Matt: 0:14

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

0:22

This is our 12th episode on Gödel Ascher Bach by Douglas Hofstadter.

matt_ch9: 0:30

Hey, John, how's it going?

jon_raw_pt_2: 0:32

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

matt_ch9: 0:35

Feeling all right? Are you feeling enlightened today? Do you feel like you've achieved enlightenment?

jon_raw_pt_2: 0:41

What's the opposite? Endarkened? I feel a little endarkened. This book, yeah, this book makes me feel endarkened.

matt_ch9: 0:50

That's, that is very I think you're the first person to ever, to ever say that, say those words. So, good on you. What, what, what's a, what's an example of in darkening?

jon_raw_pt_2: 1:02

You know, it's interesting because this book is enlightening, but it's enlightening in a way that it's, it feels like it's giving you a glimpse into this like entire universe, but then it closes, you know, it's like this little, this little tear opens and then it kind of quickly heals.

matt_ch9: 1:24

Oh, interesting.

jon_raw_pt_2: 1:26

Are you like, what is this guy talking about? I, I guess

matt_ch9: 1:30

A tear, it's like, a tear opens and you can see into another

jon_raw_pt_2: 1:36

yeah, you can see into this world of like deep knowledge and truth, but then it, it quickly kind of goes away, sort of like when Biden is giving a speech.

matt_ch9: 1:48

Oh God.

jon_raw_pt_2: 1:49

Okay, nevermind, I don't even want to talk about Biden, uh, but it's just this weird feeling that I've gotten many times from this book where, like, like, for instance, he spoke briefly, or in this chapter, he, he gave a lot of Zen stuff, which we should definitely talk about. But one of the things he talked about was he mapped the MIU system, or the MU system, which we learned about long, long ago, I think it was chapter one. He mapped that system back to typographical number theory. And basically what, you know, what he was, or the reason he did that was so that he could use the entire tool chest of typographical number theory to now work with the MIU system Prove things about it, basically. Um, so, and I guess this is, what is it? Isomorphic when you take, you're taking this one system and you're mapping it onto this other system. And it made me think about how, there's a thing that's commonly talked about with Einstein's relativity, where as he was working with relativity, he was really struggling to like, make it work. Like he just kind of couldn't. I'm not sure how you would figure out the math, basically, because Einstein was really bad at math, which, which is kind of a joke, but it's also something that he constantly said about himself, so there's like a weird type of truth to it, but anyway, he eventually stumbled on hyperbolic geometry, which had been developed, I don't know, hundreds of years ago, or, yeah, maybe just 100 years ago, I think hyperbolic geometry is in the 1800s. But hyperbolic geometry was just these guys who are kind of screwing with Euclid's elements, Euclid's five postulates. They were basically like throwing out one of those five postulates and then just seeing what would happen. And so they developed this whole new form of geometry, which at the time that they developed it, it was useless. But then when Einstein discovered relativity, it was very useful for describing relativity. So it was kind of this case where just working with math created raw knowledge, almost like unapplied knowledge. And then later on in human history, we learned how to now apply that knowledge,

matt_ch9: 4:15

Right.

jon_raw_pt_2: 4:16

which is just an immediate proof of the argument that like humans should be striving to do crazy things just for the sake of it. You know, it's like, uh, like, Another thing I was looking into for some strange reason in this chapter is the Riemann hypothesis. Because I was just wondering, this is something I wonder like every year or so, like, what would actually happen if we like solve the Riemann hypothesis one way or the other? And I was just sort of trying to understand, you know, what

matt_ch9: 4:45

Can you, can you, uh, give a, give a quick, uh.

jon_raw_pt_2: 4:49

Uh, yeah. Yeah, sorry. You can't just throw out Riemann hypothesis without, I guess, describing it a little bit. So the Riemann hypothesis is this conjecture that this particular function called the Riemann zeta function has its zeros in very particular places, namely the even integers and complex numbers that have a real part of 5, which a complex number is just a real part and an imaginary part. It's a number that is comprised of two parts. Um, So I guess that doesn't really describe the Riemann hypothesis. You sort of have to dig down into these things. A zeta function is an infinite, uh, summation of, like, if you take the zeta function of a variable, say S, it's just one plus one over two squared plus one over three squared, or I'm sorry, if, if S is two, one over two squared plus one over three squared plus one over four squared, it's just If S is three, then it would be one over two to the third power, one over three to the third power, yada, yada, yada.

matt_ch9: 5:57

Okay.

jon_raw_pt_2: 5:58

so anyway, it's saying that that function has its zeros where with complex numbers, with a real part of 0. 5, uh, which is crazy. First of all, that's like the first crazy statement, but the reason this is important and I'm realizing this is like a massive tangent.

matt_ch9: 6:17

No worries.

jon_raw_pt_2: 6:18

the,

matt_ch9: 6:18

podcast is about. This podcast is one big tangent.

jon_raw_pt_2: 6:22

Yeah, the reason this is important is because Euler discovered that the zeta function is actually, and this honestly is like the most mind blowing thing I've learned about math in a while. This is one of those things I sort of relearn every now and again. But that zeta function that I just described, which is a summation, is equal to the product of all primes. But it's, it's the primes where it's, uh, uh, it's hard to like read a math formula out loud, but I'm going to do it anyway. So it's one over the quantity of one minus one half to the S. Remember S is just like a, a thing we're passing to the Zeta function. So one over one half to the S, and you'll notice that two is a prime number times. One minus one third to the S, three is a prime number, times one over one fifth to the S. So it's basically all those denominators are prime numbers. That is equal to this summation that's just one plus one half to the S plus one third to the S plus one fourth to the S. So that's like the first crazy fact is that those two things are equal to each other.

matt_ch9: 7:42

Mm

jon_raw_pt_2: 7:42

But then the next crazy fact is just that if we figure out that the Riemann hypothesis is true, that all of those zeros lie with complex numbers, where the real part is 0. 5, then we learn about the distribution of prime numbers, which are these things that are sort of famously unknown what their distribution is. Um, and so basically, if we were to figure out that the Riemann conjecture was true, Then we would have the math to describe the distribution of primes, which would mean that all of encryption, as we know it today, just doesn't work anymore

matt_ch9: 8:21

break down. Interesting.

jon_raw_pt_2: 8:22

because all of encryption uses these massive prime numbers. And if people can calculate them very easily, then encryption breaks very easily. But anyway,

matt_ch9: 8:31

conjecture, like, it sounds like the only thing we're missing is, is, is proof though. Like,

jon_raw_pt_2: 8:37

well, right. And, and that's the thing that I think a lot of people miss about the Riemann hypothesis, breaking encryption is it's not the conjecture itself that, you know, it's not like if you prove the conjecture you've broken, uh, encryption, it's the math that would be required to prove the conjecture would be the same math that you could use to calculate primes and break encryption. So it's, and I'm so sorry. This feels like a super tangent now that I'm like doing it. But the reason I was thinking about that is because working on the Riemann hypothesis today, you are working in raw math, like there.

matt_ch9: 9:19

There's no physical counterpart that, you know, there's no real use for it.

jon_raw_pt_2: 9:25

Exactly, you are like deep in a mine of just pure numbers and like insane, like even a complex number itself is like something that literally doesn't exist. Like it's called an imaginary number. So it's just all these completely abstract things that you're working with. And yet it's already known that the math required to solve it would break encryption. Who knows what else that math would do. Maybe it would be the math required to describe quantum mechanics. Who knows? That's the whole thing. It's like when we were developing hyperbolic geometry, we had no freaking idea what it would do. And then it went on to describe relativity. So that, to me, is just insane and why, you know, he's talking about these formal systems breaking down. But I, I'm almost seeing the opposite when I'm reading this material. Like I'm just thinking about how these formal systems have basically built up humanity instead of broken down anything.

matt_ch9: 10:21

Well, right. And the other question is like, why are we reading this book? It's like, am I, it's like, we have to, we have to tell ourselves that eventually it's going to be useful for something.

jon_raw_pt_2: 10:33

Yeah.

matt_ch9: 10:34

Um, but no, uh, so, um, so we talk, there's, there's a bunch of these Zen koans in this, uh, in this chapter. And they're kind of inscrutable in a lot of ways. Um, but. Like, and I guess maybe what we should say is like the deal with Zen koans is like they basically challenge the reader to like not try to they basically are attempting to be paradoxical in a way that gets the person to stop attempting to Find one interpretation of them Yeah

jon_raw_pt_2: 11:21

contradictory or they'll be like impossible to derive any meaning. It's like one of those two.

matt_ch9: 11:32

Um, and I think we then at the end of this chapter go on to have, uh, start to talk about, uh, girdle numbers. So this is, this is another really important part of girdle theory of incompleteness is he is able to express a statement as a number.

jon_raw_pt_2: 11:52

Right.

matt_ch9: 11:53

And so, um, so we start to. map all of the constructs from, so, you know, we, we talked in the last chapter about, um, typographical number theory. I mean, it has a bunch of like characters and letters and, and all of these different things. Um, and so now we're trying to attach numerical representations to all of those characters and then any statement in In typographical number theory can actually be expressed as this enormous number and it's just

jon_raw_pt_2: 12:30

right.

matt_ch9: 12:31

like It's it's just a number and like it's funny because it starts to lose all meaning because they're literally the most enormous numbers You've ever seen in your entire life

jon_raw_pt_2: 12:39

Yeah.

matt_ch9: 12:40

it's like we don't even have words for them because the number is like a hundred characters long, but it It's really just mapping on to like a sentence where it's like, okay Well, you know we deal with sentences that are hundreds of characters long all the time

jon_raw_pt_2: 12:54

Yeah. And it's, this was interesting to me. Maybe you can help me like understand this, but he took, he took our old Muse system, I guess we already said this, but like he took the Muse system and he replaced the characters of the Muse system with numbers

matt_ch9: 13:11

Yes.

jon_raw_pt_2: 13:11

and then he used typographical number theory. To examine that earlier problem that was, uh, thrown out in the first chapter, which is, can you generate M I I, um, and I think he actually used typographical number theory to prove that you couldn't, like he was saying, you know, he was saying, like, uh, you know, all statements have an eye in them. He used like one to represent the eye, and then he said something along the lines of all statements must have like a multiple of three. Which means that you can never have two I's. And he was, he was able to use typographical number theory to prove that. Um, so I think his point there was saying like, Hey, the minute you take a formal system and convert it into typographical number theory, you can like prove anything about it. But then he like took typographical number theory itself. And this is what, what you were saying where it was like, he was taking the individual symbols of typographical number theory and converting those Siri or those symbols into numbers, like the upside down a, that means like for all X or for all, he was taking that symbol and converting it into like six to six.

matt_ch9: 14:28

Right.

jon_raw_pt_2: 14:29

And then he could take a statement like. for all x, x plus zero equals x, and convert it into this epically long integer, like 626943792. But then he could take those numbers and use typographical number theory itself to work with

matt_ch9: 14:49

Well, that's what we're going to wind up doing. I don't think he actually, he, uh, he didn't actually get there. I don't think because like the one, one minor distinction and I could be wrong about this. I don't, I don't know that he ever actually stuck the like. The. Mu system, the M I U system into typographical number theory. I think what he did do was he, he created this like, this arithmetic mapping and like, so basically he mapped M and I and U two numbers. And then the other thing that he did was he mapped all of the rules to. Arithmetic operations that had the same effect. You know, you, you were like, Oh, okay. Like if, you know, rule number two is if we have something of this, you know, form in terms of a number, if we have a number that's like three times 10 to the M plus N, then you can perform this crazy mathematical operation, um, and so, which was, which was a direct quote, you know, it directly correlated with doing that same string like replacement.

jon_raw_pt_2: 16:01

Exactly. Like, like, changing M I to M I I I is the same as Multiplying 3, 1 by 10 squared plus 11 or something. Like he was just able to, yeah, take, take the statements and represent them as math.

matt_ch9: 16:19

And so I think that. So I'm not sure. I'm not sure. Maybe he did. Maybe he did. Then go on to take those mathematical operations and stick those into typographical number theory. And so maybe, maybe that is, you know, that's where you,

jon_raw_pt_2: 16:34

Well this is, this is where I guess I probably just don't understand is like, cause I, I completely understand this concept of like, you know, taking something that's like, or taking some system like the MIU system and converting it into math. So that you can then use math to work with it. Like that, that makes sense. But is typographical number theory the ideal system for, for that? Like is, is typographical number theory sort of the lingua franca of working with logic? Or is it more of just a concept?

matt_ch9: 17:20

I think, I think typographical number theory is Hofstadter's system. Like, I think he came, came up with it to make it easy to work with. Like, and he, he derived a bunch of, you know, uh, inspiration from all these other systems, but,

jon_raw_pt_2: 17:38

Well, so, and typographical number theory is very similar to the system that, um, was being come up with in Principia Mathematica,

matt_ch9: 17:46

That's my understanding. Yeah.

jon_raw_pt_2: 17:48

Okay, so it's, this is his way of almost like creating a minimal, like, logical system that can be discussed in this book in order to like,

matt_ch9: 17:59

That's like sufficiently complex. Yeah. Like it's kind of like turn complete. He created a turn complete system that, you know, meets that minimum bar.

jon_raw_pt_2: 18:12

right. Okay. Okay. I think that makes sense. I think what I was, I think what I was assuming is that typographical number theory was like this existing system that he was just describing, but I don't

matt_ch9: 18:23

it's, it's possible, but he has called out that, like, Hey, like I know that typographical number theory is like really awkward to work with. Um, so I do think it's like, you know, it's just useful because it's just, it's got the smaller set of fundamental, like principles, I guess. Um, but so what he hasn't done Is create mathematical equivalence to the rules. Like he had, you know, he has showed us the mapping. He showed us how the, you know, a particular statement maps into a number, but unlike with the mu system where he showed us what the like mathematical rule for, like, I don't know, like your generalization or specification is. So that's really the part that I think it's going to be really hard to understand. Because, because this is going to have to turn into a series of mathematical operations.

jon_raw_pt_2: 19:25

Yeah. So I guess he's going to do that in part two.

matt_ch9: 19:29

Yeah, I guess so. But yeah, so if you can believe it, we're only halfway through. We're less than halfway through this book. Um, but this was the last, yeah, this was the last chapter in this, um, You know, in this series, but so yeah, so, but just to, just to reiterate, and I may have already, you know, apologies if sounding like a broken record, but the idea here is you have these numbers and then, and then you're going to be able to have the system, like the system at its core is able to work with numbers. So once you have a system, like once you have a number that is a statement in the system, then you can have the system, like operate on. itself, you know, reference itself. Um,

jon_raw_pt_2: 20:17

get brain breaking. Um, but I think this is. This is what Gertl was able to do, to basically completely, uh,

matt_ch9: 20:28

yeah, to get back to like the Zen, like, I think that that's part of why he was doing that is to try to like, just introduce the idea Something can, can mean two different things in two different contexts. There's not just like one way to interpret something. Um,

jon_raw_pt_2: 20:52

yeah, I think there was another reason he got into that Zen thing too, which I feel like a lot of times in this book, he'll be describing a formal system

matt_ch9: 21:05

yeah,

jon_raw_pt_2: 21:06

and then he'll describe something that as a reader, you can sort of immediately know about the system. But that the system itself isn't able to describe because the system has these existing rules and none of the rules are able to like describe that thing. So there's sort of this concept of logic being this like trap, like it being this sort of structure that's like binding you and preventing you from like making these, I think he calls it M type, I want to

matt_ch9: 21:42

yeah, yeah. Well, M is mechanistic and then I is intelligent

jon_raw_pt_2: 21:47

Oh, okay. So this is,

matt_ch9: 21:49

fun mode. Yeah.

jon_raw_pt_2: 21:50

Oh, wait, sorry. Can you say that again? I was like talking over you.

matt_ch9: 21:53

Yeah, sorry. M is mechanistic. I is intelligent. And then you is on mode, which he's still kind of like teasing.

jon_raw_pt_2: 22:00

Yeah. Yeah. But in any case, like, uh, so if you're stuck in this sort of M mode, you can't do certain things. And I think this is what, this was another reason he was throwing out all this Zen is to just sort of break out of this mode, like break out of this mode where you're like. You know, understanding these sentences or, or break out of this mode where these sentences have to be meaningful and gain this ability to sort of like think about things at a meta level.

matt_ch9: 22:33

It's, it's interesting because. The mechanistic mode. Like, I think what's nice about the mechanistic mode or mechanical mode is that it gives you some guarantees. I think like. It's, you've, you've taken all these teeny little tiny steps. And so like, you're kind of, you have a stronger argument that like, it's consistent or, you know, you have, you've proven that these rules apply when, when you apply them in this, in such a way that they produce true statements. So, because he talks, he talks in the last chapters about, um, derivation versus proofs. And like the, the distinction here is like with derivations, you're working in this, this mechanical mode where you're saying like, you're making all these tiny little steps and it just, it just, at every point it's obvious. So, uh, whereas a proof is more, it's kind of expressed more in that like intelligent layer where, It's expressed in English and anyone who's reading it who can consider things intelligently can just know that it's true you know what I mean because like they're able to think intelligently and then like so so in some ways I think I I don't know like I think trying to avoid the ambiguities of natural language is like the goal But I don't know, like, is what you're saying, like, we should be trying to break out of M mode, or?

jon_raw_pt_2: 24:13

Yeah. I think, I mean, part of what I took this Zen section to be is like, it's almost kind of a statement about language itself. You know, it's like, it's, it's saying like, language is only like, so able to represent thought

matt_ch9: 24:35

Yeah.

jon_raw_pt_2: 24:35

and, and like, in a way I think some of these koans, like they defy understanding in the same way that I think like art had this like Dada movement, for instance, where it's like, people would just do crazy stuff. You know, they would have like a broken toilet in the corner or something. And it's like, it doesn't really mean anything, and it sort of defies description, but that's like why it exists, is to like, to break us out of like this whatever current mode we're in, and to sort of think about the meta space, you know? So I don't know, like, like I get your interpretation of the Zen thing, and I agree, but I, I, I also Thought there was this other interpretation that's sort of like, you know, Zen is this way of, of like, I'm not even going to try to decipher the meaning of, meaning of this because I'm like above it. You know, I've broken out of that, that space.

matt_ch9: 25:34

Yeah, and this is another thing where it's like, I feel like the Zen mode, like, are Zen masters, like, building cool things? Like, are they getting anything done? I feel like the Zen, it's like, I've achieved enlightenment, so I just sit in a cave. And I don't do anything all day. And it's like, I'm not sure that that's what I want to do. Like, I'm not sure that I'm going to think too hard about these koans because I want to like do stuff.

jon_raw_pt_2: 26:00

There was an interesting statement that I think was part of the Zen section. It might have been like a Zen master referring to another Zen master where he was saying, Your death is like a snowflake dissolving. Which I thought was this really cool, this really cool statement about sort of how things organize briefly and then they, and then entropy happens and they get sort of disorganized, which is kind of what all the human races, you

matt_ch9: 26:28

Oh yeah.

jon_raw_pt_2: 26:28

the, the second law of thermodynamics is entropy. The, the universe will tend to randomness. Eventually we will just be like a bunch of energyless particles floating around in a void. But for now. For some strange, strange reason, the universe is organized at this one point in space and formed the frickin human race, which is, which seems like this super organized thing, I mean, where these weird biological creatures that can do all this crazy stuff, but that's within this larger entropy, which I think is so crazy and kind of poetic and kind of spiritual. Um, and I think that, that whole, your death is like a snowflake dissolving is very, very thematic.

matt_ch9: 27:16

Yeah. Yeah. There's, there's this, this term or like, there's this conception of like people as like waves where like compressions of particles, because it's like, our bodies are constantly renewing. So it's like. Who are you? You know, it's like if, if every single, every single atom in your body is different after seven years or whatever it is,

jon_raw_pt_2: 27:41

Yeah.

matt_ch9: 27:42

it's like, you really are just a compress, like a compression of particles that are just like self sustaining, but it's like you're, that is kind of, that exists at another level, like a level above particles, you know what I mean? And this is kind of where intelligence starts to come in, where it's like, There's this thing that exists that even though you know ship of theseus style every piece of it has been replaced

jon_raw_pt_2: 28:09

Yeah.

matt_ch9: 28:10

the item exists actually at a higher layer than than that, so

jon_raw_pt_2: 28:14

Dude, I think this is actually, I, I have this, uh, note in my, uh, or I have this, this thing in my notes. I think this is why Hofstetter wrote this book, is to sort of describe how weird it is. That humans are like representing knowledge, like just this whole concept of like, there's these like very inexpressible things that, that humans are trying to represent. And they always tend into these like. You know, layer upon layer of like recursive reasoning. And that's, that's sort of like what the human condition is. Uh, and I think, I think that's part of what Hofstadter is getting at this, which is why I sort of like this Zen section, because I feel like the Zen section really gets into this, just being very, uh, or just really calling into question, like, the meaning of meaning, you know, just getting hyper meta. Uh, and I think that's kind of what this book does. is on a certain level.

matt_ch9: 29:15

Yeah. Yeah. Yeah for sure. Well, I I feel like I could I mean like we talked about we could talk about this forever because it's almost like um, you know when there's a he talks about this in the first chapter when you have a Contradiction You can say anything you want about it. You can prove anything.

jon_raw_pt_2: 29:30

Yeah.

matt_ch9: 29:31

that's, that's why Zen, and people can talk about Zen forever. Cause it's like, well, it's just makes a series of contradictory statements. So you can just take anything you want from it. Um, but so, um, but yeah, I will, uh, I will leave it there. Um, this was, this was a weird one, but, uh, it's good. It's, uh, you know, interesting exercise.

jon_raw_pt_2: 29:50

Definitely. All right. Well, I will see you in the next one, man. Woof!

matt_ch9: 29:54

the next one.