‘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’ 😀 😀 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! 😀 😀

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 😀 😀

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) 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). 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’.

Mathematics vs Physics

January 13, 2010

My ‘mathematician’ friends call me a ‘physicist’ and my ‘physicist’ friends call me a ‘mathematician’!. However, I don’t want to be a ‘mathematical physicist’ :D. This conflict between the so called ‘mathematical’ and the so called ‘physical’ approach had always existed in my mind. Now, I feel there is no room for such a conflict for, I have concluded that there is no real difference between them.

A person who propagates a new idea, or a new religion (or anything! ), has a two circles of followers around him. The 1st circle, is the immediate circle around him. It consists of people who follow his ideas, and understand them, to a certain extent. The second circle is often larger, consisting of people who merely follow him. They dont understand his ideas; they just appreciate them. They just get a feel of it. The real difference between the 1st and the 2nd circle is in the ability to defend the idea. The 1st circle is capable of defending the idea. The propagator is responsible for the idea and hence is able to defend it. The second circle, is often characterized by people shifting sides. One can easily jump from the 2nd circle of one idea to the 2nd circle of a different idea(most commanly the contrasting idea! :D). People in the 2nd circle are brought by just convincing them of the validity of the idea.

On the lines of the above story, a theory too has a propagator, a 1st circle and a 2nd circle. The so called physical approach, seldom lands anyone in the 1st circle. It is a good tool just to get convinced of the theory. Most of the so called physical reasonings are worthy observations, but fail in providing philosophical insights in to nature. There is nothing called the mathematical approach. Presence of large number of equations is not mathematics; There is just one approach, and we could call it the rational or, the logical approach!

On Randomness

November 3, 2009

Most people know me as a studious kid. But I will now uncover some of the hidden truths of my life. I was never able to do things that I hate. An early example comes when I was 11 years old. I had to study a 3rd language- Hindi. I hated it like hell. I was very good in all subjects, but ordinary in hindi. In my highschool, my marks sheet looked unusual, with high 90’s in math and science, and low 60’s in social science. Once I failed in biology, in 12th. People around me, were emphasizing a lot on ‘orderliness’ and stuff…which I was never able to acheive; however, those suggestions, initiated an intellectual conflict in my mind. However, the first time I got the freedom to experiment on this conflict was when I came to IIT.

I planned a systematic waste of time in IIT. (good utilization of freedom!) As it turned out, all sensible and useful works that I did, was during the hours I planned to waste! It was the effect of the freedom given to the flow of thought. Finally, I understood that, the world is governed by randomness. Getting ‘ordered’ is a challenge to randomness. And hence, it failed..atleast in my case. These thoughts are deeper than they appear. They dont actually mean that one should live like a lazy drunkard.

The Story Of The Dog And The Monkey

In the last summer, back at home, I didnt feel like sleeping, one night. Towards the end of the night, one thought occured to my mind. It’s the story of the dog and the minkey. A dog, never denies his masters order. It does exactly what the master wants it to do. On the other hand, a monkey, is known for it’s restlessness. It does a lot of random stuff. The dog, cannot do what his master can’t think of . A monkey, is capable of that. It goes beyond the span of it’s master’s mind, simply because it is more random in its approach. That is where is the hidden power of randomness. It’s the monkey’s attitude, which brings out something new to the world; not the dog’s attitude. More precisely, it is the respect given to randomness in the monkey’s attitude, which does the trick. So, if being notorious is useful, be notorious!.

However, the conflict is still on, in my mind. Confusion, which persists, looks to be a very essential entity. Thought says, life is not about getting out of such confusions quickly; it is the conflict of contradicting thoughts in the mind. Again, these thoughts too are in the conflict!(:D :D…the last one was a joke, a self reference)

The story of billiard balls

September 29, 2009

This blog is my first one, based on my thoughts, as mentioned at the end of the blog. I conceived this thought and the idea of blogging while travelling in a train, back from Kanpur to Bangalore.

A little kid, who does not know what ‘color’ is, comes across a bag of colorful billiard balls. what would be his reaction? His childish curiosity drives him towards the bag. He is attracted by the appearance of the balls…I mean, the colors, though he doesn’t know what it is….

Most children stop there. But, imagine, an extraordinary( hypothetical, if you feel so) child, who can proceed further. The next thing he would do is, look for similar balls (balls which look like each other) The child has a way to tell whether two balls look like each other or not, by visual inspection. (looking like each other in our language means same color).
Next, the child can divide the bag of balls in to groups of like balls. Every pair of balls within a group would look alike. Hence, each group can be represented by a single ball. If the child is given new balls, he can easily put them in to respective groups. Or, if it doesn’t look like any of the group representatives, it makes a new group.
Now, he is close to defining color. The representatives of each group are not balls, they are colors. He can name each group at his will. This is precisely what man did, over generations. The names he gave were red blue green et al.
Now, a formal look at the procedure adopted by the child. An important comment to be made at this point is about the way the child decides if two balls are alike. If balls 1 and 2 are alike , and balls 2 and 3 are alike, inevitably, balls 1 and 3 will fall in the same group. Hence, they should look alike. Formally speaking, his definition of alike should be trasitive. By using ‘two balls look alike’, and not ‘one looks like the other’, I have already meant that the like is symmetric. A little more thought will convince you that the like needs to be reflexive as well, for successful classification. Thus, it gives a reason to the mathematical definition of equivalence relation. Formally, those groups arising out of the equivalence relation are called classes. Once the classes are made, a mathematician may do various things with his classes….order them etc.
This is how most of the seemingly undefinable terms like color, size, mass, charge, cardinality, etc are defined(?).
The purpose of this blog is to express my thought flow, which at the moment says, mathematics is a way of thinking, formalized.


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