Matt: 0:00

Dude, I just gotta, uh, tell you something that I find very amusing. I use script to record this, and when I start recording, instead of counting down 3, 2, 1, it counts down 2 1 0 And I'm sure that's just an off by one error, but it just amuses me every time.

Jon: 0:20

man, you see that could have disastrous results though. Like this is why the meaning of numbers is so important because like, imagine a spaceship liftoff. Like what if the guy engaged the engines at one, but he was supposed to wait till zero?

Matt: 0:49

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

Jon: 1:08

Hey Matt, uh, I'm doing very well. this is, this is the second half of chapter four. we ended chapter four, first half with an unsatisfying addition to the PQ system.

Matt: 1:24

dude, are you just going to whip this out right now? You're just going to

Jon: 1:27

No.

Matt: 1:28

Oh, okay. Okay.

Jon: 1:29

I was just going to remind people that we sort of left off on this, you know, a bit of a cliffhanger. We added this, we added to the PQ system, which, if you'll recall, I guess we've described, yeah, we destroyed it. I mean, we really, we made it like completely uninteresting. Um, but he basically starts the second half of chapter four by talking about Euclid. Who, Euclid, is just A champ to me, like I absolutely love Euclid and he talks about how Euclid was one of the first people to, to basically deal in rigor, which is, you know, Euclid came out with Euclid's elements, which was essentially like the math textbook for like, I don't know, centuries. And in Euclid's elements, what he does is he starts with these, he calls them postulates, which are basically these statements that are. You know, true. They're, they're sort of like indivisible. They're true. And he uses those statements to build new statements and the Euclid's elements is just this slow progression of like, okay, here's some true statements. I am now going to use those to reveal a new true statement. I am now going to use that to reveal new truths. And it's

Matt: 2:49

Yeah, it's just like

Jon: 2:51

Exactly.

Matt: 2:53

right? Where we have these axioms, and then, um, Although, uh, Hofstadter was throwing a little bit of shade, uh, at Euclid. He was being a little bit like, like, they were pretty good, but they weren't that good.

Jon: 3:08

Yeah. Well, he mentions kind of a fatal flaw and I don't know, I'm kind of, I kind of feel weird calling it that because obviously Euclid's contributions to math is like so incredibly monumental that it's hard to even describe, but Euclid was using words to describe these postulates or to lay these postulates out. You know, he was using words like. Two points make a line. But if you really think about it, like, what is a point? What is a line? You know, it, you think of a line as straight, but what if you're traveling along the surface of the earth? Like that line is now a curve. So it's, you know, Euclid probably thought he was giving these postulates, which are only interpretable in one way, or interpretable. I don't know what the word is. But in actuality, a lot of mathematicians interpreted these postulates differently. And we'll see throughout the course of this chapter how that led to a lot of mistakes, a lot of wasted years of effort, but also new forms of math.

Matt: 4:18

Yeah, no, very, this is a very fascinating subject, this leads to these very unintuitive people or listeners have probably heard of non Euclidean geometry. And it's funny because non Euclidean geometry. So you talked about these, uh, these five postulates. Yeah.

Jon: 4:41

Yeah.

Matt: 4:41

Non Euclidean Geometry still uses the first four postulates. It's just the fifth postulate, which is kind of like the red headed stepchild, which But it feels a little bit, like, rude to Because, like, I kind of feel like all of them are Euclidean Geometries. And it just so happens that this last one, like, it's more of a knob than a rule. You know what I mean? Uh

Jon: 5:05

like, if you read the five postulates, I don't know. I just found it slightly funny because the postulates are these very clear sort of crystalline statements. Like, here's an example of one of the first four postulates. To describe a circle with any center and distance. It's like, boom. That is a very simple statement. You can describe a circle as a center and radius, basically. Or at least I think that's what that means. But in any case, it's a very simple, you know, cut and dry statement. The fifth postulate is, If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. It's like, come on, dude. All your other postulates are like six words. And this postulate is so incredibly verbose. And it's also like, I mean, I needed to, I had to read it like, five or six times in order to even understand what it's saying.

Matt: 6:13

It's funny because I think when you read it, you're like, what is going on? But if you sit down with a paper and a pencil, it actually is pretty like, as you can draw along with it, it immediately makes sense. I think,

Jon: 6:26

Yeah, it's, it's definitely one of those things where visually it's so clear, it's almost like visually it's very similar to the first four postulates cause it just has this like immediate meaning if you see, if you see it drawn out.

Matt: 6:41

yeah. There's just a construction. And I think that's what it suffers from is like, you just need to build up this little scene.

Jon: 6:48

yeah,

Matt: 6:48

So he almost just should have drawn it, you know what I mean? Like, instead of doing it with words, use a

Jon: 6:57

he probably should have cause that's probably how he came up with it in the first place.

Matt: 7:00

Yeah.

Jon: 7:01

But in any case, you mentioned that the fifth postulate was this sort of odd man out. Um, and part of the reason for that is because in Euclid's Elements, he doesn't even use the fifth postulate for like the first, uh, I don't, I don't know how many, but, you know, the, the book is like a list of whatever, a couple, or a bunch of postulates.

Matt: 7:23

Propositions, yeah.

Jon: 7:24

Yeah, propositions. And for the first many, many propositions, he does not even use the fifth postulate. So a lot of mathematicians 20, okay. I'm so glad you had that number handy. So a lot of mathematicians were like, do we even need this fifth postulate? And, and that sort of entered us into this age of math where people were basically trying to prove, you know, the 29th proposition, like without using the fifth postulate. Um, this guy, uh, Sacchieri,

Matt: 7:57

Yeah. Mm

Jon: 7:59

did this very interesting thing where he assumed the fifth postulate was false. And then he tried to basically go through Euclid's elements and like see how far he could get by assuming the fifth postulate was actually false. And this was a really fascinating thing. Basically this guy did this, he published it, and then he died. But what this work was. showed was that these new forms of, of geometry existed. Like you just mentioned non Euclidean geometry.

Matt: 8:30

Well, it's funny though, because His assumption was that he arrived at a contradiction.

Jon: 8:37

Yeah.

Matt: 8:38

So, but the, but this goes back to what you were saying before about the imprecision of language because he, he has this hilarious phrase, which is he reached a proposition, which was quote unquote repugnant to the nature of the straight line.

Jon: 8:55

Yeah.

Matt: 8:56

And so, and this is the thing. It's like, cause he used the term straight line in, in these, uh, these postulates, I mean, it's all over the place. Every single one talks about straight lines aside from all right. Angles are congruent in the, in the first four.

Jon: 9:12

Yeah.

Matt: 9:12

Uh, but when you, when you break that fifth postulate, uh, lines are no, they don't, they no longer operate like you're talking about, you know, when you you know, the shortest path across the earth is curved. So you wouldn't say it's a straight line in the, in the common sense understanding of the term.

Jon: 9:34

Right. Which is basically elliptical geometry, right? Where it's like points and lines lie on a sphere and, and you can have a, you know, sort of quote unquote, straight line. But that line lies on the sphere. So in actuality, like if you think of a three dimensional space, it is a curved line. Um, but these are valuable forms of geometry. Like they're very useful for sort of exploring other areas of math, but they're just not compatible with, with Euclidean math, which is why they're called non Euclidean geometry.

Matt: 10:08

Right. Right. Um, and I did want to talk about this. Uh, so there's another formulation of that fifth rule called the parallel postulate So my understanding is these are isomorphic with one another, um, so it says given any straight line and a point not on it, there exists one and only one straight line. which passes through that point and never intersects the first line, no matter how far they are extended.

Jon: 10:41

right,

Matt: 10:42

so this is kind of the opposite point where it's like, he's saying, well, if these two lines are slightly angled towards each other, eventually they will intersect. This is kind of making the opposite claim, which is like, if you have a point, not on a line, there does exist one line, which is parallel to it.

Jon: 11:01

this is the postulate that mathematicians were sort of breaking in order to come up with like hyperbolic geometry. Right.

Matt: 11:12

right, exactly. And so like the knob here is how many lines, like Euclid's fifth says there's exactly one. But if there are no lines that are parallel, then, uh, you have elliptical geometry, like you're saying. And if you have more than one, at least two of those parallel lines, then you have hyperbolic geometry.

Jon: 11:40

Right. Yeah. Uh, which is kind of a, a brain breaking idea.

Matt: 11:47

Yeah, I still don't feel like I have an intuitive sense. I have seen mind bending visualizations of hyperbolic geometry., Like, you have a video game that takes place in hyperbolic geometry, it's very hard to reason about.

Jon: 12:02

Yeah. Oh, totally. I mean, elliptical geometry to me makes sense. I think because there's such an obvious. Parallel to the Earth itself.

Matt: 12:12

right. I did want to, I did want to call out one point about elliptical geometry, which was like, when I was reading this, I was a little bit confused. Cause if you think about the, like the earth, latitude and longitude, um, all of the latitude lines are parallel to one another. And I was like, wait a minute, like you can have parallel lines with one another. So, um, I did want to call that out as something that confused me. I was like, parallel lines exist, but in this system, a line is always, uh, what is referred to as a great circle. So it always is the largest, uh, you know, possible circumference

Jon: 12:55

like the equator.

Matt: 12:57

like the equator or like one of the, uh, you know, one of the meridians. like the prime meridian. Um, and, and, and the reason for that is like, like slightly unintuitive. If you consider, uh, two points on a circle, you can actually come up with a bunch of different, you know, a bunch of different arcs that pass through both of those points. I think you have infinitely many.

Jon: 13:26

Yeah, yeah.

Matt: 13:28

Right. Because you can have a circle of various diameters, but then there's only one great circle that, uh, that passes through both of those points.

Jon: 13:38

Ah, right. So in order for the first postulate to be true, everything has to be great circles.

Matt: 13:45

You have to define a line as being a great circle.

Jon: 13:49

Right, because the first postulate is to draw a straight line from any point to any point.

Matt: 13:55

I wonder if there's more there because you wind up with. More conceptions of lines because obviously you have a ring around the circle. But like, what is that?

Jon: 14:04

Yeah,

Matt: 14:06

So, um, anyway, I'll stop. I'll stop there. But this was one thing. I just wanted to raise that as like a point of confusion for myself because it seemed it seemed wrong based on if you just considered latitude circles as an example.

Jon: 14:22

right. Right. yeah, so he, he goes on to get into this concept of consistency and inconsistency, which he sort of starts describing these different forms of consistency, which I found pretty interesting. He talks about internal consistency, which is, you know, a system is internally consistent. if none of its theorems are incompatible with each other, or if, in other words, if all of its theorems are compatible with one another, uh, basically doesn't contain any contradictions. Um, he talks about logical consistency, which is another way of saying there's no contradictions. Then there's like mathematical consistency, which is basically just that it obeys math. Uh, he gets into this concept of like, Other worlds or other universes or hypothetical worlds where, you know, maybe you can have a hypothetical world where, what was the example he used? It was something like green is not green. Just these like hypothetical worlds where, you know, basically inconsistencies exist.

Matt: 15:34

Yeah,

Jon: 15:38

this sort of philosophical debate of like, is math the same in every conceivable hypothetical world? You know, it's like, can you have a world where one plus one is three? And I think the conclusion he comes to, I don't know if he's, he comes to like a very hard and fast conclusion. But at least to me, it seemed like, you know, Even in these hypothetical worlds, you need to have some form of logic. Like, there needs to be some way of building a chain of logical thought. Like, creating a true statement and then building another statement on top of that. So, kind of like what Euclid was doing. Um, and I thought that was super interesting. It's like, you know, you can't create these hypothetical worlds where they're just completely contradictory and you can't possibly even You know, create any logical chain of reasoning.

Matt: 16:31

So it sounds like there's some basic layer that they all share, but it's very minimal. Like the only thing you can say is that you would have some set of axioms at the bottom. And things would have to have to be built up from there. But then aside from, as to the specifics of those axioms, like, nothing can be said.

Jon: 16:55

And in the, in this course of describing hypothetical worlds, he throws out Escher's relativity, which is a really cool drawing. Uh, yeah. So I, I almost want to like describe it. So relativity is basically that one where there's all these staircases kind of heading in various directions. And the world is like. topsy turvy, like it's collapsing in on itself, like one staircase heads down and one heads up, but they both go to the same place. So it's like you're heading down and up at the same time. So it's this, it's one of those drawings where the more you look at it, the more you realize that it's like this impossible world and that the stairs can't, can't possibly be real. Um, and he, it's interesting cause he talks about these different ways of interpreting that drawing. And one way of interpreting the drawing is just, like, to not even interpret it as stairs. Like, to just interpret it as, like, random lines in a jumble. And how, like, that's the only, that's the only way to interpret it that's actually, like, true. Mhm. Yeah.

Matt: 18:09

where it's like, there's, you know, There's consistency at the visual level. But this violates every conception of like physical consistency. Like in this picture, you have two people walking on the same set of stairs, but one person is using. what seemed to be the top of the stairs and, you know, as the top, and then the other person is using the front of the stairs as the top, you know, as compared to the other person.

Jon: 18:40

It's very, uh, there is no spoon from the Matrix. The Matrix. It's this like, you know, if you think of reality as just a bunch of random lines, or like a bunch of, you know, pixels or something in an image, then it loses all meaning. And, you know, stairs are not stairs.

Matt: 19:01

I think, I think you could come up with a consistency in this universe where the laws of physics are a function of. Your past experience. Um, because I feel like you could come up with a video game that that has physics like this. And you have some internal state as like a character. That the universe knows about and I'm not saying like a real actual physical universe could ever do this, but I think you could come up with a description of a world that could behave this way.

Jon: 19:40

like where, if you're walking up the side of a staircase, like, that is now the, that is now the ground,

Matt: 19:47

Exactly. So like you have a certain state and you reach a particular part and it's always like, it's always consistent relative to like your, whatever you like, your current value of up is,

Jon: 20:01

Yeah.

Matt: 20:02

but you could have these transition points where you like, or maybe, maybe for some of these people, like their personal down is always in one direction, but then like different people have different ups and downs.

Jon: 20:15

Dude, crazy. Can you imagine like a Quake deathmatch in like a Asher painting?

Matt: 20:21

Dude, wait, I feel like, I feel like you could, you could do that. Like every player is down is a different direction. And then you do have to have these very crazy levels that look like a sherry and staircases.

Jon: 20:38

the railgun sniping would be very difficult. Because it'd be hard to know like, what people's trajectories are.

Matt: 20:44

Yeah. Do your bully, I guess your bullets are like, maybe you wouldn't simulate gravity and on your bullets.

Jon: 20:51

This section made me think of terryology. Are you, are you familiar?

Matt: 20:56

Oh, what is Terry ology?

Jon: 20:57

This is such a dumb thing to bring up, but, uh, do you remember that actor? Do you remember the movie? Um, was it called hustle and flow or was that just the name of the song in the movie? Uh,

Matt: 21:10

The, like that phrase sounds familiar, but I don't, I don't have it attached to anything.

Jon: 21:16

okay. Anyway, there's this actor named, I think his name is Terrence Howard. Um, and he was kind of a famous actor, like maybe a decade ago. He was in one of the Iron Man films. He kind of had this, you know, very upward slope to his, to his career trajectory, but then he, I mean, I shouldn't be laughing because I think he, I think he just had like a mental break or something. He came out with this new form of math called Terry ology where one plus one was not to, I think it might've been one. It was like one plus one is one. Um, yeah. And I guess he's gone on to, like, continue working on Terryology, like, he's basically stopped acting and now he's working on, like, you know, he's trying to sell some kind of, like, new form of particle physics to, like, some African government.

Matt: 22:03

oh, okay. I mean, what cool things do, can you do in Terry ology?

Jon: 22:08

I mean, I wish I knew there, there was like an article about it at one point and it was, it was moderately interesting. Um, I feel like there was something, something to do with the square root of two. Uh, that he, you know, some, some quote unquote breakthrough he had with the square root of two. Um,

Matt: 22:27

It's funny because. This was something I was thinking about as we were reading about, uh, about Euclid is you had some guy assembling this compendium 2000 years ago, and people are still debating the things that he said, like, just imagine the genius that you have to be. Because he, he set up this, this set of rules so precisely that for literally 2000 years, like no people couldn't like decide whether or not something was true about it.

Jon: 23:04

Yeah, no, I mean, it's, it's, it's a testament. He, he literally invented, like, this whole new branch of, uh, and it, this is like a top level branch of math. Like, if you think about the taxonomy of math, this is a kingdom of math. Um, And yeah, he, he kind of put it down in whole cloth in Euclid's elements. Like it's completely insane, the level of achievement.

Matt: 23:28

All of that is to say that maybe Teriology is onto something. Yeah.

Jon: 23:37

because you reach a point where you realize math is just so incredibly abstract and you can have, you can have Euclidean geometry, but then all of a sudden some mathematician is like, Oh, wait a second, here's non Euclidean geometry, and then you find out that that actually has. I want to feel like a little bit of experience, you know, from just this, this very valuable insight and you can like learn new things based on it. and you realize that numbers are just these tools and may be thinking of numbers completely differently like gives you a new tool to like explore a new area. There was a really, really awesome Veritasium video on p adic numbers.

Matt: 24:14

Yeah. I'm so glad you're bringing this up.

Jon: 24:16

Yeah, like I, I totally loved that video. So I would recommend it to anyone. He basically describes this new way of thinking about numbers that like, unlocked, you know, basically unlocked a new knowledge. Um, and I just think that's so interesting. It's like, you're kind of moving around these abstract ideas and like, boom, all of a sudden I got a CD player, you know, it's like it, it's in this area of just complete and utter abstraction, but then all of a sudden it touches on like a piece of reality that you can then use for a practical purpose.

Matt: 24:50

No, that, that video more than I think any other YouTube video has really like, it just made me realize how, yeah, like you say, numbers are just symbols and you can set up symbols with like completely different rules.

Jon: 25:06

Right. Yeah. No, it's cool.

Matt: 25:09

um, did you have anything else? There was this stuff about, uh, sufficient, sufficient, complex, sufficient power. For a system for girdle theory of incompleteness.

Jon: 25:22

Yeah, my, my last note, uh, was on completeness and it actually wraps back around to why I found the new conception of the PQ system so unsatisfying.

Matt: 25:33

nice.

Jon: 25:35

And that is that you realize quickly that in the PQ system, there are many expressible truths. Which aren't theorems in the system, which at least for me, it just has this immediately unsatisfying feeling. And I think that that's what, uh, what is it? Russell and Whitehead,

Matt: 26:01

Hmm.

Jon: 26:02

you know, so actually I need to define completeness. Uh, so completeness is when all statements, which are true. And which can be expressed as well formed strings of the system are theorems, meaning that the rules of the system can produce all statements which are true, which is kind of an insane idea. I mean, it sounds like a relatively simple thing, but, and this is what Russell and Whitehead were so caught up on is, you know, mathematicians kept devising these systems. But the systems had all these big gaps, you know, it's like, okay, we can use geometry to talk about, you know, solving these problems. But if you're talking about working in this hyperbolic space, geometry doesn't work. So it's like, okay, now we need a new, a whole new system to even talk about those things. And this is what I didn't like about the PQ system. Uh, sort of the re, like, definition of the PQ system is that it just has so much truth. That is not even representable in the system that it was like, unsatisfying.

Matt: 27:12

Yeah. Yeah. Yeah. Um, and I mean in, but that's actually a feature because the fact that it's not a very good system means that it's not susceptible to girdles. Wiley devices.

Jon: 27:27

Right. It's a low fidelity system.

Matt: 27:30

player. Okay, yeah, I don't think I had, I had anything else. Um, I agree. This PQ system, let's just throw it in the bin.

Jon: 27:40

Yeah. It's trash. We'll move on to other systems, but man, I mean, this chapter, I think you said this in the last episode, but it really opens up, like this is starting to get very interesting, you know, the plot is thickening. My brain is hurting from all this new information and I'm, I'm pretty stoked to keep reading this.

Matt: 28:02

I think our journey to become like those mysterious white haired men who just like burst out with incomprehensible statements is well underway from, from having read this book.

Jon: 28:15

Yep.

Matt: 28:17

All right, well, I will see you next time for chapter five. Bye.