Matt: 0:14

Hello everyone And welcome to the switch statement podcast It's a podcast for investigations into miscellaneous tech topics Hey, John, how's it going?

Jon: 0:33

Matt. How are you?

Matt: 0:35

I, uh, I could answer that question by telling you how I am, or I could say every way I'm not and tell you that I'm the inverse of that. So which, what's, what do you, what do you want?

Jon: 0:52

Uh, let's go with the ground.

Matt: 0:55

Yeah, we'll go with, we'll go with the ground. Well, I'm not bad, not bad, I would say.

Jon: 1:02

Excellent. So therefore you're good and everything else besides not bad, which includes you're a criminal. Um, and you should go to jail. But anyway,

Matt: 1:14

heard it here first.

Jon: 1:15

did you read the intro to this?

Matt: 1:19

The, are you saying like the, uh, the sonata for unaccompanied Achilles?

Jon: 1:25

Yes. Yeah.

Matt: 1:27

I'm, I'm sticking with my, uh, sticking with my policy of not reading the dialogues.

Jon: 1:33

Dude, that's a strong policy because these dialogues are insane. And I almost feel like they're getting weirder and weirder. Like this one was literally just, or I don't know if literally is the right word, but he, it seemed like he had written out a full dialogue with like several characters. And then he just deleted all the characters except Achilles. So it was like a ton of non sequiturs because Achilles would be like responding to something that someone else had said. Um,

Matt: 2:03

He probably did. I would not put that past him.

Jon: 2:05

yeah, he probably did. but in any case, I feel like reading these intros is like,

Matt: 2:13

well, it's interesting because I guess, I mean, I'm sure this is the point, but you are defined by the app, what, you know, the absence of the other things. So I guess that's like, that's the point of the exercise is like, even when something's not there, that still provides some definition to, you know, the other things. Um,

Jon: 2:35

I guess in this case, it didn't provide a lot of definition.

Matt: 2:39

it's, it's still like kind of unpleasant.

Jon: 2:41

yeah, but it was very thematic for the chapter. Um, so this chapter opens up with kind of a cool idea. He introduces yet a new system, which I'm super excited about all these systems. I'm like, I'm loving it. Uh, it's a TQ system, which in the previous chapter he had introduced the PQ system,

Matt: 3:01

a little cagey at the beginning.

Jon: 3:03

yeah, yeah, but I, I saw through his shenanigans immediately TQ thing was going to be like multiplication, uh, because the PQ thing was addition. So, hyphen, hyphen, P, hyphen, hyphen, Q, hyphen, hyphen, hyphen, hyphen, that's two plus two equals four. So the TQ system works similarly, where you just kind of multiply the values of the two hyphens.

Matt: 3:27

Well, we start off, I mean, but the, but the, the larger goal is we want us a typographical system to generate primes.

Jon: 3:38

Right. And, and the TQ system is a proposal to try to do that.

Matt: 3:44

It's kind of in service of this larger prime goal.

Jon: 3:48

Yeah. Um, which, I think we quickly figure out that it's not really possible with the TQ system. Or at least that was sort of my understanding of this section.

Matt: 4:02

Well, yeah, I mean, he, he gets it. It's funny because it's basically, so basically like just very quickly, he, his proposal is, and, and I won't go into too much detail about the, I don't think it makes sense to spend too much time on it, but essentially he does come up with a way to generate like multiplicative numbers. You're, you know, you come up with these rules. They are, they can generate strings that. Are multiple, you know, multiples of one, you know, hyphen, hyphen T hyphen, hyphen, hyphen Q. And then you do have a valid, like multiplication result on the other side of that. So,

Jon: 4:41

Right.

Matt: 4:42

and then what he does is he uses that to generate all of the composites. he tries to just say, well, All of the strings that are not in the composites are the primes,

Jon: 4:55

Yeah,

Matt: 4:55

which, which I was like, yeah, I don't know. It seems, seems legit. It's like you, you just generate all of the set of numbers and then you just like remove the, the, the primes. But I guess that really doesn't, he says you have to work outside of the system in order for that to, that to work

Jon: 5:14

Right, it's like within the system, you can't refer to things that aren't in the system.

Matt: 5:20

exactly. And even It seems like by the time you've generated a string, like the act of generation is inclusion in the system. So there isn't a way as far as I'm aware, and this is how I'm reading it. There isn't a way to like generate the string and then to like, remove it somehow. Right.

Jon: 5:53

Yeah.

Matt: 5:54

Because, because you, you could imagine a system where it is like, okay, you generate all the composites and then you generate all the numbers and then you do like a lookup and then you're like, well, I've generated it, but I looked it up in this blacklist of, uh, you know, of numbers and it's not there, you know? And it is in the blacklist. So like, we have to boot it out, but like, there's no such booting out. Like you need to have all of these like positive generation rules that. There's no way to even generate the number in the first place. If it doesn't fit in the system,

Jon: 6:30

Right, exactly. Um, this, this led me to a thought that I think is just so powerful, which is like figuring out the right abstraction to discuss a problem and to work on that problem.

Matt: 6:44

Mean like just hypergraphical strings is not the most efficient way to generate the primes.

Jon: 6:49

well, and at the time that I wrote this note, I thought that this is what Douglas Hofstadter was getting at is like, you know, humans develop all of these different systems for working with things. But some systems don't work that well with certain problems. Some systems can solve certain problems, but they're just, it's just very difficult. And every once in a while we come up with a system that just works phenomenally well to solve certain problems. And I think about like humans understanding of the universe where, you know, obviously there's this huge history to this, but you think about like Newtonian mechanics, which was Isaac Newton proposed these set of fairly simple rules for understanding how physical objects move around. And as we learned more and more about the universe, we quickly realized that for certain types of issues, in particular, like the movement of, you know, stars and planets and stuff like that, that system breaks down. And so we had to develop this new abstraction. Which was basically Einstein's, you know, theory of relativity, which added a bunch of new components to, to Newton's original, rules on, on how objects move. And, and eventually some of Einstein's rules weren't fully solving problems. So then we developed, uh, you know, the wave function and all quantum mechanics, which I don't understand at all, just to be super clear about that. But, but I guess my point is like. It feels like throughout history, we've developed all of these new abstractions for talking about reality. And it's just, it's, it's super interesting to me because I feel like one day we'll develop an even newer abstraction, which will basically unlock all these new ways for understanding our own universe and basically unlock superpowers. But it's not like we've, it's not like we've like, you know, found vibranium or something like that. We've just. We've discovered a new way to like think about things and that in itself, just that new way of thinking about things has like unlocked new abilities, which I just think is awesome.

Matt: 9:00

Oh yeah. I mean, I guess, like, I was, I was thinking about this in the context of, uh, like software engineering where it's like, Newton's theories worked really well in like, In all of the cases that mattered up until, like, you know, the turn of the century, like, the 20th century, basically. And they did start to see, they had these, like, predictions that were, like, not, didn't match. And, like, they're like, okay, well, that's weird. Like, I don't know what's happening there. But, um, but, Then, you know, for these very like high fidelity cases or like these very specific cases, like, yeah, you do, did need like to onboard all of this additional complexity and, and which along with it came with all these other crazy, like black holes and all this other stuff. But even today, like there's a lot of cases where Newton's You know, Newton's rules or Newton's like theories of gravity, gravity is still are perfectly fine. And like, sometimes I feel like, and you know, those tie back to software engineering is it's like, sometimes I feel like I'm working with people who want to like, like implement, you know, Einstein's general theory of relativity in this like system. And it's like, dude, like we barely need no Newtonian gravity for this system in terms of like the complexity of the problem that we're solving.

Jon: 10:26

Oh, my God. A hundred percent. Yeah. And, and this is why, I mean, if you look at a video game, like physics engine, they're not, well, typically they're not implementing like Einsteinian, you know, motion and things like that. Uh, because as you're saying, like Newtonian physics does an amazing job. Of representing reality. And it's like incredibly simple. There's like five formulas that are like three variables. So it's, you know, it's an amazing abstraction and it's so useful and effective, but there's, there's just these mysterious nooks of the universe that it doesn't really work with.

Matt: 11:02

Right, right, right. Um, but anyway, so to get back to, um, or so did you have like a larger, like a

Jon: 11:11

No, I,

Matt: 11:11

with, yeah, right.

Jon: 11:15

that this is where Douglas would be. Was heading is you sort of have to figure out the right formal system in order to even approach a problem. And I, I feel like there's this thing that happens often in human history where we just don't have a system to even touch a problem. It's like we can never, you know, with the current systems that we have, we will never make progress. And I don't know, I just, I thought that that's where he was going, but he proceeds to discuss intelligent mode versus mechanical mode, which is these ways of working within a system. And I guess mechanical mode is just applying the rules of the system. Um, and I wrote in my notes, it's hard to determine if something is not a theorem in a system this way. In fact, you can't determine if something is not a theorem. A theorem in a system this way, because all you can do is just generate new theorems that do exist. And in the case of the TQ system, like you were saying earlier, in order to find out the primes, you have to find out what's not a theorem in the system. Um, and mechanical mode is just not, not capable of that.

Matt: 12:26

Right. And so he, it sounds like what he's saying is you don't even realize it, but trying to define it as a negative requires I mode, you know, intelligence mode. Right.

Jon: 12:38

Right,

Matt: 12:39

And we want a system that generates primes without having to break out of the system.

Jon: 12:45

exactly. Cause I mode is, is illegal, basically. Yeah.

Matt: 12:52

Um, but, uh, yeah. And, and like a lot of what he goes into is, and I think we've made allusions to this, but figuring ground. And if you're not familiar with like the artistic meaning of these terms, figure is. Like the subject of your painting and ground is like the background. Uh, and so, and, and the, the parallel or the isomorphism, if we're using the term from last episode, is that the figure is the theorems, these strings that we're generating, and you're not allowed, like, I mean, and this is what we just said, you're not allowed to. define e theorem as like, well, this is just the ground of the, of the theorem.

Jon: 13:36

Yeah. Yeah. Yeah. And he, he discusses a couple things that I thought were interesting. He talks about cursive figures versus recursive figures,

Matt: 13:44

I hated this.

Jon: 13:46

yeah, it's a, it's a term he coined, I think. Um, and basically a cursive figure is like a normal drawing, you know, like if I draw a stick figure, that's a cursive figure. A recursive figure means that the drawing itself lies in the negative space. So he gave a few Escher examples where he had sort of drawn these, like, you know, I think there were birds, he drew these cool birds, but like in between the birds, there was also birds. And it was like the negative space consisted of all these birds just because of, you know, the shapes that were there, um, kind of an interesting concept. But then what I thought was more interesting was figuring ground in music. So, um, yeah. Yeah. Which again, he brought up our, our hero of the book, Bach. And he talked about how Bach often combined melodies. I mean, we discussed this before in the previous chapter, but there's a technique Bach used called contrapuntal melodies, where you basically have multiple melodies that are interwoven. and this could mean like, you know, your left hand is playing a melody. Your right hand is playing a melody. It could mean that like one of your thumbs is, is playing a melody while You know, your hands are doing something completely different. it's, it's one of the reasons why Bach is sometimes very, very hard to play because you can't just kind of turn your mind off and play some rhythmic thing. You have to like be focusing on multiple melodies. and then he also mentions this notion of like on beat versus off beat and music, which again, Bach would often put melodies in, you know, in a way that's In the offbeat, he'd hide one melody in the onbeat and one melody in the offbeat. Um, and I, I wrote down another example of this, which I wasn't a hundred percent sure was a good example, but there's a piece called Moonlight Sonata by Beethoven. It's like a relatively famous piece. but there's a section in the song where, you know, your pinky is kind of playing this, like, singing note. Like it's like, dun, dun, dun. And meanwhile, you're, the rest of your fingers are like playing this kind of droning, like dirge. Really cool, really cool piece. I'm not making it sound that cool, but it sort of struck me as like maybe an instance of this where you're combining these two motifs and creating this tapestry.

Matt: 16:04

Yeah. Um, and on, on this point, I mean, one, one thing that I'm not sure if this like jives well with the point he's trying to make, but just in music, like silence. Like playing with listeners expectations and sometimes like removing something can have its own, its own effect.

Jon: 16:29

Yeah.

Matt: 16:29

And that is very interesting where it's like actually the absence of sound where the listener would expect sound is its own sort of interesting, you know, figure from like the absence of anything. Um,

Jon: 16:49

Like 12 hours of silence.

Matt: 16:51

yeah, four 33, uh, no, that's almost too extreme. I mean, that's all literally all ground. That would be like, you know, I mean, I guess, you know, I've seen it in MoMA where it's like, there's just a big white canvas and it's like, yep, this is, this is my artist that my artwork, uh, but, um, yeah, that I think maybe takes that's all, all ground and I think that that's maybe like, It's funny because I guess I, I, cause what's the alternative all figure? Like what would that, what would that be? I guess just like a bunch of humans, just like cramped together. Um,

Jon: 17:31

Uh, he mentions this really, uh, or I guess a factoid that I thought was super interesting. There exist formal systems whose negative space, which is the set of non theorems, is not the positive space of any formal system. Which I thought was kind of like. Kind of blew my mind a little bit. So basically what that's saying is like, there are, well, it's hard to say what it's saying without just repeating it, but I realized like a complex sentence. So there's these systems where the theorems that aren't in the system can't possibly be created by another system. Which is like, I feel like I almost worded that in a more confusing way, but it's just a wild fact. It's, it's, it almost goes back to Godel's incompleteness where it's like, there's just things that you can't express. Um, and that just seems, I don't know. It's like a hard fact of the universe, I

Matt: 18:32

he's yeah. He says, he says that it's, you know, it's of the same depth as girl Syria of incompleteness, uh, in terms of like the complexity.

Jon: 18:43

I guess it's sort of similar to how, like, you know, we don't have some way of like generating primes. Like we, we literally generate primes by like brute forcing and it's kind of, it feels kind of similar where it's like, I guess we don't have you know, some mathematical statement that just like expresses all primes.

Matt: 19:07

I mean, he does, he does come up at the, at the very end of the chapter. He does propose a system that, that does generate, uh, primes as like a positive system,

Jon: 19:17

Right. Right. But it's like an algorithm.

Matt: 19:20

yeah, yeah, yeah. I'm trying to think, but like, I think what he's saying though, is that there are certain, yeah, there are certain negative space, like negative spaces where you can't even Write an algorithm that would generate all of the, all of the members.

Jon: 19:42

Yeah. Yeah. I think that is what he's saying. So maybe primes is, is a bad example,

Matt: 19:47

It's really hard to, it's really hard to wrap your head around him because he, even he himself is saying like, it's not intuitive why that would, how that could possibly be the case. Um, so I'm not sure that, like, I certainly don't understand what would even be an example of something that couldn't be generated.

Jon: 20:04

yeah. I want an example. Super interesting.

Matt: 20:06

It feels like it's getting into like the halting problem or like, you know, enumerate the set of programs that will complete in, you know, five steps or in an N number of steps. Like these are, you know, just like very arcane systems or something like that.

Jon: 20:25

Yeah. Yeah. So he talks about, uh, recursively

Matt: 20:29

Oh gosh, I'm just going to have to tap out, tap out of this one. If you know, if you were able to figure out, uh, what, what these, these men. I'll have more power to you. Okay.

Jon: 20:43

of recursively enumerable is it's a set which can be generated according to typographical rules. So like everything in the Mew system, which is, you know, infinite items that those are all recursively enumerable because you have this simple set of typographical rules that can generate all of them. But I guess at least so far in this chapter, he has not shown a way that primes are recursively enumerable. Because he introduced that TQ system, but it doesn't actually work to generate primes. It just generates everything except the primes. But luckily he quickly introduces, Oh, Oh, actually, before I talk about that, um, he, he had a little puzzle. He had like a series of numbers and he was like, can you figure out what this is? And I wrote down. I mean, I spent like two seconds thinking about this, so this is probably completely wrong, but it seemed like he just did. He started with one, three, and seven, and then he sort of started adding the numbers that he hadn't, that he had skipped starting with five,

Matt: 21:57

Oh,

Jon: 21:58

makes sense, which is, I mean, it sounds so incredibly arbitrary that it felt like it had to be wrong. But it was like one, three, seven, then he added five, then he added six, then he added eight because he had already written seven, then nine, then 10, then 11. Um, anyway, I just thought that was a funny little puzzle.

Matt: 22:18

Yes. Yeah. That's interesting. Yeah. Go for it.

Jon: 22:20

but after the puzzle, he finally introduced a new system that can represent primes, which it was actually very cool. And what he had to do was kind of flip the problem on its head a little bit. He introduced a DND, system, which basically stands for does not divide where it's basically, you know, you're saying some number does not divide into, you know, another number. So like for instance, two does not like divide into five, right? Cause you have one left over. It's almost like a, um, Like a modulo operator. And then he went on to, define this other concept called divisor free, which, um, what does divisor free mean? I guess it just meant that some number doesn't have a divisor up until another number.

Matt: 23:11

Yeah, exactly.

Jon: 23:12

Yeah. So it's like, you know, four is divisor free up until one, I guess. Because I,

Matt: 23:21

Well, wouldn't that be up until two or

Jon: 23:25

well, I'm just saying if df like four DF one would be true, right? Or it would be a valid, um, theorem. But in order to figure out if something is prime or not, you have to prove that DF is true up until the number itself.

Matt: 23:46

Right.

Jon: 23:46

Which, it's kind of like, the sieve of Eratosthenes, which I think we've described before. well, maybe I shouldn't go down that path cause that almost introduces like a new complicated thing. Uh, but it's, it's almost like this algorithm that he introduces where, you know, for each number you basically check everything up until that number. And if you find a divisor. You've proven that it's not prime, but if you can't find a divisor up until that number, then it's prime.

Matt: 24:15

Yes. Yes, yes, yes, yes.

Jon: 24:17

Yeah. So I don't know if I did a good job describing that, but that was kind of how he, he ended the chapter. And I think that what he was getting at was, you know, this earlier idea of finding a system that can represent the problem correctly so that you can solve it. if, if your system, or if, if you can't find a solution in the, in the positive space of your system, in the figure, then you might have to look in the ground, you might have to kind of think about the negative space and like, devise a system that can start representing some of that.,and I, I don't know, I thought it was interesting. I feel like he's starting to go down this path of like new ways of thinking about problem solving.

Matt: 25:01

Oh yeah. Uh, no. And I mean, I mean, I think I think it's in those spaces where you can define, define a system as, as the, you know, as the reverse and not the reverse, as the like negative of another, of another system. And like, yeah, just the process of going from defining a system negatively to coming up with a positive. Uh, definition for that space. Like that seems very interesting. And then especially if he's going to start to talk about the places where that's not even possible, like, I don't know, that sounds fascinating.

Jon: 25:36

Yeah, that'll be mind blowing, I think, so I'm excited to get to some of that stuff.

Matt: 25:41

but yeah, I think that's, I think that's all I had. This was, yeah, it's really starting to get into. hard to wrap your head around, uh, aspect. I mean, like I know we, that we've been there the whole time, but it's even getting, getting harder to, and this was, this was an especially visual chapter. So hopefully, uh, hopefully the listeners weren't too, uh, too thrown off about it.

Jon: 26:06

Yeah, yeah, yeah. Throughout this, uh, podcast, you should be looking at Escher drawings and listening to Bach and just relishing that stuff.

Matt: 26:15

Another thing I want to recommend people look up, look up the painting figure, figure by Scott E. Kim.

Jon: 26:22

Oh, yeah, yeah,

Matt: 26:23

I think I want to get this on my wall or something because it's just so amazing. And just to, just to take a, take a terrible crack at like describing it. mean, when you first look at it, it's just this, uh, black and white, uh, image where you can, you can tell that it's two equal parts where they're like, the parts are interleaved with each other. So they kind of tile together, they cover the whole thing. So. You know, the, you know, you can consider the black and the white as like figuring ground, um, or vice versa, I guess, you know, it's hard to say which is which in this context, but, um, the other more amazing thing is they actually spell out the word figure. it's really hard to read, but, uh, but you can, you can kind of make it out.

Jon: 27:08

yeah, it was kind of like a magic eye. It was like, you look at it for long enough and then. You know, the, the answer pops out. It's kind of cool.

Matt: 27:16

Yeah, but definitely, yeah, check, check it out, everyone, because it's, uh, it's pretty incredible that this guy was able to, able to do this 1975.

Jon: 27:24

Yep.

Matt: 27:25

but that's all I had.

Jon: 27:26

it's all I got.

Matt: 27:27

All right. Well, yeah, what were you gonna say?

Jon: 27:29

I was just going to say, I will see you in the next one. I don't know what chapter five is called, but I'm excited for it.

Matt: 27:35

Chapter five is called consistency, completeness and geometry. So. Be pretty, pretty good. All right. Well, I'll see you next time. Mhm.

Jon: 27:44

All right. See you, Matt.