GEB, Properties of Natural Numbers and Chamakam

I had written previously about how good I find the book Godel, Escher and Bach: The Eternal Golden Baird. Last week, I read chapters 3 and 4 of the book. In these chapters, Dr. Douglas Hofstadter speaks of Figure-Ground and its prominence in formal systems. Dr. Douglas Hofstadter goes into great detail to explain Figure and Ground could not only be identified in Escher and others' paintings but also in music and formal systems. As an example of figure - ground, try this

Fr! om the first look, one would see two faces. But a few seconds of careful look would reveal a vase. The Figure and Ground are thus defined as negation of one another. More on figure and ground here. Now, if you take the square in which the whole diagram is present as the universe, then the vase forms the Figure while the faces form the ground. In sketching and painting this is the general way figure and ground are defined. Dr. Douglas Hofstadter takes this concept a little further and tells us about how Bach used the same concept in The Musical Offering.

Dr. Douglas Hofstadter then enters the formal systems domain and tries to explain how figure and ground could be established. More on that later in the post.

In one of the boring moments @ work, I started browsing through the text of Sri Rudram. As I started reading t! he eleve nth anuvaka of Chamakam I found something very intersting. This was not the first time I was reading Chamakam. But somehow, I missed this point till today.

For those who dont know what Sri Rudram is,

Srirudram, also known as Rudraprasna, is a hymn devoted to lord Shiva. It is part of the Yajur Veda and one of the greatest of the Vedic hymns for all round benefits and to remove all doshas & difficulties. In most of the poojas and homas it recited by the Vedic pundits. Sri Rudram is in two parts. The first part, chapter 16 of the Yajurveda, is known as Namakam because of the repeated use of the word "Na! mo" in it. The second part, chapter 18 of the Yajurveda, is known as chamakam because of the repeated use of the words "Chame".
Rudram is divided into 11 sections called Anuvakas. In the first Anuvaka, Rudra is asked to turn away his Ghora rupa (fierce appearance) and to please keep his and his followers’ weapons at bay. Having been pacified, Rudra is requested to destroy the sins of those for whom it is being chanted.
Apart from being a hymn devoted Lord Shiva, Srirudram also contains may hidden secrets in coded format. For example the verses contain coded instructions for preparing various ayurvedic medicines.

In the eleventh Anuvaka of Chamakam, a series of numbers are chanted. It goes like this

ఏకాà°šà°®ే à°¤ిà°¸్à°°à°¶్à°šà°®ే à°ªంà°š à°šà°®ే సప్తచమే
నవచమ ఏకా దశచమే à°¤్à°°à°¯ో దశచమే
à°ªంà°š దశచమే సప్à°¤ దశచమే నవ దశచమ
ఏక à°µిà°—్ం శతిà°¶్à°šà°®ే à°¤్à°°à°¯ో à°µిà°—్! ;ం శతిà°¶్à°šà°®ే
à°ªంà°š à°µిà°—్ం సతిà°¶్à°šà°®ే సప్à°¤ à°µిà°—్ం శతిà°¶్à°šà°®ే
నవ à°µిà°—్ం సతిà°¶్à°šà°® ఏక-à°¤్à°°ిà°—్ం శచ్à°šà°®ే
à°¤్à°°à°! ¯à°¸్-à°¤్à°°ిà°—్ం శచ్à°šà°®ే
చతస్-à°°à°¶్à°šà°®ే உష్à°Ÿౌà°šà°®ే à°¦్à°µాదశ à°šà°®ే
à°·ోà°¡à°¶ à°šà°®ే à°µిà°—్ం శతిà°¶్à°šà°®ే
à°šà°¤ుà°°్-à°µిà°—్ం శతిà°¶్à°šà°®ే உష్à°Ÿాà°µిà°—్ం శతిà°¶్à°šà°®ే
à°¦్à°µాà°¤్à°°ిà°—్ం శచ్à°šà°®ే à°·à°Ÿ్-à°¤్à°°ిà°—్ం శచ్à°šà°®ే
à°šà°¤్à°µాà°°ిà°—్! ం శచ్à°šà°®ే à°šà°¤ుà°¶్-à°šà°¤్à°µాà°°ిà°—్ం
శచ్à°šà°®ే உష్à°Ÿాà°šà°¤్-à°µాà°°ిà°—్ం శచ్à°šà°®ే
à°µాజశ్à°š à°ª్రసవà! °¶్ ;à°šా à°ªిజశ్à°š à°•్à°°à°¤ుà°¶్à°š à°¸ువశ్à°š
à°®ూà°°్à°§ాà°š à°µ్యశ్à°¨ియశ్à°šాà°¨్‌-à°¤్à°¯ాయన-à°¶్à°šాంà°¤్యశ్à°š
à°­ౌవనశ్à°š à°­ువన-à°¶్à°šాà°§ిపతిà°¶్à°š

Transliteration in english:

Ekaa cha me tisrashcha me pajncha cha me sapta cha me nava cha ma ekadasha cha me trayodasha cha me pamchadasha cha me saptadasha cha me
navadasha cha ma eka vishatishcha me trayovishatishcha me
pamchavishatishcha me saptavishatishcha me navavishatishcha ma

ekatrishachcha me trayastrishachcha me chatasrashcha me
ashhtau cha me dvaadasha cha me shhodasha cha me
vishatishcha me chaturvishatishcha meashhtaavishatishcha me
vaatrishachcha me shhattrishachcha me chatvarishachcha me

chatushchatvaarishachcha meashhtaachatvaarishachcha me
vaajashcha prasavashchaapijashcha kratushcha suvashcha muurdhaa cha
vyashniyashcha antyaayanashcha antyashcha bhauvanashcha
bhuvanashchaadhipatishcha

The meaning of! this anuvaka is (as translated by the great english mis-translators)

Let me be granted the uneven number one, three, five, seven, Nine, eleven, thirteen, fifteen, seventeen, Nineteen, twenty one, twenty three, twenty five, twenty seven, twenty Nine, and thirty one, thirty three, and even numbers four, eight, twelve, and sixteen, twenty, and twenty four, twenty eight, thirty two thirty six, and forty and forty four, forty eight to ensure food and its production, its continuity, and the urge to enjoy, the origin of all productions, the sun, the heaven, the head of all, the infinite, the all pervading like the sky, time and the like present at the end of total consummation exists at the end of it on the earth ! as universal form, the Antaryami the immortal, the inner ruler of every thing, the omni present and omni potent.

As a side note, I could never understand how and why Sri Rudram talks about sacrifices to Lord Shiva. I could never find a word meaning sacrifice in the whole of Sri Rudram. May be my Sanskrit knowledge is not as good.

Now a cursory look might suggest that it just as a series of numbers. Lets have a closer look.

The first series of numbers is

1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33

All odd numbers.

The second series of numbers is

4,8,12,16,20,24,28,32,36,40,44

All multiples of four. So what, you may ask. I am coming to that.

Lets for a moment concentrate on the numbers missing in the two series'. The missing numbers (! I go till 34 only) are

2,6,10,14,18,22,2! 6,30,34< /blockquote>

Look at these numbers closely. If you divide these numbers by 2, you would get

1,3,5,7,9,11,13,15,17

You get back the odd numbers. Now I dont know and cant see why the first series is quoted til 33 only where as second series is quoted till 44 numbers. This needs to be investigated. But I am certain about one thing. With 2, 4 and series of odd numbers, you can define any number in the natural number system. How, you ask. This is how

If a set, X of all numbers {x}, such that x is divisble by 2 utmost once, forms a set of numbers

and

If a set, Y of all numbers {y}, such that y is divisible by 4 and y >=4, forms another set of numbers,

then

union of X and Y i.e., X U Y forms the set of natural numbers.

Dr. Douglas Hofstadter explains in Chapter 3 of GEB that you could construct Figure-Grounds in mathematics. He also explains us that it is even more interesting to construct Figure-Figures. He gives a variety of examples to explain this and finally comes to question "can theorems as well as non-theorems have a form?". Well my understanding is that if you take this anuvaka as an example, it is certain that theorems and non-theorems can have form and that too a very clearly defined form, just as the case is here. Make a note that X intersection Y is a null set here, which means X union Y covers the complete universe of natural numbers. This is the brilliance of this anuvaka. With two different forms (or Figures) we h! ave a complete picture (natural numbers).

C oming back to Figure and Ground concept, we can identify two clear figures here - odd numbers + odd numbers multipled by 2 and multiples of four in this anuvaka. A clear example of Figure-Figure definition of formal system of natural numbers.

Unfortunately, Dr. Douglas Hofstadter doesnt discuss this anuvaka in GEB. I dont blame him. Obviously, the translations, like the one quoted above, concentrate on Vedas as a part of ritual of offering sacrifices to Lord Shiva but miss the point that after all, Vedas describe the nature. Numbers are as much part of nature as we are. This anuvaka is a clear example of how well Vedas define the nature.

One may disagree with me and say that I am being partial to Vedic knowledge but one cant say that this piece of information had been embedded in the Vedas by coincidence. It is there because it is supposed to be there. Many questions could be asked like

Wh! y 33 and 44?

Why only odd numbers, 2 and multiples of 4?

Why only natural numbers?

and so on

Answers to these questions could be found by investigating into the Vedas with an open mind. A crooked and one sided analysis of Vedas would leave us with sacrificial procedures and pagan gods and godesses, whereas an open ended discussion on Vedas would lead to knowledge hitherto untouched by Science. Just imagine how many secrets of nature could be pried open if

a Sanskrit Scholar, who learnt Sanskrit from Vedic Pundits and not from western anti-Hindu pundits,

a mathematics Scholar, who can put aside the notion that Vedas are holy writ and God's word and look at Vedas as a mine of knowledge,

a geologist,

a biologist, and others from relevant fields work together, study and decipher the Vedas

The kind of knowledge represented in the Vedas n! ot only gives out great philosophies for life, but also give u! s great knowledge on the nature of universe itself. If only, India after Independence took this with some pride and enthusiasm, mankind as a whole would have benefited from the study.

Post Script: It occured to me, as I almost finished this post, that to define the numbers in this anuvaka the way they have been, one would need to multiply all natural numbers by 4 (this is the definition of set of numbers represented by Y above). GEB as had been explained here previously, is about self-reference in general, and specifically in formal systems. This anuvaka is a brilliant example of definition of self-referential formal system. There you have. One more special case of Strange Loop. Can we say that this anuv! aka is a special case to Godel's theorem? May be so. May be not. Only pure mathematicians can tell I guess.

Update: Dr. Sasidharan Commented "The numbers actually refer to mitochondrial DNA and Nuclear DNA and this 11th anuvakam pray for the well being of our DNA which is fundamental for all living thing in this planet." More on Mitochondrial DNA here. How exactly this anuvaka in Chamakam correlates to Mitochondrial DNA is something which would be interesting to look. Please post any interesting links or articles on this link in comments.

Update: Thanks agai! n to Dr. Sasidharan for the link explaining how Chamakam descr! ibes var ious aspects of nature. My favorite still is the 11th anuvaka which, as in his earlier comment Dr.Sasidharan wrote, is a representation of Mitochondrial DNA. Heres the article.

natural numbers examples