Gödel, Escher, Bach (Part II)

[The following, except where otherwise noted, is by Douglas Hofstadter, from his book Gödel, Escher, Bach.]

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“From early childhood, I’d loved the idea of closing a cardboard box by tucking its four flaps over each other in a kind of ‘circular’ fashion. I’d also always loved standing between two mirrors and seeing the implied infinitude of images as they faded off into the distance. A mirror mirroring a mirror ― what could be more provocative? And I loved the picture of the Morton Salt girl holding a box of Morton Salt, with herself drawn on it, holding the box, and on and on, by implication, in ever-tinier copies, without any end, ever.

With the Morton Salt box, you may think you smell infinite regress, but if so, you’re fooling yourself! The girl’s arm is covering up the critical spot where the regress would occur. If you were to ask the girl to (please) hand you her salt box so that you could actually see the infinite regress on its label, you’d wind up disappointed, for the label on that box would show her holding a yet smaller box with her arm once again blocking the regress.”

[Pannenkoek] “Closing a cardboard box in that way has always reminded me of the full lotus yoga pose.”

[1]

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“The first time my parents wanted to buy a video camera, sometime in the 1970’s, I went to the store with them, and we asked to see what they had. We were escorted to an area of the store that had several TV screens on a shelf, and a video camera was plugged into the back of one of them, allowing us to see what the camera was looking at and to gauge its color accuracy and such things. I took the camera and pointed it at my father, and we saw his amused smile jump right up onto the screen. Next I pointed the camera at my own face and presto, there was I, up on the screen, replacing my father. But then, inevitably, I felt compelled to try pointing the camera at the TV screen itself.

Now comes the curious fact, which I’ll forever remember with some degree of shame: I was hesitant to close the loop! Instead of just going ahead and doing it, I balked and timidly asked the salesperson for permission to do so. Now why on earth would I have done such a thing? Well, perhaps it will help if I relate how he replied to my request. What he said was this: ‘No, no, no! Don’t do that ― you’ll break the camera!’ And how did I react to his sudden panic? With scorn? With laughter? Did I just go ahead and follow my whim anyway? No. The truth is, I wasn’t quite sure of myself, and his panicky outburst reinforced my vague uneasiness, so I held my desire in check and didn’t do it.

But danger or no danger, I remember well my hesitation at the store, and so I can easily imagine the salesperson’s panic, irrational though it was. Feedback ― making a system turn back or twist back on itself, thus forming some kind of mystically taboo loop ― seems to be dangerous, seems to be tempting fate, perhaps even to be intrinsically wrong, whatever that might mean. These are primal, irrational intuitions, and who knows where they come from. One might speculate that fear of any kind of feedback is just a simple, natural generalization from one’s experience with audio feedback, but I somehow doubt that the explanation is that simple. We all know that some tribes are fearful of mirrors, many societies are suspicious of cameras, and so forth. This suspicion of loops and of making representations of one’s self just runs in our human grain.”

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“I remember being particularly drawn in by a long footnote in which the authors asserted that, of the two sentences, ‘Chicago is a populous city,’ and, ‘Chicago is trisyllabic,’ ― the former is true but the latter is false. They explained that if one wishes to talk about properties of a word, one must use its name, which is the expression resulting from putting it inside quotes. Thus, the sentence ‘“Chicago” is trisyllabic,’ doesn’t concern a city, but its name, and states a truth. The authors went on to talk about the necessity of taking great care in making such distinctions inside formal reasoning, and pointed out that names themselves have names (made using quotation marks), and so on, ad infinitum.”

“When you discuss a word or a phrase, you conventionally put it in quotes. For example, I can say the word ‘philosopher’ has eleven letters. Here, I put ‘philosopher’ in quotes to show that I am speaking about the word ‘philosopher’ rather than about a philosopher in the flesh. This is called the use-mention distinction. Suppose I were to say to you, ‘Philosophers make lots of money.’ Here, I’d be using the word to manufacture an image in your mind of a twinkle-eyed sage with bulging moneybags. But when I put this word ― or any word ― in quotes, I subtract out its meaning and connotation, and am left only with some marks on paper, or some sounds. That’s called ‘mention’.”

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The Human Condition by by René Magritte

“Of the surrealists, René Magritte was the most conscious of the symbol-object mystery (which I see as a deep extension of the use-mention distinction). He uses it to evoke powerful responses in viewers, even if the viewers don’t verbalize the distinction this way. Magritte’s series of pipe paintings is fascinating and perplexing.”

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The Two Mysteries by René Magritte

“Consider The Two Mysteries. Focusing on the inner painting, you get the message that symbols and pipes are different. Then your glance moves upward to the ‘real’ pipe floating in the air ― you perceive that it’s real, while the other one is just a symbol. But that’s of course totally wrong: both of them are on the same flat surface before your eyes. The idea that one pipe is in a twice-nested painting, and therefore somehow ‘less real’ than the other pipe, is a complete fallacy. Once you’re willing to ‘enter the room’, you’ve already been tricked: you’ve fallen for image as reality. To be consistent in your gullibility, you should happily go one level further down, and confuse image-within-image with reality. The only way not to be sucked in is to see both pipes merely as colored smudges on a surface a few inches in front of your nose. Then, and only then, do you appreciate the full meaning of the written message, ‘Ceci n’est pas une pipe,’ meaning, ‘This is not a pipe.’ But ironically, at the very instant everything turns to smudges, the writing too turns to smudges, thereby losing its meaning! In other words, at that instant, the verbal message of the painting self-destructs in a Gödelian way.”

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“We’re all familiar with the Chinese proverb, ‘A picture is worth a thousand words.’ This saying is often evoked to argue that written words are inefficient ways to communicate. However, the complexity and efficiency of words lie beneath their surface. A word can provide vast amounts of data in a small space; it can generalize, talk of great abstractions like ‘evolution’, ‘progress’, and ‘renaissance’. Attempting to convey such concepts through images, for example, proves much more difficult, for images aren’t as efficient at communicating generalizations. Thus, it seems that, in some cases, a word might be worth a thousand pictures.” [2]

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“Several years after my encounter with the video camera salesman, my appreciation of the phenomenon of video feedback deepened considerably when I decided to explore it in detail at the Stanford University television studios. It was a piece of cake to point the camera at the screen, zoom in and out, tilt the camera, change angles, and so on.”

“an ‘infinite’ corridor, a truncated corridor (or failed self-engulfment), and a ‘pulsating’ petal pattern”

“The first thing I discovered was that there was a critical angle that determined whether the regress of nested screens was finite or infinite. When I pointed the camera at the right edge instead of the center of the screen, this gave me what looked like a snapshot of the right wall of a long corridor. But I wasn’t able to peer all the way down to the end. When I slowly panned the camera leftwards towards the center of the screen, all of a sudden, at a critical moment, there was a wonderful, dizzying sense of infinity as I’d find myself peering all the way down the corridor, stretching arbitrarily far away toward a single point of convergence ― the ‘vanishing point’.

At one point, I accidentally stuck my hand momentarily in front of the camera’s lens. Of course, the screen went all dark, but when I removed my hand, the previous pattern didn’t just pop right back onto the screen, as I expected. Instead, I saw a different pattern on the screen, but this one was unlike anything I’d seen before ― it wasn’t stationary but pulsating! Where had this mysterious pulsation come from, given that there was nothing in the room that was moving? Whoops ― I’m sorry, that’s not true ― there was something in the room that was moving. Do you know what it was, dear reader? Well, it was the image itself! Now that may strike you as a trivial or smart-alecky answer, but since the image was of itself (albeit at a slight delay), it’s in fact quite to the point. A faithful image of something changing will itself necessarily keep changing. And the original motion that had set things going had been my hand’s motion, of which this video reverberation now constituted a stable, self-sustaining visible memory trace.

This situation reminds me of the of phenomenon of reverberant barking. If a jogger passes one house and triggers one dog’s bark, then neighbor dogs may pick up the barking and a chain reaction involving a dozen dogs may ensue. Soon the barking party has taken on a life of its own, and in the meantime its unwitting instigator has long since exited the neighborhood. In the same way, the pattern on screen took on a life of its own long after my hand had moved out of the camera’s view.

Even today, all these years later, the origins of such pulsation remain quite unclear, even mysterious, to me. For that reason, it’s an emergent phenomenon. In general, an emergent phenomenon somehow emerges quite naturally and automatically from rigid rules operating at a lower, more basic level, but exactly how that emergence happens is not at all clear to the observer. The most unpredictable of the visual phenomena always seemed to happen right in the vicinity of that central point where the infinite regress converges down to a magical dot. My explorations of video feedback showed me that a very simple looping of processes gives rise to a family of truly unanticipated and incredibly intricate swirling patterns, and that I’d entered a far richer domain of possibilities than I’d expected.”

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This is the closest I could find to a video of the phenomenon Hofstadter is describing. Unfortunately, I don’t think it contains anything quite like the “pulsating petal pattern”.

[3]

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“The mysterious and strangely robust phenomena that emerge out of looping processes such as video feedback will serve from here on out as one of my main metaphors as I broach the central questions of consciousness and self. From my video voyages, I’ve learned that, very often, wonderfully complex structures and patterns come to exist on the screen whose origins are, to human viewers, utterly opaque. I’ve been struck by the fact that it’s the circularity ― the loopiness ― of the system that brings these patterns into existence and makes them persist. Once a pattern is on the screen, that’s all that’s needed to sustain its staying up there.

To put it another way, feedback gives rise to a new kind of abstract phenomenon that can be called ‘locking-in’. From just the barest hint comes, almost instantly (after perhaps twenty or thirty iterations), the full realization of all the implications of this hint ― and this new higher-level structure, this emergent pattern on the screen is then ‘locked in’ thanks to the loop. It won’t go away because it’s forever refreshing itself, feeding on itself, giving rebirth to itself. Otherwise put, the emergent output pattern is a self-stabilizing structure whose origins, despite the simplicity of the feedback loop itself, are nearly impenetrable because the loop is cycled through so many times.”

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“What will happen when two video feedback loop systems (A and B) are gradually brought close enough together to begin interacting with each other? Camera A will then see not only screen A but also screen B, and so loop B will enter into the content of loop A (and vice versa). Let’s assume, as would seem natural, that camera A is closer to screen A than it is to screen B (and vice versa). Then loop A will take up more space on screen A than does loop B, meaning more pixels, and so loop A will be reproduced with higher fidelity on screen A. Loop A will be large and fine-grained, while loop B will be small and coarse-grained. But that’s only on screen A. On screen B, everything is swapped. Each of these two loops now plays a role in defining the other one, though loop A plays a larger role in its own definition than does loop B (and vice versa).

We now have a metaphor for two individuals, each of whom has their own personal identity (i.e., their own private loop) ― and yet part of that personal identity is made out of, and is thus dependent upon, the personal identity of the other individual. Furthermore, the more faithful the image of each screen on the other one, the more the ‘private’ identities of the two loops are intertwined, and the more they start to be fused, blurred, and even, to coin a word, undisentanglable from each other.”

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“You can imagine a soul as being a detailed, elaborate pattern that exists very clearly in one brain. When a person dies, their original soul is no longer around. But there are other versions of it in other people’s brains, though those copies are less detailed and more coarse-grained.”

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“Although video feedback is very loopy and flirts with infinity, it has nothing paradoxical to it ― no more than does its simpler cousin, audio feedback. Although it’s a loop and it feels a bit strange, it’s not an example of a strange loop.”

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Hofstadter gives an analogy between the thought processes of non-human animals (who only have the most rudimentary concepts, if any) and the thought processes of human beings (whose concepts can be limitlessly combined together into new ones):

“This reminds me ― and I don’t think it’s a pure coincidence ― of the huge difference, in video feedback, between a truncated corridor and an infinite one.”

I’d slightly misremembered this analogy as likening the difference between the self-reflective capability of non-human animals and the self-reflective capability of human beings, to the difference between the truncated corridor created by two almost parallel mirrors and the infinite corridor created by two parallel ones.

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Citations

  1. ~Eka
  2. Robert Rosenstone & David Staley
  3. Randall Brown

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