Surfaces and Essences: Analogy as the Fuel and Fire of Thinking (94 page)

The fluid way that a scientific notion is realized and “lives” in the minds of people who deeply understand it is a very different thing from a formal and rigid symbolic notation, which has been carefully devised to be as concise and rigorous as possible, and the two should not be conflated. As we have just seen, the notion of
continuous function
has a precise formal definition, and yet continuous functions form a mental category having blurry boundaries and members with different degrees of typicality, and such judgments will vary from one mathematician to another. A famous historical case illustrates this fact. Toward the end of the nineteenth century, functions continuous at every irrational point and discontinuous at every rational point were discovered, and this behavior was so unexpected that some renowned mathematicians labeled them “pathological” and strove to banish this “scourge” from mathematics, whereas other mathematicians were extremely excited by the revelation of such a rich new area for research. This episode hardly jibes with the image of mathematics as a frozen body of precise and absolutely objective knowledge. In the following chapter we’ll see that even a category as familiar as
number
has blurry boundaries and members with varying degrees of typicality, and seasoned mathematicians can hold different opinions and can argue vehemently about what is and what is not a member of the category.

The unfortunate but widespread conflation of austere formal definitions with the psychological reality of concepts in human minds has had worrisome repercussions in education. One consequence is that it tends to make the educational establishment lose sight of what ought to be its primary goal — the construction of useful, reliable categories. Another consequence is that it tends to favor teaching methods based on prickly formalisms and rigorous deductions rather than methods that use analogies to build up suitable families of categories in intuitive ways.

Our view is very different from one in which logic is seen as central. Indeed, as we stressed in
Chapter 4
, expertise builds up as categories are acquired and organized. Rather than depending on formal perceptions of situations, people have the ability to treat novel situations as if they were familiar, thanks to categorization. To acquire knowledge in a domain is to build up relevant categories. Analogical thinking is the key to understanding new situations and to building up new concepts, and this holds at all levels, ranging from the shakiest beginner to the most fluent expert. The difference between those two is not their style of thinking — logical for the expert and analogical for the beginner — but the repertoire of categories that they have at their disposition, and the way those categories are organized.

For example, in the case of continuity of a function at a given point
x
₀
, do experts really think in terms of Greek letters and flipped and rotated roman letters? Hardly. Instead, they have vivid visual imagery about the concepts concerned. The epsilons and deltas in the formula are merely helpers in translating that imagery. Thus if one wants the value of
f
(
x
) to be very close to that of
f
(
x
₀
) — in particular, if one wants their discrepancy not to exceed the tiny number
ɛ
(this desideratum is expressed by the notation “|
f
(
x
) —
f
(
x
₀
) | <
ɛ
”) — then continuity requires that there should exist some little zone centered on the point
x
₀
and whose two edges are at a distance of δ from the center (this zone is expressed by the notation “ |
x
—
x
₀
| < δ ”), and throughout this zone the desideratum should be the case.

Thus if one has a mathematically trained mind and is thinking about what continuity means, one envisions a small rectangular box centered on the point (
x
₀
, f(x
₀
))
in the plane. Continuity then will mean that no matter how close one wishes
f(x)
to be to
f
(
x
₀
) (this desire translates into the image of a box whose
height
is very tiny, so that all y-values in it are
vertically
close), one can always make the box so
narrow
that the desideratum will hold everywhere inside it. The phrase “no matter how close one wishes” is the visual translation of “∀
ɛ
” (pronounced “for all epsilon”), while the phrase “one can always make the box so narrow” is the visual translation of “∃δ”(pronounced “there exists some delta”). In the end, it all comes down to the idea of zooming in on the graph of the function at arbitrarily small scales, and thus it has to do with such familiar things as magnifying glasses and microscopes and walking towards objects so that what is blurry at a distance comes into focus. These are the kinds of everyday experiences in which a mathematical sophisticate’s understanding of continuity is rooted. Thus we see that to understand the rich concept of
continuity
is to have built up this kind of imagery, and the epsilons and deltas are merely tools, used briefly and discarded quickly, which help one to realize that goal.

In sum, notions taught in schools and colleges are internalized by students not formally, but by means of naïve analogies, which is to say, by means of analogies with familiar categories. These categories can come from any domain, and they tend to come from domains that are not covered in school. Thus the presumption that a given area of knowledge is self-contained is erroneous, because students will inevitably connect every new notion in that area with experiences they’ve had in other areas of life. Moreover, a naïve analogy, once it has taken root in a beginner’s mind, will not go away as time passes; it will stick in the mind as school goes on, year after year, because everyone has a deep need for simple, basic intuitions. Indeed, some naïve analogies are so persistent that one might begin to doubt whether schooling can have any effect at all in certain areas. The goal of transmitting to average pupils subtle ideas that humanity as a whole took thousands of years to discover and absorb is admirable, but it is not something that just happens by itself.

We will see that certain notions that are usually thought of as extremely simple and that are taught in elementary school are, unfortunately, understood through naïve analogies having quite limited realms of validity, and these often-misleading analogies remain entrenched through middle school, high school, and college. Their subliminal effect lives on, long after education should theoretically have thoroughly drummed them out. Thus naïve analogies, thanks to their great robustness, lie behind the thought processes of well-educated adults, even experts; this casts doubt on the idea that education gets rid of them. The stereotypical vision, by contrast, is that people who have deeply absorbed scientific ideas swim in a soup of formally defined notions that obey logical laws; analogies play no part in this vision. That, however, is just wishful thinking. To be sure, scientific education does convey ideas that go beyond everyday experience, and in particular they tend to be more abstract, but scientific ideas are not learned any differently from other ideas — they, too, are rooted in naïve analogies, and this is a universal fact, whether one is talking of beginners, intermediates, or experts.

There is no question that the development of computers and related technologies has given rise to a major revolution on our planet. Those who remember the day when there were no computers in homes or businesses, and when there was no World-Wide Web to which to connect such devices, have children who are filled with wonderment at the idea that people could get along at all in such a primitive world. It thus makes sense to suggest that computers were
the
great innovation of the twentieth century. It also might seem that when something so new and so completely unprecedented takes over so widely, it would involve such a radical break with prior concepts that completely new categories would have to be introduced, categories not grounded in any kind of analogy at all.

If computer technologies are so hugely innovative, don’t they have to do much more than merely use various analogies to clothe familiar old notions in fancy new wrappings? Actually, the key question is the exact opposite — namely: how could
anyone imagine a great burst of creativity taking place without its pioneers building left and right on familiar notions? And how could the lay public possibly absorb the tsunami of radically new ideas without basing their new skills on analogous skills involving notions that were already deeply rooted? The simple truth is that only through homey analogies based on familiar, everyday categories can an average person relate to all the revolutionary new technologies; such analogies are anchors keeping people from getting totally lost in a huge sea of technical terms. The following memo is a good case in point.

This note, had it been written in the 1970s, would have referred to a wooden desk, some cardboard folders, documents and sheets of paper, a metal wastebasket, glass windows, and a postal address, and there couldn’t possibly have been any confusion as to what was being talked about. Today, however, the paragraph is highly ambiguous, because the supplies and furniture to which it refers have both real and immaterial embodiments, which are so closely analogous to each other that it’s impossible to tell which type of stuff is being referred to.

We thus see that computer technology — the domain that represents the greatest break between this century and the previous one — relies on scads of analogies with old-fashioned things, and these analogies are so tight that a perfectly realistic little scenario can be concocted in which there is no way to tell which century’s technology is being referred to. Actions that can be performed on physical objects have their counterparts in the virtual world. The desk, the files, the folders, the trash all exist in both worlds and just about anything that can be done electronically can also be done in the old-fashioned way. Even if people today read
documents
that cannot be crumpled, placing them in
folders
that cannot be torn, stacking those on
desktops
that are not made of wood, sending them to
addresses
that have no particular geographical identity, opening and closing
windows
that have no handles, and filling and emptying the
trash
with just a few flicks of their fingers, they are always doing so by analogy with the most familiar situations they know.

The reason that computers have revolutionized our society but not our vocabulary is that these very powerful devices have all been grafted onto familiar categories and have borrowed their verbal labels wholesale. These days, everyone speaks about virtual technologies using terms that wouldn’t in the least have disoriented our grandparents. Just as in the good old days, we open our mailbox to get our mail, we send messages to addresses, we send and receive documents, we visit sites, we chat with friends, we do searches for things we want to find, we consult pages, we make new links, we browse, we toss useless files into the trash, we open and close folders and windows, and so forth.

To be sure, there are certain new expressions that would have completely thrown anyone who heard them thirty years ago, such as “hosting a site on a server”, “surfing the Web with one’s browser”, and “installing a firewall against hackers”, but even so, these droll expressions, too, bear witness, in their own way, to the fact that all of our electronic activities are still cast in terms of the physical world around us. The humorous juxtapositions audible in these phrases reveal that as a species, we humans are totally dependent on familiar categories in order to adjust to brand-new realities. If
hackers
are using the
Web
to transmit
viruses
hidden inside
spam
, then who’s to say we shouldn’t use
firewalls
to protect ourselves from such shenanigans?

A more systematic exploration of the lexicon that has grown up around the Web and electronic technologies only serves to confirm our thesis that extremely familiar, everyday physical categories are overwhelmingly the most standard and relied-upon source of analogies for brand-new phenomena. Here, for instance, are roughly one hundred terms all of which could be found in dictionaries published long before computers played any role in society, but which, if read today, give the sense of being technological:

One might at first be inclined to suppose that many of these terms are deceptive — that is, that they are not really based on helpful analogies to older ideas, and that their new and old meanings are no more intimately related than are the two different meanings of “race” (“a running competition” and “a subspecies”) or the two different meanings of “number” (“a quantity” and “more anesthetized”). But this idea is far from the case: the semantic connections linking the new technological meanings to the meanings that originated in far older domains are in every case quite straightforward. Aside from a couple of terms such as “mouse” and “chip”, where the analogy between the everyday notion and the technical one is little more than a visual resemblance, what indisputably lies behind each of these familiar terms is an abstraction linking the new and the old uses.