Sunday, October 31, 2010
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.
Sunday, August 15, 2010
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 indivisual 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 indeed more beautiful than the precise version that I mentioned later. (and hence probably more useless as well- consistent what once of one of our instructors said: 'most of the beautiful things are useless' :D)
Sunday, 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. 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)
Wednesday, March 24, 2010
Tuesday, January 12, 2010
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!