Matt: 0:14

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

Jon: 0:22

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

Matt: 0:31

Hey John, how are you doing?

Jon: 0:32

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

Matt: 0:35

I am doing alright. ready to talk about figuring ground. I don't know, what

Jon: 0:40

the mysteries of reality. Well, we're

Matt: 0:44

Wait a minute, this is the next one. That's the next one.

Jon: 0:46

yeah, yeah. Figuring ground is in the future.

Matt: 0:48

I'm, I'm jumping. Yeah. Spoiler alert. Meaning and form. Meaning and form.

Jon: 0:53

Meaning and form. Yeah. Which, what did I just read? I mean, these chapters are dense and they're confusing and they're, they're super, super interesting, but they have

Matt: 1:05

Where.

Jon: 1:06

a density of complex ideas that really like throws me for a loop. A

Matt: 1:11

I, we're kind of ascending the levels of abstraction into this kind of ethereal realm

Jon: 1:18

Yes.

Matt: 1:18

ties to the real world. Um, so, uh, it's, uh, take some, take some getting used to.

Jon: 1:27

I once again read the, I once again read the dialogue for this one, and I was just, I was left with this feeling of like, maybe I shouldn't have read that maybe I should've skipped it.

Matt: 1:40

I am glad you're reading the dialogues because I am not reading the dialogues. And maybe, maybe I'm. Making a terrible mistake, but,

Jon: 1:48

the, the dialogue is maybe the point, maybe you can let me know, like he brought up this idea of like, do words and thoughts follow formal, formal rules or do they not? and that, I think that might've been in the dialogue. And I thought that was like one of the interesting tidbits to pull out, because I feel like there's this immediate visceral response that I have, which is like, no, like you can express anything with. With words and thoughts, but then as you start thinking about it more, you realize that you're kind of limited by what you can even conceive. And so there, it, there does seem to be this like higher level set of restrictions that govern even the thoughts that you can produce.

Matt: 2:36

well, it's interesting because it seems like what you're saying is like there, there's, there's the domain of all possible expressible thoughts, which like, yeah, you as, as a particular individual, like you can't say something that you can't conceive of, but. But you can conceive of the entire possible domain of things you could say. You know, and that's obviously going to contain things that you wouldn't conceive of. Am I, am I losing everybody? But my point is, my point is that The only restrictions to what you can say are literally like the phonemes of words, right? It's like, cause like, yeah, it's like every, I don't know. Like, obviously there's, and I guess if we're tying this back to the book. Like this base level is like, you know, if we're kind of, uh, just refreshing everyone's memory about like the mu puzzle. The, like we're operating in this domain of strings, right? Where like our theorems are strings just out of simplicity, uh, or for the sake of, uh, simplicity. And it's like, there's this underlying vocabulary. Of strings, which is like, you can put together strings of characters, which like, they don't exist in the system, but they're still like expressible thoughts. You know what I mean? Like, and that's kind of how I imagine, like, you could say like, you know, there's a, a blue idea over there. You know what I mean? It's like,

Jon: 4:32

Yeah. Well, but I guess this is where it just gets ultra, ultra philosophical, almost to the point where it's annoying because like later in the, this same chapter, he brings up the idea of is reality itself a formal system? Like you can sort of define reality as following the laws of physics, which the laws of physics are governed by. Relatively simplistic elementary particle interactions. And so there was some axiom at the dawning of the universe that was kind of this initial figure configuration of particles and everything else has basically been deterministic from that. And, and that, that's sort of a way of arguing that like we're stuck in a formal system, but it's also, you know, kind of a cheesy, annoying, ultra nerdy thing to bring up if you want to like lose at dinner parties. Then, uh, this is your game plan.

Matt: 5:30

but anyway, yeah. So I think ultimately, I mean, I think what's being expressed is you, you constantly have these layers where you're putting some restrictions on the layer below it, and then you're kind of saying, okay, well, when, once you take those restrictions into account. Then you start to ask questions. And that's really kind of what having a formal system is about is like you define these constrictions and then you like pose a hypothesis or, or ask a question,

Jon: 6:04

Right, man. When I'm, when I'm reading these chapters, I feel like I'm taking acid. But when I talk about the chapters, I feel like I'm smoking like massive amounts of weed.

Matt: 6:17

dude, we should, um, we should have the, uh, you know, the high episode, record one of these episodes.

Jon: 6:23

People wouldn't even be able to tell, it would just sound the exact same. Uh, so he brings up a new formal system. Which I'm kind of hoping he introduces a new formal system in every chapter. I think that'd be funny. But in the last chapter, he introduced them, the mu puzzle, which involved this mu system or me system or whatever, where you could create MIU combinations. In this chapter, he introduces the, I don't know if he has a name for it. Maybe it's the PQ system.

Matt: 6:54

Yeah, the PQ system.

Jon: 6:55

Okay. Yeah. And it's a set of hyphens and a single letter P. Some more hyphens, letter Q and some more hyphens. And uh, you know, he gives an, he gives what he calls an axiom schema for it,

Matt: 7:11

Yeah.

Jon: 7:12

which he describes as the mold in which axioms are cast, which I just thought was cool. A cool set of words where it's basically XP QX where X can be any number of hyphens. Um. And yeah, he, he mentions a couple other things about the PQ system. One thing that I thought was noteworthy was he talks about how it only has lengthening rules. Like you can only use that axiom schema to add more. You can't, you can't take an existing schema and break it down or reduce it. Like you could in the, in the MUSE system. Or if you had like three eyes, you could make a U or something. I can't remember exactly what the rule was, but you could

Matt: 7:57

I do. Yeah. I want to call out one, one like distinction here. Like there's there, there's the rule to define. the axioms, like there's this mold for defining axioms, but then there's the rule to get from an ax or like from an already proven hypothesis to another proven or like a theorem to another theorem. So like, it's, it, this is what's confusing is like, there is, there's the, you already have an infinite number of axioms, like bit just by the mold, but then there are certain theorems that you can't generate just by the axiom. And then in order to do that, you have to use the, like the generator role, which is, um, which basically says, let's see, uh, if, so if you already know that you have a theorem, you can, Yeah. Okay. Uh, so, so yeah, so it gets a little complicated. So basically like every, every string in this system is some number of hyphens, a P, some number of hyphens, a Q, and then some number of hyphens.

Jon: 9:14

Right.

Matt: 9:15

And so the rule for generating new theorems is You consider the number of hyphens before the P as X. The number of hyphens between the P and the Q, Y. And then the number of hyphens after the Q, Z. And then say, say you have a theorem that you've already generated with the rules. If, you know, if you already have that, then XP Y hyphen Q Z hyphen is also a theorem. So, and, and that's probably hard to, uh, like a little bit hard to wrap your head around, but basically they're saying like, okay, if you already have a string that, you know, as a theorem, you can add a string before the Q and then at the end of the string, and that is also a theorem.

Jon: 10:12

Right. Right. So this is that, this is that lengthening rule, that we just discussed. And yeah, you, you, you made the important point that the axiom schema, uh, is sort of this mold that all the theorems have to fit into, but it doesn't necessarily describe all valid theorems, right? Like it's like you could have a theorem, or, or can you actually, can you have a theorem that, that fits in that mold, but isn't valid?

Matt: 10:43

So, so the way I see it, the, the, the axiom mold that just defines a bunch of like starting points, but it doesn't, it doesn't define the structure that all, all like valid theorems need to adhere to.

Jon: 10:59

Oh, I see what you mean. Okay. So with that axiom mold, You can, you know, create all of these theorems, but then you can apply the rule that you just described to get even newer theorems that, you know, that actually don't fit in the mold because they have more hyphens between the P and the Q,

Matt: 11:16

Exactly, exactly. Because, because the, the axiom mold, uh, only allows you to generate, uh, things with one dash in between the P and the Q.

Jon: 11:28

right.

Matt: 11:29

Or one hyphen, excuse me.

Jon: 11:30

Um, so he describes bottom up versus top down. and yeah, unfortunately I only wrote a quick note about this. I just wrote that bottom up seems to be brute force. So I don't know if you can describe this better, but I think he was talking about ways of producing new theorems and with bottom up, you're just kind of taking all the theorems you have and applying all of the available rules in theorems.

Matt: 11:55

One thing, one thing that is very, like actually one part of this that, yeah, was interesting to me is. This starts to get into like, I mean, this gets pretty specific to like programming and defining. Yeah. Like recursive solution or like recursive algorithms when it's not like super clear, some recursive algorithms, it feels like it's very clear. Like what the next step is. This feels like one where it's like, you actually have an infinite number of steps because it's like, so we just defined like, Oh, this mold that we have, like it can generate an infinite number of. you know, an infinite number of theorems just from the axiom. So it's like how in a world where you have like an infinite number of steps you can take at the very beginning. Like, how do you ever explore the rules? You know, if you, if you're looking for a certain thing

Jon: 12:53

right, yeah,

Matt: 12:54

and so, so the approach they, they propose is you put the simplest possible axiom, which is just dash p dash q dash dash into the bucket. And then you apply the rule of inference to the item in your bucket. Then you put the second simplest axiom in the bucket. And then you apply the rule to every item in your bucket. And then, and then you go on from, from

Jon: 13:23

And this is bottom up, right?

Matt: 13:25

This is, this is bottom up. He, he says that the previous approach, uh, was top down. Uh, and I don't know, I'm not sure if he's saying that this is referring to the MU rules were like the way we were doing MU was top down. Um, but, but anyway, so that's, that's at the very least we, this is this, he's calling this a bottom up approach. Cause we're working our way kind of up from, but yeah, so we can, we can, uh, move on from there.

Jon: 13:54

Yeah, yeah. Let's move on. So the next part I thought was kind of the, the cool fact of this chapter. Isomorphisms, and, and this, I'm just going to reveal what the whole P Q hyphen thing is. It's, it's literally just plus. So the P stands for the plus symbol and the Q stands for equals. So if you have two hyphens plus two hyphens, That equals four hyphens. So you can have, you know, that, that would be like a valid theorem in that, system. And, and he introduces this concept of an isomorphism, which is where you can map from one system to another. So in this case, you're mapping from this, you know, typographical system where you're using hyphens and letters to math, where you're using numerical symbols. And saying things like, you know, three plus four equals seven. And it's, it's an extremely powerful concept, obviously, because the moment you can map from one system to another, you can then fully utilize that other system and all of its, Magical powers. Like obviously math is this unbelievably powerful system that humans have developed. and so if you have this crazy lexicograph, lexicographical, that's kind of hard to say, system that you're like struggling with and you can actually map it to math, then that there's a lot of power that you can sort of tap into. but he mentions this concept of a meaningful versus a meaningless. Interpretation where sometimes it seems like you might be able to make a mapping, but it's not a true mapping. Like it just, you know, maybe it maps in a few of the cases, but that mapping breaks down after a while and you can actually reach false conclusions, you know, by accidentally mapping to math and not realizing that you can't actually use math.

Matt: 15:51

Yeah, and I like what he's doing here because he's giving very precise definitions for all of these terms. And like, it seems like what he's, what he's basically saying is an interpretation is any mapping between the elements and something else, right? And it's like, You can, and he, he does this, he has this like hilarious mapping where he says, okay, P maps onto horse and Q maps onto happy and then like the dash maps onto an apple and like, it's like, yeah, like maybe there are ways to see like, oh, well, if there's more apples, the horse is more happy. Um, yeah. But, um, but he basically argues like that's, that's a meaningless interpretation, but it is an interpretation. You're, you're kind of assigning a meaning to each one of these symbols, but you don't come up with something as you kind of alluded to, that's like more useful because you've kind of mapped it onto this other system that gives you more power in this, you know, in this other domain,

Jon: 17:09

Yeah. Yeah. And he mentions this notion of like active versus passive meaning. And he says that meaning must remain passive, which I think what that means is if you're creating an interpretation of some formal system, like you don't have the right to create a new theorem based on that interpretation. Like, like using your example, like horses and apples, you know, you could say like, Oh, horses really like apples. Therefore I can just put seven apples after the horse and that's therefore a new theorem, but like, no, that's a, that's a meaning that you just came up with and it doesn't actually apply to this formal system. So that's kind of the risk of these isomorphisms is you might, and I think this happens in math from time to time where it's like, you think like, Oh, I can use You know, I don't know. Logarithms. I'm like terrible at math. So I'm just using a crazy example, but like I can use logarithms for this problem and then you try to apply it to some value that you haven't used yet. And it's the wrong answer because you've made this mistake in interpretation.

Matt: 18:19

He gives, yeah, he gives a good example of this where he says, okay, we've, you know, just take a step back. Like, imagine you haven't figured out. And like, I didn't realize that this was like plus, uh, before, like I was just operating at the string level. So like you're generating these strings. If someone were to, like, before you understand that, you know, that you have this isomorphism with. Plus and math in this string system. If someone were to show you an arbitrary string of like dashes and P's and Q's like, and, and were to ask you, like, is this a valid string? You'd be like, I have no freaking idea. Like, I, I, I don't know. I would have to like generate a string to, to like use the rules until I could see if you were right, like if it had the right number of, um, uh, strings. But that's why having this isomorphism is so powerful because you can kind of like Dip into this other system, you you know, you translate it into this other system You do math and then like you translate it back and you're like, oh, okay Yeah, this is a valid system because I know that it has an isomorphism with this. Um, you know with this other system, uh, but so he says, um He says the danger there is like there is a mathematical, like, it's completely reasonable to say like two plus three plus four equals, you know, whatever. Uh, and then, but that it's, you're not allowed, the system doesn't allow you to say like dash dash P dash dash dash P dash dash dash dash P, you know, Q, whatever, like, and that's an example of like. forcing an isomorphism, like extending the isomorphism to an extent that the system actually doesn't allow.

Jon: 20:17

Right. Yep.

Matt: 20:18

so that's, that's an example where you're like trying to, yeah, you're just over like, I guess that's active. Uh, uh, what, what is the word?

Jon: 20:28

Active meaning. Yeah.

Matt: 20:30

Active

Jon: 20:30

Active meaning. Yeah. Uh, meaning must remain passive. Yeah. Um, yeah, then he mentions, oh, he goes on to this little tangent that we sort of already talked about, like, is reality itself a formal system? Um, which I do find to be an interesting idea. but, I think we can move on from that. He mentions this concept of ideal numbers, like formal versus practical numbers. Um, and I don't know, can you remind me what these are?

Matt: 21:03

Yeah, I'm, I mean, I think the theory here is like it's, yeah, it's easy to think of numbers as these very well defined things and they just always behave, but in the real world, it's like nothing actually. behaves as precisely as, as, you know, these ideal numbers do. Um,

Jon: 21:25

Yeah, yeah, Um, I can't remember the examples he was using, but this was actually a really interesting part of the chapter. because he was, he was basically talking about how mathematics is so rigorous and formal, but it often kind of like doesn't work in reality. I kind of want to find an example of that because it was very cool.

Matt: 21:46

yeah, it's, he doesn't give, like, he doesn't, or at least I'm not seeing him give like a really good, um, example of how he, like, of this, like, ideal number versus, I mean, he talks about this image where You know, one of these, paintings by Escher, there's all these birds and like, there's this point where lines between the, the birds are like blurred and you kind of can't tell like where one bird ends and the other bird starts. And like, this is, you know, he's kind of drawing a parallel there where reality is fuzzy in this way where there is this like transition, there's these transition regions, which are not very clearly defined in the real world. So, but yeah, I think I'm talking past my, uh, my familiarity with the, with the subject.

Jon: 22:39

Yeah. Yeah, no, it was, it was a cool section though, and I'm actually hoping he touches on it more, throughout the book, which I'm assuming he will. I feel like a lot of these early chapters are just introducing these concepts and just very much grazing, scratching the surface of them. Um, and I'm, I'm actually kind of excited to see him dig a lot deeper in these. But he ends this chapter with a really, really cool proof, Euclid's proof, uh, basically proving that there's always a larger prime number. Like there's no limit to the number of, of primes. Um, and the way he proves it is very cool.

Matt: 23:19

What it boils down to is, let's say you found a prime up to N

Jon: 23:26

Yeah,

Matt: 23:27

you, you could always multiply all of those numbers together

Jon: 23:33

right.

Matt: 23:33

and then, yeah, exactly. N factorial and then add one. And then you absolutely for sure know that like, it's not divisible by any of those prior numbers because you are like. You've already multiplied them all together, and then when you add one, it's like, you're not adding it by a multiple of any of the prior numbers. So it can't be divisible by any of those.

Jon: 24:01

Yeah. Which basically proves that there's a prime in between N and N factorial plus one. Right.

Matt: 24:09

Oh really?

Jon: 24:11

Yeah. Isn't that what it, what it's saying? So either it itself is prime or its prime divisors are greater than N. Meaning that n factorial plus one may not be prime, but it's prime divisors are greater than n, but in either case, we've shown there must exist a prime above n,

Matt: 24:31

right,

Jon: 24:32

which is awesome. Like, that's just such a cool proof. And this is what I was trying to talk about in the first chapter, where what I'm hoping this book helps me with. Is this sort of like way of thinking where you can even conceive of something like that, you know, like obviously Euclid was like a genius and I'm not like ever going to be thinking on the level of Euclid, but just the idea that you can kind of wrap your head around a super simplistic set of rules that prove definitively that there's no limit to the number of primes. Like, I just feel like my brain does not really have that muscle.

Matt: 25:14

Well, this is, I mean, and this is his very last section is called getting around infinity and it makes, it makes me think back to, uh, induction. And I mean, there's so much, like it feels like, and I'm not sure if this is true, but it feels like computer science. Textbooks that teach about induction. They're all like basically, I mean, maybe they're deriving it from other sources that Hofstadter was referring to, but it feels like very of a piece with all the stuff that they're talking, you know, he's talking about in this book where it's so powerful to be able to say, like, assume you've solved the problem for N, you know, whatever N is. Like, it doesn't matter what N is. Assume you've solved the problem for N. Yeah. I can show you that I can solve the problem for n plus one. And then it's like, boom. Well, it's like, that just, that never ends. That

Jon: 26:05

Yeah, That covers

Matt: 26:06

true. Yeah. So like, that's just like so powerful.

Jon: 26:10

Yeah, it's, it's amazing. And I hope that, um, you know, I hope I can gain this ability or maybe not fully gain it, but just like start to gain it.

Matt: 26:21

Yeah. No, I think we're gonna, we're gonna be infinite soon.

Jon: 26:24

Yep. We're going to be Euclid's.

Matt: 26:27

All right. Well, I think that, uh, did you have anything else you wanted to talk about in that chapter?

Jon: 26:32

No, that was all my notes.

Matt: 26:34

Alright, well, I will see you next time for really figuring ground, uh.

Jon: 26:40

All right, I'll see you in the next one, Matt.

Matt: 26:43

Bye.