Surfaces and Essences: Analogy as the Fuel and Fire of Thinking (85 page)

The Copycat domain is filled with cases where the encodings of the two stories involved in a me-too reminding episode are not identical but are
analogous
to each other. In other words, even when one jumps up to a fairly high level of abstraction, ridding oneself of nearly all of an event’s details and thereby arriving at a tiny, compact summary of its essence — nothing but a gist — the two gists linked by a me-too reminding episode are usually not
identical
, but only
analogous
to each other. And if one insisted on having
identical
conceptual skeletons for two analogous events, one would often be forced to leap to such an artificial level of abstraction that the conceptual skeleton applying to
both
stories would be an absurdly strange-sounding legalistic formula that no one would ever concoct in real life. Furthermore, such a skeleton could never arise as the spontaneous encoding of the first event alone, before the second event had been encountered. That would require clairvoyance.

The profound mystery of how human remindings take place is not solved by positing that we always encode events with marvelously clairvoyant conceptual skeletons that anticipate all possible other events that might ever be analogous to them, even many years down the pike. Something much subtler is involved in the act of encoding memories than just manufacturing “clairvoyant” conceptual skeletons on the fly, since that notion is a chimera.

A key ingredient so far missing from our account is the ability of a crux to evoke
analogous
cruxes in memory. This ability is what allows us to see connections between events that are too dissimilar on their surfaces to have been encoded identically, and yet that still are deeply similar. In short, the secret of making good analogies involves making good but
more abstract
analogies — analogies between encodings, or conceptual skeletons. This may sound like an infinite regress, and thus a hopeless conclusion, but since analogies between cruxes are
more abstract
than analogies between original stories as wholes, “kicking the problem upstairs” to the level of finding analogies between cruxes is in fact a genuine simplification.

Next we will explore one favorite example of a me-too event in the Copycat domain. It turns out that the string
xyz
was among those listening to abc’s tale. What experience was it reminded of? It all comes down to figuring out how to answer this question: “What is ‘the
c
’ of the string
xyz
, and how should it change?” Well, quite obviously the
c
of
xyz
is the
z
— what else could it be? — and so one’s first natural impulse is to change the
z
into its alphabetical successor. Here, though, we run into a snag, since
z
has no successor; it’s the last letter of the alphabet. End of the line; everybody off!

However, some people — many, in fact — are not in the least stymied by this snag; undaunted, they immediately propose
a
as z’s alphabetical successor, giving the answer
xya.
Now where does this idea come from? Are we explicitly taught in school that the alphabet is a circular, wraparound structure? Of course not. However, as we grow up we all learn about “circular sequences”, such as the days of the week, the months of the year, and the hours of the clock. There are also decks of playing cards, where the ace not only is the lowest card but is also higher than the king. Structures very much
like
a circular alphabet are “in the air” all around us, and thanks to their unconscious influence, the answer
xyz ⇒ xya
is easily found. In short,
xya
results from importing the concept of
circularity
from various familiar external sources into the microdomain whose alphabet is
not
circular. In so doing, therefore, one tampers with the nature of the microdomain. We might even say that when people carry out such a conceptual importation, they “corrupt” or “contaminate” the pure and pristine Copycat domain by throwing in alien ideas that are extraneous to it.

Despite this caveat, we’ll accept
xya
as a legitimate reminding for
xyz
to have had — but what if
xyz
had never had that experience? In that case, what other event(s) in its memory might the
abc ⇒ abd
story have triggered? Stated otherwise, if
z
has no alphabetical successor, then what event(s) in the life of
xyz
might be analogous to the event
abc ⇒ abd
? And in fact there is an answer that strikes most people, once they have seen it, as being far more elegant than
xya
.

Everybody wants to change the
z
into something else; the question is, into
what
? Well, since taking
z
’s successor seems to be at the heart of what is giving us trouble, we might try to go back and explore alternate interpretations of what happened to the
c
, interpretations not involving the concept of
successorship.
For instance, instead of saying that the
c
changed into its successor, we could say that it changed into a
d.
In that case,
xyz
could perfectly reasonably be reminded of the time when it got changed into
xyd.
That’s one possible answer; however, because of the intrusion of the literal
d
into the
xyz
world, it’s not very appealing. Are there other more appealing alternatives?

Well, under this situation’s unique combination of pressures, we might try to reperceive what happened to
abc
and say, “The letter
c
got replaced by a
d
”, where by “the letter
c”
, we now literally mean “the instance of the letter
c”
, rather than “the string’s rightmost letter”. In that case, we certainly don’t have to worry about the pesky
z
any more. Instead, we want to scour
xyz
for one or more instances of the letter c, and then, if and when we find one, we will change it to a
d.
A moment’s scouring of
xyz
reveals, however, that there is no
c
in it, and hence no letter to change to
d.
And so, one possible reaction on
xyz’s
part would be to say, “Ah, yes — abc’s story reminds me of a memorable occasion one time long ago, when nothing at all happened to me…”

We’re far from having exhausted the possibilites. Another thought might be to recall how
mrrjjj
was perceived on an abstract level as 1–2–3, which then, in analogy to
abc
’s becoming abd, became 1–2–4. Well, then, why couldn’t
xyz
be seen on an abstract level as 1–1–1, meaning three groups of length 1? In that case, it could turn into 1–1–2 on that abstract level, yielding the answer
xyzz
back on the literal level. This doubling of a letter is reminiscent of how stadiums and large theaters often extend the alphabet,
going from single occurrences (“A, B, C, D, ……, W, X, Y, Z”) to double occurrences (“AA, BB, …”). But even so, it feels like a bizarre evasive maneuver. If
abc’s
story had been
abc ⇒ abcc
, then of course for the story
xyz ⇒ xyzz
to bubble up out of dormancy would seem like a perfect me-too. But as we know, that wasn’t the story that
abc
told.

In short, no solution given so far seems pleasing, let alone elegant.

In trying to take the successor of
z
, we repeatedly stubbed our toe, and this repeated annoyance focused our attention more and more on the fact that
z
has no successor — otherwise put, that
z
is the last letter of the alphabet. Now focusing on the alphabet’s
last
letter is but a stone’s throw away from focusing on its
first
letter. (As we pointed out in
Chapter 5
, slippages between opposite concepts are both natural and frequent, and can sometimes give rise to fascinating errors, such as a confusion between
reading
and
writing
, or between
being born
and
dying
, or between
grandparents
and
grandchildren
, and so forth.) Now inside
abc
there is an
a
staring us in the face. What more natural act, then, than to link the first letter of the alphabet, on
abc
’s left side, with the last letter of the alphabet, on
xyz
’s right side?

Thanks to this fresh new analogy between the concepts
first
and
last
, we have uncovered a new perspective on the connection between the two strings — a charming symmetry that is not so easy to spot but that, after the fact, seems as plain as day. This symmetrical mapping of the
a
in
abc
onto the
z
in
xyz
is reminiscent of the finger-twiddling challenge, where, in order to imitate on your left hand the twiddling of your right hand’s rightmost finger (its thumb), you decided not to twiddle the
rightmost
finger (the pinky), but the
leftmost
one (also a thumb).

If the
left
end of
abc
maps onto the
right
end of
xyz
, that makes it natural — indeed, compelling — for us to use the left–right reversal consistently, by mapping the strings’
other
ends onto each other as well — so that the
c
of
xyz
, rather than being the obvious, once-irresistible choice of z, now becomes the
x.
Notice that this happy choice means that we won’t stub our toe. After all,
x does
have a successor — namely,
y
. Lucky us! And indeed, replacing the
x
by a
y
will give us
yyz.
Now there’s a sweet new answer!

And yet… Would
xyz
call this event “exactly the same thing” as what happened to
abc
? We just saw how the two changes might be called “exactly the same”, but still, something smells fishy. After all,
yyz
very saliently has two identical letters right next to each other, whereas
abd
contains no such pair. In that sense, the two changes seem glaringly unlike each other. It’s almost as if the
yy
pair is serving as a warning signal, a red flag, hinting that something crucial was overlooked. And indeed, the insightful idea of left–right reversal, though it was pushed somewhat, was not carried far enough.

Once we’ve chosen to map
abc
onto
xyz
with their physical directions reversed, then
forwards
motion in the alphabet is implicitly being mapped onto
backwards
motion in the alphabet, and thus implicitly, as part and parcel of this mapping, the concept of
successor
is being mapped onto the concept of
predecessor.
Another way to put this is that when
abc
is read c-wards, starting at the a, then its fabric is one of
successorship
, whereas when
xyz
is read
x
-wards, starting at the
z
, then its fabric is one of
predecessorship.
And so, from the simple act of seeing the
a
and the
z
as each other’s counterparts, a tight little cascade of conceptual slippages has flowed in a natural and (almost) irresistible manner. At the outset we saw
first
slipping to
last
, and then we saw
left
slipping to
right.
And now we’ve hit the final slippage in the cascade — the slippage from
successor
(the fabric of the
abc
world) to
predecessor
(that of the xyz world).

This final conceptual slippage means that instead of wanting to take the
successor
of the
x
in xyz (an overly rigid thing to do), we’ll want to take its
predecessor.
All of this will give us, in the end, the string
wyz
. This is a surprisingly pretty answer to the question “Suppose
abc
changed to
abd
; can you make
xyz
change in the same way?” Several coordinated slippages have led us to a
fluid
analogy. And all this insight came from “stubbing our toe” on the letter
z
— that is, from the snag due to
z
’s lack of successor.

This symmetry-based answer is, in a certain sense, an even stronger and more convincing analogy than when
pqrs
recalled turning into
pqrt.
That is,
xyz ⇒ wyz
is arguably more like
abc ⇒ abd
than
pqrs ⇒ pqrt
is, strong though that resemblance is. Why is this? Because both
abc
and
xyz
are strings wedged at their respective ends of the alphabet. There’s a perfect symmetry here, and the two changes are like perfect mirror images of each other. No flaw is perceivable anywhere in this analogy, and this is what makes it so appealing. Although
pqrs
has quite a bit in common with
abc
, it’s not as similar to
abc
as
xyz
is (what can rival a mirror for making symmetry?), even if
pqrs
is clearly much more similar to
abc
than
tky
is.

In
Chapter 5
, we cited the striking conceptual-proximity slippage made by a grandfather who, as he and his son drove by a cemetery, observed, “This is where all four of your grandkids were born.” There we pointed out the very human drive towards internal consistency, pushing the grandfather towards making
two
slippages between opposite concepts, rather than just one. We even suggested that either of the slippages could have brought the other one along “on its coattails”. Such a “coattails” characterization would also describe the way in which slippages coordinatedly cascade in this Copycat analogy between
abc ⇒ abd
and
xyz ⇒ wyz
, although we are obviously talking about
esthetic
coattails rather than electoral ones.