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Hello everyone And welcome to the switch statement podcast It's a podcast for investigations into miscellaneous tech topics
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This is our sixth episode on Gödel, Escher, Bach by Douglas Hofstetter.
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Hey John, how are you doing?
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Hey Matt, I'm doing pretty good.
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How are you?
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I'm doing alright, I'm just having all of meaning just broken for me by this chapter of Girdle Escher Bach.
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For real.
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Yeah, no, this guy, Douglas Hofstadter, he really knows how to like make you reinterpret the entire universe.
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So I have to say I've been trying to skirt the dialogues in this book, probably to my own detriment, but But anyway,
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But this one you can't skip.
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This one was vital.
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He references it in the chapter several times.
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well, I've been getting pretty good at just ignoring words that I don't understand in this book.
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So I could have just walked past contra across Ticus, kiss uh,
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Cross Depunctus.
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Contra Costus.
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Which, by the way, is that a word?
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I, I, like, didn't look it up or anything.
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I looked it up it's not a word.
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Oh, okay, okay.
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so contrapunctus is like the German word for counterpoint, he references, uh, Bach's art of fugue
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Yeah, yeah.
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Contrapuntal melodies.
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That's broken down, the sections of that are broken down into Contrapunctus 1, and maybe it's Latin, I don't know.
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But anyway, Contrapunctus 1, Contrapunctus 2, to Contrapunctus Like 30 or something.
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It's very long, it's like an hour and a half.
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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.
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One word in the middle of another word.
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Um, so he put the word acrostic in there and I don't know, did you see this?
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The whole dialogue was an acrostic.
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What is an acrostic?
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Okay.
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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.
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Oh, holy shit.
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Like when you're resigning from a job, but you want to say, like, fuck you to the whole team.
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So you, like, start every sentence with F U C K.
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Okay.
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Nice.
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Dude, what does it spell?
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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.
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What?
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I know like, let me, uh,
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Because it doesn't spell Bok, JS Bok backwards.
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There must be some deeper meaning to that.
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Let me see if I can, uh,
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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.
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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.
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But this one.
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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.
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So basically tortoise this ultimate jerk destroying this guy's record players.
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Yeah.
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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.
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Uh, and yeah, he just keeps doing that to crab and breaking his record players.
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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.
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S.
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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.
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H is a note.
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So, you know,
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that's cool.
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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.
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I might be getting that slightly wrong, but in any case, you can actually spell Bach's name with musical notes.
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And he was describing how one of Bach's final pieces, he.
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put his own name in the piece.
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This whole entire book is about self reference and here Bach goes like referencing himself in his final piece.
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Um, anyway, I'm taking way too long to tell this.
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No, that was a perfect amount of time.
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Um, because I just figured it out.
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All right.
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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.
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It says Hofstadter's contra cross to Punticus, uh, cross tickly backwards spells J S Bach,
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Acrostically,
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cross tickly backwards means.
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maybe it itself is an acrostic or something.
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I don't know.
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I don't know.
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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.
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Yeah, everything Douglas Hofstetter does has to reference itself and be recursive.
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Right.
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Okay.
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So, but then let's just look at this.
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So what he's saying is, so the first word is Hofstadter starting with an H.
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The second word is contra acros de punticus.
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The third word is acrostically.
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Fourth word is backwards, so it's,
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J.
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S.
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H, H, C, uh, A, B, and then, yeah, spells, is S, and then J, S, Bach.
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Hofstetter, this clever weasel.
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They're gonna, when he dies, they're gonna open his coffin, it's gonna be like another little coffin inside it.
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Another Coffin inside, yeah.
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anyway, so I was just getting to the very end of the the dialogue.
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So at the very end, they put this record on from Bach and the record destroys the goblet.
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So it's got this nice little twist at the end.
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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.
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Like this sounds like a really expensive thing and he just keeps breaking them over and over.
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Well, I mean, this is just, you know, it's the hubris of the Of the crab.
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It's like, come on, crab.
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what we find, is the tortoises, is actually girdle.
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yes, exactly.
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So that was the hidden kind of like, this is a funny thing about this chapter.
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Cause like, you know, I read this dialogue and I was like, cool, far out.
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But then he, he continually references the dialogue within the contents of the chapter.
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And there was so much hidden meaning in this dialogue.
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Like you just described the whole acrostic thing.
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Literally all of that went over my head.
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I thought this was just like a funny story, but it's actually a decent allegory for Gödel's incompleteness.
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I think we've raised the concept of isomorphisms before.
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But this chapter.
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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,
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Yeah.
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you know?
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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.
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Like it's an axiomatic system for number theory.
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exactly.
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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.
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We spent so much money on this, so much time creating this axiomatic number theory,
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it's, high fidelity.
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He uses the terms high fidelity and low fidelity, which I think high fidelity was to describe like a complete system.
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that can basically represent all true statements as theorems in the system.
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Right.
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Exactly.
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Exactly.
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And then there's an out where you can make it, you can make a phonograph that doesn't break.
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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.
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exactly.
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Yeah.
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Like you have lower fidelity, so the sound won't come, come out clearly, which Literally all of that.
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Like I did not, I did not get, but it was kind of cool to have it revealed within the chapter.
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Cause it was, you know, I kind of had these like aha moments.
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Yeah.
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Yeah.
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Yeah.
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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.
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He talks about how human language itself is kind of an isomorph where people attribute meanings to words.
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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.
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The words are just this, like, weird medium that we attempt to convey meaning through.
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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.
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Like it was like, Whoa, words don't actually have meaning.
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Like within, you know, within the combination of words, we're like conveying these complex meanings and it becomes very important later.
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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.
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And then there's the isomorphism between like our, Like throat meat and like the sound
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throat meat.
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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
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you know, just to bring it back to modern reality, big thing these days is large language models.
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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.
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And the embedding represents meaning.
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So let's say I have an image of a rabbit.
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That image will have an embedding in the large language model.
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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.
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You will now have a directional embedding that represents transitioning from rabbit to dog.
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So let's say you have a different picture of a rabbit.
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And you apply that same directional embedding, you will, you will basically transition that picture into a dog.
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So say you have a rabbit jumping, you can get a dog jumping.
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Say you have a rabbit swimming, you can get a dog swimming.
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So it's, it's just this completely insane, you know, true thing that's happening today.
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And I, I feel like it kind of connects to this book.
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Cause he's talking about how the words themselves aren't conveying the meaning.
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There's some underlying weird meaning that none of us really get.
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And it's kind of like an embedding within a large language model.
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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.
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You like, you look at one of those numbers or even you look at the whole array of numbers.
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Yeah,
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It's like that array does not mean anything aside from its relationship to everything else, you know?
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And it's like, where is the meaning?
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right.
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I guess Hofstadter would say that the meaning is in the mapping.
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Oh, the meaning is in the mapping, dude.
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I think you're onto something.
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That's what I'm, I'm getting from it.
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Like nothing has any meaning aside from the establishment of an isomorphism between its parts and something else.
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Um,
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gives a brief story of Bach.
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Which I just love learning a little bit more about Bach because I absolutely love Bach.
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I think he's one of the greatest composers to ever live.
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Um, but he tells kind of the sad story of Bach.
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Like apparently Bach was losing his eyesight and so he got this operation, but the operation basically made him completely lose his eyesight.
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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.
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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.
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Bach was dictating to Carl Philip Emanuel.
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Um, but then evidently, Bach did regain his vision at a certain point.
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But then very shortly after he regained his vision, he had a stroke.
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And then ten days later, he died.
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Literally a few hours after he regained his vision.
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So
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Yeah.
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about, talk about just like brutal, yeah.
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already, and now you're gonna make Bach blind?
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Like, come on.
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Although, I just mixed up the chronology, but you get what I'm saying.
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Yeah, there was a, there
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there's a section called problems caused by girdles result.
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Maybe I can just ask you live on the podcast.
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Uh, he mentions like Gödel's reasoning methods can't be encapsulated.
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And he mentions that it's a difference between mechanical and human reasoning.
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Yeah.
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And I feel like I kind of get that, but I don't know.
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I don't know if you have something, you know, that you would want to add to that or like, uh, yeah.
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I mean, I think it's actually necessary for.
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Like it's a necessary aspect of his theory of incompleteness.
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Yeah.
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That it has to not be able to be encapsulated because if it could, it wouldn't work.
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You know what I mean?
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Yeah, so does, does that refer to it being human reasoning?
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Because mechanical reasoning would be fully encapsulated within the system.
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So you kind of have to make this like, quote unquote, human leap.
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Yeah, I guess maybe I'm, maybe I'm wrong about that, but that's where I kind of was lost.
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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.
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I just don't understand how that, how that could be possible.
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Yeah, no, I had the exact same misunderstanding, and I think, I think that's what he's referring to as human reasoning.
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But like, I think maybe the claim is like human reasoning is not a formal system.
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Right, exactly.
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Like it's this larger thing.
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Like it's this more basic thing that just arises out of neurons.
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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.
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Yeah, probabilistic.
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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.
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That was at least my understanding of it.
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It's like, Gödel's method's not being able to be encapsulated within the system.
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is like an instance of him having this like more human style reasoning that almost like by definition breaks out of the system.
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Yeah.
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I might be like, uh, reaching a bit on that.
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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.
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But like, I guess you can continue the pattern, you know, like you could just keep on doing that.
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Yeah.
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And that's what happened to Russell and Whitehead, right?
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They had all these layers and like, so.
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you're always going to be able to like break out of the system, I think.
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Right.
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Yeah.
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Which I think Gödel has, has proven.
00:19:27.132 --> 00:19:42.979
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.
00:19:43.038 --> 00:19:45.219
So you had a certain number of dashes.
00:19:45.308 --> 00:19:48.078
You had the letter P you had a certain number of dashes.
00:19:48.078 --> 00:19:54.118
You had the letter Q and then you had the sum of the numbers of two, the two strings of dashes.
00:19:54.969 --> 00:19:58.229
So basically P is plus and Q is equals.
00:19:58.868 --> 00:19:59.398
Exactly.
00:20:00.179 --> 00:20:00.648
Um,
00:20:00.911 --> 00:20:04.941
But he introduced a new, uh, uh, rule, or
00:20:05.368 --> 00:20:06.449
he modifies it.
00:20:06.509 --> 00:20:08.328
So it's actually a replacement.
00:20:08.328 --> 00:20:12.628
It's not, um, I don't think it's like an addition to the previous system.
00:20:12.808 --> 00:20:15.618
Oh, right, because it kind of violates the previous system in a way.
00:20:15.618 --> 00:20:23.519
But in any case, he, he updates the system so that Q means less than, right?
00:20:23.519 --> 00:20:24.568
Less than or equal to?
00:20:24.568 --> 00:20:24.627
I don't know.
00:20:27.479 --> 00:20:36.012
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.
00:20:36.373 --> 00:20:37.182
We're like, Oh shoot.
00:20:37.212 --> 00:20:43.262
Like it's now producing like incorrect, uh, theorems.
00:20:44.732 --> 00:20:45.282
Oh, right.
00:20:45.292 --> 00:20:49.482
Because it's like one and one is like three or something,
00:20:50.198 --> 00:20:50.567
yeah.
00:20:50.567 --> 00:20:53.468
So did we, did we say what the new rule was?
00:20:53.657 --> 00:21:07.488
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.
00:21:09.347 --> 00:21:14.298
And so, but this allows you to create thing, create strings that.
00:21:15.692 --> 00:21:20.732
You know, for which, the previous edition isomorphism doesn't hold anymore.
00:21:21.603 --> 00:21:22.022
right?
00:21:23.377 --> 00:21:28.587
but he goes on to say that like, this is, this is an invalid way to look at, at it.
00:21:29.817 --> 00:21:38.248
Um, he basically says, as soon as you're modifying the system, you kind of have to discard any previous isomorphism.
00:21:39.597 --> 00:21:40.637
Yeah, yeah.
00:21:41.018 --> 00:21:46.688
And that's the point where he comes up with this new isomorphism that Q is no longer equals.
00:21:46.748 --> 00:21:48.887
It is now less than or equals.
00:21:50.282 --> 00:21:51.913
Right, exactly, exactly.
00:21:51.923 --> 00:22:05.432
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.
00:22:05.968 --> 00:22:06.288
Yeah,
00:22:07.542 --> 00:22:09.303
But that's just not a valid approach.
00:22:10.258 --> 00:22:11.498
right, right.
00:22:12.067 --> 00:22:17.097
Um, I feel like there was a similar notion in one of the previous chapters.
00:22:18.083 --> 00:22:44.292
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.
00:22:44.607 --> 00:22:47.887
An idea that I thought was really cool from one of the previous chapters.
00:22:48.798 --> 00:23:01.178
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.
00:23:02.528 --> 00:23:05.218
So he said that there might be a.
00:23:06.147 --> 00:23:09.097
An urge to, like, go back the other direction.
00:23:09.853 --> 00:23:15.432
And, And, make a string that's like, Oh, you say, Oh, I can make an addition statement.
00:23:15.432 --> 00:23:18.982
That's like two plus two plus two equals six.
00:23:19.643 --> 00:23:27.833
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.
00:23:28.823 --> 00:23:35.202
But he, his point was like, that is, that's an active meaning.
00:23:35.202 --> 00:23:36.532
I think that that was the distinction.
00:23:36.532 --> 00:23:37.492
There's like a passive.
00:23:38.393 --> 00:23:39.782
meaning and an active meaning.
00:23:39.782 --> 00:23:49.873
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.
00:23:50.117 --> 00:23:50.788
exactly.
00:23:50.807 --> 00:23:56.407
And you're generating a theorem that can't actually be produced with the rules of the system.
00:23:56.962 --> 00:24:00.712
He mentions that your meaning has to always remain passive.
00:24:01.262 --> 00:24:04.893
Like it's, it's a mistake to move into this active meaning mode.
00:24:05.998 --> 00:24:11.891
Um, did you have to ever have to do those things in your computer science classes where you.
00:24:12.089 --> 00:24:14.859
Had to prove something NP complete.
00:24:16.464 --> 00:24:17.674
No, no.
00:24:18.640 --> 00:24:19.950
Um, Oh, that's crazy.
00:24:20.000 --> 00:24:28.579
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.
00:24:29.295 --> 00:24:29.884
Really?
00:24:30.180 --> 00:24:36.329
The technique that they proposed was, you know, someone out there proved.
00:24:36.704 --> 00:24:41.184
Some like series of problems, NP complete at some point in the past, right?
00:24:41.734 --> 00:24:53.605
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.
00:24:54.994 --> 00:25:01.894
Demonstrate that you could solve one of the existing NP complete problems.
00:25:03.525 --> 00:25:13.954
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.
00:25:14.154 --> 00:25:30.224
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.
00:25:31.595 --> 00:25:39.115
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.
00:25:39.894 --> 00:25:40.244
Yeah,
00:25:40.835 --> 00:25:42.654
And then you would have to.
00:25:43.684 --> 00:26:09.785
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.
00:26:10.444 --> 00:26:12.045
new proposed problem.
00:26:12.045 --> 00:26:15.654
Anyway,
00:26:15.724 --> 00:26:16.785
had done this in
00:26:17.035 --> 00:26:17.454
so,
00:26:17.634 --> 00:26:18.335
fascinating.
00:26:18.335 --> 00:26:18.404
Yeah.
00:26:18.545 --> 00:26:30.505
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.
00:26:30.505 --> 00:26:31.404
It's like, wait a minute.
00:26:31.404 --> 00:26:36.162
shouldn't I be proving that I can use the NP, the known NP complete thing to solve the.
00:26:37.057 --> 00:26:43.906
Other problem, but no, it's like you need to, you need the direction to like flow the other, the other way.
00:26:43.997 --> 00:26:48.196
So anyway, that was all a very long and potentially, uh,
00:26:48.507 --> 00:26:49.336
Well, it's interesting.
00:26:49.336 --> 00:26:59.106
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.
00:26:59.346 --> 00:27:01.267
yeah, it's proving an isomorphism.
00:27:01.317 --> 00:27:08.936
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.
00:27:08.936 --> 00:27:11.326
Yeah.
00:27:11.457 --> 00:27:13.797
this episode, just to wrap up my notes.
00:27:14.832 --> 00:27:21.521
I found this, like, PQ system redefinition to be very unsatisfying when I read it.
00:27:22.031 --> 00:27:25.842
Like, it sort of, you know, the original PQ system I was kind of happy with.
00:27:25.862 --> 00:27:30.821
It felt like this, you know, system that worked really well and was kind of bulletproof.
00:27:31.471 --> 00:27:34.882
But I didn't really understand why it was unsatisfying to me.
00:27:35.432 --> 00:27:39.751
But he sort of describes later in the chapter why it's unsatisfying.
00:27:40.477 --> 00:27:42.849
Um, and I'll leave, this episode with that cliffhanger.
00:27:43.599 --> 00:27:51.795
Alright, well, you'll have to tune in next week to know why is the revised PQ system unsatisfying?
00:27:52.279 --> 00:27:55.730
alright, well, I'll see you next time for the second half of this chapter.
00:27:56.490 --> 00:27:57.359
See you next time, Matt.
00:00:14.619 --> 00:00:21.339
Hello everyone And welcome to the switch statement podcast It's a podcast for investigations into miscellaneous tech topics
00:00:21.879 --> 00:00:25.759
This is our sixth episode on Gödel, Escher, Bach by Douglas Hofstetter.
00:00:31.359 --> 00:00:32.649
Hey John, how are you doing?
00:00:33.185 --> 00:00:34.744
Hey Matt, I'm doing pretty good.
00:00:34.744 --> 00:00:35.325
How are you?
00:00:36.659 --> 00:00:45.451
I'm doing alright, I'm just having all of meaning just broken for me by this chapter of Girdle Escher Bach.
00:00:45.511 --> 00:00:46.131
For real.
00:00:46.390 --> 00:00:53.289
Yeah, no, this guy, Douglas Hofstadter, he really knows how to like make you reinterpret the entire universe.
00:00:53.685 --> 00:01:06.594
So I have to say I've been trying to skirt the dialogues in this book, probably to my own detriment, but But anyway,
00:01:06.709 --> 00:01:07.989
But this one you can't skip.
00:01:07.989 --> 00:01:08.989
This one was vital.
00:01:09.000 --> 00:01:11.459
He references it in the chapter several times.
00:01:11.674 --> 00:01:16.234
well, I've been getting pretty good at just ignoring words that I don't understand in this book.
00:01:16.265 --> 00:01:24.215
So I could have just walked past contra across Ticus, kiss uh,
00:01:24.739 --> 00:01:26.090
Cross Depunctus.
00:01:26.795 --> 00:01:27.965
Contra Costus.
00:01:28.814 --> 00:01:30.775
Which, by the way, is that a word?
00:01:30.775 --> 00:01:33.575
I, I, like, didn't look it up or anything.
00:01:33.894 --> 00:01:35.605
I looked it up it's not a word.
00:01:36.165 --> 00:01:36.935
Oh, okay, okay.
00:01:37.194 --> 00:01:47.334
so contrapunctus is like the German word for counterpoint, he references, uh, Bach's art of fugue
00:01:47.584 --> 00:01:48.105
Yeah, yeah.
00:01:48.394 --> 00:01:49.644
Contrapuntal melodies.
00:01:50.290 --> 00:01:56.040
That's broken down, the sections of that are broken down into Contrapunctus 1, and maybe it's Latin, I don't know.
00:01:56.109 --> 00:02:02.150
But anyway, Contrapunctus 1, Contrapunctus 2, to Contrapunctus Like 30 or something.
00:02:02.280 --> 00:02:03.879
It's very long, it's like an hour and a half.
00:02:04.359 --> 00:02:17.270
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.
00:02:17.919 --> 00:02:19.979
One word in the middle of another word.
00:02:20.500 --> 00:02:26.650
Um, so he put the word acrostic in there and I don't know, did you see this?
00:02:26.830 --> 00:02:29.439
The whole dialogue was an acrostic.
00:02:30.185 --> 00:02:31.485
What is an acrostic?
00:02:31.729 --> 00:02:32.060
Okay.
00:02:32.060 --> 00:02:51.465
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.
00:02:52.645 --> 00:02:53.854
Oh, holy shit.
00:02:53.865 --> 00:02:58.145
Like when you're resigning from a job, but you want to say, like, fuck you to the whole team.
00:02:58.145 --> 00:03:00.844
So you, like, start every sentence with F U C K.
00:03:01.164 --> 00:03:01.814
Okay.
00:03:02.645 --> 00:03:03.335
Nice.
00:03:03.365 --> 00:03:04.754
Dude, what does it spell?
00:03:05.145 --> 00:03:16.715
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.
00:03:17.405 --> 00:03:17.764
What?
00:03:18.425 --> 00:03:20.724
I know like, let me, uh,
00:03:21.074 --> 00:03:23.905
Because it doesn't spell Bok, JS Bok backwards.
00:03:24.199 --> 00:03:26.129
There must be some deeper meaning to that.
00:03:26.139 --> 00:03:27.330
Let me see if I can, uh,
00:03:27.944 --> 00:03:33.784
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.
00:03:33.794 --> 00:03:42.965
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.
00:03:42.965 --> 00:03:43.735
But this one.
00:03:44.455 --> 00:04:07.895
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.
00:04:08.185 --> 00:04:14.104
So basically tortoise this ultimate jerk destroying this guy's record players.
00:04:14.449 --> 00:04:15.210
Yeah.
00:04:15.634 --> 00:04:27.394
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.
00:04:28.084 --> 00:04:32.334
Uh, and yeah, he just keeps doing that to crab and breaking his record players.
00:04:32.615 --> 00:04:46.550
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.
00:04:46.550 --> 00:04:46.730
S.
00:04:46.740 --> 00:05:07.730
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.
00:05:08.449 --> 00:05:10.079
H is a note.
00:05:10.800 --> 00:05:11.379
So, you know,
00:05:11.660 --> 00:05:12.180
that's cool.
00:05:12.629 --> 00:05:25.139
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.
00:05:25.139 --> 00:05:31.220
I might be getting that slightly wrong, but in any case, you can actually spell Bach's name with musical notes.
00:05:32.120 --> 00:05:36.120
And he was describing how one of Bach's final pieces, he.
00:05:36.855 --> 00:05:39.745
put his own name in the piece.
00:05:40.154 --> 00:05:45.384
This whole entire book is about self reference and here Bach goes like referencing himself in his final piece.
00:05:46.154 --> 00:05:48.964
Um, anyway, I'm taking way too long to tell this.
00:05:49.375 --> 00:05:51.194
No, that was a perfect amount of time.
00:05:51.475 --> 00:05:53.975
Um, because I just figured it out.
00:05:53.995 --> 00:05:54.324
All right.
00:05:54.324 --> 00:06:00.214
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.
00:06:00.355 --> 00:06:11.404
It says Hofstadter's contra cross to Punticus, uh, cross tickly backwards spells J S Bach,
00:06:13.084 --> 00:06:14.444
Acrostically,
00:06:14.555 --> 00:06:16.254
cross tickly backwards means.
00:06:18.495 --> 00:06:23.105
maybe it itself is an acrostic or something.
00:06:23.154 --> 00:06:23.694
I don't know.
00:06:23.855 --> 00:06:24.324
I don't know.
00:06:24.867 --> 00:06:36.687
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.
00:06:37.101 --> 00:06:42.322
Yeah, everything Douglas Hofstetter does has to reference itself and be recursive.
00:06:42.776 --> 00:06:43.146
Right.
00:06:43.146 --> 00:06:43.427
Okay.
00:06:43.427 --> 00:06:44.776
So, but then let's just look at this.
00:06:44.776 --> 00:06:48.716
So what he's saying is, so the first word is Hofstadter starting with an H.
00:06:49.257 --> 00:06:53.317
The second word is contra acros de punticus.
00:06:54.507 --> 00:06:57.726
The third word is acrostically.
00:06:59.026 --> 00:07:01.036
Fourth word is backwards, so it's,
00:07:01.172 --> 00:07:01.372
J.
00:07:01.372 --> 00:07:01.512
S.
00:07:01.526 --> 00:07:11.567
H, H, C, uh, A, B, and then, yeah, spells, is S, and then J, S, Bach.
00:07:12.052 --> 00:07:14.482
Hofstetter, this clever weasel.
00:07:15.141 --> 00:07:19.572
They're gonna, when he dies, they're gonna open his coffin, it's gonna be like another little coffin inside it.
00:07:19.656 --> 00:07:21.007
Another Coffin inside, yeah.
00:07:21.288 --> 00:07:26.048
anyway, so I was just getting to the very end of the the dialogue.
00:07:26.298 --> 00:07:32.538
So at the very end, they put this record on from Bach and the record destroys the goblet.
00:07:34.327 --> 00:07:38.538
So it's got this nice little twist at the end.
00:07:38.627 --> 00:07:47.697
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.
00:07:47.708 --> 00:07:51.298
Like this sounds like a really expensive thing and he just keeps breaking them over and over.
00:07:52.223 --> 00:07:58.187
Well, I mean, this is just, you know, it's the hubris of the Of the crab.
00:07:58.485 --> 00:07:59.295
It's like, come on, crab.
00:07:59.625 --> 00:08:02.697
what we find, is the tortoises, is actually girdle.
00:08:03.773 --> 00:08:04.632
yes, exactly.
00:08:04.653 --> 00:08:10.423
So that was the hidden kind of like, this is a funny thing about this chapter.
00:08:10.423 --> 00:08:14.122
Cause like, you know, I read this dialogue and I was like, cool, far out.
00:08:14.653 --> 00:08:20.333
But then he, he continually references the dialogue within the contents of the chapter.
00:08:20.333 --> 00:08:23.362
And there was so much hidden meaning in this dialogue.
00:08:23.423 --> 00:08:25.622
Like you just described the whole acrostic thing.
00:08:26.247 --> 00:08:28.517
Literally all of that went over my head.
00:08:28.528 --> 00:08:37.097
I thought this was just like a funny story, but it's actually a decent allegory for Gödel's incompleteness.
00:08:37.807 --> 00:08:41.398
I think we've raised the concept of isomorphisms before.
00:08:42.738 --> 00:08:43.967
But this chapter.
00:08:44.822 --> 00:08:54.982
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,
00:08:55.597 --> 00:08:55.957
Yeah.
00:08:55.972 --> 00:08:56.363
you know?
00:08:57.013 --> 00:09:12.692
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.
00:09:12.923 --> 00:09:15.393
Like it's an axiomatic system for number theory.
00:09:15.582 --> 00:09:16.373
exactly.
00:09:16.812 --> 00:09:24.903
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.
00:09:24.903 --> 00:09:29.493
We spent so much money on this, so much time creating this axiomatic number theory,
00:09:29.535 --> 00:09:30.936
it's, high fidelity.
00:09:31.135 --> 00:09:38.826
He uses the terms high fidelity and low fidelity, which I think high fidelity was to describe like a complete system.
00:09:39.515 --> 00:09:43.875
that can basically represent all true statements as theorems in the system.
00:09:44.515 --> 00:09:44.926
Right.
00:09:45.255 --> 00:09:45.946
Exactly.
00:09:46.015 --> 00:09:46.676
Exactly.
00:09:46.765 --> 00:09:52.895
And then there's an out where you can make it, you can make a phonograph that doesn't break.
00:09:52.926 --> 00:10:02.960
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.
00:10:03.250 --> 00:10:03.950
exactly.
00:10:04.000 --> 00:10:04.330
Yeah.
00:10:04.330 --> 00:10:11.490
Like you have lower fidelity, so the sound won't come, come out clearly, which Literally all of that.
00:10:11.519 --> 00:10:17.250
Like I did not, I did not get, but it was kind of cool to have it revealed within the chapter.
00:10:17.279 --> 00:10:19.710
Cause it was, you know, I kind of had these like aha moments.
00:10:20.429 --> 00:10:20.840
Yeah.
00:10:21.100 --> 00:10:21.379
Yeah.
00:10:21.379 --> 00:10:21.690
Yeah.
00:10:21.919 --> 00:10:30.799
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.
00:10:31.169 --> 00:10:41.460
He talks about how human language itself is kind of an isomorph where people attribute meanings to words.
00:10:41.830 --> 00:11:00.879
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.
00:11:01.240 --> 00:11:07.350
The words are just this, like, weird medium that we attempt to convey meaning through.
00:11:08.110 --> 00:11:17.745
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.
00:11:17.835 --> 00:11:21.205
Like it was like, Whoa, words don't actually have meaning.
00:11:21.215 --> 00:11:30.414
Like within, you know, within the combination of words, we're like conveying these complex meanings and it becomes very important later.
00:11:30.980 --> 00:11:47.140
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.
00:11:47.630 --> 00:11:56.342
And then there's the isomorphism between like our, Like throat meat and like the sound
00:11:56.811 --> 00:11:57.561
throat meat.
00:11:58.081 --> 00:12:18.015
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
00:12:18.701 --> 00:12:25.081
you know, just to bring it back to modern reality, big thing these days is large language models.
00:12:25.860 --> 00:12:36.461
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.
00:12:37.191 --> 00:12:39.990
And the embedding represents meaning.
00:12:40.510 --> 00:12:43.130
So let's say I have an image of a rabbit.
00:12:44.071 --> 00:12:49.431
That image will have an embedding in the large language model.
00:12:50.160 --> 00:12:59.681
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.
00:13:00.000 --> 00:13:06.821
You will now have a directional embedding that represents transitioning from rabbit to dog.
00:13:07.791 --> 00:13:09.900
So let's say you have a different picture of a rabbit.
00:13:10.510 --> 00:13:18.250
And you apply that same directional embedding, you will, you will basically transition that picture into a dog.
00:13:18.681 --> 00:13:21.321
So say you have a rabbit jumping, you can get a dog jumping.
00:13:21.380 --> 00:13:24.160
Say you have a rabbit swimming, you can get a dog swimming.
00:13:24.801 --> 00:13:30.931
So it's, it's just this completely insane, you know, true thing that's happening today.
00:13:31.750 --> 00:13:33.870
And I, I feel like it kind of connects to this book.
00:13:33.880 --> 00:13:36.591
Cause he's talking about how the words themselves aren't conveying the meaning.
00:13:36.591 --> 00:13:40.000
There's some underlying weird meaning that none of us really get.
00:13:40.321 --> 00:13:43.120
And it's kind of like an embedding within a large language model.
00:13:44.296 --> 00:13:53.546
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.
00:13:53.735 --> 00:13:57.196
You like, you look at one of those numbers or even you look at the whole array of numbers.
00:13:57.250 --> 00:13:57.541
Yeah,
00:13:58.216 --> 00:14:05.525
It's like that array does not mean anything aside from its relationship to everything else, you know?
00:14:05.725 --> 00:14:07.495
And it's like, where is the meaning?
00:14:07.553 --> 00:14:07.943
right.
00:14:08.298 --> 00:14:11.198
I guess Hofstadter would say that the meaning is in the mapping.
00:14:13.114 --> 00:14:16.774
Oh, the meaning is in the mapping, dude.
00:14:17.874 --> 00:14:18.874
I think you're onto something.
00:14:19.844 --> 00:14:21.714
That's what I'm, I'm getting from it.
00:14:22.014 --> 00:14:29.561
Like nothing has any meaning aside from the establishment of an isomorphism between its parts and something else.
00:14:29.561 --> 00:14:31.154
Um,
00:14:36.639 --> 00:14:38.788
gives a brief story of Bach.
00:14:39.499 --> 00:14:44.229
Which I just love learning a little bit more about Bach because I absolutely love Bach.
00:14:44.229 --> 00:14:47.219
I think he's one of the greatest composers to ever live.
00:14:48.028 --> 00:14:50.798
Um, but he tells kind of the sad story of Bach.
00:14:51.229 --> 00:15:01.109
Like apparently Bach was losing his eyesight and so he got this operation, but the operation basically made him completely lose his eyesight.
00:15:02.028 --> 00:15:17.119
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.
00:15:17.719 --> 00:15:26.119
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.
00:15:26.119 --> 00:15:28.739
Bach was dictating to Carl Philip Emanuel.
00:15:29.509 --> 00:15:34.288
Um, but then evidently, Bach did regain his vision at a certain point.
00:15:34.688 --> 00:15:38.158
But then very shortly after he regained his vision, he had a stroke.
00:15:38.754 --> 00:15:40.953
And then ten days later, he died.
00:15:41.678 --> 00:15:45.308
Literally a few hours after he regained his vision.
00:15:45.448 --> 00:15:45.568
So
00:15:45.573 --> 00:15:46.053
Yeah.
00:15:46.109 --> 00:15:54.182
about, talk about just like brutal, yeah.
00:15:54.349 --> 00:15:56.869
already, and now you're gonna make Bach blind?
00:15:56.869 --> 00:15:57.298
Like, come on.
00:15:57.839 --> 00:16:00.967
Although, I just mixed up the chronology, but you get what I'm saying.
00:16:00.967 --> 00:16:02.793
Yeah, there was a, there
00:16:07.621 --> 00:16:10.802
there's a section called problems caused by girdles result.
00:16:16.698 --> 00:16:20.119
Maybe I can just ask you live on the podcast.
00:16:20.119 --> 00:16:25.818
Uh, he mentions like Gödel's reasoning methods can't be encapsulated.
00:16:26.489 --> 00:16:32.908
And he mentions that it's a difference between mechanical and human reasoning.
00:16:33.264 --> 00:16:33.744
Yeah.
00:16:34.278 --> 00:16:36.958
And I feel like I kind of get that, but I don't know.
00:16:37.479 --> 00:16:43.879
I don't know if you have something, you know, that you would want to add to that or like, uh, yeah.
00:16:44.114 --> 00:16:48.494
I mean, I think it's actually necessary for.
00:16:50.563 --> 00:16:54.974
Like it's a necessary aspect of his theory of incompleteness.
00:16:54.974 --> 00:16:54.994
Yeah.
00:16:55.399 --> 00:17:01.599
That it has to not be able to be encapsulated because if it could, it wouldn't work.
00:17:02.688 --> 00:17:03.339
You know what I mean?
00:17:03.698 --> 00:17:09.328
Yeah, so does, does that refer to it being human reasoning?
00:17:09.419 --> 00:17:15.798
Because mechanical reasoning would be fully encapsulated within the system.
00:17:15.808 --> 00:17:19.709
So you kind of have to make this like, quote unquote, human leap.
00:17:20.898 --> 00:17:25.909
Yeah, I guess maybe I'm, maybe I'm wrong about that, but that's where I kind of was lost.
00:17:26.618 --> 00:17:38.587
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.
00:17:39.299 --> 00:17:41.509
I just don't understand how that, how that could be possible.
00:17:42.079 --> 00:17:49.750
Yeah, no, I had the exact same misunderstanding, and I think, I think that's what he's referring to as human reasoning.
00:17:50.630 --> 00:17:56.490
But like, I think maybe the claim is like human reasoning is not a formal system.
00:17:57.579 --> 00:17:58.930
Right, exactly.
00:17:59.180 --> 00:18:01.359
Like it's this larger thing.
00:18:01.819 --> 00:18:06.940
Like it's this more basic thing that just arises out of neurons.
00:18:07.880 --> 00:18:22.670
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.
00:18:22.670 --> 00:18:22.690
Yeah, probabilistic.
00:18:23.210 --> 00:18:32.059
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.
00:18:32.559 --> 00:18:34.039
That was at least my understanding of it.
00:18:34.049 --> 00:18:39.542
It's like, Gödel's method's not being able to be encapsulated within the system.
00:18:40.063 --> 00:18:47.643
is like an instance of him having this like more human style reasoning that almost like by definition breaks out of the system.
00:18:48.292 --> 00:18:48.682
Yeah.
00:18:49.290 --> 00:18:51.901
I might be like, uh, reaching a bit on that.
00:18:52.310 --> 00:19:05.746
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.
00:19:05.885 --> 00:19:10.945
But like, I guess you can continue the pattern, you know, like you could just keep on doing that.
00:19:11.496 --> 00:19:11.836
Yeah.
00:19:12.256 --> 00:19:14.385
And that's what happened to Russell and Whitehead, right?
00:19:14.385 --> 00:19:16.836
They had all these layers and like, so.
00:19:17.211 --> 00:19:20.760
you're always going to be able to like break out of the system, I think.
00:19:21.445 --> 00:19:21.865
Right.
00:19:22.125 --> 00:19:22.375
Yeah.
00:19:22.375 --> 00:19:26.195
Which I think Gödel has, has proven.
00:19:27.132 --> 00:19:42.979
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.
00:19:43.038 --> 00:19:45.219
So you had a certain number of dashes.
00:19:45.308 --> 00:19:48.078
You had the letter P you had a certain number of dashes.
00:19:48.078 --> 00:19:54.118
You had the letter Q and then you had the sum of the numbers of two, the two strings of dashes.
00:19:54.969 --> 00:19:58.229
So basically P is plus and Q is equals.
00:19:58.868 --> 00:19:59.398
Exactly.
00:20:00.179 --> 00:20:00.648
Um,
00:20:00.911 --> 00:20:04.941
But he introduced a new, uh, uh, rule, or
00:20:05.368 --> 00:20:06.449
he modifies it.
00:20:06.509 --> 00:20:08.328
So it's actually a replacement.
00:20:08.328 --> 00:20:12.628
It's not, um, I don't think it's like an addition to the previous system.
00:20:12.808 --> 00:20:15.618
Oh, right, because it kind of violates the previous system in a way.
00:20:15.618 --> 00:20:23.519
But in any case, he, he updates the system so that Q means less than, right?
00:20:23.519 --> 00:20:24.568
Less than or equal to?
00:20:24.568 --> 00:20:24.627
I don't know.
00:20:27.479 --> 00:20:36.012
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.
00:20:36.373 --> 00:20:37.182
We're like, Oh shoot.
00:20:37.212 --> 00:20:43.262
Like it's now producing like incorrect, uh, theorems.
00:20:44.732 --> 00:20:45.282
Oh, right.
00:20:45.292 --> 00:20:49.482
Because it's like one and one is like three or something,
00:20:50.198 --> 00:20:50.567
yeah.
00:20:50.567 --> 00:20:53.468
So did we, did we say what the new rule was?
00:20:53.657 --> 00:21:07.488
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.
00:21:09.347 --> 00:21:14.298
And so, but this allows you to create thing, create strings that.
00:21:15.692 --> 00:21:20.732
You know, for which, the previous edition isomorphism doesn't hold anymore.
00:21:21.603 --> 00:21:22.022
right?
00:21:23.377 --> 00:21:28.587
but he goes on to say that like, this is, this is an invalid way to look at, at it.
00:21:29.817 --> 00:21:38.248
Um, he basically says, as soon as you're modifying the system, you kind of have to discard any previous isomorphism.
00:21:39.597 --> 00:21:40.637
Yeah, yeah.
00:21:41.018 --> 00:21:46.688
And that's the point where he comes up with this new isomorphism that Q is no longer equals.
00:21:46.748 --> 00:21:48.887
It is now less than or equals.
00:21:50.282 --> 00:21:51.913
Right, exactly, exactly.
00:21:51.923 --> 00:22:05.432
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.
00:22:05.968 --> 00:22:06.288
Yeah,
00:22:07.542 --> 00:22:09.303
But that's just not a valid approach.
00:22:10.258 --> 00:22:11.498
right, right.
00:22:12.067 --> 00:22:17.097
Um, I feel like there was a similar notion in one of the previous chapters.
00:22:18.083 --> 00:22:44.292
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.
00:22:44.607 --> 00:22:47.887
An idea that I thought was really cool from one of the previous chapters.
00:22:48.798 --> 00:23:01.178
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.
00:23:02.528 --> 00:23:05.218
So he said that there might be a.
00:23:06.147 --> 00:23:09.097
An urge to, like, go back the other direction.
00:23:09.853 --> 00:23:15.432
And, And, make a string that's like, Oh, you say, Oh, I can make an addition statement.
00:23:15.432 --> 00:23:18.982
That's like two plus two plus two equals six.
00:23:19.643 --> 00:23:27.833
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.
00:23:28.823 --> 00:23:35.202
But he, his point was like, that is, that's an active meaning.
00:23:35.202 --> 00:23:36.532
I think that that was the distinction.
00:23:36.532 --> 00:23:37.492
There's like a passive.
00:23:38.393 --> 00:23:39.782
meaning and an active meaning.
00:23:39.782 --> 00:23:49.873
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.
00:23:50.117 --> 00:23:50.788
exactly.
00:23:50.807 --> 00:23:56.407
And you're generating a theorem that can't actually be produced with the rules of the system.
00:23:56.962 --> 00:24:00.712
He mentions that your meaning has to always remain passive.
00:24:01.262 --> 00:24:04.893
Like it's, it's a mistake to move into this active meaning mode.
00:24:05.998 --> 00:24:11.891
Um, did you have to ever have to do those things in your computer science classes where you.
00:24:12.089 --> 00:24:14.859
Had to prove something NP complete.
00:24:16.464 --> 00:24:17.674
No, no.
00:24:18.640 --> 00:24:19.950
Um, Oh, that's crazy.
00:24:20.000 --> 00:24:28.579
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.
00:24:29.295 --> 00:24:29.884
Really?
00:24:30.180 --> 00:24:36.329
The technique that they proposed was, you know, someone out there proved.
00:24:36.704 --> 00:24:41.184
Some like series of problems, NP complete at some point in the past, right?
00:24:41.734 --> 00:24:53.605
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.
00:24:54.994 --> 00:25:01.894
Demonstrate that you could solve one of the existing NP complete problems.
00:25:03.525 --> 00:25:13.954
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.
00:25:14.154 --> 00:25:30.224
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.
00:25:31.595 --> 00:25:39.115
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.
00:25:39.894 --> 00:25:40.244
Yeah,
00:25:40.835 --> 00:25:42.654
And then you would have to.
00:25:43.684 --> 00:26:09.785
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.
00:26:10.444 --> 00:26:12.045
new proposed problem.
00:26:12.045 --> 00:26:15.654
Anyway,
00:26:15.724 --> 00:26:16.785
had done this in
00:26:17.035 --> 00:26:17.454
so,
00:26:17.634 --> 00:26:18.335
fascinating.
00:26:18.335 --> 00:26:18.404
Yeah.
00:26:18.545 --> 00:26:30.505
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.
00:26:30.505 --> 00:26:31.404
It's like, wait a minute.
00:26:31.404 --> 00:26:36.162
shouldn't I be proving that I can use the NP, the known NP complete thing to solve the.
00:26:37.057 --> 00:26:43.906
Other problem, but no, it's like you need to, you need the direction to like flow the other, the other way.
00:26:43.997 --> 00:26:48.196
So anyway, that was all a very long and potentially, uh,
00:26:48.507 --> 00:26:49.336
Well, it's interesting.
00:26:49.336 --> 00:26:59.106
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.
00:26:59.346 --> 00:27:01.267
yeah, it's proving an isomorphism.
00:27:01.317 --> 00:27:08.936
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.
00:27:08.936 --> 00:27:11.326
Yeah.
00:27:11.457 --> 00:27:13.797
this episode, just to wrap up my notes.
00:27:14.832 --> 00:27:21.521
I found this, like, PQ system redefinition to be very unsatisfying when I read it.
00:27:22.031 --> 00:27:25.842
Like, it sort of, you know, the original PQ system I was kind of happy with.
00:27:25.862 --> 00:27:30.821
It felt like this, you know, system that worked really well and was kind of bulletproof.
00:27:31.471 --> 00:27:34.882
But I didn't really understand why it was unsatisfying to me.
00:27:35.432 --> 00:27:39.751
But he sort of describes later in the chapter why it's unsatisfying.
00:27:40.477 --> 00:27:42.849
Um, and I'll leave, this episode with that cliffhanger.
00:27:43.599 --> 00:27:51.795
Alright, well, you'll have to tune in next week to know why is the revised PQ system unsatisfying?
00:27:52.279 --> 00:27:55.730
alright, well, I'll see you next time for the second half of this chapter.
00:27:56.490 --> 00:27:57.359
See you next time, Matt.
