# Santi-Paath or The Introductory Verse from the Isha Upanisad

Below is the introductory verse of the Isha-Upanisad:

ॐ पूर्णं अदः पूर्णं इदं पूर्णात पूर्णं उदच्यते
Om; PurnamAdah Purnam Idam Purnaat Purnam Udachyate
Om; That is Full; This is Full; From the Full emanates the Full;

पूर्णस्य पूर्णं आदाय पूर्णं एव अवशिष्यते
Purnasya Purnam Adaaya Purnam Ev Avashishyate
When Full is taken out of the Full, What remains is still the Full.

The above translation is from standard sources. It should be self-evident that this verse is discussing the Infinite. So right way, we feel amazed. Mind you, this is at the beginning of philosophical or religious thought. The idea that creators of the Upanisad thought about the concept of Infinite thousands of years ago is itself both exhilarating and intimidating.

So far, all the interpretations we have read of this verse come from Scholars of Philosophy, Religion or Language. That is why they interpret the word पूर्णं = Purnam as Full or Infinite

Scholars have interpreted this verse as an expression of the “inherent contradiction” of the concept of Infinite. They were probably influenced by Immanuel Kant, the German Philosopher, who also discussed the contradiction inherent in the concept of Infinite.

After all, how can you extract Infinite from Infinite? And even if you do, how does Infinite remain when you extract Infinite out of Infinite? This is why the earlier translations were changed from Infinite to Full.

But this really does not solve the problem.

A Mathematical Interpretation

Colloquially, the word Purnam means complete. We think this is also the most accurate translation of Purnam from a Philosophical, Metaphysical and Mathematical sense.

Let us first consider a counterexample to the use of Infinite or Full for Purnam. Look at R, the infinite space of Real numbers. It consists of Q, set of rational numbers (numbers that can be written as p/q with numerator p and  denominator q) and I, set of irrational numbers, or numbers such as $\sqrt{2}$ which cannot be written as fractions.

Now, R, Q & I are all Infinite or Full spaces. If you take out Q from R, what remains is I. While these are still Infinite or Full, what you take out =Q and what remains =I are very different in character and properties from what you started with = R. So in our judgement, this little counterexample shows that neither Infinite or Full are appropriate interpretations of Purnam from the Introductory Verse or Santi-Paath.

This problem is solved when you consider the Mathematical concept of Complete Spaces. For example, Q may be Infinite or Full, but Q is NOT Complete. Because you can have an infinite sequence of fractions in Q that get infinitesimally close to the value of $\sqrt{2}$  . In other words, you can have an infinite sequence within Q whose limit, in the calculus sense, does not reside within Q. Therefore, Q is NOT a Complete Space. In contrast, R, the space of all real numbers, is a complete space.

Intuitively, consider a Complete Space to be a space in which every boundary point of the space also resides within the space. In other words, the end point of any journey begun in the Complete Space also resides in the Complete Space. This is what we think the Creators of the Isha-Upanisad wanted to convey. So our initial interpretation of the Santi-Mantra is:

That is Complete; This is Complete; From the Complete Arises the Complete;
When you take out Complete from Complete, What Remains is Complete
.

But what could they mean by “take out” in the Santi-Paath?

Let us consider a simple example. Every reader of this Blog probably knows the concepts of x & y co-ordinates of a point. For example, the point (2,3) is 2 units to the right of (0,0) on the X-axis and 3 units above the (0,0) on the Y-axis.

This is an Infinite space R2.  It is a Complete Space. How do you extract or take out a Complete Space from this Complete Space? Consider the projection of R2 onto the X-axis. In other words, you are taking out the y-coordinates and collapsing the R2 onto the X-axis which is a line or R. You do so by mapping each (x.y) coordinate to just the point x. As an example, points (2, 3), (2,4), (2,5) etc, all collapse onto to the point 2 on the x-axis.

So in this case, you have taken out a Complete Space R (the Y-Axis) fromComplete Space R2 = , the geometric plane and what remains is the Complete Space (or the X-axis). You can see that this concept fits the Introductory Verse perfectly.

Let us take it one step further. Take the 3-dimensional space R3, which geometrically has 3 axes, x-axis, y-axis and z-axis. Now project this 3-dimensional space R3 onto the two-dimensional plane R2. You do so by taking out R = z-axis.

Again, when you take out or extracted Complete Space R (z-axis) from the Complete Space R3,  what remains is the Complete Space R2 or the cartesian plane. You can see this multi-dimensional concept fits the Santi-Paath or the Introductory Verse perfectly.

In this context, the extracted portion or the Z-axis can be called the Kernel and the remaining or the result can be called the Image. These are both standard Mathematical concepts.

Because this interpretation fits the Santi-Mantra perfectly, we feel emboldened to offer it as our own interpretive translation of the  Santi-Mantra of the Isha-Upanisad:

ॐ पूर्णं अदः पूर्णं इदं पूर्णात पूर्णं उदच्यते
Om; Purnam Adah Purnam Idam Purnaat Purnam Udachyate
Om; That is Complete; This is Complete; From the Complete Emanates the Complete

पूर्णस्य पूर्णं आदाय पूर्णं एव अवशिष्यते
Purnasya Purnam Adaaya Purnam Ev Avashishyate
When a Complete Kernel is Factored Out from the Complete; the Resultant Image is just as Complete

Yes, the ever-vigilant reader would point out that we have not provided the Completeness interpretation of the second part of the first line of the Santi-Mantra – the उदच्यते or Emanates/Arises part. They would be correct. The second line is about आदाय or Extraction from the Complete while the उदच्यते or Emanates part of the first line is about Arising to another Complete.

From our standpoint, the आदाय or the second line is about Projection or Surjection in the Mathematical interpretation while उदच्यते or the second part of the first line is about Injection. The discussion about Injection is as valid but even more complex than the above discussion of Projection. So please allow us to defer it.

In conclusion, we think that scholars who see any inherent contradiction in the Introductory Verse of the Isha-Upanisad are not versed in the concepts of Completeness, Kernel & Image. Their simplistic philosophical models fail to grasp the brilliance of the Santi-Paatha. That is their lack of rigor and not the fault of the Isha-Upanisad.

When we apply our interpretation to the Santi-Paatha of the Isha-Upanisad, we see a glorious concept of Infinite Completeness developed at the beginning of philosophical thought, a concept that received its Mathematical interpretation over 2,000 years later.

Editor’s PS:

• The Mathematical concept of Projection discussed above is similar to the Projection concept discussed in our earlier article Origin of the Concept of Many Gods.
• In that article, we discussed the concept of projecting the attributes of the Supreme Entity onto a known Image, which could be a phenomenon, object or concept and then using the known properties of this visible, understandable Image to look back and ponder the powers of the Supreme Entity.
• The more we reflect on Eternal-Dharma, the more amazed we become at the ease with which philosophical concepts developed at the beginning of time in India can be applied to Mathematical concepts developed in the past couple of centuries.