Surfaces and Essences: Analogy as the Fuel and Fire of Thinking (84 page)

Let’s return to the case of
iijjkk ⇒ iijjll.
Instead of seeing the original string
iijjkk
as a string of six letters, most people tend to see it as a string of
three pairs
:
ii–jj–kk.
This interpretation invites us to map these three pairs onto the three letters of
abc.
Here something subtle happens. Because of the perceptual chunking that took place, the rightmost “letter” of
ii–jj–kk
is no longer the single letter
k
but the
group
of two
k
’s. In other words, the meaning of the word “letter” is extending naturally outwards under contextual pressure. In this case, we recoil at the idea of merely changing the single rightmost
k
into an
l;
instead we change
both
of the
k
’s simultaneously into
l
’s, yielding the string
iijjll.
In sum, we changed the rightmost “letter” to its successor, but in so doing, we instinctively took the notion of
letter
fluidly and abstractly.

The story behind the answer
iijjll
will not be a surprise to you, but the real point is how intimately and inextricably wound up in all of this our sense of esthetics is, and how often people see eye-to-eye on the esthetic qualities of different answers.

Let us be even more explicit here. In a strict sense, there is nothing
wrong
with the answer
iijjkl
(it clearly has a certain kind of logic to it), nor for that matter is there anything wrong with
iijjkd
(“replace the rightmost letter by
d
”), nor is there even anything wrong with the answer
abd
(“replace the entire string by
abd”).
We are not concerned with
rightness
or
wrongness, validity
or
truth
here. This is not a black-and-white domain in which there is just one right answer and all other possibilities are dead wrong; rather, there is a spectrum of possible answers, and each different answer has its own logic, and answers vary in their degree of appeal.
Appeal
, not truth, is the name of the game. Rather than being
wrong
, the various answers discussed above might be said to have differing “scores” of subtlety or finesse, depending on how much or how little abstraction was involved, as well as on an elusive “sense of essence”. Especially the final pair (
iijjkd
and
abd
) are out of tune with how humans tend to see their world.

To shed some perspective on all this, we propose a simple analogy challenge. First please hold up your two hands, with their palms facing you, and wiggle your right thumb. Now we’ll simply ask your left hand to “Do the same thing.”

So you wiggled your left thumb? Good move! But you certainly could have wiggled your left hand’s little finger instead, since it is that hand’s
rightmost
finger, and since your right thumb is your right hand’s rightmost finger. However, what you
could have done
doesn’t necessarily have much to do with what people in fact
do
, because we humans are ceaselessly categorizing the world, and we try to do so efficiently and even
elegantly.
We don’t usually see a little finger as being analogous to a thumb, unless there
are strong pressures to do so — but when they are strong enough, we gracefully yield and make the conceptual slippage, often without the slightest thought. Thus, for instance, if you happened to have rings on your right thumb and on your left little finger, and no other rings anywhere on either hand, it’s very possible that you wouldn’t have wiggled your left hand’s thumb but the finger with the ring on it. And the more similar and salient the two rings were, the more likely you would be to move your little finger. Under sufficient pressure, concepts slip into other, related concepts.

Let’s return to the Copycat domain and its miniature me-too stories. It happens that
mrrjjj
was among the group of friends, and it, too, exclaimed, “Exactly the same thing happened to me!” What me-too story do you suppose
mrrjjj
then told?

Most people guess that
mrrjjj
’s story was that one fine day, it was changed into
mrrkkk.
What leads people to make this guess? Everyone agrees that
mrrkkk
beats its “rivals”
mrrjjd
and
mrrjjk
and
mrrddd
hands down (and we won’t even mention
mrdjjj
, let alone some others). None of these “rival” answers is
wrong
, no more than it would have been
wrong
for Leonardo to have put a mustache on the Mona Lisa; these are esthetic decisions. Each answer is justifiable in its own way, and each will satisfy some people while leaving others dissatisfied. Their justifications reside at different levels of literality. For example,
mrdjjj
is hands down the most literal of the just-cited answers, as it replaces the
third
letter by a
d
— a level of literality so extreme that we have never seen anyone come up with this answer. Answer
mrrjjd
is slightly less extreme, as it involves replacing not the
third
but the
rightmost
letter by a
d.
Answers
mrrddd
and
mrrjjk
are yet less literal, since for the former, the final
group
, instead of the final
letter
, is replaced by a set of
d
’s, while for the latter, the rightmost letter is replaced by its
successor
rather than by a
d.
Each time one moves to a higher level of abstraction, one finds hidden structures and patterns that tend to be more esthetically pleasing. As the old dictum says, there’s no arguing over tastes — but luckily, at least in the tiny Copycat microworld, people tend to agree on what is in good taste and what is not. And most people suggest that the best event for
mrrjjj
to recall would be when it got changed to
mrrkkk
.

And yet what’s curious is that
mrrjjj
did not, in fact, recall that event. It did
not
say, “It’s just like when I got changed into
mrrkkk!”
It certainly
could
have told this story, but that’s not the memory that came to its mind. The story
mrrjjj
told was this: “It’s just like when I got changed to
mrrjjjj”
(with four copies of
j).
Now why would this be “exactly the same thing” as
abc
’s story of having gotten changed to
abd?

Clearly
mrrjjj
, just like
iijjkk
, can be broken up into three natural parts:
m–rr–jjj.
Now, just as with
ii–jj–kk
, a mapping leaps out at the eye between this new tripartite structure and
abc.
And there are some interesting things to observe in
m–rr–jjj
, such as the different sizes of the three pieces. Indeed, there’s a group of length 3 at the right end, a group of length 2 in the middle, and… Could that be a group of
length 1
at the left end? Isn’t that
m
sitting all by itself a perfectly valid group of length 1? No one could disagree. And so we have a situation whose essence resides no longer at the letter
level — the literal level, quite literally — but at the
numerical
level. That is, we are coming to see the essence of string
mrrjjj
as the hidden pattern “1–2–3”, and this essence has little to do with the string’s quite arbitrary component letters
m, r
, and
j
, which simply acted as a
medium
carrying us the
message
“1–2–3”. In this context, we ignore the letters — the surface is irrelevant here — and we focus on something deeper.

In mapping the tripartite structures
abc
and “1–2–3” onto each other, we will of course see the “3” as the
c
of “1–2–3”, and we’ll change that “letter” to its successor. Several slippages take place at the same time here, yet all happen effortlessly. We are not looking at the rightmost
letter
of
mrrjjj
(let alone at an instance of the letter
c
) but at the rightmost
number
of “1–2–3”, and we are not changing that “letter” to its
alphabetical
successor but to its
numerical
successor, which is of course “4”. That is the reason that
mrrjjj
said to the assembled crowd, “And so I got changed to
mrrjjjj
, just like
abc
got changed to
abd.”
This esthetics-drenched answer exemplifies conceptual fluidity.

Some people, when asked what might have happened to
mrrjjj
, suggest the answer
mrrkkkk
, based on changing not just the final group’s length but also all of its letters, in one fell swoop. What about this double-barreled, perhaps “superfluid”, answer?

Well, what would you think if someone, in answering the thumb-wiggling puzzle, simultaneously wiggled both their left thumb and their left hand’s little finger? Would

that constitute a sensible me-too? Would that amount to “exactly the same thing”, in the world of the left hand, as wiggling the thumb alone, in the world of the right hand? We would say no. Taking two rival answers to a single question (in one case, wiggling the left hand’s thumb
and
little finger, or in the other case, changing both
j and
3 to their successors) and blurring them together is, in a word, confused. The answer
mrrkkkk
may have an initial razzle-dazzle, but on further thought it is an incoherent trap. There is no more reason to combine these two answers into one answer than there is to combine French fries and orange sherbet into one dish, simply because one is fond of each of them. To put it another way, there is no good reason to gild the petals of a lily. It’s often said that it’s pointless to argue over taste, since taste is so personal, but it is also a fact that in some contexts, there is a strong consensus that certain combinations of tastes or of ideas make no sense and are displeasing. Sometimes taste is nearly universal.

There is no such thing as a
proof
that an analogy is good or bad, whether in the Copycat microdomain or in the far wider real world. And so, in our attempt to explain why we see the answer
mrrkkkk
as confused and unsatisfying, we didn’t seek an ironclad
logical argument
but instead resorted to a trio of
caricature analogies
(the first involving finger-wiggling, the second involving mixing gustatory delights, and the final one involving a famously fatuous flower-furbishing act), hoping readers would agree with our subjective sense of their relevance to this situation. We sought convincing
esthetic
reasons for rejecting
mrrkkkk
by thinking carefully about the idea of combining two actions, each of which, taken on its own, does a fine job, but which, if fused together, yield something silly. Does our “superfluid” answer not seem superfluous, now?

Something noteworthy took place when
mrrjjj
recalled being changed into
mrrjjjj.
When hearing
abc
tell its
abd
story, it unconsciously encoded that story as “the rightmost letter got changed to its successor”. On the other hand, much earlier,
mrrjjj
had its own experience of being changed into
mrrjjjj
, and at that time it encoded this experience in an abstract fashion as “the rightmost group got extended by one unit”. As is obvious,
mrrjjj
’s encoding of its own experience is similar to, but far from identical with, its encoding of abc’s story. Life would be very simple if every me-too retrieval episode were due to the two events having been encoded
identically
, but that is a naïve hope.

Let’s look at another example of the subtleties of encoding in this domain. It’s very unlikely that on first hearing the story of
abc
being changed into abd, you would think of the
c
as “a group of
c
’s of length 1”, but if you were to do so, then you would probably encode the story
abc ⇒ abd
as follows:

The rightmost group got changed into another group of the same length, with all the letters inside it being replaced by their alphabetic successors.

Although this encoding is undeniably
correct
(provided one is willing to bend over backwards to see each of the three letters inside
abc
as constituting a
group
on its own), it is an unnatural, bizarre, and topheavy description of what happened to abc, and no one would ever come up with such a strange description.

But now recall
iijjkk
being changed to
iijjll.
The unnatural encoding just displayed for
abc ⇒ abd
now seems perfectly natural for
iijjkk ⇒ iijjll.
But this doesn’t make it a good encoding for the event
abc
⇒
abd
, because it contains extraneous ideas that are irrelevant to that event. A single letter is not perceived as a “group of length 1” unless the perceiver is under intense pressure! Therefore, the strong, natural analogy between the events
abc ⇒ abd
and
iijjkk ⇒ iijjll
cannot have been mediated by their sharing the exact same encoding. No; different encodings were created at different times (one involving the rightmost
letter
, the other involving the rightmost
group
), but even so, the latter story activated the former, because on some abstract level their gists were sufficiently similar (in each case, one changes the “rightmost thing” into an abstract kind of “successor”, whether that “thing” is a letter or a group).

Now let’s turn to string
ace.
What came to its mind when it heard abc’s story? Hint: it wasn’t the time when
ace
became
acf
, nor the time when it became
ade.
Of course either of those events
could
have come to
ace
’s mind, but in fact
ace
recalled the time when it turned into
acg
— when its rightmost letter was replaced by its
double
successor.
This
memory, for ace, was “exactly the same” as what happened to
abc.
Now where did the curious concept
double successor
come from? From, of course, the internal texture of the string
ace
, which is analogous to the internal texture of
abc.
Namely, where
abc
is a short chain made out of successor bonds (
a–b
and
b–c
),
ace
is a short chain made out of double-successor bonds (
a–c
and
c–e
) — and so this prompts a natural slippage from the concept of
successor
to the closely related concept of
double successor.
But surely no one
could have anticipated this esoteric slippage when looking at the event
abc ⇒ abd
in isolation — and yet in this special
ace
context, the pressures to make that slippage are very strong.
Not
to make the slippage (and thus to insist on the greater appropriateness of
ace ⇒ acf
) would seem like an unreasonably rigid stance. Once again this is a case where the natural encodings of the two analogous stories are similar but not identical.