|As hands are to our motions, models are to our thoughts. The nature of our models affects our thoughts just as the nature of our limbs affects our motions. But unlike our limbs, we have choices about the models we use, choose, devise, or share. Because of this, it is helpful to think about the models we are using.||
This book is about the models we think with. Models are stories about how our thoughts and actions are connected to the world. These stories explain how and why models go wrong when our thoughts and actions collide with unexpected and unavoidable reality. If you have ever thought about how you or others think, this book may matter to you.
Whenever we touch the world, we use models. Whether it is the world of colors and shapes and textures, of stars and trees and earth, or of love and sorrow and hope, our connections are always based on indirect models of one sort or another. Only through models do we understand these worlds and our actions in them, making models as important to our thoughts and dreams as hands are to our motions and actions.
Our models are obvious when we look at a plastic see-through automobile or struggle with arcane equations and measurements to describe an apple falling to the earth. They are less obvious when we tell stories with words or pictures, using or avoiding words or images which highlight or hide things we saw or felt. And our models are all but invisible when we see with our eyes or hear with our ears, but these senses also hide and highlight aspects of the world around us according to ancient purposes implicit in eons of evolution.
It is helpful to think about the nature of our models in many different contexts: the machines we build, the computers we program, the organizations we design, the individuals we teach, and the lives we compose. All of these rely on models of various sorts, whether we see them or not. Problematically, good models are are so reliable and robust that they are nearly transparent; just as it is awkward to grasp objects or turn knobs while thinking about the geometry and placement of our fingers, it is difficult to think about the models we are thinking with. But thinking about our models as models may be vital when we seem to be stuck, because the strengths and limits of models are often connected.
This book teaches a particular way of thinking about models: a model of models, one might say. Rather than trying to define or invent "what" models are, it asks "how" and "why" models work. I think that asking "how" and "why" questions (the questions an engineer or scientist might ask) may be more revealing that the "what" or "who" questions which a philosopher or historian might ask. But looking at models as an engineer involves assumptions which I should make clear. I assume that models have purposes and functions. I assume that we can better understand our models by better understanding these purposes and functions. I also assume that models have a structure which we can describe. And I believe that looking for this structure can help us think better about their purposes and functions.
This book is full of examples of different kinds of models. At times, it may seem confusing as to whether it is a book about computers or physics or biology or mathematics. In fact, it is "about" none of these areas but uses each of them to teach a way of looking at the models used by computers, by physicists, by biologists, and by mathematicians. I picked models of things which I thought would be interesting, but this book is not about them directly, but about the models used in thinking about them.
You may also notice that the examples also lean towards traditional "sciences" --- like biology or physics --- for two somewhat different reasons. First, the traditional sciences are historically a source of clear shared models, collaboratively constructed by generations of individuals trying to understand both the world and each other. Second, it is an opportunity to clarify the popular but murky conception that science is less influenced by its models than other realms.
The rest of this chapter introduces three very different examples
of what I mean by the word "model". Each example describes several
models side-by-side, using them to expose one another and show a
little about how models work. My descriptions, like nearly all
descriptions, will themselves be models. If they seem incomplete in
troubling ways, I urge you to think about what is missing and why it
might be important. You may be seeing something which I have
inadvertantly missed (please let me know) or I may be "missing" it
because ignoring certain details is important to this "model of
The Simplicity of Models
Models make it easier to think about certain things. For instance, consider the following numerical game, which two people might play on a rainy day when the television and computer are both broken:
The goal is to collect a set of numbers of which a subset adds up to 15. For example, in the following sequence of turns, player B wins:
Player Move A has: B has: A Picks 9 9 B Picks 5 9 5 A Picks 4 9 4 5 B Picks 2 9 4 5 2 A Picks 6 9 4 6 5 2 B(wins) Picks 8 9 4 6 5 2 8
The game of 15 is a little bit like the card game blackjack, but without the element of chance and with a relaxed rule that any selection of the player's choices (rather than all of them together) must add up to a particular sum. In the game we just saw, player A could have blocked B's winning move by taking 8 himself, and the game could have gone on further. With some thought, a math whiz could figure out an ideal strategy for this game, which would lead to a draw when both players use the strategy. A computer could use a strategy like this one, expressed in terms of arithmetic, to play the game perfectly. However, a human might have a harder time with it. Fortunately, we can also redescribe the game differently to make it easier for human players
This model organizes the numbers into a "magic square" where each row, column, and diagonal adds up to 15.
This magic square is actually an "extra magic" square because there are no other ways to add up elements to get 15. Because of these two components, we can play the "number game" by simply playing the familiar game of tic-tac-toe on our grid:
For a human being, tic-tac-toe is a more useful model of the number game than its original form. However, for the computer, the numerical version is easier to work with than the graphical model of the tic-tac-toe grid. Humans have an easier time telling if things "line up" than telling if they "add up," while computers have an easier time telling if things "add up" than if they "line up." A clever programmer writing a tic-tac-toe program might write a "number game" program and then describe its moves as tic-tac-toe moves.
The point of this example is that models simplify problems, but
simplification is always relative to a particular sort of
user with a particular set of
purposes. Change the purpose or change the user (for
instance, from a computer being programmed to a child learning the
game) and simplification becomes befuddlement.
The Choices Models Make
The "tic-tac-toe" example was one example of how the right model can make a problem much simpler for certain purposes and certain kinds of "users" (human or machine). However, I hope you weren't convinced by it, because it was itself a little too simple.
First, the "game of 15" with which it started was not all that interesting or compelling. In fact, it was constructed for the exact purpose of solving tic-tac-toe on the computer and not the other way around.
Second, the arithmetic actions on the numerical model exactly mirrored the geometric actions on the tic-tac-toe board. Models are not usually so perfectly aligned and part of what makes models so useful and complicated is that it is seldom possible to accomplish the kind of perfect translation we saw above. Normally, we use several models together rather than work with translations into any one model.
For a more complex example of model, we turn to maps. Maps are descriptions of the physical spaces we move through, settle in, and think about. Different maps have different purposes. If you look at any reasonable world atlas, you will find maps describing physical features, political features, economic and population figures, etc. Depending on the kind of question you want to ask, you use a different map. What are the major mountain ranges on the North American continent? Is Greenland bigger than Texas? Where does one find the most uniformly high per-capita income? Where are people the most densely concentrated?
Each of these questions would be answered by a different kind of map. Some questions, like the size of regions, are better answered on a globe than on a flat projection, while for others, such as which countries border which countries, a flat map is just fine. But why does the kind of map matter? Can't we just have one map with all the information? A sort of "Swiss Army Map" that serves all purposes?
To look at why not, let's look carefully at one very stylized map of the subway system in Boston:
which shows four subway lines and their 38 stops. (The green one is the oldest subway line in the United States). It is actually a pretty accurate map for purposes of planning ones' transfers. In particular, for this limited purpose, it is a much better map than this one:
which shows the physical location and context of each station. This map, though more geographically accurate, is both harder to read and harder to use for planning one's transfers. One reason for this is that the second map gives us a lot of information which we don't need to make our way on the subway. Because the subway runs underground or on reserved tracks, we don't have to worry about intervening streets, highways, forests, or bodies of water. But even if we take all of this information out, the map still wouldn't be that useful because it tells us many things which don't matter:
For all of these reasons, the map above isn't the best map for planning one's travels by subway. In the idealized map, these features are all ignored. Of course, this map would be nearly useless for navigating around Boston by foot or automobile. For example, on the stylized map it looks like the Ashmont and Alewife stations (the two ends of the Red Line) are only twenty times as distant as the Park Street and Downtown Crossing stations (which are separated by a block). In fact, Ashmont and Alewife are well over 200 times further apart (they are on different sides of the city) than Park Street and Downtown Crossing (which are separated by a city block).
For the purpose of planning your transfers on the subway, the
idealized map is just great. But it's nearly useless if you are
travelling by foot or car to need to change your commute to work
around a rail strike or track damage. This is an obvious, but
profound point: descriptions are dependent on purposes.
The Complexity of Models
Models hide, highlight, transform, and reorganize whatever they are describing. In the case of a map and a territory, this is clear, since maps were designed for particular purposes. The pedestrian who goes out with a subway map or the subway commuter who looks at a geographic map are both (mis-)using models across purposes. But sometimes, the purposes behind the models we use are not so clear. Above, some of the purposes were part of the physical context (the fixed tracks), some were part of the political context (paying to enter the system), while some were part of the organizational context (the coordination of schedules with transfer points). To illustrate this complexity of models further, let me tell you a story about models of the flow of money in two very different worlds.
When I was a graduate student at MIT, I was also on the volunteer staff of a soup kitchen called Haley House in Boston's South End. We served breakfast and lunch to the poor and homeless out of a corner row house on Dartmouth Street. We also tried to provide a haven of peace in the not very kind environment of Boston's streets.
One of our logistic challenges was figuring out how much food to make for lunch. This would also affect what we made and when we started, as we always had different amounts and combinations of materials on hand and they took different amounts of preparation. And predicting the demand could be tricky because the number of guests who actually showed up could vary widely.
For example, many of our guests did poorly paid temporary work when it was available. If some construction site in Boston needed a lot of temporary help, guests would be "on the job" at lunchtime and not eating the food we prepared. These complications were unpredictable and made meal planning difficult, though we tried to keep our "ear to the street" to plan things better. However, one more predictable heuristic for meal planning was something we called "check day."
Many if not most of our guests at Haley House were veterans of various wars and many received disability or pension checks on a regular basis. These checks generally arrived on the first Tuesday of the month and were often immediately converted into cash. Many of our guests would take advantage of their sudden wealth to eat at another establishment (which charged money). Sadly, another contingent of the temporarily "wealthy" would use their money to buy booze to help them not think (for a while) about months past and months to come. The first folks we didn't see at all and the second we didn't let into the kitchen; we knew from experience that they were likely to be disruptive or violent or both, though it was often hard to turn them away.
"Check day" was a useful rule for deciding how much food to make based on the flow of money into and out of the lives of our guests. It was, I suppose, a certain island of certainty in a pretty uncertain world.
At the same time that I was a graduate student and on the staff at Haley House, I was also consulting for a major bank in Boston on a computer program (an "expert system") to support the chief trader on the bank's Money Market desk. It was quite a transition, leaving my room at the soup kitchen (I was on the "live-in" staff at the time) to go downtown and ride the elevator up to the 15th floor of a financial district high-rise. My three-piece suit provoked good-natured ribbing as I headed out through the soup kitchen on the ground floor.
The job of the money market desk was to borrow --- on a day to day basis --- the money which would cover (over the course of the month) the percentage of deposits which the bank was required to have "on hand" by the Federal Reserve. This was completely legitimate and big banks --- which have almost all of their funds "loaned out" at any particular time --- generally borrowed from smaller banks (on a daily basis) to both sustain their "minimum balance" and pay their operating expenses. (When you have a "money market fund" in a bank, your bank is lending that money out for just such short term loans to other banks). As long as the longer term loans did better than their borrowing, everything worked out fine.
Just as with longer term loans, banks have to pay an interest rate for the money they borrow for the day. That interest rate fluctuates and can be thought of as the "price of money" on a given day. The trader's job on the money market desk was to borrow money every day on the "money market" in order to make its "on hand balance" come out right at the end of the month. And as with the attendance at the meals we prepared at Haley House, there were numerous outside factors affecting the price of money.
The trader explained the ebbs and flows of money through the nation's banks and its effects on the price of money which he was trying to foresee. One of the chief events was a sharp rise in the "price of money" at the beginning of each month. This was due to the fact that the United States government sent out pension and disability checks early in each month. In order to send out those checks, the government must move billions of dollars from "savings" into "checking" and once the money had left the federal savings account, it could no longer be lent to banks on a day-by-day basis. This led to a temporary rise in the "price of money" in general until the overall supply was replenished by deposits into other banks either directly by check recipients or by the institutions (like restaurants or shops) where their money was spent.
As the trader explained this to me in a dark-panelled office high above the financial district, a little bell went off in my head: "check day!" I was seeing the same phenomenon which we used in Haley House meal planning from "the other side." The motion of millions of small checks and amounts in individual lives revealed itself as the motion of billions of dollars around the nation's economy and banking structure.
In both the soup kitchen and the conference room, we were thinking about the flow of money in the economy. We had models of that flow, but because our context and our purposes were so different, the models were very different. In the soup kitchen, our models revolved around the individual lives of our guests and their needs and resources. In the bank, our models revolved around the blocks of cash available around the country for short-term money market loans. Unlike the tic-tac-toe example, there is no simple translation between the transactions in each of our guests' individual lives and the massive motions of money among banks. Unlike the subway map example, the purpose of each model is not nearly as clear as riding rails from one station to another. There is a strong constraint, of course: everything has to add up. But the models we had in the soup kitchen and at the bank were very different, even though they were describing the same motion of money in the economy.
Because it makes sense to have multiple models of the same thing, looking at models is important. This book introduces a way of analyzing and understanding models which addresses both their strengths (they make the world easier to deal with for particular purposes) and weaknesses (they complicate the world when purposes change).
Models are inescapable. When we "see" or "hear" the world around us with our own eyes and ears, we are using models of external opportunities and risks selected over eons of biological evolution. When we apply names or categories or expectations to the information of our senses, we are using models of objects and kinds of objects invented over millenia of cultural evolution. And when we interpret or explain our impressions to one another, we are using models learned in our personal evolution from creative children to understanding adults.
In many different ways, our models make our interactions with the world safer, more manageable, more reliable, and more coherent to ourselves and others. Like hands, they allow us to affect things at "arms length," to open puzzling objects for examination and learning, and to separate parts from wholes. However, our models can also limit us, stunt our development, and confuse others attempting to understand us. Hands can fumble in ways that a grip between toothy jaws cannot.
This double-edged property of models is part of what has inspired this book. I have spent the last decade trying to understand how humans and computers use models and to improve the ways in which computers use and change their models of our human world. In these studies and projects, I have been continually struck by how much of the apparent intelligence or stupidity of computers is a product of the models they use to describe the worlds we share with them. And at the same time, I have been amazed at the power and insight which humans gain by their use of the right model at the right time.
Computer models are designed by human beings and emerge from a series of tradeoffs between their tasks and numerous other factors: efficency, flexibility, robustness, lifetime, and the background of human knowledge. When computers appear stupid, it is typically because one of the tradeoffs made in their design is no longer appropriate in practice.
In looking at human models (through different means than reading the programs!), we see that they also make tradeoffs based on purpose. The failures of computer models are not as spectacular as the dismal failures of machine models. One reason is that the tradeoffs they make are often less sharply defined than in carefully designed computer models. Another is that humans nearly always use several models and switch between them as readily as they change the focus of their eyes, giving an appearance of seamless flexibility.
Tradeoffs in human models instead become clearest when we look at human creativity. In everyday activity, the tightly knitted patchwork of human models hides the tradeoffs at their edges. Only when human models get "to the edge" and start to change can we see these tradeoffs clearly. At this edge, creativity comes into play and we see that tradeoffs are being identified, changed, or left behind. As a consequence, the creative individual or community achieves things it has never achieved before and adds the new model, with its tradeoffs, to the apparently seamless fabric of its understanding.
This book is about models and their powers and limitations. It is a goal of this book to change the way people think about the way people think and computers operate. Humans and computers are both always working with models. These models contain knowledge and assumptions which influence what they see (and don't see), how they make decisions, and the ways in which they act. These unarticulated facts and assumptions are often more influential than the facts and assumptions which the model explicitly identifies.
It is not possible to get away from models entirely, but by being aware of how models are used in thinking, we can better understand the "when," "why," and "how" of their limitations. This understanding (which must itself be limited) may in turn help us seek creative new ways to change and transform those limitations.
Though models are our tools, they function best when they are as transparent to use as our hands. For this reason, models vary immensely with regard to their actual complexity without actually "seeming" more awkward. The examples in this book teach how to unearth the amazing complexity beneath this outward simplicity. It is not possible to think without models, but it is possible to think and choose among the models we might use in everyday life, in scientific exploration, and in shaping our individual lives.