The Edge of the Object Corresponds to the Euclidean Line

The main information system of lives is the genetic information system.  The remarkable feature of genetic code is the serial alignment. That is,  it has a 1D structure. Furthermore, two major information systems, the language and the computer code, are also isomorphic to DNA. Presumably,  human beings might have a tendency to make a similar information system to DNA.

Next, all of these three systems are digital information systems. The important advantage of digital information is that the element of digital information can be the direct target of  natural selection (1). DNA consists of four kinds of bases: A (adenine), T (thymine), G (guanine), C (cytosine). Bases are lined sequentially. Each base of DNA can be the direct target of natural selection.

Similarly, the element of the language  also determines life and death of human beings. For an example, a phoneme can be critical for the survival of a human. We assume a platoon advancing in a battlefield in the night. In this situation, the distinction between r and l is vitally important. When a soldier found the enemy on the right side, he said “right.” If another soldier hears light instead of right and then he put the light, he will be shot. In such a case, the distinction between r and l determines life and death.

Meanwhile, Plato assumed the ideal one. So, Plato’s one is insensible and has three properties: invariability, indivisibility and equality (2). The natural number is the set of Plato’s one. Furthermore, the ideal digital information corresponds to the natural number. If we can use the ideal digital information for communication, our communication will be completely correct because Plato’s one is equal to each other. However, the complete equality does not exist in the real world (3).

Instead, Plato’s one exists only in the human brain. Necessarily, digital information must be close to the natural number as much as possible. We shall consider the phoneme.  Firstly, the phoneme of the spoken language was converted into the letter of the alphabet. Next, print technology was developed. Furthermore, technology has progressed rapidly after the Renaissance.

Now, the text is converted into the text file on the computer. It consists of 0 and 1. That is, human beings have been trying to transform phonemes into signals close to the natural number during the history because Plato’s one is the base of all cognition of all living organisms.

However, our brain created the number, which cannot be represented by natural numbers. Perhaps, our basic cognition system, which consists of sets of Plato’s one, may be insufficient  for  the cognition of the moving object. Our visual cognition system was developed in order to compensate this fault.

We shall consider this problem. Firstly, There are two kinds of image formats in the computer: the raster image and the vector image. Thea raster image consists of pixels, which corresponds to the natural number. In contrast, the vector image does not consist of pixels.

The basic element of the vector image is the Euclidean line, which corresponds to the equation. The scaling of a vector image  is free. It is very important for pursuing a moving object because the size of the moving object’s image changes with respect to the distance from the sensor. Thus, our brain chosen the vector image.

Figure 1. The projection of a cuboid to a plane.

Figure 1 shows the idealized schema of the projection of a cuboid to a plane. To simplify, we shall binarize the reaction of the sensor. The edge of the 3D object corresponds to the boundary of the shadow.  The extraction of the edge from the retinal raster image is the main task of the first step of our vision. The edge is extracted in the primary visual cortex (4).

Why the extraction of the edge is important? When we pursue a fast moving object, we must identify it quickly. In order to save the amount of the calculation, our brain must identify the shape of its 2D shadow. Then, we can directly identify it from the 2D image. 

Figure 2. The expanded schema of the borderline.

Figure 2 shows the expanded schema of the borderline between the light area and the dark area. As a result of the binarization, a pixel must belong to the light side or the dark side. That is, the borderline between pixels is the borderline between light and dark. This borderline is just the Euclidean line. According to Definition 2 in Book 1 of Euclid’s Elements, a line is breadthless length.

Figure 3. The Euclidean Line. 

Figure 3 shows the Euclidean line. This concept binds the figure to Plato’s one. Furthermore, the simple cells in the primary visual cortex extract the Euclidean line, which corresponds to the edge of the 3D object (4). A simple cell in the cat’s primary visual cortex responds to the line of the special direction.

Next, we shall consider the coordinate system of our vision. Because our eyes and our head can move, our viewpoint is always moving. That is, the coordinate system of our vision is not fixed. Hence, our brain can extract the edge of various directions. Figure 4 shows the extraction of Euclidean lines from edges.

Figure 4. The Extraction of Euclidean lines

The Ancient Greeks elaborated the edge in our brain, and then they created the Euclidean line, which has only length and no width. It is the basic element of the Euclidean geometry. Because it has length, they thought that shapes and numbers were integrated. Though Pythagoras discovered the irrational number, Euclid defined the real number by Definition 5 in Book 5 of  Euclid’s Elements. This definition is equivalent to the Dedekind cut. However, this definition is indirect.

Why the Ancient Greeks could not be directly defined the irrational number? Basically, we can output only digital information. That is, our nervous system can output only muscle movement. Its minimum unit is the motor unit, which is indivisible. So, any muscle movement is digital. Even, the experienced craftsmanship is essentially digital. Furthermore, our main outputs are language and DNA, they are digital information. That is, we cannot directly transfer our subjective experiences to other persons.

More generally, the basic information system of life is digital. So, bacteria can transmit almost only digital information. Where does the Euiclidean space come from? Euclidean space was constructed by the Ancient Greeks based on both digital information and the qualia. According to Wikipedia, qualia is a term used in philosophy to individual instances of subjective, conscious experience. We cannot transmit the sense of red itself. We can only say that I see the red color because the human can transmit only digital information.

Figure 5. We recognize the real world through digital information.

Figure 5 shows the relationship among the real world, digital information, qualia and the Euclidean space. We cannot directly recognize the real world. We recognize the real world only through digital information. Thus, we cannot directly access the real world. That is, we can only conjecture the real world.

Next, the Euclidean line is the borderline between pixels. However, there is nothing between adjacent two pixels. Hence, the Euclidean line is created by our brain. Furthermore, we cannot output itself. These features are identical to the qualia.

The Ancient Greeks elaborated the qualia of the edge, and then they constructed the Euclidean geometry. Because they embedded the non-existent line in Euclidean geometry, many problems arose. If any Euclidean line can exist, the size of the pixel of the Euclidean plane must be zero. That is, Euclidean plane does not consist of pixels.

However, even if we use any writing tool, we can draw only a line, which consists of pixels. Hence, we cannot draw any Euclidean figure. This difficulty means that any Euclidean figure cannot be constructed. That is, the world, which we can handle, is only the digital world.

However, this fact does not mean that the real world is the digital  world. Rather, only digital information may not enough to recognize the true world. Our ancestors would get qualia in order to compensate the shortage of the digital information. Furhthermore, the Ancinet Greeks created the Euclidean geometry based on the qualia of the edge. 

References

1. What Is Digital Information?
2. Book 7 in The Republic by Plato.
3. Phaedo by Plato.
4. Simple Cells in David Hubel’s Eye, Brain, and Vision.

 

The Biological Foundation of Mathematics 1

Any scientific theory cannot be absolute. Even Newtonian mechanics were corrected by physicists. The theory of relativity and quantum mechanics replaced it. There is no absolute truth in science. That is, all scientific theories are hypotheses. However, scientists have considered that mathematics is the absolute truth. Is this belief sure truth? Probably, if the human were the creature of God, mathematics could be the absolute truth, because it is a product of the human brain. However, the molecular biology provides many evidences, which support Darwin’s theory of evolution (1). Now, most biologists accept his evolutionism.  So, they consider that species have evolved by natural selection and random mutation.

The evolution usually chooses the ad-hoc solution. Surviving for the time being is important for a living thing. Necessarily, the structure of the human brain is the great piles of ad-hoc solutions. It is not elegantly designed by the great engineer. The pioneer molecular biologist François Jacob said “Evolution is a tinkerer, not an engineer (2).” David Linden described in The Accidental Mind that the human brain has the primitive systems in our distant evolutionary past (before mammals) and that have been supplemented by newer, more powerful structures. Furthermore, Dean Buonomano described in Brain Bugs as follows:

Human beings, of course, not the only animal to end up with brain bugs as a result of evolution’s kludgy design process. You may have observed a moth making its farewell flight into a lamp or the flame of a candle. Moth uses the light of the unreachable moon for guidance, but the attainable light of the lamp can fatally throw off their internal navigation system (3). Skunks, when faced with a rapidly approaching motor vehicle, have been known to hold their ground, perform a 180-degree maneuver, lift their tails, and spray the oncoming automobile. These bugs, like many human brain bugs, are a consequence of the fact that some animals are currently living in a world that evolution did not prepare them for.

When the human brain cannot be trusted, what should we trust? Descartes tried to answer this question. He doubted everything in DISCOURSE ON THE METHOD. Surely, all our experiences during the waking period can be replicated in dreams. Moreover, we cannot distinguish them. Hence, all realities may be illusions. However, Descartes described ” 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.”  His statement is  the starting point of all science. The surprising point of it is the independence from the brain bugs. Even if all sensory inputs are illusions, it is true.

Why could it have the surprising universality? I shall consider profoundly. Firstly, the important premise of  it is the unity of the self, which depends on the indivisibility of a human. If a man is divided, probably he will die. Even if he survived, he lost a part of his body. However, he cannot be divided into two persons. Similarly, animals cannot be divided. Although many plants are divisible, the cell is indivisible. As I mentioned in A Life Is the Original Form of the Natural Number One, this principle is due to the unity of a life. Secondly, the important feature of Descartes’s statement is the easiness of acceptance. Everyone thinks that it is applied to oneself. This means that the self has common properties. Since all human beings are 99.9 percent genetically identical, naturally the basic structure of the self is constant. Probably, the common structure of the human self is the base of declaration of Descartes.

However, the declaration of Descartes is based on the language and the logic. When we consider the self of an animal or a human, who don’t have the ability of the language, we need the more universal definition of the self. German psychiatrist Jaspers defined the self-awareness clearly in general psychopathology (4).

 Jaspers lists four formal aspects of self-awareness: the feeling of activity, i.e. the awareness of being active, the awareness of unity, the awareness of identity, and the awareness of being distinct from the outer world.

Jaspers’ definition of the self is practical in psychiatry. The most severe state of the schizophrenia is called the catatonic stupor. When a schizophrenic patient falls in it, he cannot move, talk and eat. Because his self is disturbed, his ability of the movement is lost. So, Jaspers’ definition of the self is applicable to the schizophrenic, who cannot speak the language. As pointed out by Vittorio Gallese and Francesca Ferri, Japers’ self contains the bodily self, which might be the base of the motion of the human body.

Firstly, when a human wants to move, he must be active. Secondly, if the right side and the left side of a human body moved to opposite directions, the human would be divided. Hence, the unity of the whole body is necessary for the motion. Thirdly, when a human walks forward to take a food, he has to move forward consistently. That is, he, who found a food, is identical to himself, who walks toward food. Thus, the identity of the self is required for the coherent　movement. Fourthly, in order to exercise correctly, a human must recognize the boundary between oneself and the external world. Therefore, the intact self defined by Jaspers is required for the movement.

This principle can be applied to not only animals but also unicellular organisms. Especially, the paramecium has been well studied. When a paramecium swims to a direction, it moves all cilia systematically. Even if there are foods and blockades, the paramecium will choose the most appropriate direction. It seems that  the paramecium has the intact self defined by Jaspers. Furthermore, he defined the self so that the intactness of the self could be judged only by the objective observation. Hence, we have no choice to accept that the paramecium has the self defined by him.

Paramecium

Because the self defined by Jaspers is conserved from paramecia to human beings, the basic structure of the self is identical. The basic part of the self is invariable and equal to each other. Furthermore, the self is indivisible. While, Plato described three important properties of the natural number one in the Republic. Socrates said “there is a unity such as you demand, and each unit is equal, invariable, indivisible.” Then, we can regard the basic structure of the self as the prototype of the natural number one. The natural number one is the foundation of all science.

References

1.  Alberts B, Johnson A, Lewis J, Raff M, Roberts K, Walter P. Molecular Biology of the Cell 4th ed. New York, Garland Science; 2001 How Genomes Evolve
2.  Jacob F. Evolution and tinkering. Science. 1977; 196: 1161-66
3. Dawkins R. The God Delusion. New York, Bantam Press; 2006
4. Vittorio G, Francesca F. Jaspers, the Body, and Schizophrenia: The Bodily Self. Psychopathology. 2013; 46(5): 309-19

A Point Is a Blank

Descartes created the analytic geometry. However, the premise of the analytic geometry is problematic. In order to materialize the analytic geometry, a curve must be a set of points. However, Euclid did not say that a line is a set of points. He defined a point and a line in Elements as follows:

Definition 1: A point is that which has no part.

Definition 2: A line is breadthless length.

Definition 3: The ends of a line are points.

Adversely, a point is defined as an end of a line. I shall consider what is the point.

Figure 1

In the beginning, figure 1-1 shows a circle and radii. There are point A, point B, point C and point D on the circumference, and point O is the center of the circle. Line AO, line BO, line CO and line DO are radii. When they are taken out, four lines arise. Figure 1-2 shows them. Their length are equal. According to the definition 3 of Elements, two ends of a line is points. As a result, the point O is divided into four points. This result is natural because the length of a point is zero. Of course, zero is divisible. Inversely, the sum of any number of zero is zero. That is, if the length of all points on the circumference are added, the sum is zero.

0=0+0+0+0+0+…+0

Next, I shall consider the validity of this division.

Figure 2

Consider a hyopthetical pizza and an ideal pizza cutter. We assume that the ideal pizza cutter can cut the hypothetical pizza without loss of the body of the pizza. Figure 2 shows the quadrisection of the circle. As shown in the figure 2, since the central point is blank, the pizza cutter can pass through the central point any number of times. When the cleavage line corresponds to the radius, the central point corresponds to the blank.

Next, I shall consider our eyes for clarifying the biological foundation of the above discussion. The first step of our vision is the detection of the light. The second step of it is the construction of the two-dimensional image. Because our eyes are very similar to the digital camera in these two steps, I explain the portion common to both. Firstly, the photodetector detects photons. Secondly, one  photodetector corresponds to one pixel, and then the two dimensional image consists of pixels.

Figure 3

 When the two-dimensional image is constructed from the light, the  photodetector must be the center of the pixel and the borderline of two pixels must be the straight line. Figure 3-1 shows the borderline of two pixels and central  photodetector. Thus, the shape of the pixel must be the triangle, the square or the hexagon. Figure 3-2 shows a plain tiling by them.

Finally, I shall consider why a point is a blank. The purpose of our vision is not the two-dimensional image but the three-dimensional.  When we construct the three-dimensional image, our brain extracts the edge of the object from a 2-D image. This fact is discovered by Hubel and Wiesel. The details of their findings are in David Hubel’s Eye, Brain and Vision. I shall consider the extraction of the edge of the object using an example.

Figure 4

Figure 4 shows a flower of hibiscus.  The edge of it corresponds to the borderline between the red area and the green area  because the edge of the object corresponds to the borderline between the color of the object color and the color of the background.

Figure 5

The simple cell in the primary visual cortex reacts the straight edge. Figure 5-1 shows the magnified view of the idealized schema of the straight edge. It consists of red pixels and green pixels. Ideally, the edge is the borderline between red pixels and green pixels. Hence, it is the blank. Probably, the straight borderline between pixels is just the prototype of the Euclidean line. Firstly, it has no width. Secondly, it consists of units of the borderline. Figure 5-2 shows the unit of the borderline. Thus, the straight edge corresponds to the Euclidean line.

Consequently, the Euclidean line and the Euclidean point are the blank. Because a line has a length, it exists. However, a point has no length and  no width. Hence, I cannot regard that it exists. Rather, it arises as an end of a line.

Figure 6

Figure 6 shows an example. When a line is cut, two points arise. The total length of them is equal to that of the original line. Furthermore, points can emerge without limit because a line can be cut any number of times. Therefore, a line is not a set of points. That is,the foundation of  the analytic geometry is logically problematic. This logical defect can be compensated by the double contradiction.

A Life Is the Original Form of the Natural Number One

The modern science began from the statement of Descartes: I am thinking, therefore I exist. He said in Discourse on the Method as follows:

Lastly, I decided to pretend that everything that had ever entered my mind was no more true than the illusions of my dreams, because all the mental states we are in while awake can also occur while we sleep and dream, without having any truth in them. But no sooner had I embarked on this project than I noticed that while I was trying in this way to think everything to be false it had to be the case that I, who was thinking this, was something. And observing that this truth I am thinking, therefore I exist was so firm and sure that not even the most extravagant suppositions of the sceptics could shake it, I decided that I could accept it without scruple as the first principle of the philosophy I was seeking.

His statement means that anyone cannot deny the existence of oneself. The premise of his statement is the unity of the self. That is, the self has properties of the natural number one.

The above equation is the starting point of all science. Then, I consider more deeply.

What is the natural number one? Ancient Greeks quested this problem most profoundly since the dawn of history. 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. Socrates said “there is a unity such as you demand, and each unit is equal, invariable, indivisible.” The natural number one has these three properties.

Next, I shall compare the self to the object with respect to three properties of the natural number one. Firstly, any object can be divided. Even, a diamond is crackable. Secondly, any object is not eternal. Geographical features of the geosphere have always changed through the history of the earth. Even, continents moved. Thirdly, there is no pair of objects, which are completely equal to each other. In contrast, the self has properties, which are close to  the natural number one. Firstly, 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. That is, humans instinctively regards that selves are equal to each other. Secondly, a human usually keeps ego identity from birth to death. That is to say, the self of is invariable. Thirdly, the self cannot be divided. If a self of a person is disturbed, the person might have a mental illness: schizophrenia or multiple personality. A self of a human, who is mentally healthy, is indivisible. Therefore, the self has similar properties of the natural number one.

I shall pursue the origin of the similarity between the self and one. When I phylogenetically trace back to the origin of the self, I reach the last common universal ancestor. Most biologists believe that life on Earth arose only once. Hence, all lives are descendants of the first single cell. The basis of the dogma is as follows. All living things are composed of cells, which have common features. All cells have DNA, plasma membrane, cytoplasm and ribosomes. Additionally, the  basic biochemical system is identical in all lives. DNA preserves genetic information. Next, it is transcribed  into RNA. Subsequently, RNA is translated into the amino acid sequence of the protein. Especially, The most astonishing fact is the universality of the genetic code of all lives. Genetic code is the code, which is required for the translation of the genetic information to the amino acid sequence. Thus, the human insulin gene can be put into E. coli genome. Now, human insulin is made by E. coli.

When we accept the universality of all lives. we can abstract the equality of all lives. Also, basic featureｓ of lives have been invariable since the arising of the first life. Furthermore, if a cell is divided by force, the cell will die. A cell is indivisible. The death of a life is the only irreversible process on the Earth. Therefore, a life is the original form of the natural number one.

What Is Continuity?

What is the continuity? The question is very difficult to answer. However, according to findings of the cognitive psychology, the continuous movement of the object is  created by our brain. If you think profoundly, this task is harder than it looks. We recognize the motion through the visual sense. However, as mentioned in “The Solution of the Arrow Paradox“, our visual system can recognize only still image.s That is to say, animals cannot recognize the motion only with eyes. How the animal nervous system resolved this problem? This question is elucidated by the cognitive science. Donald Hoffman says that your brain creates the motion. He shows many examples in Visual Intelligence, which is the excellent book of the visual cognition. So, we cannot directly recognize the motion. Instead, our visual brain create the motion from still image.

Figure1. Frames of the Animation

Animation 1. The Animation

Above figures show an example of the simple animation. Figure 1-1 shows frame 1, in which the black ball is at the left side. Figure 1-2 shows frame 2, in which it is at the right side. Animation 1 shows the animation, which consists of frame 1 and frame 2. It seems as if the black ball is continuously reciprocating. However, only two still pictures, frame 1 and frame 2, are alternatively shown in it. The continuous movement does not exist anywhere. That is, it is created by our brain.

Figure 2. The Stopping Arrow

Figure 2 shows the stopping arrow. As mentioned in “The Solution of the Arrow Paradox,” our eyes can capture only still images. That is, we cannot directly recognize the motion. Instead, our brain brain creates the motion. The continuity is the property of the motion, which is created by our brain. If we assume that the motion of the object is continuous, the space and the time must be continuous. This only means that our brain regards space and time as continuous. It is another problem whether the real space and the real time are continuous or not. This problem is very difficult to solve.

Rather, it is amazing that the continuous movement is created by the brain from nothing. Perhaps, the ability of creating the continuous movement is advantageous for survival. However, Richard Dawkins says that digital information is crucial for the evolution in River Out of Eden. So, from the point of viewpoint of evolution, we can regard a living organism as a digital machine. Nevertheless, it is astonishing that the brain created the continuity.

The Approximation of Napier’s Constant

According to the Wikipedia,  e is the unique real number such that the value of the derivative of the function  f(x) = e^x at the point x = 0 is equal to 1. Let us consider the right side gradient and the left side gradient of f(x) = e^x at x = 0.When h is an arbitrary positive number, the right side gradient is the gradient of f(x) = e^x in the interval from x=0 to x=h and the left side gradient is the gradient of f(x) = e^x in the interval from x=-h to x=0. Obviously, the right side gradient is larger than 1 and the left side gradient is smaller than 1.

Left side inequalities are equivalent to right side inequalities.

The next inequality represents the definition of the double contradiction. It is true with respect to any positive number h.

When we substitute an arbitrary natural number n for 1/h, the next inequality is formed.

We can approximate Napier’s constant using the above inequality.

We have determined upper and lower limits of Napier’s constant. Next, we shall calculate the ratio of the upper limit to the lower limit.

We know the range of the approximation of Napier’s constant. Additionally, we get the double precision value of Napier’s constant. Furthermore, consider the difference of the upper limit and the lower limit. The next equation is obtained.

Furthermore, consider the sum of the upper limit and the lower limit. The next equation is obtained.

The next table shows approximate values. When n equals 100000, they are calculated with three approximate equations

The average of the upper limit and the lower limit is the best approximation of them.

The Definition of Napier’s Constant by the Double Contradiction

I want to apply the double contradiction to the exponential function. We need to define Napier’s constant at first. There are many definitions of it. Among them, a common definition is written in the Wikipedia as follows: e is the unique real number such that the value of the derivative of the function  f(x) = e^x at the point x = 0 is equal to 1. Following this definition, we shall calculate the value of e.

As Δx decreases, the above fraction approaches 1. Then, we define the derivative of f(x)=e^x  at the point x = 0 equals 1. However, when Δx isn’t zero, the value of the above fraction isn’t 1. While, if Δx equals 0, it is indeterminate. The derivative in the traditional calculus is a mysterious concept. Alternatively,we use the double contradiction to solve the mystery. Firstly, we calculate the right side gradient m(h). Consider an arbitrary positive number h. Then, we shall calculate the gradient of f(x)=e^x in the interval from x=0 to x=h.

Because m(h) is monotonically increasing with respect to h, the next inequality is true.

When h is any positive number, the above inequality is true. Next, we calculate the left side gradient m(-h). Let us calculate the gradient of f(x)=e^x in the interval from x=-h to x=0.

Because m(-h) is monotonically decreasing with respect to h, the next inequality is true.

Finally, we have gotten the new definition of Napier’s constant without the limit.

The above inequality is true with respect to any positive number h. It is the definition of Napier’s constant by the double contradiction.

The Meaning of the Double Contradiction

I shall consider the meaning of the double contradiction, which is developed by ancient Greeks to solve the difficulty of handling the curve in the Euclidean geometry. In the beginning, Euclid defined a line in Elements Book 1  as follows.

Definition 2: A line is breadthless length.

According to the above definition, the length of the straight line is identical to the length. Inevitably,the length of the curve can be only approximated by length of lines. Similarly, it is difficult to calculate the area enclosed by the curve. Ancient Greeks developed the double contradiction to solve the problem. For an example, we shall consider the are under the graph of y=x² in the interval from 0 to 1 (5). It is named S.

S is approximated  by n thin rectangles. The width of each rectangle is 1/n. The left figure shows the left sum.

The right figure shows the right sum.

The next inequality determines the area S.

For any natural number n, the above inequality must be true. Hence, S must be 1/3 in order to establish it. If S is larger than or smaller than 1/3, a contradiction occurs.

The above logic is the  same as reductio ad absurdum in “The Integral Based on the Double Contradiction“. This time, we shall consider the meaning of the above inequality. Because n is an arbitrary number, we can approximate S as correct as we want. Nevertheless, we cannot achieve the true value of S. No matter how large n is, neither left sum or right sum reach 1/3. That is, we cannot calculate S by a normal method in the Euclidean geometry. Thus S is determined by the double contradiction, which is developed by ancient Greeks to compensate the defect of the Euclidean geometry.

As just described, we cannot directly handle the curve quantitatively in the Euclidean geometry. Furthermre, the existence of the curve in the Euclidean space is suspected as mentioned in “The Curve Is the Aerial Line“. However, we often want to handle the curve quantitatively. For example, we have to treat circles and ellipses in the astronomy. Hence, ancient Greeks had been trying to adjust the Euclidean geometry to the reality for long years. Finally, Archimedes filled the gap of the Euclidean geometry using the  double contradiction.

The New Definition of the Differential

The student, who study mathematics, will be confused by the differential. Let us consider the differential of y=x² as an example.

We regard Δx as a positive number in the above equation. However, we regard Δx as zero in the next equation.

It is natural that a normal student feels strange. Perhaps, ancient Greeks might not allow the above logic. In order to solve this problem, I propose the better definition of the differential. Firstly 2x+Δx means the average gradient of the graph of y=x² in the interval from x to x+Δx. I name it the right side gradient m(Δx).

Let us consider the new definition of the differential together. Secondly, we shall calculate left side gradient m(-Δx), which is the average gradient of the graph of y=x² in the interval from x-Δx to x.

Thirdly, we shall calculate the gradient at x.

It is indeterminate. However, because the gradient with respect to Δx is monotonically increasing, we can use the double contradiction.

If m(0) is larger than or smaller than 2x, a contradiction occurs. That is, m(0) must be equal to 2x.

Finally, we shall define the derivative of y=x² at x as m(0). This definition will solve the confusion of math students. Furthermore, ancient Greeks might accept it. The orthodoxy of it will be discussed in another post.

The Integral Based on the Double Contradiction

The calculus is particularly mysterious in mathematics because the curve is the aerial line. When we calculate the area under the curve, we seek the area, which doesn’t exist. However, Archimedes calculated the area enclosed by the curve. He used double contradiction. His proof is called reductio ad absurdum or a double proof by contradiction. When we use it, we can calculate the aerial area. Let us consider an example of it.

The above figures show the graph of y=x².Let us calculate the area between the curve of y=x² and x-axis in the interval from 0 to 1. It is named S. Firstly, the left figure shows the left sum.

The next equation is the expansion of the above sum.

Secondly, the right figure shows the right sum.

The next equation is the expansion of the above sum.

The next inequality determines the area S.

If S is larger than or smaller than 1/3, a contradiction occurs. That is, S must be equal to 1/3.

Finally, we have obtained the area under the curve by the double contradiction.  Then, we can exclude the limit and the infinitesimal. They are mysterious concepts. In contrast, the double contradiction is the clear logic.