The Western mathematicians claim that they have lifted mathematics to great heights and boast of a long succession of mathematicians. Yet, none of the Western mathematicians defines scientifically what a number is or what mathematics is. The test of validity of a mathematical statement is its logical consistency. Modern mathematics is not mathematics in the strict sense as it is logically not consistent. The various branches of ancient mathematics were developed to meet the requirements of explaining various physical phenomena from the mathematical angle. However, mathematicians in Europe did not confine their numerical system only to real objects. Some say that mathematics allows statements to be made unambiguously, in a manner that is value-free and culturally independent. Only when hypotheses can be stated precisely in this way can they be compared and tested experimentally or observationally. Yet, now-a-days there is rarely a mathematical statement of physical phenomena, which is value-free or culturally independent. These aspects are given fancy names to camouflage their values or the culture of incomprehensibility that are associated with most modern mathematical statements. In modern science, the technical terms are said to be defined mathematically and the words (terms) are used subsequently as a shorthand representation of the mathematical concept. Yet, such descriptions lack the unambiguous precession while retaining clarity of expression that is associated with mathematical statements.
Thus, while no mathematician defines number scientifically, different mathematicians interpret mathematics variously. Some divide the modern views on mathematics into four categories. According to Platonism, mathematicians discover the concepts such as number, dimension, groups, sets, etc., and not invent it. The π is in the sky – they say. According to Conceptualism, popular among Sociologists, mathematics is a product of the human mind. We create an array of mathematical structures, symmetries, patterns and force the world into this mould as we find it compelling. According to Formalism, mathematics is the manipulations of Symbols according to specified rules. Here the focus is on the relation between entities and the rules governing them rather than the question of whether the object being manipulated has any intrinsic meaning. According to Intuitionism only the simplest intuitive ideas could be used. Anything outside our experience must be constructed from the simplest ingredients by a sequence of intuitively familiar steps. Here the attention is restricted to measurable quantities in order to avoid introducing “obvious” concepts like simultaneity, which may turn out to be experimentally meaningless.
According to Vedic theory of numbers, Number is a quality/property (such as colour, spread, etc.) of all substances, by which we differentiate between similars. It is capable of manifesting itself independently without the help of any other condition or agency (द्रव्येषु भिन्नव्यवहारः पृथक्त्वारिक्तभेदवशात् संख्यावशाच्च). Number is a separate property, as the cause that is responsible for perception of the number of substances is different from other properties of the substances (such as electromagnetic radiation for colour, dimension for spread, etc). An object can be perceived only through the differentiation of its inherent structure and its position from other similar objects (आन्तरालिकावयव संस्थानगतभेद). Number is the cause for the perception of differentiation among objects (भेदगणनात्मकत्वं) just like distance variable (परत्व) and proximity variable (अपरत्व) are causes for differentiation of time and space. When we cannot distinguish the substance from other similar substances, we call the substance as one (एक). If we can distinguish, we call them many (वहु). Many can be 2,3,4, ….n (not infinity) depending upon the quantum of sequential differentiation. Mathematics is the process of conversion of one to many and vice-versa as well as many to other many. Before we proceed further, it is necessary to define “matter” and “properties” as number has been defined as a property of substances, i.e., matter.
We may perceive the same substances of different colour, shape or smell, yet the number of substances may not change. This distinguishing property of substance is called number. The number can be of one object or of many objects. The expressions one and many in the above statement can be used for individual or a group of substances. The number ‘one’ is universal and associated with fundamental particles (एक इता सङ्ख्या । इता अनुगता सर्वत्र या सङ्ख्या सा एक). However, the numbers 2,3………n are associated with created substances only. These numbers are not mere collection of many “one” substances, but it has independent existence as a number like the number “one”. Since “one” can be associated with the fundamental particles as well as the created substances with any number of parts, the fractions are not treated as fractions of “one”, but the fractions itself form one or more parts of a unit, of which the whole is “one” multiple. Each individual member of the whole set is treated as the “one” and the set is treated as a combination of many one’s individually or with other numbers.
The numbers higher than one are perceived in sequences of ones such as “this is one”, “this is one” etc. Each such perception takes place at different times, even though they are a small fraction of a millisecond. Their amalgamation is physically impossible as objects exist only in present. In the present, the past objects exist as the effect, from which we could infer their past existence. Similarly, in the present, future objects exist as the cause, from which we can infer their future existence. The amalgamation of higher numbers thus transcends the barriers of time. Since higher numbers are real, even though in the interim, their amalgamation is possible only in mind. Anything that does not have a physical body (द्रव्य), but is perceptible to the mind, has to be either a property (गुण), or an action (कर्म), or both, and would be related with each other in either of the three different ways; viz., inherence (समवाय), generic attributes (सामान्य) or special characteristics (विशेष), which can only be inferred from the perception of the substances, their properties or actions. The number is a property, which is inferred through inherence with the object. As long as the object remains, a number associated with it remains
Since all objects are ever changing (which are action sequences) even though at micro-level, number is inferred due to inherence through such change. These changes, which mean a sequence of coupling and decoupling with other substances that have the same generic attribute or special characteristics, are thus, perceptible to the mind through a combination of inference and memory. Based on the mechanism of perception, the numbers that follow one in a sequence are called two (द्वौ द्रुततरा सङ्ख्या), three (त्रयस्तीर्णतमा सङ्ख्या), … eight (अष्टावश्नोते), nine (नव न वननीया नावाप्ता वा), etc. This also explains why after nine the sequence changes. The perception of such change in number in the sequence is mathematics. The exact mechanism of perception of higher numbers is described below:
When our eyes conjunct two objects of the same class or two objects of different classes, the first impression in our consciousness (brain is only a processing machine, consciousness is the actual user), generates the knowledge of “this is one”, “this is one”. Like a computer that processes only one information at one step, the eye or the mind can accept only one object (here the property of numbers), at one step and at one time. The consciousness, which is the actual repository of knowledge, cannot interact with the object directly just like the user of a computer cannot directly approach data, but has to approach through the key board and the CPU. Similarly, the number “one” is accepted through the eyes and processed through the mind with the instrumentalities of the brain, to create the knowledge of “one” in the consciousness. The vital energy of the body called “प्राण” provides the energy input for the operation through mind, which acts as the CPU. “प्राण” acts on the same principle that stabilises the Earth in its orbit. Thus, we can have the memory of “one” and we can retrieve the saved data from memory whenever required.
Since consciousness is the repository of all knowledge, the knowledge of “one” can be mingled with all other related knowledge to create the knowledge of repeated ‘one’s of the same object or many objects perceived at different times. This is because the number of times the number “one” is perceived, it gives rise to an equal number of “भावना” or inertia of creative contemplation. Since “भावना” is a conscious process that arises only in the intellect, all such “भावना” can be integrated in the intellect and reflected on the consciousness at the same time. Thus, there can be the knowledge of the common property of “one as a class”, i.e., the knowledge of “एकत्व”.
The knowledge of two ‘one’s leads to the creation of the knowledge of “two” in our consciousness. In this process, the two objects possessing the property of “oneness” are the inherent cause, the two ‘one’s are the non-inherent cause, and the conscious process that integrates the ‘one’s of the two objects after it is triggered by some agent (called अपेक्षाबुद्धि – incidental intellect), is the effective cause. (The different causes mentioned above have been explained separately).
The number next to one is called two (द्वि) as its perception after one is very rapid (द्वौ द्रुततरा सङ्ख्या). The next number in the sequence is called three (त्रिणी) as its perception is caused by the increase in the number two (त्रयस्तीर्णतमा सङ्ख्या). Though two is also cognised as increase in the number one, cognition of three is different as two, which increases to become three is not a naturally cognitive number like the number one. The next number in the sequence is called four (चत्वार) as its perception is highly mobile (चत्वार चलितसमा). After the cognition of three, the cognition pattern changes rapidly. At the macro level, it represents the controlling principle of creation, which will be discussed in the chapters relating to cosmology and dynamics of mind. The next number is called five and so on.