What Is the Natural Number One?
What Are Numbers?
What are numbers? This question has been discussed from the era of ancient Greece. However, there is no satisfying answer. Stanislas Dehaene, who studies cognitive psychology of number processing (1), claims that we should address more focused questions such as where specific mathematical objects like sets, numbers, or functions come from (2). Then we can choose the basic question: What is the natural number one?
The Difficulties of Counting Objects
In the beginning, we shall consider the difficulties of counting objects. Counting objects may be the most basic usage of numbers. For example, it seems easy to count stones. Suppose that children play the assumed game. The rule of the game is simple. Each child collects stones in the field and then they count each child's stones. Finally, the child, who has the largest number of stones, is the winner. If children try to count the number of stones accurately, they would face many difficulties. As an example, if children are going to determine the boundary of a stone and a rock, they would dispute it. Similarly, the boundary between a stone and a sand grain would become a problem, too. Thus, it is hard to determine the range of the size of the stones. These difficulties are due to vague boundaries of stones in relation to various properties, since the stone is a loosely-defined term. After a heated discussion, they would have to use the textbook of geology. Subsequently, they would be able to define strictly the size, the color, and the shape, etc. of the stones. However, other than these difficulties, the most serious difficulty is due to the fact, that any stone can be divided into pieces. Suppose that they finished completely counting the stones. Then, if one stone were cracked into pieces, they would be confused by the question, whether they should re-count pieces or not. Because many objects are divisible, this difficulty is universal. We shall return to this point later.
Human Beings Are Easy to Count
By contrast, it is easy to count human beings. No matter who counts the number of human beings, the same result will be obtained. There is no room for argument. Why are human beings easy to count? First, even though there are various people on the earth, everyone is a human. This is the basic principle of democracy. Everyone is equal in the sense that we identify everyone as a human. Second, a human usually keeps ego identity from birth to death, and split personality is a mental illness. That is to say, a human, who is mentally healthy, is invariable. Third, a human cannot be divided. If a human were divided by force, the human would be injured or killed. The death of a human is irreversible. There have been no report of reviving a dead body since recorded history. A human is indivisible.
What Is the Natural Number One?
Let us now return to the question: What is the natural number one? There were great discussions about this issue in Ancient Greece. Among these arguments, Plato described three important properties of the natural number one in the Republic as follows: "O my friends, what are these wonderful numbers about which you are reasoning, in which, as you say, there is a unity such as you demand, and each unit is equal, invariable, indivisible, --what would they answer?" These three properties are the basic properties of the natural number one. They are consistent with properties of a human being. Therefore, I propose that a human being is the prototype of the natural number one. Additionally, we shall call the natural number one Plato's one.
The Paramecium's Self
Everyone regards oneself as having the basic properties of Plato's one. Let us consider the evolution of the self. In the beginning, most animals can move, while most plants are fixed. However, branches of a tree extend out in different directions. In contrast, an animal must choose only one direction. So an animal needs the unity of the whole body. If the unity of an animal were broken, the animal could not move. If each part of the animal moved various directions, the body of the animal would be disrupted. To avoid such a situation, animals have developed the nervous system. It is uncertain when the present form of self arose. However, the primitive form of self is found in a paramecium which is a well-studied protozoa. This type of self is recognized by observing movement. A paramecium has a slipper shaped body, which is covered by cilia. If someone observes a paramecium, he cannot find paramecium's nervous system, but he may recognize paramecium's self. First, a paramecium swims skillfully using altogether cilia. The integrity of the whole body is required for the skill. That is, paramecium's self is indivisible. Second, the observer will find the consistency of its behavior. That is, paramecium's self is invariable. Thus, paramecium's self has two properties of Plato's one. Therefore, the primitive self is the minimum requirement for movement.
The Equality of Selves
The equality of selves is a controversial point. Human beings seem to have different selves. However, Schrodinger says in the epilogue of "What Is Life?" as follows:
Yet each of us has the indisputable impression that the sum total of his own experience and memory forms a unit, quite distinct from that of any other person. He refers to it as 'I' and What is this 'I'? If you analyse it closely you will, I think, find that it is just the facts little more than a collection of single data (experiences and memories), namely the canvas upon which they are collected. And you will, on close introspection, find that what you really mean by 'I' is that ground-stuff upon which they are collected.
Schrodinger says that differences among selves are the differences of experiences and memories. We shall consider the differences of selves based on his idea. First, let us think newborn babies. The differences of personalities of them are only differences of genes. Because genetic diversity is the result of the history of human beings, we shall go back to the dawn of humanity. There might be the common ancestor of present human beings. If the common ancestor had been actually existing, the newborn common ancestor's self would be the white canvas. Then, selves of present human beings are paintings on the white canvases.
A Cell as the origin of Plato's One
Next, we shall consider the relationship between a life and Plato's one. Schrodinger says in chapter 6 of "What Is Life?" as follows:
What is the characteristic feature of life? When is a piece of matter said to be alive? When it goes on 'doing something', moving, exchanging material with its environment, and so forth, and that for a much longer period than we would expect of an inanimate piece of matter to 'keep going' under similar circumstances. When a system that is not alive is isolated or placed in a uniform environment, all motion usually comes to a standstill very soon as a result of various kinds of friction; differences of electric or chemical potential are equalized, substances which tend to form a chemical compound do so, temperature becomes uniform by heat conduction. After that the whole system fades away into a dead, inert lump of matter. A permanent state is reached, in which no observable events occur. The physicist calls this the state of thermodynamical equilibrium, or of emaximum entropy'.
Now, the property of a system, which maintain its internal environment stable, is called homeostasis. If homeostasis of a living organism is destroyed, it will die. After it dies, the dead body will rapidly decompose in the ordinary environment. Homeostasis is one of the essences of life. Multicellular organisms are composed of many cells, but a multicellular organism can be divided into a cell. A living cell is a unit of life. A cell membrane, membrane proteins, enzymes, etc. are required for keeping homeostasis in a cell. So almost the whole cell is necessary for life to keep homeostasis. In addition, the integration of the cell is also a requisite for maintaining homeostasis. That is to say, a cell is indivisible. Moreover, a cell is equally a unit of life and is mostly invariable from a cell division to the next cell division. Consequently, a cell is the origin of Plato's one.
The Irreversibility of Death of a Life Is the Premise of Natural Selection
The death of a life is the irreversible process. If a life dies, it cannot be revived and the dead body rapidly decays. This process is crucial for natural selection. I say that natural selection protects information against entropy. Following the second law of thermodynamics, any information storage will deteriorate and will be finally lost. However, only genetic information is conserved for a long period, because natural selection irreversibly eliminates fatal mutations. According to the neutral theory of molecular evolution, the amino acid sequence of the protein, which is important for the survival, is highly conserved. Thus, it is necessary for natural selection that death is irreversible. Additionally, the rapid decomposition is important. It prevents the reverse process, and decomposed products are nutrients for other living organisms.
Next, we shall consider whether inanimate objects are divisible or indivisible. For example, the diamond, which is the hardest substance, can be divided. If a diamond were divided repeatedly into carbon atoms, even then a carbon atom could be divided. If we could use enough energy, we can reverse the process at any point of time. Furthermore, there will be various opinions as to when the diamond died. An ordinary man says that the diamond died when it was cracked into pieces. A jeweler says that the diamond died when it was scratched. A physicist says that the diamond died when its carbon atoms are destroyed. So, anyone can make an arbitrary definition of the death of the diamond. As just described, nonliving objects are divisible.
From a Life to a Self.
Because a life has the basic properties of Plato's one, we can apply Schrodinger's idea to the common ancestor of all life. However, our knowledge about the origin of life is very poor. Instead, we use the concept of the minimal cell. It has the minimal gene set, which would be necessary and sufficient to keep a cellular life under the most favorable conditions. The minimal cell project is trying to construct a minimal cell capable of replication and evolution. If the minimal cell were synthesized, we could assume the minimal cell as the white canvas of Schlodinger's idea. Then, we could actually recognize the continuous evolution from a life to a self. Even now, many evidences indicate the continuity of evolution. Continuity means that all lives are essentially equal to each other.
The Starting Point of All Science
Now, we shall consider our thought based on the self. The self is the point of origin of rational thought. In "Discourse on the Method", Descartes says as follows:
Accordingly, seeing that our senses sometimes deceive us, I was willing to suppose that there existed nothing really such as they presented to us; and because some men err in reasoning, and fall into paralogisms, even on the simplest matters of geometry, I, convinced that I was as open to error as any other, rejected as false all the reasonings I had hitherto taken for demonstrations; and finally, when I considered that the very same thoughts (presentations) which we experience when awake may also be experienced when we are asleep, while there is at that time not one of them true, I supposed that all the objects (presentations) that had ever entered into my mind when awake, had in them no more truth than the illusions of my dreams. But immediately upon this I observed that, whilst I thus wished to think that all was false, it was absolutely necessary that I, who thus thought, should be somewhat; and as I observed that this truth, I think, therefore I am (COGITO ERGO SUM), was so certain and of such evidence that no ground of doubt, however extravagant, could be alleged by the sceptics capable of shaking it, I concluded that I might, without scruple, accept it as the first principle of the philosophy of which I was in search.
The existence of the self is the first principle of the modern philosophy. Before that, the premise of Descartes' claim is that the self has the basic properties of Plato's one. If the premise is true, oneself is equal to oneself. Then, we can abstract Plato's one from the self:
This equation is the base of the logical thinking. If the integration of the self is broken by the mental illness, the thought of the patient will be illogical. For example, the disturbance of the self is the crucial symptom of schizophrenia. The schizophrenic patient, having the disturbance of the self, usually has the disturbance of thought.
Next, a person can regard another person's self as the prototype of Plato's one. Then, we can abstract two from the set of two persons:
These equations are the starting point of all science.
References
1. S. Dehaene, The Number Sense, Oxford University Press, 1997
2. S. Dehaene, Edge 28, http://www.edge.org/documents/archive/edge28.html
Table of Contents