Matt: 0:14

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

matt_geb_ch_14: 0:30

Hey, John, how are you doing?

jon_raw: 0:32

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

matt_geb_ch_14: 0:35

I'm doing okay. I'm wondering if you've ever had any supernatural experiences in your life. Have you ever experienced like a, like a glitch in the matrix?

jon_raw: 0:47

nothing seems supernatural to me until I have very, very hard definitive evidence. So I would say no.

matt_geb_ch_14: 0:54

So you're just, you're just a big

jon_raw: 0:57

I'm a, I am a huge buzzkill and it's my dad's fault. my dad ruined, yeah, because my dad was always like, Just poo pooing everything that was interesting.

matt_geb_ch_14: 1:07

I, I, I want to believe. I want to believe. I don't, I don't believe.

jon_raw: 1:12

In fact, I watched X Files religiously. And

matt_geb_ch_14: 1:17

that's kind of an ironic phrasing, I feel like.

jon_raw: 1:20

Yeah, and I remember my dad would be like, I'm not watching X Files. So I had to watch it with my mom. Cause he was just very, I don't know, he's very like, you know,

matt_geb_ch_14: 1:29

Does he consume fiction at all? He's like, those people weren't actually there. That's not, that didn't actually happen.

jon_raw: 1:37

No, he definitely consumes fiction and also he's, you know, he's able to like entertain fantasy fantastical ideas and stuff. I think the, I think the issue he had with X Files is, well, number one, it was kind of a silly show, but I think also like the show was about like believing in supernatural stuff, sort

matt_geb_ch_14: 2:00

Which he just took umbrage with.

jon_raw: 2:01

Yeah. I think he was like just not interested.

matt_geb_ch_14: 2:04

to him.

jon_raw: 2:05

May, I don't know. Umbridge is, is strong. I think he just wasn't interested is what it came down to, because he was really, he loved Star Trek, also. We don't have to talk about my dad too much, but he was like, I think the next generation is probably his favorite show ever. And

matt_geb_ch_14: 2:20

Douglas Hofstadter does talk a lot about your dad in this chapter. So, um, you know, I think it's, I think it's, it's useful.

jon_raw: 2:27

I had a question for you though. Did you ever do coins when you were in. Computer science.

matt_geb_ch_14: 2:33

I'm really glad that you're bringing this up because I, I, I don't think I've ever written a quine there. There there's like the, there's the dumb trivial quine, which is if you don't enter, if you just like press a return in the terminal, like that's technically a quine, uh,

jon_raw: 2:51

I remember there was like a group of kids, because I just literally did not have the ability to like write a quine when I was in college. But I remember there was a group of kids that would like, I almost said quids. There was a group of kids. They should have called themselves quids. But they would like challenge, like quine quids. They would challenge themselves to write quines. And it was kind of sweet. I remember. I don't know. They were like geniuses. They were like doing different languages and, you

matt_geb_ch_14: 3:19

we should, uh, I guess we should define what a quine is. Right. So it's a computer program. Well, quine, I guess has a, has several definitions, but in computer science, right, it's a program that prints its own source code. Is that right?

jon_raw: 3:33

Exactly. Yeah. And, and there's the trivial one that you talk about. There's also the trivial one that like literally reads itself as a file and then prints, you know, prints the contents of the source code file. But then there's also quines that are like, you know, they look utterly indiscernible, like crazy. Yeah. Just utterly obtuse. But they end up printing themselves and people try, you know, challenge themselves to write quines in different languages, just to, just to make life hard, I guess.

matt_geb_ch_14: 4:02

This is, this gets back to something I was saying before about how I feel like all of this self reference stuff is kind of fundamentally uninteresting to me because it just feels like I, I don't know. It's, it's interesting to think about for like five minutes, but the idea of spending any appreciable amount of time writing a program that printed out its own source code just feels like a complete waste of, waste of my time when there's like a lot of very interesting problems, like real world, uh, problems to solve.

jon_raw: 4:34

Oh, yeah. No, I think you and I are are similar. Like, I love software engineering, but I've always stopped just short of, like, getting really heavy into theory. You know, like I love a good data structure. I love a good, you know, merge sort algorithm, but like, unless it has practical value, it, it just becomes uninteresting to me. And that's actually like, well, I, I feel like I'm jumping ahead a little bit, but like a massive, massive problem that I have with Godel's incompleteness is like, I need a concrete example. Like, I understand the power of the idea. Well, I don't even know if I understand the power of the idea to be perfectly honest, but it would help me so much. If you could just tell me like, here is a truth that cannot be represented like a real truth, but I think it's, I think it's just the type of thing where like, you can prove that you can't do that, but then you can't show an example of it. And for me, it's kind of like, I don't know. I mean, for one, it's tough for me to understand, but it's also just loses a lot of its power to me.

matt_geb_ch_14: 5:46

It's funny because it's almost like a magic spell that Girdle, uh, or Girdle cast, and it was able to stop Russell and Whitehead from wasting a bunch of more time, uh, doing, you know, working on this,

jon_raw: 6:02

Yeah. Principia, Principia Mathematica and stuff,

matt_geb_ch_14: 6:07

yeah, just this, this, this futile quest.

jon_raw: 6:10

right?

matt_geb_ch_14: 6:11

So that's kind of the real, I mean, maybe that's like a meta, a meta effect of that, but, um,

jon_raw: 6:17

Well, it's funny too, because like, I do feel like what Russell and Whitehead were doing was very, very useful. Like they had, there was a tremendous amount of practical value to like being able to represent. Truth as these like statements, like, Oh, for all like chickens, chickens, lay eggs or whatever. Uh, and then be able to like build on truth by basically performing operations against those statements. Like there's a tremendous amount of practical value in that. And it is kind of funny that like Gödel by, you know, throwing this very abstract idea out into the world, he kind of like. Completely put the kibosh on what they were doing.

matt_geb_ch_14: 7:01

Yeah. and, and ideally they, that would allow, that would kind of free them up to work on things that were actually useful than trying to produce this system that, for which every single true statement was provable. You know, they probably had a bunch of other ideas that they kind of came across along the way, but they were like, okay, this is the big one. We really want to do this. And then they were able to do things that were more useful to other, other mathematicians. But, but he talks, he goes on to talk about this at the, at the end of the chapter. But maybe, maybe it makes sense for us to take a step back and actually talk about what the, at a high level, kind of what this chapter is about. So. This title, this is one of these titles where my eyes just start to glaze over. Um, and I just stopped being able to parse the words, but it's called on formally undecidable propositions of TNT and related systems. This is chapter 14. Uh, And this is, this feels, I mean, I feel like I keep on saying this every chapter, but it really feels like this is actually where. This is the meat of it,

jon_raw: 8:12

Yeah, this is actually the definition of Gödel's incompleteness.

matt_geb_ch_14: 8:16

Which, I don't know, it feels very cool, uh, to finally have gotten to this point.

jon_raw: 8:22

yeah, no, absolutely, this chapter felt very powerful. It is a truly, like, astounding idea.

matt_geb_ch_14: 8:30

Yeah, yeah, and So uh, mean, Gertl, joined together these two ideas, which was, The like self, self reference, I mean, let me make sure I'm, I'm saying these two, these two ideas, cause it feels like there's a bunch of different. Okay.

jon_raw: 8:45

So his Godel numbering was one of them.

matt_geb_ch_14: 8:47

Well, yeah, that was, I mean, actually, so the, that's, that's kind of like a. Base level assumption that, uh, that we're already working with. Right. So girdle girdle numbering is just a way to translate from a proposition. Like, uh, all, you know, there exists some number which is equal to two squared or whatever, you know, that's that you take that proposition and you can translate it into an arithmetic number. and so the first, so, so we, that's kind of a prerequisite for this chapter, but the first, uh, idea that, that, uh, Hofstadter talks about is this idea of proof pairs, which is like, You have, uh, some proposition and then you also have the set of steps to prove that it's true. You know, you start from one of the axioms and then you can, uh, you know, show that it's true. And he says that you can check if that's true in, you know, in a finite number of steps.

jon_raw: 9:53

Right, like you could write a bloop program to verify a proof pair.

matt_geb_ch_14: 9:57

Right. And this is really where it feels like we're starting to pull all of it. And this is basically what we just said, but pull all these things. Like it's starting to make sense why he has talked about all of these different things. Like, it's like, why is he talking about bloop? Like, I just don't know

jon_raw: 10:10

Yeah, he really, he really buried the lead. But, and he, he talked about this other concept too, which is like, yes, it's trivial to test if MN is a proof pair and when I say MN M is like this entire theorem. And then N is like the thing that's proven. So it's trivial to test that MN is a proof pair and you could even write a blue program to do so, but it's hard to find M given N. So if you have some statement that you think is true. You can't, like, calculate, uh, like a, a theorem or set of theorems that will show that it's true.

matt_geb_ch_14: 10:52

Right. And this, this was so subtle to me that I feel like you really needed to. Read very closely because when you're reading it, he, he says, you know, cause basically what he has is this function for all intents and purposes called TNT proof pair,

jon_raw: 11:09

Mm hmm.

matt_geb_ch_14: 11:10

which accepts the list of instructions and then the conclusion. And it tells you whether or not it's, uh, whether or not it's true. But then later on, he, he creates like a generalization of that, which says like, there exists, you know, given, you know, given some number, there exists some proof that it's true. And then he says, like, that's not, that's not primitive recursive, or you could not write a bloop program. And, but it's just like, when you're reading it, they're so similar, they look so similar. But I, it's, it took me a while to figure out like, wait, I thought you just told me that this could be computed, but it's the difference between checking an answer and finding an answer, which immediately took me back to P versus NP, so, uh, so Yeah. so then the next, the next idea is this idea of like substitution where you have a formula and then, you know, specifically of one free variable, he basically comes up with like a mathematical expression that, or it's like a mathematical function, which accepts a formula of one free variable. Or, or, and this is what, this is what's a little bit confusing, and let me know if this confused you too, is the way he's framing all of these things, he's always framing them as like predicates, where you're testing whether or not it has this attribute. Um,

jon_raw: 12:39

When you say predicate, do you mean he's like using that on, on or upside down a thing, like he's saying for all X, you know, there's some X squared equals X times X or

matt_geb_ch_14: 12:49

not, not exactly. So, so when he talks about substitution, the way he frames the, the way he, like the, the shape of the function that he writes is Uh, you know, some girdle number, uh, which you're, that has one free variable, a number that you are substituting into that thing, and then the third number, which is what the result of that is. And basically the, the, the function is going to return true if they're equivalent. Do you see what I mean? So. it's just a predicate. It's just something that accepts stuff and returns true or false. And

jon_raw: 13:27

right. So. Yeah, yeah. It's like an equality and yeah, like, depending on what you pass into the variable, like that equality is going to resolve into true,

matt_geb_ch_14: 13:39

right? I think the part that, that felt confusing to me is I would just think of it as a function that accepts a good old number for some, you know, with a one free variable and accept something to substitute in, and then returns you what the result of that number is, because that is a, that is something that you can do, or that's an operation that you can do with a bloop program, right?

jon_raw: 14:05

I might be, I might be losing you a little bit.

matt_geb_ch_14: 14:07

So, okay. So maybe, maybe like by way of example, like you say, the, the formula is, Uh, X squared equals Y.

jon_raw: 14:18

right.

matt_geb_ch_14: 14:19

And then, oh, and let's even say, like, actually the, the formula is two squared equals Y. And now you have one, you have one. Unbound or one free variable.

jon_raw: 14:33

Yeah. Why?

matt_geb_ch_14: 14:35

Why now you can, if you have the good old number for that, you know, that expression,

jon_raw: 14:43

Yes.

matt_geb_ch_14: 14:44

that in and then you can substitute in a number for, you can pass another number in and it will create you the good old number for the thing with the subs, the number substituted in.

jon_raw: 14:58

I think I see what you're saying. And I think where I lost you is like, so he, he introduces that TNT proof pair concept. And, and then he goes back to quantification, which quantification was kind of introduced in the dialogue. It's basically where you have, like, a, I mean, we talked about it just a second ago, like a program that can write itself, but it can even exist in, like, a language where you can have a sentence that, like, reference, references its own self. So he introduces this concept called arithmoquantification.

matt_geb_ch_14: 15:33

Yeah.

jon_raw: 15:34

Which is an amazing word where basically you take a, you take a formula, you convert it into a good old number, which is just this tremendous number. And then you pass that good old number into the free variables of the formula, which I think you've said already, but I just kind of like, I lost you a little bit.

matt_geb_ch_14: 15:57

Right. Exactly. Exactly. So the, the, the core part is you have a formula that has a free variable and then you take the, you take the good old number. And this is the, this is the brain breaking part of this that I feel like I still don't like. Don't fully understand, but you have this crazy formula. You've turned it into a good old number and you're able to create a new. So this is what, this is what, where I get confused because you create a new good old number with it self substituted,

jon_raw: 16:32

Yeah, yeah, yeah. And I also get confused here, by the way. But yeah, you're basically taking a Godel number, and you're replacing, you're replacing part of that Godel number with its own self. Uh,

matt_geb_ch_14: 16:47

good old number that does not have any. Free variables anymore, because it's, because it's been bound by its own Gödel number.

jon_raw: 16:57

Yes, yeah, because the free variable is, is replaced with this tremendous number.

matt_geb_ch_14: 17:01

Right. Right. That's the other hilarious thing. These numbers have like, probably have like millions of digits or something. It's, or I don't, I don't know if it's millions, but like an unbelievable number of, of digits.

jon_raw: 17:13

yeah, But yeah, so, and then he goes through a couple, kind of, numeric gymnastics, which honestly were, were confusing to me, uh, and he ends up with a statement that basically says, Well, he creates this statement called G, which is like, I guess this is Gödel's string, and he's able to substitute in G into this other theorem. In order to say that G is not a theorem of typographical number theory. So basically, you have a statement which disproves its own self. Which goes all the way back to the whole, like, this statement is false stuff that were introduced in the very beginning of the book. But this is like, you've used this rigorous statement in order to construct a new statement that like defies the entire system.

matt_geb_ch_14: 18:11

And, and it, he uses, he uses the, we were just talking about that proof pair where basically it's like, okay, you have some string of numbers or, you know, you have something that has all of the steps to prove something. And then you have like the resulting conclusion.

jon_raw: 18:28

Yeah.

matt_geb_ch_14: 18:29

And, and the, the little trick that he plays is basically like there's a statement. There exists some proof. And so this is where, this is where it starts to, okay, so there does not exist two numbers, A and A prime. And this is like, I should just say A and B. Uh, there does not exist two numbers, A and B, such that there is some proof of its truthness. Otherwise, you know, in other words, it's, it's provable in, uh, in typographical number theory.

jon_raw: 19:06

Right.

matt_geb_ch_14: 19:07

And there is an arithmoquine for it. And this is what you're saying, arithmoquinification. It just means that there's some number that, uh, you know, that represents this, this formula.

jon_raw: 19:22

Yeah.

matt_geb_ch_14: 19:24

And so the way this works is we are able to create an arithmoquinification of that number. So the other, the other claim that there is a proof for it must be false. And so we've kind of shown that there cannot exist a proof for there. There, it's not possible to come up with a proof for this theory, because we already know that there is an arrhythmia quantification based on the way we've structured the, this truth statement. Cause

jon_raw: 19:58

So would it like, would it like create an infinite loop or something? Because we'd have to constantly be arithmoquintifying the new,

matt_geb_ch_14: 20:07

This, I think, was what I was getting stuck on was the looping of like, do he, like, but you can't, you can't arithmo quinn, quinnify more than once. I don't think because then there's no, there's no free variable.

jon_raw: 20:23

yeah. I think that's the key is like once you've substituted the free variables, like, there's nothing more to arithmoquintify.

matt_geb_ch_14: 20:31

Yeah, So I think now you're just passing something in, but it already. yeah. And, and, and it, it does feel like you're making a statement about something else at that point, but I don't think it matters actually, because there you're staying, you're stating something about it itself. Like, you know,

jon_raw: 20:54

Yeah, yeah. I mean, I'll just say like, so this was very hard for me to understand. It's still hard for me to understand. I would highly recommend people just look at these formulas. I mean, first of all, I would highly recommend people read this book because it's awesome. But yeah, just read through these formulas and try to grok it for yourself because I'm not smart enough. But, but there is, there's sort of this like, like I, I think I understand the conclusion. Yeah. Which is that within this formal system, you're basically able to prove that the formal system can't accomplish certain goals. And that was the breakdown of Principia Mathematica, Russell and Whitehead's whole entire life goal. That Gödel just, you know, destroyed. But yeah, oh,

matt_geb_ch_14: 21:43

yeah, I think the craziest part of this that, that it's just based on this, but all of like, in order to prove these steps, every single one of those like proof steps is a mathematical operation that you can like validate, which is crazy to me. It just feels wild. Um,

jon_raw: 22:06

Yeah. It's, it's very cool, because it, it really is like a formalization of that, this statement is false concept.

matt_geb_ch_14: 22:16

Yes, because that's, that's basically what this statement is saying at the bottom. And he goes through and he does a pretty good job. I mean, like, I think I'm at the limits of my ability to comprehend what is going on, basically. But he does a really good job of starting from this arcane representation, which you, you understand. This kind of gets back to something we talked about really early in the book, which is like I mode, which is like intelligent mode versus mechanical mode. And he starts in this very mechanical place, which you understand that, okay, we're not breaking any of the rules. And step by step, he, he shows you how the higher level interpretation of this kind of getting to the I mode is that it's making a statement about itself. And what it's saying is you can't prove that this is. That this is true, and if you reject, if you, if you say that it's false, you need to like, he, cause there's a logic here where it's like, if you, if you argue that it's false, then it leads to a contradiction. Do you remember? Do you remember exactly? I'm cause. Okay.

jon_raw: 23:27

Ooh, I, this I don't remember. Was this in the chapter?

matt_geb_ch_14: 23:30

I think, I think he referenced he read. Yeah, he did reference it where it's like, If you assume, cause it makes a statement, which, and, and this gets back to like, this statement is false, where if you say that it's, uh, if you say that it's false, then, you know, it, this statement is false as a contradiction in both directions. Um, this is different because if you assume that it's true, then there's no contradiction. It just like, so you kind of are led to the conclusion that it must be true because otherwise. If it were false, then it would lead to a contradiction.

jon_raw: 24:06

I see what you mean. Yeah, yeah. So it's like, the, the whole, this statement is false, regardless of how you interpret it, it's broken. But, but the The concept introduced in this chapter, you can interpret it as true, but it's just, it just proves that you can't, uh, you can't formulate certain truths.

matt_geb_ch_14: 24:33

right, right. So it says, it says, uh, G is not a, so like the high level interpretation of G is G is not a theorem of TNT.

jon_raw: 24:45

Mm.

matt_geb_ch_14: 24:45

then if you, if you attempt to say that that's false, That's immediately a contradiction because in order for something to be a theorem of TNT, it needs to be true. Like, being a theorem of TNT means that it's true.

jon_raw: 25:04

Yeah. Yeah. Exactly. Cause it,

matt_geb_ch_14: 25:06

And so,

jon_raw: 25:07

right. Like as long as you've followed the rules of TNT, which this. This does because it is a theorem like it's like in order to be a theorem. You have to have followed the rules of TNT

matt_geb_ch_14: 25:19

Exactly, exactly. And, and their point is everything, like truth, truth is defined as the set of things, like kind of the thing, well, and I guess this is, this is what they're attempting to do. is that truth exactly maps onto following the steps. So if you just follow the steps, like, then it means that it is defined to be true. And everything that can't be proven if you follow the steps is false. But this is what Gödel is messing around with, is we have something that must be true, because if it's not true, it leads to a contradiction. But also it's saying that it cannot be proven in TNT.

jon_raw: 26:00

Right, okay. Well, well my brain hurts I think He that he actually talks about more stuff in this chapter, which I thought was really really interesting one thing he talks about which I kind of didn't understand was super natural numbers and and it felt like super natural numbers were almost a way of, like, fixing this. Like, it's basically a new, an entirely new number system that, like, enables you to fix this, he calls it W inconsistency.

matt_geb_ch_14: 26:36

Yeah, yeah. Uh, I think that's an omega. That should be pedantic.

jon_raw: 26:42

Yeah, yeah, omega inconsistency. It seemed as though These supernatural numbers were like, infinitely large or something like that, but I didn't really get, I kind of didn't get this section, but it reminded me of much earlier in the book where he talks about like, breaking out of the system, like he introduces a few, you know, kind of crummy systems, which really struggle to represent things. And he introduces this concept of like, breaking out of those crummy systems in order to like. You know, solve problems. And so this is the, this is the example of breaking out of TNT, introducing these supernatural numbers, and I guess somehow fixing Godel's incompleteness? Uh, but,

matt_geb_ch_14: 27:30

it, like, the, the parallel is if you think about imaginary numbers, Right. Like, I think this actually is a super useful idea because you say, okay. You take the square root of a negative number and then you're like, no, there's no, there's no such thing like it breaks and then you could kind of think of, of a mathematical system before, without, uh, imaginary numbers as being incomplete and then, cause, cause you, you're like, Oh, it breaks. We do square root of negative one and it breaks. And then it's like, Oh, like imaginary numbers swoop in and kind of like patch over that weirdness. Um, and you basically just define away the problem. And that's kind of what he's attempting to do, or I don't know if this is him, or he's just kind of like referring to what other mathematicians have done. My problem is I As, as poorly as I understand imaginary numbers, like I really have basically zero grasp of what Like, what the effect of supernatural numbers is.

jon_raw: 28:37

Yeah. No, exactly. Like. And he has a funny quote, I think it was a Lincoln quote, but someone asks, I think it was Lincoln, how long should a man's legs be? And Lincoln said, long enough to reach the ground. And I think that was his way of saying, like, Supernatural numbers are just a thing that, like, fills that gap. You know, it's like, we don't really know what they are. It's exactly like I. And he even mentions, I thought this was hilarious, like, I actually kind of, I, like, very nearly laughed out loud at this, where he was talking about how you could have defined I wrong. You know, cause, like, I, basically the square root of negative one is I. But a square root always has two answers, right? It's either, like, the square root of twenty five. Uh, it could be negative 25, um, or sorry, negative five. So the square root of negative one could be negative I. So he was just introducing this funny anecdote, like, Oh, maybe we defined I incorrectly, like maybe I should have been negative I. Um, but I think, I think his, the reason he was doing this is just to show that like, it's almost just a placeholder. Like it's, it doesn't really matter what it is. It's just like a thing that like fills this gap that you have.

matt_geb_ch_14: 30:00

right.

jon_raw: 30:00

often you can sort of like, like once you've defined that placeholder, then you can sort of define ways of like getting out of it, you know, like for I, if you multiply an I by another I, you've sort of gone back into normal numbers.

matt_geb_ch_14: 30:16

Yeah, yeah.

jon_raw: 30:17

And so it's just this, it's basically just a tool, uh, in order to like work with crazy mathematical stuff.

matt_geb_ch_14: 30:26

And, and, just to explain a little bit more about how we are getting to supernatural numbers. So, a core part, and at the risk of wading back into the insanity of before, like a core part of what, what G said, which is that crazy formula we were talking about. Is that there does not exist some number for which like, there's a, there's a pair. And one of the ways you fix that is to say, Oh, there is, there is a number that. There does exist a number that satisfies the constraints that that sets up. And, you know, before that, that, that was a contradiction because there wasn't a natural number that, that satisfied it. But basically what they do is they say, there is a number that satisfies it, and it's a supernatural number. It's kind of like, before, there was like, there is no number that satisfies square root of negative one. And then they were like, oh, no, actually, there is, and it's I. And it's like,

jon_raw: 31:25

oh, dude, you're like, you're, this is a revelation. You're like basically filling this gap that I had in this chapter.

matt_geb_ch_14: 31:33

Um, but like, I have no idea how to, like, I, like, it feels like, I, that's all I've figured out. And it's like, well, what does that, what does that number mean, or how does it react, or what, like, all of that is unclear to me.

jon_raw: 31:46

Well, and, and now this kind of makes more sense why he talked about this other thing. I really liked this part where he was basically talking about learning about the real world, which is, you know, the real world is this very messy place. We don't actually know how it works. Like, we have quantum mechanics, which is a very close approximation of, of how it really works, but who knows if that's the, the truth. and he talks about how, I guess, in his master's thesis or PhD thesis or whatever, he was investigating crystals and magnets, but he was using this, you know, hyper idealized, like, structures, like, basically these perfect geometries in order to investigate that problem. And, you know, without going into too much detail, by using those, you know, hyper idealized geometries, he was able to find these, themes, you know, these high level truths that would have been very hard to find were you actually investigating like these real world examples, which are often way messier. So it's almost like, by dropping into this level of abstraction, Where, you know, you're probably losing a lot of detail. You're still able to glean these, like, very interesting pieces of information. And, you know, I think that was, like, sort of his, his follow up to, like, Why do we introduce these things like I? Well, it's in order to have an abstraction so that you can work with it.

matt_geb_ch_14: 33:17

Yeah. We recently learned about something in law school called The Coase Theorem. Which, um, This is, it's kind of hilarious that I'm bringing this up because it's this very confusing topic and I'm just gonna throw this whole new thing in there, but It basically says that like, you know, if, if you're deciding, so, you know, let's say someone is, uh, you know, a factory is polluting and the nearby people like are suing the factory to stop polluting. The Coase theorem basically says it doesn't, it actually doesn't really matter what the. You know, who wins in that case? Because, uh, the parties are going to negotiate based on their interests. This is kind of like an economic theory, where it's like, as long as the transaction costs are zero, and property rights are clear, there, you know, it's not going to matter how you do it, because they're going to just negotiate to kind of arrive at Whoever, you know, values the amount the most is going to bargain with the other side and the specifics of it are not super important, but the high level takeaway is it's based on things that are not true, that are never true, like transaction costs are never zero, uh, but, um, but it still provides this useful, useful framework for, for thinking about like, okay, well, who should, who should bear the property rights or should the judicial system kind of try to actually Um, so anyway, uh, so that, that, but, but just, just the, the point being that you can have these systems, which are actually like very wrong, uh, in their assumptions, but they still are useful frameworks for, for, you know, understanding a system.

jon_raw: 35:14

right. Yeah, I thought this was interesting because we've discussed before like, you know, you have Newtonian mechanics, which Isaac Newton developed. But there's a point where Newtonian mechanics kind of stops, like basically if you're trying to calculate relativistic motion or something, Newtonian mechanics just breaks down. It's unable to represent that. But this is almost like a reverse, where you're taking something that is complicated and you're like, creating this simpler, more idealized system to work with it. I mean, it's a similar notion. Uh, but, I don't know. I just thought it was interesting. It felt like a, like, slightly different from that move from, like, Newtonian mechanics to Einsteinian mechanics.

matt_geb_ch_14: 36:02

Interesting. What's the. Are you saying, is the parallel here the move from, from TNT to these like supernatural numbers?

jon_raw: 36:16

Yeah, I guess, like, you know, if I'm thinking about trying to understand motion, Newtonian mechanics is this thing that, like, works for 99. 5 percent of all cases. And anything that's happening on Earth. Like, it pretty much works, but then you reach this point where, okay, there's all these new cases that we're observing that this system just does not represent. And so you have to develop this new system that has, you know, it's basically Newtonian mechanics plus all of this other stuff. So that's almost like, that to me feels like a graduation of a system into like a bigger system, you know, that's more complicated, but can do more.

matt_geb_ch_14: 37:00

Yeah.

jon_raw: 37:02

This concept, at least to me, it, it feels like this weird, like lateral move where you've, you have, and maybe it's similar, maybe it's like the exact same thing where it's like, you know, you have a system and then you hit a square root of negative one. And so you're just introducing more complexity in order to handle that. Uh, but

matt_geb_ch_14: 37:22

It's, it's funny cause I think, I think the, I think the comparison is actually really apt, but it's, it feels like it's occurring backwards, because, cause, When Einstein, and actually this may not be right. but when Einstein was working on general relativity, there were problems, there were wasn't, there were like physical problems that he was trying to solve, uh, like there were, you know, whereas this feels like the reverse where it's like, you're adding this complexity without a corresponding physical phenomenon that you want to explain. And I guess I'm a little bit like, Okay, well maybe we just leave that, let this problem remain unsolved and let's not create supernatural numbers, but I think, I think there are many cases actually in mathematics where someone was just like, I don't know, doing random things in the corner and it didn't, it didn't come to light until later why that would be useful. I think hyperbolic geometry is actually a perfect example of that.

jon_raw: 38:27

Well, right. Which helped Einstein. figure out

matt_geb_ch_14: 38:31

yeah. Yeah, actually, that's a perfect, I guess that's for it all. Yeah, like hyperbolic geometry.

jon_raw: 38:38

But yeah, there's a statement he has towards the end of the chapter where he says it's essential that rival geometries should exist, which I thought was cool. Cause it's, you know, it's like for any given problem, there may be many, many levels of abstraction to approach that problem. And all of them might offer some value or some trade off.

matt_geb_ch_14: 39:00

Yeah. Yeah, Yeah, I did have one tiny little trivia, which is, Douglas Hofstadter coined the term Quine. He was the original, like, person, there was a guy named Quine, but his name was not used to refer to the process of of putting something, you know, putting something in front of it. He called them auto reps, which were computer programs that produced themselves. Um, but only after Douglas Hofstadter's book, Gert Luscher Bach, was, did it actually become used as a term used to refer to this, like, self reference. I thought that was very funny.

jon_raw: 39:39

wild. That's, that's amazing. Yeah, Douglas Hofstetter, man.

matt_geb_ch_14: 39:43

He just, he just is the originator of like so many of these ideas. It's pretty wild.

jon_raw: 39:49

truly original. Yeah, this book is, it's, it's insanely original.

matt_geb_ch_14: 39:54

Oh yeah, but I think That's all I got. for this chapter.

jon_raw: 39:58

That's all I got.

matt_geb_ch_14: 40:00

Alright, well I will See you next time. for chapter 15.

jon_raw: 40:03

See you next time.