ONTOLOGICAL ANALYSIS OF NUMBERS
Ontological analysis of numbers was done in ancient India, which has been forgotten now. Number is a quality of all substances by which we differentiate between similars. Quality is not an abstract term, but must be related to objects. Quantity is one of the qualities of objects. Like shape, color, smell, etc. Since it is so common with every object, we generally overlook this quality as distinct like other qualities. When we say color, we differentiates objects by their different colors. When we say quantity, we differentiates objects by their number. When discussing ontology, we must go deep and not be superficial.
If there are no similars, it is one. If there are similars, it is many, which can be 2,3,4,....n, depending upon the sequential perception of one's. For this reason, in Vedas, one is called Eka (एक), which means that which cannot be used in an "if-then" format (एकत्वं केवलान्वयीति). We perceive one number at a time only. As had already written, we perceive two initially as one and one. The object neutral concept of the number two which we recognize when we see one and one is called two-ness (द्वित्व). Similarly, three is perceived initially as two and one, etc. Thus, it can be written in a format: “IF there is 2 and one more, THEN it is three”. But this format cannot be used in the case of one. This concept is applicable only in the initial perception of the number. Hence generally we overlook this. After perceiving one, the perception of one and one as two is immediate. This has been established in many experiments.
The next number two is called Dwi (द्वि) because its perception after one is immediate (द्वि द्रुततरासंख्या). The next number three is called Tri because it also comes floating to be easily perceived (त्री तीर्णतमा संख्या). Even babies and some animals perceive numbers up to three. The next number four is called Chatwaara because it is a mobile number (चत्वारसमा). From this onward, one has to exercise their mind to perceive it. Zero is called Shoonya because it it is about something that does not exist at here-now, so that it cannot be perceived like other numbers (आतिशयेन ऊन - शुनँ गतौ॑). Infinity is not a big number, but like one - without similars, with one exception. Whereas the dimensions of one are fully perceptible, the dimensions of infinity are not perceptible. Hence like other numbers, it cannot be used in quantitative mathematics. Negative numbers imply exchange of ownership without adequate consideration. And so on. The paper admits that the tribal people find problem only with four onwards. This refutes the claim that some cultures have language without numbers. They have at least numbers up to three. The people live in natural environment without the complexity of the modern world. Hence mentally, they are like children. The paper rightly concludes "there are undoubtedly cognitive commonalities across all human populations, our radically varied cultures foster profoundly different cognitive experiences".
1) Floating is not the exact equivalent of तीर्णतमा, which means coming naturally to a conclusion easily. Even children or animals easily perceive three and it has been established in many experiments. The paper also admits this.
2) Four onwards are not perceived effortlessly like one, two or three, but one has to exercise the brain intelligently to perceive these. The paper also admits this. For this exercise the word Chatwaara has been used, which I have loosely translated as mobile numbers.
3) a) Infinity is not a big number n, because then there will always be n+1. The difference has already been explained. Numbers are qualities of objects, and the dimensions of objects are fully perceivable, because they can be displaced due to application of energy. Displacement means leaving its position and moving to the adjacent place once or in a sequence. It is a common property of all objects. Since by definition, the dimension of infinity is not perceived, and since it cannot leave its position and move to the next position by application of energy, it is not a number at all.
b) "asaṃkhyeya" is a big number, which is difficult to perceive. Gita uses this word in this meaning. One example of it is the hierarchy problem in dark energy research, where theory and observation differ by a factor of 10 ^120. It is dubbed the biggest mismatch in physics. We cannot perceive such a big number. Normally Vedic numbers are precisely defined as has been shown from 1 to 4 above. Usually, these are restricted to 10^17 (Paraardha), though in some cases 10^24 (Vikaathi) have been used. Some other ancient texts go up to 10^54 (Tallaakshana) and others go up to 10^140. "ananta", on the other hand means a special case of infinity. When a number is divided by zero, the result can be called “khahara” in some cases, which is not exactly infinity, though it has resemblance with infinity. When dealing with fields, division by zero leads to “khahara”, which is broadly the same as renormalization except for the fact that here only non-linear multiplication and division are considered, whereas renormalization considers linear addition and subtraction by the counter term. Renormalization is a procedure in quantum field theory by which divergent parts of a calculation leading to nonsensical infinite results are absorbed by redefinition into a few measurable quantities yielding finite answers. For example, in quantum electro-dynamics the electron’s ability to constantly emit and absorb “virtual” photons meant that its total energy and mass are infinite. Thus, the mass of the “bare” electron was redefined to include these virtual processes and setting it equal to the measured mass. This way, the infinity is removed. The electroweak theory incorporates the weak force together with the electromagnetic force after renormalization. But as pointed out above there are big question marks on these virtual particles. Khahara can be visualized as something of a class that is taken out completely from the field under consideration. However, if a number is first divided and then multiplied by zero, the number remains unchanged. When a number signifying an object is divided by zero, the number is treated not as infinity, but the number is considered as unchanged. I have written about it in these forums and elsewhere sometime ego elaborately with proof.
4) Can anyone have a sense of one without observing the dimensions of an object? Can there be an abstract entity called number one without associating dimensional objects? When sitting on a beach and looking at the sea, or looking at the vast night sky, can we perceive number one? If so, you must precisely define what a number is and what is number one. If objects are necessary for perception of numbers, dimensionality automatically comes in.
5) Mathematics is the quantitative aspect of Physics and quantity is related to dimensional numbers. Since infinity has similarities with one, which is a number, but is not confined like objects with dimensions, I have used the term “unlike other numbers”. There are only four infinities. These are time (काल), space (आकाश), coordinates (दिक्) and consciousness (आत्मा). Objects being constituted of fermions follow the exclusion principle (विष्टम्भकत्व). Infinities can co-exist like bosons. Since they are all pervasive (विभु), they are immobile and cannot take part in any interaction. Everything happens in space and time arranged in different coordinates. But these do not interact with any reaction. Consciousness is the observer (साक्षी, द्रष्टा), without which nothing would be perceived – hence become meaningful. Yet Consciousness or Observer does take part in any interaction.
6) In a relational number, the object with the number has an inherent relationship (अन्योन्याभाव) with the number. For example, in the statement “A has five apples” there is a relationship of conjunction between A and the apples, and the apples have an inherent relationship with the number, whose conjunction is perceived as five. As long as the apples remain intact, the number remains unchanged. Any change in the number is possible only if something happens to the lot. This relationship of the number to the apples is called तादात्म्य सम्बन्ध. Suppose B does not have any apple, then he has an inherent relationship of non-existence with the apples. Now, if A gives these apples to B without adequate consideration, then the relationship of conjunction of A with the apples would temporarily become non-existent. Though B now has a relationship of conjunction with the five apples, his inherent relationship of non-existence with the apples remains as he is not the owner of the apples. The original inherent non-existence of apples of B has now spread to A as non-existence of the apples through the relationship of conjunction. This relationship is called exchanged relational non-existence (अन्योन्य अस्मिन् अन्योन्यस्याभावः) and is the cause of negative numbers.
The apples with B are denoted by the negative sign (ऋण) to distinguish the ownership of A, which is denoted by the positive sign (धन). The above principle has important implications for modern mathematics and physics. The difference between the positive sign and the negative sign is one of proprietary relation and not physical non-existence. Thus, a negative number can be accumulated or reduced. Accumulation or reduction can be of two types: linear accumulation (reduction) or non-linear accumulation (reduction). Linear accumulation can be between objects that are similar in dimension as the dimensions of the quantities in both sides of the equation have to be the same. In case of non-linear accumulation or reduction through multiplication or division, the condition is that the dimensions of the objects must be related. For example, area is mathematically valid as both length and breadth are related to the field. Thus, we can prove the equation (a+b)2 = a2+2ab+b2. (a+b)2 = a2+2ab+b2 . The above facts can be better explained with some example.
Let us consider two consecutively joined straight lines of lengths a and b respectively. Total length of the two lines will be a+b. The square of the quantity represented by a+b will be equal in quantity to the area of a square field whose sides are a+b. Thus, the result is, a2+2ab+b2. The formula (a+b)2 = a2+2ab+b2 has a real meaning which can be verified from actual observations, i.e., the arithmetic of experience.
Similarly volume is valid, which can be shown graphically. All classical mathematical formulae for volume such as a cube could be physically explained. For finding the value of (a + b)3, we may calculate the volume of a square field, whose sides represent (a + b). In the above example for (a + b)2, let us consider another forward dimension equaling the length a + b. The volume above the area representing a2 will be equal to a3 and a2b. The volume above the two areas representing ab will be equal to a2b and ab2 each. The volume above the area representing b2 will be equal to b3 and ab2. Thus the total volume will be a3 + 3a2b + 3ab2 + b3, which is equal to (a + b)3.
It is because the square represented summation of two non-linear quantities (®úVVÉÖ ºÉ¨ÉÉºÉ) spread in different directions and the cube represented summation of three non-linear quantities spread in three different cardinal directions. Since we have spread in three spatial directions only (the so called spatial dimensions – right-left, up-down, forward-backward), we could go up to square and cube. But higher dimensions are physically unrealizable. Hence, Vaidic science discarded equations involving powers above that of the cube.
According to the laws of indices of algebra, if a certain number ‘a’ is multiplied ‘m’ times in succession, then the continued product so obtained is called the ‘m’th power of ‘a’ and is written as am. Thus, am = a x a x a x … to m factors. Here ‘a’ is called the base of am and m is called the index or exponent of am. Mathematically, such operations are possible and writing the continued product of such operations as am is permissible as a notation. But what does the notation am mean? Does it represent a number or does it indicate only the product? In later pages it has been shown that spatial dimension is nothing but spread in a given direction. It will also be shown that this spread can only be in three cardinal directions and both the so called compactified and non-compactified dimensions are nothing but projections within these three dimensions. Thus, multiplication beyond cube is not mathematically valid. The higher powers can be used as a mathematical tool for calculation by restricting numbers to these three dimensions.
The other implication is that, no number can exist below one. Thus, reduction of one is not possible. Decimals are valid only as a fraction involving many ones. A fraction is valid as the composite object with the number is first reduced to divisions equal to the denominator to make each fraction as the “new one”. Out of this, number equal to the numerator is taken to represent the fraction of the “old one”. Since reduction of one is not possible, square root or cube root of one is one, though dimensionally reduced. Since negative numbers exist, though with a different relationship, they cannot be reduced below one. Negative one cannot be reduced to have the imaginary numbers. Thus, the concept of complex or imaginary numbers is mathematically invalid. Elsewhere we have discussed the mathematical fallacies in the Argand Diagram, which is used to describe the geographical representation of the complex numbers to conclusively prove that the so called theory is a fraud. This is the reason why imaginary numbers cannot be used in IT programming where binaries, which represent the two fundamental forces of dispersal and contraction as explained earlier, are widely used.