tile

The above figure shows three types of tiling the plane with regular polygons: the square, the equilateral triangle, the regular hexagon. Only these three regular polygons can be the pixel of the digital image. That is, curves cannot exist in the digital plane (1). They must be created by our brain. That is, they are fictitious. It is natural to think that they don’t exist. Why do our brain create aerial lines?

The answer is simple from the viewpoint of the evolution. The evolution  has given priority to utility, and it often ignored the logical consistency. Gary Marcus says as follows (2).

Nature is prone to making kluges because it doesn’t “care” whether its products are perfect or elegant. If something works, it spreads. If it doesn’t work, it dies out. Genes lead to successful outcomes tend to propagate: genes that produce creatures that can’t cut it tend to fade away: all else is metaphor. Adequacy, not beauty, is the name of the game.

More specifically, our visual system ignores the logical coherence. Rather, our visual system recognizes patterns, which are important for survival. Probably, the ability of recognizing a circle would be advantageous in the struggle for survival. So, shapes like a circle are categorized as a circle. Similarly, the shape like a human face is recognized as a face. The human being excels in the recognition of the face (2). So, we can easily recognize the face of a specific person (3). However, a face does not have a specific geometrical meaning. Therefore, the evolution created curves without concerning the logical consistency.

1. The Origination of the Revolution of Mathematics

2. Kluge: The Haphazard Evolution of the Human Mind

3. Invariant visual representation by single neurons in the human brain

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The Equality Is Required for Accurate Communication

November 8, 2012 Leave a comment

The language is the basic tool of our thought. Furthermore, it is the base of the cognition. That is , it is important for an individual. However, its main role is the tool of the interpersonal communication. For example, suppose that a primitive man encountered a tiger. Then, he wanted to convey the information to other persons. If there were no language, he would have to simulate the voice of the tiger by his voice and the motion of the tiger by his gesture. That is, he would need great labor to convey information. It would spend longer time than the verbal communication, but it might be incorrect. Hence, the nonverbal communication is  inefficient. By contrast, if language is used, he could easily transmit information accurately. Why the language is so efficient? Let us consider the feature of the language.

The language is composed of basic units. So, any spoken language can be divided into phonemes. The phoneme is the basic unit of the speech. Furthermore, any written language consists of finite characters. For example, the number of Chinese characters is very large, but it is finite. The character is the basic unit of the written language.

Surprisingly, the basic unit of the language is equal to each other. However, an object isn’t the same as another object.  Socrates said in Phaedo “Then we must have known absolute equality previously to the time when we first saw the material equals, and reflected that all these apparent equals aim at this absolute equality, but fall short of it?” That is, the complete equality cannot achieve in the reality. Notwithstanding, we regard the basic unit of the language being equal. For example, we consider that a letter “A” completely equals another letter “A.”

Why is the basic unit of the language equal to each other? The words of Socrates indicate that the concept of the absolute equality must be inherited from our distant ancestors. Ultimately, our roots may be traced back to the last universal common ancestor (LUCA). LUCA is the most recent common ancestor of all lives on the earth. LUCA must have the elaborate DNA replication system because almost all living organisms can accurately copy DNA. The common features of all living organisms might be that of LUCA. So, LUCA might keep its genetic information stable. Hence, the equality is originated from LUCA.

The materiality of the genetic information is DNA, which consists of four bases: adenine (A), thymine (T), guanine (G), cytosine (C). DNA extremely resembles the language. Both are the sequence of finite characters. A three base sequence of DNA corresponds to an amino acid. In the similar fashion, the word is a sequence of letters, and it is the smallest unit of the language, which has the meaning. Furthermore, both are read one by one in serial order. Therefore, DNA and the language are isomorphic. It seems that the language was modeled on DNA.

DNA is the original form of digital information. Each base of DNA can be the direct target of natural selection, as I say in “What Is Digital Information?” Thus, the critical base of the important DNA sequence for survival is highly conserved according to the neutral theory of molecular evolution. It is important that the deleterious mutation is eliminated by natural selection. Especially, the individual, which has the fetal mutation, will die soon. In fact, natural selection enhances the equality of the important base of DNA. It is always copied accurately because its miscopy is removed irreversibly by natural selection.

Let us return to the first example. When a primitive man encountered a tiger, it would be important to convey the position of the tiger to other primitive men. That is, the information of the position of the tiger is life-threatening. Suppose that he went to the right and met the tiger, and the left side was bright and the right side was dark. In this case,  a man, who mistook right for light, would be exposed to the risk of the death. The misconception of the phoneme can be fatal. Hence, the equality of the phoneme is necessary for accurate verbal communication.  Furthermore, the accuracy of the communication influences the survival rate of the individual. That is to say, natural selection enhances the equality of the phoneme. The higher the extent of the equality of the phoneme is in a group, the higher the survival rate of an individual of the group is. Therefore, the spoken language and the writing language and DNA are digital information. They consist of finite elements. Each element is equal to each other. Natural selection has strengthened the equality of the element.

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The Origin of Digital Information

June 8, 2012 Leave a comment

The second half of the 20th century was the era of the computer and DNA. Their common feature is digital information. As mentioned in “Properties of Digital Information“,  it has properties of the natural number one: equality, invariability and indivisibility. Let us seek the origin of digital information.  The computer is invented in the 20th century. In contrast, scientists discovered 419 million-year-old DNA intact. It is the known oldest digital data.

Let us consider DNA. DNA is a chemical compound. By contrast, the natural number one is the concept. There is the definitive difference between them. Plato emphasized the distinction between the real matter and the abstract concept. He described in Phaedo “Then we must have known absolute equality previously to the time when we first saw the material equals, and reflected that all these apparent equals aim at this absolute equality, but fall short of it? ” That is, DNA cannot have ideal properties. However, lives try to make DNA close to the ideal digital information.

For example, a large set of DNA repair enzymes are required for the storage of genetic information. Even under normal cellular conditions, each nucleotide of DNA is modified by spontaneous  reactions: oxidative damage, hydrolytic attack and uncontrolled methylation by the methyl group donor S-adenosylmethionine.  Furthermore, DNA undergoes modifications by many other reactions. Hence, the life evolved the DNA repair system for preserving genetic information. This example shows that digital information is produced by life for survival. That is, the life can change the grade of the completeness of digital information. However, because the mutation is necessary for evolution, DNA will never be identical to ideal digital information.

As I mentioned in “What is digital information?“, we can define digital information as the substantial entity, which is the replica of the Platonic number and has the possibility of determining life or death of a living organism. The most important thing is that digital information determines the survival rate of  a life. How long a DNA sequence is close to the ideal digital information is determined by its importance for survival.

Let us consider the maintenance of DNA sequences. The mutation rate of bacteria is 1 nucleotide change per 109 nucleotides per cell generation. On the other hand, under laboratory conditions, E. coli divides about once every 30 minutes. Under these conditions, more than half nucleotides will change about 40,000 years. That is, if all mutants survived, all genetic information would be rapidly changed. This change speed is good for evolution, but is inappropriate to preserve genetic information. However, natural selection excludes disadvantageous mutations in reality. Thus, the rate of evolution varies according to a gene. Because the mutation of the important gene for survival is likely to fatal, its rate of evolution is slow. In contrast, if mutations within a gene are likely to tolerate, it will rapidly evolve.

Finally, let us consider an extreme example. The most conserved gene is the ribosomal RNA. Especially, strongly conserved regions of 16s rDNA are shared by all known cells. Nucleotides in them have close properties to the natural number one: equality, invariability, indivisibility. They might be the nearest real thing to the natural number one on the Earth. Because properties of the natural number one are abstracted from life, their properties are come from the life itself.

If you want to know the more detailed discussion, please read “Natural Selection Protects Information against Entropy”

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The Origin of the Natural Number One

June 3, 2012 Leave a comment

The natural number one is the most fundamental concept of mathematics.  Certainly, the number anyone knows very first is one, which is equal, invariable and indivisible.  Moreover, as I say in “Properties of Digital Information“, these properties are the basic property of digital data. Let us consider the origin of these properties. Firstly, Plato says in Phaedo as follows.

But what would you say of equal portions of wood and stone, or other material equals? and what is the impression produced by them? Are they equals in the same sense as absolute equality? or do they fall short of this in a measure?
Yes, he said, in a very great measure, too.
And must we not allow that when I or anyone look at any object, and perceive that the object aims at being some other thing, but falls short of, and cannot attain to it-he who makes this observation must have had previous knowledge of that to which, as he says, the other, although similar, was inferior?
Certainly.
And has not this been our case in the matter of equals and of absolute equality?
Precisely.
Then we must have known absolute equality previously to the time when we first saw the material equals, and reflected that all these apparent equals aim at this absolute equality, but fall short of it?

So, we have never seen absolutely equal objects. Hence, we might be inherited the concept of the absolute equality from our ancestors.  However, probably equal objects have never existed. If we want to seek for the origin of the complete equality, we must trace back to the evolutionary tree without limit. Therefore, I think that life might have the built-in equality.

Let us consider the drastically simplified life.

The above figure represents the simplified cell. A circle represents a cell, and capital letters represent the base sequence of DNA. The left circle represents a cell with DNA, and the right circle represents a cell without DNA. Almost all features of a cell are encoded in DNA. However, the most remarkable feature of  life might not be encoded in DNA. It is the irreversibility of the death of a life. When any life dies, it cannot revive. This is the only absolutely irreversible process, which we know. As mentioned in “Natural Selection Protects Information against Entropy“,  this is the necessary property for preserving information. Another important fact, when a cell dies, its whole body is dead.

Next, let us consider a thought experiment. If we removed DNA and products of DNA from a cell, there would be an empty container, which consists of the lipid bilayer. The function of it is the separation of the inside of the cell from the outside.  That is, it gives unity to a cell.

According to the cell theory, Virchow said that every cell originates from another existing cell like it (Omnis cellula e cellula). Hence, common properties of every cell might be inherited from the first cell on the earth. Among them, the property of this hypothetical container is important. It is identical to the natural number one. Plato described “there is a unity such as you demand, and each unit is equal, invariable, indivisible” in Book Ⅶ of the Republic. These three properties are the basic property of the natural number one.

Let us consider each property. Firstly, the indivisibility is the most remarkable feature of life.  A cell can be divideｄ into two cells only in the final stage of the cell division. Usually, a cell is indivisible. If a cell is divided severely, it would die.  If it is cut off a part, it would repair the part. This principle can be applied to a human being. If a man is divided, he would die or survive with serious injury. However, one human being is never divided into two human beings. Thus, we can regard one generation of a cell corresponding to an entire life of a human. So, both a human and a cell correspond to the natural number one.

Next, let us consider the equality and the invariability. Because the difference of cells is due to the variety of DNA, the common feature of all cells might be inherited from the first cell. Especially, properties of the hypothetical container might have been invariable. It is necessary for the unification of the whole cell. Furthermore, if it is broken, the cell dies. If we consider the abstract container, it is indivisible, invariable and equal to each other. Therefore, the natural number one is abstracted from properties of the cell.  If you want to read the more detailed description, please read “What Is the natural Number One?”

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Properties of Digital Information

May 28, 2012 2 Comments

The digital computer can recognize only 1 and 0. Because it distinguishes only two signals, the intermediate signal cannot exist. This is the remarkable feature of digital information. According to Wikipedia, historical digital systems are the written text, Morse code and etc. They are the serial sequence of restricted characters. They can be converted into each other. The alphabet can be converted into 1 and 0.

The above figure is the example of the conversion. The upper row represents alphabets, and the lower row represents corresponding binary digits according to the ASCII code. Why is the conversion possible? Each character of any kind of digital information has common features, which are properties of the natural number one. As I say in “What Is the Natural Number One?“, it is indivisible, invariable and equal to each other. These properties are the basic property of  digital data. However, there are some differences between digital data and natural numbers.

Digital data consists of multiple characters. In contrast, the natural number is the set of the natural number one only. Furthermore, we can add two natural numbers. However, we cannot apply the calculation rule to the ASCII code. Rather, 0 and 1 of the ASCII code are regarded as mere symbols. Especially, the digital data 0 and the number 0 are very different. Though digital data 0 is indivisible, the number 0 is divisible.

0=0+0+0+•••+0

Thus, the digital data and the number is different. Still, they can be converted reciprocally. Especially, as mentioned in “the concrete basis of the number theory“, the ordinal number has the basic property of digital data, which is the basis of the conversion of digital data. Therefore, the basic property of digital data is necessary for the interconversion of the digital data and the interconversion between the digital data and the number.

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