Surfaces and Essences

by Douglas Hofstadter

  Unwitting Default Assumptions: A Pitfall of Categorization

  For a quite different example, consider the story of the father who dies in a serious traffic accident, and whose son, a passenger in the same vehicle, is taken by ambulance to the hospital in critical condition. An emergency operation is needed to save his life. The surgeon on duty comes quickly into the operating room, suddenly goes white as a sheet, and exclaims, “I can’t operate on this boy — he’s my son!”

  It’s extremely common for people to read and reread this story many times, always bumping into the irreconcilable problem that the boy cannot possibly have two fathers. What is going on? Has the surgeon misperceived the boy’s identity? No. Or was there perhaps not just a biological father but also an adoptive father? No. Did the father somehow get resuscitated and miraculously make it to the hospital before his wounded son? No. In fact, all these questions and answers, despite giving correct information, serve only to further entrench the problem, which is caused by categorical blinders. There is in fact a perfectly reasonable explanation of this situation, which we are confident that every reader, after sufficient reflection, will find.

  The manifold ways in which we are continually blindsided by our categories cannot be overemphasized. Take, for instance, the story that you just read about the traffic accident. Did it at any point occur to you that the vehicle involved was a bus? Virtually no one envisions it that way, even though nowhere in the short paragraph recounting the story does it say “car”. It simply never crosses our mind that our initial categorization could have been wrong. The default assumption that comes along with the concept of traffic accident is that it involves a car, as opposed to, say, a bus, a truck, a motorcycle, a bicycle, a mobile home, and so forth. Such default assumptions are at times very deeply entrenched, and in some cases they are nearly impossible to detect and overthrow. That is why it is so very hard for many readers — even today, when women have made so many strides along the pathway to social equality with men — to figure out how the surgeon’s remark makes perfect sense.

  Flitting from One Essence to Another

  Categorizations that lead to a mistaken understanding of a situation can have major consequences for someone trying to solve a problem. Indeed, a poor categorization can make a simple problem difficult, if not unsolvable. An external observer might take someone’s being stumped by a simple problem as a symptom of incompetence, when actually the would-be solver is merely trying to solve a problem other than the one that was posed. Just as E.’s biases led her to misconstrue the word “essence” in the lecture title, so the solver has misconstrued the problem because of biased categorization.

  Various studies in cognitive psychology have brought out the nature of such mistaken categorizations and are helping to overturn the stereotype that says that if someone fails to solve a problem, it’s only because they didn’t find a successful strategy. That would indeed be the case if the person had perfectly understood the problem statement but didn’t know how to proceed from there. But research has shown that one of the most frequent sources of trouble in problem-solving is a misunderstanding of the problem statement, meaning that the wrong categories are mobilized. Once the proper categorization is found, finding the solution is often quick and easy.

  An example will help make this clear. The study in question involves the famous Towers of Hanoi problem, studied by many psychologists. In a very simple version, there are three disks — one small, one medium-sized, and one large — with holes in their centers, which have been stacked in a pile on one of three posts — post A — and which must be moved to post C while respecting three constraints: (i) only one disk can be moved at a time; (ii) only the uppermost disk in a pile can be moved; and (iii) a larger disk can never be placed on a smaller one.

  Many experiments have been done on the topic of how people tackle this challenge and other “isomorphic” challenges (so called because they are in fact just superficial disguises of the same problem, and exactly the same technique will solve them all — for example, the disks can be replaced by acrobats, and the third constraint then says that no acrobat can ever stand on the shoulders of a smaller acrobat). We’ll now shine a spotlight on some interesting results. Cognitive psychologist Jean-François Richard has shown that a significant percentage of elementary-school children, at around age eight, solve this problem with a very large number of moves — around thirty or so — whereas only seven moves will suffice if one knows exactly what one is doing. However, the strategy adopted by these children is anything but random. It seems that they throw in one extra rule that isn’t part of the official statement of the problem. This rule, which they tacitly assume without realizing it, states that one cannot move a disk directly from post A to post C (or vice versa). In other words, these children don’t allow themselves to jump from A to C or the reverse, requiring instead that any disk moved from either A or C must always have B as its landing-spot. If you tackle this alternate challenge yourself, you will soon discover that the addition of this rule slows the solution process up considerably: the minimal solution jumps from 7 moves all the way to 26.

  Why would so many children impose this extra rule on themselves, essentially tying one hand behind their back? It all comes down to the concept of motion, which seems to be perceived somewhat differently by adults and children. Children often implicitly suppose that moving from one post to another means passing through each intermediate position, whereas adults have a more sophisticated conception of motion, which doesn’t depend on the way one gets from point A to point C. When someone says, “Next Friday I’m going to Arkadelphia”, it could be by car, by bus, by train, or by plane, and the details of the route make no difference. Adults have no problem distinguishing between a trip as a state change (“Today I’m in Bloomington and on Friday I’ll be in Arkadelphia”) and a trip as a process (“My Bloomington–Arkadelphia jaunt will carry me through Indianapolis, Little Rock, and several other towns”).

  The tendency to see a motion as a process rather than as a state change is closely related to the idea discussed in Chapter 4, according to which various stages of abstraction of a concept involve stripping away one after another of its less crucial aspects (such as removing from the concept of desk the notions of weight, volume, and substance, in the end leaving just the core notion of workspace, or removing from the concept of key the notions of metallic, long, thin, and irregular in shape, in the end leaving just the core notion of transportable entity that performs unlocking, such as the magnetic cards that one gets in hotels). It appears from the aforementioned experiments that a good number of eight-year-olds have trouble separating the abstract notion of a move from the more concrete notion of the route by which the move is accomplished. The former doesn’t care about how it took place and thus allows direct jumps from A to C or back, while the latter insists on passing through all the salient intermediate points. Just as Mica, in the previous chapter, could not help hearing in the word “hump” an earlier collision that had caused the hump, so many young children cannot help hearing in the word “move” the idea of passing through all intermediate stations.

  Precisely the same phenomenon can be seen in adults when one “dresses” the Towers of Hanoi problem in other “clothing”, thereby yielding one of its isomorphs. Jean-François Richard and his colleague Évelyne Clément studied an isomorph of the Towers of Hanoi in which, rather than moving the disks, one can change their size. In that situation, adults tend to behave exactly as do the children who move the disks. That is, a good fraction of them unconsciously add the extra rule that one can’t change a small disk into a large one (or vice versa) — one has to pass through the intermediate size. In order to jump from size A to size C, they feel they need to take two “steps”, stopping momentarily at size B. The cause of this unexpected mental rigidity, which markedly slows these adults up in their solution of the problem, is the same as for the children: they use an overly concrete (and thus naïve) notion of the concept size change,
not separating the final size from the process or “route” taken in getting to it. For most people, the prototype for the concept size change is that of biological growth: one grows from babyhood to adulthood in passing through childhood and adolescence. This model of the concept of size change is naïve, though, and adults who get stuck in this trap, unconsciously borrowing the naïve model in tackling the Towers of Hanoi isomorph, are fated to take much longer than those who do not.

  It thus appears that one single mental phenomenon is at work both in the “Sources of Essence” trap and in the Towers of Hanoi trap — namely, memories of analogous situations impose themselves without consulting the person who matters — the person in whose head all this is happening. The standard excuse that one gives is along the lines of “I didn’t quite catch what they told me”, or “They sure didn’t make it clear what it was all about”, or “I didn’t pay close attention” — but in fact a more accurate explanation is that one unconsciously succumbed to an inappropriate categorization.

  When the entire process is unconscious, as in the cases we’ve been discussing, only through very meticulous research will the hidden analogies be revealed. Sometimes the existence of categories shared among people in a culture or subculture furnishes a particularly clear demonstration of the way categories influence the perception of a situation, since people who do not belong to this community would be amazed by a point of view that strikes them as extremely odd.

  The following example of this phenomenon will speak directly to readers who live in large metropolitan areas, and may intrigue those who live in less traffic-congested locales. It shows why the loveliest spot in a city isn’t necessarily what one might think.

  What is San Francisco’s Loveliest Spot?

  Union Square? Chinatown? Twin Peaks? The Great Highway? The Cliff House? Pacific Heights? The Golden Gate Bridge? Fisherman’s Wharf? Golden Gate Park? The Presidio? The Marina? The Palace of the Legion of Honor? The Top of the Mark? Coit Tower? The Ferry Building? Lake Merced? West Portal? Russian Hill?

  Surely, for a non-resident of the City, one of the above would fill the bill, but a true San Franciscan sees things differently. Finding a place to park one’s car in the City without worrying about getting an astronomical fine or having to go pick it up at the pound can verge on the miraculous, especially in certain areas and at certain times of day. Thus it is not infrequent that one finds oneself driving up and down steep hills, back and forth on broad avenues, crisscrossing one’s prior pathway umpteen times in desperate search of a parking spot, knowing full well that the likelihood of finding one is microscopic. This plight is so common and so upsetting that San Francisco drivers tend to be powerfully drawn to vacant parking places, finding high esthetic value in simple concrete rectangles as long as it is legal to park in them. The rarity of such a spot turns it into a precious entity.

  It’s not uncommon to hear one local say to another, as they walk down the street, “Just look at that beautiful spot! Wow!” The intimate relationship that San Francisco drivers have with untaken parking spaces thus engenders the category of lovely spots. As anyone who’s driven in the City can testify, spotting a lovely spot when one doesn’t need to park in it always evokes a tempting counterfactual scenario of chancing upon exactly that empty spot just when one needs it desperately, and thus a sense of “if only” or “too bad” is triggered.

  The abstract category of lovely spots deeply affects how these people perceive the physical spot. This example shows how powerfully categories impose a view of the world. If Kazimir Malevich’s famous White Square on White Background is a member of the category works of art, then surely Market Street’s Gray Rectangle on Gray Background is a member of the category lovely spots. The intense feeling of longing that the gray rectangle inspires in so many people shows how irresistible is the psychological force that pushes for categorizing it in that fashion. And yet such a categorization, for all its emotional intensity, doesn’t drive people into paroxysms of irrationality. So far as we know, no one has yet succumbed to the siren song of a lovely spot by suddenly throwing their Saturday-evening plans out the window, screeching to a halt, and parking their car then and there, fatally seduced by the lovely spot that was winking at them.

  Certain categorizations, however, have very powerful influences on our thought and behavior. For example, people’s perception of the October 11th crash was very different from what it would have been had the event occurred a few years earlier.

  The Irresistible Strength of Analogies: The 10/11 Crash

  In the middle of the afternoon on October 11, 2006, an airplane crashed into a tall building in New York City. On first hearing this breaking news, no one, unless they had spent the last few years on a remote island, could fail to think of the destruction of the World Trade Center on September 11, 2001. One would automatically assume it was a terrorist attack, and one would retain a lingering feeling that this must be the case even after learning that it was only a four-seater private propeller plane, not a huge jet, that the only deaths were of the pilot and copilot, that the fire in the building was quickly extinguished, and that the building was never in danger of collapsing.

  The analogy with the events of September 11th sweeps in immediately, profoundly coloring one’s perception of this event. It would thus be hard for anyone to imagine, on first hearing about this event, that it was simply a random accident of the sort that takes place frequently and in many different ways all over the world — an unfortunate incident, to be sure, but with limited consequences, and without any link to religious fundamentalism or terrorism. That’s a most unlikely first thought! As a matter of fact, the Dow Jones average declined for a short while after the collision was announced.

  The September 11th events cannot fail to be evoked front and center when one hears about the October 11th crash. Because of September 11th, the broad category of terrorist attack is instantly activated, and so is the more specific category of September 11 attack, perhaps even more strongly so. Thus one tends to think, “Oh, no — September 11th all over again!” or “Is this another September 11th?” Together, the broader and the narrower category provide the inevitable framework for understanding an event that would have been perceived totally differently had September 11th never occurred.

  Someone might object, saying “Hold on, now — ‘September 11th’ isn’t the name of a category but of an event!” We would counter this claim by reminding the objector of categories such as popes, bibles, meccas, Bachs, Einsteins, Picassos, and Rolls-Royces, discussed in Chapter 4, which show that apparently single-member categories, no matter how unique or sui generis they might seem to be (even the super-specific Château Beaucastel Hommage à Jacques Perrin 1998), will be quite effortlessly pluralized by the similarity-driven human mind, when the proper situation shows up.

  Other clear signs of the plural nature of the category September 11th (more often called “9/11”) are the standard use of such phrases as “India’s 9/11” (the bombings of many buildings in Mumbai, including two luxury hotels, in late November of 2008), “Russia’s 9/11” (which has been exploited by various political groups to denote various massacres of innocent people), “Pakistan’s 9/11”, “Spain’s 9/11”, and so forth. In particular, the bomb attacks that took place in Madrid on March 11, 2004, claiming the lives of over 200 people, soon acquired the label “Spain’s 9/11”, and Spaniards very understandably quickly took to using the analogy-drenched nicknames “11-S” (for the attacks on New York and Washington) and “11-M” (for those on Madrid).

  On the Web, one can also find such phrases as “the 9/11 of the seventeenth century”, “the 9/11 of New Orleans”, “the 9/11 of World War II”, “the 9/11 of the Scriptures”, “the 9/11 of rock”, “the 9/11 of hormone replacement therapy”, “the 9/11 of commercial shipping”, and on and on. Moreover, one easily finds scads of Web sites that refer, using explicit plurals, to such categories as “the 9/11’s of the future”, “the 9/11’s of the 1960’s”, “the 9/11’s of
history”, and so forth.

  In brief, the concept of September 11th is now a common category having many members of different strengths (Pearl Harbor being a fairly strong member — and, amusingly enough, in a sad way, 9/11 itself being a fairly strong member of the category Pearl Harbors). Just think how many birthdays and wedding anniversaries were sabotaged in the years following 2001 simply because, many years earlier, someone happened to be born on the 11th of September, or because a couple had perfectly reasonably picked that date for their nuptial celebration. And for quite a while after 2001, all sorts of receptions, openings of shows, and other public events were carefully scheduled so as to avoid being stigmatized by the irrepressible association with the “radioactive” date September 11th, which poisoned everything that came close to it.

  One Thing Changes and Everything Changes

  As we have seen, analogies constrain one’s perception of situations. Whether we’re talking about members of the category of lovely spots or that of 9/11’s, analogies constitute filters through which the world is seen. This statement may seem surprising, since it’s easy to forget that whoever one is, one sees the world through filters that powerfully control the flow of one’s thoughts. But the lessons learned from the preceding categories apply throughout everyday life, and they apply to any idea, however small or large, that occupies center stage for a while.