Poorna from Poorna
Poorna from Poorna: Is that possible?
पूर्णस्य पूर्णमादाय पूर्णमेवावशिष्यते- declares the second line of one Shanti Mantra1. This Shanti Mantra is associated with the Svetasvatara2 and Isha Upanishads3 . In English, the mantra translates to: from the poorna (पूर्ण) when when you take out poorna, you are left with poorna. Poorna is a Sanskrit word, which is usually roughly translated as ‘whole’. In the context of the Shanti Mantra, translators2,3 have interpreted poorna to be a synonym for Brahman (ब्रह्मन्) and the statement to imply that “Brahman is still full, although the whole universe has come out of it”2.
Let us take a different route and pretend that we do not know what this statement or the word poorna
implies. Instead, we translate the shloka into the following equation:
𝑥-𝑥 = 𝑥, where 𝑥 is this entity to be determined. The Shanti Mantra
declares that there exists an entity called poorna that satisfies this algebraic equation. Keeping with our pretense of not knowing what poorna implies, at this stage the variable as an entity that needs to be discovered.
Having taken the daring first step and reduced the Shanti Mantra to its mathematical
equivalent, the next obvious question is whether entity ‘𝑥’ actually exists. The first obvious
solution is shoonya (शून्य) or zero, which satisfies the above equation by its very definition. But, are
there other mathematical solutions that could satisfy the equation?
Let us now investigate the properties of the set of natural numbers, S1 = {1,2,3, … , ∞} and
see if they can be of any assistance to us. Set S1 is an example of a countably infinite set4
defined as having an infinite number of countable elements which, even if the counting
takes forever, will arrive at a particular element in a finite amount of time. Using set S1, we
will take the set of natural numbers and delete all the odd numbers to create set S3. This is
equivalent to subtracting from S1 the set S2 = {1,3,5, … , ∞} to create another set S3.
Set S3 now contains all even numbers i.e. S3 = {2,4,6 … , ∞}. At this point one might intuit
that set S3 is approximately ‘half an infinity’, if there is such a thing. But take a closer look at S3 and you will see that it looks very similar to set S3, only the corresponding elements are
doubled. In fact, you can divide each element of S3 by ‘2’ and you will recover set S1. The
entire operation is depicted in the figure below:
To express this in a more rigorous mathematical statement we say that sets S3 and S1 are
both countably infinite and are actually ‘equivalent’. In fact, it can be shown that the set we
subtracted from the original set, S2, also is equivalent to the two sets. Mathematically, S1~S2~S3, where the symbol ‘~’ denotes equivalence. The above is not a formal proof, but a
more rigorous proof for the same can be provided using the methods of mathematical
analysis.
Voila! we took an infinite set and deleted half of that set and we find that the deleted set and
remainder are both the same as the set we started out with. Roughly speaking we took
ananta (अनंत; meaning - infinity) from ananta and were left with ananta.
This weird or non-intuituve
behavior occurs because the sets are infinite and would not have
occurred if the sets were
finite. In a strict mathematical sense we cannot write ∞ − ∞ = ∞, but
rather we can say that
certain infinite sets seem to satisfy the property that the Shanti
Mantra indicates. Strictly
speaking infinity is not a number, and infinite sets are known to have
paradoxical properties.
The infinite hotel paradox, attributed to the famous mathematician David
Hilbert (1862-
1943), states – a hotel that has infinite rooms, which are all occupied,
can accommodate
any number of additional guests despite being fully occupied. This
paradox is just another
version of the above, where one can add two countable infinite sets and
end up with the
same countable infinite set. There also exist examples of non-countable
infinities, such as the set of all positive real numbers, which can be
set up to result in similar paradoxes. Since
this article is not a mathematical treatise, we’ll leave that discussion
for another time.
Thus, we realize that our equation does indeed have multiple solutions: shoonya and ananta.
At this point, one could ask several questions. The most obvious of which would be whether
the authors of the Upanishads were aware of the profound truth behind this shloka or this
just a co-incidence? The question is not easy to answer and would take us into a narrative
based on exhaustive research into the chronology of the various shlokas, etc. Therefore, we
will not attempt to answer this question, but instead take an indirect approach. If this deep
mathematical and philosophical understanding was more than an accident, then it should
show up elsewhere in the strands of Hindu thought in a logically consistent manner.
If the word, poorna is meant to imply Brahman, and poorna can stand for either shoonya or
ananta, then all three of these words should be admissible as synonyms for the Brahman.
Indeed, a perusal of the Vishnushahastranama5 reveals that amongst, the thousand names
of Vishnu are ananta, poorna and shoonya. Thus, it seems that architects of Hindu thought
have maintained a semblance of logical consistency in this particular context.
In conclusion, we see that the Shanti Mantra does indeed allude to a truth that can be
examined with our current mathematical tools.
Notes and References:
1. The entire mantra reads
ॐ पूर्णमदः पूर्णमिदम् पूर्णात् पूर्णमुदच्यते |
पूर्णस्य पूर्णमादाय पूर्णमेवावशिष्यते ||
ॐ शान्तिः शान्तिः शान्तिः ||
2. Svetasvatara Upanishad, Tr. By Swami Tyagisananda, Sri Ramakrishna Math Printing Press (2010)3. The Upanishads, Tr. By Swami Paramananda, Indian Heritage Books (2015)
4. Principles of Mathematical Analysis, Walter Rudin, McGraw-Hill (1976)
5. http://www.drikpanchang.com/hindu-names/gods/lord-vishnu/1008-vishnu-names.html ; Ananta (659) Shoonya (743) Poorna (685).
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