What Are Numbers?

Japanese Version

by Kazuhiko Kotani

Table of Contents

  1. Main Text
    1. The Origin of Numbers Is a Life
    2. DNA Is Digial Information
    3. The Importance of Natural Selection
    4. The Empty Bag
  2. The Constructive Geometry
  3. The Archive of My Knols
    1. Number
      1. What Is the Natural Number One?
      2. The Concrete Basis of the Number Theory
      3. A Negative Times a Negative Is a Positive
    2. Infinity
      1. An Objection to Cantor's Diagonal Argument
      2. Counting Real Numbers
      3. What Is Infinity?
      4. Is the Infinite Decimal Available
    3. Information
      1. What Is Digital Information?
      2. Natural Selection Protects Information against Entropy
      3. Sense vs. Digital Information
      4. The Action Potential as the Universal Coin
      5. Life as One
    4. Geometry
      1. Area Is Essential for Euclidean Geometry
      2. The Biological Foundation of Geometry
      3. The Re-evaluation of the Euclidean Number System
      4. The Integration of Mathematics
      5. Pi Is Not a Fixed Value
    5. Motion
      1. What Is Real?
      2. The significance of Zeno's Paradoxes
      3. The Motion and the Differential Calculus
  4. Appendix
    1. Natural Selection Enables the Preservation of Information

    What are numbers? This question has been discussed from 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 shall choose the basic question: What is the natural number one?

1. The Origin of Numbers Is a Life

    Let us consider the question: what is the natural number one? In the beginning, we consider the difficulties encountered in counting objects. Counting objects may be the most basic usage of numbers. For example, it seems easy to count stones. Suppose that children are collecting stones and competing for the number of the stones. If children try to count the number of stones accurately, they will face many difficulties. As an example, if they are going to determine the boundary of a stone and a rock, they will dispute it. Similarly, the boundary between a stone and sand becomes a problem, too. Thus, it is hard to decide the range of the size of the stones. These difficulties are due to vague boundaries of range of stones in relation to various properties, since the stone is loosely-defined term. After heated discussion, they would have to use the textbook of geology. Subsequently, they would be able to strictly define the size, the color, and the shape etc. of the stones by using it. 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 puzzled by the question whether they should re-count pieces or not. Because many objects are divisible, this difficulty is universal.

    By contrast, it is easy to count the number of human beings. Even if anyone counts the number of people, 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 being. This is the basic principle of democracy. All human beings are equal in the sense that we identify every human as a human being. Second, a human being usually keeps ego identity from birth to death, and split personality is a mental illness. As mentoned above, a human being, who is mentally healthy, is invariable. Third, a human being cannot be divided. If a person were divided by force, he would be injured or killed. A human being is indivisible, and two human beings never fuse together into one human being.

    Let us now return to the question: What is the natural number one? There was a great discussion about this issue in Ancient Greece. Among these arguments, Plato described three important properties of the natural number one in The Republic [3] as following quotation: "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 of the natural number one 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 these three properties Plato's principles of the natural number one.

    The origin of these properties of human beings can be phylogenetically traced back to the unicellular organisms. It is a single living cell, and if it isn't just before cell division, it cannot be divided into two unicellular organisms. In contrast, multicellular organisms are composed of many cells, and it can be decomposed into single living cells. So it can be cultured. However, we cannot disassemble the single living cell. If it were divided by force, it would be injured or killed. If it died, it could not come back to life. That is, a single living cell is an indivisible unit of life. Consequently, life is indivisible. On the contrary, 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. Therefore, any object is divisible. In conclusion, only a single living cell is indivisible, because the death of a single living cell is absolutely irreversible. In addition, all cells are equal in the sense that all cells are descendent of a single common ancestor cell. It is suggested that these properties of a single living cell are the origin of properties of a human being as the natural number one. Therefore, the origin of numbers is a life because the natural number one is the element of numbers.

2. DNA Is Digial Information

    Next, let us consider why a life has similar properties of the natural number one. We must search the core of the life to solve the problem. When we consider the essence of life, it is beneficial to refer Schroedinger's book: "what is life?" [4]. So he is the theoretical founder of the molecular biology. He pointed out two fundamental features of life: self-replication and homeostasis in the book. The first feature was the main target of the molecular biology. That is, early molecular biologists quested for the mechanism of the inheritance. Primarily, they had sought the molecular entity of the gene. In 1953, Watson and Crick discovered that it is DNA [5]. The discovery is the revolution of the biology.

    The base sequence of DNA consists of only four nucleotides: adenine(A), thymine(T), guanine(G), cytosine(C). DNA is the blue print of life. The configuration of a life is determined in the order of bases. The most prominent feature of our genetic system is that DNA is digital information [6]. That is, each base must have similar properties to the natural number one. So, there is no intermediate signal between adenine and guanine. Minor modification of a nucleotide will not be allowed. Even though the DNA bases have many properties, the most important feature of them is equality: A=A, T=T, G=G, C=C. Because they have the equality, the base sequence of DNA is easy to copy.
    However, there is no true equality in the real world. For example, an object must be different from another one. Although a hydrogen atom may be likely identical to another one, we cannot see it. So we have to use measuring instrument. As a result, there are errors of measurement. Why do we know the true equality? It is difficult to answer this question. Let us refer to dialogue of Socrates in Phaedo [7].
Well, now, he said, what do we find in the case of the equal sticks and other things of which we were speaking just now? Do they seem to us to be equal in the sense of absolute equality, or do they fall short of it in so far as they only approximate to equality? Or don't they fall short at all?
They do, said Simmias, a long way.
Suppose that when you see something you say to yourself, This thing which I can see has a tendency to be like something else, but it falls short and cannot be really like it, only a poor imitation. Don't you agree with me that anyone who receives that impression must in fact have previous knowledge of that thing which he says that the other resembles, but inadequately?
Certainly he must.
Very well, then, is that our position with regard to equal things and absolute equality?
Then we must have had some previous knowledge of equality before the time when we first saw equal things and realized that they were striving after equality, but fell short of it.
Amazingly, words of Socrates seem to have already known DNA about 2300 years before the discovery of it. He says that we have known the equality before the birth. That is, it is implemented in our nervous system. The origin of it can be phylogenetically traced back to bacteria. A bacterium tries to copy itself precisely. So the self-replication is the first feature of life. The equality of bases of DNA is necessary to attain the accuracy of the copy. Rather, the concept of the equality is come from them. If there weren't the equality of them, life would have gone extinct long before. Why do they have the equality?

3. The Importance of Natural Selection

    Darwin says that natural selection drives evolution [8]. His finding is the great progress of science. However, the mechanism of the heredity was unknown in his days. In the beginning, Mendel found Mendel's law of inheritance, and he published his work in 1866 [9]. This is the start of genetics. After that, many scientists had continued to study genetics. Lastly, Watson and Crick reached the big break through: the double helix of DNA. From that time, the molecular biology branched from genetics has developed greatly. Now, the mystery of heredity is almost solved. Then, let us consider the true role of natural selection. The importance of it becomes clearer in the present age.

    Lives digitize the environment by the number of them. As a simple example, let us consider bacteria. If all of them had the same genetic information, they would easily become extinction. Hence, the variation of gene is necessary. For example, if they are starved, they will try to survive. Some of them move for seeking nutrients. Some of them will come to rest, and then they will lower the rate of the metabolism. Some of them will gain the ability to metabolize the matter, which they couldn't metabolize. It depends on the situation which method is advantageous. When the environment deteriorates locally, moving might be advantageous. On the contrary, when the environment deteriorates widely, they should be still. In any case, only survivors of them can produce offspring. That is, survivor's DNA sequence is adapted to the environment. This process is natural selection. We can regard it as the digitization of the environment. From a standpoint, the environment is digitized by the number of bacteria, which have the particular DNA sequence.

    Although DNA is a highly stable molecule, the DNA of a cell is frequently damaged everyday. Ultraviolet rays and the harmful chemical substance injure DNA. Furthermore, even under normal cellular conditions, many bases of DNA spontaneously change everyday. This is the serious problem for living organisms because only the mutation of a single nucleotide can cause the cancer. Thus, natural selection evolved the DNA repair system. Even so, the DNA sequence cannot be conserved completely. This is a natural result of the law of entropy. In short, the DNA of a cell deteriorates certainly. With this, a cell is destined to die. In fact, all animals must die. Because errors are accumulated in DNA of their somatic cells, they will age and eventually die. The most important point is that natural selection is incomplete in the animal's body. If there isn't natural selection in the world, all lives will become extinct. It is difficult to escape the domination of the second law of thermodynamics. Only natural selection can counteract it.

    The most remarkable advantage of natural selection is that fatal mutations can be eliminated by the death of mutants [10]. Let us consider bacteria, which multiply by cell division. The mutation rate in bacteria is roughly 10−8 per base pair per generation. That is, the probability of the complete copy of bacteria's DNA is less than one.
    While bacteria proliferate smoothly, it is all right, since precise copies increase more quickly than miss copies. However, the proliferation must stop because the earth is limited. Then, if there weren't natural selection, bacterial DNA would deteriorate. This is the necessary result of the law of entropy. But in reality, bacteria are surviving. If the saturated condition of bacteria is achieved, struggle for survival will become severe. The mutant bacteria, which have disadvantageous gene, may be extinguished. However, if a bacterium gain advantageous gene by a mutation, it will proliferate. As a result, offspring of it will overcome other bacteria. That is, natural selection evolves them. This seems as if it violates the second law of thermodynamics.

    However, the second law of thermodynamics can coexist with natural selection. Because we can deal with only thermodynamic equilibrium states in the classical thermodynamics, we have to make a simple closed system using a heat insulating container. When the equilibrium is achieved in it, the rate of the forward reaction and the reverse reaction are equal but net change is zero. There is no irreversible process. Because classical thermodynamics is the study of equilibrium, a system including an absolutely irreversible reaction is out of range.

    The irreversibility of death is the key point of natural selection. The second feature of life, which was pointed out by Schroedinger, is homeostasis [4]. It is the property of life, which keeps the internal environment stable. Although it has been modest compared with the double helix of DNA, it is equally important to the self-replication. Schroedinger says that a non-living system rapidly approaches to the equilibrium.
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.
However, living organisms keep homeostasis and avert the equilibrium. For example, an Eskimo dog can keep its body temperature in the frigid Antarctica. If the non-living matter of the same weight and temperature as it is placed in the environment, it will freeze immediately. Far from it, the dog pulls a dogsled. Its ability is impossible by a non-living thing.

    Meanwhile, death of life is irreversible. Because Schroedinger says that a living organism avoids rapid decay into the equilibrium, the dead cell would be rapidly decomposed. This process is difficult to reverse. Hence, the difficulty of the reanimation may be equivalent to that of abiogenesis. Actually, there have been no report that the dead life revived, after Pasteur denied abiogenesis in 1859. Perhaps, abiogenesis might have occurred only once. As another possibility, although it happened several times, only the present form of life survived. In either case, the environment on the earth at that time is greatly different from the present environment. Hence, now life doesn't occur spontaneously. Above all things, if an incomplete life were generated, it would be immediately eaten by bacteria. Because the probability of abiogenesis is too small, we can regard it as zero. Therefore, the law of entropy cannot be applied to natural selection. Additionally, although the law of entropy cannot be applied to natural selection, it is applicable to living organisms.

    Furthermore, natural selection endows crucial nucleotides of DNA with similar properties of the natural number one. If a nucleotide is critical for the survival of the living organism, it is conserved by natural selection. According to the neutral theory of evolution [11], the DNA sequence of the important gene for survival is highly preserved. As a result, the vital nucleotide in DNA comes to have the character near the natural number one: indivisibility, invariability, equality. Of course, if the mutation of it is always fatal, it is indivisible. Furthermore, it is permanently conserved. The famous example is the homology between bacterial 16s rRNA and eukaryotic 18s rRNA [12]. The indispensable nucleotides of them is identical. That is, they have been conserved from the divergence between the prokaryotes and the eukaryotes. Because the time of the branching is estimated about 3 billion years ago [13], we can regard them invariable and indivisible. Naturally, they gain equality. Therefore, important nucleotides of DNA have come close to the pure natural number one during the evolution.

4. The Empty Bag

    Let us consider what is the number, again. We shall consider a single cell for the simplification. Thus, the syncytium and multicellular organisms are excluded. Firstly, the death of a life is only known irreversible process. That is, death is specific to life. In contrast, ordinarily vital activities are the combination of chemical reactions. They are reversible, and they proceed to the equilibrium. That is, the second law of thermodynamics dominates them. Of course, chemical reactions involving DNA aren't exceptions. Secondly, the integration of a cell is necessary for keeping homeostasis. Hence, the death must be the doom of the whole cell. It keeps internal environment: pH, electrolyte profile, osmotic pressure and so on. If its homeostasis is broken, many enzymes will lose activities, and then many structural proteins will be denatured. As a result, it will die. That is, it is indivisible.

    The cell has two specific features: the indivisibility and the irreversibility of the death. If we exclude other properties of it, an empty bag will remain. It is indivisible, and the death of it is irreversible. They are common properties of all single cells. Moreover, they are the premise of natural selection. Now, we regard other properties of cells as the modification of an empty bag. Then, it is equal to each other and invariable. That is, even though cells have been changing their properties during the evolution, they are consistently indivisible, and the death of them is irreversible. Namely, it satisfies Plato's principles of the natural number one: indivisibility, equality, invariability. Therefore, we can regard the empty bag as the natural number one.

    Let us go back to the primeval age. The universal ancestor must have the properties of the empty bag. Perhaps, the origin of it may be older than the universal ancestor. I think that the origin of it may be a simple bubble, which separates the inside and the outside. The separation may be the origin of the homeostasis. Probably, the empty bag was generated at the genesis of the cell. At that time, the evolution started.

    If the empty bag is compared to the white paper, genes will be compared to letters. Genes has added many features to the cell. The DNA sequences of contemporary living organisms are documents, in which the history of the evolution is written. Every feature of living organisms corresponds to a gene. That is, the difference of lives is due to the diversity of genes. Even if the appearance of the cell has changed drastically, essential features of it are permanent. Hence, the empty bag can be abstracted from any cell. It is just the original form of the natural number one, and it is the indispensable trait of a cell. Next, let us compare it with DNA. Although the critical base of DNA is an ordinary chemical compound, natural selection gives it properties of the natural number one. Accordingly, the vital nucleotide itself doesn't preserve. Instead, it has been copied repeatedly, and miss copies have been eliminated by natural selection. Therefore, the cell's character as the empty bag is the nearest entity to the natural number one.

1. S. Dehaene, The Number Sense (Oxford University Press, New York, 1997).
2. S. Dehaene, Edge 28 (http://www.edge.org/documents/archive/edge28.html).
3. Plato, The Republic (http://classics.mit.edu/Plato/republic.html).
4. E. Schrodinger, What is Life (Cambridge University Press, New York, 1944).
5. J. D. Watson F. H. C. Crick, A Structure for Deoxyribose Nucleic Acid. Nature 171, 737-738 (1953)
6. R. Dawkins, River Out of Eden: A Darwinian View of Life (Basic Books, New York, 1995).
7. Plato, Phaedo (http://classics.mit.edu/Plato/phaedo.html).
8. C. R. Darwin, On the Origin of Species by Means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life (http://darwin-online.org.uk/content/frameset?itemID=F373&viewtype=text&pageseq=1)
9. G. J. Mendel, Experiments in Plant Hybridization (http://www.esp.org/foundations/genetics/classical/gm-65.pdf)
10. K. Kotani, Natural Selection Protects Information aginst Entropy.
11. Motoo Kimura, Evolutionary rate at the molecular level. Nature 217, 624-626 (1968).
12. D. R. Samols, O. Hagenbuchle, L. P. Gage, Homology of the 3' terminal sequences of the 18S rRNA of Bombyx mon and the 16S rRNA of Escherichia coli. Nucleic Acids Research 7, 5, 1109-1119 (1979)
13. A. M. Poole, What is the Last Universal Common Ancestor (LUCA)?



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