Models are natural interfaces between systems in the world. Just as a cell membrane is an interface between the structures and operations of the cell and a complex environment, models are interfaces between the complexity of the world and the structure of our understanding. Models

(Chapter II)
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What is a model, exactly? Rather than proposing a precise definition of "model" to use in the following chapters, I'm going to introduce a framework for thinking about models which starts --- like most natural ways of thinking --- with an analogy. The models we use when we think are like the membrane of a cell or the skin of an organism. They work by limiting certain kinds of connections between our minds and the world and by facilitating others. They are interaces which serve purposes.

For instance, the numeric model for the game of tic-tac-toe uses an association of the digits one to nine under addition with the squares on the tic-tac-toe board and their visual axes of alignment. There are many ways of associating numbers with board locations, but only some of these associations will readily serve the purpose of winning at either fifteen or tic-tac-toe. Those interfaces which serve purposes are models, picking out the models from the non-models.

Interfaces involve both separation and connection and I will use the usefully vague word "system" to describe what is being separated and connected. The natural world is full of distinct systems separated by interfaces which serve purposes. Boundaries between systems are neither rigid nor unique but they are definitely not arbitrary either. Sometimes, as in the membranes of cells or the bark of trees, the boundaries have direct physical manifestations. In other cases, like the zones of a natural ecology or the patchwork spread of forest over meadows, the boundaries are harder to identify and change with time. Indeed, it is the permeability and change of these boundaries which makes them interfaces rather than mere barriers.

There are good reasons for systems to be separated and good reasons for them to be connected. The tension between these reasons is what leads to the structure of interfaces. And these same reasons lead to similar tensions and structures in the models which we and our machines continually use as interfaces with one another and the world.

Looking at models as interfaces is a novel step because it treats them as a natural phenomena which happen to involve human minds, cultures, and artifacts. The most common alternative is to describe models metaphysically, treating them as part of a "non-physical" reality which somehow connects to the physical world. In addition to philosophical problems posed by this invention, it hides the fact that the models have their own logic and reality distinct from whatever they are modelling. This logic and reality reflects our own purposes regarding the world as much as it does the nature of the world "itself."

It is important to be clear that though I am using the noun "model" in my descriptions, I am describing a process or an action: models are what interfaces do. Thinking about models as interfaces requires some getting used to and deserves some discussion. In this chapter, I will begin by talking about interfaces in the natural world and their purposes and then extend the discussion to the models engaged, consciously and unconsciously, in the ways we perceive, think, and act.

Dividing up the World

It is a property of the physical world that the influence between objects and events may be diminished by a wide range of factors: distance, physical barriers, noise, and time (to name a few). Objects or events can be grouped into collections based on the strength of mutual influence, naturally dividing the world into overlapping "systems" with clear but "grey" boundaries based on those degrees of influence. A virus introduced into a biological cell has more immediate effects on the activities of the cell than it does on the activities of the other cells around it. A particularly powerful idea introduced at one university will have more immediate consequences for that university's faculty and students than on those at other universities. The division of the world into these overlapping and interacting systems is pervasive and profound.

There is not neccessarily any single way to divide the world into systems. For instance, biological systems can be divided at the level of biological cells, the level of organs, the level of organisms, or the level of populations. In this case, the organization at one level is based on the organization at other levels.

The intertwining of systems can be complicated. The human body is made up of many different organs and organ systems but some of these, like the circulatory system, run throughout the body. And interfaces, though they are sometimes based on direct physical boundaries, can rely on intermediary systems like these for their interaction.

Systems need not even be in particular locations, but can be distributed. The human immune system includes some identifiable organs (like the bone marrow) but also consists of millions of cells spread throughout the body, learning individually and collectively to identify and contain harmful influences from outside the skin (which is another pervasive organ). A system is defined not by a location or even a set of components, but by how the components are connected to one another. It is these patterns of connection which group components into systems.

Systems are also connected to one and other and interfaces are how they are connected. An interface implies a separation, since it constrains the ways in which two systems can interact. In the absence of such a separation, constraint would be impossible and the two separate systems might as well be one. But an interface also implies certain connections. To make this more precise, let's look at one of the most common yet amazing interfaces in nature, the wall of a living cell.

The Walls of the Cell

The single biological cell is a revealing example of a complex system with a variety of interfaces to the environment around it. All cells have membranes that perform various complex functions. By looking at these more closely, we can understand some of the roles played by interfaces in complex natural systems.

Biological systems are complex chemical machines developed over millenia of evolution in many different contexts. The chemical processes which are the basis of these systems are, however, remarkably similar in all earthly organisms. The cells in biological organisms are a watery soup of complex molecules constantly in motion. Their activities rely, for the most part, on accidental moments of interaction between these complex molecules. The miracle of life within the individual cell relies on the laws of statistics and the preferences of certain chemical reactions to occur rather than others.

Because of this dependence, the integrity of the cell relies on the mix of chemicals within the cell and the wall or membrane of the cell plays an important role in maintaining that mix. Both the inside and the outside of cells consist of mostly water and the starting point of all biological membranes is the waterproof boundary created by layers of lipid (fat) molecules. Just as visible drops of fat congeal about themselves in water, layers of this same material spread into a wall can isolate the chemical context of a living cell from the substances around it.

But if isolation were the only role played by lipid layers, cell membranes would be barriers rather than interfaces. In nearly every case, lipid membranes are permeated with different kinds of structures for providing particular kinds of connections between the inner activity of the cell and the context around it.

For example, some membranes are permeated by holes of carefully constrained size, allowing only the simplest of molecules through. At other points, membranes may be pierced by complex proteins whose dynamic action actually pumps materials from one side of the cell wall to another. In many cells, another kind of channel provides for the flow of charged particles --- ions --- across the layer to energize the cell's internal activities.

The much-studied E. Coli bacteria has a complex membrane of several layers which actually digest material coming from the environment. Food particles from around the bacteria --- though only of a particular size --- enter through the outer layer through holes and are broken down into simpler molecules by enzymes in the gap. Some of these simpler molecules are then passed through the inner layer into the body of the cell itself. The membrane not only isolates and connects but transforms the organism's connection with its environments.

Connections go in both directions. The same pumps which move particular substances into the organism from the environment move other substances out. The holes in the membrane that let certain molecules in also let other molecules out. Sometimes, as in E. Coli, changes in the chemistry inside the membrane can actually lead to the energetic physical motion of complex molecular machines outside the membrane; these external flagella propel and turn the bacterium in its quest for chemically richer environments.

Safety, Consistency, Simplicity, and Cooperation

What is the cell membrane for? Understanding purposes can be important because structure is shaped by purpose, whether by design, evolution, or their interaction. But purposes are often interconnected, so it may be difficult to separate them or to isolate "real" purposes. There is seldom a unique "why" for any aspect of a complex design or decision, where different aspects are the product of many different reasons supporting one another.

But listing reasons can be useful because it is the starting point of conversations about how purpose relates to structure. I will pick out four purposes of cell membranes which I will also bring to the discussion of models. There might be other words to use or other ways to divide up the purposes of the membrane, but we need to start somewhere.

Safety is the first purpose served by the interface of the cell wall. Safety is provided by the presence of a persistent barrier, protecting the cell's chemical context from the range of chemical and physical challenges of its environment. When the cell's natural environment moves or changes, the complex machinery inside the cell moves as a whole, protected by the cell membrane. Without the membrane, the turbulence of the environment would tear this molecular machinery apart, separating the molecules which depend on chance interaction to function.

The membrane also shields the internal environment of the cell from chemical challenges. Molecules cannot normally pierce the lipid layer without specific provision for letting them through, letting the mix of reactions in the cell proceed undisturbed by outside chemical influences.

Consistency is the second purpose served by the cell membrane. While safety relies only on the presence of a barrier, consistency requires an active barrier maintaining the cell's internal environment by letting particular elements in as well as keeping unwanted elements out. The dependence of the cell's activity on both particular chemicals and particular distributions of chemicals means that the membrane needs to allow certain sorts of passage in both directions as well as keeping most influences out of the cell.

Many of the cell's internal chemical reactions require extra energy which is provided by free-floating ATP (adenosine tri-phosphate) molecules in the cell's molecular soup. The production of these molecules includes the cell's membrane in interesting ways. In one mechanism, the membrane actually defines a chemical battery whose flow of current creates ATP molecules within the cell. The mechanisms by which the membrane maintains consistency may be varied, but the purpose is itself consistent: to ensure that the cell's chemical mix will sustain the cell's activities.

Simplicity is a special sort of consistency served by the cell membrane. Though the functioning of the cell relies on very complex molecular machinery, most of the complexity comes from molecular blueprints and instructions in the cell's nucleus. The membrane limits the complexity of the molecules which enter the cell's internal environment in order to serve the complexity of the molecules already there.

This censorship is important because there are many more ways of being complex than there are of being simple. Because of this, there are many molecular machines in the external environment which "almost work;" permitting such machines within the cell may hamper or undo the work of those machines which do work for the purposes of the cell.

It is also worthwhile distinguishing two particular sorts of simplicity, both of which are provided by the cell membrane acting as an interface. Molecular simplicity is a restriction based on the chemical properties (number of molecules and kinds of shapes) of individual molecules. Varietal simplicity restricts the number of kinds of molecules which enter the cell. Certain complex molecules do make it through the cell membrane (for example, chemical signals of certain sorts) but the membrane still restricts the number of different kinds of complex molecules permitted through, maintaining statistical simplicity while permitting a degree of molecular complexity.

Cooperation is a fourth purpose provided by the cell membrane as an interface. Cooperation is really a "catch all" purpose for different functions which emerge because there are many cells in the environment rather than just one. For example, some of the substances passed out by the cell membrane are explicitly for use by other cells. This is even more true of subsystems within complex cells, where one part of the cell emits a complex molecule which is unpacked and processed upon arrival at another part of the cell. On the larger scale of entire organisms, the membranes of individual cells in complex organisms are usually dotted with patterns of proteins which identify the cell as "legitimate" and keep it from being attacked and consumed by the organism's immune system.

Models as Interfaces

In the previous chapter, we saw three very different kinds of models: the tic-tac-toe grid (for playing Fifteen), the subway map (for planning travel and transfers), and the models of federal payments used for planning in different contexts. We can see each of these different kinds of models as interfaces by thinking about the systems they connect and the purposes they serve by that connection. As the models get more complex, the systems they connect get more complex and our understanding of their interaction becomes more incomplete. We can probably understand the connection between Fifteen and Tic-Tac-Toe in its entirety, while we can only glimpse some of the connections between federal support payments and our models of them in different contexts.

The game of Fifteen is one system consisting of its pieces (the digits 1 to 9), its rules (turn taking and addition), and its goals (summing one player's numbers to fifteen). The Tic-Tac-Toe grid serves as a model of this system for a different system (usually a human being) which knows how to play tic-tac-toe based on the layout of squares and their visual alignment. The interface is the mapping of numbers to locations which converts "add up" to "line up." Though there are many possible mappings of numbers to locations, only some of them work as interfaces for a system which can play tic-tac-toe. Not every possible interface is really an interface nor is every possible connection between systems a model. Purpose and context separate models from non-models just as the complex machinery of the cell distinguishes cell membranes from mere lipid layers laced with impurities.

The subway map is an interface between a socially and technologically constructed system and human riders using their visual skills and abilities to make their way from point to point in the system. The system on the "outside" of the interface consists of the cars and rails, schedules and fees, workers and riders. The system on the "inside," the rider who is using the map, can find paths from point to point (through hubs) but requires that the display of stations and connections be clear and legible. The reason the subway map distorts space is that it can safely sacrifice some structure from the outside (such as exact distances or physical proximity) to serve the purposes of the mechanisms (such as visual ease or spacing) on the inside.

The models of financial transactions are far more complicated. The actual system being described by the models is incredibly complex, consisting of millions of transactions affected by many diverse factors and resulting from many different legislative, executive, and individual decisions. This mass of complexity is reduced, by both the model of the soup kitchen and the trading desk, to just what is needed to solve the scheduling and resource problems of those peculiar contexts. When and how to expend resources are the problems in both contexts, but because the scale and kind of resources are so different, the models of the financial contexts are very different.

It is useful to understand models as interfaces and to look at how the purposes of models constrain their structure and application. In the rest of this chapter, we take the four purposes served by the cell wall and to look at them in the context of models in general. Safety, consistency, simplicity, and cooperation are also purposes served by models. By keeping this in mind, it allows us to think about the different ways in which our models serve us and fail us and about the goals which it is important to keep in mind while thinking about how our models evolve and change.

Models and Safety

Models provide safety by acting as substitutes for or simulations of potentially risky situations. With a model, we can experiment in a context where consequences are both ephemeral and reversible before we risk ourselves in a world where consequences are often persistent and final. Before brave men and women are flung into orbit in rocket ships, they spend hours in simulators living through models of their missions. If a mistake gets made or an unexpected event occurs, the consequences and reactions can be discussed over coffee or beer instead of tragically over coffins and in review boards. It is from this that the astronauts and crews learn gently and safely how to deal with an environment that might so readily end their lives or derail their mission.

At a greater remove but of equal importance, it is the mathematical models of mass and accelaration, of fuel and combustion, that ensure astronauts and mission planners that their vehicle will be able to reach orbit and safely return without countless trials and failures. If the astronauts fly simulated missions hundreds or thousands of times, calculations of fuel consumption, velocity, and orbital stability have been made thousands or millions of times from Galileo's and Newton's first observations and predictions to the reams of analysis pouring from NASA computers. Success might be possible without these models, just as bridges can built without a detailed understanding of physics. But we are safer when we take our first steps disengaged from a complex and unforgiving world.

Models also provide more than direct physical safety. For instance, financial institutions use very sophisticated models of the investment packages they offer their customers. The models of these packages include estimates of the risk to which they expose assets, the win/lose possibilities of various outcomes, and the sensitivity of these estimates to the popularity of the investment package itself. These models are not for the sake of the customer (at least directly) but are for the sake of the sponsoring institution whose corporate financial security depends on effectively managing both sales and risks. Whenever changes are potentially costly and irrevocable, we can find safety by first working within a model.

But even though models mitigate the various risks of acting in the world, they cannot remove it entirely. As we know too well, careful plans can collapse during their execution. Napoleon once wryly admitted "No plan of battle survives the first encounter with the opponent." The reason for this is that models fill many other roles, including the cloaking of incompleteness, the reduction of complexity, and the support of communication among different individuals. Each of these roles, for its own important reasons, ends up introducing a gap between the model and whatever it is representing. And though the risk of being hurt is reduced by working in a model, the risk of being wrong may be amplified by the gaps which the model introduces.

Models and Consistency

Suppose I were to show you this picture of a chair behind a table and ask you to draw just the chair. You would probably produce something slightly better than the sketch to the right of the picture.

Your sketch, like this one, would include parts of the chair which you could not actually see but might be able to guess at because of your past experience with chairs. For instance, the seat is not visible in the picture, but you would know that chairs have seats and that those seats are often made of the same material as their backs. This allows you to draw a picture which is more "complete" than what you could actually see.

Completeness is a certain kind of consistency, since it ensures that our descriptions of similar objects have similar forms despite the accidents of how we happen to see them at any moment. Part of the reason we use models is that they can make our experience more "complete" by filling in details which we have not actually seen but can reasonably expect. This completeness means that we do not have to learn different things about "chairs we only see the back of" than about chairs in general.

Just as the cell membrane ensures consistency within the cell through a mix of synthesis, transformation, and isolation, models acheive consistency by both augmentation, interpretation, and omission. Our model of the half-seen chair includes the fabric of the seat we do not see, interprets the shadings of the fabric as a repeated texture, and ignores apparent accidents of damage to the fabric or frame.

The fact that models "fill in" blank spots in our experience or transform remembered events introduces the possibility of error. While Shakespeare reminds us that to err is human, a more general statement might be that to err is to model! To use a model is to introduce the possibility of error; but errors in models are peculiar because they are asymmetrical.

In his profound book Intentionality, the philosopher John Searle introduces the notion of "direction of fit" for models which assigns "blame" for errors of description. When the world does not agree with a model, we do not normally say that the world is at fault. Instead, we say that the model is at fault, indicating an important asymmetry in the character of model.

Searle's distinction reveals a directionality to models which we can add to the characterization of models as interfaces. Not only are our model/interface's causal connections to the system they represent less intense than connections within the system, they are asymmetric: changing the model may be less likely to change the system than changing the system is to change the model.

One of the biggest problems with the models which we use in our computers is that they have little recourse when errors of these sorts occur. When the fit of a human model fails, it may be noticed and prompt a shift to a different model or a change of the model being used. When the fit of a computer model fails, it blindly continues to omit things which are important or add things which are in error, asking the same question over and over again until interrupted or reprogrammed by its human developers or maintainers.

The acronym GIGO (Garbage In/Garbage Out) describes what happens when a computer (or person) blindly uses a model which doesn't "fit". The problem is not just with the particular model or with models in general, but is with the coordination of the model and its context and purposes of application. As I'll discuss towards the end of the book, the difference between computers and human beings is not that one uses models and the other does not, but what each does when its models fail to fit its purposes and situation.

Models and Simplicity

Another important purpose of models is the reduction of external complexity. Working in a model is often simpler and faster than working with a "real thing." To pick a simple but striking example, consider adding the following two numbers:

+  3,448,117

Done by hand using conventional methods, this overall addition takes 18 additions of single digits and 5 carries. If it takes 1 second for each such operation, we can easily add these numbers in a little under half a minute.

Now suppose that instead of this convenient digital (base 10) model, we used an "analog" model of the numbers as `tallies' as in this addition of five and seven:


combining the tallies (12 operations) and counting the result (another 12) to add the above numbers with 24 operations which will also take roughly half a minute. However, if we use tallies to add larger numbers (like 1,538,945 and 3,448,117), we would require nearly 10 million operations which --- at 1 second an operation --- would take almost four sleepless months to complete.

The advantages of models in reducing complexity are not always this remarkable. Often, because a model is really just representing another model, the second model may not improve so drastically on the simplicity of operation provided by the first. But in this case, it may often provide some other advantage, in terms of either safety, consistency, or cooperation.

An Example: Copernicus' Cosmology

The simplifying role of models often plays a role when models change. For the creative people changing models, the desire for simplicity may be partially pragmatic and partially aesthetic. A case in point is the model implicit in the Copernican revolution, which proposed that the earth orbited the sun rather than the sun (and other planets) orbiting the earth.

Copernicus, like star-gazers for millenia before him, was interested in predicting the motions of the stars and particularly of the planets (which were much less reliable, at least on the scale of a human lifetime, than the stars). The dominant theory at the time, Ptolemy's epicycles, imagined that the planets moved in circular paths around points and that these points themselves were moving in circular paths around other points and that eventually all of these points were moving around the earth. The first two levels of this are shown in this figure: (Place holder image from, also see the movie.)

By adding new points and cycles (called epicycles) to this sequence, it would be possible to increase the precision with which the predictions for planetary motion could be made.

At the time that Copernicus proposed his theory, the improved quality of observations was forcing the epicyclic model to grow more and more complicated with the addition of further epicycles and --- significantly --- the introduction of elliptical, rather than circular, cycles. For both of these reasons, calculations were growing practically much more complicated. The introduction of these elliptical cycles was also aesthetically troubling to Copernicus who thought that God would have made things work with the more perfect circular patterns. He proposed a radical reorganization --- making the Earth into a planet --- which would recover the circular orbits. Ironically, further data eventually made the Copernican orbits as elliptical as the epicycles he was fleeing, but other evidence soon validated Copernicus' general rearrangement of the heavens.

The shift to a helio-centric (sun centered) cosmology made the calculation of planetary orbits easier just as the use of a digital model makes arithmetic addition easier. While we might not strictly argue that the helio-centric shift confers a logarithmic advantage like digital addition, increasing the precision of observations in the helio-centric model requires only that we increase the precision of our orbital calculations. The epicyclic model, on the other hand, would demand the addition of more epicycles to increase the precision. In the epicyclic model, more precision requires more calculations; in the helio-centric model, more precision only require more precise calculations.

While a helio-centric cosmology makes it easier to calculate the paths of planets, it makes it harder to calculate the positions of stars or to use the positions of the stars to calculate our own position on the earth. For practical navigation on earth or even in near-earth space, the earth-centered model is still used. Even astronomers, when they describe the position of a planet on a given night, will say that the planet is "in Orion" as though the planets were moving over the vault of stars swinging around the Earth. The reason is that on a stellar scale, the orbital motion of the Earth is nominal and --- in most cases --- it is easier to ignore the variation than attempt to include it.

This again illustrates how effectiveness depends on the purposes to which the model is being put. One of the purposes might be solving problems, another might be recording facts, and another might be changing in particular ways to solve new kinds of problems. And because humans and machines solve problems together with other humans and machines, this leads us to another purposes of models: cooperation.

Models and Cooperation

For most of human history, most external models have been used to connect human beings to one another and create the community that enables cooperation. While we particularly value individuality, especially in the West, it is clear that not one of us would be able to prosper or survive without the community that brought us up from infancy. Human infants are unique in their vulnerability and dependence; from infancy onward we are connected to one another by a rich web of shared and related models. Spoken language, unspoken signals and conventions, the myths that give us roles with respect to one another, and the written words that span miles and ages all play a role in creating and maintaining our comfort and power.

Though I will not talk extensively in this book about the role of models in human cooperation and community building, it is worthwhile being clear about how human models differ from most current computer models. Human models --- ranging from maps to recipes to this book in your hands --- rely on sophisticated interpretation by the user of the model. Even though I have tried to be excruciatingly careful in not assuming too much of a shared context and knowledge in writing this book, it would be impossibly confusing if we did not share some culture, language, and experience. Likewise, even if a map is very carefully drawn and designed, it might be hard to use the map to specify a journey without being in or knowing the place it describes. Think of the awkward (but not insurmountable) complications in using a map of a mutually unfamiliar region to guide someone over the telephone.

These human models all provide degrees of safety, completeness, and simplicity in ways appropriate to their individual purposes. But a human using a model has a degree of "play" or "variation" which most computer programs lack. Later on, in discussing consciousness, I will return to this theme. Human creativity cannot be separated from the play and variation which humans routinely apply to otherwise rigid models.

Life Without Models?

Looking at models is interesting because they are the foundation for human intellectual power and worldly success. Looking critically at models is vital for our time because so much of our individual and corporate activity and decision making relies on powerful but un-examined models. Though most of the examples in this book are simple models from physics or computation, the same purposes, structures, and assumptions apply to the complex models we humans have (and use) of social and economic structures, family and society, personality and identity.

Some workers in artificial intelligence (my own discipline) argue for machines without models (which are called "representations" in artificial intelligence), built like robust and responsive insects rather than clever and articulate human beings. However, when we look at the specialized ways that insects perceive the world, we see that they too use models of a sort. A fly looking out at the world identifies only certain features and responds to only certain combinations of features. This is a model, even if it doesn't look like a language or a drawing or anything which we would normally call a model. Talking about machines without models is really talking about machines without some particular kinds of models. And this makes a great deal of practical and scientific sense, since we can't give our machines every kind of model.

We might ask whether it is possible for human beings to do without models, sacrificing their admitted advantages in order to avoid the confusion and entrapment they sometimes generate. However, it is not a choice we have. Our perceptual systems, our use of language, our entire understanding of the world, are all models of a sort which we cannot escape. Though some Eastern thought advises us to seek just such an escape, it also admits that we can never really acheive it. It is one thing to be critical and aware of models --- that is part of what this book is about --- but we cannot do without them. Models are our tools, but they are also part of us, and the best we can do is understand, suspect, critique, improve, and then embrace them again.

Ways of Being Wrong

Though this book is about looking at models as interfaces (among other things), I would not mind if this view were wrong in many ways. In fact, I am sure that any attempt to divide minds and ideas along rigid boundaries is sure to leak like the proverbial sieve. The boundaries or interfaces I describe are gray and permeable rather than sharp and solid, but drawing and describing them is still useful. And that is my goal: I wouldn't mind being wrong but I would deeply regret being useless. So, enjoy, learn, and use.

Copyright (C) 1997, 1998 by Kenneth Haase
Draft, not for citation or circulation
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