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?”
About Kazuhiko Kotani
The Revolution of Mathematics 