If we assume that facts are only those things which we can believe in with 100% certainty then very few things are facts. That the sun rises in the east and sets in the west is not a fact. That carbon produces carbon di-oxide on burning is not a fact. The law of conservation of energy is not a fact. We only have an enormous amount of evidence in favor of all these things. We only know that thousands of experiments being done world-wide have all been consistent with the above statements, that if these statements were wrong, then the results of those experiments wouldn’t make sense. But we don’t know that these statements are true with 100% certainty. In principle it is possible that someone does an experiment some day that violates the law of conservation of energy. In principle, it is possible that someone finally builds a perpetual motion machine. It will shake the foundation of physics, it will prove that everything we thought we knew about the universe was wrong and so on. But it is still possible.

You would say that it’s depressing. It probably is. But that’s how things are. It’s impossible to make a hypothesis about something that exists in the real world and then gather evidence that establishes that the hypothesis is correct beyond doubt. If you believe in any such statement beyond doubt then there is something fundamentally inconsistent going on in your belief system and it very well deserves some repair work.

The situation is not that grave though. There are some statements whose truth is established beyond doubt. That happens when we are not dealing with real world objects. For example, the fact that there are an infinite number of prime numbers, or the fact that the square root of 9 is 3 or the fact that the ratio of the area of a circle to its circumference is exactly equal to half its radius. These are all true. The reason why it is possible to establish these statements beyond doubt is that we are the ones who have defined what a circle means or what radius means or what number, infinite, prime, square root, area or circumference mean. The definition of a circle is “a closed curve which has all its points equidistant from one given point” and not “the shape of the cross-section of coconut trees”. If circle was defined as the shape of the cross-section of coconut trees, then to check if the statement “ratio of the area of a circle to its circumference is exactly equal to half its radius” was true, one would need to cut down lots of coconut trees and measure the area, circumference and radius of the cross-section and see if the ratio was half the radius or not. But that wouldn’t at any point establish this beyond doubt, even if you checked all coconut trees on the planet and found that the statement was true for all of them. There would always be the possibility that a new coconut tree would grow up in some corner of the world that violated this statement. But with the definition that it has, the fact can be verified beyond doubt.

It is the difference between trying to figure out the rules of the game some other people are playing and trying to argue about a game whose rules have been defined by you. If some other people are playing a game whose rules are unknown to you, you can never figure them out with 100% certainty just by observing their moves.  If you have designed the game yourself though, you are completely sure of what the rules are and so you can say things about the rules with 100% certainty.

But why would one define things arbitrarily and talk about them? Isn’t that a completely trivial exercise? No, it’s not. If you have defined the rules of the game, it might be easy for you to state the rules, but it may be extremely non-trivial to figure out, say, what would be the best move for the next player. For a game with rules as simple as chess, for example, we are still not capable of calculating the absolute best move for a player given the state of the board in a reasonable amount of time. And that’s just one thing that’s non-trivial. There can be several other things about those rules that are very difficult to decide. And that’s what happens in math. The whole fact that we should find it difficult to decide something about a set of rules that we have defined ourselves is fascinating itself. But very often it also happens that these things that we are trying to decide are not just difficult to decide, but are also extremely surprising. The fact that things that we define ourselves are not our slaves, but in fact, to the contrary, they are capable of confusing us and surprising us is extremely fascinating. Mathematics is full of such examples. For example, taking the number 5 and adding it 9 times is the same as taking the number 9 and adding it 5 times. Or, the fact that the rate of increase in the area under the curve representing a function is the same as the function. Or, when you take the ratio of the circumference of a circle to its diameter, multiply it with the square root of -1 and raise the product to the power of the inverse of the probability that you end up placing all letters into wrong envelopes if you do it blindly, the result that you get is exactly equal to -1.

That’s not all by the way. Surprising and confusing us about things that we have defined ourselves is not the only role mathematics has to play. Very often it so happens that one finds something in the real world that, for some reason, follows the rules of one of the games that we had designed. And that’s when all the exploration we had done, all the confusing and surprising things we had found about the set of rules come in handy. We discover that the thing in the real world doesn’t just follow the rules of this game that we had designed, but it satisfies this bunch of surprising properties as well. And that gives new insight. For example, it’s just a mathematical fact that if something is going around a circle with a constant speed then that means it’s accelerating towards the center. This can be proven with some elementary vector calculus. But since we have also empirically observed that the earth pulls things towards itself, we can use this empirical observation and the mathematical fact above to say that the reason why the moon revolves around the earth is because of earth’s gravity. We can also use these two to send communication satellites into space with that exact velocity that makes them revolve around the earth in an orbit.

So, to put it in a nutshell, you can be completely sure about the truth of a statement only if the statement is about a game whose rules are decided by you. Also, to decide whether such a statement is true or not is very often non-trivial even though you are the one who decided the rules. Besides being non-trivial, many of these statements are also very surprising and sometimes they might even play some role in broadcasting the latest episodes of your favorite sitcoms on your television.

Let’s say you work in a hospital. One random day, I come to your office and ask, “Is it going to rain today?” You look outside and see that it’s not particularly cloudy but it’s not one of those bright sunny days either. You have seen days like this pass without a single drop of rain and you have seen days that start like this and end in a flood. So you say, “I am not sure.”

Since I like asking questions, I don’t stop here and instead, I say, “What’s the probability that it will rain today?” You think for a while. You try to remember the number of days which started like this in your life and the fraction that ended up in rain. But to your disappointment, you don’t have that sharp a memory. You try to estimate this fraction, but very soon realize that you have absolutely no clue what it is. So you say, “I am not sure.”

Now let’s pause here and ponder for a while. Whenever someone says ‘I am not sure’ for any question, it is only reasonable to ask them to assign a probability distribution to the set of possible answers to the question. For example, if it is a yes/no question, then the natural thing to ask in reply is, “So what’s the probability that the answer is yes and what’s the probability that it’s no?” If the set of possible answers is, say {1, 2, 3}, that is, the set comprising the numbers 1, 2 and 3, then you will want to ask, “What’s the probability that the answer is 1, what’s the probability that it’s 2 and what’s the probability that it’s 3?” The set of answers may not be countable, that is, it may occupy a continuum. But you can still ask to assign a probability distribution to the set of answers. If you don’t understand how, then this post will not make any sense to you.

Anyway, the point is, when I ask you to assign a probability to the event that it rains today and you say you are not sure, I can ask you to assign a probability distribution to the different values of probability with which it might rain today. This means that I can ask you for the probability that the probability that it rains today lies in a small interval around x, where x is a real number between 0 and 1. And if you claim you are still not sure, you know what I can ask next. Let me still state it, just for the kicks. The question I will ask next is this – “What is the probability that the probability that the probability that it rains today lies in a small interval around x, where x is a real number between 0 and 1?”

I am going to state something really interesting in this paragraph. But before that, I will need to number these questions so that I am able to state the interesting thing in a precise way. So let’s say the question “Is it going to rain today?” was my zeroth question. The question “What’s the probability that it rains today?” was the first question and so on. So here comes the interesting thing. If you choose to answer my nth question, instead of just saying that you are not sure, then from the values that you give me, I can calculate your answer to the (n-1)th question (with a simple integration), and from there, I can calculate the answer to the (n-2)th question and so on till the first (not the zeroth) question. Or in other words, if you are sure about the nth question in the series for any value of n, then you must be sure about the first question as well, or else you don’t understand probability. Or, stated in the contrapositive, if you are not sure about the first question, then you cannot be sure about any of the questions in the series. Or, finally, to summarize everything, you either know the answer to all questions in the series (the set of questions from the first question to the last question) or you don’t know the answer to any one of them.

Some time ago, I was thinking about these things and was trying to figure out what was wrong when I realized something that made it all clear. What I realized was as follows. When I ask you what’s the probability that a certain event will happen, I am not asking you to give me some magic number so that if I perform a certain experiment in similar conditions a hundred times, then the event in question will happen that many number of times. What I am actually asking you is to give me an estimate of your own uncertainty about the event, based only on the information that you currently have. Different people might assign different values to the probability. But it won’t mean that any one of them is wrong. The probability that one assigns to a certain event’s occurrence shows his own uncertainty about it.

This is why Bayes’ Theorem makes sense. It gives you a precise way to update your own uncertainty about something based on the knowledge you receive and the evidence you assimilate.

So, to sum it up, the problem I tried to explain in the post was that you either know the answer to an infinite number of questions or you don’t know the answer to any one of them. The way it is resolved is that you actually know the answer to all the questions, because the reason the questions sounded unanswerable earlier was that you had not understood them correctly.

This is good understanding for the start. I plan to read E.T. Jaynes’ Probability Theory: The Logic of Science some time soon. I hope I will be wiser after that.

Suppose you take a test and pass it along with nine other people. Initially you are only told that you have passed and hence you are all too happy about it. But later, you discover this webpage where a list of passed candidates is given. The list is numbered and your name is mentioned at the first place. However, nowhere is it mentioned that the list is sorted according to the scores obtained in the test. So may be, it is just arranged in the order in which the papers were checked. Or may be, the order is just random. The question is, should this new information make you happier than before?

Suppose you had never seen this list and were just told that you had passed. In that case, it would only be fair to assign equal probabilities to you being at any of the positions from one to ten. Your expected position, therefore, would be .

You should be happier than before if the new information raises your expected position to something above However, since you have no idea about the order in which the list is sorted, all permutations of the present order are equally likely and since there are equal number of permutations with you occupying position for any from to , you are once again, equally likely to occupy any of the ten available positions. So this new information has absolutely zero information content and your expected position is still exactly equal to .

Now let’s say, that a friend of yours comes and tells you that the list is either sorted in the increasing order of scores obtained in the test or in the decreasing order and he seems quite confident about it, should you be happier than before now? The answer, once again, is ‘no’. This is because your expected position is still In fact, this holds even if you were at position instead of 1 in the list given on the webpage. Your expected value would be

I find this analysis interesting because the above situation occurs quite often and in most cases, one feels tempted to draw conclusions about his actual position in the test from the position in the list provided. Also, increasing order of merit and decreasing order of merit are two most likely orders that come to mind.

Let’s say you kill someone and then the police comes to your house to arrest you. Let’s also assume that both of you are completely rational, which means that the police will arrest you only if they are able to prove to you that arresting you is the best thing to do and that if you argue correctly, you may escape the arrest. So then one of the things that can happen is as follows.

Police Guy (PG): I have an arrest warrant for you. You have commited murder, so you will have to spend some time in jail.

You: Jail? What’s that?

PG: Oh, it’s this place where we keep all those people like you who cause troubles to other people in the society so that there is peace in the society.

You: Really? That does sound like a nice thing to do, but I really didn’t know about it.

PG: What do you mean you didn’t know? It’s very clearly written under Section <<some arbitrary three digit number>> of the Indian Constitution.

You: Ok, so what’s written there exactly?

PG: It says people who kill others will be arrested.

You: But I have not read the book. How am I supposed to know that?

PG: You are supposed to know that. Everyone’s supposed to know that.

You: But I didn’t know I was supposed to know that. Is that also written in some book?

I have no clue what could the police guy say after this. There are a few possibilities though. For example, he might say that yes, it’s written in the same book – the Indian Constitution. I don’t know whether that’s true, but if it is, then it’s a very stupid thing to do. If you write in a book that people are supposed to read it, then people can’t know that before reading it and once they have read it, they have already read it and so your instruction solve no purpose. In any case, it will not be surprising if people don’t read it even though you have clearly instructed them to, in the book.

The other possibility is that he says yes, it’s written in a book, which is called, may be, the Indian Meta-Constitution. But then you can use the same reasoning with this book – you didn’t know you were supposed to read it! It’s not just with books though. Let’s say the police guy tells you that it was announced on TV for four weeks continuously. You can still say that you did not know you were supposed to watch it.

So the bottomline is basically this – in any sytem of law, there always exists a rule that is not written anywhere but everyone is somehow supposed to follow it. For example, in the Indian Constitution, the rule is “You are supposed to know that. Everyone’s supposed to know that.” The question is regarding people who break this particular law. Aren’t they innocent, simply because the law was not written anywhere? And as it is clear from the paragraphs above, any crime can be claimed to be a violation of just that one law; you just have to replace ‘murder’ with that crime in the above conversation. So doesn’t this mean that we are all innocent all the time?

Now to some people, this might look like just a stupid analysis of a situation that will never really arise. The policeman, after all, will not entertain questions such as “Jail? What’s that?” He will probably say something like “Come with me, I will show you.” and get along with it. So then why bother about it if it’s not going to be of any help ever? Let me try to present this whole issue from a different point of view, which will hopefully make it look more realistic.

Let’s say I sit with some friends over dinner and prepare a book of laws. We fabricate our own laws that promise to ensure that the world turns into exactly what we want it to be. For example, because we like blue cars, we make it against our law to travel in a car that’s not blue. Then, we hire some strong and sturdy people who go around in public punishing people who own non-blue cars. Now read the above conversation again with the police guy replaced with one of these sturdy people and murder replaced with owning a non-blue car, and may be, jail replaced with doing hundred situps on the footpath. This is something that can definitely happen. In fact people do this every once in a while. For example, people who crave to be in a world where women do not go to pubs and always wear sarees. They have this fabricated book of law which they follow. But it’s somehow very clear to (most of) us that what they are doing is wrong.

Now the point is, even the Indian Constitution, or the law-book of any group of people must have had to pass a similar period, that is, a period when no-one knew about it just like no-one knows about our fabricated book of laws right now. Imagine being in that period and then try to read the conversation above. It will probably make more sense.

I attended Pandit Divyang Vakil’s Tabla performance yesterday. It was a part of Spring Fest 2009. The show was unique in the sense that there were actually four tablas kept on the stage, which were to be played by four different people. I had not seen this happening before. Generally there is either just one person doing a solo or there are two doing a jugalbandi. But four looked exciting. Four different people playing simultaneously is equivalent to one super-human with eight hands playing alone, and this super-human seemed to have a lot of potential. However, when the show started, it wasn’t very clear whether they were really achieving anything by having four different people. Or in other words, would it be possible for less than four people to produce the same effect as them? This kept me occupied for the rest of the show.

The whole performance could be divided into two different modes – one, in which one of them was playing something alone and two, in which more than one people were playing but they were playing the exact same thing, that is, hitting the instrument at the exact same moments at the exact same points. Now the times they were in mode one, could be easily replaced by just one player. I was a little doubtful about mode two. Isn’t four people playing the same thing simultaneously, the same as one person playing it four times louder?

Although it seemed to be true, it did not sound true. I mean, it was very easy to close your eyes and figure out the number of people who were playing simultaneously just from the sound. However, I later realized that it was happening just because of the fact that the sounds coming from two different tablas cannot be the same even though the two players are trying to play the same thing. There has to be some difference, in the exact times at which they are hit, and even in the sound quality. And this difference between the two sounds is what makes four players playing simultaneously different from one player playing alone.

But anyway, that’s not a very big problem. We can still replace such a group of four by just one person if we have some elecronic circuitry at our disposal. All we have to do is receive the sound produced by this one person at one end and then make four copies of it and add some small and random differences in the copies. The result, I think, should sound exactly the same to the audience as yesterday’s show.

So in effect, it seems that three of the tabla players were pretty redundant. A better thing to do for them would be to have the four people playing different things on their respective tablas. That would be something that just cannot be achieved by one player. For example, one single player can never play, let’s say, a Tin and a Na at the exact same moment, even if he makes use of extremely complicated electronic circuitry, because (most probably) both the sounds require the same fingers. But, two different people can do this.

This is so awesome.

I think I felt the General Theory of Relativity very recently. Felt. That’s different from understanding it, or solving a difficult numerical problem from a text book. Now I am not an expert of the general theory. In fact, I don’t even understand it very properly. So I am not really sure whether what I felt was the general theory. However, I remember reading about the famous elevator experiment where Einstein demonstrated that it was not possible for a person (Bob) closed in a spaceship to differentiate between gravity and the pseudo force caused by acceleration of the spaceship in different directions. So if he observes that a ball when dropped falls down, that can be because the spaceship is accelerating upward or because there is a planet below it. Bob has no way to find out which one of the above is true. The elevator experiment is what I had a recent experience with. I felt as if I was Bob.

It happened in Madras, at the airport. I was in a plane that had just landed and was soon to take off for Kolkata. I had not slept properly the last night, so I was half asleep, or, since I have my GRE in three days from now, I was somnolent. You know, the kind of state where your consciousness is transformed into a strange amalgam of dreams and reality, the state where you want to be in the world of dreams but reality keeps disturbing you by constantly running in the background. But why was I half asleep, you would ask, why not full? Because I was sitting, on a chair shaped thing, with my back almost vertical and head unsupported and gravity pulling my lungs towards my butt. This, in general, is a particularly uncomfortable position for going to sleep. Hence the result.

But then something weird happened. Between all this shifting from dreams to reality and back, from the cute girl from school who used to sit next to me to the pretty female crew members in the plane and back, I felt a small, very small period of relief, about a minute long. All this while I was cursing that chair shaped thing I was forced to sit on, I was cursing the fact that I was not going to be able to recline horizontally on a bed for at least six more hours, when suddenly, for that one minute, I felt as if that stupid chair shaped thing I was sitting on, got transmuted into a comfortable bed, as if someone tilted it and tilted it even more until it became almost horizontal, or, better yet, as if someone took the earth and placed it right behind the plane for a while. It was a relief. That one minute was the best sleep I had in the whole journey. Yes, right, the plane was accelerating for the take off.

I am sure that the phrase “jack of all trades and the master of none” was not invented to be used for a person you were pleased with, for if you were, you wouldn’t, for sure, call him jack. You would definitely use something more euphemistic. For example, “someone who is more than decent in several fields but hasn’t really gained the expertise in many of them.” So no doubt, the phrase has a pejorative connotation. But this leads to the following important question – Is being a jack of all trades really inferior to being the master of one (or more)? I believe that it’s not and I will try to explain why in the following paragraphs.

Let me begin my argument by asking this – What exactly is a “trade”? We understand the notion of a trade with the help of our gut feeling. We know, for example, that physics is a trade. We know this because we have seen people acknowledging physics as a separate entity – we have done courses named “physics”, we have seen books with the word “physics” in their names, we know that Nobel Prizes are given to people working in physics, we know their are physics professors in universities and so on. Similarly, we know that cycling is a trade because we have heard of famous cyclists, we know that cycle races are conducted at different parts of the world every now and then and so on. So basically, we decide whether something is a trade or not on the basis of its place in the society, the propensity with which other human beings are willing to recognize it etc., which by the way, is very highly dependent on the way the society has evolved. A society evolved in a different way might have had a completely different set of accepted trades. It is not difficult to imagine a society where, for example, trying to predict a person’s future from his handwriting is considered a trade. Oh wait, that’s our society, but I think I have made the point nonetheless.

But then, the question is – Isn’t this weird? Shouldn’t the concept of a trade and the way the society has evolved be as uncorrelated as possible? A quote from Feynman’s Lectures on Physics is enlightening. At one point in the book, Feynman remarks –

A poet once said “The whole universe is in a glass of wine.” We will probably never know in what sense he meant that, for poets do not write to be understood. But it is true that if we look at a glass closely enough we see the entire universe.

Then he goes on to show how a glass of wine contains the entire universe and adds –

If our small minds, for some convenience, divide this glass of wine, this universe, into parts — physics, biology, geology, astronomy, psychology, and so on — remember that Nature does not know it! So let us put it all back together, not forgetting ultimately what it is for. Let it give us one more final pleasure: drink it and forget it all!

That Nature doesn’t know it is indeed an intelligent observation, which further strengthens my hypothesis that the definition of a trade should be as independent as possible from what the society thinks about it. And yet, we find that there have been enough people to make the phrase “jack of all trades and master of none” a phrase. We can’t really blame all of them, for it definitely seems that there is one way in which a jack of all trades is inferior to the master of one, which is this. A human being depends so much on the society that what it thinks of him might have an effect on what he gets to eat, and in fact, even on whether he gets to eat. It may seem, therefore, that in order to have a happy and contented life, it is safer to keep the society happy, which will be the case if you do what the society recognizes as a trade, even though Nature doesn’t bother about it. The final step of my argument is to prove that even this is not completely true.

I will quote Scott Admas this time. This post on his blog gives a little career advice. Although reading the whole post will be informative, I am copy pasting the most relevant part from it.

If you want an average successful life, it doesn’t take much planning. Just stay out of trouble, go to school, and apply for jobs you might like. But if you want something extraordinary, you have two paths:

1. Become the best at one specific thing.
2. Become very good (top 25%) at two or more things.

The first strategy is difficult to the point of near impossibility. Few people will ever play in the NBA or make a platinum album. I don’t recommend anyone even try.

The second strategy is fairly easy. Everyone has at least a few areas in which they could be in the top 25% with some effort. In my case, I can draw better than most people, but I’m hardly an artist. And I’m not any funnier than the average standup comedian who never makes it big, but I’m funnier than most people. The magic is that few people can draw well and write jokes. It’s the combination of the two that makes what I do so rare. And when you add in my business background, suddenly I had a topic that few cartoonists could hope to understand without living it.

QED

Some interesting things happened as a result of my previous post. For example, several people did a dude-i-have-a-better-method-than-yours with me on Gtalk and also, Robin Kothari claimed that you can actually spend 50 hours in a single day on earth, as opposed to 48. I am going to elaborate upon these two things in the next few paragraphs.

The people who said they had a better method, had the following method in mind. You sit just east of the IDL for one complete day, and a very small number of seconds before midnight, step to the west, the place where the day you are currently at the end of, is actually about to start. So then after spending 24 hours in a day, you are once again at the start of the same day and hence, have another 24 hours in hand. This method has a shortcoming, which, I realized after answering a few buzzes on Gtalk, should have been mentioned in the post itself. The reason is simple. If you step to the west x seconds before midnight, you will first have to spend x seconds in a wrong day. If you step x seconds after midnight, you will again spend x seconds in a wrong day, only this time before stepping over the line. So the only way to achieve our aim would be to cross the line exactly at midnight. This can be accepted as a solution depending on the sloppiness you want to allow. This won’t fit in if you are a perfectionist.

The point raised by Robin is quite interesting. As it turns out, the IDL is not a straight line, as one would have expected. It has kinks in it, the biggest one being directed eastward at a place called Kiritimati. Until about 1994, the line bisected the group of islands, but then, the people got annoyed because the eastern part of the republic was always 24 hours ahead of the western part and hence there were only four days in a week in which official business could be conducted between the two parts. Hence they pushed a part of the line towards east, which led to the following two very weird things, which I quote directly from wikipedia –

1. “However, for two hours every day—Coordinated Universal Time (UTC) 10:00–11:59, there are actually three different days observed at the same time. For example, at UTC time Thursday 10:15, it is Wednesday 23:15 in Samoa, which is eleven hours behind UTC, and it is Friday 00:15 in Kiritimati, which is fourteen hours ahead of UTC.” And also,
2. “Because the earliest and latest time zones are 26 hours apart, any given calendar date exists at some point on the globe for 50 hours. For example, April 11 begins in time zone UTC+14 at 10:00 UTC April 10, and ends in time zone UTC-12 at 12:00 UTC April 12.”

(If the term UTC scares you, you can assume for the present post that UTC is actually GMT.)

Although it seems that wikipedia has directly answered our original question (just keep traveling to the point where it is still April 11), several other questions have been raised in the whole process. For example, “What do you mean by shifting a part of the IDL towards east?”, “Shouldn’t the IDL always be a straight line?”, “Who decides what the IDL should look like?”, “Can we arbitrarily give it any shape whatsoever?”

I am going to try to answer all these questions and hence fit everything into place now.

Suppose for the moment that we do not have such discrete time zones as we have now and that each longitude has its own individual time. If you are unclear about how to assign time to different longitudes, just assume that we say it is 12 noon at a particular longitude whenever the sun is directly above it and we say it is 12 midnight when the sun is directly opposite it. Since the earth completes one rotation in 24 hours, there will be a difference of 24 hours between two consecutive midnights (or noons) at a particular longitude. Also, at any given moment, different longitudes will have different times. So now we have a scheme to assign a unique time to each longitude. Note that by counting the number of times it has been 12 midnight at a particular place, we can assign unique dates to each longitude too. So initially, all the longitudes except the one exactly opposite to the place the sun is currently pointing at, have seen zero midnights. So all the places except this one place has the date 0. As the sun moves towards west, more and more places start seeing midnights and hence their dates increase to 1. After one complete rotation, all the places have seen exactly one midnight, except, of course, the longitude we started with. So this particular longitude has the date 2 and the rest have date 1.

Since we are going to refer to this longitude again and again, let’s give it a name, for example, The Longitude. Now consider a snapshot of the earth taken at a time when, say, it is 11 am of day 3 at The Longitude. The time at a place a bit west of it at this moment will be a little less than 11 am, a bit more to the west, a bit more less than 11 am and so on until you reach a place where the time is 12:01 am of day 3. If you travel further to the west, you will find yourself in day 2 and when you are just short of completing a full round of the earth, you will be at a place where it is just before 11 am of day 2. So whereas it is 11 am of day 3 at The Longitude, it is just less than 11 am of day 2 at the point immediately east to it. So notice that The Longitude, apart from being the place where each calender date arrives first, also happens to be a line, which, if you cross at any point of the day, you will encounter a sudden jump in the date, which will be an increment if you have travel from west to east and a decrement otherwise. Notice that we have a reference to the time when earth started rotating in the definition of The Longitude. Evidently, the earth started rotating a really long time ago, at least much before life became complicated enough that one would require the concept of date and time. So now, when we do require these concepts, we are free to assume that a particular longitude is the one where the sun shone first. This longitude is an approximate version of what we call the International Date Line. However, we are not done yet.

The above scheme would work really well if the only inhabitants of earth were dead cows in a pond. Unfortunately, there are several other occupants of this planet, most notably, living human beings. These are creatures with emotions and (hence?) a large tendency to screw things up. They came up with the idea of segregating the whole of their population into groups they called “countries” and since it was “inconvenient” for people in one country to follow different times, they decided to form discrete time zones. So people formed groups, called time zones and assigned a longitude to each of them, so that everyone in a particular time zone would follow the time at the assigned longitude. Also, for the sake of convenience, they came up with a naming scheme for the time zones. They named a particular longitude (and the time zone which followed its time) the Greenwich Mean Time, or GMT, and named each time zone on the basis of the number of hours each were ahead, or behind of the GMT. So a time zone which was called GMT + 5, was the one which was 5 hours ahead of GMT, and so on. Also, it just so happened, that the longitude they called GMT, was the one exactly opposite the one they called IDL. As a result, the time zones on earth all lied between GMT – 12 and GMT + 12, because if it was GMT + 13, it would be on the other side of the IDL and hence, because of the jump of date, would be called GMT – 11.

We can now answer the questions we started with.

What do you mean by shifting a part of the IDL towards east?

Let’s first understand what it means by shifting the whole of the IDL towards east. It just means that we are ascribing the name IDL to a new longitude. And if we don’t change the position of the GMT, it means we want to divide the earth into time zones from, say, GMT – 10 to GMT + 14, instead of the usual GMT – 12 to GMT + 12. So what does it mean by shifting only a part? It means there will be one small part of the world that will believe that the earth is divided into time zones from GMT – 10 to GMT + 14 and the rest will believe its GMT – 12 to GMT + 12.

Shouldn’t the IDL always be a straight line?

It would at least be simpler that way, but because of problems such as the one faced by the people at Kiritimati, let’s keep it as it is until someone thinks of a more elegant solution.

Who decides what the IDL should look like?

I don’t really know, but this place may give you some idea.

Can we arbitrarily give it any shape whatsoever?

Yes. We are the ones who defined it after all. In fact, it may be fun to think on the following – Is there some shape we can assign to the IDL so that it is always your birthday at some point or the other on earth?

I have to say that 4th May, 2008 was the longest day of my life till now, not just because it never seemed to get over, but also because I actually spent more than 24 hours (28.5, to be precise) in it. Well, the reason, of course, was that I started and ended the day in two different countries (India and Scotland, respectively). However it did make me think on two rather interesting questions – 1. Can one spend a day longer than this? 2. What would be the length of the longest day one can possibly spend being on earth? Obviously, the answer to the first one should be “yes”. It would be too much of a coincidence to unknowingly do something which cannot be outdone as long as you are on earth.

In any case, I did think on the aforementioned questions and after a cup of coffee and some discussions with Eliot Gehrt Setzer, it turned out that the interesting question had a simple answer. The answer, first of all, is 48 hours. Secondly, there are several ways to spend a 48 hours day, an approximate but intuitive way being as follows –

Go to this place called the International Date Line, which, according to wikipedia, “is an imaginary line on the surface of the Earth opposite the Prime Meridian which offsets the date as one travels east or west across it.” Wait there till the start of a new day and start traveling towards west as soon as a fresh day arrives. Remember to keep your speed equal to the speed of the sun (figuratively speaking, of course). This will ensure that its always midnight in the country you are presently in. After just a little less than 24 hours, you will be at a place just a little east of the International Date Line, where it will still be midnight and the date, surprisingly, the same as the one it was when you started. So now you stop and spend your next 24 hours there. At the end you will have spent 48 hours in the same day.

Starting at midnight isn’t really a necessity. The key idea is that you should spend some 24 hours of your 48 hours day without changing the time. This happened to be your first 24 hours in the previous case. It can be the last 24 hours too, or it can be just spread randomly across the day.

Starting at the IDL is a necessity though. That’s because if you don’t start there, then at some point of time you will end up crossing it, resulting in a sudden change of date. Of course, you will have progressed by only one day at the end of your 48 hours in this case too, but you won’t spend all of those 48 hours in the same day.

The above is definitely a good solution as long as you do not mind occasionally traveling to the past. Since the time zones change in a discrete way rather than continuously, even though you are traveling with the sun, so to speak, you will not always remain at midnight (in case you decided to choose the first of the ways described above). You will, depending on the width of the time zones you travel through, keep going back in time by a few hours every now and then. You can however, choose only very narrow time zones to keep such occurrences to the minimum.

Another interesting question arises if you restrict your idea of a “day” to only the time between a sunrise and the next sunset or a date change, whichever happens first. Realize that with this new definition of the day, normal days are less than 24 hours long. They can easily be made equal to 24 hours though, by being at one of the poles during summer. Can we do better? By being just south of the north pole (or north of the south pole) and doing the circle starting at the IDL will do the trick. Being exactly at the pole may lead to some elegant trick, but thinking about the date and time at the pole is scary and I will refrain from getting into it.