Surfaces and Essences

by Douglas Hofstadter

  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.

  Gilding the Lily

  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?

  Seeing a More Abstract Gist than the Gist that was Encoded

  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.

  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 othe
r. 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.

  The Zaniness of the Letter “Z”

  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.

  The Snag Triggers a More Satisfying Reperception

  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.