Surfaces and Essences

by Douglas Hofstadter

  The notion of debugging a computer program is yet another fertile source of imagery for everyday life. Thus a salsa dancer says, “I’m working on debugging my Latin hip motion — my hips always move in the wrong direction”, while a Chinese teacher says, “You really have got to debug your tones — they’re all mixed up!”

  And finally, a few miscellaneous examples of the insidious manner whereby computer terminology worms its way into everyday discourse. One business executive confides to another: “I’d really like to have your input on this matter.” A medical-school student sighs: “Every so often, I just need to disconnect from this crazy routine.” An advertisement exhorts: “Reboot your brain with a caffeine nap!”

  Taken together, these examples provide another pillar of support for the thesis of the first two chapters that word choices are made via analogies, and that word-choice analogies are usually (though not always) made unconsciously. Take the computer-science term “multitasking”, for instance. In the first few years of its existence, the idea of applying it to some kind of human behavior was a choice available only to computer-savvy people, and for them the computer-to-human analogical bridge was built with deliberateness and delight. Pushing a word’s range outwards is fun (because making inventive analogies is fun). Eventually, however, this extended sense of the term leaked, by osmosis, into the much broader community of non-computer-savvy people, and at that point, using the term didn’t require building an analogical bridge linking a human behavior to an arcane computational trick (after all, these speakers knew nothing of that trick); the analogical bridge was simply to human behaviors that had been labeled by the catchy new term. But no matter which set of speakers we focus on — technically savvy or technically non-savvy — we see that it’s always analogy in the driver’s seat, always analogy that is handing words to speakers, and often handing them words on such convenient silver platters that, to them, their word choices seem to have taken place instantly, naturally, effortlessly, and without any help from analogy-making. That silver platter was just magically there. Of course, that’s an illusion — just another case where the human mind, not surprisingly, fails to be aware of the seething activity constantly going on below the surface of its familiar linguistic behavior.

  Technomorphism — an Analogue to Anthropomorphism

  Sometimes it’s not the choice of a computer-oriented word that betrays the tendency of technical ideas to slip into everyday thinking, but simply a computer-oriented habit that unconsciously pops up and tries to insinuate itself into an unfamiliar new situation. After all, when one is constantly dealing with technology and using the Web on a day-in day-out basis, one can’t help starting to see this familiar old world with fresh new eyes. This can happen, for instance, when one is trying to find a favorite passage in a book. While randomly flipping through its pages, seeking the desired passage much like someone seeking a needle in a haystack, one can easily grow very frustrated, since one is painfully aware that if one only had an on-line version of the book, it would be a piece of cake to find the desired passage.

  The following anecdotes illustrate the very human and very natural — indeed, irresistible — tendency for computer-based concepts to pop to mind in everyday situations as if they applied, when in fact they don’t apply at all (or at least not yet!).

  A woman was driving her ten-year-old son, a video-game addict, to another city. After several hours, the boy started complaining of being bored to death. His mother replied, “Would you please stop your whining? I’ve got to concentrate on the road.” Her son shot back, “But it’s not fair — you get to play the whole time!” For the boy, who had virtually driven virtual cars for several years, the act of real driving was exactly like playing a video game, and thus to him, it seemed that his mother had the good fortune to be having lots of fun while he was unable to do so.

  A little girl walked into a room where two digital photo frames were sitting on a buffet, each of them periodically flashing one image, then another, then another, and so on. Between them was an old-style frame containing a standard still snapshot. She commented, “Look, Daddy, the frame in the middle is broken!” For her, the old-style frame was automatically seen as a member of the category digital photo frame, and as such, it was clearly a defective member of the category.

  An eight-year-old girl was in her family’s car when a heavy rain started to come down as they were driving through the countryside. She remarked, “Watch out, Dad, you can’t see anything on the screen any more!”

  A computer addict confessed that when he was sitting at his computer and a fly landed on the screen, his knee-jerk reflex was to try to get rid of it by clicking on it and then dragging it off the screen.

  When you’re at home and can’t find an object, you often wish you could simply search for it just as one can search for things on one’s hard disk or on the Web, or for that matter on one’s cell phone. If only you could type in a couple of key words, you feel, you could instantly retrieve any lost article! Take Alice, for example. She was very groggy when she woke up, and badly needed her morning coffee. While drinking it, she got a phone call from her mother, so she set the cup down. When she hung up, she’d forgotten where she’d placed it, so she blithely picked up her cell phone and started typing the word “coffee” in the “Search” slot of her list of contacts. Then there’s Bob (another example we found on the Web), who for five straight days had been working all day long at his computer, trying in desperation to finish a grant proposal before the deadline. At some point, he misplaced his glasses and realized that he couldn’t squelch the desire to type a couple of words into a search engine to find them. Examples of this sort are a dime a dozen on the Web.

  Unfortunately, no tool at that level of sophistication exists to help us locate lost items in the physical world, and for that reason the on-line world’s concept of search, which by now is second nature to nearly everyone, has become the source of analogies in the physical world, rather than the other way around. It is very tempting to think that objects that we’ve misplaced over and over again, such as our keys, our wallet, our checkbook, or our glasses (or their case), should be just as easily coaxed to chime out their presence to us as is our misplaced cell phone. In short, we’d like to be able to summon anything that we can’t find and have it instantly chirp back at us, telling us just where to find it.

  By now, the Web and cell phones have given us the sense that pretty much anything should be within easy reach at any time. Wherever we are located, just about anything that matters to us is within clicking range, emailing range, cell-phoning range, or text-messaging range. This gives us the illusory feeling that anything of any sort should be available to us instantaneously.

  The great ease of obtaining things from afar these days has had the effect of reinforcing the sense of presence of people who are in fact absent. Some people, for example, have started to feel ill at ease about making the slightest critical comment about anyone, any time their cell phone is nearby — as if the person being criticized might somehow overhear the comments even without any call having been initiated. This scenario is a bit reminiscent of living in an apartment with very thin walls, where one never knows for sure whether one might be overheard by others in neighboring apartments. Along much the same lines, certain people, when in a chat room on the Web, often start whispering to physical people who are physically near them, as if the virtual occupants of the virtual room could overhear every word they say aloud. In short, some computer addicts develop the jittery feeling that the physical world all around them is populated by virtual people who are able to overhear conversations about themselves.

  One very useful feature of computers is that they offer us all sorts of chances to undo mistakes that we have made. Thus when we are using a word processor or a photo editor, it seems only reasonable that we should instantly be able to reverse any action that we’ve done — including massive deletions carried out by accident — and we get very used to such luxuries, taking them for gr
anted. The habit then spills over into other domains of life and becomes an expectation, and when we realize there is no “undo” button in nearly any of them, we become frustrated. Here are two anecdotes illustrating this tendency:

  A student who was cooking a cake had put too much flour in her dough, and she would have liked to go back a few moments in time and undo her mistake. All of a sudden, she caught herself wishing that she could just push the “undo” key that she was so accustomed to.

  A teen-ager who was plucking hairs from her eyebrows suddenly realized that she’d done more than she intended, and she thought to herself, “Oh, no problem, I’ll just revert to the older version.”

  Not only the act of “undoing” but many other types of frequent computer actions can become such strongly ingrained habits that they wind up shaping the way people see and behave in the material world, as the following set of anecdotes demonstrates:

  A teen-ager was looking at a photo in a magazine in which she recognized some people but not others, and she said, “My first reflex was to wonder, ‘How come they’re not tagged?’ Then I remembered I was looking a magazine, not a Facebook page.”

  A man reported that one time when driving he found himself using two fingers on his rear-view mirror in an attempt to blow up the image. A woman replied that “the same thing” had happened to her but on her flat-screen television, and yet another said “exactly the same thing” had happened to her when she was looking at her reflection in her bathroom mirror.

  An avid moviegoer said that she wanted to go to the restroom during a film and for a split second she found herself trying to grab her mouse to stop the film momentarily.

  Another film buff confessed, “When I was in the movie theater, I tried to jiggle my mouse a little bit to see how many minutes were left in the movie.”

  A schoolboy said that during a quiz, he wasn’t sure whether he had spelled a certain word right, so he waited for a few seconds to see whether what he’d just written in pencil on paper was going to get underlined in red.

  A college student described how she was cramming for an exam and had finally, with great effort, figured out how a certain complex biochemical reaction worked. At that point, she was sitting in front of her computer, and she mindlessly hit the pair of keys that she always would hit when saving a file.

  These days, amusing mental contaminations due to these types of crossed wires involving computer concepts and pre-technological concepts are a dime a dozen, and there are Web sites aplenty “where” (if we may analogically borrow this concept from the physical world) people gleefully report and laugh at their own gaffes of this sort.

  Some Equations Are More Equal than Others

  Now that we have taken a careful look at the ways that naïve analogies originating in recent technology have insidiously invaded our lives, we can turn back to the field of education, focusing specifically on the role that naïve analogies play in how children pick up basic mathematical concepts in school.

  Is the equation “3 + 2 = 5” completely clear? Is there just one way to understand it? Do all educated adults understand the equals sign in the same way? Theoretically, an equation symbolizes a perfect equivalence or interchangeability; that is, an equals sign tells us that the two expressions flanking it stand for one and the same thing. The notion of equality, when described this way, seems so simple and straightforward that it would seem hard to imagine any other way of interpreting it. And yet there is another side to the notion of equality, and it comes out of a naïve analogy that we will call “the operation–result analogy”.

  In this alternate interpretation, the left side of an equation represents an operation, while the right side is the result of the operation. This is a naïve analogy in which equations are tacitly likened to processes that take time, and it crops up in situations that have nothing to do with school or mathematics, and which influence everyone, including young children. For instance:

  point at + cry = obtain a desired object

  vase + knock over = shards of glass on the floor

  mud + hands = mess

  DVD + DVD player + remote control = watch a movie

  chocolate + flour + eggs + mix + bake = cake

  cheese + lettuce + tomato + bread = sandwich

  3 + 2 = 5

  Here, the equals sign is a symbol that links some sort of action in the world to its outcome, and it can be read as “gives” or “yields” or “results in”. When seen that way, “3 + 2 = 5” is not the statement of an equivalence at all; rather, it expresses the idea that the process of adding 3 and 2 results in 5.

  The ideas of interchangeability and operation–result are different. The second point of view clearly embodies an asymmetrical conception of equations, in which the two sides play different roles, one side always standing for a process and the other always representing its outcome. To write “5 = 5” would be incompatible with this viewpoint, since no process is indicated. Likewise, writing “7 – 2 = 8 – 3” is also troublesome, since now there is no result. And lastly, writing “5 = 3 + 2” would be disorienting, because the operation and its result occupy the wrong sites. Indeed, many first- and second-graders understand equality in just this fashion, insisting that “5 = 3 + 2” is “backwards”, and that “7 – 2 = 8 – 3” makes no sense because “after a problem there has to be an answer, not just another problem”. Some even balk at “5 = 5”, replacing it with something such as “7 – 2 = 5”.

  The operation–result naïve analogy guides children before they encounter the concept of equivalence, because the notions of a process and its result are familiar even to toddlers. These notions are close cousins to the notions of cause and effect, as well as to the idea that certain means have to be used to reach certain ends.

  Although today’s children may acquire a fairly decent understanding of equality in elementary school, coming up with the symbol “=” took a long time for humanity as a whole. A symbol for equality in mathematics first appeared only in the year 1557, in a book by the Welsh mathematician Robert Recorde. He wrote:

  I will sette as I doe often in woorke use, a pair of paralleles, or Gemowe lines of one lengthe, thus : ===, bicause noe 2. thynges, can be moare equalle.

  The word “gemowe” means “twins”, and the “twinnedness” of the upper and lower horizontal lines was intended to symbolize the general idea of equality. The fact that a symbol for equality took such a long time to occur to anyone, even though mathematics had existed for at least two millennia, reveals that it is far from a self-evident notion.

  Although for many adults today the idea that “equality equals equivalence” may seem obvious in a mathematical context, it doesn’t follow that the operation–result view of equality has disappeared from their minds. In fact, people often write down, and read aloud, equations in a way that reflects their unconscious understanding. For instance, in reading “4 + 3 = 7”, many people will say “four plus three makes seven”, whereas for “7 = 4 + 3” they might say “seven is the sum of four and three”. If education always resulted in equations being seen as statements of interchangeability, then by the end of high school, the operation–result view of the equals sign would surely have disappeared for once and for all. The order of the two sides in an equation would be completely irrelevant, and both ways of writing an equation down would elicit exactly the same commentary. However, this turns out not to be the case. Let’s take a look at some specific cases, starting out with some that are very remote from mathematics.

  Naïve Equations in Advertisement

  It’s standard practice for advertisements to appeal to the child inside each of us rather than to the budding mathematician. Here are a few examples of “equations” culled from real ads:

  buy two items = 50% off on the second one

  buy a pair of glasses = get a pair of sunglasses for free

  buy a loyalty card = free home delivery for a year

  buy a TV set = a DVD player for just $1

  buy any pizza = get
another one free

  Just to convince ourselves that interchangeability is not the idea behind these equations, let’s flip them around. As you will see, the resulting “equations” sound utterly silly, even nonsensical.

  50% off on the second one = buy two items

  get a pair of sunglasses for free = buy a pair of glasses

  free home delivery for a year = buy a loyalty card

  a DVD player for just $1 = buy a TV set

  get another one free = buy any pizza

  As these examples reveal, people’s first glimmerings of understanding of the equals sign come from the naïve analogy of an operation followed by a result, and even if the concept of interchangeability gains some ground in the course of twelve years of education, the operation–result viewpoint is never fully eradicated. It can always be coaxed out of dormancy when the right cues are presented. We thus see that education does not eliminate the first naïve ideas about a mathematical notion — even one that we tend to think of as completely trivial because it is taught in elementary school, when children are only six or seven years old. In childhood and even when one is fully grown, the naïve view coexists with a different view, which is instilled at school but which is also dependent on a pre-existent and familiar notion: that of same thing (that is, identity). The transition from the earliest viewpoint (operation–result) to the more sophisticated one (identity) does not obliterate the earlier viewpoint, which remains triggerable, and on which one frequently relies on a day-by-day basis, sometimes even in scientific contexts, such as mathematics or physics, as we shall now see.

  Of Equations and Physicists

  For physicists, the most fundamental formula of classical mechanics is doubtless Newton’s second law, which describes how a force affects the motion of an object. The basic idea of this celebrated law is compatible with the naïve analogy that says that one side of an equation should represent a process, with its other side representing the result of that process. In this case, the process (ideally occupying the left side of the equation) would be the action of a force of size F on a mass of size m, and the result (ideally on the right) would be an acceleration of size a imparted to the mass. Rendered symbolically, this yields the equation “F/m = a”. Unfortunately, though, Newton’s law is virtually never written this way. Instead, it is almost always cast as follows: “F = ma”. This famous formula is quite confusing to many students, since neither side of it cleanly symbolizes either the process or the result. The alternative notation “F/m = a” encodes the naïve analogy much more clearly, and would therefore be easier for students to relate to, but it is seldom if ever found in textbooks. From a purely logical standpoint, these two versions of Newton’s law are completely equivalent and interchangeable, but from a psychological and pedagogical standpoint, they certainly are not.