Hidden in the games

July 20, 2013

Few days back, I was playing a carrom game on android. My machine opponent is an expert at striking. With its accuracy, it can score a coin placed almost anywhere on the board, in a single strike. But it cannot score in a single strike when the coin is cluttered with other coins, or when its path towards the hole is obstructed by other coins. In such cases, it is theoretically impossible to score in one strike. It needs two- one to separate the clutter and one to score.

I am not perfect, unlike the machine; but I can score most well placed coins on a single strike, unless I get unlucky(only on android, not in real carrom :P). But the machine is not tactical.



Once at the climax of a game[fig-1], I was at the WHITE end, ready to play my turn. I could have finished the game in this turn, if only the two whites were separated; but they are cluttered. I cannot score them in one turn- I need two: one to separate them, and one to score.

If I separate them this turn, the machine finishes the game in the next turn- just the red and the black left, both positioned conveniently. I need to do something before the machine gets its turn, to prevent it from winning. I can score the red and retain my turn, but next, I have to separate the whites, so I cant retain my turn further. So the machine gets its turn with the board in almost same sate- red back on the board at the center and black at the same location, and it wins if I do so.

But, it is not yet check-mate. There was still a way out. I used the machine’s turn to separate my white coins!. I pushed the red towards the whites, cluttering it along with them :P [fig-2].



The machine can’t finish the game in one turn now- red can’t be scored in one turn. So, being un tactical, it used its turn to separate the red from the whites, as it has been programmed. In the process, invariably, the two whites also got separated. I got my turn with the red and the two whites uncluttered, all wide apart :P. I scored these three, one after the other, and won the game.

The red coin played a key role in above strategy. Red is the only link between me and the machine- both of us can strike it directly. Look at the rules of the game- there is a red coin, no player can win the game while it is still on the board. But, successfully capturing it doesn’t still guarantee victory. If you capture it well before the climax of the game, it adds no credit to you, instead, it clears one obstacle towards the victory of your opponent!. With these rules, the value of red lies in such strategies.

Why is there even a red coin? Why are the rules constructed in a way so that it ought to be used in strategies like the one above?. The rules facilitate a tactical game over a purely skilled game.

If the game were a strike out(like a T20 bowl out :D), i.e., a direct contest of who scores better, with just one or a fixed number of strikes for both of us, there would be no way to beat the machine. But the rules of the game make it a play of tactics rather than skill. So, although I cannot beat the machine in a strike-out, I can beat it in a complete game. The game, as a whole is different from an individual strike. It comprises of several individual strikes, but they are not to be perceived as independent strikes; there is a longer process connecting them all. The machine’s skill at individual strikes remains underutilized, since it treats them independent.

The strike out is the most plausible ancestor of the game in today’s form. It would have started as a fun-game, involving skills of striking alone. Later, gradually it would have grown in to a mind game, with the rules designed to consistently depart the game from a skill-game to a tactical game.

Look at the very pattern of arranging the coins to start a game- all the coins are cluttered in a hexagon at the center, alternating whites and blacks. If I separate my coins, or score a lot of them, in the process, I will have inevitably separated out my opponents coins too!, spreading them out over the board. This makes it easy for him to score in a rally. This is why we often see our opponent scoring a rally right after we do so. Therefore, it is a prudent practice to avoid clearing the center and to strike on particular coins. The game would have been different if the pattern of arrangement was to cluster whites and blacks separately, in two halves of the hexagon, with the red at the center.

You get to retain your turn after scoring a coin. Even if you are left with all 9 of your coins and your opponent is left with just one, it is still possible for you to win the game. So, you can afford to make mistakes in scoring. If it were a one-chance to you and one-chance to your opponent game, mistakes in scoring would have
been expensive. A rule common to carrom and other related games(snooker, billiards, pool) is, there are two types of coins, and each player is assigned one. Your opponent cannot gain from scoring your coins which you brought near the hole. A slight imperfection or a misfortune can leave a coin you struck just before the hole, making it an easy score in the next turn. This rule ensures that such a coin is not credited to your opponent in the next turn. These two rules significantly reduce the emphasis on perfection of a single stroke and fortune.

Evolution of the rules of a game over time tells us about what is adorable and enjoyable to the human mind!. It seems, ultimately, every game is played in the mind :).

Does every game, over time, mature into a mind game? Do the rules develop over years to make it so?. Cricket is now a mind game. Most games are mind games or tactical games, built upon a basic fun game. Learning how to use a striker is learning just the language in which the carrom game is played; likewise, learning how to bat/bowl, is learning just the language of playing the cricket. The real game is always a mind game.

While learning, all of us believe, carrom is about aim, bowling is about line and length, batting is about timing and footwork. Yet, we accept the rules, which facilitate a mind game over all these skills, as the supreme rules. Understanding the rationale behind these rules requires a certain level of expertise in the game, but we accept them nevertheless.

Waves don’t drown the boat

March 23, 2014

I started writing this post four months back. But in between, I was stopped by an existential question associated with both the contents of this post and the act of posting it O_o. While the process of overcoming this hurdle was a struggle, it also generated content for many future blog posts :P. To begin the way I intended:

“As our boat was rowed between the two islands in the ocean, huge waves appeared to engulf it. While we screamed in fear, the locals in the boat were smiling(and laughed at us silently). The waves were much higher than our boat, but, they didn’t engulf us. They lifted the boat higher up than themselves, while they passed underneath. It was a common part of a boat ride. The boat is after all, on the water.”

This is a tourist’s(not me) experience, at the Andaman and Nicobar islands. I don’t remember who was it, but his description was crisp enough to leave a permanent picture of the waves lifting up the boat as they approached. I remember even the exact words he used to describe it. And I am able to use this picture to understand something totally unrelated to the boat and the waves. :P

One ingredient of intelligence is, a deep understanding of the world(or equivalently, the mind :D ). By deep understanding, I mean a knowledge of how different elements of the world, both physical elements and nonphysical, but thinkable elements are inter connected. The world includes three things: the bare physical world(without people), the people in it, and finally all nonphysical thinkable objects in it. An Understanding of the world would include observations on how two people interact, how many people come together to form a society, and how two societies interact. These are indeed observations on the fundamental properties of the human mind. Understanding the bare world is actually, observing how our mind reacts to it. Finally, the non physical thinkable objects form the biggest part of our world. They include all mathematical objects, games, stories, music etc. Such objects are constructed out of a language of communication between people. So much so, that they can be regarded as a part of the language. They are created out of the mind and hence bear fingerprints of the mind. In effect, exploring them is exploring the mind. So the question I want to ask is, on the whole, how does one develop such an understanding?

It is natural to assume that a man, privileged from childhood, has better opportunities to attain a deep understanding of the world. But sometimes, I felt that privilege leaves little to explore, so it is more advantageous not to be privileged. Now, I have concluded that, the deepest insights about the world are attained, neither by the one who was born privileged, nor by the one who was born in destitution, but it is the one who traveled between the two ends(either of the ways ;) ). Traveling the socioeconomic spectrum(not as a tourist!) provides the deepest insights.

Problems are like waves in the ocean; they can do both: drown a boat or rise it high up. A new insight is gained out of the experience of overcoming the problem. All non trivial insights are gained through solving a non trivial problem. There are two questions to be answered here: One, is there any value to the insight that is developed in this process? or, is the insight useful only to solve that particular problem? In that case, it would be redundant; However, a very deep insight is also very general in its applicability. It contributes significantly in enhancing the overall understanding of the world.

Secondly, what determines whether the wave drowns the boat or rises it up? This is a hard question. At this moment, I don’t have an answer more precise than saying that it depends on how many unintended observations we made in the past. Naturally, all the way, we make a lot of random irrelevant observations without intending to do so. Such observations made in the past are put together in overcoming the problem.

Obviously, one cannot intentionally make unintended observations :D. Any conscious attempt to make irrelevant observations would result the set of observation being one-sided, since they are powered by an intention. Moreover, the mind tries to put together all conscious observations to make a ‘conclusion’, and there after it systematically remembers the conclusion with most clarity.

Though this was a simple post, the process of writing it, strangely, was one such boat ride over a wave :P. I believe I went over the wave, and that, at least, generated more posts to come :D.

Conceiving and convincing

May 30, 2012

Imagine, we are given a task to fill up as many pages as possible on word, in a given time, with the letter ‘A’. One way is to copy ‘A’ into the clip board and keep pressing Ctrl V. This is the AP(Arithmetic Progression) way. There is another way; we can copy ‘A’ and keep pressing ctrl C, Ctrl A, ctrl V in sequence. This is the GP way. Mathematics tells us that although Ctrl V appears once in three buttons, the GP method is faster. This is not obvious for a common man at the first sight. And most of the public, are insensitive to logical arguments. Nevertheless, one can convince anyone of this fact, simply by demonstrating it.

The above is an example of what I call as an operationally testable statement. However there are statements which are not operationally testable. The man on the platform, says “the train is moving”; while, the man in the train says “the platform is moving”. Usually, a common man assumes that the man on the train is wrong; he knows the ‘truth’- the train is moving. The profound realisation is that, neither of them are wrong. But there is no way to demonstrate it! This concept of relative motion is operationally un-testable. So much so, that this un-testability was responsible for the Galileo affair. (besides religious concerns)

In fact, most of the statements with profound reasoning are operationally un-testable. For instance, the counter intuitive results of cantor, like the number of points on a side of a cube, the number of points on a face and the number of points inside its volume, are all equal; it is impossible to trisect an angle using a straight edge and a compass. A common man certainly has problems with accepting it. And unfortunately, there is no operational way to convince him of this fact; i.e, a person who assumes the contrary will not be punished for being wrong. :D Hence it is apparent that there is no way to convince the public of these facts.

To digress a bit, I often say utilising an object is to do something with it, which cannot be done without using it :D. By that token, reasoning should be used to conceive (currently)un-testable facts. Because, operationally testable facts can be conceived even without reasoning. Hence, real utilisation of reasoning is to conceive operationally un-testable facts.

How do I convince a common man of such facts? In the first place, should one care to convince someone who is not sensitive to logic? To answer these questions, I shall consider examples from the history where the task of convincing the public has been accomplished.

The earth is not flat, but spherical, and further, it is not at rest, it is rotating and revolving. These two are among the most profound, but operationally un testable realisations. However, they are widely accepted by the public!. Let us examine how were the public convinced of these. Aristotle conceived that the earth is spherical. At that time this would have been counter intuitive and operationally un testable; So, he would have had a great trouble in convincing people about it. It is clear that he did care about convincing people about it; why else would he list down the common fallacies in logic committed by people :D (see ‘Aristotle’s 13 fallacies’). And the way he did it, was to impose it as a belief. This is clear from how people believed everything that Aristotle said.

Most of the public today, believe that the earth is in a complicated motion. They just believe- they don’t really know the reasoning which led to this fact!. In fact, to really go through the reasoning, one has to understand relative motion. This was the major trouble with accepting Galileo’s arguments; he was asked to prove that the earth is moving (for which he gave a wrong argument :P). And it is apparent that most of the public don’t really appreciate relative motion. So, it is clear that they have been convinced of the heliocentric theory, just by imposing it as a belief. This, is not very different from religion!. Isn’t it unjustified for an intellectual to impose a belief?

Majority of people are insensitive to logical reasoning; ie, if the result of a logical reasoning is against their intuition or religious or any other concerns, they cease to accept it. Therefore, it is impossible to propagate the picture of moving earth, through reasoning. If it was not propagated as a belief, the public would have accepted a different picture of the earth, still as a belief!. Hence it is not unjustified, to propagate a belief, if it is necessary to convince them of these facts.

It is clear that whether or not a statement gets propagated as a belief among the public doesn’t depend on whether the statement is based on a sound reasoning or not!. It depends on the ability to impose a belief among the public, of the person who conceived it. This means, almost anything can be propagated as a belief!. That is a little disturbing :D. There ought to be a fundamental difference between conceiving a statement out of rigorous logical reasoning, and claiming without a strong logical background. I guess this difference is brought out in the confidence: the confidence attained by conceiving a fact through thorough reasoning is stronger. I guess(hope :P) this difference can be utilised to beat the propagation of unsupported claims.

Finally, I come to the question I postponed to the end. Is it necessary to care about convincing others? Again let’s ask (old)people :D.Usually, mathematicians don’t care about the public; after conceiving a result, they wouldn’t worry about convincing. Kepler, who went a long way ahead of Galileo, at the same time, didn’t care to convince everyone; that is why he doesn’t have an affair attached to his name, unlike Galileo :D. Apparently, he was able to go that far simply because he didn’t care about convincing people. It is clear that Galileo and Aristotle cared about convincing people. If none of the physicists and mathematicians care about convincing others, their next generation to be physicists and mathematicians will find it hard to see the facts amidst misconceptions. Avoiding this is the only possible motivation for a physicist/mathematician to get in to the job of convincing, as far as I can see right now. This post is the longest one so far, and has crossed 1K words :D and so I stop here :D :D

The Car

March 23, 2012

Ever since one of our profs said ‘most of the beautiful things are useless’, I was disturbed by the fact that in mathematics and physics, most of the intellectually deeper works don’t have practical value. This means, there is no value associated with the ability to do such a deep work. Also many of my friends keep asking me what is the point of all the hard core theory in physics, and why do I study them? The best way to analyse it is to look at the history.

‘His’ story
When he(mentioning who ‘he’ is, is irrelevant :P ) was on his way back from his work, his car broke down. He went to a mechanic, got it repaired, and reached home. At home, relaxing on a chair, he was thinking about his car’s history. Half an hour ago, it was at the mechanic shop. The mechanic is an important person in the car’s and its owner’s immediate history. If he hadn’t done his job, the man wouldn’t be home by now. His work has had immediate effects on the car and it’s owner. However, the job wasn’t a high skill-demanding one; in fact, with a little experience, anyone could have done that job. Also, the guy is not remembered; the man paid him and forgot about him. That completes the first layer of the car’s history.

Where was the car before this? The next interesting part in it’s history is when it’s model was designed. At this stage, it is not just it’s history, it is the history of all cars of it’s model. This was about a decade ago. The car was on paper, on the desk of the man who designed it. This man, is another person who influenced the car’s future. His job, unlike the mechanic’s, didn’t have an immediate impact. If he hadn’t done his job, that would have probably gone un noticed after all! The effect of his effort would have taken a couple of years to come out. But this guy is actually skilled; any arbitrary person cannot be trained to do this work. One needs to be a little talented to be able to learn to do such a job. And, at least people in his company will remember him for designing that model. So That’s the second layer.

The third layer is over 250 years ago.(that’s exponential in time! 30 mins-10yrs-200yrs). This car and several other machines had their common point in history, on the notebooks of the guys who discovered the laws of thermodynamics. Now there is a trend!. This piece of work takes 100s of years to yield its value!. At the time they did it, no one could have imagined that someone will make an auto-mobile out of it, 200 years later!. Coming to the skill required, even a considerably talented person cannot be trained to do such a job. It requires a rare capability. And after 200 years, we still remember them for their work!

I have spoken about three quantities-the time scale in which the work will be utilised, the skill levels required and the reward in terms of people remembering the man who did it. And, the trend is clear :D. That summarises all I have to say about the value of hard core theory.

However everything that looks deep and, useless at the moment is not necessarily going to be useful some 100 years later :D. In fact, most of them are so, which is to be understood from G H Hardy’s A Mathematician’s apology, where he justifies the work of a mathematician saying they are harmless, rather than useful :D. To foresee what could be useful in the long run is unimaginably non trivial!. It is possible that a great mind can foresee it; but they usually work for the fun of it, rather than its impact on the society in the long run. It seems to me, that Newton might have foreseen the impact of his laws of motion-the industrial revolution, although this impact was none of the reasons why he did all this work. But I believe he did not foresee the giant impact(we are able to watch TV today!) of his law of gravitation.

Mystery is a guide to hope

January 14, 2012

Once, I was walking down along with my prof, discussing a result I had just managed to prove. He said “well, you have managed to prove it, but you should also understand your result”. We generally believe we certainly understand something that we developed ourselves. For the first time, I was talking to someone who had, in his mind, a genuine definition of understanding. I knew how I arrived at the result, but it was counter intuitive. That means, there is a flaw in our mental picture of the subject. So, reconstructing our intuition, based on this result is a more important part of understanding it. In fact, this is the value attached to the result.

The above instance speaks also about the value hidden behind a mystery. A mystery points at something that we don’t understand. A counter intuitive result is also a mystery. The moment we encounter a mystery, we can hope for something radical, coming out of resolving the mystery. In fact, look at almost any radical change in human thinking, there would have been a mystery out of which it emerged! Quantum mechanics was itself found hidden behind a mystery. Also, the theory of relativity was born out of a mystery!. Another classic example is quantum computation. This example is too technical to be written in a blog, but I prefer to mention it because, I myself solidified the thoughts behind this post using this example. It all started with Einstein pointing out an apparent contradiction in quantum mechanics, and Bell clarifying that this was no contradiction, but it was only a mystery. They called this mystery ‘entanglement’. It took a while for people to understand it; but a revolutionary looking idea did emerge out of it. That was quantum computing.

The examples I gave are of a huge magnitude :D, and very specific(to physics). One may not expect to encounter mysteries of this magnitude, but the idea works at all scales. A big mystery leads to a big revolution; a small one to a small one :D. So when we confront a mystery, we can expect something new coming out of resolving it. However, every mystery need not lead to something fruitful, but we can hope!.

‘Sneak’ into the Gaps

September 3, 2011

While walking on the roads of the campus, if someone casually asked me ‘where are you going?’ I would reply ‘Nowhere..I am just moving parallel to the road’ :D :D I always believed that most of our path is decided by the road, we just move along the road!; It’s only at the turnings where we get to make our choice. Yet, we get complete freedom to choose our final destination!. I was always amused at it. Probably because, I used time/length as a measure of domination. When I say we always walk along the road, and make a choice only once in while, I mean, our walk is dominated by instances where the road decides the path. Invariably I am comparing the chosen part of the travel and the predetermined part of the travel in terms of the time spent or the distance travelled. This is clearly an incorrect measure to quantify and compare how much do we get to choose and how much is predetermined.

Recently I came across a formal study of similar properties in a language. A language has got some rules in the form grammar etc.. As in, once I start writing, I don’t have complete freedom to decide the next character. There are restrictions to it. There are choices left to us and, in between the restrictions, we make use of the choices to give the meaning at our will. This is beautifully captured in what is known as redundancy of a language. This is a number between 0 and 1. A completely random language has zero redundancy; any character can appear after any character. It is completely up to our choice. A language where the user gets no chance to choose is a completely redundant language; every character is uniquely decided by its previous character. Anything in between these two extremes is assigned a number between 0 and 1(there are more beautiful results; the redundancy of a language is related to existence of arbitrary infinite n-dimensional cross word puzzles!). I was so fascinated that, in fact I started using the word ‘redundancy’ almost whenever I feel like! :D :D

Rules might seem like they are restrictions on our freedom. However, even in a very general abstract system, rules are very much necessary. I started comparing the rules and the freedom in a system to those of a game. There is no game without rules; also, there is no game with the moves completely determined by the rules. It’s a proper combination of freedom and redundancy which makes the game interesting. There are rules to be followed while playing a game. However, the real playing happens in the free region; If I am spending all his energy to merely be religious in following the rules, and doing nothing with the freedom, I am not playing at all! To play is to find gaps in between the rules and sneak into them!

So to play a game is to sneak in to the gaps between rules to get our job done. We might expect it to be easier to do so with lesser rules. But it is just the opposite!. The reason is, rules just give us a platform to work on; we are actually working in the free region. So, larger the free region, difficult it is. This is quite the reason why simplifying assumptions are made to begin with a new theory. We just don’t know how to work with very few constraints!. The extra assumptions give us the guidelines to work. That seems to be the purpose of rules. This post is abstract..probably because, the thoughts are so :D :D

The Chase!

February 7, 2011

I have decided to blog after quite some time. This time is rather a strong thought. This post is about a chase, a chase of safety. Let ,me begin(as usual :D) with a small story.

Things are not normal in a refugee camp. People’s freedom is severely restricted. Their life is highly insecure. They are always in the safety zone; if not, hoping to get in to the safety zone soon. In short, they don’t look around and say “where does that road lead you to?”; instead, they are wondering, “how long is it, before I get back to my safe home?”. Imagine, the refugee camp sustains for a long time. Long enough so that there are third generation refugee’s, whose parents and grandparents too are refugees. Such people, wouldn’t have seen anyone living a normal life. When they get back, out of the refugee camp, they are still ill fated to run behind ‘safety’. They try to think progressively; they everything required for a successful rehabilitation. However, a large part of their motivation is inevitably lost in chasing safety. such is the effect of their inbuilt insecurity. This is the inbuilt insecurity I am talking about. This insecurity, runs through generations, as a practice; Of course, it’s effects decay over generations.

I see a rough shade of the above story in our country. The first post independent generation in India appear to be highly motivated. But, unfortunately, it’s all lost in chasing a secure life. Government jobs gained popularity just because, people were behind a ‘secured future’. What disappoints me is, the whole lot of people, considered to be ‘successful’ by their near neighbourhoods, are all successful just in chasing down a secure future- a clear effect of the inbuilt insecurity. (What also disappoints me is, this post is turning out to be serious, without even a single PJ :D :D) While people now are not behind government jobs, but behind the not-so-secure software jobs (an optimist would call that a ‘decay’ of the feeling of insecurity, from running behind a completely secured job to just a job :D) To that (small!) extent, we are now comfortable with the non-deterministic nature of future.

While it is clear that such a feeling of insecurity cuts down the freedom of an individual, sometimes I feel I am rather overvaluing the importance of freedom. In fact, to me, ‘For every living organism, the job of hunting for food is basic; it is respectable only when it does it on its own, in the natural way; and it should be given the freedom to do it by itself‘. This thought has been quite strong within me. Also, I see remote connections between freewill and intelligence or creativity. Hence, freedom or freewill, deserves a supreme position. What I would like to point out here, is this freedom is eaten up by ‘the chase’.

I don’t have much more to say. That was a blur, but strong thought flickering in my mind. Further, I don’t want this post to be chasing down something! :D, although I feel, it is some or the other chase which keeps us busy.


October 31, 2010

The idea of giving such a title to this post isn’t entirely mine. It is inspired by someone else’s idea of naming something else( :D :D). What it means is something the reader has to imagine after reading the post :P. This post is not related to the original PvsNP problem; but it is certainly inspired by that problem. It is about the question of questions.

A question, has a shape. It has two components.

1.) A solution space. (i.e., one should know how the answer ‘looks like‘)

2.) A verifiable condition.

The question is to look for some element in the ‘solution space’, which satisfies the condition. As an example, the question 2-3 = ? can given a shape.
1.) the solution space is the set of integers
2.) the condition is, 2+x=3.
That’s rather numerical. But the scope of this shape of a question is much wider than questions related to numbers. The space and the condition are much more abstract in many useful cases. An essay; everyone knows how to check for the condition. But what is the solution space? You could use “the set of all essays”, if the meaning of essay is known. Or, the “set of combinations of the 26 letters, the space( ) and the other symbols used” :D. That looks awkward. However, the point is that one should know what the answer looks like or what are we looking for. That is the job of the solution space. The two examples should be read and forgotten, the crux of the story is yet to come.

So, why not always take the so called universal set as the solution space, and reduce the structure of a question to just a condition? (which is what most of us think of a question as). Well, the universal set, if it exists (no, it doesn’t!) doesn’t tell us anything about how the answer looks like. A solution space can be any big. But it must tell us what we are looking for. By the way, for those who were surprised at my earlier remark, the universal set does not exist. One can not create something out of nothing. Assuming that there is something which contains everything results in a paradox, called the Russell’s paradox. All it means is, ‘you cannot put all thinkable objects in a single set’.

Constructing the solution space turns out to be the major issue in building a question. Most questions which seem to be unanswerable are so simply because they don’t have a solution space(I mean, we don’t really know what we are looking for!). Just an attempt to construct a solution space resolves many of such queries. So, whenever a perplexing query comes to mind, one has to stop and think what am I looking for

As it turns out, it is a very non-trivial job to build such a structure to the queries of the human mind. As a matter of fact, the problem of finding such structures is itself a structured question!. However, in this case, the verifiable condition is given by the satisfaction of the mind. That makes it somewhat different from ordinary questions. In fact, it makes it interesting(=less boring :D). Figuring out what our mind is looking for forms the core of thinking.

What does one do after structuring the query? nothing! :D. “The real job of a mathematician is to get equations, not to solve them!”. Solving them is the job of a computer. whatever needs to be done next is too ordered to interest the human mind. However, it seems ‘finding’ the answer turns out to be either too trivial or unimportant. So, before asking “how can a man pass through a wall?” one has to stop and think what exactly is our mind looking for, and in many cases, such an attempt alone can resolve the query.

What Is Intelligence?

August 15, 2010

In this post I am not defining or characterising intelligence. I don’t think I’m close enough to characterising an attribute attached to the mind. The only thing I have done here is throw out what my mind says about what we call as intelligence. These thoughts are quite old. I do remember thinking somewhat on these lines, almost 10 years ago.

Intelligence, to me, refers to ‘The ability to connect oneself well with the world’. World here does not mean what it does in most contexts. By world, I mean the surrounding system or influencing system. It doesn’t have to be physical. It can be as abstract as ‘physics’, it can be ‘mathematics’, it could be a cricket match, or even ‘music’. It can even be the ‘IITK campus’–finally something physical :D, and another crucial example–’our own mind’. Well, let me call it as any system, with which one can interact. If you are still curious to know what the hell do I mean by ‘…connecting well’, you’ll probably read the next paragraph more carefully.

So, a more precise version: ‘The ability to perform an undirected sequence of experiments with the system and draw inferences’. This is why, the system needs to be interactive. The term undirected is the key for the first half of the above statement. It means, without prior instructions or, on one’s own. The second half of the statement essentially banks on the ability to recognise similarities and differences in experiences. The word experiment shouldn’t make people think of LAB :D.

It is appropriate to talk about the Mumbai masala tea here. On my first visit to Mumbai, I was at the NSC(Nehru Science Centre), attending the astronomy Olympiad camp. Once, when I slipped out of the NSC campus(‘slipped out’ because we were not supposed to go out alone :D) to make a phone call, I saw an old man making masala tea (I somehow attach masala with tea, because, the first time I had tea was not a normal tea).I remembered, I had heard that those people are making tea for a long time and they can judge what how good the tea is just by looking at its colour. This is what I meant in the above definition. No one told him to keep an eye on the colour-taste relation. This was an understanding coming out of undirected experiments. Also it demands a high ability to recognise similarities and differences in the colour. So, this was what I meant by recognising similarities and differences. Two seemingly similar objects might have subtle differences which become clear over time.

One word here: The Mumbai masala tea story does not imply or justify anything. Nothing can be deduced by that. My sole intention was to clarify the meaning of what I stated. An example can do nothing to a general statement(except, probably disprove it!). I have seen people deducing things from individual instances. That was the reason why I wanted to make this point.

When I say undirected, I already mean we don’t know it’s mechanism!. This is the reason why theories which get internal about the mind aren’t that beautiful :D. I always feel it is better to treat the mind as a black box for this reason. That way, I am more towards the 1st statement I made about intelligence, though it has ambiguous terms. It is more concise than the precise version that I mentioned later.

Why is a mirror image upright?

June 6, 2010

I am blogging on this topic rather unwillingly. My thoughts upon this problem are nearly two years old. I was reluctant to blog this one for two reasons: one, this is an old and well known question; hence I expected a handful of articles addressing this one on the net, providing answers close to mine. Surprisingly, I found no answer close enough to mine. And two, this topic is technically way too specific to appear in my blog. However, I have tried my best to use the problem just to illustrate what I want to say.

Well, let’s begin with the question. “why is a mirror image laterally inverted and not vertically inverted ?” I have observed that quite a few people, after a second’s reflection, don’t even realise that there actually is some trouble with the image. The question as such is not clear and hence needs to be defined properly. Here, I have described the ‘trouble’ with the mirror image in a slightly different language.

The mirror has a plane. And it has an axis, perpendicular to the plane. The human body has three directions intrinsically defined along three axes. Feet to head defines the directionality of the vertical axis. Back to front defines the directionality of one of the horizontal axes. Left to right defines the directionality of the other horizontal axis. What the mirror does is, it reverses the directionality of the axis pointing towards the mirror, i.e, the axis perpendicular to the mirror. It does not change the directionality of the axes parallel to the surface of the mirror(at least this is what one would expect). Let’s look at what the mirror does to the human body. The front-back of the image is opposite to that of the object, as one would expect. The top-down of the image is same as that of the object, again as one would expect. However, the left-right of the image is not same as that of the object. This is the trouble that we are referring to in the problem. It’s a serious violation of symmetry between the two directions parallel to the mirror.

One must reflect for a while and convince themselves that above problem is same as the one we have in our mind when we say ‘why is a mirror image laterally inverted and not vertically inverted?’ for the case when the object is a human being (Let us take up the cases of the other objects later). Once formulated this way, it is immediately solved. The left-right is fundamentally different from the other two directions. The top-down and the front-back are defined through asymmetries in the appearance of the human body. While, it is impossible to define left-right just by appearance! They look perfectly alike. But still, we manage to unambiguously define left-right. How? We live in a 3 dimensional space-this is the answer. Any object with two directionalities defined, can be given a third directionality, arbitrarily, as a convention. Once this convention is set, it can be followed unambiguously, since we live in 3 dimensional space. This definition of the third directionality uses the two already defined directionalities. Mathematically bent people can, with a little reflection, convince themselves that this is indeed, ‘defining of the cross product‘. In fact, the ‘left-right = back-front X bottom-top, once the cross product is defined the way it is defined now. Now, why is the left-right reversed in the image? The image preserves the bottom top, but the back-front gets reversed, hence, the left-right, which is defined based on these two directionalities also gets reversed.

About the case of non-human objects, even if the object is perfectly assymmetrical, we attach our left-right to everything (we read and write from the ‘left’ etc) All those troubles are closely related to this. I don’t want to get in to those, since all of them involve just one thing: defining the problem properly.

Now the crux; as said earlier, the purpose of this blog was not to solve the mirror problem(:D). It was to illustrate that mathematics is simply thinking. Putting the problem in a clear language is what we call ‘formalism’.


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