Matt: 0:14

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

geb_matt_ch1_p2: 0:31

Hey, John, how are you doing?

jon_raw: 0:33

Hey, Matt. How are you?

geb_matt_ch1_p2: 0:36

I am doing all right. I've got my, um, abstraction helmet on. I'm ready to just dive in heads first into the the punishing world that is, uh, Gödel's theory of incompleteness.

jon_raw: 0:50

Yeah, man. This, this theorem beats me up. I, as many times as I read it, I still don't understand it. Like, I think my brain is just not smart enough to understand this. Yeah.

geb_matt_ch1_p2: 1:09

to explain it in a tremendous amount of detail, later in the book. So, so this is only going to be, I mean, we're going to, we're going to keep on strangely looping back to this, to this point, uh, you know, throughout this series. So this is just going to be a little, like just a gentle introduction to, uh, to this, this concept, I suppose,

jon_raw: 1:30

So gentle, but it feels like it just punched me in the face.

geb_matt_ch1_p2: 1:34

Yeah, I think we just need to get you get used to get used to that. But, um.

jon_raw: 1:39

Yeah. Yeah,

geb_matt_ch1_p2: 1:41

I think we kind of start off with the basically. The core what I think is kind of the core of all these paradoxes. I think they all kind of boil down to this statement is false with more steps, you know, and he calls this the. At the amenities. Paradox, which that's one of those words that I read and I don't ever like Figure out how you actually are supposed to say it

jon_raw: 2:08

Yeah, I'm like, is it Epimenides? I don't know.

geb_matt_ch1_p2: 2:11

I could see that I could see that.

jon_raw: 2:13

No, I like the way you pronounced it. So this statement is false. So it's, it's the type of statement where if it is false, then that makes the statement itself true, which is obviously a paradox. If the statement is true, then what it's saying can't possibly be true. Okay. So anyway, it's just a statement that like, doesn't function like

geb_matt_ch1_p2: 2:37

Yeah, it makes it makes a claim about its own truth Value In in such a way that It's actually like it, it would always contradict whatever assignment of truth, you know, basically it's like this statement contradicts with itself

jon_raw: 2:54

yes.

geb_matt_ch1_p2: 2:55

and yeah, so, so, and, and again, like this, this self reference, I mean, self reference, which we're going to keep on getting back to, um, it's kind of at the heart of girdle theory of incompleteness.

jon_raw: 3:09

Yeah. And there is like, I'm sure we're going to discuss a trillion examples of like this very similar construction he calls them, but there's a very similar one that I also like called Russell's paradox. And this one is a little bit more, um, Abstracts because it's a, it's in set theory, but I think it's to the point where a normal human being can still understand it, but Russell's paradox is, uh, basically he describes it as like, you know, most sets don't contain themselves, like you think of the set of all gorillas, like the set of all gorillas doesn't contain the set of all gorillas because set is this abstract mathematical concept. It's not a gorilla. Therefore that set does not contain itself. Some sets do contain themselves, like the set of all things except gorillas. So that set actually has its own self in it because it is not a gorilla. Therefore it is in itself. Um, but if you think of a set of all sets that don't contain themselves, that set does contain itself, but its very definition is that it shouldn't contain itself. Because it's the set of all things that don't contain themselves. So I don't know if I explained that very well, but that feels like a very, you know, it's a very mathy version of the same epinides or epiminides,

geb_matt_ch1_p2: 4:36

Yeah.

jon_raw: 4:37

paradox. But I sort of, I sort of liked that one cause I can still sort of understand it. And there's, there's almost like a visual or physical way of understanding it. Yeah.

geb_matt_ch1_p2: 4:49

you know, because obviously in the physical world, something can't contain itself.

jon_raw: 4:53

Yeah.

geb_matt_ch1_p2: 4:54

So.

jon_raw: 4:55

Well, I think it just gets into this, you know, cause I view all of these things as like, almost like linguistic puzzles where, you know, reality has, is governed by the laws of physics and the laws of physics can't be broken. And so you can't create some construction in reality that like, is a paradox. But obviously when you start creating. Representations or, you know, if you start getting into like the written word, then the rules in that realm are very, um, they're very arbitrary. They're not, you know, it's not like the laws of physics where you can't possibly break them. Like you can just write a sentence that breaks them. And so I've always viewed all of these things as sort of fitting into that category where it's like. You know, it's, it's just that you've devised a system that enables you to express something that like breaks the system. I feel like I, I actually, let me take one step back because he kicks off this section with a sentiment that I really like. And, and this really resonates with me. He sort of, he says something to the effect that like, we have become more used to jankiness in science. Where, like in the modern age, we have like Heisenberg's uncertainty principle. You know, we have like a probabilistic understanding of reality. You know, there's all these experiments like the, um, the slit, the double slit experiment experiment where it's like the results are just very contradictory and weird. And I think we've sort of gotten more used to like the reality just being strange and like hard to. Hard to describe in some ways. So I think we're less concerned about these like hyper formal systems that have to be able to like, you know, represent the entirety of reality and never, never break, but during this time in history, the thinking was very different, you know, everyone wanted these very formal systems where, you know, all truth could be represented and it was impossible to, you know, to break and. It was complete, meaning, you know, completeness, meaning it could fully represent reality. but that's just not the case. You just can't do that.

geb_matt_ch1_p2: 7:24

And this is, you know, you, you brought up the Russell paradox. Bertrand Russell and I forget Whitehead's first name. Um, and I think we've talked about, uh, Principia Mathematica before, maybe not. This was kind of a, this was an exercise. I mean, this was a saga apparently in trying to come up with this system, you know, a formal system that. Allows you to prove everything that is true in mathematics and the way he got around, you know, so the way he solved Russell's paradox was you're only allowed, like, there's basically a hierarchy of sets where the base set is only allowed to contain things that are not sets or objects. And then the next level of sets is only allowed to. the base level of sets or objects. So, and then you kind of, now you break this self reference because it's not even a lot, like it's an invalid. It's not a valid set.

jon_raw: 8:30

Exactly. Yeah. And I think this is going to be another strong theme in this book, which, which is that Russell and Whitehead sort of broke reality into tiers where, you know, this lowest tier is these super simple statements, like the set of all gorillas. You know, that's like a relatively simple statement. It's sort of easy to understand. Then you have maybe this higher order thing, like, I don't know, like the set of all things that aren't gorillas, which is like a little bit more abstract and weird, um, and in this way of presenting these tiers. You've sort of given yourself an out. You know, you've sort of said like, Oh, you've constructed an invalid thing. Well, that's just in a higher tier and we're not going to talk about that.

geb_matt_ch1_p2: 9:09

Yeah. Yeah. The challenge for girdle becomes. How do I, how do I get, like, create a loop? They, you know, uh, Russell, uh, and Whitehead have attempted to forbid these circular references. And this is ultimately what, you know, this is kind of the core of what Gertl does is he is able to create a, you know, and, and, and correct me if this is stupid. You have a different understanding, but basically he's able to create a claim and represent it as a number, thereby putting it at the bottom of the hierarchy. You know, it's like you have, and this is kind of where the loop comes in, where he comes up with these translations from logical statements to numbers.

jon_raw: 10:05

Right.

geb_matt_ch1_p2: 10:06

And then anything about, and then that's where the loop comes in. It's like, okay, once you can translate logical statements to numbers, then everything, all the other rules apply.

jon_raw: 10:15

Yeah.

geb_matt_ch1_p2: 10:16

And so that means that, like, you know, and this is where I think my, my understanding starts to break down is like, what does the fact that you've put a. represented a claim as a number. How does that allow you to break, the possibility of completeness? Like that's, it's still not clear to me.

jon_raw: 10:40

Yeah. Now, and this is like extremely difficult for me to understand. I mean, I don't understand it, but yeah, I think everything you're saying was accurate, like girdles whole thing. is he basically proved that this whole tier system doesn't work because you can take these impossible expressions and express them in those lower tiers. Like he's basically saying you can convert between tiers by, and the example he does is he converts all statements into numbers, which means that, you know, they exist in this low tier, just like you're saying, I'm like repeating what you just said,

geb_matt_ch1_p2: 11:18

Is it number an object in this system? I just want to like, make sure I'm understanding. So a set can contain numbers. Is that correct?

jon_raw: 11:28

Uh, I think so. I mean, yeah, I'm not going to be, I don't really know the answer, but I think so. I think that's the point is like numbers are these elementary building blocks. And so what Gödel is saying is like, you know, you can take this, this statement is false and you can represent it in whatever fundamental building block that your formal system has. And therefore there's no formal system that can possibly represent all truth. You know, there's no, there's no such thing as a quote unquote, complete formal system

geb_matt_ch1_p2: 12:01

Right. And.

jon_raw: 12:02

will, yeah.

geb_matt_ch1_p2: 12:03

And this is the crazy claim that this book is making, is Gödel's theory of incompleteness. Like it doesn't just break Principia Mathematica, it manages to prove that there cannot exist a system of I don't know what it is, like a logical system such that everything is, everything that is true is provable.

jon_raw: 12:27

Yes. Yeah. So it's, it's basically, you know, that whole era that I was just describing where, you know, the entire math world was sort of working towards this fully, uh, you know, fully defining or, or having a formal system that could be able to fully define reality. For the rest of time, which would have been this, you know, huge, huge discovery. And, and this is what Russell and Whitehead were like trying to do. That's what the Principia Mathematica was trying to show. It was like, Hey, here's a formal system that actually does it. You know, Godel came down with a complete atom bomb and just annihilated that whole way of thinking he sort of, I mean, he sort of Whitehead. Like they had this whole book, but he was like, nah, I'm You know, he dropped, dropped Mike on them and like made them look like jerks. And, and basically showed that that whole way of thinking was a dead end. You know, it's like, we got to move on guys. Like incompleteness exists. Just forget about it.

geb_matt_ch1_p2: 13:29

Which in a way, I mean, it's kind of a. Kind of a blessing, you know, to be shown that it it's not possible. So you can just spend your time doing something else.

jon_raw: 13:41

Yeah. Yeah. And incompleteness, I don't fully understand this relationship, but there's, there's actually, Like one complaint I've heard about incompleteness is like, yeah, but where does this actually manifest itself? Like in a way that actually affects humanity. Like, it just seems like this hyper, hyper abstract thing. And it is, but there's a few manifestations of it that are a little bit more practical, I would say. And one of them is the halting problem, which is something that Turing introduced that basically says, given a program, like a computer program. And its input, you can't prove whether the program will continue running forever or we'll stop at some point. Um, and this is actually like an important fundamental principle in software engineering. So it's kind of a, it's kind of a tie over that is a little bit more, a little bit more real, I would say.

geb_matt_ch1_p2: 14:40

I think this ties so much to, yeah, to programming and especially like recursion and, you know, um, This is gonna come, come back a ton, I'm sure. And he even, he even starts to, he introduces computers. but actually before we get onto that, there was this point, this, this claim that he made, which I'm, I wanted to look for the exact wording, it's, it was something, something to the, to the, to the effect of that, Provability is weaker than truth, essentially, that it's like, or, or rather, the set of things that are provable is a strict subset of the things that are true in any logical system.

jon_raw: 15:20

Yeah.

geb_matt_ch1_p2: 15:20

which I don't know. That seems, that, that seems like the coolest, the like coolest formulation of what Girdle's Theory of Incomplete

jon_raw: 15:28

Yes. Yeah. Yeah. That's, yeah, I'm glad you brought that up.

geb_matt_ch1_p2: 15:32

Because that's the thing, it's like the statement is ultimately true. The statement, and I don't think we said it, um, the statement that, uh, that Gödel translates into numbers is, uh, this statement of number theory does not have any proof in the system of Principia Mathematica. So it's not that. It's, it doesn't, it's slightly different than the, like, oscillating nature of the, this statement is false. Um, it's just true, but it forbids its own proof. Um, so, uh, which I think that is, in, in a lot of ways, I think that makes it a more interesting statement because the, it feels like it's doing real work.

jon_raw: 16:20

Right. Yeah. It's basically saying that there will always be these nooks and crannies of math that will kind of be unapproachable, like you're not going to be able to create a system that will be able to encapsulate those things. So it's a super interesting side effect of incompleteness.

geb_matt_ch1_p2: 16:39

um, at the end of this, he starts to, uh, talk about computers and, uh, and Babbage, uh, actually, uh, former, former topic, uh, of

jon_raw: 16:50

Oh yeah, I love, love Babbage. And he, he throws out a factoid about Babbage that I'd never heard that I, that I totally loved. Apparently during Babbage's day, there was these organ grinder guys, basically these guys that would, you know, they'd sit there with their little organ that, where you turn the crank and it like plays a song. And they would, I guess they were like panhandlers, basically, like, you know, trying to get coins from people and apparently Babbage would like run these guys out of town because they were like, they were like infuriating him, which I don't know, it just, it seems so Babbage to me because I just see Babbage as this like curmudgeonly old man.

geb_matt_ch1_p2: 17:27

Yeah, this is grim. Um, but so he starts to, he very quickly kind of pivots towards, uh, machine intelligence.

jon_raw: 17:36

Yes. Yeah, he mentions Ada Lovelace. Yes.

geb_matt_ch1_p2: 17:40

Oh yeah, yeah, he definitely, he talks, talks about Ada Lovelace and, um, And yeah, she, I don't know, did we want to like, uh, talk about her, uh,

jon_raw: 17:49

Yeah, yeah. I just, I mentioned her cause like, um, she was one of the first people, if not the first person to sort of realize that with this, these algorithms, you know, these logical sequences of statements, you can do anything like she was one of the first people to realize that the implications of that go way beyond logic, way beyond math. And I think that that's just such a huge, huge mental breakthrough.

geb_matt_ch1_p2: 18:17

Yeah. Well, my understanding is, she is considered the first programmer, like, Babbage didn't, he thought of it as being able to be used for like math, but she came up that you could like actually sequence like these programs together or something like, I don't know. That's, that's my rough understanding.

jon_raw: 18:37

Oh yeah, no, I think you're absolutely right. And it's, if you think about it, like that breakthrough is. Our whole entire modern world is kind of based on that breakthrough and that sort of extension of these logical sequences of statements. Um, so I dunno, I just, that's one of, like, I really love Ada Lovelace and that's one of the reasons I think she's like a pretty cool figure in computer science.

geb_matt_ch1_p2: 19:07

Oh yeah. They should, are there two TV series that follow her life? I feel like they should, they should have that.

jon_raw: 19:14

They should, cause she's, she also just has like a very interesting life. Like her dad was Lord Byron. You know, the Famous poet who was like running all over the, running all over Europe, like having sex with everyone. Um, and she had kind of an interesting upbringing. Like she had a very solitary childhood. She was sick for long periods of her early life. So she was kind of by herself. I just listened to a podcast where they talked about how she had a project where she was trying to duplicate bird's wings, except for human size.

geb_matt_ch1_p2: 19:45

Ooh, nice.

jon_raw: 19:47

and I guess she was like reaching out to these anatomists who could like describe bird wing anatomy to her. It's like, she's like this 14 year old girl, like doing these things and it's just very cool.

geb_matt_ch1_p2: 19:59

Dude, so, you know, you can't say who is gonna who's gonna catch that, that itch where you're just like trying to solve problems like it would be so cool to fly and she's like, Hey, let's, let's just go about, you know, figuring it out. Yeah.

jon_raw: 20:14

She had sort of a courage, sort of an out of box thinking that I just think is so cool where, just like you're saying, it's like, you know, some goal that seems completely impossible. She's like, Oh yeah, I'm just going to like, Give it the old college try going to reach out to a bird anatomist and try to duplicate bird wings.

geb_matt_ch1_p2: 20:31

so moving on from at a loveless, she, uh, he talks about what, what a machine is. Wouldn't need he's saying that the user would all be necessary, but maybe not sufficient per se for to have an intelligent machine And man, it feels like we are we are already checking like checking all of these boxes

jon_raw: 20:52

Oh yeah. Yeah.

geb_matt_ch1_p2: 20:53

Yeah, so it says respond to situations very flexibly take advantage of fortuitous circumstances that's actually an interesting one because I feel like I'm not sure that we want a machine that's taking advantage of fortuitous circumstances.

jon_raw: 21:05

Yeah.

geb_matt_ch1_p2: 21:06

Make sense out of ambiguous or contradictory messages. Recognize the relative importance of different elements of a situation. Uh, find similarities between situations. So, I mean, there's like, there's a bunch of them, but like, um, you know, synthesize new concepts by taking old concepts and putting them together in new ways, and then come up with ideas which are novel, like I think chat GPT is doing all of those things today.

jon_raw: 21:31

Yeah. Yeah, it sure is. I mean, I, I wonder, I mean, Douglas Hofstadter is going to have to write a new forward for this book soon because. I mean, he even has chapters on like AI, and I'm very curious to read those chapters and sort of, you know, the, what the level of thinking would, would have been back when he originally wrote this book because we've come a long way.

geb_matt_ch1_p2: 21:54

yeah. And yeah, just how, how he sees, is this a consciousness? Cause he's, he has already kind of criticized the concept that there's something special about human intelligence. Like,

jon_raw: 22:07

Yeah.

geb_matt_ch1_p2: 22:08

is there something? You know, different at the physical layer with our brains, like, no. And he's, he kind of says, there's no reason. And he talks about this, like logical systems can convey a certain sort of like self awareness. Uh, so why can't, why can't a machine? So I think he would probably be of this, of the side that like, Actually it is. You know, chat GPT is approaching general intelligence.

jon_raw: 22:37

Yeah, no, I, I agree. If not, you know, surpassing it in a lot of ways. Uh, or at least surpassing human, you know, human's abilities. Um, he mentions a couple of things, a couple of other things in this chapter that I really liked. One thing he talks about is how, you know, like, like you were saying earlier, mathematicians tried to move onto systems where like these types of contradictions just couldn't exist. But he mentions that having these types of contradictions and sort of holding them together in one, you know, Sort of in, in your brain is at the core of intelligence, like being able to deal with this type of nuance where like, okay, these things don't sort of fit perfectly together, but maybe I can make practical use of them. You know, being able to think of all of these rules and meta rules and all of these different layers of reasoning about a system, like those are all absolutely essential to understanding reality. And I found that to be very interesting. So it's almost like if we were to have a system that basically didn't go into this metaphysical space, like we'd only be able to get so much mileage out of that system

geb_matt_ch1_p2: 23:52

Yeah, we wouldn't exist, you know,

jon_raw: 23:54

So,

geb_matt_ch1_p2: 23:55

But yeah, I, I, maybe this is like, um, hubris, but I feel like the solution to the like paradoxes of these, like truth statement assignments is kind of like. It's just like you think about, you know, and I'm coming from the computer science background, but it's like, you just kind of trace it through time. Like you add this element of like iteration of evaluation and it's like, it feels like, Oh, like it doesn't have a valid truth assignment. And it's like, technically, yeah, that's true, but it's actually, you can kind of think of it as just like this infinite, and this is kind of like a halting problem thing where it's like. You think of it like a computer program and its truth assignment is actually this, you know, it's this infinite, evaluation that's just oscillating back and forth.

jon_raw: 24:46

yeah.

geb_matt_ch1_p2: 24:47

that actually is its truth assignment. Um, you know, and, and when, if you allow these, if you allow the progression of some semblance of time, maybe it's like, you know, logical ticks, then. It stops feeling as like paradoxical. And it more just feels like, you know, when you allow that, that like time dimension that it's just like, yeah, most things you evaluate them and then in wine time step, they have a value, but sometimes you evaluate them and then you need to, and then like in a recursive function, like that's one where. It's calling it, it's referring to itself multiple times, but it stops at a certain point.

jon_raw: 25:29

Yes. Yeah. Yeah. This is why I think that that sort of connection to programming is so fundamental to this book. Yeah. Cause yeah, if you think of these mathematical statements as like a logical sequence of, of statements that are evaluated in order, you know, where that evaluation just sort of like has a go to in it and it goes to an earlier part of the statement, like, yeah, I think that's actually a really cool way to think about this and kind of demystifies it a little bit.

geb_matt_ch1_p2: 26:00

Right. So, um, but yeah, I don't think, I don't think I had anything else. Did you have any other thoughts about this, this chapter?

jon_raw: 26:09

No, that was all I had. I'm, I'm simultaneously excited about this book, but also terrified by it because I think there's going to be a lot of things that will be very difficult for me to like talk about.

geb_matt_ch1_p2: 26:22

Oh yeah.

jon_raw: 26:23

But I think it will be fascinating. Like this first chapter was really interesting and it was, it felt like it was just scratching the surface. Like it felt like he was intentionally pulling his punches. So I'm, I'm very excited to get deeper into this.

geb_matt_ch1_p2: 26:45

The Mew Puzzle

jon_raw: 26:46

Oh yeah. The Mew puzzle. All right, Matt, I'll see you next time.

geb_matt_ch1_p2: 26:50

All right, see you next time