Goals and Subgoal: The most powerful way we know for discovering how to solve a hard problem is to find a method that splits it into several simpler ones, each of which can be solved separately.
Much research in the field called Artificial Intelligence has been concerned with finding methods machines can use for splitting a problem into smaller subproblems and then, if necessary, dividing these into yet smaller ones. In the next few sections we'll see how this can be done by formulating our problems in terms of goals.
Using Knowledge: The most efficient way to solve a problem is to already know how to solve it. Then one can avoid search entirely.
Accordingly, another branch of Artificial Intelligence research has sought to find ways to embody knowledge in machines. But this problem itself has several parts: we must discover how to acquire the knowledge we need, we must learn how to represent it, and, finally, we must develop processes that can exploit our knowledge effectively. To accomplish all that, our memories must represent, in preference to vast amounts of small details, only those relationships that may help us reach our goals. This research has led to many practical knowledge-based problem-solving systems. Some of these are often called expert systems because they're based on imitating the methods of particular human practitioners.
A curious phenomenon emerged from this research. It often turned out easier to program machines to solve specialized problems that educated people considered hard — such as playing chess or proving theorems about logic or geometry — than to make machines do things that most people considered easy — such as building toy houses with children's blocks. This is why I've emphasized so many easy problems in this book.
There is an old and popular idea that we learn only what we are rewarded for. Some psychologists have claimed that human learning is based entirely on reinforcement by reward: that even when we train ourselves with no external inducements, we are still learning from reward — only now in the form of signals from inside ourselves. But we cannot trust an argument that assumes what it purports to prove, and in any case, when we try to use this idea to explain how people learn to solve hard problems, we encounter a deadly circularity.
This circularity was no great problem when Ivan Pavlov studied conditioned reflexes nearly a century ago, because in his experiments the animals never needed to produce new kinds of behavior; they only had to link new stimuli to old behaviors. Decades later, Pavlov's research was extended by the Harvard psychologist B. F. Skinner,
who recognized that higher animals did indeed sometimes exhibit new forms of behavior, which he called operants. Skinner's experiments confirmed that when a certain operant is followed by a reward, it is likely to reappear more frequently on later occasions. He also discovered that this kind of learning has much larger effects if the animal cannot predict when it will be rewarded. Under names like operant conditioning and behavior modification, Skinner's discoveries had a wide influence in psychology and education, but never led to explaining how brains produce new operants. Further- more, few of these animal experiments shed much light on how humans learn to form and carry out their complex plans; the trouble is that other animals can scarcely learn such things at all. Those twin ideas — reward/success and punish/failure — do not explain enough about how people learn to produce the new ideas that enable them to solve difficult problems that could not otherwise be solved without many lifetimes of ineffectual trial and error.
The answer must lie in learning better ways to learn. In order to discuss these things, we'll have to start by using many ordinary words like goal reward, learning, thinking, recognizing, liking, wanting,
imagining, and remembering — all based on old and vague ideas. We'll find that most such words must be replaced by new distinctions and ideas. Still, there's something common to them all: in order to solve any hard problem, we must use various kinds of memories. At each moment, we must keep track of what we've just done — or else we might repeat the same steps over and over again. Also, we must somehow maintain our goals — or we'll end up doing pointless things. Finally, once our problem is solved, we need access to records of how it was done, for use when similar problems arise in the future.
Much of this book will be concerned with memory — that is, with records of the mental past. Why, how, and when should such records be made? When the human brain solves a hard problem, many millions of agents and processes are involved. Which agents could be wise enough to guess what changes should then be made? The high-level agents can't know such things; they scarcely know which lower-level processes exist. Nor can lower-level agents know which of their actions helped us to reach our high-level goals; they scarcely know that higher-level goals exist. The agencies that move our legs aren't concerned with whether we are walking toward home or toward work — nor do the agents involved with such destinations know anything of controlling individual muscle units. Where in the mind are judgments made about which agents merit praise or blame?
One thing is sure: we always find it easier to do things we've done before. What happens in our minds to make that possible? Here's one idea: In the course of solving some problem, certain agents must have aroused certain other agents. So let's take reward to mean that if agent A has been involved in arousing agent B, the effect of reward is, somehow, to make it easier for A to arouse B in the future and also, perhaps, to make it harder for A to arouse other agents. At one time, I was so taken with this idea that I designed a machine called the Snarc, which learned according to this principle; it was composed of forty agents, each connected to several others, more or less at random, through a reward system that, when activated after each success, made each agent more likely to rearouse the same recipients at later times.
We presented this machine with problems like learning to find a path through a maze while avoiding a hostile predator. It quickly learned to solve easy problems but never could learn to solve hard problems like building towers or playing chess. It became clear that, in order to solve complicated problems, any machine of limited size must be able to reuse its agents in different ways in different contexts — as See must do when involved in two concurrent tasks. But when the Snarc tried to learn its way through a complicated maze, a typical agent might suggest a good direction to move in at one moment, then suggest a bad direction at another moment. Later, when we rewarded it for doing something we liked, both those decisions became more likely — and all those goods and bads tended to cancel one another out!
This poses a dilemma in designing machines that learn by reinforcing the connections between agents. In the course of solving a hard problem, one will usually try several bad moves before finding a good one — for this is virtually what we mean by calling a problem hard. To avoid learning those bad moves, we could design a machine to reinforce only what happened in the last few moments before success. But such a machine would be able to learn only to solve problems whose solutions require just a few steps. Alternatively, we could design the reward to work over longer spans of time; however, that would not only reward the bad decisions along with the good but would also erase other things that it had previously learned to do. We cannot learn to solve hard problems by indiscriminately reinforcing agents or their connections. Why is it that among all the animals, only the great-brained relatives of man can learn to solve problems that require many steps or involve using the same agencies for different purposes? We'll seek the answer in the policies our agencies use for accomplishing goals.
You might argue that a beaver goes through many steps to build a dam, as does a colony of termites when it builds its complex castle nest. However, these wonderful animals do not learn such accomplishments as individuals but use the procedures that have become encoded in their species' genes over millions of years of evolution. You cannot train a beaver to build a termite nest or teach termites to build beaver dams.
Suppose that Alice, who owns a wholesale store, asks the manager, Bill, to increase sales. Bill instructs the salesman, Charles, to sell more radios. Charles secures a large order, on profitable terms. But then the firm can't actually deliver those radios, because they are in short supply. Who is to blame? Alice would be justified in punishing Bill, whose job it was to verify the inventory. The question is, should Charles be rewarded? From Alice's viewpoint, Charles's actions have only disgraced the firm. But from Bill's viewpoint, Charles succeeded in his mission to get sales, and it wasn't his fault that this failed to accomplish his supervisor's goal. Consider this example from two perspectives — call them local reward and global reward.
The Local scheme rewards each agent that helps accomplish its supervisor's goal. So Bill rewards Charles, even though Charles's action served no higher-level goal.
The Global scheme rewards only agents that help accomplish top-level goals. So Charles gets no reward at all.
It is easy to invent machinery to embody local learning policies, since each assignment of credit depends only on the relation between an agent and its supervisor. It is harder to implement a global learning scheme because this requires machinery to find out which agents are connected all the way to the original goal by unbroken chains of accomplished subgoals. The local scheme is relatively generous to Charles by rewarding him whenever he accomplishes what is asked of him. The global scheme is much more parsimonious. It dispenses no credit whatever to Charles, even though he does as his supervisor requests, unless his action also contributes to the top-level enterprise. In such a scheme, agents will often learn nothing at all from their experiences. Accordingly, global policies lead to learning more slowly.
Both schemes have various advantages. The cautiousness of the global policy is appropriate when mistakes are very dangerous or when the system has plenty of time. This can lead to more responsible behavior — since it could make Charles learn, in time, to check the inventory for himself instead of slavishly obeying Bill. The global policy does not permit one to justify a bad action with I was only obeying the orders of my superior. On the other side, the local policy can lead to learning many more different things at once, since each agent can constantly improve its ability to achieve its local goals, regardless of how they relate to those of other portions of the mind. Surely our agencies have several such options. Which ones they use may depend, from moment to moment, upon the states of other agencies whose job it is to learn, themselves, which learning strategies to use, depending on the circumstances.
The global scheme requires some way to distinguish not only which agents' activities have helped to solve a problem, but also which agents helped with which subproblems. For example, in the course of building a tower, you might find it useful to push a certain block aside to make room for another one. Then you'd want to remember that pushing can help in building a tower — but if you were to conclude that pushing is a generally useful thing to do, you'd never get another tower built. When we solve a hard problem, it usually is not enough to say that what a certain agent did was good or bad for the entire enterprise; one must make such judgments depend, to some extent, on the local circumstances — that is, on how the work of each agent helped or hindered the work of related agents. The effect of rewarding an agent must be to make that agent react in ways that help to accomplish some specific goal — without too much interference with other, more important goals. All this is simple common sense, but in order to pursue it further, we'll have to clarify our language. We have all experienced the pursuit of goals, but experience is not the same as understanding. What is a goal, and how can a machine have one?
Whenever we talk about a goal, we mix a thousand meanings in one word. Goals are linked to all the unknown agencies that are engaged whenever we try to change ourselves or the outside world. If goal involves so many things, why tie them all to a single word? Here's some of what we usually expect when we think that someone has a goal:
A goal-driven system does not seem to react directly to the stimuli or situations it encounters. Instead, it treats the things it finds as objects to exploit, avoid, or ignore, as though it were concerned with something else that doesn't yet exist. When any disturbance or obstacle diverts a goal-directed system from its course, that system seems to try to remove the interference, go around it, or turn it to some advantage.
What kind of process inside a machine could give the impression of having a goal — of purpose, persistence, and directedness? There is indeed a certain particular type of machine that appears to have those qualities; it is built according to the principles below, which were first studied in the late 1950s by Allen Newell, C. J. Shaw, and Herbert A. Simon. Originally, these systems were called general problem solvers, but I'll simply call them difference-engines.
A difference-engine must contain a description of a desired situation. It must have subagents that are aroused by various differences between the desired situation and the actual situation. Each subagent must act in a way that tends to diminish the difference that aroused it.
At first, this may seem both too simple and too complicated. Psychologically, a difference-engine might appear to be too primitive to represent the complex of ambitions, frustrations, satisfactions, and disappointments involved in the pursuit of a human goal. But these aren't really aspects of our goals themselves but emerge from the interactions among the many agencies that become engaged in pursuit of those goals. On the other side, one might wonder whether the notion of a goal really needs to engage such a complicated four-way relationship among agents, situations, descriptions, and differences. Presently we'll see that this is actually simpler than it seems, because most agents are already concerned with differences.
When we watch a ball roll down a slope, we notice it seems to try to get around obstacles that lie in its path. If we didn't know about gravity, we might be tempted to think that the ball has the goal of moving down. But we know that the ball isn't trying to do anything; the impression of intention is only in the watcher's mind.
When we experiment with Builder we also get the sense that it has a goal. Whenever you take its blocks away, it reaches out and takes them back. Whenever you knock its tower down, it rebuilds it. It seems to want a tower there, and it perseveres until the tower is done. Certainly Builder seems smarter than the rolling ball because it overcomes more complicated obstacles. But once we know how Builder works, we see that it's not so different from that ball: all it does is keep on finding blocks and putting them on top of other blocks. Does Builder really have a goal?
One ingredient of having a goal is persistence. We wouldn't say that Builder wants a tower, if it didn't keep persisting in attempts to build one. But persistence alone is not enough — and neither Builder nor that rolling ball have any sense of where they want to go. The other critical ingredient of goal is to have some image or description of a wanted or desired state. Before we'd agree that Builder wants a tower, we'd have to make sure that it contains something like an image or a description of a tower. The idea of a difference-engine embodies both elements: a representation of some outcome and a mechanism to make it persist until that outcome is achieved.
Do difference-engines really want? It is futile to ask that kind of question because it seeks a distinction where none exists — except in some observer's mind. We can think of a ball as a perfectly passive object that merely reacts to external forces. But the eighteenth- century physicist Jean Le Rond d'Alembert showed that one can also perfectly predict the behavior of a rolling ball by describing it as a difference-engine whose goal is to reduce its own energy. We need not force ourselves to decide questions like whether machines can have goals or not. Words should be our servants, not our masters. The notion of goal makes it easy to describe certain aspects of what people and machines can do; it offers us the opportunity to use simple descriptions in terms of active purposes instead of using unmanageably cumbersome descriptions of machinery.
To be sure, this doesn't capture everything that people mean by having goals. We humans have so many ways of wanting things that no one scheme can embrace them all. Nevertheless, this idea has already led to many important developments both in Artificial Intelligence and in psychology. The difference-engine scheme remains the most useful conception of goal, purpose, or intention yet discovered.
We naturally admire our Einsteins, Shakespeares, and Beethovens — and we wonder if machines could ever create such wondrous theories, plays, and symphonies. Most people think that accomplishments like these require talents or gifts that cannot be explained. If so, then it follows that computers can't create such things — since anything machines do can be explained. But why assume that what our greatest artists do is very different from what ordinary people do — when we know so little about what ordinary people do! Surely it is premature to ask how great composers write great symphonies before we know how ordinary people think of ordinary tunes. I don't believe there is much difference between normal and creative thought. Right now, if asked which seems the more mysterious, I'd have to say the ordinary kind.
We shouldn't let our envy of distinguished masters of the arts distract us from the wonder of how each of us gets new ideas. Perhaps we hold on to our superstitions about creativity in order to make our own deficiencies seem more excusable. For when we tell ourselves that masterful abilities are simply unexplainable, we're also comforting ourselves by saying that those super-heroes come endowed with all the qualities we don't possess. Our failures are therefore no fault of our own, nor are those heroes' virtues to their credit, either. If it isn't learned, it isn't earned.
When we actually meet the heroes whom our culture views as great, we don't find any singular propensities — only combinations of ingredients quite common in themselves. Most of these heroes are intensely motivated, but so are many other people. They're usually very proficient in some field — but in itself we simply call this craftsmanship or expertise. They often have enough self-confidence to stand up to the scorn of peers — but in itself, we might just call that stubbornness. They surely think of things in some novel ways, but so does everyone from time to time. And as for what we call intelligence, my view is that each person who can speak coherently already has the better part of what our heroes have. Then what makes genius appear to stand apart, if we each have most of what it takes?
I suspect that genius needs one thing more: in order to accumulate outstanding qualities, one needs unusually effective ways to learn. It's not enough to learn a lot; one also has to manage what one learns. Those masters have, beneath the surface of their mastery, some special knacks of higher-order expertise, which help them organize and apply the things they learn. It is those hidden tricks of mental management that produce the systems that create those works of genius. Why do certain people learn so many more and better skills? These all-important differences could begin with early accidents.  Everyone can praise the first child's castles and towers, but no one can see what the second child has done, and one may even get the false impression of a lack of industry. But if the second child persists in seeking better ways to learn, this can lead to silent growth in which some better ways to learn may lead to better ways to learn to learn. Then, later, well observe an awesome, qualitative change, with no apparent cause — and give to it some empty name like talent, aptitude, or gift.
Finally, an awful thought: perhaps what we call genius is rare because our evolution works without respect for individuals. Could any tribe or culture endure in which each individual discovered novel ways to think? If not, how sad, since the genes for genius might then lead not to nurturing, but only to frequent weeding-out.
We often talk of memory as though the things we know were stored away in boxes of the mind, like objects we keep in closets in our homes. But this raises many questions.
How is knowledge represented? How is it stored? How is it retrieved? Then, how is it used?
Whenever we try to answer any of these, others seem to get more complicated, because we can't distinguish clearly what we know from how it's used. The next few sections explain a theory of memory that tries to answer all these questions at once by sug- gesting that we keep each thing we learn close to the agents that learn it in the first place. That way, our knowledge becomes easy to reach and easy to use. The theory is based on the idea of a type of agent called a Knowledge-line, or K-line for short.
Whenever you get a good idea, solve a problem, or have a memorable experience, you activate a K-line to represent it. A K-line is a wirelike structure that attaches itself to whichever mental agents are active when you solve a problem or have a good idea. When you activate that K-line later, the agents attached to it are aroused, putting you into a mental state much like the one you were in when you solved that problem or got that idea. This should make it relatively easy for you to solve new, similar problems!
In other words, we memorize what we're thinking about by making a list of the agents involved in that activity. Making a K-line is like making a list of the people who came to a successful party. Here is another image of how K-lines work, suggested by Kenneth Haase, a student at the MIT Artificial Intelligence Laboratory who had a great deal of influence on this theory.
You want to repair a bicycle.  Then every tool you need to use will end up with red marks on it. When you're done, just remember that red means ‘good for fixing bicycles.’ Next time you fix a bicycle, you can save time by taking out all the red-marked tools in advance.
If you use different colors for different jobs, some tools will end up marked with several colors. That is, each agent can become attached to many different K-lines. Later, when there's a job to do, just activate the proper K-line for that kind of job, and all the tools used in the past for similar jobs will automatically become available.
This is the basic idea of the K-line theory. But suppose you had tried to use a certain wrench, and it didn't fit. It wouldn't be so good to paint that tool red. To make our K-lines work efficiently, we'd need more clever policies. Still, the basic idea is simple: for each familiar kind of mental job, your K-lines can refill your mind with fragments of ideas you've used before on similar jobs. In such a moment, you become in those respects more like an earlier version of yourself.
Suppose once, long ago, you solved a certain problem P. Some of your agents were active then; others were quiet. Now let's suppose that a certain learning process caused the agents that were active then to become attached to a certain agent kP, which we'll call a K-line. If you ever activate kP afterward, it will turn on just the agents that were active then, when you first solved that problem P!
Today you have a different problem. Your mind is in a new state, with agents Q aroused. Something in your mind suspects that Q is similar to P — and activates kP.
Now two sets of agents are active in your mind at once: the Q-agents of your recent thoughts and the P-agents aroused by that old memory. If everything goes well, perhaps both sets of agents will work together to solve today's problem. And that's our simplest concept of what memories are and how they're formed.
What happens if the now active agents get into conflicts with those the K-line tries to activate? One policy might be to give priority to the K-line's agents. But we wouldn't want our memories to rearouse old states of mind so strongly that they overwhelm our present thoughts — for then we might lose track of what we're thinking now and wipe out all the work we've done. We only want some hints, suggestions, and ideas. Another policy would give the presently active agents priority over the remembered ones, and yet another policy would suppress both, according to the principle of noncompromise. This diagram shows what happens for each of these policies if we assume that neighboring agents tend to get into conflicts:
The ideal scheme would activate exactly those P 's that would be most helpful in solving the present problem. But that would be too much to ask of any simple strategy.
Many modern scientists think it quaint to talk about mental states. They feel this idea is too subjective to be scientific, and they prefer to base their theories of psychology on ideas about information pro- cessing. This has produced many good theories about problem solving, pattern recognition, and other important facets of psychology, but on the whole it hasn't led to useful ways to describe the workings of our dispositions, attitudes, and feelings.
Is this because, as many think, our feelings are inherently more complicated than the things we more easily describe in words? Not necessarily: our memories of attitudes and feelings could come from relatively simple K-line mechanisms — yet still be inexpressible. This is because K-lines can easily record relatively widespread and diffuse activities and, later, reactivate them all at once. This helps explain a familiar psychological phenomenon:
The experiences we find easiest to recollect are often just the kinds we find the hardest to describe.
For example, a novice can remember how it felt to be at a concert. A more proficient amateur can remember more of the music itself — the rhythms and the harmonies and melodies. But only skilled musicians can recall the smaller details of timbre, texture, and arrangement. Why do we find it easier to recollect our attitudes and feelings than to describe what actually took place? That's just what we should expect from memories of the K-line kind. Suppose that a certain sentiment or disposition involved the activities of many different agents. It would be easy to construct a huge K-line with which we could, later, make ourselves approximately reexperience that complicated state — simply by rearousing the same activities. But this would not automatically enable us to describe those feelings, which is another matter entirely, because it would require us to summarize that huge, dispersed activity in terms of some much more compact arrangement of verbal expressions.
We cannot always judge the complexity of our mental states by how easily we can express them in words. A certain state of mind might involve a mass of information simply too enormous and diverse to express in any small number of words, yet not be very complicated in any interesting sense. Furthermore, the things we can express in words are, to a large extent, constrained by the social process through which we learn to use those words. In order for a word to have a predictable effect on other persons, we must maintain strict, public discipline on how that word is used — whereas each individual's private, internal signals need not be so constrained. The signals that come from our nonverbal agents can have K-line connections that branch out very rapidly to arouse other agents. If each member of such a society were to arouse a mere hundred others, then in only three or four steps the activity of a single one of them could affect a million other agents.
Once we think in terms of K-line memories, it becomes easy to imagine, at least in principle, how a person could recall a general impression of a complex previous experience — but it becomes hard to understand how a person can so easily comprehend a specific statement like John has more candy than Mary. If this theory is correct, the traditional view must be upside down, which regards it as easy to understand how minds can deal with facts and propositions, but hard to see how minds could have diffuse, hard-to-express dispositions.
We make our new ideas by merging parts of older ones — and that means keeping more than one idea in mind at once. Let's oversimplify matters for the moment and imagine that the mind is composed of many divisions, each involved with a different activity, like vision, locomotion, language, and so forth. This pattern repeats on smaller scales, so that even the thought of the simplest ordinary object is made up of smaller thoughts in smaller agencies. Thinking about a small white rubber ball could activate some divisions like these:
We'll need some way to talk about the states of many agencies at once. So, in this book, I'll use the expression mental state or total mental state when talking about the states of all of one's agents. The new phrase partial mental state is for talking about the states of smaller groups of agents.  We shall assume that each agent in our society, at each moment, is either in a quiet state or an active state. Why can't an agent be partially aroused, instead of only on or off ? They could indeed, but there are technical reasons why this would not make any fundamental difference to the issues we are discussing here. In any case, this assumption allows us to be precise:
A total state of mind is a list that specifies which agents are active and which are quiet at a certain moment. A partial state of mind merely specifies that certain agents are active but does not say which other agents are quiet.
Notice that according to this definition, a mind can have exactly one total state at any moment, but it can be in many partial states at the same time — because partial states are incomplete descriptions. The picture above shows a mind-society made up of several separate divisions, so we can think of each division's state as one partial state, and this lets us imagine that the entire system can think several thoughts at once, just as a crowd of separate people can. When your speech division is being occupied with what your friend is saying while your vision division looks for a door to exit through — then your mind is in two partial states at once.
The situation is more interesting when two K-lines activate agents in the same division at the same time: imposing two different partial mental states on the same agency can lead to conflicts. It is easy to think of a small white ball because this activates K-lines that connect to unrelated sets of agents. But when you try to imagine a round square, your agents for round and square are forced to compete to control the same set of shape-describing agents. If the conflict is not settled soon, noncompromise may eliminate both — and leave you with the sense of an undefined shape.
The basic idea is simple: we learn by attaching agents to K-lines, but we don't attach them all with equal firmness. Instead, we make strong connections at a certain level of detail, but we make weaker connections at higher and lower levels. A K-line for a kite might include some properties like these:
Whenever we turn on this K-line, it tries to activate all these agents, but those near the fringes are attached as though by twice used tape and tend to retreat when other agents challenge them. If most of the kites you've seen before were red and diamond-shaped, then when you hear about Jack's kite, those weak connections will lead you to assume that Jack's kite, too, is red and diamond-shaped. But if you should hear that Jack's kite is green, your weakly activated red-color agent memories will be suppressed by your strongly activated green-color agents. Let's call these kinds of weakly activated memories assumptions by default. Default assumptions, once aroused, stay active only when there are no conflicts. In psychological terms, they are things we assume when we have no particular reason to think otherwise. Later we'll see that default assumptions embody some of our most valuable kinds of commonsense knowledge: knowing what is usual or typical. For example, they're why we all assume that Jack has hands and feet. If such assumptions turn out to be wrong, their weak connections allow them to be easily displaced when better information comes to mind.
We sometimes think of memory as though it could transport us back to hear the voices of times gone by and see the sights of the past. But memory can't really take us anywhere; it can only recall our minds to prior states, to visit what we used to be, by putting back what was in the mind before. We introduced the level-band theory to provide a way for a memory to encompass some range or level of detail of descriptions, as when, in remembering that kite experience, certain aspects were recorded firmly and others weakly or not at all. The concept of a level-band can be applied not only to descriptions of things, but also to our memories of the processes and activities we use in order to achieve our goals — that is, the mental states we re-create that once solved problems in the past. The problems we have to solve change with time, so we must adapt our old memories to our present goals. To see how level-bands can help with that, let's now return to Play-with-Blocks — but this time let's suppose that our child has grown to maturity and wants to build a real house. Which agents from the old building-society can still be applied to this new problem?
The new house-building agency can certainly use many of Tower Builder's skills. It certainly will need Add's lower-level skills like Find and Get and Put. But House Builder won't have so much use for Tower Builder's highest-level agents like Begin and End — because these were specialized for making towers. Nor will it have much use for Builder's lowest-level skills, like those in Grasp, because picking up such small blocks isn't the problem. But most of the skills embodied in Builder's middle level-bands will still apply. These seem to embody the sort of knowledge that is most broadly and generally useful, whereas uppermost and lowest level-bands are more likely to be based on aspects of the problem that are specific to an older goal or to the particular details of the original problem. But if our memory machinery has been designed so that the contents of those distant fringes can be easily detached, the extra knowledge stored in them will rarely do much harm and can often be helpful. For example, Tower Builder's fringe details could tell us what to do in case our house should grow very tall or require a high chimney.
We started out by using level-bands for describing things — but we ended up using them for doing things! In the next few sections we'll see that it is no accident that level-related ideas play many different roles in how we think.
It's hard to recognize a thing when you're presented with too much detail. To know that you are seeing a kite, it helps to look for paper, sticks, and string. But if you were to use a microscope, what you'd perceive would not be properties of kites at all, but merely features of particular bits of paper, sticks or string. These might allow you to identify a particular kite but not to recognize any other kite. Past a certain level of detail, the more one sees, the less one can tell what one is seeing! The same applies to memories; they should weaken their attachments at lower levels of detail.
Lower Band: Beyond a certain level of detail, increasingly complete memories of previous situations are increasingly difficult to match to new situations.
To explain why K-lines need an upper-level fringe, let's return to that example in which our child originally learned how to build a tower — but now desires to build a house. Here, we could have another kind of difficulty if we remembered too much about our previous goals!
Upper Band: Memories that arouse agents at too high a level would tend to provide us with goals that are not appropriate to the present situation.
To see why our K-line memories should weaken their attachments above a certain level of detail, consider this most extreme form. Suppose some memory were so complete that it made you relive, in every detail, some perfect moment of your past. That would erase your present you — and you'd forget what you had asked your memory to do!
Both fringing effects serve to make our memories more relevant to our present purposes. The central level-band helps us find general resemblances between remembered events and present circumstances. The lower fringe supplies additional details but does not force them upon us. We use them only by default when actual details are not supplied. Similarly, the upper fringe recalls to mind some memories of previous goals, but again, we're not forced to use them except by default, when present circumstances do not impose more compelling goals. Seen this way, we can think of the lower fringe as concerned with the structures of things, and we can think of the upper fringe as involved with the functions of things. The lower levels represent objective details of reality; the upper levels represent our subjective concerns with goals and intentions.
How could the fringes of the same K-line lie in two such different realms? Because in order to think, we need intimate connections between things and goals — between structures and their functions. What use would thinking be at all, unless we could relate each thing's details to our plans and intentions? Consider how often the English language employs the selfsame words for things and for their purposes. What tools would you use, when building your house, to saw and clamp and glue your wood? That's obvious: you'd use a saw and a clamp and some glue! Behold the wondrous force of those meanings : no sooner do we hear the noun form of a word than our agents strain to perform the acts that correspond to it as a verb. This phenomenon of connecting means with ends is not confined to language — we'll see many other instances of it in other kinds of agencies — but language may allow such linking with the least constraint.
Yesterday, you watched Jack fly his kite. How do you remember that today? One answer would be, Remembering it is much like seeing it again. But yesterday, when you recognized that kite, you didn't really see it as something wholly new. The fact that you recognized it as a kite yesterday means that you already saw that kite in terms of even older memories.
This suggests two ways to make new memories of what you saw a moment ago. One scheme is shown to the left below: you simply connect a new K-line to all the agents that were recently active in your mind. The other way to make that memory is shown in the diagram to the right below: instead of attaching the new K-line to that whole multitude of separate agents, connect it only to whichever of your older K-lines were active recently. This will lead to a similar result since those K-lines were involved in arousing many of the agents that were active recently. This second scheme has two advantages: it is more economical, and it leads to forming memories as organized societies.
Consider that when you realized Jack was flying a kite, this must have involved the use of K-lines — for Jack and Fly and Kite — that had been formed at earlier times and were aroused by the sight of Jack flying his kite. When those three K-lines were activated, each of them in turn activated hundreds or thousands of other agents. (Your state of mind, when seeing that scene, resulted from combinations both of agents aroused directly by your senses and of agents aroused indirectly by your recognitions.) Now, our left-hand memory-scheme would need an enormous number of connections to link all those agents to the new K-line. But our right-hand scheme would obtain much the same effect by attaching the new K-line to only three old K-lines! Yet when you reactivate that K-line at some later date, it will, in turn, arouse the same K-lines for Jack, Fly, Kite, and whichever other recognitions were involved. As a result, you will reexperience many of the same recognitions as before. To that extent, you will feel and act as though you were back in the same situation again.
To be sure, these two types of memories would not produce exactly the same results. Our trick of connecting new K-lines to old ones will not recapture so many of the scene's precise, perceptual details.
Instead, the kinds of mental states that this hierarchical type of memory produces will be based more on stereotypes and default assumptions than on actual perceptions. Specifically, you will tend to remember only what you recognized at the time. So something is lost — but there's a gain in exchange. These K-line memory-trees lose certain kinds of details, but they retain more traces of the origins of our ideas. These memory-trees might not serve quite so well if the original circumstances were exactly repeated. But that never happens, anyway — and the structured memories will be much more easily adapted to new situations.
If each K-line can connect to other K-lines, which, in turn, connect to others, then K-lines can form societies. But how can we make sure that this can serve our purposes, instead of becoming a great, disordered mess? What could guide them into representing useful hierarchies like these?
To keep things orderly, we'll now apply that level-band idea again. Remember that we first invented K-lines to link older agents together; then we invented level-bands to keep those K-lines from filling up with too much useless, unrelated stuff. Now we have the same problem again: when connecting new K-lines to old ones, we must keep them from including too much inappropriate detail. So why not try the same solution? Let's apply the level-band idea to the K-line trees themselves!
When making a new K-line memory, do not connect it to all the K-lines active at the time but only to those that are active within a certain level-band.
It might be supposed that this idea would be hard to apply unless we specify what level means. However, something like this will happen automatically, simply because the new K-line societies will tend to inherit whatever hierarchy already existed among the original agents that become connected to those K-lines. We've actually seen two different ideas about this. In our Kite example, we talked about a description's level of detail. That is, we regarded it as more elevated to talk about a sheet stretched across a frame than to discuss the paper or the sticks themselves. In our Builder example, we talked about goals and considered the Tower Builder agent itself to be a level above the agents it exploits to solve its subproblems — agents like Begin and Add and End.
This policy of connecting new K-lines to old ones must be used in moderation. Otherwise, no new agents would ever be included in our memories. Furthermore, it should not always be required to produce simple, orderly hierarchy-trees; for example, in the case of Builder, we found that both Move and See will often need one another's help. Eventually, all of our knowledge-structures become entangled with various sorts of exceptions, shortcuts, and cross-connections. No matter: the level-band idea will still apply in general, since most of what we know will still be mainly hierarchical because of how our knowledge grows.
Isn't it interesting how often we find ourselves using the idea of level? We talk about a person's levels of aspiration or accomplishment. We talk about levels of abstraction, levels of management, levels of detail. Is there anything in common to all the level-things people talk about? Yes: they each appear to reflect some way to organize ideas — and each seems vaguely hierarchical. Usually, we tend to think that each of those hierarchies illustrates some kind of order that exists in the world. But frequently those orderings come from the mind and merely appear to belong to the world. Indeed, if our theory of K-line trees is correct, it would seem natural for us to classify things into levels and hierarchies — even when this does not work out perfectly. The diagram below portrays two ways to classify physical objects.
These two hierarchies split things up in different ways. The birds and airplanes are close together on one side, but far apart on the other side. Which classification is correct? Silly question! It depends on what you want to use it for. The one on the left is more useful for biologists, and the one on the right is more useful for hunters.
How would you classify a porcelain duck, a pretty decorative toy? Is it a kind of bird? Is it an animal? Or is it just a lifeless piece of clay? It makes no sense to argue about it: That's not a bird! Oh, yes, it is, and it is also pottery. Instead, we frequently use two or more classifications at the same time. For example, a thoughtful child can play with a porcelain duck as though it were a make-believe animal, yet at the same time treat it carefully, in its role as a delicate object.
Whenever we develop a new skill or extend an old one, we have to emphasize the relative importance of some aspects and features over others. We can place these into neat levels only when we discover systematic ways to do so. Then our classifications can resemble level-schemes and hierarchies. But the hierarchies always end up getting tangled and disorderly because there are also exceptions and interactions to each classification scheme. When attempting a new task, we never like to start anew: we try to use what has worked previously. So we search around inside our minds for old ideas to use. Then, when part of any hierarchy seems to work, we drag the rest along with it.
According to our concept of memory, the K-lines of each agency grow into a new society. So, to keep things straight, let's call the original agents S-agents and call their society the S-society. Given any S-society, we can imagine building memories for it by constructing a corresponding K-society for it. When we start making a K-society, we must link each K-line directly to S-agents, because there are no other K-lines we can connect them to. Later we can use the more efficient policy of linking new K-lines to old ones. But this will lead to a different problem of efficiency: the connections to the original S-agents will become increasingly remote and indirect. Then everything will begin to slow down — unless the K-society continues to make at least some new connections to the original S-society. That would be easy to arrange, if the K-society grows in the form of a layer close to its S-society. The diagram below suggests such an arrangement.
If arranged this way, the layer pairs could form a curious sort of computer. As S-agents excite K-agents and vice versa, a sort of spiraling activity would ensue. Over time, the location of that activity might tend to drift upward or down and might also tend to spread out; without some control, the system might soon become chaotic. But it would be hard to control the system from within, nor would that serve the purposes of other agencies. However, we can easily imagine how yet another, third agency could confine and control the K-S system's activity — by specifying which level-band should remain active and suppressing all the rest. Indeed, that is precisely the sort of coarse control that a B-brain might exercise, since it could do all this without needing to understand the fine details of what is happening inside the A-brain. The third agency might simply look on and say impatiently, This isn't getting anywhere: move up to take a higher-level view of the situation. Or it might say, That looks like progress, so move farther down and fill in more details.
Is there any essential difference between the K-and S-societies? Not really — except that the S-society came first. Indeed, we can imagine an endless sequence of such societies, in which each new one learns to exploit the last. Later we'll propose that this is how our minds develop in infancy — as sequences of layers of societies. Each new layer begins as a set of K-lines, which starts by learning to exploit whatever skills have been acquired by the previous layer. Whenever a layer acquires some useful and substantial skill, it tends to stop learning and changing — and then yet another new layer can begin to learn to exploit the capabilities of the last. Each new layer begins as a student, learning new ways to use what older layers can already do. Then it slows its learning rate — and starts to serve both as subject and as teacher to the layers that form afterward.
 Did you like that play? Do you like Wagner? Did you enjoy your year abroad?
What makes us want to compress so much into such inexpressive summaries as like, prefer, and enjoy ? Why try to reduce such complex things to simple values or amounts of pleasurable quality? The answer is that our measures of pleasure have many uses. They help us make comparisons, compromises, and choices. They are involved with the communication signs that we use to signify various degrees of attachment, satisfaction, and agreement. They show themselves not only in words, but also as gestures, intonations, smiles and frowns, and many other expressive signs. But we have to be careful not to accept those signs at their face value. Neither the state of the world nor that of the mind is ever so simple that it can be expressed in a single, one-dimensional judgment. No situation is ever completely satisfactory or entirely disagreeable, and our reactions of pleasure or disgust are only superficial summaries of pyramids of underlying processes. To enjoy an experience, some of our agents must summarize success — but other agents must be censuring their subordinates for failing to achieve their goals. So we ought to be suspicious when we find ourselves liking something very much, because that might mean some of our agencies are forcefully suppressing other possibilities.
The surer you are that you like what you are doing, the more completely your other ambitions are being suppressed.
To choose between alternatives, the highest levels of the mind demand the simplest summaries. If your top-level feelings were too often mixed, you would rarely be able to make a choice to decide which foods to eat, which paths to walk, or which thoughts to think. At the level of action, you're forced to simplify right down to expressions like Yes and No. But these are not informative enough to serve the lower levels of the mind, where many processes go on at once, and every agent has to judge how well it is serving some local goals. At lower levels of the mind, there must be hosts of smaller, coexisting satisfactions and annoyances.
We often talk as though we ought to be controlled by what we want. Indeed we scarcely distinguish between wanting something and potentially obtaining pleasure from it; the relation between these two ideas seems so intimate that it actually feels odd to mention it. It seems so natural to want what we like and to avoid what we don't like that we sometimes feel a sense of unnatural horror when another person appears to violate that rule; then we think, They surely wouldn't do such things unless, deep down, they really wanted to. It is as though we feel that people ought to want only to do the things they like to do.
But the relation between wanting and liking is not simple at all, because our preferences are the end products of so many negotiations among our agencies. To accomplish any substantial goal, we must renounce the other possibilities and engage machinery to keep ourselves from succumbing to nostalgia or remorse. Then we use words like liking to express the operation of the mechanisms that hold us to our choice. Liking's job is shutting off alternatives; we ought to understand its role since, unconstrained, it narrows down our universe. This leads to liking's artificial clarity: it does not reflect what liking is but only shows what liking does.
We all know how accomplishment can bring satisfaction, and we tend to assume a direct connection between them. In very simple animals, where satisfaction means no more than meeting simple, basic needs, satisfaction and accomplishment must indeed be virtually the same. But in a complex human brain, a great many layers of agencies are interposed between the ones that deal with body needs and those that represent or recognize our intellectual accomplishments. Then what is the significance, in these more complicated systems, of those pleasant feelings of accomplishment and disagreeable sensations of defeat? They must be involved with how our higher-level agencies make summaries.
Suppose that you once had to send a present to a friend. You had to choose a gift and find a box in which to wrap it. Soon, each such job turned into several smaller ones — like finding strings and tying them. The only way to solve hard problems is by breaking them into smaller ones and then, when those are too difficult, dividing them in turn. So hard problems always lead to branching trees of subgoals and subproblems. To decide where resources should be applied, our problem-solving agents need simple summaries of how things are going. Let's suppose each agent's summary is based on other summaries it gets from the agents it supervises. Here is a pathological example of what could happen if every such summary were based on a simple majority decision:
When all is done, if someone asked if you enjoyed the whole experience, you might say that it was fun or terrible. But no such summary can say very much of what your agencies actually learned. Your knot-tying processes learned which actions worked and failed, your paper-folding and gift-selecting processes had other failures and accomplishments; but your overall assessment of the experience cannot reflect all those details. If the entire episode left you unhappy, you might be less inclined to give presents in the future, but that should not have much effect on what you learned about folding paper and tying string. No single sense of good or bad can reflect much of what went on inside all your agencies; too much information must be concealed. Then why does it seem so satisfactory for us to classify our feelings into positive and negative and conclude that on the whole the net effect was bad or good? True, sometimes feelings are more mixed and everything seems bittersweet, but, as we'll see, there are many reasons why we have to oversimplify.
So far, we've talked mostly of learning from success. But consider that when you succeed, you must already have had the necessary means within your grasp. If so, then making changes in your mind might only make things worse! As people often say, You shouldn't argue with success. For whenever you try to improve an already working procedure, you risk damaging whichever other skills depend on that same machinery.
Accordingly, it may be more important that we learn from how we fail. What should you do if some well-established method — call it M — has failed to reach a certain goal? One policy would be to alter M, so it won't make the same mistake again. But even that might be dangerous because it might cause M to fail in other ways. Besides, we might not know how to change M to remove the error. A safer way to deal with this would be to modify M by adding special memory devices called censors and suppressors (we'll discuss this in detail later), which remember particular circumstances in which M fails and later proceed to suppress M when similar conditions recur. Such censors would not tell you what to do, only what you shouldn't do; still, they prevent your wasting time by repeating old mistakes.
Learning has at least two sides. Some parts of our minds learn from success — by remembering when methods work. But other portions of our minds learn mainly when we make mistakes, by remembering the circumstances in which various methods failed to work. Later we'll see how this can teach not only what we shouldn't do, but also what we shouldn't think! When that happens, it can permeate our minds with prohibitions and taboos of which we're entirely unaware. Thus, learning from success tends to aim and focus how we think, while learning from failure also leads to more productive thoughts, but in a less directed way.
We would not need to deal with exceptions and censors if we lived in a universe of simple, general rules with no exceptions, as in the lovely mathematical worlds of arithmetic, geometry, and logic. But perfect logic rarely works in the real worlds of people, thoughts, and things.
This is because it is no accident that there are no exceptions to the rules in those mathematical worlds: there, we start with the rules and imagine only objects that obey them. But we can't so willfully make up the rules for objects that already exist, so our only course is to begin with imperfect guesses — collections of rough and ready rules — and then proceed to find out where they're wrong.
Naturally, we tend to prefer learning from success rather than from failure. However, I suspect that confining ourselves to positive learning experiences alone leads to relatively small improvements in what we can already do. Probably, there is no way to avoid at least a certain degree of discomfort when we make substantial changes in how we think.
Why do children enjoy the rides in amusement parks, knowing that they will be scared, even sick? Why do explorers endure suffering and pain — knowing that their very purpose will disperse once they arrive? And what makes ordinary people work for years at jobs they hate, so that someday they will be able to — some seem to have forgotten what?
There is more to motivation than immediate reward. When we succeed at anything, a lot goes on inside the mind. For example, we may be filled with feelings of accomplishment and pride, and feel impelled to show others what we've done and how. However, it is the fate of more ambitious intellects that the sweetness of success will swiftly fade as other problems come to mind. That's good because most problems do not stand alone but are only smaller parts of larger problems. Usually, after we solve a problem, our agencies return to some other, higher-level cause for discontent, only to lose themselves again in other subproblems. Nothing would get done if we succumbed to satisfaction.
But what if a situation gets completely out of our control — and offers no conceivable escape from suffering? Then all we can do is try to construct some inner plan for tolerating it. One trick is to try to change our momentary goal — as when we say, It's getting there that's all the fun. Another way is looking forward to some benefit to future Self: I certainly shall learn from this. When that doesn't work, we can still resort to even more unselfish schemes: Perhaps others may learn from my mistake.
These kinds of complications make it impossible to invent good definitions for ordinary words like pleasure and happiness. No small set of terms could suffice to express the many sorts of goals and wants that, in our minds, compete in different agencies and on different scales of time. It is no wonder that those popular theories about reward and punishment have never actually led to explaining higher forms of human learning — however well they've served for training animals. For in the early stages of acquiring any really new skill, a person must adopt at least a partly antipleasure attitude: Good, this is a chance to experience awkwardness and to discover new kinds of mistakes! It is the same for doing mathematics, climbing freezing mountain peaks, or playing pipe organs with one's feet: some parts of the mind find it horrible, while other parts enjoy forcing those first parts to work for them. We seem to have no names for processes like these, though they must be among our most important ways to grow.
None of this is to say that we can discard the concepts of pleasure and liking as we use them in everyday life. But we have to understand their roles in our psychology; they represent the end effects of complex ways to simplify.
The psychologist Jean Piaget was one of the first to realize that watching children might be a way to see how mind-societies grow. In one of his classic experiments, he showed a child two matching sets of eggs and cups — and asked, Are there more eggs or more egg cups?
Then he spread the eggs apart — before the child's eyes — and asked again if there were more eggs or more egg cups.
One might try to explain this by supposing that older children are better at counting. However, this can't explain another famous experiment of Piaget's, which began by showing three jars, two filled with water. All the children agreed that the two short, wide jars contained equal amounts of liquid. Then, before their eyes, he poured all the liquid from one of the short jars into the tall, thin one and asked which jar had more liquid now.
These experiments have been repeated in many ways and in many countries — and always with the same results: each normal child eventually acquires an adult view of quantity — apparently without adult help! The age at which this happens may vary, but the process itself seems so universal that one cannot help suspecting that it reflects some fundamental aspect of the child's development. In the next few sections we'll examine the idea of more and show that it conceals the workings of a large, complex Society-of-More — which takes many years to learn.
What do those egg and water jar experiments say about our growth from infancy? Let's consider several explanations.
QUANTITY: Perhaps the younger children simply don't yet understand the basic concept of quantity: that the amount of liquid remains the same.
In the next few sections I'll argue that we don not learn one single, underlying concept of quantity. Instead, each person must construct a multileveled agency, which we'll call the Society-of-More, that finds different ways to deal with quantities.
EXTENT: The younger children seem unduly influenced by the larger extent of space taken up by the spread-out eggs and taller water column.
That cannot be the whole story because most adults, too, judge that there's more water in the taller jar — if they merely see the final scene, without knowing from where the water was poured! Here are a few other theories about the younger child's judgment:
REVERSIBILITY: The older children pay more attention to what they think remains the same — while younger ones are more concerned with what has changed. CONFINEMENT: An older child knows that the amount of water stays the same, if none was ever added or removed or lost or spilled.
LOGIC: Perhaps younger children have not yet learned to apply the kinds of reasoning that one would need to understand the concept of quantity.
Every one of these explanations has some truth in it, but none reach the heart of the issue. It is clear that the older children know more about such matters and can do more complex kinds of reasoning. But there is ample evidence that most younger children also possess enough of the required abilities. For example, we can describe the experiment without actually doing it at all or we can perform it out of the child's sight, behind a cardboard screen. Then, when we explain what is happening, quite a few of the younger children will say, Of course they'll be the same.
Then what is the difficulty? Evidently, the younger children possess the ideas they need but don't know when to apply them! One might say that they lack adequate knowledge about their knowledge, or that they have not acquired the checks and balances required to select or override their hordes of agents with different perceptions and priorities. It is not enough to be able to use many kinds of reasoning;
one also must know which to use in different circumstances! Learning is more than the mere accumulation of skills. Whatever we learn, there is always more to learn — about how to use what was already learned.
Let's try to explain the water jar experiment in terms of how a child's agencies deal with comparisons. Suppose the child begins with only three agents:
Tall says, The taller, the more. There's more inside a taller thing. Thin says, The thinner, the less. There's less inside a thinner thing. Confined says, The same, because nothing was added or removed.
How do we know children have agents like these? We can be sure that younger children have agencies like Tall and Thin because they all can make these judgments:
It is harder to know whether younger children have agents like Confined, but many of them can indeed explain that something remains the same when one pours a liquid back and forth. In any case, there is a conflict because the three agents give three different answers — more, less, and same! What could be done to settle this? The simplest theory is that the younger children have placed their agents in some order of priority.
Such a scheme can be extremely practical, since placing all the agents in order of priority makes it easy to know which to use. For example, we often compare things by their extents — by how far they reach in space. But why put Tall ahead of Wide? People do indeed seem most sensitive to vertical extents. We do not know whether this is built from the start into our brains, but in any case, the bias is usually justified because more height so frequently goes along with other sorts of largenesses.
Who's bigger — you or your cousin? Stand back to back! Who's the strongest? Those adults looming way above! How to divide a liquid into equal portions? Match the levels!
No other agent seems so good as Tall for making everyday comparisons. Still, no priority-scheme will always work. In the situation of the water jar experiment, Confined ought to come first, but the younger child's priorities lead to making the wrong judgment. One might wonder, incidentally, whether Tall and Short, or Wide and Thin, should be considered to be different agents. Logically, just one of each pair would suffice. But I doubt that in the brain it would suffice to represent Short by the mere inactivity of Tall. To adults these are opposites, but children do not work so logically. One child I knew insisted that knife was the opposite of fork, but that fork was the opposite of spoon. Water was the opposite of milk. As for the opposite of opposite, that child considered this to be too foolish to discuss.
What should one do when different kinds of knowledge don't agree? It sometimes helps to place them in some order of priority, but as we've seen, that can still lead to mistakes. How can we make our system sensitive to different circumstances? The secret is to use the principle of noncompromise and look for help from other agencies! For help with comparing quantities, we'll need to add new administrative agents to our Society-of-More.
The new Appearance administrator is designed to say more when the agent Tall is active, to say less when the agent Thin is active,
and to say nothing at all when something appears both taller and thinner. Then the other new administrator, History, makes the decision on the basis of what Confined says.
This explanation of the difference between the older and younger children was first proposed by Seymour Papert in the 1960s, when we first started to explore society of mind ideas. Most previous theories had tried to explain Piaget's experiments by suggesting that children develop different kinds of reasoning as time goes by. That certainly is true, but the importance of Papert's conception is in emphasizing not merely the ingredients of reasoning, but how they're organized: a mind cannot really grow very much merely by accumulating knowledge. It must also develop better ways to use what it already knows. That principle deserves a name.
Papert's Principle: Some of the most crucial steps in mental growth are based not simply on acquiring new skills, but on acquiring new administrative ways to use what one already knows.
Our two new middle-level managers illustrate this idea: Appearance and History form a new, intermediate layer that groups together certain sets of lower-level skills. The choice of agents for those groups is absolutely critical. The system will work quite well if we group Tall and Thin together, so that Confined can take control when they conflict. But it would only make things worse if we were to group Tall and Confined together! Then what decides which groups to form? Papert's principle suggests that the processes which assemble agents into groups must somehow exploit relationships among the skills of those agents. For example, because Tall and Thin are more similar in character to one another than to Confined, it makes sense to group them more closely together in the administrative hierarchy.
Think how many meanings more must have! We seem to use a different one for every sort of thing we know.
More red. More loud. More swift. More old. More tall. More soft. More cruel. More alive. More glad. More wealthy.
Each usage has a distinct sense, involving different agencies. How could all these ways to make comparisons get grouped into just one society? Here's a Society-of-More a child might use to deal with that egg cup problem.
This society has two main divisions. In its Appearance division, a Spatial subdivision considers both the increased extent occupied by the spread-out eggs and also their thinned-out appearance or reduced density. In the case of those spread-out eggs, these conflict — and the Spatial agency withdraws. Then, if the child can count, Numerical decides; otherwise the History division applies some agents that use memories of recent happenings. If some of the eggs were rolled away, Confined would say that their amount is no longer the same; if the eggs were merely moved around, Reversible would claim that their amount cannot have changed.
To solve the water jar problem, the Society-of-More would need other kinds of lower-level agents:
You might complain that even if we needed these hordes of lower-level agencies to make comparisons, this system has too many middle-level managers. But those mountains of bureaucracy are more than worth their cost. Each higher-level agent embodies a form of higher-order knowledge that helps us organize ourselves by telling us when and how to use the things we know. Without a many layered management, we couldn't use the knowledge in our low-level agencies; they'd all keep getting in one another's way.
Although Piaget's experiments about conservation of quantity have been confirmed as thoroughly as any in psychology, we can appreciate why many people are skeptical when they first hear of these discoveries. They contradict the traditional assumption that children are much like adults, except more ignorant. How strange it is that in all the centuries of history, these phenomena went unnoticed until Piaget — as though no one had ever watched a child carefully! But it has always been that way with science. Why did it take so long for our thinkers to discover such simple ideas as Isaac Newton's laws of motion or Darwin's idea of natural selection? Here are some frequent challenges.
Parent: Couldn't it be that younger children use words in ways that do not mean the same things to adults? Perhaps they simply take Which is more? to mean Which is higher? or Which is longer?
Careful experiments show that this can't be entirely a matter of words. We can offer the same choices, wordlessly, yet most younger children will still reach out for taller, thinner jars of orange juice or stretched-out rows of candy eggs.
Critic: What happens when Appearance and History conflict? Won't that paralyze your whole Society-of-More?
It would indeed — unless More has yet other levels and alternatives. And adults have other kinds of explanations — like magic, evaporation, or theft. But indeed, stage magicians find that making things disappear does not entertain the youngest children; presumably they are too used to encountering the unexplainable. What happens when More cannot decide what to do? That depends upon the states of other agencies — including those involved in dealing with frustration, restlessness, and boredom.
Psychologist: We've heard of recent evidence that, despite what Piaget said, very young children do have concepts of
quantity; many of them can even count those eggs. Doesn't that refute some of Piaget's discoveries? Not necessarily. Consider that no one disputes the outcomes of those jar and cup experiments. What is the significance, then, of evidence that the young children do possess methods that could give correct answers — and yet they do not use those abilities? As far as I can see, such evidence would only further support the need for explanations like those of Papert and Piaget.
Biologist: Your theory might explain how some children could acquire those concepts about quantities — but it doesn't explain why all children end up with such similar abilities! Could we be born with built-in genes that make brains do this automatically?
This is a profound question. It is hard — but not impossible — to imagine how genes could directly influence the higher-level ideas and conceptions that we eventually learn. We'll discuss this in the appendix at the end of this book.
In learning their Societies-of-More, children learn various skills for comparing different qualities and quantities, like number and extent. It is tempting to try to summarize all that by saying that the children are learning something; we could call it the concept of quantity. But why do we feel we have to think of what we learn as things or concepts? Why must we thingify everything?
What is a thing? No one doubts that a child's building-block is a thing. But is a child's love for its mother also a thing ? We're imprisoned by our poverty of words because even though we have good ways to describe objects and actions, we lack methods for describing dispositions and processes. We can scarcely speak of what minds do except as though they were filled with things that one could see or touch; that's why we cling to terms like concepts and ideas. I don't mean to say that this is always bad, for thing-ifying is indeed a splendid mental instrument. But for our present purpose, it is disastrous to assume that our minds contain some single concept of quantity. At different times, a word like more can mean many different kinds of things. Think about each of these expressions.
More colorful. More loud. More swift. More valuable. More complicated.
We speak as though these were similar, yet each of them involves a different, hard-earned web of ways to think! The phrase more loud might seem at first to be merely a matter of magnitude. But consider how the sound of a distant gong seems louder than a whisper near the ear — no matter that its actual intensity is less. Your reaction to what you hear depends not only on its physical intensity, but also on what your agencies conclude about the character of its source. Thus you can usually tell whether a gong is loud but distant, rather than soft but close, by unconsciously making assumptions about the origin of that sound. And all those other kinds of more engage equally subtle sorts of expertise.
Instead of assuming that our children come to crystallize a single concept of quantity, we must try to discover how our children accumulate and classify their many methods for comparing things. How do agents like Tall, Thin, Short, and Wide get formed into subagencies? To an adult, it seems natural to associate both being taller and being wider with being larger. But what prevents the child from inventing senseless concepts such as being Green and Tall and having recently been touched ? No child has the time to generate and test all possible combinations to find which ones are sensible. Life is too short to do that many bad experiments! The secret is: always try to combine related agents first. Tall, Thin, Short, and Wide are all closely related, because they are all concerned with making comparisons between spatial qualities. In fact, they probably involve agencies that are close to one another in the brain and share so many agents in common that they'll naturally seem similar.
Such lessons just don't seem to work very well. Given enough explanation and encouragement, and enough drill and practice, we can make children appear to understand — yet even then they don't often apply what they've learned to real-life situations. Thus it seems that even when we lead them along these paths, they remain unable to use much of what we show to them until they develop inner signposts of their own.
Here's my guess about what goes wrong. Presumably the child senses that the spaced-out eggs are more because they stretch across a longer span. Eventually, we want that sense of greater length to be canceled out by the sense that there's more empty space between the eggs. In the more mature Papert hierarchy, this would happen automatically — but for now, the child could learn this only as a special, isolated rule. Many other problems could also be solved by making special rules for them. But to simulate that multilayer society, complete with middle-level agents like Appearance and History, would involve so many special rules, and so many exceptions to them, that the younger child would be unable to manage so much complexity. The result is that educational programs allegedly designed according to Piaget often appear to succeed from one moment to the next, but the structures that result from this are so fragile and specialized that children can apply them only to contexts almost exactly like those in which they were learned.
All this reminds me of a visit to my home from my friend Gilbert Voyat, who was then a student of Papert and Piaget and later became a distinguished child psychologist. On meeting our five-year-old twins, his eyes sparkled, and he quickly improvised some experiments in the kitchen. Gilbert engaged Julie first, planning to ask her about whether a potato would balance best on one, two, three, or four toothpicks. First, in order to assess her general development, he began by performing the water jar experiment. The conversation went like this:
Gilbert: Is there more water in this jar or in that jar? Julie: It looks like there's more in that one. But you should ask my brother, Henry. He has conservation already.
Gilbert paled and fled. I always wondered what Henry would have said. In any case, this anecdote illustrates how a young child may possess many of the ingredients of perception, knowledge, and ability needed for this kind of judgment — yet still not have suitably organized those components.
Parent: Why are all the agents in your societies so competitive? They're always attacking each other. Instead of making Tall and Thin cancel each other out, why can't they cooperate?
The first part of this book has given this impression because we had to begin with relatively simple mechanisms. It is fairly easy to resolve conflicts by switching among alternatives. It is much harder to develop mechanisms that can use cooperation and compromise — because that requires more complex ways for agencies to interact. In later sections of this book we'll see how higher-level systems could make more reasonable negotiations and compromises.
How could a brain continue functioning while changing and adding new agents and connections? One way would be to keep each old system unchanged while building a new version in the form of a detour around or across it — but not permitting the new version to assume control until we're sure that it can also perform the older system's vital functions. Then we can cut some of the older connections.
We could use this method to form our hierarchical Society-of-More:
Now let's draw this in another form, as though there were no room to fit new agents in between the older ones.
As we accumulate more low-level agents and additional intermediate layers to manage them, this grows into the very multilevel hierarchy we've seen before.
The nerve cells in an animal's brain can't always move aside to make more room for extra ones. So those new layers might indeed have to be located elsewhere, attached by bundles of connection wires. Indeed, no aspect of the brain's anatomy is more striking than its huge masses of connection bundles.
What possible kind of brain-event could correspond to anything like the meaning of an ordinary word? When you say red, your vocal cords obey commands from pronouncing agents in your brain, which make your chest and larynx muscles move to produce that special sound. These agents must in turn receive commands from somewhere else, where other agents respond to signals from yet other places. All those places must comprise the parts of some society of mental agencies.
It's easy to design a machine to tell when there is something red: start with sensors that respond to different hues of light, and connect the ones most sensitive to red to a central red-agent, making corrections for the color of the lighting of the scene. We could make this machine appear to speak by linking each color-agent to a device that pronounces the corresponding word. Then this machine could name the colors it sees — and even distinguish more hues than ordinary people can. But it would be a travesty to call this sight, since it's nothing but a catalog that lists a lot of colored dots. It would share no human notion of what colors come to mean to us, because without some sense of texture, form, and very much more, it would have few of the qualities of our human kinds of images and thoughts.
Of course no little diagram can capture more than a fragment of any real person's thoughts about the world. But this should not be taken to mean that no machine could ever have the range of sensibilities that people have. It merely means that we aren't simple machines; indeed, we should understand that in learning to comprehend the qualities of vast machines, we are still in the dark ages. And in any case, a diagram can only illustrate a principle: there cannot be any compact way to represent all the details of full-grown mind-society. To talk about such complex things, we can only resort to language tricks that make our listeners' minds explore the worlds inside themselves.
The brain is imprisoned inside the skull, a silent, dark, and motionless place; how can it learn what it's like outside? The surface of the brain itself has not the slightest sense of touch; it has no skin with which to feel; it is only connected to skin. Nor can a brain see, for it has no eyes; it only is connected to eyes. The only paths from the world to the brain are bundles of nerves like those that come in from the eyes, ears, and skin. How do the signals that come through those nerves give rise to our sense of being in the outside world? The answer is that this sense is a complicated illusion. We never actually make any direct contact with the outside world. Instead, we work with models of the world that we build inside our brains. The next few sections try to sketch how that could come about.
The surface of the skin contains countless little touch-sensing agents, and the retina of the eye includes a million tiny light detectors. Scientists know a good deal about how these sensors send signals to the brain. But we know much less about how those signals lead to sensations of touch and of sight. Try this simple experiment:
Touch your ear.
What did that feel like? It seems impossible to answer that because there's scarcely anything to say. Now try a different experiment:
Touch your ear twice, in two different places, and also touch your nose.
Which two touches feel most similar? That question seems much easier to answer: one might say that the two ear touches feel more similar. Evidently, there is scarcely anything that one can say about a single sensation by itself, but we can often say much more when we can make comparisons.
Consider the analogy to how mathematics treats a perfect point. We shouldn't speak about its shape; it simply doesn't have a shape! But since we're used to things as having shapes,
we can't help thinking of points as round, like very tiny little dots. Similarly, we're not supposed to talk about the size of a point — since mathematical points, by definition, have no size. Still, we can scarcely help but think, in any case, they're very small. In fact, there's absolutely nothing to be said about a single point, except how it relates to other points. This is not because such things are too complicated to explain, but because they are too simple to explain. One cannot even speak about where a point is, by itself — since where has meaning only in relation to other points in space. But once we know some pairs of points, we can relate these to the lines that connect them, and then we can define new, different points where various pairs of lines may intersect. Repeating this can generate entire worlds of geometry. Once we understand the terrifying fact that points are nothing by themselves but exist only in relation to other points, then we can ask, as Einstein did, whether time and space are anything more than vast societies of nearnesses.
In the same way, there is little that one could say about any single touch — or about what any single sense-detecting agent does. However, there is much more to be said about the relations between two or more skin touches, because the closer together two skin spots are, the more frequently they'll both be touched at the same time.
The reason our skin can feel is because we're built with myriad nerves that run from every skin spot to the brain. In general, each pair of nearby places on the skin is wired to nearby places in the brain. This is because those nerves tend to run in bundles of parallel fibers — more or less like this:
Each sensory experience involves the activity of many different sensors. In general, the greater the extent to which two stimuli arouse the same sensors, the more nearly alike will be the partial mental states those stimuli produce — and the more similar those stimuli will seem, simply because they'll tend to lead to similar mental consequences.
Other things being equal, the apparent similarity of two stimuli will depend on the extent to which they lead to similar activities in other agencies.
The fact that the nerves from skin to brain tend to run in parallel bundles means that stimulating nearby spots of skin will usually lead to rather similar activities inside the brain. In the next section we'll see how this could enable an agency inside the brain to discover the spatial layout of the skin. For example, as you move a finger along your skin, new nerve endings are stimulated — and it is safe to assume that the new arrivals represent spots of skin along the advancing edge of your finger.
Given enough such information, a suitably designed agency could assemble a sort of map to represent which spots are close together on the skin. Because there are many irregularities in the nerve-bundle pathways from skin to brain, the agencies that construct those maps must be able to tidy things up. For example, the mapping agency must learn to correct the sort of crossing-over shown in the diagram. But that is only the beginning of the task. For a child, learning about the spatial world beyond the skin is a journey that stretches over many years.
We've seen that touching nearby spots of skin will usually give rise to similar sensations: this is because the corresponding nerves run in parallel courses and thus cause similar activities inside the brain. The reverse is also usually true: the more similar two sensations are, the closer their origins in the skin. This has an important consequence:
The nerve pathways that preserve the physical nearness relations of our skin-sensors can make it easy for inner agencies to discover corresponding nearnesses about the outer world of space.
Moving your hand across an object tells you something about that object's shape. Imagine what must happen when a very young infant moves its hand across some object: each continuous motion produces a sequence of skin-sensor signals. Over time, various mapping agents can first use this information to learn, simply, which skin spots are nearest one another. Later, further layers of mapping agents could learn which skin spots lie between which others; this should be easy, too, because most small-scale motions tend to go in nearly straight lines. But then, since space itself is just a society of nearness relations between places, this is all the information we need to reconstruct the spatial structure of the skin. All this is in accord with a basic principle of mathematics:
Suppose you were lost in some unknown space — and could only tell which pairs of points were close to one another. That would be enough for you to figure a great deal about the space. From that alone, you could deduce if you were in a world of two dimensions or three. You could tell where there were obstacles and boundaries, holes and tunnels and bridges, and so on. You could figure out the global layout of that world from just those local bits of information about nearnesses.
It is a wonderful fact that, in principle, one can deduce the global geography of a space from nothing more than hints about which pairs of points lie near one another! But it is another matter to actually make such maps, and no one yet knows how the brain does this. To design a machine to accomplish such tasks, one could begin with a layer of correlation agents, one for each tiny patch of skin, each engineered to detect which other skin spots are most often aroused at nearly the same times; those will then be mapped as the nearest ones. A second layer of similar agents could then begin to make maps of larger regions, and several such layers would eventually assemble a sequence of maps on various scales, for representing several levels of detail.
If brains do something of this sort, it might illuminate a problem that has troubled some philosophers: Why do we all agree on what the outer world of space is like? Why don't different people interpret space in different, alien ways? In principle it is mathematically possible for each person to conclude, for example, that the world is three-dimensional — rather than two-or four-dimensional — just from enough experience with nearby pairs of points. However, if the wires from the skin to the brain were shuffled and scrambled around too much, we would probably never get them straightened out because the actual calculations for doing such things would be beyond our capabilities.
Our ways to think depend in part on how we're raised. But at the start, much more depends upon the wiring in our brains. How do those microscopic features work to influence what happens in our mental worlds? The answer is, our thoughts are largely shaped by which things seem most similar. Which colors seem the most alike? Which forms and shapes, which smells and tastes, which timbres, pitches, pains and aches, which feelings and sensations seem most similar? Such judgments have a huge effect at every stage of mental growth — since what we learn depends on how we classify.
For example, a child who classified each fire just by the color of its light might learn to be afraid of everything of orange hue. Then we'd complain that the child had generalized too much. But if that child classified each flame, instead, by features that were never twice the same, that child would often be burned — and we'd complain that it hadn't generalized enough.
Our genes supply our bodies with many kinds of sensors — external event-detecting agents — each of which sends signals to the nervous system when it detects certain physical conditions. We have sensory-agents in our eyes, ears, nose, and mouth that discern light, sound, odors, and tastes; we have agents in the skin that sense pressure, touch, vibration, heat, and cold; we have internal agents that sense tensions in our muscles, tendons, and ligaments; and we have many other sensors of which we're normally unaware, such as those that detect the direction of gravity and sense the amounts of various chemicals in different parts of the body.
The agents that sense the colors of light in human eyes are much more complex than the redness agents of our toy machine. But this is not the reason that simple machine can't grasp what Redness means to us — for neither can the sense detectors in our human eyes. For just as there is nothing to say about a single point, there's nothing to be said about an isolated sensory signal. When our Redness, Touch, or Toothache agents send their signals to our brains, each by itself can only say, I'm here. The rest of what such signals mean to us depends on how they're linked to all our other agencies.
In other words, the qualities of signals sent to brains depend only on relationships — the same as with the shapeless points of space. This is the problem Dr. Johnson faced when creating definitions for his dictionary: each separate word like bitter, bright, salt, or sweet attempts to speak about a quality of a sensory signal. But all that a separate signal can do is announce its own activity — perhaps with some expression of intensity. Your tooth can't ache (it can only send signals); only you can ache, once your higher-level agencies interpret those signals. Beyond the raw distinctiveness of every separate stimulus, all other aspects of its character or quality — be it of touch, taste, sound, or light — depend entirely on its relationships with the other agents of your mind.
How do we learn about the real, three-dimensional world? We've seen how certain agencies might map the layout of the skin. But how could we progress from that to learn about the world of space beyond the skin? One might ask why infants can't simply look around to see what's really going on. Unfortunately, the easy-sounding phrase simply look conceals too many hard problems. When you look at an object, some light from it shines into your eye and stimulates some sensors there. However, every motion of your body, head, or eye makes drastic changes to the image in your eye. How can we extract any useful information when everything changes so rapidly? Although it should be possible, in principle, to design a machine that could eventually learn to relate those motions to the resulting changes in the images, this would surely take a long time, and it appears that our brains have evolved with special mechanisms that help us compensate for motions of the body, head, and eye. This makes it easier for other agencies to learn to use visual information.
Later we'll discuss some other realms of thought in which we use analogies and metaphors to change our points of view. Perhaps those wonderful abilities evolved in similar ways, since recognizing that an object is the same when seen from different views is not so different from being able to imagine things that are not in view at all.
In any case, we really do not understand how the child learns to understand space. Perhaps we start by doing many small experiments that lead to our first, crude maps of the skin. Next we might start to correlate these with the motions of our eyes and limbs; two different actions that lead to similar sensations are likely to have passed through the same locations in space. A critical step would be developing some agents that represent a few places outside the skin. Once those places are established (the first ones might be near the infant's face), one could proceed to another stage: the assembly of an agency that represents a network of relationships, trajectories, and directions between those places. Once this is accomplished, the network could continue to extend to include new places and relationships.
However, this would be only the beginning. Long ago, psychologists like Freud and Piaget observed that children seem to recapitulate the history of astronomy: first they imagine the world as centered around themselves — and only later do they start to view themselves as moving within a stationary universe, in which the body is just like any other object. It takes several years to reach that stage, and even in their adolescent years, children are still improving their abilities to envision how things appear from other viewpoints.
It would be wonderful if we could classify all behavior into two types: built-in and learned. But there simply is no clear-cut boundary between heredity and environment. Later, I'll describe an agency that is sure to learn one particular thing: to recognize human beings. But if such an agency is destined to end up with a certain particular behavior, is it reasonable to say that it learns? Since this type of activity appears to have no common name, we'll call it predestined learning.
Every child eventually learns to reach for food. To be sure, each different child lives through a different history of reaching-act experiences. Nevertheless, according to our theory of nearness models of space, all those children will end up with generally similar results because that outcome is constrained by the nearness relations of real-world space. Why make the brain use a tedious learning process when the final outcome seems so clear? Why not build in the answer genetically? One reason could be that learning is more economical. It would require an enormous store of genetic information to force each separate nerve cell to make precisely the right connections, whereas it would require much less information to specify the construction of a learning machine designed to unscramble whatever irregularities result from a less constrained design.
This is why it isn't sensible to ask, Is the child's conception of space acquired or inherited? We acquire our conceptions of space by using agencies that learn in accord with processes determined by inheritance. These agencies proceed to learn from experience — but the outcomes of their learning processes are virtually predestined by the spatial geometry of our body parts. This kind of mixture of adaptation and predestination is quite common in biology, not only in the brain's development but in that of the rest of the body as well. How, for example, do our genes control the shapes and sizes of our bones? They may begin with some relatively precise specification of the types and location of certain early cells. But that alone would not be adequate for animals that themselves have to adapt to different conditions; therefore those early cells must themselves be programmed to adapt to the various chemical and mechanical influences that may later be imposed on them. Such systems are essential for our development, since our organs must become able to perform various tightly constrained activities, yet also be able to adapt to changing circumstances.
Perhaps the growth of the Society-of-More is another instance of predestined learning, for it seems to develop in every normal child without much outside help. It seems clear that this complex agency is not built directly by inborn genes; instead, we each discover our own ways to represent comparisons — yet we all arrive at much the same final outcome. Presumably, genetic hints must help with this by supplying new layers of agents at roughly the right times and places.
Let's do one more experiment: touch one ear and then touch your nose. They don't feel very similar. Now touch one ear and then the other. These touches seem more similar, although they're twice as far apart. This may be in part because they are represented in related agencies. In fact, our brains have many pairs of agencies, arranged like mirror- images, with huge bundles of nerves running between them.
The two hemispheres of the brain look so alike that they were long assumed to be identical. Then it was found that after those cross-connections are destroyed, usually only the left brain can recognize or speak words, and only the right brain can draw pictures. More recently, when modern methods found other differences between those sides, it seems to me that some psychologists went mad — and tried to match those differences to every mentalistic two-part theory that ever was conceived. Our culture soon became entranced by this revival of an old idea in modern guise: that our minds are meeting grounds for pairs of antiprinciples. On one side stands the Logical, across from Analogical. The left-side brain is Rational; the right side is Emotional. No wonder so many seized upon this pseudoscientific scheme: it gave new life to nearly every dead idea of how to cleave the mental world into two halves as nicely as a peach.
What's wrong with this is that each brain has many parts, not only two. And though there are many differences, we also ought to ask about why those left-right brain halves are actually so similar. What functions might this serve? For one thing, we know that when a major brain area is damaged in a young person, the mirror region can sometimes take over its function. Probably even when there is no injury, an agency that has consumed all the space available in its neighborhood can expand into the mirror region across the way. Another theory: a pair of mirrored agencies could be useful for making comparisons and for recognizing differences, since if one side could make a copy of its state on the other side then, after doing some work, it could compare those initial and final states to see what progress had been made.
My own theory of what happens when the cross-connections between those brain halves are destroyed is that, in early life, we start with mostly similar agencies on either side. Later, as we grow more complex, a combination of genetic and circumstantial effects lead one of each pair to take control of both. Otherwise, we might become paralyzed by conflicts, because many agents would have to serve two masters. Eventually, the adult managers for many skills would tend to develop on the side of the brain most concerned with language because those agencies connect to an unusually large number of other agencies. The less dominant side of the brain will continue to develop, but with fewer administrative functions — and end up with more of our lower-level skills, but with less involvement in plans and higher-level goals that engage many agencies at once. Then if, by accident, that brain half is abandoned to itself, it will seem more childish and less mature because it lags so far behind in administrative growth.
That fascination with those left-right halves, on the part of both the lay and scientific populace, is nothing really new. It is a symptom of how we acquire various pairs of words that each divide some aspect of the world into opposing poles.
Such divisions all have flaws, but often give us useful ways to think. Dividing things in two is a good way to start, but one should always try to find at least a third alternative. If one cannot, one should suspect that there may not be two ideas at all, but only one, together with some form of opposite. A serious problem with these two-part forms is that so many of them are so similar. This leads us into making false analogies. Consider how the pairs below, in which each self is neatly split into two parts, lead everyone to think they share some common unity.
The items on the left are seen as neutrally objective and mechanical, and only found within the head. We think of Thought and its associates as accurate, but rigid and insensitive. The items on the right are seen as matters of the heart — as vital, warm, and individual; we like to believe that Feeling is the better judge of the things that ought to matter most. Cool Reason, by itself, seems too impersonal, too far from flesh; Emotion lies much closer to the heart, but it, too, can be treacherous, through growing so intense that reason gets completely overwhelmed.
How marvelous this metaphor! How could it work so well, unless it had some basic truth? But wait: whenever any simple idea appears to explain so many things, we must suspect a trick. Before we're drawn into dumbbell schemes, we owe it to ourselves to try to understand their strange attractiveness, in order that we not be deceived, as Wordsworth said, by
. . . some false secondary power, by which, In weakness, we create distinctions, then Deem that our puny boundaries are things Which we perceive, and not which we have made.
 Let's call it a Block-Arch.
Block-Arch seems to cause a strange new phenomenon: when you push the car through it, your arm gets trapped! Then, in order to complete that action, you must release the car — and reach around to the other side of the arch, perhaps by changing hands. The child becomes interested in this Hand-Change phenomenon and wonders how Block-Arch causes it. Soon the child finds another structure that seems similar — except that Hand-Change disappears because you can't even push the car through it. Yet both structures fit the same description!
But if Block-Arch causes Hand-Change, then this can't be a block-arch. So the child must find some way to change the mental description of Block-Arch so it won't apply to this. What is the difference between them? Perhaps this is because those standing blocks now touch one another, when they didn't touch before. We could adapt to this by changing our description of Block-Arch: There must be two standing blocks and a lying block. The standing blocks must not touch. But even this does not suffice, because the child soon finds yet another structure that matches this description. Here, too, the Hand-Change phenomenon has disappeared; now you can push the car through it without letting go!
Again we must change our description to keep this from being considered a Block-Arch. Finally the child discovers another variation that does produce Hand-Change:
Our child has constructed for itself a useful conception of an arch, based entirely upon its own experience.
What is learning, anyway? That word is certainly hard to define. The child in our Block-Arch scenario has found one way to learn one sense of what some adults mean by arch. But we can't assume that the same kinds of processes are involved when we learn to recite a poem, to use a spoon, and to tie a shoe. What happens when a person learns to read, learns to add numbers, learns a new language, learns to anticipate the dispositions of a friend, or learns to build a tower that will stand? If we tried to find a single definition for learning to span so many kinds of processes, we'd end up with some phrase too broad to have much use — like this:
Learning is making useful changes in the workings of our minds.
The problem is that we use the single word learning to cover too diverse a society of ideas. Such a word can be useful in the title of a book, or in the name of an institution. But when it comes to studying the subject itself, we need more distinctive terms for important, different ways to learn. Even that one Block-Arch scene reveals at least four different ways to learn. We'll give them these new names:
Uniframing combining several descriptions into one, for example, by observing that all the arches have certain common parts.
Accumulating collecting incompatible descriptions, for example, by forming the phrase block or wedge.
Reformulating modifying a description's character, for example, by describing the separate blocks rather than the overall shape.
Trans-framing bridging between structures and functions or actions, for example, by relating the concept of arch to the act of changing hands.
These words will be explained in the sections that follow. It seems to me that the older words used in psychology — such as generalizing, practicing, conditioning, memorizing, or associating — are either too vague to be useful or have become connected to theories that simply aren't sound. In the meantime, the revolutions of computer science and Artificial Intelligence have led to new ideas about how various kinds of learning might work, and these new ideas deserve new names.
Our Block-Arch scenario is based on a computer program developed by Patrick Winston in 1970. Winston's program required an external teacher to provide the examples and to say which of them were arches and which were not. In my unprogrammed version of this, the teacher has been replaced by the concern of some agency inside the child to account for the emergence of that mysterious Hand-Change phenomenon: why do certain structures force you to let go of the toy car, while other structures don't? We thus assume that the child is led to learn for itself in order to account for strange events. One might complain that it only makes learning harder to explain, to make it depend upon the child's curiosity. But if we are ever really to understand how our minds grow, we must first face reality: people just don't learn so well unless they're interested or concerned. The older theories of learning and remembering never got very far because in trying to oversimplify, they lost essential aspects of the context. It wouldn't be much use to have a theory in which knowledge is somehow stored away — without a corresponding theory of how later to put that knowledge back to work.
The child in our Block-Arch scene examined several different arrangements of blocks — yet ended up describing them as all the same! The great accomplishment was in discovering how to describe all the different instances of arch with the selfsame phrase, a top supported by two standing blocks that do not touch. I'll use the new word uniframe for this — a description constructed to apply to several different things at once. How does a person make a uniframe?
Our child's Block-Arch uniframe was constructed in several steps — and each step used a different learning-scheme! The first step dissects the scene into blocks with specific properties and relationships; some were lying down or standing up, and some were touching or supporting other ones. Next, we required our uniframe to insist that the arch top must be supported by the standing blocks: let's call this enforcement. Then, we required our uniframe to reject structures in which the two standing blocks touch one another;
we could call this prevention: a way to keep from accepting an undesired situation. Finally, we required our uniframe to be neutral about the arch top's shape in order to keep from making distinctions we don't consider relevant. Let's call that tolerance.
How does a person know how to choose which features and relations to enforce, prevent, or tolerate? When we compared the two structures below, we enforced the relation that A is supported by B and C. But think of all the other differences we could have emphasized instead.
Was it wasteful to use only one of these facts when we could have used them all? Should we learn to exploit all the information we can get? No! There are good reasons not to notice too much, for every seemingly essential fact can generate a universe of useless, accidental, and even misleading facts.
Most differences are redundant. Most of the rest are accidents.
For example, suppose that we already know A is supported by B. There is then no need to remember that A touches B or that A is above B — because these are things that we can figure out. For a different kind of example, suppose we knew that A was not supported by B. It then seems unnecessary to remember that A was to the right of B. Common sense can tell us that if A is not on B, it must lie somewhere else. However, at least in the present context, it does not matter whether that somewhere else is to the right; another time it might just as likely lie to the left. If we stored such details too recklessly, our minds would get cluttered up with useless facts.
But how can we judge which facts are useful? On what basis can we decide which features are essential and which are merely accidents? Such questions can't be answered as they stand. They make no sense apart from how we want to use their answers. There is no single secret, magic trick to learning; we simply have to learn a large society of different ways to learn!
Suppose an adult watched our child and said, I see you've built an arch. What might the child think this means? To learn new words or new ideas, one must make connections to other structures in the mind. I see you've built an arch should make the child connect the word arch to agencies embodying descriptions of both the Block-Arch and the Hand-Change phenomena — since those are what is on the child's mind.
But one can't learn what something means merely by tying things to names. Each word-idea must also be invested with some causes, actions, purposes, and explanations. Consider all the things a word like arch must mean to any real child who understands how arches work and how they're made, and all the ways one can use them! A real child will have noticed, too, that arches are like variants of many other things experienced before, like bridge without a road, wall with door, tablelike, or shaped like an upside-down U. We can use such similarities to help find other things to serve our purposes: to think of an arch as a passage, hole, or tunnel could help someone concerned with a transportation problem; describing an arch as top held up by sides could help a person get to something out of reach. Which kind of description serves us best? That depends upon our purposes.
Among our most powerful ways of thinking are those that let us bring together things we've learned in different contexts. But how can one think in two different ways at once? By building, somewhere inside the mind, some arches of a different kind:
Is that a foolish metaphor — to talk of building bridges between places in the mind? I'm sure it's not an accident that we so often frame our thoughts in terms of familiar spatial forms. Much of how we think in later life is based on what we learn in early life about the world of space.
Many things that we regard as physical are actually psychological. To see why this is so, let's try to say what we mean by chair. At first it seems enough to say:
A chair is a thing with legs and a back and seat.
But when we look more carefully at what we recognize as chairs, we find that many of them do not fit this description because they don't divide into those separate parts. When all is done, there's little we can find in common to all chairs — except for their intended use.
A chair is something you can sit upon.
But that, too, seems inadequate: it makes it seem as though a chair were as insubstantial as a wish. The solution is that we need to combine at least two different kinds of descriptions. On one side, we need structural descriptions for recognizing chairs when we see them. On the other side we need functional descriptions in order to know what we can do with chairs. We can capture more of what we mean by interweaving both ideas. But it's not enough merely to propose a vague association, because in order for it to have some use, we need more intimate details about how those chair parts actually help a person to sit. To catch the proper meaning, we need connections between parts of the chair structure and the requirements of the human body that those parts are supposed to serve. Our network needs details like these:
Without such knowledge, we might just crawl under the chair or try to wear it on our head. But with that knowledge we can do amazing things, like applying the concept of a chair to see how we could sit on a box, even though it has no legs or back!
Uniframes that include structures like this can be powerful. For example, such knowledge about relations between structure, comfort, and posture could be used to understand when a box could serve as a chair: that is, only when it is of suitable height for a person who does not require a backrest or room to bend the knees. To be sure, such clever reasoning requires special mental skills with which to redescribe or reformulate the descriptions of both box and chair so that they match despite their differences. Until we learn to make old descriptions fit new circumstances, our old knowledge can be applied only to the circumstances in which it was learned. And that would scarcely ever work, since circumstances never repeat themselves perfectly.
Uniframing doesn't always work. We often try to make an everyday idea precise — but just can't find much unity. Then, we can only accumulate collections of examples.
It certainly is hard to find any properties that all these share. Coins are hard and round and flat. Bills are thin and flexible. Bullion has unusual weight, and credits aren't even physical. We recognize them all as media of trade — but that won't help us recognize the things themselves. The situation is the same for furniture. It's not so hard to say what furniture is for — things that equip a room for living in. But when it comes to the objects themselves, it's even hard to find a uniframe for chair. Again, its function-role seems clear — a thing one can sit upon. The problem is that one can sit on almost anything — a bench, a floor, a tabletop, a horse, a stack of bricks, a rock. Even defining Arch has problems, since many things we recognize as arches just don't match our Block-Arch uniframe:
All these shapes could be described as shape with hole or blocks that bridge across a gap. But those descriptions would also admit many things we don't want to regard as arches. The simplest way to learn, when one can't find a uniframe, is to accumulate descriptions of experiences.
At first it may seem simpler to accumulate examples than to find more uniform ways to represent them. But later there's a price to pay for this: when we try to reason about things, accumulations can be nuisances — because then we'll be forced to find a different argument or explanation to justify each separate example. Most likely, different parts of our brains have evolved to use both kinds of strategies. Accumulations need not take longer to manipulate if all the examples can be handled at the same time, by separate agents that don't interfere with one another But once those processes begin to need each other's help, the whole society's efficiency will decline rapidly. Perhaps that slowing-down itself might be the stimulus that makes us start to try to unify — at least for processes we use frequently.
A simpler theory of when we start new uniframes would be that in the brain, there is an architectural constraint on how many K-lines are directly accessible to various types of agents. For example, the agents in a certain agency might be able to accumulate no more than about seven branches for each classification in a certain hierarchy. When more than that accumulate, the agency would be forced either to merge some examples into uniframes or to turn for help from somewhere else.
Let's make a dumbbell theory of some people's personalities.
Uniframers disregard discrepancies in favor of imagined regularities. They tend to be perfectionists but also tend to think in terms of stereotypes. This sometimes leads to recklessness because they have to reject some evidence in order to make their uniframes. Accumulators are less extreme. They keep collecting evidence and hence are much less prone to make mistakes. But then they're also slower to make discoveries.
Of course these imaginary personalities are only caricatures, and everyone blends both extremes. Most people find some reasonable compromise, though a few of us lean more in one direction than the other. I'm sure we all use mixtures of different learning strategies
— accumulations of descriptions, K-lines, uniframes, or whatever. On the surface, it might seem easier to make accumulations than to make uniframes — but choosing what to accumulate may require deeper insight. In any case, whenever an accumulation becomes too large and clumsy, we try to replace some groups of its members with a uniframe. But even when we succeed in finding a suitably compact uniframe, we can expect it, too, to accumulate exceptions eventually, since first descriptions rarely work for all our later purposes.
For example, when a child first encounters dogs, an attempt might be made to create a uniframe that catalogs those animals' parts — eyes, ears, teeth, head, body, tail, legs, and so on. But the child will eventually have to learn that even here there are exceptions.
Furthermore, that uniframe won't help answer the child's most urgent questions about any one dog in particular: Is it friendly? Does it have a loud bark? Is it the kind that tends to bite? Each such concern could require building a different kind of hierarchy-tree.
This leads to an inescapable difficulty. Our various motives and concerns are likely to require incompatible ways to classify things. You can't predict a dog's bite from its bark. Each of the classifications we build must embody different kinds of knowledge, and we can rarely use more than a few of them at once. When we have a goal that is simple and clear, we may be able to select one particular kind of description that makes the problem easy to solve. But when goals of several types conflict, it is harder to know just what to do.
When should you accumulate, and when should you make uniframes? The choice depends upon your purposes. Sometimes it is useful to regard things as similar because they have similar forms, but sometimes it makes more sense to group together things with similar uses. At one moment you may wish to emphasize a similarity; at the next moment, you may want to emphasize a distinction. Often, we have to use both uniframes and accumulations in combination. In Block-Arch, for example, we found that there could be two different kinds of arch tops — the block and the wedge. Accordingly, when we used the phrase block or wedge, we actually inserted a subaccumulation into our uniframe.
Accumulations rarely seem quite satisfactory because we feel ideas should have more unity. We wouldn't have a word for chair or arch or currency if they meant nothing more than lists of unrelated things. If each did not involve some unifying thought, we'd never think to make those lists in the first place! Why is it so hard to describe their essences? In the next few sections we'll discover a number of reasons for this. Here is one of them:
Many good ideas are really two ideas in one — which form a bridge between two realms of thought or different points of view.
Whenever we build a bridge between structure and function, one end of that bridge may represent a goal or use, while the other end describes what we might use to gain those ends. But it is rare for those structures to correspond neatly to those functions. The problem is that we usually find many different ways to achieve any goal. This means that we'll find an accumulation on the structural side of the bridge. For example, if you want to reach something high up, you can stand on a chair, reach with a stick, or ask someone taller to get it for you. Similarly, an accumulation of functions or goals can be found for any structure. My colleague Oliver Selfridge once wrote an entire book entitled Things to Do with a Stick.
Our different worlds of ends and means don't usually match up very well. So when we find a useful, compact uniframe in one such world, it often corresponds to an accumulation in our other worlds.
We encountered this problem earlier. When we classified birds as animals while classifying airplanes as machines, we thereby forced disunity upon the class of things that fly. Later, when we come to theories about metaphors, we'll see that such problems are almost inevitable because we know only a very few — and, therefore, very precious — schemes whose unifying powers cross many realms.
What should one do with a law or rule that doesn't always work? We saw one way when we developed our uniframe for the Block-Arch. We simply kept changing it to fit each new example. But what if, after all that work, there still remain exceptions that don't fit?
The Exception Principle: It rarely pays to tamper with a rule that nearly always works. It's better just to complement it with an accumulation of specific exceptions.
All children learn that birds can fly and that animals that swim are fish. So what should they do when told that penguins and ostriches are birds that cannot fly, or that whales and porpoises are animals that swim but aren't fish? What should the children do with uniframes that no longer work so well? The exception principle says: Do not change them too hastily. We should never expect rules to be perfect but only to say what is typical. And if we try to modify each rule, to take each exception into account, our descriptions will become too cumbersome to use. It's not so bad to start with Birds can fly and later change it into Birds can fly, unless they are penguins or ostriches. But if you continue to seek perfection, your rules will turn into monstrosities:
Birds can fly, unless they are penguins and ostriches, or if they happen to be dead, or have broken wings, or are confined to cages, or have their feet stuck in cement, or have undergone experiences so dreadful as to render them psychologically incapable of flight.
Unless we treat exceptions separately, they'll wreck all the generalizations we may try to make. Consider why the commonsense idea of fish is so useful. It is an accumulation of general information about a class of things that have much in common: animals that live in the water, have a certain sort of streamlined shape, and move by wriggling their bodies and fanning the water with various finlike appendages. However, a biologist's idea of fish is very different, being more involved with the origins and internal mechanisms of those animals. This leads to emphasizing aspects less evident to the eye: if whales have lungs where trout have gills, they must be uniframed in different ways. Children are disturbed to hear that whales are not fish because they are usually more concerned with uses and appearances than with origins and mechanisms. They're more likely to want classifications like these:
What do those animals do? Where do they live? Are they easy to catch? Are they dangerous? Are they useful? What do they eat? How do they taste?