By Basudeba Mishra
Number (सङ्ख्या) is a property of everything that exists, by which we differentiate between similars (भेदाभेद विभागो हि लोके सङ्ख्या निबन्धन). It is not the same as the object (mass – गुरुत्व or form – रूप) or the effect of energy (action), and is different from all other properties (अन्वयव्यतिरेकाभ्यां सङ्ख्याभ्युपगमे सति). If there are no similars, it is one (एकत्वं केवलान्वयीति). If there are similars, it is many (बहु), which can be 2 (द्वौ द्रुततरा सङ्ख्या), 3 (त्रयस्तीर्णतमा सङ्ख्या), 4 (चत्वारश्चलितसमा), … n, depending upon the sequential perception of one’s at a time. Infinity is not a very big number, as for every n, there is n+1.
Infinity is the concept of something unlimited – without bound and endless. The mathematical, the physical, and the metaphysical infinities are covered by this definition. By definition, it does not define a limit, as is commonly believed. For this reason, it cannot have any mathematical operation except as a field where everything exists and every action takes place. It is an analog unit, of which, we use digitized segments without disturbing the whole. Infinity (विभू) is like one; with a difference. It has no similars (hence like one – यत् पृथक्त्वमसन्दिग्धं तदेकत्वान्नभिद्यते); but whereas the dimensions of ‘one’ are clearly confined (एक इता सङ्ख्या), the dimensions of infinity (परममहत् परिमाण) are not confined – hence cannot be measured (यदेकत्वमसन्दिग्धं तद्पृथक्वान्न भिद्यते). There are only four infinities in the universe that are revealed by themselves (पृथक्त्वेकत्वरूपेण तत्त्वमेव प्रकाशते): space (आकाश), time (काल), coordinates (दिक्) and information (आत्मा).
The present day mathematicians consider zero as the identity element of the infinite Abelian additive group of integers. If in an integral domain (more so in a field) a product is equal to zero, then at least one factor of the product is zero. There is another concept, according to which, zero is the value of a variable for which a function is equal to zero. For example, “A polynomial of degree n has n zeros”, or “The Riemann zeta function has all its complex zeros in the strip 0 < real parts < 1”. This view neither treats zero as a number nor a value of a variable in a scientific manner.
Zero as a number signifies non-existence of something, which existed earlier or exists elsewhere. When it signifies the positional value, it actually shows a void for that position, which is construed as helping in identifying the positional value of the other digits, if any, signifying that number. When we make a statement that; “in an integral domain a product is equal to zero, then at least one factor of the product is zero”, we actually mean that “something in a system no longer exists which forces a change in the domain or the field so that it ceases to be in the same way it used to be”. Also we express our inability to identify that “something”. A number is not a number unless we perceive it as such. Then it remains indeterminate. Thus, the above view is not a mathematical statement, as all mathematical statements are deterministic. It is because numbers are associated with objects and mathematics is the science of changes of scaling up or down the numbers.
Similarly, the statement that “zero is the value of a variable for which a function is equal to zero” is not mathematically correct, as zero signifies no values. The examples cited in this regard are abstract ideas devoid of any real mathematical properties. Since the number one only is associated with substances and the other numbers are emergent, all numbers are the accumulation of individual “one”s. The emergent properties are perceived after activation by an external agency which consists of perceptible substances. Absence of perceptible substances cannot generate emergent properties. Thus, we feels a void which is called as शून्य – zero.
Zero (शून्य) is not a number, but a transition of state of something from being perceived directly at here-now to being inferred from memory about something that does not exist at here-now (शुनँ गतौ॑, अतिशयेन ऊनः). If we say “I have zero apples”, it means “I had apples, which are no longer available at here-now” or “I never had apples that other people possess and I still do not possess it at here-now”.
Since all operations – mathematical or otherwise – are done at here-now, and since Zero is nonexistence at here-now, mathematical operations involving zero in linear ways is not possible (addition or subtraction involving zero leaves the number unchanged). But non-linear accumulation (multiplication) takes part of the number to cross over from here-now to an unknown state. Thus, multiplying a number by zero makes the number zero.
Since zero is not at here-now – hence cannot be directly perceived – its value is indeterminate. For example:
n + 0 = 1
n – 0 = 1
1 x 0 = 0
5 x 0 = 0
n x 0 = 0
In all the above examples, we cannot determine the value of zero. Whether addition to or subtraction from a number, the value remains same, as if no operation has taken place. Whether 1 or 5 or n, when multiplied by zero, the result is same. Since 1, 5, and n multiplied by the same factor zero leads to equal results (zero), reducing the same zero by the same factor (zero) would make the result equal. Thus, zero cannot be used in mathematics except for its notational value. Yet, there is an exception, which will be discussed later here.
If: a x b = c, then a can be recovered as a = c/b as long as b ≠ 0. Division by zero is the operation of taking the quotient of any number c and 0, i.e., c/0. The uniqueness of division breaks down when dividing by b = 0, since the product a x 0 = 0 is the same for any value of a. Hence ‘a’ can’t be recovered by inverting the process of multiplication (a = c/b). Zero is the only number with this property and, as a result, division by zero is undefined for real numbers and can produce a fatal condition called a “division by zero error” in computer programs. Even in fields other than the real numbers, division by zero is never allowed.
A negative number (ऋण सङ्ख्या) is all about a relation – and not about existence (अन्योन्य अस्मिन् अन्योन्यस्याभावः). “I have -3 apples” means, “I had 3 apples, but now someone else has them for a consideration paid to me in exchange”. Here only some consideration (may or may not be monetary) is responsible for the operation and not reduction. Hence the statement “+1 to -1 through 0” is mathematically wrong, because mathematics deals only with the quantitative aspect of accumulation or reduction linearly (addition/subtraction) or non-linearly (multiplication/division) of objects that exist at here-now.
Exponentiation involves repeated operation or reduction of by a base factor. When we say a^n we mean a multiplied by itself n times. When n = 0, it implies no operation at here-now, which leaves the number unchanged. It will still be 2. The is no reason to believe or deduce that a^n means 1 x a x a .. n times. Because this will require every number to first become 1 before exponentiation. This can only imply the object (exponent) being considered as a whole as one unit. In that case, the base factor will also be treated as one unit. This will be a linear accumulation or reduction (addition or subtraction) and not multiplication or reduction. Hence 2^0 is 2 or n^0 is n only and not 1.
India taught the world calculus long before Newton or Leibnitz. In fact, the Newton’s laws can be seen in much more details in the ancient Indian text Padarth Dharma Samgraha, which has also discussed relativity and rejected it as apparent – not real.