Like rotation, spatial alignment appears in other tasks. When people are asked to select which of a pair of maps is correct, the correct one or a world map in which the United States and Europe are more aligned, the majority select the incorrect but more aligned version. The same error shows up for North and South America. When asked to select which of a pair of maps is correct, the correct one or one in which South America is pulled eastward (or North America westward) to be more or less under North America, the majority select the incorrect map that is more aligned.
People make the same rotation and alignment errors when remembering new fictional maps or even meaningless blobs that are not interpreted as maps. They incorrectly remember new fictional maps and blobs as more aligned, and they incorrectly remember “cities” located on the fictional maps as more aligned. When sketching maps of their own familiar surroundings, surroundings they have traversed many times, they align streets, sketching them as more parallel than they actually are. These errors are widespread and persistent in the face of experience.
Hierarchical organization
You may have noticed that grouping, perceiving similar things as a group, is essentially placing them into a category. Although geographic space is mostly flat, we humans group geographic entities into categories such as continents and subcategories such as countries, then states, then cities, then neighborhoods. Out of flat space, minds—or governments—construct hierarchies of spatial categories. Spatial hierarchies are partonomies, not taxonomies like the categories of objects, events, and scenes, discussed in Chapter Two. Partonomies are hierarchies of parts; taxonomies are hierarchies of kinds. Cities are parts of states and states are parts of countries just as fingers are parts of hands and hands are parts of bodies.
We saw earlier that categories and hierarchies of categories allow reducing the amount of information in the world. Instead of thinking about each and every apple or dog or outing to the grocery store individually, we can think about apples and dogs and grocery outings in general. We can go up a level and think about fruit and domestic animals and shopping as categories. Categories are efficient for another reason. They allow broad inferences. If I learn that a kinkajou is an animal and I know that animals breathe and reproduce and move about, then I know that kinkajous do all those things (in case you are wondering, kinkajous are mammals with enormously long tongues that live in Central and South America). This kind of reasoning is especially important for new information. If you encounter a mangosteen and a grocer tells you it’s a fruit, you know it is likely to grow on trees; have seeds, peel, and edible pulp; and so on. Categories are an efficient way to organize and store familiar facts and to learn new ones.
Like taxonomies, partonomies allow inferences, but inferences of containment, not of properties. If a knee is part of a leg and a leg is part of a body, then a knee is part of a body. Direct measurements of distance and direction, the kind surveyors do, use flat space, so spatial partonomies are irrelevant. But people, who don’t store zoomable maps or, for that matter, any maps in their heads, use them as proxies, in the absence of direct measurements. Here’s an example, another quiz. Note that, as before, it’s the systematic errors that demonstrate the phenomenon.
Is Reno east or west of San Diego? Hard. Again, there are no complete and accurate maps in the mind that can be consulted. But, wait, there’s an easy way to figure it out. Use hierarchical reasoning, the same kind of reasoning that allows us to infer, on being told that a kinkajou is an animal, that kinkajous breathe, eat, and reproduce. Here the hierarchy is spatial. And here’s how people seem to reason. Reno is in Nevada, San Diego is in California. Nevada is east of California. So, Reno must be east of San Diego. Good reasoning but wrong answer. To make things worse, or maybe better, a wrong answer that the majority agrees with. The inference from location of state to location of city, from larger category to smaller, would be correct in many cases, but not in the case of Reno and San Diego. The problem is that the southern part of California cuts east, so that Reno, northwest in Nevada, is actually east of San Diego, far south in California.
This brings us to the Seventh Law of Cognition: The mind fills in missing information.
Spatial categories are used as proxies for distance estimates as well as for direction estimates. Distances of pairs of locations that are within a spatial group like a state or a country are estimated to be smaller than distances of pairs of locations between spatial groups, even when pairs are deliberately chosen so that the between-group distance is smaller than the within-group distance.
You might think that because hierarchical organization can lead to systematic error only novices would rely on hierarchical organization and experts would not. In fact, experts, namely, experienced taxi drivers, have more refined and better hierarchical knowledge of the environments they traverse than novices. This is not to say that experts make more errors than novices. Experts have also developed ways to counter the errors. Hierarchical knowledge is not the only method to determine routes or estimate distances, and experts have many methods at their disposal.
Significantly, the bias that within-group estimates are smaller than between-group estimates occurs for groups that are defined functionally rather than spatially, groups that have no spatial integrity at all. In Ann Arbor, the home of the University of Michigan, buildings belonging to the university are interspersed among commercial buildings that have no association with the university. Students in Ann Arbor give smaller distance estimates to pairs of university buildings or to pairs of commercial buildings than to pairs formed from one university building and one commercial building. Similarly, Israelis, both Palestinian and Jewish, estimate the distances between two Jewish settlements or two Palestinian settlements to be smaller than the distances between one Jewish and one Palestinian settlement even when the latter are in truth smaller than the former. In both cases, then, within-group distances are perceived as smaller than between-group distances, yet here the groups are functional or political rather than spatial.
Even more support for the spatial foundations of abstract thought comes from studies on similarity within and across social groups. Similarity is distance in conceptual space. Just as politically similar settlements are judged to be closer in space than politically different settlements, so people who happen to be in the same social or political group are judged to be more similar to each other even on an irrelevant dimension than people who happen to be from different social or political groups. Inferences that are natural but that can lead to trouble.
So far, we have described three mechanisms that are used for judgments of space, rotation, alignment, and hierarchical organization. Each acts as a proxy or heuristic for judging distance or direction or selection of the correct map in lieu of direct measurement. Each biases judgments systematically in ways that would be difficult to show unless you knew how to find them. Rome and Philadelphia, Berkeley and Stanford, Reno and San Diego were deliberately chosen for the geography quizzes. There are other heuristics spatial reasoning takes. There is straightening. Even long-time residents of Paris straighten the curvaceous Seine in their sketched maps and in their mental maps. One can only imagine the numbers of hours of extra walking and liters of gasoline this distortion has cost. Next, we turn to systematic errors due to reference points and perspective.
Reference points
Yet another proxy for estimating distance is using landmarks. Every city seems to have them, and they often symbolize the city. Think of the Eiffel Tower in Paris or the Empire State Building in New York. Landmarks frequently serve as reference points. Because they are familiar, other places are located with respect to them. I might approximately locate a restaurant for you by saying it is near the Piazza del Duomo in Milan or the Pantheon in Rome.
Spatial reference points are larger than themselves. This isn’t really a paradox: landmarks are themselves, but they also define neighborhoods around themselves. In a paradigm that has been repeated on many campuses, researchers first coll
ect a list of campus landmarks from students. Then they ask another group of students to estimate the distances between pairs of locations, some to landmarks, some to ordinary buildings on campus. The remarkable finding is that distances from an ordinary location to a landmark are judged shorter than distances from a landmark to an ordinary location. So, people would judge the distance from Pierre’s house to the Eiffel Tower to be shorter than the distance from the Eiffel Tower to Pierre’s house. Like black holes, landmarks seem to pull ordinary locations toward themselves, but ordinary places do not. This asymmetry of distance estimates violates the most elementary principles of Euclidean distance, that the distance from A to B must be the same as the distance from B to A. Judgments of distance, then, are not necessarily coherent.
Cognitive reference points work the same. Cognitive reference points are convenient: a composer might be compared to Bach or Beethoven and an artist to Picasso or Pollack. Somebody’s the new Babe Ruth or David Bowie or Spike Lee. Like spatial reference points, cognitive reference points are larger than themselves; they serve as genres or prototypes. Cognitive reference points create asymmetries of judgments of similarity just as spatial reference points create asymmetries of judgments of distance. Similarity, after all, is a measure of conceptual distance. People judge magenta to be more similar to red than red to magenta. Red is the prototype, it stands for a category of colors that are red-like in the way that the Eiffel Tower stands for a neighborhood. Magenta stands only for itself. Ditto Pierre’s house. Moving from the perceptual to the cognitive, people judge a son to be more similar to his father than a father to his son. They see North Korea as more similar to the People’s Republic of China than the PRC to North Korea. Fathers and the PRC are like red or the Eiffel Tower, they are prototypes, endowed with a broader set of qualities than sons or North Korea. Another set of examples of the Fifth Law of Cognition: Cognition mirrors perception.
Perspective
When you are high on a mountain or a tall building taking in the vast panorama in front of you, you might notice that things that are far from you seem closer together, more crowded, than things closer to you. The dispersion of close things and crowding of distant things occurs for imagined perspective as well. Students from a university somewhere between the coasts of the United States were asked to imagine themselves either in San Francisco, on the Pacific coast, or in New York, on the Atlantic coast. Then they were asked to estimate the distances between several cities scattered along an east-west axis between San Francisco and New York. Those who imagined themselves in San Francisco gave a longer estimate for the distance between San Francisco and Salt Lake City than those who imagined themselves in New York. Conversely, those who imagined themselves in New York gave a longer estimate for the distance between New York and Philadelphia than those who imagined themselves in San Francisco. It’s as if distances were telescoped, larger when close, smaller when far. Remember that the students were actually in neither place. What’s remarkable is that this distortion happens solely with imagined perspectives, irrespective of where people are actually located in space.
That same distortion, finer discrimination for things that are close to us than for things that are far, occurs for judgments of people on social dimensions. People judge members of their own social group, a close group, to be more different from each other than members of other social groups, faraway groups. Individuals in our university, our political party, our country are quite different from each other, but those in the rival university, another political party, and surely another country, they’re all alike. It’s natural; we’ve had far more experience with our own social groups than with others.
Expertise matters. Bird experts and car experts make far greater distinctions in their areas of expertise than the rest of us. It seems inevitable that we discern more differences for the close and the familiar, and see the distant and less known in generalities. But then, there are times we need to think in generalities, to see the broad strokes rather than the fine detail. Those cognitive trade-offs again.
COGNITIVE COLLAGE
Cognitive map is an old notion. It came from Tolman’s pathbreaking work on rats: he showed that when rats solved mazes they made spatial inferences, they took shortcuts when given the opportunity, as if they had map-like representations in their brains. As does at least one young child, despite being blind from birth. Yet it seems that whatever people have in their minds that they use to estimate distances or directions or to sketch maps are constructions of the mind, from pieces, often from different experiences and different kinds of information. Because people’s minds lack direct measurements and lack map-like mental representations of the world, they bring together whatever information seems to be relevant for the task at hand. That information is useful in general, but piecemeal and approximate. Many cities in Nevada are east of many cities in California, but not Reno and San Diego. The United States is generally aligned east-west with Europe, but Rome in southern Europe is actually north of Philadelphia in the northern United States. The Bay Area is closer to running north-south than east-west just like the state of California, but inland Stanford happens to be west of coastal Santa Cruz. Pierre’s house can’t be closer to the Eiffel Tower than the Eiffel Tower to Pierre’s house. These proxies are independent for the most part, and so are the consequent errors. Because they are independent, the errors can cancel each other, so sketching a map of many spatial relations is one way to increase accuracy. The more constraints that are fulfilled, the greater the accuracy.
Yet, ultimately, these proxies—alignment, rotation, hierarchical thinking, perspective, landmarks, and more—can’t be resolved in a flat Euclidean map. Rather than cognitive maps, the mind seems to have cognitive collages.
Why do these errors persist? Because the mind doesn’t have ways to correct them, short of measuring the world, and when accuracy is crucial, the world gets measured. Measurements can overcome many biases in natural judgments, not just these. Because in many cases, the errors are not large and don’t matter. Because when we’re actually navigating on the ground, the environment itself can correct us.
Even if incomplete, ambiguous, inconsistent, and biased, our mental spatial frameworks play crucial roles in our lives and in our imaginations. They allow us to envisage other worlds, worlds we have not seen, that no one has seen, even impossible worlds. Metaphoric worlds where places are replaced by any kind of entity or idea and paths by the relations among them. The worlds of fiction, of the arts, of science.
CHAPTER FOUR
Transforming Thought
In which we distinguish representations of thought from transformations of thought, then analyze spatial transformations and what they are good for (plenty!) followed by spatial ability and how to get it.
NOW WE LEAVE THE BODY IN THE WORLD AND ENTER THE MIND. We’ve populated the mind (and the brain) with the things that are central to our lives: faces, bodies, objects, scenes, events. We’ve put them on the mind’s stage for the mind to use, to ponder, to perform, to transform. Once the players, any sort of object or thing, are on stage, we can play with them. We can turn them into symbols in mathematics, words in poetry, particles in physics, molecules in chemistry, buildings in a neighborhood, dancers on a stage. We can change their shapes and sizes and properties. We can change where they are and what they do. That stage is the stage of imagination, and it can carry us far.
Thinking starts somewhere, with an idea or a problem, vague or precise. Then you do something to the idea, you transform it some way, and voilà! a new idea. After that, you can begin to work on the new idea. And so on, until you’re done or reach a dead end or are simply worn out. Sometimes thinking comes with instructions: multiplication and piano music and waltz steps and chemistry problems. That doesn’t mean that following the instructions is trivial—instructions aren’t always clear. It can take concentration and thought to follow instructions, but there are instructions. There’s a script that tells you what to do at each step. Like follow
ing a recipe or assembling Legos or furniture. Recipes and assembly instructions are a sequence of actions on real objects that transform them step-by-step into something else. The right sequence of actions on butter and sugar and eggs and flour and baking powder takes glop and transforms it into cupcakes. Pieces of wood can be turned into a desk or a bookcase. Lego blocks, into houses and robots.
Assembly is actions on real objects. Thinking is mental actions on mental objects—ideas. Actions on ideas that transform them into something else. That’s how we talk about thinking, as if it were actions on ideas. We put ideas aside or turn them upside down or inside out. We split them into parts or pull them together. We arrange and rearrange, enlarge, stretch, reverse, join, copy, add, scramble, subtract, lift, glue, push, fold, mix, toss, embellish, separate, nail, scatter, bury, eliminate, turn, elevate, and poke holes in both real objects and mental ones. Intriguingly, we will see soon and in the chapter on gesture that performing the actual actions helps the mental ones, the thinking.
Of course, not all cooking or assembly follows scripts. Chefs, professional and hack alike, improvise new recipes on the fly or invent them studiously. Rauschenberg famously took junk he found on the streets and turned it into art. Le Witt made art by systematically and methodically arranging lines and boxes and cubes. If you or I tried to do what they did, alas, it probably wouldn’t be art. No script for Rauschenberg or even the more methodical Le Witt. Instead, trial and error, practice, skill, expertise, and, yes, talent. Same for thinking. No script for creating a good novel or a catchy melody or a gripping film or an elegant teapot or an unreturnable tennis return. No script for figuring out what you want to do on your visit to Rome, no script for figuring out why an election turned out this way instead of that way, no script for rearranging the furniture in your living room or deciding your next move in chess. There are schemas, constraints, rules of thumb, educated hunches: what spices work together, what makes a balanced composition, what chess moves are likely to succeed. The truly interesting and challenging kind of thinking can’t follow a script; there isn’t any.
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