A good part of my student life was spent in learning mathematics and of course therefore, learning examples and dreading counter-examples to whatever I may have claimed. I shifted to the study of economics after my formative years were done. Some may say that I continued to dabble with mathematics even then. Be that as it may, I have learnt that there is a distinct shift in the role of examples and counter-examples from the discipline of mathematics to their role in the social sciences in general. In mathematics, these examples play a major role; in fact, I recall a book that we would often consult was one titled “Counter-examples in Analysis”. A book, which a fellow student (who would politically become very important in later life) described as a book containing counter-examples to all theorems! While this was sheer nonsense, imagine writing a book in the social sciences titled Anecdotes, for that is what they are. Amongst all the social sciences, sociology may perhaps tolerate such a title and content, the others specially economics would sniff and dismiss this as being merely anecdotal. This perhaps is as inane as the claim that Counterexamples in Analysis contained counterexamples to all theorems in analysis.
We try first to exhibit the role that examples and counterexamples play in understanding mathematical propositions. Consider the claim for real valued functions of a single variable, continuity need not imply differentiability of the function. And all that is required in defense of this is the function say y=|x|, the absolute value of the variable x. At x= 0, the function is continuous but lacks a derivative. This simple example suffices; to provide the killing blow of a function which is continuous everywhere but lacks a derivative everywhere ( the Weirstrass function) is only for masochists. But that too is there. Consider what an important role that simple function plays. No one would sniff at this as being anecdotal surely? Or consider the claim that continuity is sufficient condition for differentiability. The same example would destroy this claim completely. No one would ask for another example. One is enough.
I am going to present two examples or anecdotes, which I have used sometimes in lectures but have never written them up. In fact, while writing about them, the differences associated with such examples and the modulus of the variable x arose in my mind and I thought that it may be of some interest to see these differences, if any.
During the academic year 1988-89, I was teaching at the University of Tsukuba in Japan. No, I did not have to teach in Japanese; all my lectures were in English. The University had given visiting foreign teachers housing which was located some distance away. The bus fares being expensive, the University also provided an allowance for the fare. The problem was with the bus schedules; there was no direct bus. One had to go to the City Centre and the buses from the University campus were very frequent. From the City Centre, there was just one bus route which went past our housing campus, and that one was quite infrequent, one every hour almost. On cold evenings, my target was to catch the bus from the City Centre which left at 640 pm which meant I could be home by 7pm and just in time to catch the news in English. There was hardly any one catching this bus from the City Centre, except one other person, also from the University. He and I got off at the same stop too. But we never talked while waiting for the bus; this was of course quite common between foreigners and Japanese. I need to specify what happens on Japanese buses then. While getting up, one got in from the middle of the bus; and inside the bus at the top, just facing the entrance there was a board which showed the next four stops; the catch was that these stops were written in Japanese. The third stop was our stop and I had by then started recognizing some of the characters. One day the bus came to our appointed stop, doors opened; I looked in and failed to recognize the characters describing my stop, withdrew. My fellow passenger did the same. The bus closed doors and was on its way. After a few minutes, when I was deciding what to do, my fellow passenger came up to me and asked in reasonably clear English, whether I read Japanese. I said of course no but I do recognize the characters which make up Teshirogi Denchi and I thought these were not indicated. I could be wrong of course. The gentleman confirmed I was indeed correct and wondered where our bus was. It was past the time the bus left; it was almost 645 pm by then. He went towards the small kiosk which housed the bus company office and returned smiling. He explained that the bus driver had forgotten to fix the board before starting out and so it was the driver’s mistake. The bus which had come and left was our bus. And I must have said that Oh no, we have to wait another hour. One might as well walk, I told him. Waiting with a cold wind blowing was not comfortable. And my fellow traveler said, wait they are arranging alternative transport. As he finished speaking two taxis drew up. He and I decided to share a taxi which took us to the Teshirogi Denchi stop. When we tried to pay, the driver said something and my fellow passenger told me that it is paid for by the bus company. The fact of putting up a schedule meant evidently that the bus company will provide a bus service at the appointed time. If for some problem on their side, they cannot meet the stated time they will try to compensate. And they did so that day to the two of us. Thereafter, whenever we met at the City Centre, we would nod to one another with a smile but still we never talked.
This example shows how trust is built, for sure. I do not think anything else needs be said. If you do as you say, come what may, people will trust you. I have used this example many times, once even in Japan. It is of some interest to report a question from a senior faculty who asked “which year was this?”. On hearing the year 1988-89, he said, “that was before; now it won’t happen”; the Japanese economy then was in some trouble thereafter, given the asset price bubble. And the person went on to say how trust was the foundation to the success of all economic activity. Trust was being taken to be a pre-condition for economic success.
The second example follows. In the year 2001 December my father fell down and fractured his femur badly. The doctor in Patna where my father stayed decided that an operation was required and a pin had to be put in; but given that he was old, and was known to have a heart condition, the surgeon would need the presence of a requisite machine (heart-lung machine if I recall correctly) and that we had better shift my father to Kolkata for the operation. Since my father was in pain this was done on an emergency basis and we shifted him by air. He was in no condition to sit down so he was lying on a stretcher which was fixed to the floor of the aircraft cabin and we had to pay for the three seats which had to be removed apart from the tickets for those who accompanied my father. It was done and he was in Kolkata, operated upon and the pin was put in and a month passed when the hospital released him. He was fit to travel back to Patna by air. He could now sit up for about an hour or so but the flying time from Kolkata to Patna was about 40 minutes. If he did not have to wait too long at the airport, it was feasible. Those days, security concerns were less and my idea was that if he could be lying down in the ambulance till the flight was ready to depart and then check in and go directly to the aircraft, it was done. So we landed up at the airport well in advance, negotiated with the Traffic Police to park the ambulance close by. I went ahead to check all of us in, and told the airlines concerned about my father’s predicament. And there met a complete roadblock to my plans. The person at the counter sent some one to check on my father; he came back and reported that my father was lying down in the ambulance and that was the end! We were not allowed to check in and there was some talk of my father not being allowed to fly. The fact that he had a certificate to being fit to fly seemed only to delay the inevitable. I had no other plan but to be the last to check in with my father and hope the flight would not be delayed. Meanwhile the Traffic Police started to complicate matters by saying that the ambulance has to be moved away from where it was parked, since VIP traffic was headed that way. Now as it happens, some senior persons in the administration were my batchmate in college, and I was getting ready to approach whoever VIP it was that was travelling asking for help. What happened thereafter was really a miracle. The person who got down from the VIP car was Professor Amartya Sen. Seeing me he waved and said “Anjan what are you doing here?”. I asked what he was doing, in return, and found he was going to Chennai and the conversation could not have lasted more than a few minutes. And then Professor Sen was on his way. I discovered that a small crowd had gathered by then; Amartya Sen had been awarded the Nobel Memorial Prize in Economics and was very well known; I saw the official who had said that we would not be allowed to board the flight was hurrying towards us. He came up to me and said in Bengali, “Sir, you are known to Professor Sen, please tell me how we can help you?”. I was flabbergasted and first I repeated what I had wanted in the first place and he readily allowed everything and we were allowed to be the last to board and the flight left almost immediately. I should add that this was not Indian Airlines. I wrote to Professor Sen later telling him how his presence had helped.
That’s the story or anecdote. What does it show? In India, the quality of service that one may receive varies tremendously; from the first class first world service to no service at all. It all depends on who is at the receiving end. Friends of Nobel Laureates are special of course and therefore provided quality service; ordinary persons are not to be entertained. Often you see extreme behavior from people asking “Do you know who I am?”, in order to establish some right to service; this is the reason why loudmouths appear to rule. And if you are in the market to purchase something expensive, you ask around amongst your contacts and friends whether anyone knows some one and could put in a word. The airlines people instead of helping me and my father had decided that it was not worth their while to help us. The fact that I may make matters unpleasant for them by complaining (they realized that they were not doing their job!!) given my perceived closeness to an important person, made them change their behavior.
The first example must be pretty unique; the second, must be familiar to everyone at least in India. Together they identify situations where economic activity thrives and where, no matter what, it does not. As conclusively as the example of modulus of x shows the connection between differentiability and continuity.
This too has been privately circulated earlier.
Anjan Mukherji 