Mind and Matter

Foundations of Mathematics II

2010-03-26T16:48:00.010-05:00

Okay...the last post needs some clarification. I did not mean the notation '3' for the number '2+1' is unjustified. That is perfectly okay. We can have the symbol '4' for the number '3' or anything for that matter and the math would just be fine and consistent as it is now. So, denoting '3' for '2+1' is understandable and it is not the issue. The point was that how one can assume the existence of a set that satisfies the Peano's axioms? The set that satisfies the conditions of Peano's axioms could well be empty, isn't it? Thus natural numbers are not captured by those axioms rather the axioms hold for what we all know as natural numbers. That is we have to agree on such a basic set in order to move forward.

This points us to the fundamental question on the existence of numbers as basic as the natural numbers. Do they exist at all?

In my view, natural numbers are mathematical representation of the notion of duality that we all perceive, if not take it as a basic philosophical position. The mere fact that we see more than one thing is enough to justify the existence of numbers. In other words, numbers exist and they are real as much as you believe in reading this piece of writing.

Foundations of Mathematics I

2010-03-22T10:41:00.015-05:00

Everybody knows what are natural numbers. Or is it so? If I ask, is 2+1 a natural number, everybody (including me) would rush to say it is so - the reason being 3 is a natural number. I agree that 3 is a natural number, but why 2+1? Isn't it silly to question 2+1 = 3? If not, what is 2+1 after all?

The Italian mathematician Giuseppe Peano knew in 1900's that there would be such silly minds like me who will want to question even basic (and trivial) things. So he introduced a set of axioms that would define precisely what those numbers are, in an attempt to capture the notion of a natural number. Here are the axioms:

1. There is a natural number 0.
2. Every natural number a has a natural number successor, denoted by S(a). Intuitively, S(a) is a+1.
3. There is no natural number whose successor is 0.
4. Distinct natural numbers have distinct successors: if a ≠ b, then S(a) ≠ S(b).
5. If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers (This postulate ensures that the proof technique of mathematical induction is valid).

It will be very easy to see that the natural numbers satisfy these axioms. But do they define natural numbers or capture the idea of what natural numbers are? The answer is clearly no. I still haven't got the answer why 2+1 should be 3. None of these axioms say it is so. To take a closer look why it is so, consider an alien coming to the earth who don't have the number 3 in their world. He doesn't know what 3 is and in their system 2+1 equals 4. All they have is the set {1, 2, 4, 5,...}. Now, how does one explain our concept of natural numbers to him. Note that these axioms would still be valid in their system.

So Peano hasn't convinced me yet! If one says that there could be some other set of axioms that would precisely define what natural numbers are - then go please find that and let me know. I will give you million dollars, okay 10 dollars. All one can say is that any other set that satisfies these axioms will be isomorphic to the natural numbers. In other words, the word isomorphism here means, I don't know what natural numbers are, but can we all agree there is one so that if you have a different system of numbers then either agree to our agreed definition or go to hell!

Looks like natural numbers are not so natural at all! More on this later.

Why Mathematics Fails?

2010-03-20T16:14:00.007-05:00

Albert Einstein said "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality". Let us examine why it is so. Consider these two statements (or instances of propositional symbols):

P: I wake up late
Q: I go to office late

The deductive apparatus in formal logic consists of two things: A set of premises (called axioms) and the rules of inference. The rules of inference could be anything like Modus Ponens (MP) or Modus Tollens. MP basically says, the statement if P then Q taken together with P will imply Q. Which kinda looks true in our case: If I wake up late, then I go to office late and I wake up late would mean I go to office late. Among the basic ideas in formal logic is the idea that the deductive apparatus or syntactics should be kept aside from the semantics of the language. One may then naturally ask what is the relation between the syntactics and the semantics of the language. It is here the idea of axioms come in. In general, in mathematics, the main goal is to come up with the set of premises so that the syntactics is matched with the semantics. In other words, we keep the rule of inference fixed and come up with our axioms that deduces statements that are true or false. Although we can change the rules of inference as much as the axioms, mathematicians don't attempt that. If P then Q and P need not always make Q true. For instance, take

P: We men going to mars
Q: Saturn is full of chocolates

Now, if P then Q and P is true does not make Q true - for sure we know that right, although P is a real possibility (and not true now). Mathematicians ignore the general notions of truth and reduce them to validate their statements by building axioms rather than equally exploring the rules of inference. Thus the notion of truth or falsity in mathematics may not conform to the real world. Here is where temporal logic and relevance logic come to rescue!

Immortality Pill.

2010-02-20T20:36:00.008-06:00

During our usual lunch conversation, we came across a topic in which we were talking whether the medical science is in the right direction. I was in the position that even if it is in the right direction it is not in the right interests. My friend asked me what if the medical science can discover an immortality pill which can keep us physically alive eternally. The topic shifted direction when I told him that it is possible to be alive for a long time using some yogic methods - with total control of prana. And then I told him that there are yogis who live for more than 200 years in Himalayas and the right approach for the medial science to understand immortality is not through fragmentary and divisive studies but through a unified understanding of what consciousness means.

On retrospect, I think that we don't need to go too far on the idea of immortality and the magic pill the medical field can possibly discover. Even if the medical community discovers a pill that could make us immortal, how does it help from the social conditioning that we find ourselves today? I can still shoot a person at will, can't I?

Vernon Howard

2010-02-19T21:37:00.002-06:00

Vernon Howard - It was only last week that I came across this wonderful person. He is one of the enlightened masters and his philosophy looks similar to Jiddu Krishnamurthy's. Here are some YouTube links in which he speaks.

Right Ideas for a Right Day

Way to True Command (3 part series)

Inner Power for Today (3 part series)

Sense of self.

2010-02-16T11:12:00.003-06:00

Ultimately everything depends on from where we derive our sense of self. It could be from our culture, religion, political affiliation, nationality, our dress, new hair style and all things external to us - the world of forms. Or it could be from deep within, from our very existence or the feeling of I. In this case, our sense of self is not actually derived from an external entity but it is the feeling of being or completely present. Even the thought can be considered as something external since its away from the feeling of being. Ego is nothing but the derivation of the sense of self from anything away from the complete presence. Being in thought is being in ego. There is nothing wrong to be in ego (we will be forced to) but we will just miss the wonderful feeling of being present, of being alive.

Big Bang in Your Closet?

2010-02-14T16:04:00.010-06:00

For long I knew that all the problems of the mind cannot be solved in the level of the mind. For instance, the question of death. Mind as I has envision is like a field which is not algebraically closed just like how the real numbers are insufficient to solve the equation x*x + 1 = 0. However, I did think that the mind is capable of solving almost all analytical questions. By analytical questions I mean the ones that are put forth by science. But yesterday, I had this incredible revelation about a question in science that the mind won't be able to arrive at a solution.

Modern science tells us that there was Big Bang from which the visible universe has expanded from a primordial hot and dense singularity at some finite time in the past. Now we can ask a legitimate question within the domain of science on where was that singularity present - I mean the spatial coordinates of the dense particle. Once we pose this, we realize that the question depends inherently on the notion of distance. Now the concept of distance is a relative measure - we need at least two objects have a distance measure. When all the objects are condensed as a single small package, where does the concept of distance come in? In other words, the Big Bang singularity can never be traced to a specific location in space. Assuming such a thing happened, all we can say is that the singularity could have been anywhere in space. Who knows, it could have been in your closet too.

The Eternal Warrior

2010-02-02T13:45:00.002-06:00

Charma Slokam.

2010-01-31T21:36:00.006-06:00

sarva-dharman parityajya
mam ekam saranam vraja
aham tvam sarva-papebhyo
moksayisyami ma sucah

Yesterday I was in a lecture by a ISKCON Maharaj and he spoke about this famous Charma Slokam in the Gita (since I requested). I found that the interpretation is different from the one in the Sri Sampradayam. In the ISKCON phylosophy, the word dharman refers to duties of all kinds including any religious obligations one may have. However, in Sri Sampradayam, the word dharman (in Pravritti maargam) refers to all prayaschitta-dharmas, or all obligations taken to compensate for our wrong doings, but the essential dharmas should not be renounced. Bagu (short for Bagavaan) takes care of all the prayaschittas, thus paving the way for bhakti to emerge. In light of Eckhart Tolle, the Sri Sampradhayam interpretation seems right, at least when following the Pravritti maargam.

On Growing Old.

2010-01-19T20:01:00.007-06:00

I think I heard this idea somewhere as almost everything for me now looks familiar. Anyway, I will come to the point. The reason for our physical growth is because of the flexibility of our body to cope up with the mind's energy. As soon as our bodies become stiff and inflexible, aging starts, as our mind still works with the same energy. Probably that's why yoga-aasanas helps in slowing the aging process!

Sunset House.

2010-01-06T20:54:00.002-06:00

My new painting - (paint by numbers, of course). The original doesn't have the sun like bright spot in the middle - that's the flash glare. It looked nice and I left it alone.

Truth is a Pathless Land - Really?

2009-12-31T20:02:00.027-06:00

Happy New Year Folks...

For sometime I have been reading Jiddu Krishnamurthy 's works (and watching his videos) and have been terribly impressed by his philosophy and his views. He is an amazing person. To summarize, the core of his teachings are as follows:

Truth, which can be thought of as something beyond time and cannot be measured (or divisible), is a pathless entity. It cannot be reached by any intellectual analysis or introspection. Nor it can be attained by any self-effort. No religion, dogma, spiritual practices or rituals (and equally political, cultural, social alliances) can lead to this entity. To make the matters worse, these personal affiliations will put oneself into more misery by "conditioning" the mind. The human mind has already been conditioned through several centuries and we are adding "more stuff" to it by involving in such false security measures - religion, politics etcetera. In a nutshell, only when the mind is "empty" of its contents, can the real Truth be known ('known' as in observed and not as in understood).

I love this philosophy and I think one has to leave aside his/her prejudices to see that this is true. And I respect this great man for his wisdom who never claimed himself to be a spiritual teacher, guru or even a philosopher. He was nobody in his opinion and he dismissed the Order of the Star under his leadership.

However, to all the people who are influenced by JK and value his teaching, I have a question. Is JK conditioned by his thoughts on Truth? This I think is a mind boggling self-loop on what could be one of the greatest theories on Truth (remember, mind has to empty its contents to verify the theory's validity!). If JK is conditioned by his own thoughts, there is nothing to argue - his teaching can only be appreciated like a poem or a painting and that he himself has fallen into the trap (on his defense, he never claimed anything and could have equally dismissed his teaching). On the other hand, if he is not conditioned by his thoughts, there cannot be any core principles in his teaching. Having a fixed set of ideas - devoid of any vitality is a conditioning by itself! In fact, we do see some central message in his teaching. Is he conditioned by his ideas? How do we break from this self-loop that his teachings create?

In my opinion, there is no way one can be out of this loop and JK fell in his own trap (unknowingly, of course). This is not surprising, after all, we all know (and it is his thesis) that the process of thinking is limiting and fragmentary. His teachings being a product of thought has to contradict itself at some point. In that case, what is the validity of the statement that Truth is a pathless land? - which I think is true. The only way to keep this thesis intact is to allow some pointers in the direction of that (Truth) land. The land may not be accessible through any path, however there could be some sign posts saying the land is there. Well, what are those sign posts? - read Eckhart Tolle.

Melting Pot and Salad Bowl.

2009-08-21T12:02:00.005-05:00

Yesterday I was watching Charlie Rose interviewing Ramachandra Guha and Guha was explaining the main difference between the heterogeneity found in India and in the US. He was saying US is like a melting pot where several cultures mix and try to find a unique identity among themselves whereas India is like a salad bowl where all the cultures co-exist without losing any of their identity. Thought this was a cool analogy.

Google Book Search Result.

2009-05-13T19:15:00.001-05:00

This was interesting.

My Recent Painting

2009-05-04T08:14:00.003-05:00

Of course, the usual paint-by-numbers.

Furniture Making Workshop

2009-04-25T19:43:00.005-05:00

The 10 week course got over and I managed to complete this.

Photostich from SD630

2009-04-05T10:41:00.005-05:00

A view from my office window. I am not happy with the shots though (the real view looks beautiful as opposed to this).

Digital Macro on Canon SD630

2009-03-28T21:05:00.001-05:00

Sketch of the Day

2008-07-30T17:14:00.004-05:00

Eckhart Tolle

2008-07-14T13:39:00.005-05:00

I am really amazed by the clarity of Eckhart Tolle speeches and writings (for speeches see YouTube videos searching for Eckhart Tolle). It has a transforming effect. His ideas are not new - for that matter no ideas can be new. But, it is the best synthesis of Eastern philosophies (especially Buddhism and Vedanta) I have come across. It's practical - very, very practical. Long live Eckhart Tolle.

Survey of the Quantum World - 2

2008-04-25T18:25:00.011-05:00

In the previous post, I had written about three experiments, namely spraying bullets (several at a time) through the double slit and recording the pattern on the wall, keeping a candle instead of the machine gun and recording the pattern and in the third experiment, shooting a bunch of electrons (in this case we may need a detector instead of the wall) and observing the effects on the detector. Not surprisingly, in the case of the first experiment (with the machine gun bullets), we get the following pattern in which the bullet marks are around the places opposite to the two slits. i.e., we get two distributions on the wall exactly around the places corresponding to the slits.

But, in the case of the second and the third experiment, we get a pattern which is not discrete as above but shades of dark and bright bands like the following figure shows.

One can imagine this kind of dark and bright stripes correspond to the different levels of water that reach the shore on the beach. This is a clear indication that the light and the electrons do not come in discrete packets like the machine gun bullets (otherwise we would be getting two distributions) but they have some sort of a continuous ("wave-like") behavior. Let us now leave the candle light for some time and just concentrate on the electrons. We know that electrons are particles. Numerous experiments show that electron behaves like particles. But then, how do we have dark and bright bands instead of two dark bands like the one we got for the bullets? How do we test the difference between bullets behavior and electrons behavior? That's very simple. Check the behavior bullet by bullet and electron by electron. Its like coding a difficult software. When you don't understand the logic, just break it into modules. Execute it line by line and see where you are going wrong. So, we do the first experiment again this time by just shooting one bullet at a time. Again no surprises here - same pattern with two distributions correspond to the slits. But what about electrons? What will happen if we shoot them one by one and see what pattern it creates when the time progresses? To our surprise, we don't have any surprise - we still get the same dark and white bands, same as what we got when we shoot a bunch of electrons. How can that be possible? This is the beginning of one of the beautiful areas of physics - the Quantum Physics which gives life and meaning to the Hilbert space theory, which otherwise would have lived only to satisfy the whims of the so called "pure mathematicians".

To be continued.

Survey of the Quantum World - 1

2008-04-18T09:26:00.018-05:00

In my previous blog, I had talked about several things, namely, Copenhagen interpretation of the Quantum Theory, Schrödinger's cat paradox, the famous Einstein quote "God does not play dice" and the collapse of the wave function. Frankly speaking, until recently this Einstein's quote was over and above my head although I kept saying the quote several times to myself just because it sounded "high funda" (I know it had something to do with the probability but nothing more!). Similarly, the Schrödinger's cat paradox was no paradox to me - just some strange words beyond the scope of my understanding. By this time, I guess I don't need to say anything about my ignorance of the wave function collapse or the Copenhagen interpretation. But when I managed to sneak into the world of Quantum Mechanics (QM), some of the things were clear as a crystal - especially the Einstein quote and our cat. Of course, there are several things that are still hazy and they are bound to be. Even the great practitioner of QM, Richard Feynman had said "No body really understands Quantum Mechanics". I will discuss later in what spirit he had said that but in the meantime, in the series of blogs to continue, I want to discuss QM. The idea behind writing these series of blogs are two fold:

1. To share the joy and excitement that I have had so far had in my little understanding of this amazing world.

2. To travel more closer to this world and have a 'aerial look' of the much nicer things that are abound.

I was careful when I said 'aerial look'. I will not land into this world to look at the landscapes and sunrises. That job is for the experts. They are specifically for the people who go by the Feynman words "just shut up and calculate" and I am positive that these people will have a nicer (and a closer) look of the beauties that escape the aerial survey. Having said that, by aerial survey, I don't mean the philosophical speculation of the QM alone. Although any thought out QM discussion is bound to contain quite an amount of philosophy (I am tempted to say QM itself is a philosophy, but let me not go into that right now), I don't want to ignore the beautiful white clouds of mathematics (especially, the infinite dimensional Hilbert space theory) that envelope the rich quantum world. So, without any more ado let me begin my journey. Oh and by the way, by QM I mean the traditional way (or Copenhagen interpretation) as it is typically understood.

At the heart of QM lies this famous experiment called the double-slit experiment. It is one of the most amazing and a remarkable experiment in the history of physics. In September 2002, it was voted as the "most beautiful experiment" by readers of the Physics World. It is indeed. Quite fairly one can say that this one experiment shaped the modern theory of Quantum Mechanics and still keeps baffling physicists. The philosophical implications of this experiment are abound. Feynman had remarked that this single experiment contains everything that one needs to know about Quantum Mechanics. I will briefly describe the experiment here and give the statistical interpretation (or postulates) of Quantum Mechanics completely based on this. This is the crux of the Copenhagen interpretation that I had mentioned in this blog and the Schrödinger's cat paradox is the outcome of this interpretation.

The Double-Slit Experiment

The origin of this experiment dates back to 1801 when Thomas Young wanted to study the behavior of the light and determine whether the light is a wave or a particle. The experiment is quite simple. There is a machine gun which can spray either bullets or electrons, a barrier with two slits and a wall (or a detector) which can record the place where the bullets or electron hits the wall. Something like the one shown in the figure. We now conduct three experiments.

Experiment 1: We spray the bullets from the gun and record the pattern on the wall.

Experiment 2: Instead of the gun, we keep a candle and record the pattern on the wall.

Experiment 3: We spray electrons instead of bullets and record the pattern on the wall.

To be continued...

Collapse of the Wave Function.

2008-04-17T12:03:00.007-05:00

I was contemplating about the collapse of the Schrödinger's wave function and trying to see what it could really mean while I happen to engage in a conversation with a friend of mine. He is a physicist and he asked me two questions.

1. Suppose I have an upright glass containing water and I tilt the glass to a 90 degree so that the glass is horizontal and the water is now on the floor. Why is that we don't expect the water to get back to the glass when leaving the glass (horizontal to the floor) and the water as such? By the way, I am not holding the glass and it rests on the floor so no gravity business.

2. Why is that all the air molecules around my location move out to somewhere else so that I die by not able to breathe in any oxygen?

His point was that these things don't happen just because the probability of these things happening is very small. In other words, nature by itself is probabilistic. So does that mean the Copenhagen interpretation of the Quantum Mechanics and the collapse of the wave function is justified? How does it then answer the Schrödinger's cat paradox and the realist views of scientists such as Einstein, who famously declared "God does not play dice".

Philosophiæ Doctor

2008-04-12T20:41:00.021-05:00

Finally the day (03/31/08) ended with my committee members and my advisor congratulating me for successfully defending my dissertation (don't know if they had any time to go through my work!). Oh by the way, it was only recently that I found out that a PhD. work is called "dissertation" in the US and a "thesis" is relatively smaller work (like the Masters' work) while the opposite is true in Europe and in India. There is a funny anecdote: When I wanted a recommendation letter from my prof in India, he gave me a open reco which had some words like "his dissertation" (referring to my Masters' work). When I asked him if that should be a "thesis", he told me "no, a thesis is for a much larger and detailed work, as in a PhD". I am wondering why this hasn't caused any major confusion so far. Anyway, coming to my defense, not too many questions, but some pointers which were not central to my focus of research. I would say its not a great work, but a decent piece of manuscript that can be potentially used in the simulation of N-body problems to reduce the computation time to a significant level. My advisor says it has good prospects but that can happen only if I have the time, patience and energy to publish the ideas. I believe earth shattering thesis like the ones produced by de Broglie and Lebesgue will have a fair share of extreme intelligence along with the researching abilities (not to undermine the hard work that went along). I have learnt/unlearnt several things along the way - the key point being: the more one studies, the more one understands academics is abyss. Standing on the shoulders of the giants would first require the ability to stand!

P.S: A faculty in a renowned institution told his students in the class "why do you guys think a PhD is a wonderful thing, all my colleagues have them, after all!"

Hello World !

2008-04-11T11:45:00.003-05:00

Just wanted to see if there is any response...

Do You Agree ?

2006-12-21T00:15:00.000-06:00

"If an artisan were certain that he would dream every night for fully twelve hours that he was a king, I believe that he would be just as happy as a king who dreams every night for twelve hours that he is an artisan."

Good and Evil.

2005-11-02T20:50:00.000-06:00

An angry serpent shows its evil fangs just to defend her offspring. A terrorist bombs the railroads to get back his old freedom from the opposing force. Good and evil are not opposites. They are not even mere set of relatives. They are the siblings of the same mother. They spring from a single source. No, even this undermines their association. The emotionless evil is instigated by the incandescence of the inherent good. A refracted ray from the glittering good on the mirror of 'me and mine' marks the emergence of evil. It is the gaiety in the glaring goodness which garbs the attire of affliction to emanate as evil. Evil can never be the source of something good, but it is always the other way around. It is the spark from this goodness when passing through the crystal of ego, results in the spectrum of virtues and vices, which we call 'nature'. It is from this good that God originates, devoid of a 'o' - representing the origin of evil.

Revolutionary Evolution.

2005-10-31T14:16:00.000-06:00

Okay, this gives enough fodder for a post. I really don't understand why it is so difficult to accept the theory of evolution. My understanding fails if someone says evolution is against the theory of intelligent design. I don't see how they combat with each other. Why should natural selection be considered as an unguided process ? For me, evolution is just metamorphosis of things from one state to another. While it is true that natural selection doesn't pre-suppose any intelligent engineer, it doesn't rule that out as well. The silence of Buddha misconstrued as a nod for atheism marked the alienation of Buddhism in India. Similarly, holding evolution and intelligent design in opposition will only divorce science from its honest inquiry. In my opinion, any consistent doctrine should involve both - the theory of evolution and the theory of intelligent design.

Acceptance.

2005-10-29T00:00:00.000-05:00

After sometime, life becomes just acceptance. No, not submission or defiance but acceptance. Submission involves active surrender with passive resistance and defiance is active resistance and passive surrender. But mere acceptance is just taking what comes by, no agitation whatsoever. Soon enough, you realize there is no need for one to be in conformity or holding challenges in the toils of life. Survival of the fittest seems to be the law of the jungle. So does the pangs of the subjugated. You become indifferent to defense and deference. Acceptance is more like the process of thinking than the conclusions of thought. Conclusions of thought generally results in submission, defiance or confusion, but the process of thinking just keeps one going. Acceptance is similar to the silence of death after a chaotic bloody war or a calm before a storm, living neither in the fear of the past nor in the insecurity of the future, but in the present, nay, it is verily the present.

Untying the Knot.

2005-10-27T00:10:00.000-05:00

Renunciation is not difficult. It is quite possible to stay away from sense pursuits. But, its just another form of indulgence. We indulge in things which makes us stay away from the sense objects. A man of renunciation is a man yet to find the tranquility of the mind. The silence of the graveyard is not to be confused with the equipoise and stillness of the mid-ocean. So, any conscious effort just binds us to some thing else. The Gita says, "Only when the mind gets rid of all its affiliations, the vision of the Supreme dawns on human consciousness". How do we reconcile between these two thoughts ? How do we untie the knot in the mind which fixes on something even with the slightest conscious effort ? Buddha questions this to his disciple. When the disciple is not able to find answers, Buddha says "Simple, you can untie the knot when you know how the knot is made on the first place." To observe how the mind operates is by itself the beginning of the mind's de-conditioning. To me, distractions are as important as concentration during meditation attempts. When we know how distractions are caused, we know how to get rid of them as well. That's when we come in face to face with the stark naked Reality.

Familiarity.

2005-10-25T10:49:00.000-05:00

I like the word 'familiarity'. If there is any word that gives meaning to the process of understanding, I think it is this word. I believe there is no new knowledge anywhere in this world. Its all just re-discovery and becoming familiar with things. Things or ideas exists all by itself. Its just the manifestation or "becoming familiar with" we call as gaining or imparting knowledge. I am sure, if one doesn't understand things, even if he mechanically goes through it for some time, he'll be an expert in the topic. No wonder we gave utmost importance to srutis. Swami Vivekananda boldly proclaims "Religion is the manifestation of divinity already in man" and "Education is the manifestation of perfection already in man." How true! To be spiritual and intellectual we just need to get used to our already existing divinity and perfection. Von Newmann was amazingly right when he said "Young man, in mathematics you don't understand things, you just get used to them." I think this applies not only to math but also to life, in general.

(1/Epsilon) and Epsilon.

2005-10-22T11:17:00.000-05:00

Yesterday I saw the documentary "N is a Number" by George Csicsery. Its a nice piece of work about Paul Erdos (pronounced as "air-dish"), one of the greatest mathematicians of the century. He surpassed any other mathematician by the sheer volume of papers-about 1600 with the co-authors all around the globe. To me, Erdos presents himself as a truly spiritual person. Just that his religion is Mathematics and his God-Combinatorics. He thought about mathematical problems 24-7. The very characteristics of a great spiritual person at the height of realization was present in him. He was not able to stand physical pleasure and didn't marry. His possessions where just two half-empty suitcases. He never stayed in a place, roamed all around the world giving lectures and gave whatever money he got as an endowment. Totally dispassionate life. More than these things, a childlike behavior found in spiritual maturity was found in him (he was very pleased with the emergence of velcro footwear as he found buckling his sandals difficult). He just loved kids and called them 'epsilons' ('epsilon' usually denotes a very small quantity in math). He passed away in 1996 with the great amount of collected works yet to be amassed. He never strived to be a mathematical god, he was just one by himself.

Blogs and Identity.

2005-10-19T10:47:00.000-05:00

I think blogs represent the personality of the blogger at least quite a bit. A maxim goes "One can fool some people for sometime, he can even fool all the people for sometime, but never all the people all the time". So, I am wondering how far blogs speak about people's nature. It would be interesting if some study is taken on this. On a different note though, given that a blog is a great medium for self-expression and has a potential to influence readers thoughts, it is quite likely that a blogger with good repute can be approached by a management for publicity. If such politics enters the blogsphere, the credibility of the blogs would go for a ride. Hope such things doesn't happen.

(Mis) Management ?

2005-10-12T15:47:00.000-05:00

You guessed it right ! It's about IIPM. The mis-management is setting up the blog world ablaze (those who are not aware of the issue, please type in "IIPM" in Technorati and read posts). The crux of the matter is, an IBM employee Gaurav writes about the credibility of IIPM in his blog and gets a legal notice and protests from them. I am really curious to know how this whole thing would turn around. Since it has been noticed by the MSM (Main Stream Media), it would be interesting to watch whether the issue fizzles out or hits loud with a bang. Whatever happens it is definitely going to test the integrity of the bloggers. A.Sandip, IIPM's all-India dean says "We are not concerned about the blog, and in no way has the written matter on the blog affected us, but we are going to take legal action against the blogger for defamation. The person is identifiable. It is a legal notice against the person and not the blog." If any of you guys find some logic in these words please let me know !

I Exist.

2005-10-10T22:30:00.000-05:00

In the figure, there are numbers from 1 through 6 horizontally and vertically. For each horizontal number is it possible to choose one and only number in the vertical column (and hence the selection can be represented by a dot), so that the line joining the corresponding dots do not intersect the diagonal ? For instance, one such selection, (1,5), (2,6), (3,4), (4,6), (5,6) and (6,6) is shown here and they are represented by magenta dots, but the line joining them meets the diagonal in (6,6). We can observe that no such selection is possible (which doesn't meet the diagonal). The point where the line joining the dots meets the diagonal is called the 'fixed point' [here it is (6,6)]. This interesting result is called the Fixed Point Theorem in mathematics. The dots in the diagonal have the same values like (1,1), (2,2) etc.

Now, consider the fact that we always think 'I exist' in every moment. Whatever state we are in, it can always be said 'I exist doing so and so'. For example, 'I am angry' means 'I exist in an angry state', 'I am writing this post' can be rephrased as 'I exist writing this post' etc. Also, when we say 'I exist' we mean 'I' and 'existence' are different ('I' is the subject and 'exist' is an object). Assuming that 'I' and 'existence' are functions of time, we can plot every moment of our living like a graph where the horizontal axis would mean the existence in time and the vertical axis would be the notion of 'I' in time. This is shown in the figure. Just as how the lines joining the dots had a fixed point, our continuous stream of living should have a fixed point where the notion of 'I' coincides with our 'existence'. Probably, that fixed point is the samadhi state (state of undifferentiated being) experienced by yogis.

P.S : It may be a valid question to ask what happens in deep sleep. Probably, the notion of 'I' just remains (identical with the y-axis). We can never express the state that "I am in deep sleep now".

Contempt ?

2005-10-01T11:19:00.000-05:00

This is really interesting. Karunanidhi says yoga keeps him going. Yoga is a philosophical school established by Sage Patanjali in third century BC. Sage Patanjali was a hard core theist. It is interesting to note that people are furtive enough to use the best aspects of Hinduism while showing great aversion to the system that preserved them.

God-A Convenience Theory ?

2005-09-28T21:08:00.000-05:00

Update : See comments below.

Somedays back, I had been asked a question about the existence of God. The question was whether the existence of God is a convenience theory and I thought about it now. Does existence of God needs logical demonstration ? I would say it needs logical demonstration for people who are bent solely on logic. By demonstration, I don't mean a proof. Proving the existence of God may be next to impossible. But, it can be logically argued that God exists. Consider this simple argument which treats God from a dualistic point of view (not necessarily my view and Vedantic view).

1. The only thing that is permanent is change.
2. Change cannot be perceived without something which is changeless.
3. Our thoughts, feelings and emotions change.
4. The changeless entity which perceives our changing thoughts, feelings and emotions is God.
5. Therefore, God exists.

Of course, one can counter the above arguments by saying all is subject to change and so what I have written is neither true/not true. Other than this, is there any logical flaw in this argument ? Any of you guys who has studied philosophy of religion can you please tell me what type of argument is this (like ontological, cosmological etc.) ?

Update : Points (1) and (2) needs little clarification. (1) and (2) is based on the emprical world whereas (3) and (4) is for the personal world. This means, (1) and (2) is true with respect to our senses while (3) and (4) is validated because of (1) and (2). Also, I have changed the order in (1) and (2) although it doesn't matter much.

Scientific Inaccuracy

2005-09-25T09:30:00.000-05:00

Hurricane Rita merged into the eternal clouds after registering not even a single drop of rain in our place and I began to wonder the scientific accuracy we have got here. Few hours before its arrival, there wasn't an iota of doubt that its going to bring heavy rains to our neighborhood. The weather station announced the chance of rain to be 100 %. It makes me only laugh about the predictions. I think accurate prediction (even in hours) is almost impossible with the present computational resources. First, I am not sure whether the simulation models are exact. Second, even assuming an exact simulation model, the precision we have is insufficient for the cause. Even if we are able to approximate the real time by millions of bits, there are more than millions left to cause inaccuracy. Probably quantum computers may be able to make a significant contribution but exact prediction may be next to impossible from what I see. A funny scene from 'Swadesh' comes to my mind. Shahrukh Khan, a NASA engineer tries to explain his position to a layman in an Indian village that he predicts rains with satellites. The layman immediately looks at the skies and says there is no rain in another 2 days and asks our hero why he is complicating things.

In my opinion, its better if meteorologists start to search for a rational basis for astrology than rejecting it as a mere pseudo-science. When we have such a huge scientific inaccuracy, it makes me wonder whether philosophical accuracy (accurate depiction of two similar philosophical ideas through a common language, like a common representation of Vedanta and Quantum Mechanics) is possible at all in near future.

P.S : I am not discrediting the acheivements of meteorology. Its just that I think it has a very long way to go from here.

It Rains

2005-09-22T20:40:00.000-05:00

Now that Rita is going to torment us like anything, let me pose this question.

When we say 'it rains', what does 'it' refer in the phrase ? Rita rains may be ?!!

Brahman and Objective Reality.

2005-09-18T18:00:00.000-05:00

Following an interesting debate that took place in Anand's post recently, I thought I'll post a view on Brahman and objective reality. Rene Descartes, the father of modern philosophy talks about three types of realities-eminent reality, formal reality and objective reality. The definitions are as follows:

(1) Objective reality: x is objectively real if x is the object of an idea (this comes close to what in modern terminology would be subjective reality - existing 'in' the mind).
(2) Formal reality: x is formally real if x has actual existence in the world represented by an idea (i.e. exists 'outside' the mind).
(3) Eminent reality: x is eminently real if x exists without limitation or imperfection, as contained within a higher level of being (from the standpoint of the world in which we exist, that which is eminently real exists potentially, not actually).

For example, when we think of the sun, we think of it as a two dimensional disc. This idea of the sun which exists in the mind is the objective reality. The actual existence of the sun as a three dimensional sphere with hot gases is the formal reality. The eminent reality is something like God which exists potentially but cannot be comprehended actually (by all).

Also, Descartes gives a nice hierarchy of these realities. He says, eminent reality contains formal reality and objective reality, formal reality contains objective reality (three dimesional sun exists which appears as two dimensional disc) and the objective reality can stand alone. I think this is analogous to three types of realities mentioned in Visishtadvaita. The philosophy of Visishtadvaita is based on Tatvatriya as mentioned in this post. Tatvatriya is the classification of realities into insentient (Achit), sentient (Chit) and Brahman which corresponds to objective reality, formal reality and eminent reality respectively. The objective reality (idea of insentient things) exists in the mind, the formal reality (like consciousness or sentient things) is outside the mind and it can comprehend objective reality and thus resulting in a kind of inclusion. Similarly, Brahman or eminent reality includes formal reality and objective reality. Descartes attributes God to eminent reality. A note should be made here that Visistadvaita doesn't suppose any kind of inclusion between achit and chit, where as Descartes assumes such inclusion.

Anyway, coming back to the question of Brahman and objective reality, we can conclude something from the above observation. This view reconciles the debaters views that Brahman is not the objective reality and Brahman is the basis for the objective reality.

The Meta-Complex Brahman

2005-09-12T21:42:00.000-05:00

A tribute to Sri Adi Sankara

Man discovered counting. As a crude mortal he was naturally satisfied with the elementary counting 1,2,3,...etc. He was living with just food and water, the gross annamaya kosha (anna means food so layer (made) of food) till one day he felt the intense need for energy (prana) to hunt and solve equations like x+2 = 0. These equations were not solvable in the domain of natural numbers (this needs x = -2). Thus, pranamaya kosha (layer of energy) gave raise to negative numbers. Did he feel happy ? No. His wandering higher mind (manas) wanted to solve larger system of equations like 2*x = 3. The integers were insufficient for this cause (this requires x = 3/2). Hence, manonmaya kosha (layer of mind) gave raise to fractions. Okay, is that all what he wanted ? How can he ever be satisfied with fractions which doesn't solve the equation x*x = 2 ? His intelligence (vignanam) urged even more. The vignanamaya kosha (layer of intellect) found the real numbers like sqrt(2) satisfied his cause. How can a man ever be happy ? Isn't this a basic contradiction in life ?There are still some complex equations like x*x = -1, which has no solution in the domain of real numbers. He confuses ananda for happiness while it denotes bliss. He wants happiness at any cost and invents complex numbers. The anandamaya kosha (layer of bliss) drives him to complex numbers. He is now not in the plane of ordinary mortals. He is full of bliss and boldly proclaims this is the largest system of numbers possible. He stops there saying there is no higher system than complex numbers**, no higher state than ananda, no higher being than Brahman. He proclaims Brahman is complex or Brahman is ananda, beyond the state of intellect, beyond the system of real numbers. But, he is not able to conceive the idea put forth by a genius of the highest state, a genius of a much higher evolution, a genius who can put all mathematicians to shame and a genius who can claim Brahman is more than complex, Brahman is more than ananda*^ and Brahman is beyond the highest being. That is Sri Adi Sankara.

** There is a proof for the algebraic closure of complex numbers which is essentially saying there is no system higher than complex numbers (in terms of solving polynomial equations) similar to saying Brahman is ananda.
*^ Sri Adi Sankara says anandamaya kosha is also a kind of maya and hence not really real - courtesy: Radhakrishnan Sarvepalli, Indian Philosophy, Pg 167-168 .

Tatvatriya Vs States of Existence.

2005-09-10T20:35:00.000-05:00

Often I used to wonder about dreams. To many people, dreams are unreal. One of the reasons that I could think of for the unreal nature of the dreams, is that, sleep constitutes 1/3 rd of the day which is less compared to the waking state. I think if we spend equal time in sleeping as we do in the waking state, then dream state could be as real as being awake. Anyway, one of the nice thing about Vedanta philosophy is that it takes into consideration of all the three states of existence, namely, the waking state, the dream state and the dreamless state. There is another state of existence called turya or the super-conscious state which we'll leave it for now.
The philosophy of Visishtadvita identifies three principles known as Tatvatriya. They are achit, chit and Brahman. Insentinent things are classified as achit, sentinent things are chit and Brahman is the one who possesses both achit and chit as an organic whole. These principles are similar to the three states of existence. In our waking state we are in contact with the insentinent things or achit, in the dream state we know our sentinent being alone which is chit and in the dreamless sleep we are one with Brahman**.

** This idea of being with Brahman in deep sleep is an Advitic view which I have used freely.

Reductio ad Absurdum

2005-09-06T00:02:00.000-05:00

UPDATE : For people who want to skip the theorem can look at an easier illustration of Reductio ad Absurdum in the bottom of the post.

Reductio ad Absurdum or Proof by Contradiction is one of the usual techniques followed by mathematicians to prove a theorem in mathematics. To give you a flair of this technique, let me illusrtate the celebrated classical theorem of Euclid.

Theorem : There are infinitely many prime numbers.
Proof : Assume the contrary that there are only finite number of prime numbers and let k denote the largest prime number. Thus, the prime numbers can be enumerated by 2,3,5,7,...,k. Consider the product P = 2x3x5x7x...xk. Now, P is clearly divisible by the prime numbers 2, 3, 5, etc. upto k. And P+1 is not divisible by any of these numbers and so P+1 is a prime number greater than k. This is a contradiction to our hypothesis that k is the largest prime number. Hence, there cannot be any largest prime number and so prime numbers are infinite.

This method of proof is called Reductio ad Absurdum and believe me it really works. The key idea is to assume a contradiction and we come up with another contradiction to deduce a truth. But, the other way doesn't work. If we assume the truth of a statement we can never arrive at a contradiction. We can come up with another truth but that doesn't prove anything.

Now, let's come to philosophy. There are several philosophies that are founded based on Vedas and Upanishads. Now, if we assume the authority of the Vedas/Upanishads and propound a philosophy based on our authority, then we cannot claim Vedas/Upanishads are consistent just because our philosophy is in sync with them. We have already assumed their truth. On the other hand, if we start with a philosophy and prove the validity/consistency of the Vedas/Upanishads then we can claim the true nature of these scriptures. Do any of the philosophies Advita, Visishtadvaita, Dvaita follow Reductio ad Absurdum ?

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Another illustration of Reductio ad Absurdum

Theorem : Zero is the smallest non-nagative number.
Proof : Suppose not. Let x be the smallest non-negative number other than zero. Then x/2 is a smaller number than x which is also non-negative. This is a contradiction to the hypothesis that x is the smallest non-negative number. Therefore, zero is the smallest non-negative number.

Greatest Mystery

2005-09-01T12:11:00.000-05:00

I kinda very much like the conversation that took place between Yudishtra (the eldest pandava prince) and a Yaksha (wizard).

Yaksha : What is the most interesting mystery in this world ?
Yudhistra : Even though there are nine doors (or passages) in a human body through which God who is the form of Aatman can easily pass out any time, he chooses to wait till the appropriate time comes. This is an interesting mystery to me.
Yaksha : What is the greatest mystery in this world ?
Yudhistra : Although many people are dying everyday, most of us don't realize or think about our death. Isn't the hope that we'll be living, the greatest mystery of all things ?

Can we constantly think about death and still be happy ? I think affirmative.

Krishna Jayanti Post

2005-08-26T12:57:00.000-05:00

Dedicated to அமலனாதிபிரான் (Amalanadipiran - One who is devoid of any impurity).

I was thinking of posting something dedicated to Sri Krishna on Janmashtami. Sri Krishna was born on 'Rohini' star which coincides with the star of Thiruppanazhwar (one of the 12 great saints of Sri Vaishnava tradition). So, I thought it would be fitting to write about Amalanadipiran - a beautiful composition of Thiruppanazhwar.

The charm of Amalanadipiran goes beyond description. This set of 10 short poems (paasurams) describe the beauty of Sri Ranganatha in Sri Rangam. Starting from His Lotus Feet to His Beautiful Eyes, Thiruppanaahzwar sings in praise of His Looks. When it comes to the last poem, he merges with the Lord. The following is the last poem and its translation.

கொண்டல் வண்ணனை கோவலனாய் வெண்ணெய்
உண்ட வாயன் * என் உள்ளம் கவர்தானை *
அண்டர் கோன் அணி அரங்கன் என் அமுதினைக்
கண்ட கண்கள் * மற்றொன்றினைக் காணாவே ||

Translation : He who has the form of a moisture laden cloud, who has the mouth with which He ate butter when born as a yadava, who stole my heart, who is the Lord of the nithyasooris, who rests in Srirangam which is an ornament to this world, who is nectar to me, the eyes which saw Him, will not see anything else.

The appeal of this poem is amplified if one goes through the rest of the poems where he always says "My mind", "My heart" where ever he is engrossed in the beauty of the Lord (this is not despicable, when one is enlightened). But, when it comes to the last line where he completely loses himself to Him, he says "kaNdakaNkaL", instead of "En kaNkal" or "My eyes". There is no longer distinction. Also, Amalanadipiran is unique among the works of azhwars. In the works of every other azhwar, when the compositions are coming to an end, they say so and so praises Him. But, Thiruppanazhwar forgets even that which ascribes full glory to Him. There are other interesting features in Amalanadipiran. The initial three words of first three poems are amalanAthipirAn, uvantha, manthipAy respectively. When we take their first letters we get AUM - the pranava. Similarly, there are continuous stanzas which begins as pAramAya, thundaveNpiRaiyan, kaiynAr and if we take the first three letters they give raise to Pathukai - or the Divine Feet. More than these things, the grandeur of the composition lies in places where he himself finds the Lord beyond description and thus finding it difficult to replace by words, he says "aiyO" (seyyavAy aiyO). In short, Amalanadipiran symbolizes the epitome of Bakthi that mankind can ever comprehend.

Science and Religion

2005-08-23T19:55:00.000-05:00

The theory of relativity acknowledges the existence of the relative world. Although, I haven't studied the theory, I am sure it involves highly complicated mathematical equations. These equations in math rely heavily on axioms. And axioms are self-evident truths that needs no explanation/proof. Thus, science comes up with the notion of the relative world by assuming an absolute (self-evident truths or axioms). Is it not the same case with religion which assumes an Absolute and tries to come up with explanations to the manifold of relative phenomena? Isn't God a self-evident truth ?

Being Impersonal

2005-08-20T12:35:00.000-05:00

The following is a verse from Nasadiya Sukta - the 129th sukta of the 10th mandala in Rig Veda.

i`yaṁ visṛ̍ṣṭi`ryata̍'āba`bhūva̍ yadi̍ vā da`dhe yadi̍ vā` na |
yo'a`syādhya̍kṣaḥ para`me vyo̍ma`ntso'a`ṁga ve̍da` yadi̍ vā` na veda̍ || 7 ||

The meaning is as follows :

Whence all creation had its origin,
he, whether he fashioned it or whether he did not,
he, who surveys it all from highest heaven,
he knows - or may be even he does not know.

I could appreciate something really interesting here. The verse shows the height of being impersonal and having a scientific frame of mind and not yielding to the cherished opinion that "It is God who made everything". Who ever came up with this verse, seems to be a theist. But, he never claims his God created the universe and things. He says God just surveys everything and he also may or may not 'know' who created the universe. Such an unbiased attitude. Great indeed are the seers.

Aham Brahmaasmi.

2005-08-18T11:17:00.000-05:00

I am inspired to take this post following Anand's critical review of "Aham Brahmaasmi" or "I am Brahman". He beautifully draws interesting parallels in mathematics for the mahavakya. I concur with his views and I want to put it in my way of understanding.
The statement "I am Brahman" is different from the statements like "I am a man" or "I am a woman". These are clear statements because all of us know what "man" or "woman" means. But the very word "Brahman" for most of us mean different things and the statement "I am Brahman" cannot stand alone without substantiating what "Brahman" means. So the truth of the statement lies in saying

S1 : I am Brahman
S2 : Brahman is so and so...

This is analogous to the following scenario. Consider statement T1,

T1 : The following statement is true.

This is a true statement, in the sense that it has a truth value to it. Now, I introduce another statement.

T2 : The preceding statement is false.

Consider T1 and T2. Now, what happens to the truth of the statement T1 ? Is T1 true or false ? We can see T1 is no longer true. Thus, the validity of the statement T1 is not just restricted to T1 but lies on what statements that follow T1. If we say "I am Brahman" and say "Brahman is so and so" then the validity of the statement "I am Brahman" lies on the attributes of Brahman like "Brahman is so and so" rather than the word "Brahman" itself. Or in other words, "Brahman" is inseparable from his attributes for a consistent valid statement "Aham Brahmaasmi".

Knowledge - Transcendental and Intellectual

2005-08-16T12:00:00.000-05:00

This is a passage I liked from the book "Human Being in Depth : A Scientific Approach to Religion" by Swami Ranganadananda. This is a *highly* recommended book for people interested in the synthesis of science and religion.

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In the Mundaka Upanishad [I.1,3], we find this question put by a earnest student to a great teacher : "What is that reality, O blessed One, by knowing which we can know all that there is in this manifested universe ?"
Is there such a unique reality by knowing that we can understand all the manifestations of nature, internal as well as external ? Is there a unity behind this diversity, a One behind the many ? To this the teacher gave a very significant reply:

There are two kinds of knowledge to be acquired by man; so say the knowers of Brahman. One is called transcendental knowledge (para vidya), the other is knowledge of intellectual nature (apara vidya).

Both must be cultivated. Of these, intellectual knowledge, says the teachers of the Upanishads, consists of the sacred Vedas, phonetics, the code of rituals, grammar, etymology, prosody, and astronomy. In fact it comprises what we today would call the "entire gamut of positivistic knowledge," including the secondhand knowledge of the experience of religion contained in the sacred books of all religions. Here in this Upanishad we have a scientific mind of the highest order-impersonal and detached. There is no desire to put forth a cherished opinion. Truth alone is the motive power, even if that truth goes against one's attachments and aversions. The teacher says that even the Vedas, the sacred book of the Indian people, belong to the category of ordinary knowledge. Who would dare say that one's own sacred books are ordinary, except one who is of a detached and scientific frame of mind, who is in search of truth and not dogma; who has nothing to hide, no opinion to uphold, no prejudice to defend, who just wants to know the truth and is prepared to sacrifice everything else in the bargain.
No religion except that derived from the Upanishadic tradition has practiced this bold detachment. The follower of every other religion, if asked what is ordinary knowledge, would unheisatingly reply: All the sacred books of all religions except my own. But this teacher of the Upanishads has the detachment and boldness, proceeding from the love of truth, to say that even the Vedas, held in such veneration by all, are secondary; all the sacred books and all the positive sciences and the arts are of a lower nature**.

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** - Here lower nature doesn't mean science is inferior. It just means there is a higher knowledge which is transcendental. I could really appreciate our attitude towards science. We have given equal emphasis to science as well.

Numbers and Vedanta.

2005-08-13T12:00:00.000-05:00

A set of numbers, in general, can be classified into three types in mathematics. They are as follows :

Finite Sets : These are finite in the sense that they have a starting value and an ending value. For example, {1,2,3,4,5,6}.
Countably Infinite Sets : These are infinite but can be counted, like for instance {1,2,3,4,...,}.
Uncountably Infinite Sets : These are also infinite but they cannot be counted. For example, numbers between 1 and 2 (They include fractions and non-fractions like sqrt(2)).

A simple observation behind these numbers leads to the following philosophies.

Philosophy of Finite Sets : Finite set of numbers can never become infinite.

Philosophy of Countably Infinite Sets : Countably infinite set of numbers are infinite but each number maintains their 'individuality' in the sense that there is a clear notion of separability between two numbers (i.e, two numbers are not that close, there are a unit distance away in the above example).

Philosophy of Uncountably Infinite Sets : Uncountably infinite set of numbers are infinite but two numbers can be as close as possible. There is no clear notion of 'individuality' between two numbers (for example, sqrt(2) and 1.41421356, the difference is almost zero).

Let's now see three major philosophies of Vedanta in crude terms.

Dvaita : Jiva-atmans can never become Parama-atman.
Visishtadvaita : Jiva-atmans are also Parama-atman but they maintain their individuality.
Advaita : Jiva-atmans and Parama-atman are one and the same (there is no notion of individuality).

Assuming Parama-atman to be infinite and Jiva-atmans to be set of numbers.

Dvaita : Jiva-atmans are like finite set of numbers and they can never be infinite, which is Parama-atman - Philosphy of Finite.
Visishtadvaita : Jiva-atmans are like countable set of numbers, they are infinite (Parama-atman) but they maintain their individuality - Philosophy of Countably Infinite.

Advita : Jiva-atmans are like uncountable set of numbers, they are infinite (Parama-atman) but there is no clear notion of individuality - Philosophy of Uncountably Infinite.

Oh, how similar !

On the Existence of God.

2005-08-11T10:15:00.000-05:00

Recently, I had been asked an explanation for the existence of God. It is my humble opinion that the existence of God can neither be proved nor disproved ~~at least according to Hinduism~~ . This is because of the limitaions of logic. God ~~according to Hinduism~~ is beyond space and time and whatever we come across in logic is constrained to space and time. Having said this, is it at least possible to "prove" (prove formally) that the existence of God can neither be disproved nor proved ? I definitely think yes-it can be proved. I think the crux of the proof would involve Godel's theorem. The essence of Godel's theorem is that truth is different from provability. Something which can be proved (or disproved) does not mean it is true (or false) universally. If at all anything is proved, it just means that it is true within the "context" and the truth cannot be held universal. Thus, if the existence of God is proved (or disproved) then it means that God is true (or false) within the context of space and time. But, God ~~according to Hinduism~~ is beyond space and time. So, I think it can be formally deduced (this needs rigorous research) that the proof of existence of God cannot exist (if God is taken to be beyond space and time ~~according to Hindu theology~~ ).

Sankhya and Sambhar.

2005-08-09T12:00:00.000-05:00

Sankhya is a philosophical school of thought in India that was established by Sage Kapila around 10th century BC. This philosophy talks about the origin of the universe. Initially, there were two things : primordial matter (prakriti) and pure consciousness (purusha). When they came in contact with each other, this universe began its evolution. It is important to note that purusha doesn't act on prakriti. The mere presence of purusha made prakriti to evolve just like the mere looks of a lover makes the loved one happy. This concept is also akin to the process of making sambhar. When we bring the pan with vegetables, water, spices etc. close to the flame, we have sambhar. Here too, the flame (consciousness) doesn't act on the stuffs (matter) in the pan. The very presence of the flame 'cooks' the contents. Although, the process of making sambhar was started with a good intention of making it edible, the appreciation highly depends on the person who is tasting it !

Scriptures Vs Science

2005-08-06T21:35:00.000-05:00

We have been seeing some analogies between Vedanta and Science (Godel, Darwin etc). The last post showed some parallel between Newton and Upanishad and I saw couple of comments along these lines...
"If it was said in our scriptures, why were the scriptures not popular? In general, I wonder why the content of our scriptures (scientific or spiritual) have not reached the public compared to Newton's law. Is there a basic flaw in our system? If so, why don't we work towards rectifying it? Maybe we had vision, but to complement it, we need implementation too. "

I thought these are good questions and they need a separate post.

Before I take a guess into the answers, I wish to point out that these analogies were drawn just to emphasize the existing ideas in our scriptures. It was never intended to downplay modern science or anything. I really think Newton, Godel or Darwin require honor and praise as they had put forth a great deal of effort in understanding the external world. They are sages (aka wise men) and their intellectual ability surpasses normal man. So, we should never discredit them.

First Question: If it was said in our scriptures, why were the scriptures not popular?
Scriptures are different from scientific theories in the sense that they deal more with internal science than dealing with nature (there are some exceptions to this like the sankhya philosophy that discusses nature as well). The simple reason why scriptures are not so popular as science, is that, each person perceives the object of perception in his own way although the object of perception is the same. For instance, you may look at tree and notice its leaves and I may look at its bark. While both of us are looking at the tree (the object of perception being the same), what you perceive is different from what I perceive. Scriptures essentially deal more with the subject than with the object and this may differ from person to person. Generalizing the scriptures to satisfy the masses is a very difficult task although I wouldn't say it is impossible.

Second Question: In general, I wonder why the content of our scriptures (scientific or spiritual) have not reached the public compared to Newton's law.
I think modern day science is a product of the big boom that happened during the period of renaissance when several people like Newton, Leibniz, Galileo, Tycho Brahe, Kepler etc. suddenly started working on the nature. As a result things got heated up and continues till this day. While scriptures are time tested for a long time, its a fact that things are slow if they are not sudden. Anything that has a beginning in time will also have its end and the modern day science will have its culmination. Whether the culmination would mark the merging of scientific truths with the scriptures, time will only answer.

Third Question: Is there a basic flaw in our system? If so, why don't we work towards rectifying it? Maybe we had vision, but to complement it, we need implementation too.
I don't know whether its a flaw. Irrespective of whether its a flaw or not, its always good to understand and relate things. It doesn't matter whether the scriptures are true or false as long as it appeals to logic and scientific truths. In that case, I guess we should accept it. Yes, I agree implementation is good to convince the masses, but it is our internal satisfaction that's going to last.

Newton and Prasna Upanishad.

2005-08-04T22:55:00.000-05:00

In Sanskrit, "Prana" means the life-energy that goes up and "Apana" is the one that pulls down. The following is a verse from Prasna Upanishad (to be precise Prasnopanishad 3.8) which comes under Atharva Veda.

"Adityo ha vai bahyah prana udayatyesa hyenam caksusam pranamanugrhnanah Prthivyam ya devata saisa purusasyapanamavastabhyantara yadakasah sa samano, vayur-vyanah"

The following is the translation.

"The sun rises as the external Prana, for it assists the Prana in the eye. The deity that exists in the earth, is there in support of man's Apana (down-breathing). The ether between (sun and earth) is the Samana (on-breathing), the air is Vyana (back-breathing)."

Sri Adi Sankara in his commetry for the above upanishad says "the earth has apana-sakti". Further he declares, just as an object thrown up is attracted by the earth, so prana that goes up is pulled down by apana. Did we already know gravitation ? Probably, we didn't bother about it too much.

On Bashing Gandhi.

2005-08-03T22:45:00.000-05:00

Often I have heard people bashing Gandhi. For them, I would want to tell only one thing : Try to fast for just two days and tell me how it feels like. While Gandhi as a politician is debatable, I am sure Gandhi as a human is a noble soul. The aphorisms of Patanjali (The Yoga Sutras) talk about Yamas or physical disciplines out of which Truthfulness and Bramacharya being very important virtues. Gandhi in that sense is a true yogi (and I guess that's one of the reasons for earning the title "Mahatma"). Also, it is not so easy to think of God when one is dying. Let's first try to be humans than being politicians.

Infinite and Finite.

2005-08-01T10:13:00.000-05:00

God in Man...ha...ha...you must be kidding. How real is the infinite occupying a finite position ? I would question, is it not as real as the idea of infinite contained in the finite letters i,n,f,i,n,i,t,e ? Is it not as real as the world of letters contained in the world of words? Is it not as real as the world of words occupying the world of reasons ? Or is it not as real as the world of reasons seated on the realm of thoughts ? Let's see an analogy.

Consider the infinite to be an endless straight line. Also, consider a semi-circle in some place on the line. Now, I draw tangents from the semi-circle to meet on the line, so that, each point on the semi-circle correspond to a point on the line as in the below figure.

Thus, each point on the semi-circle is just a representation of a point on the straight line. Okay, what does the point 'P' represent ? Doesn't it represent the infinity ? Lo ! We have captured the infinity. Just as how 'P' is the representation of the infinity in the straight line, God is just a representation of the highest ideal in Man. Some of you, who have studied some higher math, may identify this with the idea of streographic projection/one-point compactification. The one-point compactification (or as in the above figure, just as how we captured infinity by identifying the endless straight line with a compact semi-circle) leads us to an interesting parallel that if we make all our thoughts into a single point (thought) we can know what the infinite God is all about.

Trinity and the Yogas.

2005-07-27T10:17:00.000-05:00

In Vedanta philosophy (uttara mimamsa), Sri Adi Sankara, Sri Ramanuja and Sri Madhvacharya can be identified with the philosophies of Advita (pure monoism), Visishtadvita (qualified-monoism) and Dwaita (dualism) resp. In broad terms, the philosopher's approach to the Truth can be stated as follows :

Adi Sankara : Knowing the Supreme by overcoming the ignorance (or the veil of maya), cause I am HE.
Ramanuja : Loving the Supreme, cause I am a part of HIS existence, just like how body is a part of our existence and we all love our bodies.
Madvacharya : Serving the Supreme, cause we are related to HIM just like how a servant is related to his master.

Vedanta also talks about three ways we can know the Truth. They are Jnana Yoga, Bakthi Yoga and Karma Yoga. Broadly, these ways can be stated as follows :

Jnana Yoga : Way to the Truth through constant discrimination.
Bakthi Yoga : Way to the Truth through constant Love.
Karma Yoga : Way to the Truth through constant selfless action.

We can now see how these yogas closely correspond with the approach of the philosophers (discrimination, love and selfless action correspond with the acts of knowing, loving and serving respectively). We can also find these pattern in day to day life. Madhvas closely stick on to rituals, Vishnavites to Bakthi (adhering intensely to divya prabandam) and Shivites focus on learning Vedas. Since all these approaches lead to the Truth as Vedanta says, theoretically there shoudn't be any notion of superiority at least among these creeds. But, I hardly meet such balanced people.

Kids and Crazy Equations.

2005-07-25T10:20:00.000-05:00

Imagine a school kid who is made to learn multiplication tables by heart, imagine a scientist analyzing crazy equations to come up with something. In both these acts, the subjects are undergoing the process of gaining knowledge - the former uses only the mind and the latter uses the intellect as well. Can we assign the superiority of the scientist (who uses both intellect and the mind) over the kid (which uses the mind) ? I would on the other hand say, learning multiplication tables, by heart, greatly helps in doing equations. Superiority of the intelligence over the mind cannot be argued in general. Similarly, in any religion, there are rituals and philosophy behind the religion. It will greatly help a student (in life) to understand the philosophy, if he performs the rituals, just as learning tables helps us to do the equations. Here too, the notion of superiority cannot be assigned to a person who is questioning some aspects in religion and a person performing a ritual. Well, the student can be bright enough to understand the existence of a calculator and avoid learning tables by heart. Its the same case with religion, the calculator is just invisible !

Quote for the Day

2005-07-24T12:55:00.000-05:00

Death - An award posthumously offered by God irrespective of qualifications.

P.S : Do tell me if you have seen this before, I am thinking this is original.

The Dual Logic.

2005-07-22T22:43:00.000-05:00

I hope the last post didn't drive away fellow bloggers whom I managed to maintain for a while :-). I always believe things can be better described and I am just learning to do that. Anyways, I guess in the last post the illogic/ambiguity behind the Escher's art would have been easy to spot out. As far as the duality in logic is concerned, consider the following two statements instead of the one on the sets.

Statement 1 : The following statement if true.
Statement 2 : The preceeding statement is false.

Now are both statements true (or false) at the same time ? There is a clear sense ambiguty here similar to Escher's painting (Statement 1 is like water coming down and the Statement 2 is like water going up-both cannot happen at the same time). As far as the third analogy (Sri Krishna from Gita) is concerened, it rests on the fact that, on the surface everyone operates from the plane of Ego.

Sets and Sri Krishna

2005-07-20T10:59:00.000-05:00

Claimer : This post is a fusion of art, logic and Vedanta. You may want to call this - a confusion :-). So, take your own time in going through.

Some days back, I had written rave reviews about Godel, Escher, Bach: The Eternal Golden Braid. The author of this book Hofstatder claims that even though the book explicitly deals with meta-mathematics, art and music, the major theme is about discovering the self, or the soul. I agree with the author that the book is about soul searching. It is my feeling that the book would have been more complete with the philosophy of Vedanta incorporated. But that would have made it even more specialized and highly technical. It may also be that, the author is not aware of this profound philosophy. Anyway, what I am about to write is the result of some of the things that I was able to appreciate in GEB along the lines of Vedanta.

Let's first consider the following M.C.Escher's painting.

It should be no doubt that the painting is beautiful although it is highly illogical (water defying gravity ?!). There is a clear sense of ambiguity in the painting. A cyclic waterfall is absurd without external energy. The presence of water cycle without external energy is the cause of ambiguity here. Let's see how does this relate to the last post.

I hope some of you might have given a thought about it. (If not, I would strongly encourage you to give it a shot before proceeding). The last post asks "Is the set of all sets, a set ?". At first, it would seem like "it is a set" since it is the "set of all sets", but on the other hand we'll see that "it is not a set" because "a set of things is not that particular thing any more" as shown in some examples in the post and "thing" can be replaced by a set. This ambiguity is due to the cyclic nature of our logic that we many times think, if something is true then the negation of that is also true. If 'A' exists then 'Not A' also exists. I'll call this as duality in logic. Thus, the presence of duality in logic without any singularity is the cause of ambiguity. Again, how does this relate to the painting ? Notice that the duality is a cycle, we think something as true and after some time as non-true, again after some time as true,...so it is a cycle of assigning truth which causes ambiguity. Okay let's see what Sri Krishna says.

Everyone in this world does something or the other at any given point of time. It can be classified into two things : thinking and acting. Here I am considering sleeping as an activity too. Thus, we are constantly engaged in the cycle of thoughts and actions. In the Escher's painting the presence of water cycle can be attributed to our vision, in logic the presence of duality can be assigned to the existence of our intellect. Similarly, the presence of the cycle of thoughts-actions can be attributed to the existence of our Ego (remember, even in deep sleep, our Ego is constantly working whereas the intellectual faculty remains dormant, and that's why we say we had a good sleep). Thus, the cycle of thought and actions (caused by Ego) without realizing the Self is the cause of ambiguity in life.

So, what is the point ?

Realize the beauty in the painting by raising above the ambiguity (attributed to vision) caused by the water cycle.

Realize the beauty in the logic by raising above the ambiguity (attributed to Intellect) caused by the duality in logic.

Realize the beauty in the life by raising above the ambiguity (attributed to Ego) caused by the cycle of thoughts-actions.

Set of All Sets.

2005-07-19T19:46:00.000-05:00

We would have studied "A Set is a collection of well defined objects". This is one of the primitive definitions in high school mathematics. Thus, we can speak of "set of cats", "set of books" etc. Once we define a set we can notice that, a set of objects is not that object anymore. For instance, a set of cats is no more a cat, a set of books is no more a book. Thus, a set of things is no more that particular thing.

Is the set of all sets, a set ? Why ? Why Not ?

Darwin and Vedanta.

2005-07-18T13:08:00.000-05:00

As many people know, Hinduism and Science are not really opposed. More than that, they go together. I think this is one of the major reasons why we were indifferent to Science, unlike the church upheavals that happened during the period of the Renaissance (and it is still happening). While "The origin of species" by Darwin is receiving several controversies from the west even now, we hardly raise an issue on this theory. This is because of the following reason.
The Upanishads speak of five layers that things are made up of, and the difference in the objective world is just the difference in the degree of manifestation of these layers. These layers are called koshas and they are anna-maya kosha, prana-maya kosha, manon-maya kosha, vijnana-maya kosha and ananda-maya kosha. These sheaths correspond with domains of matter, energy, mind, intellect and bliss. And the evolution proceeds from annamaya kosha to anandamaya kosha. These koshas can also be considered as the levels of consciousness present in the things. For instance, gross matter (like atom) is annamaya kosha by itself, plant life has both annamaya kosha and pranamaya kosha (matter and energy), animal's have annamaya kosha, pranamaya kosha and manonmaya kosha (matter, energy and mind), man's consciousness is made up of all that of animals plus the intellect (vijnanamaya kosha) and saints are blessed with anandamaya kosha as well. Our ancient seers (read scientists) unlike modern day scientists were able to comprehend the wide spectrum of nature. Present day Science is an analysis through division while Vedanta is an analysis through unification. Its high time we realize that we already posses the Grand Oneness Doctrine (GOD), which Science is eagerly pursuing now in the name of grand unification theory.
Coming back to Darwin, rephrasing in the language of Upanishads, he proposed the evolution from manonmaya kosha to vijnanamaya kosha (in other words, from monkeys to man). Did he study Vedanta before ? May be, may be not.

A Note on Conjecture 2

2005-07-17T10:52:00.000-05:00

After going through the comments on conjecture 2, I believe I should do a little clarification here. First, I should define what is an idea. An idea is a subset in the world of thoughts. An idea can be an emotion or just a bunch of plain thoughts. Conjecture 2 states that "Ideas are limits of languages" in the mathematical sense, which essentially means, given any idea we can express it as close as possible using words. I don't mean any idea can be made felt by words - no, that can never be done and I am not saying that, but they can be made felt as close as possible using words taken from languages. There is more optimism in the conjecture.

To be even more precise, languages form an incomplete subspace whose closure is the power set of the space of thoughts.

Agree to Disagree ?

2005-07-16T12:33:00.000-05:00

Why do we agree/disagree ?

Agreement/disagreement happens during mutual exchange of "generalization of experiences". By this I mean, if I say "all people are mad" by seeing a mad fellow, you are sure to disagree. But if I say "man is a mortal", you'll agree. In the first case, your generalization does not match with mine while in the second, it does. If we observe, one cannot generalize things based on his experience alone while he "has to" do it only based on his experience. There's no choice. This serious limitation of the human mind causes such agreement/disagreement.

Death - An End of Existence ?

2005-07-15T11:32:00.000-05:00

Now that several thoughts are going around in blogs about cultural identity, caste system, discrimination et al., I thought this would be the best time to raise this question. When we go through them, we see that they all point towards coming up with questions like "which is good ?", "what is the best thing to do ?". That is, the notion of relativeness is inherent among all the thoughts. Is this relativeness a part and parcel of our existence ? At any given moment, we never question our non-existence, which shows that there is something absolute and not relative. But, we 'artificially force' ourself to think that we cease to exist after we die. So, the notion of relativeness (or life-death) is just a superficial thought. Why we have to force such a contradiction that creates disharmony ? Why should we create a relative world for ourself that makes us to see only the difference ? Why such a negative outlook in life ? Why can't we think we'll live forever and its only that our body perishes ? Why can't we see everything from a higher perspective which will make us all better citizens ?

Research Life ...

2005-07-14T13:27:00.000-05:00

It is a real pity to observe my intelligence when I realized only today what my advisor had been talking about for more than 2 years ! Is this what is meant by being a student ?

Kasthuri's Conjecture 2

2005-07-14T10:29:00.000-05:00

Conjecture 2 : Ideas are limits of languages.

That's given an idea, there exists a sequence of words in language(s) that convey (or converge to) the idea. For example, the idea of love can be conveyed through words like kaadhal, pyar or love. What I mean is, there may be some ideas which cannot be represented by words alone, like for instance Aatman or the idea of being in a blissful state, but they can be described as close as possible.

Any proofs ?

Self Reference and Work.

2005-07-13T11:04:00.000-05:00

I am able to relate self-reference and work. First let me tell what I mean by self-reference. We can think of self-reference as a paradox. For example, the statement S : This statement is true, is a self-reference because if we think of S as false, it will be true and if we think of it as true then S is false. So, we are referring the notion of truth and falsehood by a cycle. Pictorially,

True ------> False
True <------ False

Similarly, everyone is doing some thing or other with regard to the outside world and in turn gets some feedback from the world. And this feedback again propels us to do some act and which in turn gets some feedback. So this cycle goes on and on. This is kind of self-reference. Pictorially,

Work ------> Feedback
Work <------ Feedback

This notion of self-reference caused several problems in the development of modern day mathematics where people were unable to determine the validity of some mathematical statements. Similarly, one can think of being caught in the 'vicious' cycle of "work and feedback". To 'break' this self-reference mathematicians finally decided to stick on to the fact that we cannot break the cycle per se, so why don't we just ignore the question. After all, the development of mathematics does not depend on the validity of some statements. Similarly, doesn't it seem logical to think, only if we think that we are working and receiving feedback, we'll never be able to come out from the working world. But, from a wider perspective the development in this world doesn't depend upon our working and receiving feedbacks. But, this shouldn't stop us from working and receiving feedbacks, just as we shouldn't stop asking the validity of the statement S. I guess this is similar to the principle given in Bhagavad Gita, which says don't worry about the work and feedback, just do it without expectations.

Good and Bad.

2005-07-12T10:20:00.000-05:00

I got a comment asking 'what is the whole point that I am trying to convey ?' So, I have made a little addendum.

I just thought of combining previous two posts (one about sporting a naamam/kudumi and other about a secular person saying "one should be good more than being religious"). If we thoroughly analyze the posts, they both convey the same thing - about an average person in our society evaluating the other based on the person's culture/religion etc. Before asking why should we do such an evaluation, lemme tell you one thing. I am first of all not speaking relatively at all. By this I mean, I believe in some being which is apart from good and bad, virtue and vice. From the stand point of such a being the act of murder and the act of generous help mean all the same. It doesn't have any double standards that we mortals have. And also I am addressing to only those people who believe in such a being (for example, a theist secularist). You may call such a being as God, Christ, Perfection Absolute, Goodness Absolute or anything one wants to call. From the stand point of such a thing, nothing is relative and everything will be clear in its own terms. So, if we try to view from a higher pedestal, there should be no relativeness and everything should appeal the same thing. Being religious will be same as being secular. Sporting a naamam will be same as not sporting a naamam. No differences. If everyone tries to view or atleast attempts to view then harmony naturally sets in. No differences will be there. I know this is an old wine, but it'll be good to drink as long as IT EXISTS.
So why should we evaluate a person on his outlooks/culture if we believe in such a being ? Doesn't this contradict with our own notion of having an absolute. I have observed that in the US people never evaluate a person by what he/she sports.

"Being Good"

2005-07-11T11:57:00.000-05:00

Most often I have come across people (secularists, in particular) saying that "ultimately one has to be good" and that's what that matters more than being religious or anything. While the notion of being good is such a highly relative term, how can one have this as a basis to defend secularism and cultural unity ?

Naamam Implies Cultural Identity?

2005-07-09T18:58:00.000-05:00

Srivatsan Murali had raised a good question and we have myriad of responses to that. I am wondering why wearing a naamam or having a kudumi should be seen from the point of view of cultural identity ? When I went through some of the comments I could see some disagreements coming only when it was considered as representing the culture. I mean, having naamam/kudumi might have some scientific reasons behind it. For example, naamam could give a good cooling effect to the forehead. More than that it could make the person to remember about God, when looking at the mirrors. So having a naamam could be for one's own self rather than representing a culture. Of course, when many people do this, it becomes a cultural identity but it need not have arisen for that purpose at all. So, if one wears naamam, it may be that he is trying to find his own identity rather than indentifying him with his culture/society. I don't see any logical disagreement with this view.

Kasthuri's Conjecture 1

2005-07-09T11:30:00.000-05:00

Conjecture 1 : Any sufficiently rich article which is consistent is bound to be incomplete.

A Little Overexcitement.

2005-07-09T10:17:00.000-05:00

Overexcited by my new found conjecture that "any sufficiently rich article which is consistent is bound to be incomplete" I thought about defining precisely what is meant by 'sufficiently rich' and 'exaggerations'. I'll also give some examples of what I exactly mean by the conjecture.

I had mentioned 'rich' can be associated with 'pondering'. By pondering I didn't mean just thinking but there is an element of feeling associated with it. While we have to exert some amount of thinking over any article, I didn't mean that. That is just 'mechanical' thinking. A rich article should have much more to it. It should evoke some kind of emotional response. It should generate some sort of feeling inside us. One may look here for a precise definition of a feeling. A typical example would be the songs of Barathiyar or lectures of Swami Vivekananda.

'Exaggerations' are meaningful sentences which are within the context of an article, but they need not correspond to the real world scenario or the message given by the article. For instance, the line "Yindhu thalai pambu enbaar, Appan aaru thalai endru magan sollivittal" in "Nenju Porukudhilaye" poem by Barathiyar.

So essentially what my conjecture states is that "For any article, that has some amount of emotional appeal to it, we cannot expect all the sentences to make sense in the real world". Here, an article can also be a poem. After all, this conjecture should not be a surprise because in most cases emotional appeal is caused by emotional words. Finally, I would like to call upon this article to stress my point. Here the message (as I construe) is, an educational system that places its emphasis more on logic and reasoning is not a complete education as it doesn't deal with all aspects of man. But, the sentence "The child is taken to school, and the first thing he learns is that his father is a fool, ..." is an exaggeration that it may or may not correspond with the real world.

Godel and Exaggerations.

2005-07-08T12:37:00.000-05:00

Well, I hit upon another beauty in Godel's theorem. I am wondering whether I should name this blog as 'Mind and Matter' or "The beauty of Godel's theorem'. Anyway, here it goes.

Any article (that people write) can be looked as a formal system having a content (sequence of sentences, each of which has meanings) and a message (the meaning of the whole article). Now, I define the following:

Definition 1 : An article is consistent as long as it conveys a message.
Definition 2 : An article is complete as long as each sentence in the article has some meaning associated with it.

The terms 'conveys a message' and 'meaning associated' can differ from person to person but they cannot be radically different. Naturally, we may raise the question whether, in an article we'll have both completeness and consistency. Godel's theorem asserts that - No, we cannot have both completeness and consistency for an article, provided the article is 'rich' enough. One may question, what is 'rich' enough. I would say 'rich enough' here is that the article makes someone 'ponder'.

One must understand that Godel's theorem doesn't say we cannot have any axiomatic system which is both complete and consistent. For example, a mathematical system that allows addition on two numbers and just that alone, but doesn't have any other rules, is both complete and consistent. Similarly, an article that has for instance, "Man is a human being. And humans have life", is both complete and consistent, because one doesn't 'ponder' much over these facts.

So, my claim is that "any sufficiently rich article which is consistent is bound to be incomplete". That is, we cannot search for meanings for all the statements if we are looking for a message in the article. In other words, a rich article should allow some exaggerations if we want to convey some message through the article. Well we can have a bland article without any exaggerations, for after all it is our choice to be either inconsistent or incomplete. But, it is a fact that the world prefers incompleteness over inconsistency !

Can all thoughts be 'reasoned' out ?

2005-07-08T10:37:00.000-05:00

I could draw another interesting parallel between Godel's theorem and the world of thoughts. Assuming there is a world of thoughts and a world of reasons, we can ask the question whether are they the same. In other words, can all thoughts be 'reasoned' out. Godel's theorem says no ! Not all thoughts can be reasons. And again if we think of some thoughts having reasons, we'll have inconsistency. So, it is better to leave the world of reason as incomplete than being inconsistent. We can then define those thoughts which doesn't have any reason as 'feelings'. One can also think of the world of thoughts as the real number system and the world of reasons as the set of rationals (which is incomplete). So, thoughts are allowed be irrational as well !

Death.

2005-07-07T13:18:00.000-05:00

Now, for some time the blast in London killing around 40 people is going to have its effects on media. While, there are going to be some noises and commotion, I feel a similar event in India (with similar toll) will be having comparatively lesser response from the Indian media. Is it because of the general attitude of the people towards the society or is it because of the running philosophy behind our culture that "death is just another state of existence" ? !

Exaggeration - A Weakness ?

2005-07-06T23:58:00.000-05:00

While seeing a comment for a post, I was really taken into a thought about 'exaggeration' and 'weakness'. Why should 'exaggeration' be thought of as 'weakness' ? Can't it be the greatest expression of the spirit that God has given us. Can't it be an inspiration that came out of a deep thought ? Can't it make our hair stand erect and we feel like moving a mountain ? Did anyone find the songs of Barathi 'weak' ? Did he not show the strength in exaggeration ?

Truth and Theoremhood.

2005-07-06T14:31:00.000-05:00

A line from GEB says, "... it seems that Godel has unearthed a hitherto unknown, but deeply significant, difference between human reasoning and mechanical reasoning. This mysterious discrepancy in the power of living and nonliving systems is mirrored in the discrepancy between the notion of truth, and that of theoremhood..." Does this mean "the more we study math more we are moving away from truth" ? Hmm...point to ponder.

Godel a Vedantin ?

2005-07-05T23:40:00.000-05:00

Russell Said: If there is a man, he is sure to have his life of Ego (that he is the doer of all actions).

God(el) Said: Stating from the highest viewpoint, a man has his Self (or Aatman) different from his Ego, and if he thinks of his Self as his Ego (i.e, if he confuses his Aatman with his Ego), he'll destruct himself.

In more logical terms,

Russell Said: If there is an axiomatic system (man), it (he) is sure to have statements (life) with contradictions (Ego).

God(el) Said: If you consider a higher system (highest system), the contradicting statements (life with Ego) are no more contradictions but mere statements (mere life or Aatman or Self) and if we say those statements are theorems in the axiomatic system (identifying Ego with Self, identifying contradiction with the non-contradiction), the system will go inconsistent (self-destruction).

Is Godel a Vedantin ?

Primes and Positivity.

2005-07-05T17:59:00.000-05:00

Here is a classic that I liked from GEB. In math, composite numbers and prime numbers can be defined as follows.

Definition 1: A number is composite if it is a product of two numbers other than 1 and itself.
Definition 2: A number is prime if it is not composite.
Thus prime numbers are defined in terms of negation. Philosophically, I can attribute a negation to something negative. Thus primes are negatively defined. They are in the 'background' of composites. Can primes be positively defined ? It turns out that primes can also be positively defined. And it appeals to our logic that anything that can be negatively defined can also be positively defined.

As another example consider the following Figure 1.

Figure 1

There is moon and the background space. When I look at the moon, I say it is positively defined (as it looks like moon is the major theme of Figure 1) and the background space gets characterized. But one can reverse the roles and say the space is positively defined (by looking at the space) and the moon gets negatively characterized. It perfectly appeals to our logic that we can look the picture either way. So, it seems logic (hence mathematics) and arts are 'natural systems' in which one can define things in one way and the 'background' gets characterized. But, life is not that simple. Even within the realm of logic not all the 'background' things gets characterized if we positively define something. In other words, there are things that can be positively defined but the negative space cannot be characterized in anyway. An analogy in arts is given by this picture.

Figure 2

Can you distinguish the background in the above figure?

I could appreciate much more beauty in this. I am sure the second pic will appeal special to almost all the people as the reason being 'foreground and the background are the same'. In other words, it appeals special to us because it is both positively and negatively defined 'equally'. While our logic naturally proposes that something that can be positively defined can also be negatively defined, this picture should not have something 'special' in it. Or to put it in a different way, if the special appeal comes because of our natural logical perception, Figure 1 should be even 'more special' because it defies our logic. So take your stand, which is more 'special' - Figure 1 or Figure 2 ? ! Doesn't this draw a line between thought and reason ?

Godel, Escher, Bach : an Eternal Golden Braid

2005-07-04T23:11:00.000-05:00

Oh my God, Oh my God ! How am I going to describe the joy, rather ecstasy that I am going through when reading Pulitzer Prize Winner Godel, Escher, Bach : an Eternal Golden Braid (a metaphorical fugue on minds and machines in the sprit of Lewis Carroll) by Douglas R. Hofstadter. I just finished about 75 pages (it has about 700 and odd pages) and each page is worth million thoughts ! Hofstadter sharply draws the line between the realm of reason and the realm of thought. It is definitely a MUST READ for people interested in Philosophy, Meta-mathematics, Logic, Art and Music. Never again you will admire Escher after reading this book. Never again you'll praise Godel for his works. You'll adore them. I don't know anything about Bach but this book doesn't require the reader to know much about him (so far). I particularly like the author's view of this book. It goes like this. "But finally I realized that to me, Godel, Escher and Bach were only shadows cast in different directions by some central solid essence. I tried to reconstruct the central object, and came up with this book". This book is worth reading several times and I am sure one can never exhaust it. As John.L.Casti, Nature puts it, "In some ways, GEB, is an entire humanistic education between the covers of a single book. So, for my next visit to a desert island, give me sun, sand, water and GEB, and I'll live happily ever after". Its kinda a bliss that I am going through when reading this book. Though I should warn you that it is a pretty heavy dose of Philosophy and Logic. Finally, I would like to put Martin Gardner views on this book.
"Every few decades an unknown author brings out a book of such depth, clarity, range, wit, beauty and originality that it is recognized at once as a major literary event. [This] is such a work."

Triumph of Truth Over Provability !

2005-07-03T17:16:00.000-05:00

Wow ! Just now understood something about the incompleteness theorem. I am really amazed by the keen insight of Kurt Godel. While Principia Mathematica (PM) by Alfred North Whitehead and Bertrand Russell needed the necessity of having self-contradicting statements for the existence of a mathematical system, Godel proved the unnecessity of having such statements. Let me try to explain. Consider the following statement, S.

S : This statement is false.

If asked whether 'S' is true or false, we'll get into a paradox. These types of paradox is fundamental in any logical system we build. Although, we feel 'S' is a genuine statement we are not able to ascertain any truth to it. So Whitehead and Russel admitted these statements as an integral part of a logical system. Here is where the genius of Godel lies. He made a 'coding machine' which would transform 'S' (or any other statement) to a statement in Number Theory. He then showed the impossibility of existence of a proof for the transformed statement within the framework of PM. Thus, 'S' is 'true' but we cannot prove or disprove it within the system of PM. Hence, the present mathematical system is 'incomplete' as it is not possible to verify the truth. Moreover, the greatness about Godel is that he showed that this 'incompleteness' is just not restricted to the system of PM but to any system that tries to achieve the aims of PM.

The genius Godel differentiated the genuine 'feeling' about 'S' from examining its truth. He showed the triumph of Truth over Provability. He showed the triumph of Self-awareness over Self-contradiction. He showed the triumph of God over Man !

Thoughts on Indian Culture-Part III

2005-07-02T15:45:00.000-05:00

The real appreciation of Indian culture lies in understanding and appreciating the science of Vedanta. The major difference between India and the West is that India has excelled in the study of mental science whereas the West is well versed in the study of material science. As long as the mind is over and above the matter, India will shine in its full glory and radiate the message of Vedanta. The West may have conquered the world in science and technology, but when it comes to studying internal science and questions about man's real nature, no other nation can beat the wisdom of the ancient Indian seers. Ultimately, it's all the difference between growing inwards or growing outwards, studying the mind versus studying the matter.

It requires a philosophical mind to understand the beauty of the Indian culture in its present state of affairs. Here one must take a careful note that something is beautiful doesn't mean it is necessarily superior and this is where I guess some people make a wrong identification. India, at present has several problems. There are two ways to find a solution. One way is to see where the problem lies by comparing it with the good systems and trying to find a solution. This is a good approach but it has a negative aspect of concentrating on the weakness. On the other hand, we can constantly focus on the strength of India to overcome the weakness, if at all it has. This is an extremely positive approach. And I am sure Vedanta is the greatest strength that India possesses.

One may have a thousand criticisms of India, one may find a thousand socio-economic problems in India, but as soon as one is able to appreciate the beauty in this ancient wisdom, every problem that is seen seems to vanish like the ignorance of mistaking a rope for a snake, in the dark !

Change !

2005-07-02T15:00:00.000-05:00

Recently, I was going through "India and USA : A comparative study : Part I" in this board. The post was interesting. The last line "Only if we recognize and accept what is wrong, we can bring about a change" caught my attention. I started thinking about change. Does change presuppose recognizing and accepting what is wrong ? I believe change can also come when one is constantly thinking about something. In this context, a positive change in India can also be accomplished if one concentrates on the strong points (of India) alone. This attitude would be entirely optimistic than recognizing and accepting what is wrong which involves comparison and criticism which in turn has some pessimistic stance as well.

Thoughts on Indian Culture-Part II

2005-07-02T11:15:00.000-05:00

The message of Vedanta cannot be explained more beautifully than Swami Vivekakanda. Vedanta is more of a science than a philosophy. It is a science with a goal. The goal in Vedanta is to identify unity among the things which we perceive and don't perceive. Modern day science has its foundations in the principles of mathematics. Science never accepts anything that is not mathematically modeled. And mathematics in turn has its foundations on axioms (like the axiom of choice). Thus any scientific theory indirectly presupposes these axioms and hence it is not devoid of any assumption. On the other hand, Vedanta doesn't assume anything other than one's own existence. Of course, one might claim that this is also an assumption, but I guess for a normal person (unless one is really drunk), the reality of one's existence will be more fundamental than a belief in an axiom ! So Vedanta deals with more real things than modern day science.

However, Vedanta has no conflicts with science and technology. Even if we are going to find an extraterrestrial or set up a base in some planet, Vedanta will appeal to mankind. As long as we believe in our existence, Vedanta is going to exist. Moreover, Vedanta is in perfect harmony with science and its findings. Present day views on quantum mechanics suggests this harmony. Erwin Schrodinger claimed to have been inspired by Vedanta in his discovery of quantum theory. The medical community is rediscovering the benefits of yoga and meditation. Vedanta is in perfect accordance with the law of conservation of energy.

Vedanta teaches a systematic way to observe the condition of the mind in order to discover facts about it. Just like any other science which validates the theories through experiments, the truths in Vedanta can be acknowledged through the experiments of Yoga. Yoga is more than a set of bodily exercises. The asanas (bodily exercises) are a set of experiments that are useful in identifying the goal in Vedanta. Just like the modern science which has branches that suits each individual's interests, Vedanta gives us the paths of Jnana Yoga, Raja Yoga, Bakthi Yoga and Karma Yoga to identify the goal-which is unity in life. Asanas come under Raja Yoga. More details on the yoga sutras of Patanjali can be found here. The beauty of Vedanta is that it is not a single person's vision, but it is time tested by the wisdom of hunded's of thousands of induviduals.

Aalavandhaan

2005-07-02T08:54:00.000-05:00

Watched the movie Aalavandhan yesterday. It was okay and could be watched once I guess. Kamal as usual has attempted to do something. I couldn't understand the funda behind making some cartoons !

Ethnomathematics

2005-07-01T13:56:00.000-05:00

Its almost ten years that I undertook some serious study in mathematics and it is only today I learnt about the existence of a branch of mathematics - ETHNOMATHEMATICS

Will this help to understand the Indian culture better ?

A Theist's Rationale

2005-06-30T16:00:00.000-05:00

I liked the following Pascal's wager for believing in the existence of God, based on Christian philosophy. Here +1,0,-1 denotes positive, neutral and negative effects respectively.

If I don't believe in God and God exists, I go to hell (-1)
If I don't believe in God and God doesn't exist, nothing happens (0)
If I believe in God and God doesn't exist, nothing happens (0)
If I believe in God and God exists, I go to heaven (+1)

From above we see that the gains we get in believing in God is always greater than the gains obtained by not believing in God. So, it is always better to believe in God ! Of course, this argument has strengths and weakness, but it appeals me from probabilistic point of view.

Thoughts on Indian Culture-Part I

2005-06-29T14:23:00.000-05:00

For quite some time I had been thinking about India and its culture. It is almost 5 years since I came to the US. Several times I have heard friends (especially Indians) speaking highly about Indian culture and that they would want to go back for good at some point. While most of the time it coincided with my thoughts, I realized they were all just nostalgic feelings rather than a 'genuine' feeling born out of little understanding of the Indian culture in limelight of the west. Nostalgia should never be confused with culture and heritage. Often it happens that people exaggerate things about India and this counteracts with the fellow Indians resulting in "hey no big deal about India". Of course, India and the US cannot be compared per se on cultural grounds but the differences can be easily pointed out. And it is my belief that if one really understands the beauty underlying the Indian thought, he'll not be too bent on returning--for India teaches you the place you live is really immaterial in your growth.

Before going further to discuss about the Indian culture, I would like to say that the appreciation of Indian culture highly depends upon one's belief, experience and understanding the philosophy of religion in the right spirit. Most people make the mistake of identifying the religion with the philosophy of religion and as a result we have a mass fanaticism resulting in a great damage (both societal and personal). In most cases, not understanding the beauty of Indian culture is also a part of this identification, but they can also result from one's experience and beliefs (for example, the Vedas can be considered just as some rustic songs without associating any spiritual significance).

Every country has its own cultural basis. And this cultural basis has its origins in religion. And religion in turn has ritualistic content and its philosophical counterpart. For example, in Christianity, the ritual of Baptism has a philosophical significance of uniting an individual believer and the Christ. It is relatively easier to see this aspect in Christianity, but in Hinduism which is the cultural basis of India, it is more complex. Understanding the true philosophy behind Hinduism, which is Vedanta takes much time and patience. More than time and patience, it requires faith strengthed through experience. The philosophy of Vedanta is very profound and deep and it encompasses all other faiths and beliefs. It is a universal science which has both theoretical and experimental aspects. It is my belief that understanding Vedanta will inspire people to understand their faiths better. In the upcoming series of writings I am planning to write more about Vedanta and expound the beauty of the Indian culture and heritage.

Welcome Message !

2005-06-29T13:13:00.000-05:00

Welcome to mind and matter. Here is the place where I write things that range from mind to matter, from philosophy to physics. I am planning to put down the things that I read, enjoy and experience. Thanks for visiting.