Friday, 23 November 2012

The Meaning of ``OM'' and Purpose of Life

The sound ``OM'' is associated with Hinduism. Before I get to this topic, I should clarify that this is not a religious topic for you to shut your mind and eyes. As I have said in one of my earlier blog, ancient Indian science got intertwined with religion for political reasons. Thus, due to invasions, the way of living of the local people of  India was referred to, by the invaders, as Hindu (as an indication to its geography crossing Sindhu river/region) to distinguish it from their way of living. So keeping religion apart, the question we should address is how much of ancient science is right, sensible, useful etc. irrespective of our religion. Clash/division is human nature. Even if you had one religion, there will be clashes within the religion. Note that the division within the religion as Vaishnavites vs Shaivites, Shia vs Sunni, and Protestants vs Catholics are all politically motivated. So, please keep your religion aside. You can read this if you have a body that modern science categorises as homo sapiens.

The sound ``OM'' is in reality a reference to the three sounds ``A'', ``U'' and ``M''.  Ancient texts usually refer to it as Pranav mantra, meaning the sound of life or syllable of life. It is called so because it is the meaning of life. The verse 10.53.6 of Rig veda may be regarded as putting forth the purpose of life:

manur bhava janaya daivyam janam

which means ``be the human and create divine beings''. But what does ``becoming human and creating being'' mean? How should one understand this as the purpose of life? There is a common agreement that the human mind, by its nature, is constantly in search of eternal peace. If this is taken as the purpose of life then the Mandukya upanishad contains the answer on how to attain this purpose of life by answering the question  ``Who am I?'' and introduces and explains the notion of ``OM''. The answer to the question of ``Who am I?'' is kind of hinted at four places in the upanishads (the four important mahavaakyas or highlights):
  1.  prajnaanam brahma in verse 6.3.3 of Aitareya upanishad (Rig Veda).
  2. Aham brahmaasmi in verse 1.4.10 of brihad aranyak upanishad (Yajur Veda).
  3. Tat tvam asi in verse 6.8.4 of chandogya upanishad (Sama Veda).
  4.  Ayamaatmaa brahma in verse 1.2 of mandukya upanishad (Atharva Veda). 
However, the Mandukya upanishad explains this elaborately. The mandukya upanishad is the smallest of all upanishad, containing 12 verses, and is considered the most difficult. The difficulty is in terms of the maturity to accept, grasp and verify it. Depending on one's maturity, it may sound totally trash, like the answer 42 of Douglas Adams or might truly reveal something to you.

The first verse of Mandukya upanishad states that ``OM'' is the only thing that existed, exists and will exists. The second verse describes all these are supreme being, including us and we exist in four stages. The third verse describes our first stage as the jagrat or waking state. This state of consciousness is outward. The fourth verse describes the second stage as the svapna or dreaming state whose state of consciousness is inward. The fifth verse describes the third stage as susupti or deep sleep state where he has no experiences; state of bliss. No activity of body and mind. This state of bliss is not the same as mukti. A person has no control (at least directly) over his/her dream and deep sleep state. (S)He has some control in the waking state. I say ``some control'' because even in waking state one has no control over your heart beat or absolute control over breathing etc. The Mandukya upanishad conjectures that there is a fourth state of consciousness, a state of eternal peace and tranquility, called turiyam,  from where one is able to observe all the three states of consciousness, very similar to the way you observe dream and deep sleep state from waking state. This fourth stage is described in the seventh verse. Of course, you are aware that you dreamt only after you come back to waking state. Similarly, it is very difficult to realise the state of turiyam being in waking state. These concepts have been adapted into fictional movies such as the Matrix series and Inception.

As per the eighth verse, the three sounds ``A'', ``U'' and ``M'' represent the three states of consciousness viz., waking state (jagrat), dreaming state (svapna) and deep sleep state (susupti), respectively.  The ninth, tenth and eleventh verse explains why these three sounds are primal and related to the three states of consciousness. The twelfth verse   states that the entire concept of achieving turiyam state is encoded in a single word ``OM or AUM'' seen as combination of  ``A'', ``U'' and ``M''.  The state of turiyam represents the soundless (or still) state. 

These three basic sounds are precisely those that one can make without using tongue. The sounds are a result of the successive deformation of the lips (without moving your tongue) while creating sound from throat. Start making sound from your throat. While doing so, open lips wide apart you get the sound ``A''. Without stopping, squeeze your lips to form a ``O'', you make the sound ``U''. Still without stopping close the lips, you get ``M''. Therefore, ``OM'' is just a convolution of the natural sounds you can make by moving your mouth. Every other sound needs movement of tongue. I'm just conjecturing, you prove for yourself and negate me. I will be happy to correct myself.

The first sound ``A'' is similar to the sound of ``a'' when you pronounce ``china''. In English, the alphabet ``A'' takes two sounds. For example, when you pronounce ``Asia'', the first ``a'' has a different sound and the second ``a'' has another. The sound we are referring to is the second sound of ``a''. The sound ``A'' is the first sound to come into existence from silence (or nothing). No body has to teach a kid to utter the sound ``A''! Thus, it naturally gets connected to the supreme (or first) being and this is emphasised both in the Bhagavat Gita and the first kural of Thirukkural by Thiruvalluvar:

agara muthala ezhuthellam;
aathi bhagavan muthare ulagu.

The sound ``A"" is primal included in all other sounds that can be made by us. The ``A''  sound is made by opening your mouth (upper and lower jaw opening wide) and releasing the air from the throat. Try making this sound at a stretch, you will notice that your stomach vibrates. This sound is associated to your belly region. You may have noticed that people who have severe stomach ache, will generally make the sound ``A''. We think the person is crying out of pain. But the body naturally induces this sound to give the person the strength to overcome the pain till he receives medication.

The second sound is ``U''  similar to the ``oo'' in words like boom, cool, goose etc. This sound is made by making the shape ``O'' out your lips and then blowing air out. This is intermediary and advanced than ``A'' sound. Try making this sound at a stretch, you will notice vibration in your chest now. Start observing in future that people who have severe chest ache will make the sound ``U''. We think the person is crying out of pain. In reality, the body naturally induces this sound to help the person overcome the pain till he receives medication. This sound is associated to your chest region.

The third sound is ``M'' similar to the ``m'' in words like sum, come etc. This sound is made by closing the lips tight and making sound from throat. This represents the final sound. Silence can be achieved only after attaining this as the final sound. Try making this sound at a stretch, you will feel vibration in your head. You now know what I'm going to say. Yes, start observing that people who have head-ache will make the sound ``M''. We think he is crying out of pain. But the body naturally induces this sound to help the person overcome the pain till he receives medication. This sound is associated to your head region.

The convolution of these three sounds is what was referred to as ``OM'', also written as ``AUM'' to denote this meaning. The real meaning behind the instruction ``Chant OM everyday'', actually meant chant the three different sounds individually for your  well-being. Note that all the organ systems in our body is above the hip region. The three sounds activates the region above your hip.

If jogging, walking, running and every other sport is religion independent then why  is chanting``OM'' religious? Just because the inventors and followers of this exercise were termed Hindus? To summarise, ``OM'' is an encoding to remind oneself that we have to search within us to answer the question ``Who am I?'' and purpose of life.

Deriving the formula for EMI and amortization schedule

emi

I am going to derive a simple math formula this time. The EMI (Equated Monthly Installment) disease is spreading. No, I’m not going to talk for or against it. I’m just going to derive the formula used to compute EMI and on what basis one decides the part of EMI that goes for principal and interest.

Let us say that you borrow L INR1 amount as loan at the rate of interest i% per annum for a period of n months. Let’s say we agree to pay every month an equal amount E (called EMI) to clear the loan in n months. Then what should be the E?

Since we are going to work in terms of month, we shall convert the rate of interest per annum to per month. Thus, the rate of interest per month is (i/12)%. This means for every 100 INR of the loan amount the lender charges an extra of i/12 per month. Equivalently, for each 1 INR of your loan L, the lender charges you an extra i/(12 × 100) per month. This means that at the end of the first month you owe the lender an amount which is the sum of the original loan L and the interest imposed in a month, i.e.,

L + L


i
12 × 100



L


1 +
i
12 × 100



Lr

where we have set r = 1 + i/(12 × 100) to simplify our notation. You will pay E at the end of first month and hence will owe an amount L1 = LrE. At the end of second month you will owe

L1 + L1


i
12 × 100



L1


1 +
i
12 × 100



L1r.

After paying E, you owe

L2 = L1 r − E = (Lr −E)r − E = Lr2 − E(1+r).

Continuing this argument, we notice that at the end of n-th month, after paying E, you will owe

Ln = Lr n − E(1+r+r2+…+rn−1).

Let us set S = 1+r+r2+…+rn−1 and get a formula for S in terms of r and n. In fact, S is the sum of the first n terms of a geometric series. Every term in the sum is a r multiple of its predecessor. Note that

rS − S=r + r 2 + r3 + … + rn−1 + rn − (1+r+r2+…+rn−1) =  rn −1
S=
rn −1
r−1

and hence

Ln = L  rn − ES = L rn − E 


rn −1
r−1



.

If you want to finish your loan at the end of n-th month, you expect Ln =0. This gives that

E
L rn (r−1)
rn −1
 = 
L


1+
i
1200



n



 
 
i
1200



1 +
i
1200



n



 
 −1
.

This is the formula for the EMI that you pay for any kind of loan.

Those who have taken loan might have noticed that some part of EMI is deducted from principal and the remaining as interest. A natural question is, on any given month, how much from your EMI is deducted towards principal payment. Let us deduce this formula. Recall that at the end of first month the interest imposed is

L


i
12 × 100



L (r−1).

This interest imposed, at the end of first month, is deducted from your EMI E and the balance is used as payment towards principal. Therefore the amount that goes as payment towards your principal loan amount is EL(r−1) at the end of first month. Continuing this way, one notices that out of your k-th month EMI E, an amount of rk−1 {EL(r−1)} is deducted towards principal payment and the rest is towards interest. Notice that r>1 and hence your principal payment increases with each month. Also, as derived above, the principal loan you owe after k-th month EMI is rkLE(rk−1/r−1).


1
the unit of currency is not an issue

Friday, 16 November 2012

Something about nothing and everything

zinfty

The word finite in English is an adjective that means limited or having bounds or one that has an end. In mathematics, this adjective is used often, viz., finite sets, finite dimension etc. Further, to refer to things that are not finite we have, of course, an adjective which is the antonym to finite, called infinite. This adjective means limitless or boundless or endless. In Sanskrit and Tamil, the equivalent word for infinite is ananta. The word ant means end and anant means that which has no end. Both finite and infinite is a measure of size (large or small). Infinite is larger than all finite things.

The word infinity is a noun which refers to the state of being infinite. The perceptions of any organism on this planet, in normal circumstances, have only finite capabilities. For instance, we know today that human eye can respond to wavelengths between 390–750 nm, the audible range for human ear is between 20Hz–20kHz. Humans can distinguish six types of taste. Similarly, human smell and feel are all limited. This limitation is also true with our logical/rational thinking, usually considered the sixth sense. This attribute is, naturally, inherited to human inventions as well. For instance, computers cannot deal infinity. The number may be as large as 10500 today, but still finite! Telescopes can see farther in the sky today, still finite! Thus, in my opinion, understanding infinity may not be possible using logical/rational mind. Let me explain myself.

Consider the two words nothing and everything. It is clear that they are antonyms to each other. Thus, we need to define one of these words and the definition of the other follows naturally as its negation. But you see there is a philosophical problem in defining either of them consistently. Suppose, we say, nothing is that which is not a thing, then the fact that nothing has a definition means it is something and hence cannot be nothing. If, on the contrary, we try to define everything as that which includes or considers all the things, then does everything include itself? Also, by definition, it should include its antonym nothing. Is this the Russell’s paradox? Hold! Don’t throw things at me. I’m not yet done!

In mathematics, we do know that things that are so natural to us begins to fail in an infinite set-up. For instance, we know 1+2+3 = (1+2)+3 = 1+(2+3) = (1+3)+2 =6. Simply put, in whatever order you sum, you will get the same answer. This is called the distributive and commutative property of natural (or real) numbers. But this would fail in an infinite set-up. If not, we should have

0 =0 + 0 + 0 + …    (Infinite sum)
 =(1 −1) + (1−1) + (1−1) + …
 =1 +( −1 +1) +  (−1 +1) + (−1 + 1) + …    (rearranging the terms)
 =1 + 0 + 0 + 0 + …
 =1.

We have proved 0=1! Or have we? If you want to see more such counter-intuitive stuffs, do a course (or read a book) on basic real analysis, especially limits, sequences and series. Anyway, back to our argument of 0=1. You see the moment we carry forward notions from finiteness to infinity, we get absurd(?) things like 0=1. So, what went wrong? We had conveniently assumed, in first equality, 0 is same as sum of 0 infinite times. Was that right? May be not. If nothing has to yield nothing, however infinite, then modern Big Bang theory is a contradiction. Ancient Indian texts claim that everything comes from nothing and goes back to nothing. That is why the Indian symbol of nothing is a circle (0). Lord Shiva is considered to have no aadi (beginning) and no antam (end) and a mythological story depicts this where Lord Brahma and Vishnu go in search of His aadi and antam, respectively. One should view mythological stories in ancient Indian texts as fables for adults!1 One meaning of the word Shi Va is “that which is not or does not exist”, referring to nullity and at the same time by saying it has no beginning and no end they are referring to infinity. Thus, they seem to imply the duality of nullity and infinity. In mathematics, this duality is exhibited by the function f(x) = 1/x. Also, infinity is omnipresent. Mathematically, we know that the cardinality of [0,1] is same as the real line ℝ but [0,1] is properly contained in ℝ. Consider the analogy of ℝ being universe. I know, I know, I’m talking absurd. The power set of ℝ is bigger than ℝ. Just take the analogy. I’m just illustrating a property of infinity. You can work with the power set of ℝ or its power set or its. The property will remain true.

So, back to our analogy. If ℝ denotes the universe and every unit (or any length, just an example) interval [a, a+1] represents an organism in this universe. Then, every organism has a universe within itself. In other words, exploring universe is same as exploring within. Though you are part of the universe, you have the universe within you; something like [0,1] and ℝ. This is property of infinity. The set of even (or odd) integers is a proper subset of all integers but have the same cardinality. (I’m going to be torn apart for the contents of this paragraph!). On the other hand, it is the beauty of mathematics that discretises infinity using cardinal numbers.

Let’s ponder further on our argument that 0=1, assuming nothing yields nothing, however infinite. The second equality is not a problem, just writing each zero as sum of two terms. The third equality is rearrangement. Mathematically, this is where the main problem lies. If the rearrangement was true, then we could have also rearranged the term after second equality as

(1+1+ …) − (1 + 1 +  …) = ∞ − ∞,

difference of divergent series and that is not zero (why?). Of course, mathematics fixes such issues by studying absolutely convergent series, uniform convergence of functions, dominated convergence result of Lebesgue etc. But, in my opinion, these are just isolating particular cases where infinity behaves like finite. Axiom of Choice is nothing but imposing one property of finiteness to infinity. One of its many consequence, the Banach-Tarski paradox is again due to infinity. We conveniently call some guys, who don’t fit in our understanding, as bad guys (non-measurable sets) to come up with a consistent theory that fits with what we see as reality. Note that I said “what we see as reality”.

Yeah, yeah! I’m going to conclude. My assertion is that our attempt to study infinity using logic is like trying to study infinity using finiteness and this may be an impossible situation. So, what is the right way to handle infinity? Of course, I don’t know. Else I would be in the hall of Nobel laureates. But identifying wrong paths is the first step in right direction.


1
The ancient science took the form of religion for political reasons like, invasions and let us not digress there now.

Tuesday, 6 November 2012

To blog or not to blog?

Out of curiosity, I started this blog page in 2009 and also wrote two posts. Then, a few months later (six months, may be), I removed the posts and deleted the blog. No, not because I got hate comments! The reason was that I disliked the concept of talking to unknown faces.

 In Tamil there is an ancient expression ``idam, porul, eval arinthu sollu''. This means one has to speak minding
  1.  the place,
  2.  the content of speech and
  3.  the interest shown by your listener (by order or direction). 
None of the three parameters can be taken care of while blogging. The place where you speak is important because same speech could be right in one place and wrong in the other. As a popular Tamil orator, Arivoli, says: the hymns recited on the death-bed to help the dying person reach Vaikuntam cannot be a bad hymn. But reciting this hymn during marriage is considered inauspicious. So a good hymn is considered bad if it's recited in the wrong place. Of course, who cares about these nowadays. A mobile phone starts ringing with a happy tune when an entire house is in sorrow due to a loss!

As regards to content, any information has to be revealed/shared keeping in mind the maturity of the listener. Same information may have to be revealed in different ways to a matured adult and an immature child. Everything that we speak is, usually, interpreted the way listeners want to rather than what the speaker intends to convey.

Can one take care of these issues while writing a blog? Both a five year old child and fifty year old man may be reading my post and interpreting them differently. These compulsions made me withdraw from blog world. Do I sound paranoid? Let it be so.

Now, why did I come back? Because I think I have matured from what I was three or four years back. I now have an answer to my dilemma. I decided to stick to topics whose wrong interpretation may not cause a severe damage. Also, I have matured to understand that no one can learn anything without unlearning something.

To avoid putting pressure on myself, I shall just not care, like any other blogger on this planet, and go ahead and write about anything I wish or care about and leave the rest to my conscience, which I know will never cross certain threshold.