Quantum Reality and the Concepts of Infinity, Infinitesimal, and Zero in Mathematical and Vedic Sciences

Abstract

Absolute reality in Nature has two aspects of its existence―one is only realizable, and the other is describable with an element of realization. While the objective sciences follow the second route to understand the absolute reality, the subjective sciences, on the other hand, go mostly by the former. It will be argued in this paper that the quantum reality, manifesting through the subjects of mechanics of microscopic systems and quantum field theory, is not the final step in approaching the absolute reality. Like other cases in the history, it only offers a rung in the ladder and that too strictly in the domain of analytical description vs. accurate measurement. The understanding of quantum reality, in fact, brings in the concepts of infinity (ananta), infinitesimal, and zero (śūnya). Further, these concepts while are necessary in precise mathematical terms in objective sciences, in philosophical terms in Vedic (subjective) sciences, however, these concepts are found to have much deeper meanings. Some mathematical tools for this purpose are pinpointed here which can act as a guide for analytical studies of these concepts in Vedic literature.

Appendix A: Śūnya and Ananta in Mathematics vis-à-vis Analogy-Based Vedic Wisdom

In continuation of Sect. 6, here we pinpoint some cases from mathematical analysis relating to zero and infinity that may be useful in analyzing the analogy-based Vedic wisdom on śūnyatā and ananta.

• (a) Discrete-continuous conversion: Recall the definition of differential coefficient (df/dx) for the function f in y = f(x), viz.,

$$ \frac{df}{dx}\equiv \underset{\Delta x\to 0}{\lim}\;\left(\frac{\Delta y}{\Delta x}\right), $$

(8.A1)

where Δx and Δy are small but finite changes in the independent variable x and dependent variable y, respectively. In the limit when the change Δx→0, the ratio (Δy/Δx) becomes a continuous function (df/dx) of x. At the computation stage since the computer does not understand the philosophical idealized concept of limit or zero, one introduces the “∈−δ language” at a subtler level. Sometimes the identity

$$ \frac{\mathrm{d}f}{\mathrm{d}x}-\underset{\Delta x\to 0}{\lim}\;\left(\frac{\Delta y}{\Delta x}\right)=0 $$

(8.A2)

is also used in setting the accuracy in computation.

• (b) Algebraic equation, identity, and symbolic relation versus śūnyatā: When one talks of objects or the operations on the set of objects, then there are various ways of arriving at zero (or śūnyatā). As a naïve example, consider the following algebraic relation among (say) two variables x and y:

$$ ax+ by=0. $$

(8.A3)

Some plausible meanings of this relation are as follows:

• (viii) Adjustment of fraction a of the item/operation x and of fraction b of the item/operation y is such that the net effect of both operations neutralizes or vanishes. (A case when (8.A3) is a symbolic relation).

• (ix) For fixed nonzero parameters a and b in (8.A3), variables x and y vary in such a way that the net effect of all arithmetical operations involved on the left-hand side in (8.A3) is zero. Not only this, x and y in this case are expressible in terms of each other, i.e., they are dependent variables (A case when (8.A3) is an algebraic equation).

• (x) If x and y are not linearly dependent and left-hand side of (8.A3) has to be zero, then only way out is that a = b = 0 (A case when (8.A3) is an identity).

These interpretations of (8.A3) suggest that the concept of śūnyatā in Vedic literature can be understood in a variety of ways. Therefore, the context hinted by the given verse is important.

• (c) When studying the integration of a particular type (logarithmic derivative) of complex analytic function within a closed contour in the theory of functions of complex variable, recall that there exists a definite relation between the total number of zeros and total number of pole-type singularities (infinities) of this function (Cauchy’s theorem). Such an observation may be relevant to find a link between śūnyatā and ananta in Vedic literature.

• (d) Case of Fourier transform: Out of a large number of integral transforms defined in applied mathematics, it is only the Fourier transform that is physically realizable. It is this transform which has explored the one after the other substructure(s) deep inside the matter at microlevel in terms of structure functions or form factors. Symbolically, one writes this transform in the form

$$ g(p)=\int f(x)\;{e}^{ipx}\;\mathrm{d}x, $$

(8.A4)

where the variables x and p constitute a canonical pair in the sense of Hamiltonian dynamics. This transform connects two functions f(x) and g(p) which are the functions of their respective arguments. An interesting property of this transform is that the behavior of the function f(x) for small values of x correspondingly describe the behavior of g(p) for large values of p or the vice versa. Keeping such a property in mind, one can use the Fourier transform in psychophysics in correlating one’s behavior in the material and spiritual worlds through the meditation variable μ (see glossary of words in [11]). (Recall the behavioral fact that small μ corresponds to worldly involvement, whereas large μ corresponds to spiritual involvement.)

• (e) Case of Dirac delta-function: It is worthwhile to add another interesting case of Dirac delta function δ(x) here following the work of [14]. This function (or rather “functional”), defined as

$$ \delta (x)=\left\{\begin{array}{c}0,\kern1em \mathrm{for}\kern0.24em x\ne 0,\\ {}\infty, \kern0.9em \mathrm{for}\kern0.36em x=0,\end{array}\right. $$

(8.A5)

$$ \mathrm{with}\kern10em \underset{-\infty}{\overset{\infty}{\int}}\delta (x)\;\mathrm{d}x=1, $$

(8.A6)

was first introduced in quantum mechanics by Paul Dirac in 1928. Later, this functional has given birth to a new branch in mathematics called “distribution theory.” Prof. Pandey calls this peculiar function as “Vedic functional” and has offered a number of interpretations of Eqs. (8.A5) and (8.A6) and other properties of δ(x), which, in turn, support the immortality of soul and the theory of rebirth in Vedānta philosophy.

• (f) Null set and universal set in set theory: One can find wonderful ways to interpret śūnyatā and ananta in Vedic literature by using the concepts of null set and universal set along with various operations on them in pure mathematics.

• (g) Zero and infinity under simple arithmetical operations: In numerical analysis sometimes one comes across certain situations when it becomes difficult to extract the exact values/meanings of certain computed quantities. These quantities, often quoted in mathematics literature as “indeterminate forms,” are $\frac{0}{0}$, 0.∞, and $\frac{\infty}{\infty}$. While there are ways and means to handle these situations in mathematics, in philosophical terms, the meaning(s) of these quantities has been attached with the roles of prakṛti and puruṣa in the maintenance of world order.

What we have presented above are some representative cases from mathematics in which the roles played by zero and infinity can analogously be extended to the philosophical meanings of some verses in Vedic literature but in a judicious manner.