Become a Creator today!Start creating today - Share your story with the world!
Start for free
00:00:00
00:00:01
70.1: Episode 70.1 of Breaking Math Podcast (Self-Reference)
579 Plays2 years ago
Seldom do we think about self-reference, but it is a huge part of the world we live in. Every time that we say 'myself', for instance, we are engaging in self-reference. Long ago, the Liar Paradox and the Golden Ratio were among the first formal examples of self-reference. Freedom to refer to the self has given us fruitful results in mathematics and technology. Recursion, for example, is used in algorithms such as PageRank, which is one of the primary algorithms in Google's search engine. Elements of self-reference can also be found in foundational shifts in the way we understand mathematics, and has propelled our understanding of mathematics forward. Forming modern set theory was only possible due to a paradox called Russel's paradox, for example. Even humor uses self-reference. Realizing this, can we find harmony in self-reference? Even in a podcast intro, are there elements of self-reference? Nobody knows, but I'd check if I were you. Catch all of this, and more, on this episode of Breaking Math. Episode 70.1: Episode Seventy Point One of Breaking Math Podcast
[Featuring: Sofía Baca, Gabriel Hesch; Millicent Oriana]
Recommended
Transcript
Introduction to Self-Reference
00:00:00
Speaker
Seldom do we think about self-reference, but it is a huge part of the world we live in. Every time that we say myself, for instance, we are engaging in self-reference.
Historical Examples of Self-Reference
00:00:08
Speaker
Long ago, the liar paradox and the golden ratio were among the first formal examples of self-reference. Freedom to refer to the self has given us fruitful results in mathematics and technology. Recursion, for example, is used in algorithms such as PageRank, which is one of the primary algorithms in Google's search engine.
00:00:26
Speaker
Elements of self-reference can also be found in fundamental shifts in the way we understand mathematics, and has propelled our understanding of mathematics
Humor and Self-Reference
00:00:33
Speaker
forward. Forming modern set theory was only possible due to a paradox called Rust's Paradox, for example. Even humor uses self-reference. Realizing this, can we find harmony in self-reference? Even in a podcast intro, are there elements of self-reference? Nobody knows, but I'd check if I were you. Catch all of this and more on this episode of Breaking Math.
Meet the Hosts and Guest
00:00:55
Speaker
Episode 70.1 of Breaking Math Podcast. I'm Sophia. And I'm Gabriel. And you're listening to Breaking Math Podcast. With us and who did the intro, we have on Millie of Nerd Forensics Podcast, another podcast inside of Atrial Media. Welcome, Millie.
00:01:18
Speaker
I'm back, everybody. I am thrilled to have you, Millie. Nerd Forensics has been an absolute blast to listen to. Yes, I'm totally plugging you, but it has been. It's like a chance to let our hair down and just talk about the nerdy stuff like Star Wars or The Simpsons or whatever. So yeah, glad to have you here. Glad to have you here.
00:01:35
Speaker
I'm glad to be back. I'm actually planning on recording a few episodes this week. I know it has been a little while. It has been a while. Your intro was all, if you don't mind, I'm going to dive right in here. Your intro was all about self-reference. Now, I want to mention one thing. We trolled you all, all of you all who are listening this, yes. We trolled you a little bit. You may have noticed on your podcast player that there was episode 70 with a certain title. What was it called, Sophia?
00:02:00
Speaker
This episode intentionally left blank. And it was literally like 45 minutes of nothing but white noise.
Creative Risks in Podcasts
00:02:06
Speaker
Now, there's a reason why we did this. It's a little risky. Sometimes when you're trying to be funny or humorous or original, you have to take some risks. And we just did that. I don't know if we lost listeners or not, but we don't really care. Well, we do. But there's a reason for it as as as dare you to delete us or your podcast. Do it.
00:02:22
Speaker
Yeah, no, no, yeah, we hope you don't, but do as you wish. I remember as a young kid, I'd always open up a book in a library and the first page you'd have, you know, the publisher, then you'd have a page that actually says this page intentionally left blank. I've seen that a lot in testing packets.
00:02:38
Speaker
Yeah, yeah. And I always thought it was funny. Like, why did you do that? You're, you know, as a kid, I was so worried about recycling and wasting paper. You're wasting paper. Anyways, when I met Sophia, we had post-it notes everywhere, especially when we were planning this actual podcast. We had all kinds of post-it notes one time. And I saw that Sophia had written one post-it note with some notes on it. Then right next to it, she put this post-it note intentionally left blank.
00:03:00
Speaker
Oh yeah, wasn't that when we were doing the project, the genetic algorithm? Yeah, yeah, yeah. I thought it was hilarious because it reminded me of those books that I've written in school. So anyways, in an episode that's all about self-reference, we did that. Now, I want to mention that I took your humor, Sofia, and I had my stepdaughters do that with their dad because they always were good friends with
Philosophical and AI Perspectives on Self-Reference
00:03:23
Speaker
their dad.
00:03:23
Speaker
We always talk about dad jokes and inside jokes. So they always write each other post-it notes. So I'll have you know they stole that joke. I also stole it at work. We've got these white boards that are cubicles that we were all given. The white boards are supposed to have important notes as well as where we can be found if we're not at our desk. So I did two things. I wrote on one part of the white board. I wrote this section of the white board intentionally left blank.
00:03:48
Speaker
then beneath that I wrote if Gabe is in the chair he's here if Gabe is not in the chair he's not here
00:03:57
Speaker
Oh yeah, tautology, right? So that also brings up the point, what is self-reference? And it's a difficult question because it's kind of nebulous. It's something self-referential. I would say it's self-referential if there's an element that can be said to be part of the thing that either encodes the thing or points to the thing.
00:04:19
Speaker
is that the technical definition you'd find the thing you know read the philosophy wiki on the word thing it's like 10 20 pages long that should be punishment but you know I mean like we should actually have somebody do that it's like a dare anyways you're right self-reference when I was thinking about it myself I tried to think about how to explain self-reference and I think the way I described it is like
00:04:38
Speaker
I'll use the word thing liberally as well. You'll have to forgive me. Whenever you have a thing like, say you're going to design a little artificial intelligence robot or something, it has to map its environment around it. It has to take input data on the world around it and makes a little internal map. Whenever you can have something like that that eventually can map itself, it refers to itself.
00:05:03
Speaker
Yeah, because you have the location of where it is. Maps themselves are sort of a self-referential meta sort of thing. And anyone who is under 40 knows the expression and is sick of it by now. Pull yourself up by your bootstraps, which is pretty self-referential. If you think about it, you pull yourself up by your bootstraps, you can't do that. You'd fall on your head. Oh, no, exactly.
Pop Culture and Self-Reference
00:05:27
Speaker
And Millie, can you think of some examples of self-reference you run into?
00:05:31
Speaker
Uh, my favorite example is you are here with a big star in the middle. Oh yeah. And like malls. Yeah. Yeah. What is that thing in the Simpsons when they go to area 51A? Yeah. It's you are here. We are not. Oh yeah. That's hilarious.
00:05:48
Speaker
And before we dive into this episode, we're going to talk a little bit about, we're going to have a few guests on for a joint episode in either next or the one after that episode, I think. Yeah. So what we can tell you is this, we were planning to do it very, very soon, perhaps as soon as a week, but the details are still being finalized. So it's still maybe a ways out. However, the Breaking Math podcast is going to team up with another
00:06:09
Speaker
fabulous podcast, a hardcore science and engineering podcast called The Materialism Podcast. Now, this is put on originally by an Andrew Falkowski and Taylor Sparks. They are also joined by a Jared Duffy, who's a mechanical engineering student, and Ramsey Iplivica, a material science PhD student. Quick note here, I, this is Gabe, of course, I was very excited when I found a material science podcast because I was almost a material scientist myself.
00:06:39
Speaker
I'm obsessed with material science and I get very excited by all the stuff that we haven't discovered yet You know you can get into sci-fi about what material properties like what what have we not heart What have we not harnessed yet in terms of technology? Oh, yeah, I remember when you're doing that antenna design class, right? Yeah, there's those there's those weird problems about like spheres of like weird jelly that refracted in one way and another way electronic waves and
00:07:05
Speaker
Yeah, so so I want to encourage our listeners to check out the materialism podcast what I've also done for my co-host Sophia Oh, and I'm so sorry I forgot to mention we're gonna be joined by a friend of ours the mad scientist podcast host Chris Cogswell doctor Chris Cogswell Who is a P who has his PhD in chemical engineering? That is podcast is pretty awesome It's kind of diving into a lot of like pseudoscience in a mathematical and very scientific way
00:07:32
Speaker
Yes. So we will have the breaking math podcast, who we've got folks who are into mathematics and computer science and electrical engineering. We will have a chemical engineering podcast or host, and we will also have material scientists. And it'll be focused on material science as well as their show. And of course, our episode is the superior because everyone uses math.
00:07:53
Speaker
I love those fights. That kind of infighting is always all kinds of fun. But listen, before we move on to the next section, I recommend our listeners check out their site and look at their show notes. They have episodes on things like an interview with a company that's making biodegradable
00:08:09
Speaker
Packaging for example they have all kinds of episodes on what kind of materials can you use in? Science and medicine that your body won't reject and why and like how are these materials made it's fascinating I also have economics questions for them like what kind of things in material science like affect economics by making things cheaper and
00:08:32
Speaker
Oh yeah, either cheaper or there's even the issue of how dollar bills and stuff are made with those little almost microscopic red threads. Yeah. But yeah, we'll discuss all of that on that podcast. Yeah, yeah. So in the meantime to prepare, just check out the materialism podcast. You can just Google it and you should find it.
Self-Reference in xkcd Comics
00:08:54
Speaker
And now...
00:08:57
Speaker
Humor and self-reference. One author, and his name is Randall Munro, and he uses self-reference constantly in his comics. And we're just gonna do a couple of these. Just xkcd.com slash 33 is just three panels. And the first one is just somebody saying, I promise to never again squeeze humor out of self-reference. Says nothing in the second panel. Third panel, he says, damn it.
00:09:22
Speaker
Yeah, obviously it's referring to itself, so you get the humor. I think my favorite one we're going to save for last, but yeah, that's a good example.
00:09:30
Speaker
Then we got xkcd.com slash 688, which contains a pie chart, a bar chart, and kind of like a plot. And the pie chart says how much of the image is red. I mean, how much image is white, how much is black. The bar chart says how much ink is in each thing, how much ink is in the pie chart, bar chart, and the miniature. And the miniature of the comic just kind of is a picture of the miniature of the comic next to a few axes.
00:09:58
Speaker
What's cool about that is that every single square depends on every other square, right? So like the pipe if there's if the bar chart shows an increased thing There's more ink in that so the pie charts gonna change which means the bar charts gonna change, you know, okay That's totally hilarious I mean make a pie chart and just label just make some arbitrary slice and say this amount of the pie chart is black This amount of my chart is white like that. Yeah, it'll probably converge, you know
00:10:20
Speaker
You know what that reminds me of? That reminds me of Claude Shannon's... What is it? The Greatest... What's the name of his machine? Where you turn it on and all it does is a hand comes out of a... Oh, the useless machine. No, it's called like the ultimate machine, I think. Let's find out. Yeah, like the greatest machine ever or whatever it is. Claude Shannon, greatest machine or ultimate machine, I think. Yes, that's what it's called. Okay, yeah.
00:10:46
Speaker
All you do is you turn it on and a hand comes out of some compartment and turns it off. That's all it does. Then we got xkcd.com slash 917. xkcd917 is my absolute favorite. It is divinely funny and I chose that word intentionally. Let me go to that comic real quick and I'll explain why.
00:11:06
Speaker
So, it's a three panel comic. And the first one, it has two people talking. One of them is at a laptop. One of them is standing beside the person just talking to him. And he says, what's this? The person at the laptop says, Douglas Hofstadter's six word autobiography. After all those 700 page tomes, I guess he wanted to try for brevity. And then the other person says, huh, let's see. Then he reads it and it says, I'm so meta, even this acronym.
00:11:34
Speaker
And that's it, that's all six words. Then he says, whoa, I think he nailed it.
00:11:39
Speaker
And what are the first letters of all those words? Okay, I'll read that one more time. The first letters of every word in the sentence, I'm so meta, even this acronym. I-S-M-E-T-A. So, is meta. Is meta. So, I'm so meta, even this acronym is meta. Which, if you've read Douglas Hofstadter, who wrote Go to Leisure Bach, I think episode 17 or something was that. Yeah, yeah, yeah. I think you're right. We'll have to see it. Or whether it's called incomplete.
00:12:05
Speaker
Oh yeah. Incomplete. That's funny. Yeah. So, so obviously it's hilarious because meta is, is that, would you say the definition of meta is self-referential? Oh yeah. Meta is almost just a word that makes things self-referential. So like, you know, um, meta store would be a store that you buy stores in. Okay. I love that. Yeah, that's great. So yeah. What about a store where you buy the currency to buy stores? Uh, Oh, let's see. What would that be? Um, meta? Okay. So just self reference with extra steps.
00:12:33
Speaker
Yeah, but I think we could do this one. Let's see. So you go to the store. You're saying the word for the money that you use to buy stores from the store, right? A store that you buy money to buy the stores. OK, a store that you buy money to buy the stores. I'm thinking of conversions like when you change one currency to another. Yeah, like imagine that the store owner's only accepted Dogecoin or something stupid. So I think you call it like a meta store.
00:13:03
Speaker
meta, meta, meta store. Well, it's like you'd be like meta cash, meta store. You know what, we're gonna work on that. And we'll talk about that. Can we do like a creative, creative self referential challenge? Like, so you know, who thinks of the best plots that are self referential? I don't know.
00:13:18
Speaker
I say that anybody who gives us a concise way to define what Millie's... So I'm going to bring up another xkcd. Let me pick it up. Okay. Okay. Very good. So xkcd. So it goes, you're looking festive. And this person with a Christmas hat says, I love Christmas. Really? Doesn't seem like your kind of thing. It's our most meta holiday. How so? All of our Christmas stories now are about discovering the true meaning of Christmas.
00:13:42
Speaker
Yeah, and then sharing it with others at some point that quest itself became the true meaning Like a word in the definition is the act of looking up the definition of this word automatic logo Lex my least favorite of Sanders reindeer, okay
00:13:57
Speaker
I blame Kirk Cameron another automatic local Lex I think is What would another one be
Literary Self-Reference
00:14:04
Speaker
fourth of July? Okay, I'm not gonna lie. I think that became my favorite one now Like literally that's hilarious. Oh, yeah Auto meta loading the true many of Christmas itself became the true meaning of Christmas The quest to find the many of life became the meaning of life. I know the meaning of Christmas It's Kirk Cameron and Kevin Sorbo
00:14:23
Speaker
need to make a movie together that tells everybody about how they're wrong for writing happy holidays on things. Hey, let's make a parody on that. I love that plot. Okay, we veered far off math.
00:14:36
Speaker
Menastore numisma... I'll figure that out. Sorry, I broke her. Now I'm not going to be satisfied until it's like an itch in my brain now. And we have the book Cats Cradle by Kurt Vonnegut, in which there's a form of ice that they call ice nine, which is different than the real ice nine, which was discovered later. It's just a type of pressurized ice that has different properties than normal ice.
00:15:00
Speaker
that's produced from ice-3, uh, quickly so it doesn't produce ice-2. But anyway, ice-9 in this book, in contact with normal water, produces ice-9, which is a solid. So basically what it does is, um, some ice-9 gets out of the laboratory, and it basically turns all of the, um, ice, all the water on Earth solid.
00:15:19
Speaker
Interesting. Wow. Like a Borg type thing? Yeah. Kind of like the board, but imagine it's just chemicals. Like imagine you just have a form of water that's even more stable than water. Okay. That when it touches water, like basically it crystallizes outwards. Okay. And at the time this was like, you know, thought to be like, you know, there are substances that this does work with, um, that had this property. So catch 22. Um, so y'all have heard the expression catch 22, right?
00:15:49
Speaker
Oh, yeah, absolutely. I mean, like, almost like you're doomed if you do, doomed if you don't. Yeah, like, like you need the thing to have the thing. Okay. So and the reason why it's in, it's a book in Catch-22 is a book by Joseph Heller, who was published in 1961 and he wrote in 1953. But in the book, if a pilot is deemed insane, they don't have to fly. To be deemed insane, a pilot was requested to be evaluated.
00:16:15
Speaker
If a pilot requests to be evaluated, this demonstrates that he must be sane. Therefore, no pilot can ever be deemed insane, and no pilot can get out of flying. And the whole book is about the ridiculousness of the military.
00:16:29
Speaker
I've only read the first I've only read like the first like five chapters. I don't quite remember. It's World War Two though, right? Oh, yeah, it's World War Two. Yeah. I want to say it was the North African theater because I want to say that were Italy. Yeah, I was going to say because I know the whole Mediterranean had like the craziest like death rate for pilots in the United States. Wow. Interesting. And Milly, can you think of any pop culture references that are self-referential?
00:16:54
Speaker
the entirety of the Dark Tower. That's pretty self-referential to every other Stephen King story. Do you want to explain that? Okay, so the Dark Tower takes place in a world called Midworld that serves as the nexus between all of Stephen King's different stories.
00:17:10
Speaker
Because some stories take place in the same world. Others take place in different worlds. A good example is what is it? The silver bullet and it and a few other like pretty much any any story about like a kid who's dealing with something awful all pretty much takes place in the same world. Stephen King. I see. And then you have you have some stuff that of course takes place in like the real world and stuff. And then mid world, it's where it all links up.
00:17:39
Speaker
Okay, so everything's tight together through mid-world and didn't you say that um that that um? Was he dusty Sam or whatever the evil guy he appears in Randall flag Randall flag like dusty Sam is a good name I'm gonna use that for a villain now dusty Sam, but yeah Randall flag though. He's in children the corn
00:17:59
Speaker
It's very likely that he was he who walks between the rows. He's also the main antagonist in the stand and a ton of other stories. Eyes of the Dragon and a bunch of other stories. Yeah. And that's kind of self-referential because, you know, you have all these different characters in all these books connected and then you have all and then you have the thing that connects them all. So it's almost it's it's like a self-referential because each world refers to itself. Right.
00:18:26
Speaker
And they actually do it multiple times. Flag, though, is a great example of self-reference because it's constantly referencing him.
00:18:35
Speaker
So another major thing that self-reference does is it allows for some paradoxes that have shattered all mathematics and logic. So we'll
Mathematical Paradoxes and Self-Reference
00:18:43
Speaker
go ahead and discuss a few of those in this section. Well, we'll, yeah. And that's, and one of the earliest ones, it comes from Greek times is the liar paradox. And Gabriel, do you want to read the liar paradox? Oh, totally. Yeah. Yeah. So, so essentially these are two statements and one of them says the next statement is false. And then the next one says the previous statement is true.
00:19:05
Speaker
And so Millie, which one of these statements is true and which one is false? Well, neither of them actually. Yeah, exactly. That's the paradox, right? Because if the next sentence is false is true, then that means that the previous sentence is true, which means that this next sentence is false is true. You can make like a logical engine with that, a perpetual motion engine, you know, just by running that.
00:19:27
Speaker
Oh yeah, because the next sentence of all the previous sentence is true is false, which means that the previous sentence is actually true, which means that the next sentence is false, and it just keeps going around in a circle, right? Do you remember SpongeBob when he has his arms all tied around himself? Yeah, it wasn't even trying to figure something out. Opposite day. Oh yeah, on opposite day, yeah. It feels like that, yeah. And the original liar paradox, by the way, it wasn't two sentences, it was just one. It says, I am lying. Oh, wow. And it's like, okay, is that person telling the truth? Yes, then they're not lying. Then they are, then they're not, then they are.
00:19:57
Speaker
Yeah, I think it's a little easier, a little more, the paradox is a little more apparent in the two sentence version. Oh yeah. And now we have a Quine's paradox. So Quine's paradox to me, just from looking at it, it looks almost the exact same as the two sentence version of the liar paradox, except it uses a different punctuation. Okay. So Quine's paradox yields falsehood when preceded by its quotation yields falsehood when preceded by its quotation. The first part was in quotations.
00:20:25
Speaker
Yeah, so, yields falsehood when preceded by its quotation, yields falsehood when preceded by its quotation. Now, like you said, it's basically the same paradox as before, right? Is that apparent, Millie? And yeah, so if you think about it, if the sentence yields falsehood when preceded by its quotation, if you take that and you precede it by its quotation, it yields this sentence, which is paradoxical.
00:20:46
Speaker
because, you know, if it's true, then it's false and it's true, etc. And the thing that's really good about Guine's Paradox is that it doesn't say this sentence or like it isn't referred to anything else. It doesn't say this or that or me. It refers to itself indirectly, which is used in a lot of math like, for example, in Godel's Incompleteness Theorem. That's kind of the way that the Godel's Sentence works, is it refers to itself indirectly.
00:21:12
Speaker
Can't we name two consecutive episodes that the next episode is false, the previous episode is true? Sure. One more thing, one more thing. Hey, quick little humor thing here. Another way of using self-reference as humor. Years ago, when our first part, when our podcast first began, we were invited to do something called the New Mexico Podscape by a local podcast group called City on the Edge.
00:21:37
Speaker
And they were asking us, you know, personal questions as one does during an interview. And Sophia said, oh, and be sure to check out our other podcast through the Looking Glass in which we talk about what will be on the next week's episode. Oh, yeah. Like that's that's what the podcast is. That's you just discuss what will be on the next week's episode. Like if it's oh, yeah, each each episode is only about what's on the next episode. Yeah. Yeah. Like that's the only. So what is it then? It's a self-referential thing. It's hilarious. I loved it.
00:22:07
Speaker
All right, yeah, and exactly. And we're gonna get to the last paradox, which actually caused a huge shift in mathematical thinking. So Russell, Bertrand Russell was a logician in the early 20th century and he was studying, or actually it was 1800s, I think. Let me check. Bertrand Russell.
00:22:24
Speaker
Yeah, Bertrand Russell, he was born in 1872, so he was about the same time as Einstein, all of them. So just a reminder, a set is just an abstract mathematical concept that can contain everything, anything that you're talking about. So you can say the set of all real numbers, and every real number will be in that. Or you can say the set of Barney and Homer, and the set will just be Barney and Homer.
00:22:47
Speaker
And so you can say Barney is in the set is true. Homer's in the set is true, but margins in the set would be false. And for the set of all real numbers, like I, you know, the imaginary number does not exist within the set, but like 3.97 does. And he was studying the set of all sets.
00:23:09
Speaker
And I'm re-recording this part because I made a major mistake in the way I explained Russell's paradox. So Bertrand Russell, what he was studying was the set of all sets that have certain properties. So for example, the set of all sets that are even would have 0, 2, 4, 8, 16, whatever, all the even numbers in it.
00:23:31
Speaker
So what he was studying is not just the set of all sets, but the set of all sets that do not contain themselves. So for example, in this hypothetical set, 1, 2, 3, the set that contains just 1, 2, and 3, would be in this set because 1, 2, 3 does not contain itself.
00:23:50
Speaker
However, like a set that did contain itself, which you can actually have when you have infinite sized sets, would not be in this set. And Russell asked a very simple question of the set. Does this set contain itself? That is to say, does a set of all sets that don't contain themselves contain itself?
00:24:11
Speaker
So if the set did contain itself, then it would not be in the set because of the definition. Therefore, it would not contain itself. But if it didn't contain itself, then it would have to be in the set, which means it would contain itself. And this showed that you cannot have the set of all sets that do not contain themselves.
00:24:29
Speaker
as a set. What this meant at the time was that set theory had to be reformulated. There's certain formulations like ZFC, which avoid this, but basically what this is is kind of like a litmus test. Any theory that allows the set of all sets that do not contain themselves is an inconsistent set theory, because you could derive anything true or false within this system. And we remember from Godel's Inclin's Theorem, which we discussed on that episode entitled Incomplet,
00:24:59
Speaker
Any theory has to be either inconsistent, which means that is basically useless because you could derive anything within it, or incomplete.
Quines in Programming
00:25:07
Speaker
There are things expressible that are true within the theory that are not provable within the theory. And now, back to the episode.
00:25:17
Speaker
Now we're going to talk about quines. And quines are related to quine's paradox, the yield's falsehood with preceded by its quotation, yield's falsehood with preceded by its quotation, in kind of an indirect way. A quine is a computer program that can write its own code. So if you run the program, it'll write something out. And if you compile that, that is to say you run that in the program's code, you'll get that code again at infinitum.
00:25:41
Speaker
Yeah. Very cool. Uh, which is kind of strange because you could do it in, it's proved that you could do this in literally any programming language. This reminds me of the philosopher question about the original unmoved mover. You know what I mean? Uh, can you go into that real quick? Sure. Sure. Oh yeah. This is a very common one right now. This has done a lot in theological debates, uh, involving something called the Kalam cosmological argument, I believe, but also it goes back to, I want to say Plato if I'm not mistaken. I think it's actually, um, Aristotle. Uh, no, I think it's, um, Thomas Aquinas.
00:26:10
Speaker
Okay, but even before that, no, I think it's Play-Doh. Let's find out. Let's Google search it. But the idea is that whenever something, you know, everything has a cause. So the idea is that everything has a cause, obviously, you know what I mean? Like sun goes onto a plant and the plant gets energy from the sun and therefore produces fruit. Well, these are causal chains.
00:26:37
Speaker
the idea behind an unmoved mover is what caused everything. And it had to be something that itself didn't have a cause. And this example of a code where if you run the code, it will write its own code in sort of a self-referential thing, well, you had to have somebody run the code in the beginning to start.
00:26:59
Speaker
Now you can also get further philosophical, like we would be the uncoated coder, but we ourselves have causes as well. So I'll leave the philosophy to our listeners. Oh yeah. And yeah, the Thomas Aquinas actually did use that concept quite a bit in his protocol calculus kind of stuff. Interesting. Yeah. So Millie, is the concept of a coin, what do you call it? Clear?
00:27:21
Speaker
For the most part, you're talking about something that could program itself, essentially. Well, not yet. Pretty much, yeah. But the only program that it could do, though, is the program that spits out its code. So it's a lot like the impossible machine or the useless machine. Yeah, it's the ultimate machine. Yeah. Does Quine come from Aquinas? No. Are you sure? I mean... Positive. Oh, okay. Well, it's cool.
00:27:49
Speaker
I mean, yeah, I was actually pretty shocked when he said no because his name is literally a quine is Yeah, and I mean, you know, everybody should name everything after themselves I know Robert Fripp has been fighting for that for years Fripp what's a Fripp? So looping also known as Fripp tonics. Yeah, the guy who invented it Robert Fripp He's been trying to get it named after him for years. It's like musicians are like no, we're just gonna keep calling it looping and
00:28:17
Speaker
Oh, so yeah, you can actually talk about a multi-coin in terms of just getting back to this. Parthenogenic lizards, for example, are kind of like a coin because parthenogenesis is when something is born basically a virgin
Science Fiction Concepts
00:28:35
Speaker
birth.
00:28:35
Speaker
birth without insemination and so these lizards will produce themselves and that could actually be done if there were no DNA degradation it could be do that forever right yeah so that's kind of like a coin the coin just produces itself which produces itself which produces itself and in this case the the birthing process would be like compiling the DNA would be like the code yeah and
00:29:00
Speaker
Okay. That makes sense. Now this next section is probably my favorite section of this entire podcast. I'm just going to give you guys a little preview here. So Sophia wrote a little science fiction concept in order to illustrate exactly what we're talking about. Oh yeah. The multi-coin and right before we go into the science fiction, I'll just give the kind of technical definition. Sure. A multi-coin is a program that can produce code in a chosen language given an input that produces the multi-coin in that, in whatever language that you choose.
00:29:30
Speaker
So like, for example, a multi-coin written in C could produce the same multi-coin written in Java or C given the input Java or C. But that's kind of confusing, right? Well, sort of. I mean, so, OK, yeah, for the sake of this episode, I'll say, yes, it's confusing. An illustration would help. And for all of our breaking math listeners who are into science fiction or creative writing, check this out. This might be a good writing prompt for you.
00:29:54
Speaker
Suppose we are on the planet Zanit. There are five species of different planetary origins here. There are the greys from Alpha Centauri. There are the blues from Beta Malori. The pinks from Floridia, humans from Earth, and the shadow people from Planet X.
00:30:10
Speaker
Just as humans have DNA to represent their genetic code, the greys have ANA, the blues have BNA, the pinks have CNA, and the shadow people have ENA. As part of a futuristic pack, they produced multi-quin specimen Dolores together, a human who is the first of a multi-quin people. Dolores DNA has been altered so that she can produce ANA, BNA, CNA, DNA,
00:30:34
Speaker
or ENA out of her earwax based on brainwaves. If Dolores closes her eyes and goes into a trance, and pictures either a grey, a blue, a pink, a human, or a shadow person, her ears will produce the genetic code to produce another multi-quin specimen.
00:30:51
Speaker
For example, if she thinks about a gray during a trance, then ANA will be produced out of her ear. And if the ANA is taken and it's put into a gray's egg, it will produce another multicline specimen, say Dolores. And now if Dolores goes into a trance and pictures a human, DNA will be produced out of her ears. And this DNA when put into a human egg will produce Dolores again. Yeah. And you have like, and you'll have five different types of Dolores, right? Yes.
00:31:19
Speaker
Yeah. And this analogy, you know, DNA, DNA, DNA, DNA, and DNA are kind of like the source code of the multi-coin. The Delorei, Delores, Delorex, Deloreum, all of them, they are the compilers, right? The compilers of the program that take in code and produce output, right?
00:31:37
Speaker
Yes. And the output is the source code, the ANA, BNA, et cetera, right? Yep. Meditation is the process of compiling the code and outputting stuff, right? Yes. And the image that they choose to put in their head during the meditation is the input to the multicoin. Yeah. And so that is the story of the Lorei of Planet Zanit.
00:32:00
Speaker
That's so cool. All right. Now for the rest of the episode, sadly, I must dip out, but weird phrase. I'm sorry. I'm using that phrase. I should apologize to dip out. I don't know. Well, it was a real phrase. It is a weird phrase. So my nephew says that whenever I totally heard that before. It's totally normal. I mean, but dip, I think of like you're dipping into, how do you dip out? When I dip, you dip, I dip, we dip. Okay. Can you play the sound clip from that?
00:32:30
Speaker
Okay. Gabe's dipping and keep saying that late. I'm not going to go actually dip. That's disgusting. Okay. Anyways, I like with, uh, tobacco. Yeah. Yeah, exactly. It's a weird phrase. It makes me feel weird saying that anyways. I'll save the rest of the episode for you too. I'm sure you guys will do a great job. Uh, please follow, um, us on, on all the socials and, uh, uh, yeah, we will do some great episodes that are coming soon. So yeah.
00:32:53
Speaker
Tech Pod Gabe. And I'm Camp Pod Millie and that's Camp with a K. Okay, and we'll see you guys in the next episode.
Mathematical Systems and Recursion
00:33:01
Speaker
Awesome!
00:33:04
Speaker
Now we're going to talk about recursive definitions in mathematics. So the first example we're going to give, which we've talked about this on multiple episodes, is Peano arithmetic, which was created by Giuseppe Peano, who was born in 1858 and died in 32. The definition that it's addition for specifically integers, right? Yes. And, and, and, you know, integers, right?
00:33:29
Speaker
Uh, integers are, um, the, uh, uh, if I remember correctly, that's when you, you, when you, uh, put things in brackets, right? Actually, no, that's, um, uh, that's, I can't remember what that's called right now, but I didn't injure is just zero one to actually not integers. These are natural numbers, but integer is a whole number, like negative three or 57. Um, and a natural number, which is what piano arithmetic is for.
00:33:52
Speaker
is 0, 1, 2, 3, all the way to infinity. So negative 2 is not a natural number, right? Yeah. But 72 is. Yes. And 0 is, but negative 1 isn't, etc. Yes. So the way that addition is defined within piano arithmetic is using a function called the successor function, where the successor of 3 is 4, the successor of 5 is...
00:34:13
Speaker
Sorry, the successor of 5 is 6, right? Yep, 6. Successor of 100 is 101. So the way that addition is defined is using the successor function. And the definition is A plus the successor of B is equal to the successor of A plus B. And A plus 0 equals A. Those are the two rules for Peano addition. So let's give an example. So let's do 3 plus 4, right? Yeah. 3 plus 4 is 3 plus successor of what? Successor of 3.
00:34:41
Speaker
Yeah, and so this three plus successor of three, given this definition, is a successor of three plus three, right? Yes. So now we're gonna look at three plus three. Three plus three is equal to three plus successor of two, right? Yes. Which means that three plus successor of two is equal to the successor of three plus two. Yes. So now we have two successor functions that we're dealing with. And then the third thing is three plus two is equal to three plus successor of one, right?
00:35:07
Speaker
Yeah, which means that it's a successor of 3 plus 1 and then the next one is 3 plus 0 is a successor of 3 plus 0 and 3 plus 0 is just 3 plus 0 is 0. No, sorry 3 Sorry, I'm thinking multiplication. I'm all over the place
00:35:25
Speaker
Oh yeah, you've been getting bad sleep. Yeah, I've been getting bad sleep, so I'm a little, I'm half awake everybody, so I am not up for math today. But we'll plow through. So we have the successor of the successor of the successor of the successor of three is our answer, right? Yes. So it's the successor of three? Four. The successor of that?
00:35:46
Speaker
uh five successor of that six and then the successor of that seven seven so three plus four equals seven and that's how addition is done in piano arithmetic does that make sense yes
00:35:59
Speaker
And you might be wondering why this is defined this way. It's because Peano was trying to figure out a way to define the natural numbers formally so that they could be manipulated on paper using simple rules. But a theorem called Goldel and completeness theorem showed that there are statements that are true within Peano arithmetic that you can't prove using Peano arithmetic, which is kind of strange.
00:36:20
Speaker
Yeah. And actually that goes for any system, no matter what system you have for math, no matter how complicated it is, you will always find at least one thing in that system that you can't prove within that system that happens to be true.
00:36:35
Speaker
And it's and it's kind of fascinating and the reason why is because any formal system is either inconsistent meaning It's basically useless like like we did with naive set theory earlier That's inconsistent theory or incomplete and those are the only ones
Recursive Mathematical Concepts
00:36:51
Speaker
that are useful. No really so weird. Yeah, it is weird It's one of the most insane results in mathematics and factorials are defined the same way. Do you know what a factorial is? I
00:37:04
Speaker
Not entirely. Well, so the factorial of 5 is 1 times 2 times 3 times 4 times 5.
00:37:10
Speaker
Okay. Factorial zero has to be one. Because the reason why is because since the factorial of one is equal to one times the factorial of zero, right? Yes. And the factorial of one has to be one. So that means that factorial zero has to be one. So n factorial is defined as n times n minus one factorial. So that's just another thing. Also, the gamma function is related to it. The gamma function of n is equal to n minus one factorial. But the gamma function is defined for all numbers, not just integers, right?
00:37:39
Speaker
Yeah. And so, yeah, it's defined as the gamma function of n plus one is equal to n times gamma n. Another example is feed the golden ratio. You've heard of this before, right? Yeah, but, you know, don't ask me for any information about it. Well, what it is put simply is that let's say you have like a meter stick, right? Yeah.
00:37:59
Speaker
And let's say you want to take a piece of it such that the ratio between the entire meter stick and that piece is the same as the ratio between that piece and the smaller piece left over. And what that turns into is the definition phi is equal to one plus one divided by phi.
00:38:18
Speaker
And so if you plug that definition into phi, you get 1 plus 1 over 1 plus 1 over phi. And if you plug that in, you just get that ad infinitum, which means that phi is equal to 1 plus 1 over 1 plus 1 over 1 plus 1 over 1 all the way to infinity.
00:38:31
Speaker
And yes, self reference in this form is useful in so many in so many contexts, right? It's just it's become a fact of mathematics at this point. Yes. And do you want to ask any questions you have any at the moment? Not so much. It's just I'm I'm wrapping my head around it and I'm getting that so self reference. It's because of the fact that a lot of it is to like is to prove theory and stuff like that.
00:38:58
Speaker
Oh yeah, I've approved things. The big one is defining things. Because sometimes it's easier to define something in terms of itself than it is in terms of other things.
00:39:07
Speaker
Well, yeah, you can't define something if you're not aware of what it is. Oh, yeah. But what I mean is like fee is equal to one plus one over fee. It's like, well, how do we calculate that? You know, it's like, well, you just kind of plug it into itself forever. It's like, why would you now? It turns out that if you plug in this last fee with any number, it'll just converge to fee. If you just keep if you just keep adding one and then taking one divided by that forever, you'll get fee.
Fractals and Self-Reference
00:39:30
Speaker
Yeah.
00:39:31
Speaker
And yeah, Fi just happens to be like a nice little ratio. It's overused, though. The golden spiral is not in every Renaissance painting like everyone says. No, it is not. It's fun, though, to put spirals on things. It's still pretty. I like it. Millie, have you heard of the Cantor set? The Cantor set? No, actually, I have not. So this is the way it's made. Let's say we draw a line, right?
00:39:57
Speaker
Yes. And let's say we erase the middle third of that line. You have two lines left over, right? Yeah. Now let's erase the middle third of those. You have four lines left over, right? Yeah. If you keep doing that forever, you'll get what's called the Cantor set. Hmm. And what's funny is that the Cantor set has area of, has a length zero, and it actually has a dimension between zero and one. Interesting. But Millie, did you catch how the Cantor set is self-referential? No, actually I did not.
00:40:26
Speaker
Well, think about this, if you take the cantor set, right? Yeah. And you squeeze it into the size of one of the original lines, that will, it'll match, right? Yeah. So that's what a self-refrigerator fractal is. You can find the fractal inside the fractal. Ah. And we also have the Sierpinski triangle, and you've encountered that, right?
00:40:45
Speaker
Um, not too sure if I have actually, uh, it's like, um, you know, the tri force from Zelda. Uh-huh. I do imagine all the triangles were made out of tri forces, et cetera. So basically what it is, is, is you first take a big triangle and you make it out of smaller triangles.
00:41:00
Speaker
Yeah, you cut you cut out the triangle in the middle that turns into three smaller triangles that are equal, right? Yeah, and they cut out the middle of those three and you keep doing that forever and you get a fractal with dimension between one and two and The super ski triangle. Can you did you catch how that sell for a virtual? Because you're just creating triangles non-stop. You're just basically duplicating the original shape. Oh
00:41:21
Speaker
Yeah, you can actually define the Sierpinski triangle in terms of itself, right? You could say the Sierpinski triangle is the thing that if you copy it here, like to the bottom left, top, and bottom right, you'll get the Sierpinski triangle. And then we got the Mandelbrot set. And Millie, I'm gonna look on the picture of the Mandelbrot set for you. You've seen this, right? I have, actually, yes.
00:41:45
Speaker
And what's funny is that the Mandelbrot set is self-referential as well. It turns out that there's a lot of little circles within the Mandelbrot set. And if you take the ratio of successive circles and you go to the point that is defined by them, you'll find another Mandelbrot set.
00:42:02
Speaker
And it was created by the complete curmudgeon, Benoit Mandelbrot, who believed that he had a monopoly on everything fractal. But that's his episode for another time. That'd be a fun episode to do, jerk mathematicians. Yeah, but we'd be here like all day, I think. Oh yeah, we'd be here for a while. Honestly, not as many mathematicians as you think are outwardly jerky, but quite a few are.
00:42:28
Speaker
Yeah, I'm also, uh, physicists don't count as mathematics people, do they? Well, no, not, not my math per se, but they use a ton of math, obviously. Okay. So yeah. Yeah. Cause I was gonna say I'm more aware of physicists being. Yeah. I think that'd be a really fun episode, actually. Yeah. That sounds fun. Actually. I'd be, I'd be interested in it. Oh, um, I do know that John Nash had some very, very, very backwards views regarding women.
00:42:54
Speaker
Yeah, he was a complex character to say the least, right? Oh yeah. And John Nash for anyone not acquainted was a... Oh yeah, he was a game theorist, differential geometer, and he made a lot of contributions to partial differential equations.
00:43:10
Speaker
And he studied a lot of complex systems, but he was very schizophrenic, and the schizophrenia was treated in the worst way possible in the late 50s, early 60s. I don't recommend anyone watch the movie, A Beautiful Mind. It completely mischaracterizes him and his research.
00:43:34
Speaker
They don't even show him getting arrested once for soliciting sex from a cop. It happened multiple times, right? Yeah, it happened multiple times in John Nash's life. Like he was also he was also closeted and secretly gay. But yeah, he used to go and try to pick up men all the time and he had the worst luck ever because he would always just run into cops.
Conclusion and Podcast Promotion
00:43:52
Speaker
Oh, yeah. This back when they did like those ridiculous gay stings. Oh, yeah. Thank God things have changed.
00:43:59
Speaker
A careful analysis of self-reference has led to many fruitful results in mathematics. Over and over, we see new ways in which self-reference can be used. Nebulous and mind-bending is a concept, though it may be. Self-reference has always been around. Creative ways to reference the self, it seems, will always be around.
00:44:18
Speaker
limits of mathematics, ways of dealing with information, and the structure of mathematics itself have been proven to be dealt with often better with self-reference. Understanding consciousness itself may even be only tractable using the magic of self-reference. Self-reference is here to stay. In the future, ask yourself what asking yourself things means. Oh, by the way, is this conclusion self-referential? Nobody knows.
00:44:44
Speaker
I'm Sofia, and this has been Breaking Math. With me and Gabriel, who's not on anymore, we had Millie, who is part of Nerd Forensics. Millie? And you can find me anytime at Twitter, at Camp Pod Millie, and that's camp with a K. And otherwise, you can find me at nerdforensics at gmail.com. And as always, you can find me on Facebook at Facebook.com slash Nerd Forensics. And really quick, do you want to just plug your show what it is?
00:45:13
Speaker
So nerd forensics, we are, uh, an amorphous omnibus series that just covers everything in pop culture. Uh, we're going to have episodes ranging from conspiracy theories to cryptids to why Nickelodeon in the nineties can't be reproduced.
00:45:32
Speaker
And what was what would you say is a couple of your favorite episodes you've done that I've helped you with? Well, I produce a show, but favorite episodes that I've done. The anime one that I recorded with Jacob was really great. Oh, I really liked that one. Otherwise, I really, really, really, really enjoyed the one we did about about magicians.
00:45:54
Speaker
Oh, that one was great too. Yeah, so if you like a pop culture at all or just like cool stories, honestly, this podcast, it's really cool podcast. Check it out, Nerd Forensics, you can find it wherever you find podcasts. And just a little warning, it is an explicit podcast. There's some bleeping, but it definitely contains much more risque content than this show, right?
00:46:17
Speaker
Yes, yes it does. I have much less of a filter, but I don't consider myself an educator. Think of me as the guy you don't want to sit next to on the bus.