Switch Statement

068: Gödel, Escher, Bach - Ch. 4: Hofstadter's Recursive Coffins

• Matthew Keller

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Matt:

Hello everyone And welcome to the switch statement podcast It's a podcast for investigations into miscellaneous tech topics

Jon:

This is our sixth episode on Gödel, Escher, Bach by Douglas Hofstetter.

Matt:

Hey John, how are you doing?

Jon:

Hey Matt, I'm doing pretty good. How are you?

Matt:

I'm doing alright, I'm just having all of meaning just broken for me by this chapter of Girdle Escher Bach.

Jon:

For real. Yeah, no, this guy, Douglas Hofstadter, he really knows how to like make you reinterpret the entire universe.

Matt:

So I have to say I've been trying to skirt the dialogues in this book, probably to my own detriment, but But anyway,

Jon:

But this one you can't skip. This one was vital. He references it in the chapter several times.

Matt:

well, I've been getting pretty good at just ignoring words that I don't understand in this book. So I could have just walked past contra across Ticus, kiss uh,

Jon:

Cross Depunctus.

Matt:

Contra Costus.

Jon:

Which, by the way, is that a word? I, I, like, didn't look it up or anything.

Matt:

I looked it up it's not a word.

Jon:

Oh, okay, okay.

Matt:

so contrapunctus is like the German word for counterpoint, he references, uh, Bach's art of fugue

Jon:

Yeah, yeah. Contrapuntal melodies.

Matt:

That's broken down, the sections of that are broken down into Contrapunctus 1, and maybe it's Latin, I don't know. But anyway, Contrapunctus 1, Contrapunctus 2, to Contrapunctus Like 30 or something. It's very long, it's like an hour and a half. Um, so, but anyway, it's actually kind of like It's a portmanteau, but it's like Instead of just combining two words at like the front and back You like, jammed. One word in the middle of another word. Um, so he put the word acrostic in there and I don't know, did you see this? The whole dialogue was an acrostic.

Jon:

What is an acrostic?

Matt:

Okay. So an acrostic is a, it's not really a poem, but it's, I think you could call it a poem where the first letter of every line spells something like, so if you look at the first letter of every line of an acrostic and you put them together in sequence, it spells something out.

Jon:

Oh, holy shit. Like when you're resigning from a job, but you want to say, like, fuck you to the whole team. So you, like, start every sentence with F U C K. Okay. Nice. Dude, what does it spell?

Matt:

So it, it spells Hofstadter's Contra Acrostic, uh, Acrostipuntus backwards spells JS Bach, which I don't know what that's even supposed to mean.

Jon:

What?

Matt:

I know like, let me, uh,

Jon:

Because it doesn't spell Bok, JS Bok backwards.

Matt:

There must be some deeper meaning to that. Let me see if I can, uh,

Jon:

Maybe while you're pulling that up, I can just sort of describe the dialogue because I feel like we haven't even described it yet. So this dialogue, which is just as zany and weird as the other ones, although a little bit less weird than the last one, which was just completely off the wall. But this one. Was about the the tortoise and Achilles who are kind of the main characters of all of these dialogues having a record player and Actually crab is the one who had the record player, right and Tortoise was giving crab these records and the records were destroying His record player when he played them. So basically tortoise this ultimate jerk destroying this guy's record players.

Matt:

Yeah.

Jon:

what he's doing is he's like, you know, he's looking at the record player and he's saying, Oh, I can devise this series of sound waves that will like cause this record player to explode. Uh, and yeah, he just keeps doing that to crab and breaking his record players. But then there's a little twist thrown in where, uh, Achilles gives, uh, It's a tortoise, this goblet, it's a glass goblet, and he says it was, it was blown by J. S. Bach, Johann Sebastian Bach, um, and, and then he starts talking about this Bach choral that actually has Bach's name encoded into it, which, this was crazy by the way, like, I, I did not look up and verify that this was true, but this is so cool, like basically in German music. H is a note. So, you know,

Matt:

that's cool.

Jon:

yeah, which I, I mean, he said this in the, in the dialogue, so I don't know if this is true or not, but basically I think it was like B is H and B flat is B something along those lines. I might be getting that slightly wrong, but in any case, you can actually spell Bach's name with musical notes. And he was describing how one of Bach's final pieces, he. put his own name in the piece. This whole entire book is about self reference and here Bach goes like referencing himself in his final piece. Um, anyway, I'm taking way too long to tell this.

Matt:

No, that was a perfect amount of time. Um, because I just figured it out. All right. Well, I just went back and I confirmed there was a point, there was a word that I was missing, but I, it still doesn't make any sense. It says Hofstadter's contra cross to Punticus, uh, cross tickly backwards spells J S Bach,

Jon:

Acrostically,

Matt:

cross tickly backwards means.

Jon:

maybe it itself is an acrostic or something. I don't know. I don't know.

Matt:

Oh, oh, oh, oh, of course, because that, because it has its own, it has its own, it's a, it's a, it's a recursive acrostic.

Jon:

Yeah, everything Douglas Hofstetter does has to reference itself and be recursive.

Matt:

Right. Okay. So, but then let's just look at this. So what he's saying is, so the first word is Hofstadter starting with an H. The second word is contra acros de punticus. The third word is acrostically. Fourth word is backwards, so it's,

Jon:

J. S.

Matt:

H, H, C, uh, A, B, and then, yeah, spells, is S, and then J, S, Bach.

Jon:

Hofstetter, this clever weasel. They're gonna, when he dies, they're gonna open his coffin, it's gonna be like another little coffin inside it.

Matt:

Another Coffin inside, yeah.

Jon:

anyway, so I was just getting to the very end of the the dialogue. So at the very end, they put this record on from Bach and the record destroys the goblet. So it's got this nice little twist at the end. And also the tortoise gets his comeuppance, which for me, I mean the whole dialogue, I was just like, come on tortoise, like you're breaking your friend's phonographs. Like this sounds like a really expensive thing and he just keeps breaking them over and over.

Matt:

Well, I mean, this is just, you know, it's the hubris of the Of the crab. It's like, come on, crab. what we find, is the tortoises, is actually girdle.

Jon:

yes, exactly. So that was the hidden kind of like, this is a funny thing about this chapter. Cause like, you know, I read this dialogue and I was like, cool, far out. But then he, he continually references the dialogue within the contents of the chapter. And there was so much hidden meaning in this dialogue. Like you just described the whole acrostic thing. Literally all of that went over my head. I thought this was just like a funny story, but it's actually a decent allegory for Gödel's incompleteness.

Matt:

I think we've raised the concept of isomorphisms before. But this chapter. dives even deeper into isomorphism and just to like, cause I feel like I still don't even have a great handle on isomorphisms, but it really is a mapping,

Jon:

Yeah.

Matt:

you know? So if we're talking about the isomorphism between the story and Godel's theory of incompleteness, the phonograph in this story is, Russell and Whitehead's Principia Mathematica. Like it's an axiomatic system for number theory.

Jon:

exactly.

Matt:

And they are the crabs, you know, if we're going even deeper, you know, Russell and Whitehead are the crab where we're like, this is a great record player. We spent so much money on this, so much time creating this axiomatic number theory,

Jon:

it's, high fidelity. He uses the terms high fidelity and low fidelity, which I think high fidelity was to describe like a complete system. that can basically represent all true statements as theorems in the system.

Matt:

Right. Exactly. Exactly. And then there's an out where you can make it, you can make a phonograph that doesn't break. Like you can't play a record that will break it, but you actually like give up the ability to play certain sounds or like represent certain, statements.

Jon:

exactly. Yeah. Like you have lower fidelity, so the sound won't come, come out clearly, which Literally all of that. Like I did not, I did not get, but it was kind of cool to have it revealed within the chapter. Cause it was, you know, I kind of had these like aha moments.

Matt:

Yeah. Yeah. Yeah.

Jon:

He drops an amazing fact right out of the gate in this chapter, which is really, I mean, at least for me, it was really brain breaking. He talks about how human language itself is kind of an isomorph where people attribute meanings to words. Like, I am saying words to you right now, and it seems like the words themselves are the meaning, but in actuality it's like the meaning is this weird, you know, nebulous thing that's somehow represented in your brain and in my brain. The words are just this, like, weird medium that we attempt to convey meaning through. And again, I feel like I'm smoking right now, but it's really reading that section of the chapter, Was, was completely like mind blowing. Like it was like, Whoa, words don't actually have meaning. Like within, you know, within the combination of words, we're like conveying these complex meanings and it becomes very important later.

Matt:

it feels like the world is just like infinitesimally many handoffs of like isomorphisms, which is like, it's amazing that we can talk at all because it's like, there's the representation of a word in our brains. And then there's the isomorphism between like our, Like throat meat and like the sound

Jon:

throat meat.

Matt:

throat meat and then like yeah, the isomorphism between the sound and like your ears and they're vibrating and creating electricity and then the isomorphism between the electricity and so it's like I it's turtles all the way down basically Yeah

Jon:

you know, just to bring it back to modern reality, big thing these days is large language models. And there's a thing in a large language model called an embedding, which is essentially a tremendous, uh, you know, vector, which is essentially a list of floating point values. And the embedding represents meaning. So let's say I have an image of a rabbit. That image will have an embedding in the large language model. And let's say you, let's say you have another image of like a dog and you take the embedding of the dog and you subtract the embedding of the rabbit. You will now have a directional embedding that represents transitioning from rabbit to dog. So let's say you have a different picture of a rabbit. And you apply that same directional embedding, you will, you will basically transition that picture into a dog. So say you have a rabbit jumping, you can get a dog jumping. Say you have a rabbit swimming, you can get a dog swimming. So it's, it's just this completely insane, you know, true thing that's happening today. And I, I feel like it kind of connects to this book. Cause he's talking about how the words themselves aren't conveying the meaning. There's some underlying weird meaning that none of us really get. And it's kind of like an embedding within a large language model.

Matt:

It's so weird because when you look at any one piece of these, of this thing, and I think this is his point, it's like, it's completely meaningless. You like, you look at one of those numbers or even you look at the whole array of numbers.

Jon:

Yeah,

Matt:

It's like that array does not mean anything aside from its relationship to everything else, you know? And it's like, where is the meaning?

Jon:

right.

Matt:

I guess Hofstadter would say that the meaning is in the mapping.

Jon:

Oh, the meaning is in the mapping, dude. I think you're onto something.

Matt:

That's what I'm, I'm getting from it. Like nothing has any meaning aside from the establishment of an isomorphism between its parts and something else. Um,

Jon:

gives a brief story of Bach. Which I just love learning a little bit more about Bach because I absolutely love Bach. I think he's one of the greatest composers to ever live. Um, but he tells kind of the sad story of Bach. Like apparently Bach was losing his eyesight and so he got this operation, but the operation basically made him completely lose his eyesight. So for the last, uh, I can't remember how long it was, but for the last like period of Bach's life, he was And he had his son in law, who I think was Carl Philip Emanuel Bach, help him write music. And going back to that chorale that he mentioned during the dialogue, that was actually written, you know, the notes were written by Bach's son in law. Bach was dictating to Carl Philip Emanuel. Um, but then evidently, Bach did regain his vision at a certain point. But then very shortly after he regained his vision, he had a stroke. And then ten days later, he died.

Matt:

Literally a few hours after he regained his vision. So

Jon:

Yeah.

Matt:

about, talk about just like brutal, yeah.

Jon:

already, and now you're gonna make Bach blind? Like, come on. Although, I just mixed up the chronology, but you get what I'm saying. Yeah, there was a, there

Matt:

there's a section called problems caused by girdles result.

Jon:

Maybe I can just ask you live on the podcast. Uh, he mentions like Gödel's reasoning methods can't be encapsulated. And he mentions that it's a difference between mechanical and human reasoning.

Matt:

Yeah.

Jon:

And I feel like I kind of get that, but I don't know. I don't know if you have something, you know, that you would want to add to that or like, uh, yeah.

Matt:

I mean, I think it's actually necessary for. Like it's a necessary aspect of his theory of incompleteness. Yeah. That it has to not be able to be encapsulated because if it could, it wouldn't work. You know what I mean?

Jon:

Yeah, so does, does that refer to it being human reasoning? Because mechanical reasoning would be fully encapsulated within the system. So you kind of have to make this like, quote unquote, human leap. Yeah, I guess maybe I'm, maybe I'm wrong about that, but that's where I kind of was lost.

Matt:

Well, this is what makes it so hard to reason about is like, how have we arrived at something that we feel confident has been proven true, but a formal system can't incorporate it. I just don't understand how that, how that could be possible.

Jon:

Yeah, no, I had the exact same misunderstanding, and I think, I think that's what he's referring to as human reasoning.

Matt:

But like, I think maybe the claim is like human reasoning is not a formal system.

Jon:

Right, exactly.

Matt:

Like it's this larger thing. Like it's this more basic thing that just arises out of neurons.

Jon:

Yeah, yeah, it's like this nuanced kind of, like, I feel like formal systems sort of require kind of an element of rigorousness, where it's like everything is quantifiable, nothing is probabilistic. Yeah, probabilistic. But in human reasoning, like, my opinion on something changes every day, or like, my understanding of a concept changes or gets buttressed or whatever. That was at least my understanding of it. It's like, Gödel's method's not being able to be encapsulated within the system. is like an instance of him having this like more human style reasoning that almost like by definition breaks out of the system.

Matt:

Yeah.

Jon:

I might be like, uh, reaching a bit on that.

Matt:

No, I like my, my tendency is that like, there should, there should be a way to like have this system that undergirds the formal system that can encapsulate it. But like, I guess you can continue the pattern, you know, like you could just keep on doing that.

Jon:

Yeah.

Matt:

And that's what happened to Russell and Whitehead, right? They had all these layers and like, so. you're always going to be able to like break out of the system, I think.

Jon:

Right. Yeah. Which I think Gödel has, has proven.

Matt:

so we, returned to the PQ system and just as a reminder, um, PQ is like the previous kind of string manipulation rule that we had was an between Like a single addition operator. So you had a certain number of dashes. You had the letter P you had a certain number of dashes. You had the letter Q and then you had the sum of the numbers of two, the two strings of dashes.

Jon:

So basically P is plus and Q is equals.

Matt:

Exactly. Um,

Jon:

But he introduced a new, uh, uh, rule, or

Matt:

he modifies it. So it's actually a replacement. It's not, um, I don't think it's like an addition to the previous system.

Jon:

Oh, right, because it kind of violates the previous system in a way. But in any case, he, he updates the system so that Q means less than, right? Less than or equal to? I don't know.

Matt:

eventually we see that, but, what he's saying is we've modified the system then what we say is like, Oh, this system is inconsistent. We're like, Oh shoot. Like it's now producing like incorrect, uh, theorems.

Jon:

Oh, right. Because it's like one and one is like three or something,

Matt:

yeah. So did we, did we say what the new rule was? Uh, so basically he had his new axiom schema is if X is a hyphen string, then X P hyphen Q X is an axiom. And so, but this allows you to create thing, create strings that. You know, for which, the previous edition isomorphism doesn't hold anymore.

Jon:

right?

Matt:

but he goes on to say that like, this is, this is an invalid way to look at, at it. Um, he basically says, as soon as you're modifying the system, you kind of have to discard any previous isomorphism.

Jon:

Yeah, yeah. And that's the point where he comes up with this new isomorphism that Q is no longer equals. It is now less than or equals.

Matt:

Right, exactly, exactly. So, and I think the point being that it's easy to want to modify things and then think about it in terms of the historical system and see whether or not it's right anymore.

Jon:

Yeah,

Matt:

But that's just not a valid approach.

Jon:

right, right. Um, I feel like there was a similar notion in one of the previous chapters. Where he was just talking about how something might seem like an isomorph and you can make these logical leaps because of it, you know, like let's say a lexical system, you know, seems like it's behaving like math and then you suddenly start to use math to like work with it, like you have made an invalid logical leap and you know, the system just might not be compatible with that, which was a. An idea that I thought was really cool from one of the previous chapters.

Matt:

Yeah, I mean, I think that I think the example that he used was we established this isomorphism between the string and addition in the last one of the last chapters. So he said that there might be a. An urge to, like, go back the other direction. And, And, make a string that's like, Oh, you say, Oh, I can make an addition statement. That's like two plus two plus two equals six. So then maybe I can do dash, dash P dash, dash P dash, dash P Q, you know, dash, dash, dash, dash, or whatever it is. But he, his point was like, that is, that's an active meaning. I think that that was the distinction. There's like a passive. meaning and an active meaning. And that is you're actively like pushing meaning from this discovered isomorphism to the old system in a way that it actually doesn't support.

Jon:

exactly. And you're generating a theorem that can't actually be produced with the rules of the system. He mentions that your meaning has to always remain passive. Like it's, it's a mistake to move into this active meaning mode.

Matt:

Um, did you have to ever have to do those things in your computer science classes where you. Had to prove something NP complete.

Jon:

No, no.

Matt:

Um, Oh, that's crazy. I feel like there was one class where, um, where that was just like, all we did for a long time was prove that things were NP complete.

Jon:

Really?

Matt:

The technique that they proposed was, you know, someone out there proved. Some like series of problems, NP complete at some point in the past, right? Well, the, the technique to, to prove a new thing, NP complete is to assume you have a black box solution to the new thing. Demonstrate that you could solve one of the existing NP complete problems.

Jon:

Oh, so you're like using an existing NP complete problem to like prove that a new It's like a, it almost sounds like it would have a chicken and egg issue.

Matt:

Well, this is, this is what was so like brain breaking about this exercise is, is someone is like, Oh, like, is the problem of grouping nodes in this graph, like in sets of three and P complete. The, the like trick you would have to do is you would have to start by assuming you already have a black box algorithm to solve that problem.

Jon:

Yeah,

Matt:

And then you would have to. transform the statement of a known NP complete problem and map it onto the inputs to this new black box that you have and demonstrate that if you had a black box algorithm that could solve that problem that it's just as hard as the NP complete problem because you were able to reduce a known NP complete problem to a formulation of this. new proposed problem. Anyway,

Jon:

had done this in

Matt:

so,

Jon:

fascinating. Yeah.

Matt:

so I feel like I've lost everybody at this point, but the point, the point was that like, there's this important directionality there where like, sometimes it feels like you should be doing it the other direction. It's like, wait a minute. shouldn't I be proving that I can use the NP, the known NP complete thing to solve the. Other problem, but no, it's like you need to, you need the direction to like flow the other, the other way. So anyway, that was all a very long and potentially, uh,

Jon:

Well, it's interesting. I feel like, I feel like that idea might apply to some of the sections, some of the later sections of this chapter, which I guess we'll cover in our next episode.

Matt:

yeah, it's proving an isomorphism. You're like demonstrating that these problems are isomorphic to one another, or at this problem is at least as difficult as this other problem. Yeah.

Jon:

this episode, just to wrap up my notes. I found this, like, PQ system redefinition to be very unsatisfying when I read it. Like, it sort of, you know, the original PQ system I was kind of happy with. It felt like this, you know, system that worked really well and was kind of bulletproof. But I didn't really understand why it was unsatisfying to me. But he sort of describes later in the chapter why it's unsatisfying. Um, and I'll leave, this episode with that cliffhanger.

Matt:

Alright, well, you'll have to tune in next week to know why is the revised PQ system unsatisfying? alright, well, I'll see you next time for the second half of this chapter.

Jon:

See you next time, Matt.