Year of Economics – May 1, 2003 – 1: The Numbers Problem
Dear Abhay,
This spring I have been teaching my writing class as usual on Tuesday mornings, but for various reasons, the administration assigned me to a classroom in a different building. We meet in a building on Michigan Avenue, in a newly redesigned classroom on the 5th floor with a spectacular view of The Art Institute and Lake Michigan beyond it. I cannot help thinking of my first visit to The Art Institute of Chicago as a student at the age of nineteen, and how much I loved this part of the city even then. I remember taking a photograph of the buildings in this block. If somebody had told me on that spring day in 1979 that on a similar spring day in 2003 I would walk into one of those same buildings as a writing teacher, I never would have believed it. How does it happen that the future arrives so different from the present, and yet so similar?
We take our mid-class break at 10:35. “We’ll take fifteen minutes now,” I announced, “When we return, we will start with Adam’s written response to Steve’s story from last week. I’ll make photocopies of Adam’s two-page piece for everyone during the break.” We filed out of the classroom, and I walked toward the stairs. There is a 5th floor photocopier, two doors down the hall from my new classroom. However, since both my new classroom and the 5th floor photocopy room technically belong to the Film Department, a department with which I have little familiarity, I only recently learned that this 5th floor photocopy room exists. At the time of the event I am about to describe, the event that prompted the question that initiates our Year of Economics correspondence, I, from force of habit, used the 6th floor photocopy room, which belongs to the Liberal Arts Department, a department for which I teach a second class on Wednesday mornings, and with which I feel far more familiar. The event was this: I entered the stair well and began to walk down.
One flight of stairs later, I encountered a door with the number 4. I stared at it. I said to myself, “Has the building turned upside-down?”
A moment later, I thought, “Wait, it’s true. The numbers increase as the floor go up, and decrease as the floor go down. This has always been the case.”
I walked up two flights, and once on the 6th floor, I began photocopying Adam’s two-page response to Steve. As I did this, I contemplated the possibility that something had gone wrong in my mind. I have encountered some variation in the numbering of floors of buildings, it’s true. But this variation has limited itself to buildings in the UK, for example, which do not number the ground-level floor, or number it zero, and begin counting with 1 on the floor one level above the ground. The floor, in this case, takes its number from the number of flights of stairs one must climb to reach it. Here in America, however, we number the ground-level floor as 1. Thus, after climbing only one flight of stairs, we say to ourselves, “Already I have reached the 2nd floor.” We feel as if we have accomplished something.
I have never encountered or even heard of buildings whose numbers decrease as the floor ascend, with the 20th floor, for example, one flight above the ground, and the 1st floor just below the rooftop. Yet I apparently momentarily imagined myself in just such a building. How did this peculiar mistake come to pass?
The photocopier finished churning out and stapling its copies of Adam’s two-page response to Steve’s story from last week. As I removed them from the tray, I noticed the stack seemed oddly thick. I checked the counter: 20 copies. Why had I made 20 copies for a class with 16 students in it? Instantly I realized the answer – because my Wednesday class has 20 students in it. My Wednesday class is a Liberal Arts class, and I am in the Liberal Arts Department photocopy room. The Writing Program photocopy room is in the building on Wabash Avenue, the one in which the writing class met until it was displaced into this building on Michigan Avenue. The Wednesday Liberal Arts class meets in the building on Wabash Avenue, not far from the Writing Program photocopy room. Of course, this had been the source of the confusion. The number of copies I had made corresponded to the correct number of students but the wrong day of the week. I began to return to the classroom. I went to the stairs and stood a moment on the landing. Which direction do I go now?
Suddenly, I knew the answer.
Ramanujan, the great mathematician from India, claimed he had a distinct friendship with each of the first 100 integers. How do we form such friendships? I befriended Michigan Avenue in 1979 when, as a student I took its photograph. I befriended Wabash Avenue in 1992 when I began teaching there. I befriended the 6th floor of the Michigan Avenue building two years ago, when the Liberal Arts office moved into the new facilities there. I befriended the 5th floor only this semester, as the home of my new classroom. I am still in the process of befriending it. Somehow, it seemed the earlier friendship naturally belonged closer to ground level. As one appears to climb in life, each year like a new flight of stairs, and every preceding year existing below – like geological layers, or like the unseen stilts extending beneath the oldest guest at the party in the final pages of the final volume of Marcel Proust’s In Search of Lost Time – time stacks up.
I now understood the elusive pleasant sensation I had felt as I had headed down the stairs from the 5th floor expecting to find the 6th floor below, and the startling sense of injustice at finding the 4th floor in its place. I related this pleasantness, the sensation of minor well-being, to the sensation I had often felt in early school math classes when a problem, at first opaque, became clear. “Is that all there is to it?” I would say to myself, in algebra class, for example, when solving for the value of x by dividing both sides of the equal sign by the common denominator. “This is too simple.” Still, I had absorbed the experience. I had befriended not only a number, but a pattern, an equation, a system of mathematics, no matter how small. Maybe its smallness allowed me to absorb it. Maybe this is what people mean when they say the word, “learning.”
I can still remember the exact point when this feeling of well-being stopped arising: trigonometry. From that point on, mathematics inspired only a vague internal panic, which I managed to keep at bay until the point at which math was no longer a requirement. From that point on, I have never turned back to look at that fog-enshrouded nexus lurking behind me. Yet I know it remains, awaiting its chance to engulf me in its hazy labyrinth and call me stupid.
For a moment, on the stairs, the fog, it seemed, had lifted, and I saw the world not as it was but as it could be, wherein a personal relation to numbers aligns with the objective relations of their mathematical formulae. As if a voice had whispered to me, “This building counts its floors in an order that echoes the events of your life.” It was as if I knew these numbers and patterns so well that I had begun to dance with them, and before that clarity dispelled and the fog descended again, a secondary question arose, which I ask you now, concerning economics.
I received the textbook you wrote. Thanks for sending it. I opened to page 3, Exercise 1.1. Pencil in hand, I read the directive. I reread it. I reread it again. I turned to Section Four: Interviews with Economists, and there I have remained ever since, paralyzed by numbers, comfortable among words.
In one interview, Paul Krugman talks about long-term economic problems that have grown steadily worse in the US since the post-war years. Among the reasons for this, he prioritizes a diminished rate of productivity and a growth in poverty.
“…we have growing poverty, which is the consequence both of the fact that the overall pie is growing slowly and that the distribution of the pie is getting more and more unequal. People at the bottom have had a quite rapidly falling share of national income, and we’re pushing more and more people over the edge into a really appalling state of poverty, given that we’re still a very rich country.”
He goes on to discuss the damaging effects of poverty on the overall economy, and all the unused tools at our disposal to address the problem. This reminded me of Amartya Sen’s efforts to measure economic growth by looking at indicators such as infant mortality and literacy rates – factors associated with the poor in a society – rather than, for example, the housing market – a factor associated with the affluent sectors. Why do we fail to approach economics in such human terms? While a powerful minority may resist the logic of Sen or Krugman – that fighting poverty makes sense from both human and economic standpoints – I suspect most people would agree. What is it that so deeply inhibits our economic will? Could it be simply The Numbers Problem?
Because of the math wall that I, and I suspect most Americans, hit around the point of trigonometry, we have turned to fairer pastures in which we can feel, and perhaps even be, fairly intelligent: words, images, music, fashion, cooking, bodies. We understand stories of poor people and rich people. We tend to feel sympathy for people in stories. But the same inhibition that stopped us in math class now prevents us from understanding the real difference between one million and one billion, or from understanding the economic consequences of poverty. Math is simply not human. Without a little math, how can we understand economics? I was impressed by your analysis of valueless exchange in this regard, and because of this I see you in the lineage of Amartya Sen as one who attempts to re-frame economics conceptually and succeeds in avoiding math altogether. But another problem exists, which is that certain very human consequences cannot be understood without the abstract reasoning of mathematics, with all its difficulty. To paraphrase Euclid’s response to his complaining pupil the King, “There is no royal road to math.”
So before we can talk about economics, I feel I must ask you this question, and I believe you are the person to answer it.
How is math human?
The events I have related here perhaps point the direction of an answer, but they are very subjective events, stumbled upon by accident. As such, they only successfully provoke the question. I turn to you Abhay, the economist, to ask, does this question matter? If so, how can we systematize our responses? How can we fashion a mathematics that resembles the building that counts its floors in an order that echoes the events of our lives?
After the break, I returned to the front of the classroom, and proudly announced, “I have made an important discovery about the numbers in my life. I will explain it all to you in time. First, we will hear Adam’s response to Steve. Pass these copies around and take one. And will the last person please hand the four extra copies back to me?”
Sincerely,
Matthew
Year of Economics 2: The Mangoes Problem.
Berkeley, May 7, 2003
My dear Matthew,
1.
I was stunned by your email inaugurating our Year of Economics. It was so very intriguing that I was speechless at first. Later, as I rode my bicycle home from work I laughed out aloud. By the time I was home and read your letter to Krista, watched Krista’s eyes become big with wonder like a Japanimation character, I found myself already developing a response. It is in nine parts.
2.
I have led the students in my microeconomics class to the end of one of the two long corridors on the 2nd floor. My two volunteers, Trina and Juan Carlos start walking to the other end of the corridor. I start clapping my hands. My students take the cue and join in. A slow, resonant, rhythmic clapping soon echoes through the entire 2nd floor. I notice that my students clap using not just an arm motion, but using their entire bodies. They are enjoying themselves.
I walk, or rather skip down the corridor and catch up with Trina and Juan Carlos. I stop and position Trina just outside room 209, facing the clapping mass of students up the hallway with a sign around her neck: USA. I walk Juan Carlos all the way to room 202, which is the last room on that floor. Juan Carlos also faces the clapping students way up the corridor with a sign around his neck: China. I run and join the clapping students.
Trina and Juan Carlos are to walk up the corridor. My students see that Trina is starting way ahead of Juan Carlos. That gives her a considerable advantage over Juan Carlos in getting to us. On the other hand, once they get started, I have asked Trina to take one step every time she hears a clap, and Juan Carlos to take three and a half steps.
The experiment begins. At first it seems inevitable, even to me, even though I know the answer, that Trina will get to us sooner than Juan Carlos. We can see her approaching us, right here, perhaps twenty-five meters away from us while Juan Carlos is so far away from us that we can not even see his face very clearly. The rhythmic clapping goes on and at a certain point, although the rhythm is unchanged the clapping gets louder. Juan Carlos is fast approaching Trina.
Then it happens. Around room 213, Juan Carlos moves ahead of Trina and in a few claps is with us leaving Trina to join us a few claps later. Everyone laughs and thumps both Juan Carlos and Trina on their backs. We return to the classroom.
Each time we clapped, a year went by, I explain. For the next hour, we plunge into the intricacies of the significance of starting positions and the rate of change. By the time I am done discussing the gross domestic product and productivity, the students have little smiles on their faces as if to say: Is that all there is to it?
3.
The great economist John Maynard Keynes once wrote in an essay on Isaac Newton that people have generally misunderstood the nature of Newton’s genius. According to Keynes, Newton was a very intuitive thinker. He would ponder the questions on his mind not with methodical, or even mathematical precision, but rather with wild leaps of fancy. It was only when he was intuitively convinced of an idea that he would sit down to set it to mathematics. We do a singular disservice to Newton’s genius when we think of ‘Newtonian Physics’ as mechanical, abstract, and disembodied.
Newton was involved in solving conceptual puzzles just as Robert Lucas, Nobel Laureate in Economics and inventor of the concept of Rational Expectations was. Lucas’s students describe him as being ‘very intuitive’. In each case, scientific work begins with a conceptual puzzle, is developed intuitively, and only ends with a mathematical formulation.
4.
Matthew Goulish walks down a flight of stairs from the fifth floor and encounters the fourth floor when he fully expects to encounter the sixth, ponders this question, and finally arrives at a sophisticated, intuitive answer: Time stacks up. He lacks the mathematical rigor necessary to end his enquiry with the appropriate mathematical formulation. But to focus on what has not been accomplished is to miss a very important point: It is not so important that Matthew Goulish is lacking in the requisite mathematics necessary for the analysis of the problem he has posed. What is important is that his formulation of the problem and intuitive development of a solution is far more sophisticated in its analytical content than even a college mathematics major would be able to formulate.
The numbers problem as I see it is not that most Americans are lagging behind in their math skills but that the math that is taught in our schools is not rich enough to formalize the conceptual puzzles that most intelligent students are capable of formulating and solving.
5.
Education has increasingly diminished the role of the intelligence of the body. Over time, the arbitrary separation of mind and body has led to the creation of sterile classrooms for the exercise of the mind, and smelly, recycled-air gyms for the exercise of the body. In the great Buddhist Universities of ancient India, students would go on long walks with their teachers. Similarly in ancient Greece, the followers of Aristotle would walk about in the Lyceum while he was teaching. Learning would involve mind, body, and spirit, in no particular order or hierarchy, but as a collaborative process.
Today, millions of American students are being asked to memorize facts, learn math, and engage in economic reasoning. By all measures, these attempts have been a failure. Adding ‘extra-curricular activities’ will not help. Making them listen to Mozart will not help. What will help is making them move as they learn. What will help is encouraging children to formulate conceptual puzzles as they go about their daily lives and solving them. Then a skilled teacher could introduce the math necessary to create a mathematical formulation of the puzzle and its solution while making it clear that the math would most likely not capture the richness and intricacies of the child’s conceptual framework.
6.
Being the first faculty member hired at the new San Francisco Bay area campus of the DeVry Institute in 1988, I was one of the first to walk into the building we are housed in. I walked into the building with a small group of people. While the others stepped here and there tentatively, I walked straight ahead, took a left turn, walked up the flight of stairs, took a right and headed through the unmarked door to the series of offices by the windows, and dropped off my things in an office overlooking the mountains. No, I had never been in this particular building before, but I had spent four years teaching at the DeVry Institute in Chicago, and I hazarded a guess that economics would dictate a replication of the building we had in Chicago in all the five new campuses being built nation wide.
My colleagues started holding me in mock awe for my successful guesswork when I confidently walked into the mens washroom, found that the builder had forgotten to provide urinals, and realized only a few minutes into the conducting of my business that a washroom without urinals sometimes acts as a washroom for women.
My mixed success at acting nonchalantly familiar with the new building brought to my mind the principle of Tolerance. We recognize people and places not because they look the same every day, but because we allow a certain degree of tolerance, a margin of error. And hence we sometimes make mistakes.
7.
My view of the Numbers Problem can be illustrated with the Mangoes Problem. In R. K. Narayan’s great first book, Swami and Friends, there is a chapter in which the young boy Swami is given an algebraic word problem to solve. A problem in a form probably common around the world: If it costs 10 rupees for 5 mangoes, how much would you have to pay the mango seller for 2 mangoes? Swami agonizes over this problem endlessly. Would father clear up the point of whether the mangoes are ripe or raw? For who but a fool would pay 10 rupees for 5 mangoes that were unripe? And just because the mango seller wanted 10 rupees, would one simply agree to such a price? Would father allow Swami to run to the market to ascertain the current market price of mangoes, both ripe and unripe, to bring to the consideration of the problem at hand?
Swami is conceptualizing the Mangoes Problem in a manner that is taken in context, an approach far too rich for the algebra he has been taught. Predictably, his father, echoing the attitudes of millions of parents and school-teachers world-wide, sees Swami’s conceptualization as merely an attempt to get out of doing his work and severely admonishes him. Swami does sit down and finally reach the answer. But it is very likely that Swami, like his creator R. K. Narayan (see My Days, Narayan’s autobiography), never took up the study of mathematics beyond the compulsory stage.
8.
Matthew, you started out from the fifth floor, went down a flight of stairs and expected to reach the sixth floor. You stared at the fourth floor sign and thought, has the building turned upside down?
Now suppose you are standing on Michigan Avenue. You are then blindfolded. You are told that you will be teleported to one of the floors of the Michigan Ave. building using a very safe process as seen on Star Trek. Zap! Your blindfold is removed. You blink once, perhaps twice, and see the fourth floor sign. But you do not stare. You do not think, has the building turned upside down?
This new, imaginary situation is path independent. A choice is said to be path independent when the alternative chosen is independent of the order in which the alternatives are considered.
The situation which you actually experienced, on the other hand, was path dependent. Mathematical logic says that you must not expect the floor below the fifth floor to be the sixth floor. But you did! You fully expected to reach the sixth floor when you descended the stairs from the fifth floor because you befriended the sixth floor earlier in time. Mathematical logic often requires us to behave as if events were path independent, when it is very often the case that they are path dependent. You, Matthew Goulish, went down a flight of stairs from the fifth floor expecting to encounter your older friend the sixth floor. One may say you shouldn’t have expected that, that it was wrong to expect that. But you did. And that says as much about mathematical logic as it does about Matthew Goulish.
A similar example occurs in economics with some regularity. Economists would tell the world that sunk costs do not matter in further decision making. Once you’ve bought the expensive suit and passed the money-back return period, your decision whether to ever wear it or dump it in a heap with some old clothes is unaffected by its price. Whether you paid $800 for it or $80, your decisions of how often to wear it, whether to dry clean it or wash it at a Laundromat, etc. should be unaffected by its price which is after all a sunk cost, a done deal. The mathematical logic of the argument is irrefutably sound.
However, people often do not behave that way.
9.
Human beings are capable of an extraordinarily wide range of conceptualizations and emotions. As the philosopher J. Krishnamurti has pointed out (Last Bombay Talks, 1985), knowledge is memory. And memory comes from experience. It is fairly obvious that we humans have experienced a very limited amount of what we are capable of experiencing. How long would it take humans to experience all that they are capable of experiencing? A thousand years? A million years? A billion years? It seems to me that no amount of time will lead us to an absolute experience of, or knowledge of being human. In that sense, what it means to be human extends to infinity. Humanity is infinite.
Now, our mathematical knowledge is very limited. Knowledge, mathematical or otherwise, comes as we have seen, from experience. So part of the limitation of mathematics surely comes from the fact that human experience is limited. How long would it take us humans to learn everything there is to know about mathematics? A thousand years? A million years? A billion years? Once again it seems to me that no amount of time will lead us to an absolute understanding of mathematics. There will always be more, both quantitatively and also qualitatively, as with the development of more and more subtle and hence richer approaches to mathematical theorization. So math extends to infinity. Mathematics is infinite.
Humanity is infinite. Mathematics is infinite. But I would argue that one infinity is infinitely vaster than the other.
Consider a simple comparison of two infinities A and B. Suppose A is a set of all numbers including decimals between 1 and 10. This set includes not just 1, 2, etc. but also 1.1, 1.59, etc. Now suppose B is a set of integers 1, 2, 3 and so on into infinity. Both A and B are infinite sets. Yet the infinite set B can fit into a tiny corner of infinite set A. For example, the entire set B which consists of 1, 2, 3, …… can fit between the numbers 1 and 2 in set A: 1.1, 1.2, 1. 3, ……. so on into infinity never even reaching the number 2. The set B fits into the small space between the numbers 1 and 2, leaving the vast expanse of set A from 2 to 10 unfilled.
I believe set A illustrates what it means to be human. A seemingly modest set that can be contained within the fingers of our human hands. Set B illustrates the field of mathematics. An ever expanding additive structure like the floors of an imaginary building that extends into the sky. Though both are infinite, mathematics can capture but a small part of what it means to be human.
Sincerely,
Abhay
Dear Abhay,
This spring I have been teaching my writing class as usual on Tuesday mornings, but for various reasons, the administration assigned me to a classroom in a different building. We meet in a building on Michigan Avenue, in a newly redesigned classroom on the 5th floor with a spectacular view of The Art Institute and Lake Michigan beyond it. I cannot help thinking of my first visit to The Art Institute of Chicago as a student at the age of nineteen, and how much I loved this part of the city even then. I remember taking a photograph of the buildings in this block. If somebody had told me on that spring day in 1979 that on a similar spring day in 2003 I would walk into one of those same buildings as a writing teacher, I never would have believed it. How does it happen that the future arrives so different from the present, and yet so similar?
We take our mid-class break at 10:35. “We’ll take fifteen minutes now,” I announced, “When we return, we will start with Adam’s written response to Steve’s story from last week. I’ll make photocopies of Adam’s two-page piece for everyone during the break.” We filed out of the classroom, and I walked toward the stairs. There is a 5th floor photocopier, two doors down the hall from my new classroom. However, since both my new classroom and the 5th floor photocopy room technically belong to the Film Department, a department with which I have little familiarity, I only recently learned that this 5th floor photocopy room exists. At the time of the event I am about to describe, the event that prompted the question that initiates our Year of Economics correspondence, I, from force of habit, used the 6th floor photocopy room, which belongs to the Liberal Arts Department, a department for which I teach a second class on Wednesday mornings, and with which I feel far more familiar. The event was this: I entered the stair well and began to walk down.
One flight of stairs later, I encountered a door with the number 4. I stared at it. I said to myself, “Has the building turned upside-down?”
A moment later, I thought, “Wait, it’s true. The numbers increase as the floor go up, and decrease as the floor go down. This has always been the case.”
I walked up two flights, and once on the 6th floor, I began photocopying Adam’s two-page response to Steve. As I did this, I contemplated the possibility that something had gone wrong in my mind. I have encountered some variation in the numbering of floors of buildings, it’s true. But this variation has limited itself to buildings in the UK, for example, which do not number the ground-level floor, or number it zero, and begin counting with 1 on the floor one level above the ground. The floor, in this case, takes its number from the number of flights of stairs one must climb to reach it. Here in America, however, we number the ground-level floor as 1. Thus, after climbing only one flight of stairs, we say to ourselves, “Already I have reached the 2nd floor.” We feel as if we have accomplished something.
I have never encountered or even heard of buildings whose numbers decrease as the floor ascend, with the 20th floor, for example, one flight above the ground, and the 1st floor just below the rooftop. Yet I apparently momentarily imagined myself in just such a building. How did this peculiar mistake come to pass?
The photocopier finished churning out and stapling its copies of Adam’s two-page response to Steve’s story from last week. As I removed them from the tray, I noticed the stack seemed oddly thick. I checked the counter: 20 copies. Why had I made 20 copies for a class with 16 students in it? Instantly I realized the answer – because my Wednesday class has 20 students in it. My Wednesday class is a Liberal Arts class, and I am in the Liberal Arts Department photocopy room. The Writing Program photocopy room is in the building on Wabash Avenue, the one in which the writing class met until it was displaced into this building on Michigan Avenue. The Wednesday Liberal Arts class meets in the building on Wabash Avenue, not far from the Writing Program photocopy room. Of course, this had been the source of the confusion. The number of copies I had made corresponded to the correct number of students but the wrong day of the week. I began to return to the classroom. I went to the stairs and stood a moment on the landing. Which direction do I go now?
Suddenly, I knew the answer.
Ramanujan, the great mathematician from India, claimed he had a distinct friendship with each of the first 100 integers. How do we form such friendships? I befriended Michigan Avenue in 1979 when, as a student I took its photograph. I befriended Wabash Avenue in 1992 when I began teaching there. I befriended the 6th floor of the Michigan Avenue building two years ago, when the Liberal Arts office moved into the new facilities there. I befriended the 5th floor only this semester, as the home of my new classroom. I am still in the process of befriending it. Somehow, it seemed the earlier friendship naturally belonged closer to ground level. As one appears to climb in life, each year like a new flight of stairs, and every preceding year existing below – like geological layers, or like the unseen stilts extending beneath the oldest guest at the party in the final pages of the final volume of Marcel Proust’s In Search of Lost Time – time stacks up.
I now understood the elusive pleasant sensation I had felt as I had headed down the stairs from the 5th floor expecting to find the 6th floor below, and the startling sense of injustice at finding the 4th floor in its place. I related this pleasantness, the sensation of minor well-being, to the sensation I had often felt in early school math classes when a problem, at first opaque, became clear. “Is that all there is to it?” I would say to myself, in algebra class, for example, when solving for the value of x by dividing both sides of the equal sign by the common denominator. “This is too simple.” Still, I had absorbed the experience. I had befriended not only a number, but a pattern, an equation, a system of mathematics, no matter how small. Maybe its smallness allowed me to absorb it. Maybe this is what people mean when they say the word, “learning.”
I can still remember the exact point when this feeling of well-being stopped arising: trigonometry. From that point on, mathematics inspired only a vague internal panic, which I managed to keep at bay until the point at which math was no longer a requirement. From that point on, I have never turned back to look at that fog-enshrouded nexus lurking behind me. Yet I know it remains, awaiting its chance to engulf me in its hazy labyrinth and call me stupid.
For a moment, on the stairs, the fog, it seemed, had lifted, and I saw the world not as it was but as it could be, wherein a personal relation to numbers aligns with the objective relations of their mathematical formulae. As if a voice had whispered to me, “This building counts its floors in an order that echoes the events of your life.” It was as if I knew these numbers and patterns so well that I had begun to dance with them, and before that clarity dispelled and the fog descended again, a secondary question arose, which I ask you now, concerning economics.
I received the textbook you wrote. Thanks for sending it. I opened to page 3, Exercise 1.1. Pencil in hand, I read the directive. I reread it. I reread it again. I turned to Section Four: Interviews with Economists, and there I have remained ever since, paralyzed by numbers, comfortable among words.
In one interview, Paul Krugman talks about long-term economic problems that have grown steadily worse in the US since the post-war years. Among the reasons for this, he prioritizes a diminished rate of productivity and a growth in poverty.
“…we have growing poverty, which is the consequence both of the fact that the overall pie is growing slowly and that the distribution of the pie is getting more and more unequal. People at the bottom have had a quite rapidly falling share of national income, and we’re pushing more and more people over the edge into a really appalling state of poverty, given that we’re still a very rich country.”
He goes on to discuss the damaging effects of poverty on the overall economy, and all the unused tools at our disposal to address the problem. This reminded me of Amartya Sen’s efforts to measure economic growth by looking at indicators such as infant mortality and literacy rates – factors associated with the poor in a society – rather than, for example, the housing market – a factor associated with the affluent sectors. Why do we fail to approach economics in such human terms? While a powerful minority may resist the logic of Sen or Krugman – that fighting poverty makes sense from both human and economic standpoints – I suspect most people would agree. What is it that so deeply inhibits our economic will? Could it be simply The Numbers Problem?
Because of the math wall that I, and I suspect most Americans, hit around the point of trigonometry, we have turned to fairer pastures in which we can feel, and perhaps even be, fairly intelligent: words, images, music, fashion, cooking, bodies. We understand stories of poor people and rich people. We tend to feel sympathy for people in stories. But the same inhibition that stopped us in math class now prevents us from understanding the real difference between one million and one billion, or from understanding the economic consequences of poverty. Math is simply not human. Without a little math, how can we understand economics? I was impressed by your analysis of valueless exchange in this regard, and because of this I see you in the lineage of Amartya Sen as one who attempts to re-frame economics conceptually and succeeds in avoiding math altogether. But another problem exists, which is that certain very human consequences cannot be understood without the abstract reasoning of mathematics, with all its difficulty. To paraphrase Euclid’s response to his complaining pupil the King, “There is no royal road to math.”
So before we can talk about economics, I feel I must ask you this question, and I believe you are the person to answer it.
How is math human?
The events I have related here perhaps point the direction of an answer, but they are very subjective events, stumbled upon by accident. As such, they only successfully provoke the question. I turn to you Abhay, the economist, to ask, does this question matter? If so, how can we systematize our responses? How can we fashion a mathematics that resembles the building that counts its floors in an order that echoes the events of our lives?
After the break, I returned to the front of the classroom, and proudly announced, “I have made an important discovery about the numbers in my life. I will explain it all to you in time. First, we will hear Adam’s response to Steve. Pass these copies around and take one. And will the last person please hand the four extra copies back to me?”
Sincerely,
Matthew
Year of Economics 2: The Mangoes Problem.
Berkeley, May 7, 2003
My dear Matthew,
1.
I was stunned by your email inaugurating our Year of Economics. It was so very intriguing that I was speechless at first. Later, as I rode my bicycle home from work I laughed out aloud. By the time I was home and read your letter to Krista, watched Krista’s eyes become big with wonder like a Japanimation character, I found myself already developing a response. It is in nine parts.
2.
I have led the students in my microeconomics class to the end of one of the two long corridors on the 2nd floor. My two volunteers, Trina and Juan Carlos start walking to the other end of the corridor. I start clapping my hands. My students take the cue and join in. A slow, resonant, rhythmic clapping soon echoes through the entire 2nd floor. I notice that my students clap using not just an arm motion, but using their entire bodies. They are enjoying themselves.
I walk, or rather skip down the corridor and catch up with Trina and Juan Carlos. I stop and position Trina just outside room 209, facing the clapping mass of students up the hallway with a sign around her neck: USA. I walk Juan Carlos all the way to room 202, which is the last room on that floor. Juan Carlos also faces the clapping students way up the corridor with a sign around his neck: China. I run and join the clapping students.
Trina and Juan Carlos are to walk up the corridor. My students see that Trina is starting way ahead of Juan Carlos. That gives her a considerable advantage over Juan Carlos in getting to us. On the other hand, once they get started, I have asked Trina to take one step every time she hears a clap, and Juan Carlos to take three and a half steps.
The experiment begins. At first it seems inevitable, even to me, even though I know the answer, that Trina will get to us sooner than Juan Carlos. We can see her approaching us, right here, perhaps twenty-five meters away from us while Juan Carlos is so far away from us that we can not even see his face very clearly. The rhythmic clapping goes on and at a certain point, although the rhythm is unchanged the clapping gets louder. Juan Carlos is fast approaching Trina.
Then it happens. Around room 213, Juan Carlos moves ahead of Trina and in a few claps is with us leaving Trina to join us a few claps later. Everyone laughs and thumps both Juan Carlos and Trina on their backs. We return to the classroom.
Each time we clapped, a year went by, I explain. For the next hour, we plunge into the intricacies of the significance of starting positions and the rate of change. By the time I am done discussing the gross domestic product and productivity, the students have little smiles on their faces as if to say: Is that all there is to it?
3.
The great economist John Maynard Keynes once wrote in an essay on Isaac Newton that people have generally misunderstood the nature of Newton’s genius. According to Keynes, Newton was a very intuitive thinker. He would ponder the questions on his mind not with methodical, or even mathematical precision, but rather with wild leaps of fancy. It was only when he was intuitively convinced of an idea that he would sit down to set it to mathematics. We do a singular disservice to Newton’s genius when we think of ‘Newtonian Physics’ as mechanical, abstract, and disembodied.
Newton was involved in solving conceptual puzzles just as Robert Lucas, Nobel Laureate in Economics and inventor of the concept of Rational Expectations was. Lucas’s students describe him as being ‘very intuitive’. In each case, scientific work begins with a conceptual puzzle, is developed intuitively, and only ends with a mathematical formulation.
4.
Matthew Goulish walks down a flight of stairs from the fifth floor and encounters the fourth floor when he fully expects to encounter the sixth, ponders this question, and finally arrives at a sophisticated, intuitive answer: Time stacks up. He lacks the mathematical rigor necessary to end his enquiry with the appropriate mathematical formulation. But to focus on what has not been accomplished is to miss a very important point: It is not so important that Matthew Goulish is lacking in the requisite mathematics necessary for the analysis of the problem he has posed. What is important is that his formulation of the problem and intuitive development of a solution is far more sophisticated in its analytical content than even a college mathematics major would be able to formulate.
The numbers problem as I see it is not that most Americans are lagging behind in their math skills but that the math that is taught in our schools is not rich enough to formalize the conceptual puzzles that most intelligent students are capable of formulating and solving.
5.
Education has increasingly diminished the role of the intelligence of the body. Over time, the arbitrary separation of mind and body has led to the creation of sterile classrooms for the exercise of the mind, and smelly, recycled-air gyms for the exercise of the body. In the great Buddhist Universities of ancient India, students would go on long walks with their teachers. Similarly in ancient Greece, the followers of Aristotle would walk about in the Lyceum while he was teaching. Learning would involve mind, body, and spirit, in no particular order or hierarchy, but as a collaborative process.
Today, millions of American students are being asked to memorize facts, learn math, and engage in economic reasoning. By all measures, these attempts have been a failure. Adding ‘extra-curricular activities’ will not help. Making them listen to Mozart will not help. What will help is making them move as they learn. What will help is encouraging children to formulate conceptual puzzles as they go about their daily lives and solving them. Then a skilled teacher could introduce the math necessary to create a mathematical formulation of the puzzle and its solution while making it clear that the math would most likely not capture the richness and intricacies of the child’s conceptual framework.
6.
Being the first faculty member hired at the new San Francisco Bay area campus of the DeVry Institute in 1988, I was one of the first to walk into the building we are housed in. I walked into the building with a small group of people. While the others stepped here and there tentatively, I walked straight ahead, took a left turn, walked up the flight of stairs, took a right and headed through the unmarked door to the series of offices by the windows, and dropped off my things in an office overlooking the mountains. No, I had never been in this particular building before, but I had spent four years teaching at the DeVry Institute in Chicago, and I hazarded a guess that economics would dictate a replication of the building we had in Chicago in all the five new campuses being built nation wide.
My colleagues started holding me in mock awe for my successful guesswork when I confidently walked into the mens washroom, found that the builder had forgotten to provide urinals, and realized only a few minutes into the conducting of my business that a washroom without urinals sometimes acts as a washroom for women.
My mixed success at acting nonchalantly familiar with the new building brought to my mind the principle of Tolerance. We recognize people and places not because they look the same every day, but because we allow a certain degree of tolerance, a margin of error. And hence we sometimes make mistakes.
7.
My view of the Numbers Problem can be illustrated with the Mangoes Problem. In R. K. Narayan’s great first book, Swami and Friends, there is a chapter in which the young boy Swami is given an algebraic word problem to solve. A problem in a form probably common around the world: If it costs 10 rupees for 5 mangoes, how much would you have to pay the mango seller for 2 mangoes? Swami agonizes over this problem endlessly. Would father clear up the point of whether the mangoes are ripe or raw? For who but a fool would pay 10 rupees for 5 mangoes that were unripe? And just because the mango seller wanted 10 rupees, would one simply agree to such a price? Would father allow Swami to run to the market to ascertain the current market price of mangoes, both ripe and unripe, to bring to the consideration of the problem at hand?
Swami is conceptualizing the Mangoes Problem in a manner that is taken in context, an approach far too rich for the algebra he has been taught. Predictably, his father, echoing the attitudes of millions of parents and school-teachers world-wide, sees Swami’s conceptualization as merely an attempt to get out of doing his work and severely admonishes him. Swami does sit down and finally reach the answer. But it is very likely that Swami, like his creator R. K. Narayan (see My Days, Narayan’s autobiography), never took up the study of mathematics beyond the compulsory stage.
8.
Matthew, you started out from the fifth floor, went down a flight of stairs and expected to reach the sixth floor. You stared at the fourth floor sign and thought, has the building turned upside down?
Now suppose you are standing on Michigan Avenue. You are then blindfolded. You are told that you will be teleported to one of the floors of the Michigan Ave. building using a very safe process as seen on Star Trek. Zap! Your blindfold is removed. You blink once, perhaps twice, and see the fourth floor sign. But you do not stare. You do not think, has the building turned upside down?
This new, imaginary situation is path independent. A choice is said to be path independent when the alternative chosen is independent of the order in which the alternatives are considered.
The situation which you actually experienced, on the other hand, was path dependent. Mathematical logic says that you must not expect the floor below the fifth floor to be the sixth floor. But you did! You fully expected to reach the sixth floor when you descended the stairs from the fifth floor because you befriended the sixth floor earlier in time. Mathematical logic often requires us to behave as if events were path independent, when it is very often the case that they are path dependent. You, Matthew Goulish, went down a flight of stairs from the fifth floor expecting to encounter your older friend the sixth floor. One may say you shouldn’t have expected that, that it was wrong to expect that. But you did. And that says as much about mathematical logic as it does about Matthew Goulish.
A similar example occurs in economics with some regularity. Economists would tell the world that sunk costs do not matter in further decision making. Once you’ve bought the expensive suit and passed the money-back return period, your decision whether to ever wear it or dump it in a heap with some old clothes is unaffected by its price. Whether you paid $800 for it or $80, your decisions of how often to wear it, whether to dry clean it or wash it at a Laundromat, etc. should be unaffected by its price which is after all a sunk cost, a done deal. The mathematical logic of the argument is irrefutably sound.
However, people often do not behave that way.
9.
Human beings are capable of an extraordinarily wide range of conceptualizations and emotions. As the philosopher J. Krishnamurti has pointed out (Last Bombay Talks, 1985), knowledge is memory. And memory comes from experience. It is fairly obvious that we humans have experienced a very limited amount of what we are capable of experiencing. How long would it take humans to experience all that they are capable of experiencing? A thousand years? A million years? A billion years? It seems to me that no amount of time will lead us to an absolute experience of, or knowledge of being human. In that sense, what it means to be human extends to infinity. Humanity is infinite.
Now, our mathematical knowledge is very limited. Knowledge, mathematical or otherwise, comes as we have seen, from experience. So part of the limitation of mathematics surely comes from the fact that human experience is limited. How long would it take us humans to learn everything there is to know about mathematics? A thousand years? A million years? A billion years? Once again it seems to me that no amount of time will lead us to an absolute understanding of mathematics. There will always be more, both quantitatively and also qualitatively, as with the development of more and more subtle and hence richer approaches to mathematical theorization. So math extends to infinity. Mathematics is infinite.
Humanity is infinite. Mathematics is infinite. But I would argue that one infinity is infinitely vaster than the other.
Consider a simple comparison of two infinities A and B. Suppose A is a set of all numbers including decimals between 1 and 10. This set includes not just 1, 2, etc. but also 1.1, 1.59, etc. Now suppose B is a set of integers 1, 2, 3 and so on into infinity. Both A and B are infinite sets. Yet the infinite set B can fit into a tiny corner of infinite set A. For example, the entire set B which consists of 1, 2, 3, …… can fit between the numbers 1 and 2 in set A: 1.1, 1.2, 1. 3, ……. so on into infinity never even reaching the number 2. The set B fits into the small space between the numbers 1 and 2, leaving the vast expanse of set A from 2 to 10 unfilled.
I believe set A illustrates what it means to be human. A seemingly modest set that can be contained within the fingers of our human hands. Set B illustrates the field of mathematics. An ever expanding additive structure like the floors of an imaginary building that extends into the sky. Though both are infinite, mathematics can capture but a small part of what it means to be human.
Sincerely,
Abhay
