The Shāntī Pāth & Cantor’s Infinities

Millions of Indians chant the Shāntī Pāth or the peace invocation daily at the beginning and end of every satsangh, literally a gathering of truth-seekers. Such gatherings are commonplace among the populace where thousands assemble to listen to vedic discourses from Gurus of their choice. The Shāntī Pāth is placed at the beginning of the Iśavāsya Upanishad and in its extended form is the first shlöka of the 5th Adhyāy of the Brihadaranayak Upanishad .

! PaUNa-ma\ Ad:, PaUNa-ma\ [dma\ , PaUNaa-t\ PaUNa-ma\ ]dcyato.
PaUNa-sya PaUNa-ma\ Aadaya , PaUNa-ma\ eva AvaiSaYyato .
! Saaint: Saaint: Saaint:.

pūrnam adah, pūrnam idam, pūrnāt pūrnam udacyate
pūrnasya pūrnam ādāya pūrnam evāvaśisyate.

For our purpose here we take S.Radhakrishnan’s[1] explanation :
That is full, this is full. From fullness fullness proceeds.
If we take away the fullness of fullness, even then fullness remains.
Aum (the syllable) is Brahman (who) is the ether, the primeval ether….

Swami Chinmayanandji[2] explains it as follows :
That is Whole; this is whole; from the Whole the whole becomes manifest.
From the Whole when the whole is negated what remains is again the Whole.

Paul Deussen[3] translates it as:
That is perfect and this is perfect, out of the perfect, the perfect one is created.
If one takes out the perfect from the perfect, there still remains the perfect.
Om ! the expanse is Brahman, the wide expanse; the primaeval airspace filled with expanse !….

‘The Brhadāranyaka Upanisad’ by Ramakrishna Math is more explicit in its translation:
Om That (Brahman) is Infinite, this (universe) too is infinite. The infinite (universe) emanates from the Infinite (Brahman).
Assimilating the infinitude of the infinite (universe), the Infinite (Brahman) alone is left.
Om is the ether-Brahman – the ether that is eternal….

The word pūrna [4] has been variously translated above as full, whole, perfect and infinite. pūrna = pūr + kta[5] here pūr is the root that signifies fullness or completeness; the suffix kta is used to further stress this completion. In other words pūrna stands for something that is more than complete or beyond are imagination!

The interesting dichotomy of the dual usage of this word stresses the fact that there are further two levels of completeness that exceed our mental capacity; there is an inherent limitation. In the words of Swami Chinmayananda [6] –“This peace invocation reads as though a pleasant contradiction in terms. On a very superficial reading one is apt to consider only its word-meaning and then each sentence should certainly confuse and confound any modern student who dives into it especially through its literal translations available for him in English or in any vernacular. This is a philosophical truth-declaration, and as such unless we know something of Vedantic conclusions over the theme of the transcendental Infinite and its ‘relationship’ with the finite, this stanza should necessarily confuse any reader.”

On the mathematical side in 1870, Georg Cantor (1845 – 1918) gave us the ‘set theory’ – a brilliant thesis on the study of collections of numbers, points, objects, anything really in general. This revolutionized ‘number theory’ and gave rise to his– ‘Theory of Infinities’ or what is more precisely called ‘Cantor’s theory of Trans-infinite cardinals’. The word trans-infinite here maybe understood as a condensed form of transcendental Infinite – that which is beyond the infinite. How is this possible ? Let us, at this point, push our limited reasoning on the lines of Cantor and attempt to explain this brilliant equivalence between – mantra and math – separated by almost four millennia of human endeavor !

The ‘lazy eight’ symbol ‘’ has long been used to denote the infinite since this shape can be traversed endlessly.

Till the mid 19th century, the only concept of infinity, in the mathematical world, was linked to ‘countable numbers’ or integers or natural numbers – as they are commonly known. As we count, we go :

1, 2, 3, 4,…..n…….. ∞

Here ‘n’ stands for any countable integer. At the end of this seemingly endless sequence, the number becomes too large to count and is denoted by ‘’.

The earliest limitation of this way of thinking was probably expressed by Euclid (330 – 275 BC) when he tried to answer the question – “How many points are there in a ‘line segment’ ?” Now a line is a ‘line’; it is continuous and how can we talk about it as ‘discrete’ points ? But then, that is what mathematicians are all about; they question what seems obvious and keep building a pyramid of well-grounded higher knowledge. In fact Euclid’s axioms and theorems are even today part of elementary school Geometry. But he could not answer this one!Galileo Galilei (1564 – 1642) published in 1638 Dialogue of the Two New Sciences gives some excerpts of his arguments on this). In this work, he outlined a paradox relating to ‘natural numbers’ and their ‘squares’. He derived a geometric equivalent by arranging two concentric circles as shown in Fig. 1. and raised the question– “Are the number of points on the inner, smaller circle the same as those on the outer, bigger one ?” For every point A or B on the inner circle, there is a corresponding ray OA or OB respectively and a corresponding A’ or B’ point on the outer circle and so on. Thus, infinite rays emanate from ‘one’ center O. These diverge to an infinite number of points A,B etc. on the smaller circle and each in turn connects to a corresponding A’, B’ etc. on the outer one. The paradox explicitly is – “how can the rays emanating from the center link an identical number of infinite points on the inner and outer circle ?” The answer to our question is not obvious and it did trouble intellectuals for almost three centuries till Cantor resolved it rigorously.

Cantor carried the concept of the ‘countable infinity’, which he termed א0 (Aleph – nought) to the ‘indenumerable infinity’ which he named א1 (Aleph-one). [A very readable account of Cantor’s work is given in ‘Fractals – Images of Chaos’ [7]]. Note here that the description ‘countable infinity’ also means ‘that which in principle can be counted’ and it can also be termed as ‘denumerable infinity’.The ‘indenumerable infinity’, א1 is likened to ‘that which cannot be counted’ and is part of the continuum. Cantor developed ‘degrees of infinities’ between א0 to א1 by using one-to-one correspondence between the elements of two sets.

  • e.g. What do we make of the statement : There are as many even (or odd) numbers as there are natural numbers ? There are infinitely many of each sort and at first it seems that the even(or odd) numbers appear alternatively and therefore should be less than the natural numbers…but it is not so…for we can match them as follows :

We see the one-to-one correspondence between the two sets and we conclude that since every number has its match – both the sets approach the same degree of infinity – in this case א0.