Gödel, Escher, Bach Finale

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

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“Intricately woven together in my adolescent mind were the study of pattern (mathematics) and the study of paradoxes (metamathematics). I was somehow convinced that all the mysterious secrets with which I was obsessed would become crystal-clear to me once I’d deeply mastered these two intertwined disciplines. Although over the course of the next couple of decades, I lost essentially all of my faith in the notion that these disciplines contained (even implicitly) the answers to all these questions, one thing I never lost was my intuitive hunch that around the core of the eternal riddle ‘What am I?’, there swirled the ethereal vortex of Gödel’s incompleteness theorem.”

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“Language creates strange loops when it talks about itself, directly or indirectly. Here, something in the system jumps out and acts on the system, as if it were outside the system. What bothers us is perhaps an ill-defined sense of topological wrongness: the inside-outside distinction is being blurred. Even though the system is an abstraction, our minds use spatial imagery.”

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“The truly unusual day would be a day where nothing unusual happens.”

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“A fascinating area where hierarchies tangle is government ― particularly in the courts. Ordinarily, you think of two disputants arguing their cases in court, and the court adjudicating the matter. The court is on a different level from the disputants. But strange things can start to happen when the courts themselves get entangled in legal cases. Usually there’s a higher court which is outside the dispute. Even if two lower courts get involved in some sort of strange fight, with each one claiming jurisdiction over the other, some higher court is outside. But what happens when there is no higher court, and the Supreme Court itself gets all tangled up in legal troubles?

The irony is that once you reach this level, where you’re prevented from jumping out of the system to a yet higher authority, the only recourse is to forces which seem less well-defined by rules, but which are the only source of higher-level rules anyway: the lower-level rules, which in this case means the general reaction of society.”

When I was a little kid, my dad once asked me the riddle, “Who’s the president’s boss?” I think I accused him of giving me a trick question. But his answer was, “the people”.

It also reminds me of the chorus of the song No Church in the Wild: What’s a human being to a mob? What’s a mob to a king? What’s a king to a God? What’s a God to a non-believer?” [1]

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“Voting to decide which voting system should be used ― if there was disagreement, how could you even begin?”

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“Compromise may not always be possible. How can one compromise ‘fairly’ between right and wrong? Between fair and unfair? Between compromise and no compromise? These questions come up over and over again in disguise, in arguments about everyday things.”

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“The problem of wondering about one’s sanity ― how can you figure out if you’re sane? This is a strange loop indeed. Everyone knows that the insane interpret the world via their own peculiarly consistent logic. How can you tell if your own logic is ‘peculiar’ or not, given that you have only your own logic to judge itself? I don’t see any answer. I’m just reminded of Gödel’s second theorem, which implies that the versions of formal number theory which assert their own consistency are inconsistent.”

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“Consider the fact that you’re now experiencing death, in a curious way. Not being in Paris right now, you know what it’s like to be dead in Paris. No lights, no sounds ― nothing. The same goes for Timbuktu. In fact, you’re dead everywhere except for one small spot. Just think how close you are to being dead everywhere! And, in the same way, you’re dead in all other moments than right now. There’s just one small piece of spacetime in which you’re alive!”

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“Perhaps the greatest contradiction in our lives, and the hardest to handle, is the knowledge, ‘There was a time when I wasn’t alive, and there will come a time when I’m not alive.’ On one level, when you ‘step out of yourself’ and see yourself as ‘just another human being’, it makes complete sense. But on another level, perhaps a deeper level, personal nonexistence makes no sense at all. All that we know is embedded inside our minds, and for all that to be absent from the universe is not comprehensible. This is a basic undeniable problem of life ― it’s perhaps the best metaphorical analogue of Gödel’s incompleteness theorem.”

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Print Gallery by M. C. Escher

“A strikingly beautiful illustration of the cyclonic ‘eye’ of a tangled hierarchy is given to us by Escher in his Print Gallery. What we see is a picture gallery where a young man is standing, looking at a picture of a ship in the harbor of a small town, with its little turrets, occasional cupolas, and flat stone roofs, upon one of which sits a boy, relaxing in the heat, while two floors below him a woman ― perhaps his mother ― gazes out of the window from her apartment which sits directly above a picture gallery where a young man is standing, looking at a picture of a ship in the harbor of a small town… What!? We’re back on the same level as we began, though all logic dictates that we can’t be.

We, the observers of Print Gallery, are outside of the system. And when we look at the picture, we see things which the young man can certainly not see, such as Escher’s ‘MCE’, in the central ‘blemish’. Though the blemish seems like a defect, perhaps the defect lies in our expectations, for in fact Escher couldn’t have completed that portion of the picture without being inconsistent with the rules by which he was drawing the picture. That center of the whorl is ― and must be ― incomplete. Escher could’ve made it arbitrarily small, but he couldn’t have gotten rid of it. Thus we, on the outside, can know that Print Gallery is essentially incomplete ― a fact which the young man, on the inside, can never know.”

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[Pannenkoek] “What if Escher had drawn a bug so as to block out the center of the whorl? A bug flying through the air would make sense and look the same in any of those contexts. Wouldn’t the picture then be both consistent and complete?”

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“Could it be valid to suppose that the ‘magic’ of human consciousness somehow arises from the closing of a loop whereby the brain’s high level ― its symbol level ― and its low level ― its neurophysiological level ― are somehow tied together in an exquisite closed loop of causality? Is the ‘private I’ just the eye of a self-referential typhoon?”

“The important idea is that this ‘vortex’ of self is responsible for the tangledness of the mental processes. People have said to me on occasion, ‘This stuff with self-reference and so on is very amusing and enjoyable, but do you really think there’s anything serious to it?’ I certainly do. I think it’ll eventually turn out to be at the core of artificial intelligence, and the focus of all attempts to understand how human minds work.”

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“My feeling is that the process by which we decide what’s valid or what’s true is an art, and that it relies as deeply on a sense of beauty and simplicity as it does on rock-solid principles of logic or reasoning or anything else which can be objectively formalized.”

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“Artificial intelligence, when it reaches the level of human intelligence ― or even if it surpasses it ― will still be plagued by the problems of beauty and simplicity, and will run up against these things constantly in its own search for knowledge and understanding.”

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“It can’t be expressed with words, and it can’t be expressed without words.”

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“The drive to jump out of the system is a pervasive one, and lies behind all progress in art, technology, and other human endeavors.”

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Citations

  1. James Brown, Sean Carter, & Kanye West
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