Mathematics is just thinking: Why is a mirror image upright?

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. It was to illustrate that mathematics is simply thinking. Putting the problem in a clear language is what we call ‘formalism’.

(planning to move: http://bit.ly/bZ6st5)

Academics vs Success

"Those who excel at academics at school never succeed later in life - this is true world wide" -this was the interesting statement made by our psychology instructor, which caught the attention of all students(I'm confident it is all and not 'almost all':D). He was talking something about IQ and EQ(emotional intelligence). His point was, people with high IQ have some kind of a trouble with emotions and hence are unable to utilize their intelligence-something with which some of us(specifically, myself :D) were a little uncomfortable. The problem was with what he refered to as IQ. Strangely, this wasnt the first time that he didnt appreciate the well known claim 'school academic performance has got little to do with IQ'. However, I agree with his statement in a sense different from what he intended. Here are some of my thoughts in this issue.

The current Indian educational system, puts students in an 'Academic Enclosure' till they are 17. The system forms an 'enclosure' because, it avoids most of the conditioning forces from the real world from accessing the students in the system. It conditions the students with its internal forces (rather differently than what the forces from the real world would have done!) For instance, the internal forces of the academic enclosure condition the students to believe that 'it is extermely important to score very high marks in exams'(everyone knows how!). Exams are conditioned 'aim' of students. At the rear end of it, the system even conditions kind of 'losers'' attitude in the students. What I mean is, it conditions students 'to live as if they have been cursed to live on the earth' or 'to live the life of an indian 007 agent in ISI (:D)'. In 10th std, very few of us think of this: 'why not take 10th as 'just another year', nothing special; after all, I didnt fail/perform poorly in 9th, which was 'just another year''. Finally, there are internal forces which make students find 'satisfaction' in impressing mortals(and not 'god' :D).

The 'First Rank School Kids' always have one serious problem: They feel their first failures after 17. Till then, inside the enclosure, they would have been exposed only to success.(they might have had what is called 'failure' in the real world, but wouldnt have recognised them as failures). I have observed this happen quite a few times. These properly trained dogs get severely hurt by the monkeys, once they are out in the real world.(dogs= the '1st rank school kids'..read the post 'On Randomness' to see why they are 'dogs'). One common reaction is to go emotionally down for some time. This is what makes them unsuccessful further. In cases of stronger conditioning, they simply refuse to accept failures. They refuse to accept the 'new' definitions of success and failure in the real world.

Within the academic enclosure, students respond differently to the conditioning forces. Now, one can easily see that the academic performance of a students is a measure of how well he/she has responded to the conditioning. A good response means a strong conditioning. Hence, the performance is directly related to how strong the conditioning has worked out. Hence, an excellent academic performer is strongly conditioned to the academic enclosure. I have already said that there is a strong campatibility problem between the 'real' world and the academic enclosure. Hence strongly conditioned students after 17, find it extremely hard to succeed in the real world.(and I believe reconditioning is not so effective after 17) This does not mean that the weakly conditioned students succeed in the real world!(:D) If they are poor at responding to a conditioning force, they fail at every system. However, those who are poorly conditioned in the academic enclosure because they somehow got in touch early with the real world forces succeed well in the real world.(rare case!) Also those who are intelligent enough to understand the compatibility difference, before they get in to the real world, show the highest success rate. This was the reason for my contradiction with the IQ vs EQ thing.

However, I cant predict fate of the 1st rank kids, if they were not conditioned in the academic enclosure before getting into the real world. The reason being, the academic enclosure is an artificial system, not fitting in well in the real world. This makes the response to the real world and the response to the academic enclosure rather unrelated. Hence the two responses are unpredictable from each other.

(PS: planning to move to wordpress: http://bharaththoughtflow.wordpress.com/2010/03/24/academics-vs-success/)

Mathematics vs Physics

My 'mathematician' friends call me a 'physicist' and my 'physicist' friends call me a 'mathematician'!. However, I don't feel so much like 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

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

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.