Read_846 - Three Strangest Paradoxes of Mathematics

[00:00:00] Speaker A: Sometimes your gut feelings lead you astray, particularly in mathematics, in which one constantly comes across results that seem impossible. For example, infinity does not always equal infinity, and tortoises may outpace human athletes, at least from a certain mathematical point of view. There are also many scenarios that appear contradictory at first glance, or second or third. These paradoxes can be explained, however. They are not errors, but rather reminders that we should not rely too heavily on our intuition in mathematics. Here are three of the strangest paradoxes in the field, the best in bitcoin made audible. I am Guy Swan, and this is bitcoin audible. What is up, guys? [00:01:10] Speaker B: Welcome back to bitcoin Audible. [00:01:13] Speaker A: I am Guy Swan, the guy who has read more about bitcoin than anybody else you know. [00:01:18] Speaker B: And a shout out to everyone who supports this show, who listens on fountain, who boosts and does streaming sats and value for value, and zaps me on noster. Thank you all so very much. [00:01:29] Speaker A: We are getting right into it today. [00:01:31] Speaker B: We're starting off the week, actually, with a really fun read, and it's a little bit different. [00:01:36] Speaker A: This isn't specifically about bitcoin. [00:01:39] Speaker B: I found a couple of the logical, the kind of contradictions and, like, working through how intuition can lead us astray and where and why, like understanding the. [00:01:52] Speaker A: Nuance in the complexity of an idea or in a paradox. [00:01:56] Speaker B: And I also found that basically every single one of these concepts, these mathematical paradoxes, actually could apply very well to bitcoin. They're essentially really good lessons and ways of breaking down the granularity of an idea to understand where and why things. [00:02:19] Speaker A: Fall apart that seems like that seem like they would otherwise work. [00:02:23] Speaker B: And so I just felt like this would be a fun way to kick off the week and also just something really fun to cover, because these paradoxes. [00:02:30] Speaker A: In and of themselves are just really. [00:02:32] Speaker B: Interesting to talk about, but then how they actually kind of apply to things that are actually relevant to us, and concepts and principles of economics and of. [00:02:44] Speaker A: Bitcoin and of cryptography. It just makes these things both relevant. [00:02:48] Speaker B: And fun to dig into. So with that, we're just going to. [00:02:52] Speaker A: Go ahead and dive right into today's article. [00:02:54] Speaker B: And this is actually from Scientific American. I don't really have permission to read this, but don't tell anybody. And it was also originally posted in something called, it looks like a german spectrum. [00:03:08] Speaker A: S p e k t r u. [00:03:10] Speaker B: M. Spectrum, der Weissenschaft. So this was some other publication, is where it actually originated, and I'm reading it from Scientific American. And if nobody tells them, uh, we'll probably, it'll probably be fine. So sh ch. Nope, nope, don't say anything. But just because this article is cool, you should go over there, get them some views. Uh, check them out right there in the show notes. The link will be available. And the author, Manon Bischoff, I do not know at all, but uh, he's a theoretical physicist and has some other. [00:03:45] Speaker A: Pretty interesting looking articles. So I will actually link to his work specifically in scientific. [00:03:51] Speaker B: So you can kind of do a little rabbit hole. There's a how a secret society discovered irrational numbers. That's actually one that I've got saved. I haven't read it yet, but I want to give a shout out to the author so you can check him out if you enjoyed this one. But with that, let's go ahead and get into today's article and it's titled. [00:04:11] Speaker A: Three of the strangest paradoxes in Mathematics by Manon Bischoff. A barber shaves all men who don't shave themselves. Does he shave himself? Mathematics offers explanations for this and other curious contradictions. Sometimes your gut feelings lead you astray, particularly in mathematics, in which one constantly comes across results that seem impossible. For example, infinity does not always equal infinity, and tortoises may outpace human athletes, at least from a certain mathematical point of view. There are also many scenarios that appear contradictory at first glance, or second or third. These paradoxes can be explained, however, they are not errors, but rather reminders that we should not rely too heavily on our intuition in mathematics. Here are three of the strangest paradoxes in the field. Hilbert's hotel imagine you are traveling to a city and have forgotten to book a room beforehand. Fortunately, you come across a beautiful hotel named after the famous mathematician David Hilbert, whose work you greatly appreciate. You step up to reception and see that the hotel has an infinite number of rooms. The room numbers correspond to the natural numbers 1234, etcetera. Without ever coming to an end. The receptionist tells you that the hotel is fully booked. However. But you know your way around math, so you dont let yourself be fobbed off so easily. You know a trick that will allow you and all other endless guests to find a room too. You suggest to the receptionist that each guest move to the room numbered one higher than their current lodging. So the person from room one goes to room two, the person from room two to room three, and so on. Because Hilbert's hotel has an unlimited number of rooms available, even when fully booked, there is still room for more guests. And that's not just the case for one person, they can have brought a whole busload of people who also wanted a room. In this case, the hotel guests would have to move not just one, but several room numbers away. It gets even stranger. Even if you bring an infinite number of people to Hilbert's hotel, you can still accommodate them in the fully booked hotel. To do this, the guest in room one would have to move to room two, the guest in room two to room four, the guest in room three to room six, and so on. As each person moves into a room with a number that is twice their current room number, an infinite number of odd numbered rooms become available. German mathematician David Hilbert presented this supposed paradox during a 1925 lecture on Infinity. The example illustrates how not all concepts can be transferred from finite to infinite cases. The statements every room is occupied and the hotel cannot take any more. Guests are synonymous in the real world, but not in a world with infinitesimal the birthday paradox. The next paradox is more familiar to many. When I was at school, it was not uncommon for several of my classmates to have their birthday on the same day. In fact, I also shared a birthday with a classmate. At first, this seems like a huge coincidence. After all, a year has 365 days, or 366 in leap years. But we'll ignore that for the sake of simplicity. And a school class consists of around 20 to 30 students. Our gut feeling therefore tells us that it is unlikely that two children were born on the same day. But that's not true. In fact, the probability that two people in a group of 23 have a birthday on the same day is more than 50%. To better understand this, it helps to look not at the number of people, but at the number of pairs of people. From 23 people, 23 times 22 divided by two equals 253 pairs can be formed, and this number exceeds half of all of the days in a year. If we look at the probability that one of the pupils in a school class of 23 was born on a particular date, however, the probability is only one -365 minus one divided by 365 to the 23rd power, which equals 6.1%. The birthday paradox therefore, arises from the fact that looking at pairs of students gives you a greater number of possibilities than if you only look at the individuals. This fact has tangible effects in cryptography. For example, if you want to sign a digital contract, for example, hash functions are used. The document is converted into a character string, a hash of a fixed length when it is signed. If even the smallest change is made to the original document, the hash that is formed from it is completely different. By keeping their hash, the signatory can prove what they originally signed, making the process tamper proof. There is an extremely low probability that two completely different documents will generate one and the same hash, however, and that poses a security risk. As a rule, the length of the hash function is chosen that such collisions, where two different data records produce the same hash, are extremely rare. A hacker can carry out a birthday attack. However, they can generate many different documents and compare their hash functions in pairs. Just as a teacher compares the birthdays of classmates, instead of focusing on a specific date and a single student. In practice, a birthday attack could look like this. I first create two contracts, version one and version two. Version one is a fair contract, but version two has wording that is in my favorite. Then I change both contracts in various places. I add spaces, tabs, and line breaks to create variations of version one and version two. These changes are virtually invisible to a reader, but they drastically change the hash function of the documents. If I compare the individual hash functions of the modified contracts, version one and version two in pairs, I will find a matching hash much more quickly than if I specifically try to reproduce a particular hash, such as that of version one. If I can find a matching pair of v one and v two, I can give you the contract v one to sign, but claim afterward that you signed v two. Because both generate the same hash, the fraud cannot be detected by digital signature software. Russell's antinomy british philosopher Bertrand Russell formulated a paradox in 1901, sometimes called Russell's antinomy, a term for a statement that describes two seemingly contradictory ideas. Unlike Hilbert's hotel and the birthday paradox, Russell's antinomy is not a result that merely eludes our intuition. It contradicts the rules of logic per se. The antinomy produces statements that can neither be false nor true. There are several examples that can illustrate Russell's antinomy, but the famous case is the barber paradox. Suppose a barber shaves all the men in town who do not shave themselves, and only those men. Does the barber shave himself? If he shaves himself, then he no longer belongs to the group of people who do not shave themselves. But if he does not shave himself, then by definition he would have to shave himself, because all residents who do not shave themselves go to him. This problem arises because of poorly defined sets. At the time that Russell presented his antinomy, a set generally referred to as a collection of things, the natural numbers, for example, form a set, as does the set of all inhabitants that do not shave themselves. This also allows sets to contain themselves or refer to themselves as a whole. And these properties lead to contradictions. This antinomy, therefore led to the end of what mathematicians call naive set theory. The foundation of mathematics continues to rely on set theory. But sets in this construct are no longer mere collections and instead must fulfill certain conditions. For example, sets must be composed of already existing sets and must not refer to themselves. This rules out antinomies such as the barber paradox. [00:13:37] Speaker B: To put this in mathematical notation, people. [00:13:40] Speaker A: In town who can grow a beard and are men form a set m. That set includes the men who shave themselves and those who do nothing. Next, the set capital c includes all the barbers customers. To form c, you have to follow the rules of modern set theory. If the barber is a man with a beard or part of m, then the set of customers cannot be defined as all male residents who do not shave themselves, because in this case, the definition would refer to itself, with both the barber and the customers as part of mh. Set theory simply does not permit such a definition. But if the barber is not part of m, for instance, if the barber is a woman or unable to grow a beard, then the definition is permitted. We can now breathe a sigh of relief. The paradoxes have been solved, and mathematics is not doomed to failure. There is no guarantee that the mathematical rules will not, at some point, produce an unresolvable contradiction. However, logician Kurt Godel proved this in the 1930s, and in doing so, made it clear that there is no certainty that mathematics will work forever in a self contained way. The best we can do is hope that an unsolvable contradiction never arises. [00:15:12] Speaker B: All right, that wraps up the article. [00:15:14] Speaker A: By Manon Bischoff, three of the strangest paradoxes in mathematics. [00:15:20] Speaker B: Now, there's a reason I wanted to. [00:15:23] Speaker A: Read this piece, and it's not even. [00:15:25] Speaker B: The more simplistic idea of politics and narrative and, you know, looking for biases and stuff that we all do, and it happens all the time, and we are all viciously guilty of it. It is the natural consequence of the limited access to information and the fact that what we, how we frame the world is always going to lead us to see the things that align with. [00:15:55] Speaker A: The framing of our world. [00:15:57] Speaker B: If something appears to contradict our worldview, it's inevitably going to look obnoxious. It's going to look broken or confusing in multiple different ways, because only from. [00:16:10] Speaker A: A completely different perspective or a completely. [00:16:12] Speaker B: Different foundation does it actually align with. [00:16:15] Speaker A: The things that we think. [00:16:16] Speaker B: So it's like, literally like puzzle pieces. And so much of what happens with our information as it is funneled through things and this also includes assumptions about. [00:16:28] Speaker A: Math, assumptions about the outcomes of some. [00:16:30] Speaker B: Sort of physical reality or some event or the action of some object or something. This is why visual illusions and kind of visual paradoxes are such a thing. Because you will. [00:16:44] Speaker A: You kind of have these puzzle pieces. [00:16:46] Speaker B: For how the world works. [00:16:48] Speaker A: And they are overly simplified. They are necessarily overly simplified because what we do is we abstract away simple principles from very complex and multifaceted reality. And so when you hear a story. [00:17:01] Speaker B: Or an action or a statement from somebody or, you know, there's this personification of this group or whatever it is, you're inevitably going to. If it, like, almost fits in your. [00:17:16] Speaker A: Puzzle piece, it's a whole lot easier. [00:17:18] Speaker B: To just kind of like carve out the pieces that don't matter or the pieces that seem to contradict things so. [00:17:25] Speaker A: That it fits in your worldview and. [00:17:26] Speaker B: That you have useful information. Because otherwise the information is just not useful. [00:17:30] Speaker A: Or it messes up the whole idea. [00:17:32] Speaker B: Of your puzzle altogether, which is a nightmare. Because now you have to throw all of your pieces away in the trash. [00:17:38] Speaker A: And you have to rebuild them from scratch. [00:17:40] Speaker B: While you've spent hours, days, months, years, decades of your life building up this. [00:17:47] Speaker A: Picture from all of these different puzzle pieces. [00:17:49] Speaker B: And, you know, the end. I hadn't really thought about this ahead of time. [00:17:52] Speaker A: But the analogy of a puzzle is. [00:17:54] Speaker B: Actually a really great example. Because, you know, how do you start a puzzle? [00:17:59] Speaker A: How does anybody do a puzzle? [00:18:01] Speaker B: What do you do? [00:18:01] Speaker A: First? You find the edge pieces, right? You outline it. [00:18:05] Speaker B: You find the limits. [00:18:09] Speaker A: You find the boundaries of this worldview, of this perspective. [00:18:14] Speaker B: And then you start filling in with. [00:18:16] Speaker A: Pieces that you have seen or heard. [00:18:19] Speaker B: Or news or other mathematical elements or physics or information or evidence, whatever it is, you just start filling in this thing. And when the edges are broken or. [00:18:34] Speaker A: When you realize that something is so far out of the picture that it. [00:18:38] Speaker B: Doesn'T even fit in your outline, you realize that either your puzzle is of. [00:18:45] Speaker A: No consequence or it's a waste of. [00:18:46] Speaker B: Time because it doesn't even account for all of this stuff that's actually happening in the world. Or you just realize that things that. [00:18:52] Speaker A: You thought were edge pieces were actually. [00:18:55] Speaker B: Much finer, just had much finer teeth on it. And you don't even know now where. [00:19:00] Speaker A: The edge actually exists. [00:19:01] Speaker B: And it just destroys your grounding for reality. [00:19:05] Speaker A: And it's a super scary place to be. [00:19:07] Speaker B: That's why, you know, it's called, it's regularly referenced, referenced in like religious text and even all good writing as like a rebirth, as a death and being reborn as enlightenment, as letting go of everything that has, you know, trapped you in this world. Like, that's the metaphorical idea of having your entire worldview shattered, having your entire puzzle just destroyed from the context of how all of it actually relates to each other and what actually fits together and what doesn't. And yet our entire lives, like just an unbelievable amount of investment and time and energy and emotion is built or. [00:19:52] Speaker A: Is spent in building this, this puzzle, this, this picture of the world. [00:19:57] Speaker B: And this is why people have such, such idiotic and aggressive dismissals of something like bitcoin, because bitcoin does not even come close to fitting in all of this. In fact, everything that they have seen. [00:20:13] Speaker A: That even relates to digital points or whatever, has always been a joke. And they cannot, it looks too much like that puzzle piece that they've seen a million times. [00:20:24] Speaker B: And if you do not understand any. [00:20:25] Speaker A: Of the nuance or any of the complexity in how it works, and specifically, you don't understand any of the nuance. [00:20:31] Speaker B: Or complexity of how the thing that. [00:20:33] Speaker A: Has already put a foundation in your puzzle piece, your actual money, that is. [00:20:37] Speaker B: A core part of the outline of your puzzle. [00:20:41] Speaker A: If you don't recognize that there's all sorts of little pieces of this thing. [00:20:45] Speaker B: That don't look at all like what you thought, then, of course, this thing is never going to fit in your picture. And then, then it becomes an instantly an impossibility. It becomes a stupid thing. And then people are just retarded. And look at this thing going up. This is just proof that everybody's insane. There's no hope for humanity, and we should just kill everything. Because that's, that's the ultimate endgame of. [00:21:06] Speaker A: Humanity, is to just die in poverty and destitution and foolishness. [00:21:11] Speaker B: And shockingly, this is actually a pretty prevalent idea. And I think it's because all of. [00:21:16] Speaker A: These puzzle pieces aren't fitting together. [00:21:18] Speaker B: It's because our frame of the world, from an establishment perspective, the frame of. [00:21:23] Speaker A: The world that has been the norm. [00:21:25] Speaker B: For decades and decades, is falling apart. [00:21:28] Speaker A: And the people who are changing their. [00:21:29] Speaker B: Framing are either getting locked into their change. That's another thing that happens, is that you, when you find some sort of an evidence or your trust breaks down. [00:21:39] Speaker A: In one piece of it, and then. [00:21:41] Speaker B: You have to make that shift in your puzzle. A lot of times, what happens as. [00:21:45] Speaker A: Well is you build it up too. [00:21:46] Speaker B: Quickly and too defensively, and then anything that contradicts that is actually rejected even. [00:21:54] Speaker A: Worse, because you've already come to this. [00:21:56] Speaker B: Point where you rejected this other so. [00:21:58] Speaker A: You feel you're more emotionally, even more emotionally attached to this thing that gave. [00:22:02] Speaker B: You this sense of losing it all. So if you don't then grasp onto the truth or a better framework, you can actually cause yourself even more problems from the context of being stubborn and being trapped in a way of thinking that is purely a system of contradiction. And I don't mean that, like, contradicting itself, but in the sense that all it does is a negative. It's just always hating the establishment or always hating what's normal, and it becomes more of a reflection of some other thing rather than something in and of itself. In other words, if I wanted to put it as simply as possible, the identity becomes a negative rather than a positive thing. It becomes a, I'm oppressed, and these are the oppressors. It becomes a. [00:22:52] Speaker A: These are the evil establishment cabal. [00:22:54] Speaker B: And I'm. I'm just not that. That is evil Nazi Trump, and I'm just not a Nazi. So clearly, I'm on this side. But then there's no definitely, there's no definition, there's no policy, there's no thinking. [00:23:09] Speaker A: Like, all of these things are not defined. [00:23:11] Speaker B: They're defined by what they are the opposite of, rather than anything of themselves. And then you easily just get thrown around. So, you know, the party that's anti Trump used to be, like, the left. [00:23:24] Speaker A: The liberals in this country used to be anti war. Now they're the most aggressively pro war people out there. [00:23:29] Speaker B: And it just. It's, like, mind boggling. [00:23:31] Speaker A: They have nothing to stand for because they are only standing against something. [00:23:36] Speaker B: Anyway, that's all just a really long rant to explain that our assumptions and our biases. And I say this because I've been guilty of this. I've been falling more into the right, political right way of thinking, and then constantly I get punched in the face or slapped by somebody that I'm just like, oh, my God, that's right. They're stupid, too. And, like, Cernovich was talking about how we should make weed illegal and actually we should make alcohol illegal and that this is important. Just, oh, my God. And there's a lot of just conservatism that is so illogical. Like, all I can think is, oh, you want cartels, you want big, violent cartels, because that's how you get big, violent cartels. But sure, whatever, man. [00:24:19] Speaker A: Just FYI, having something be not socially acceptable and having something be illegal are not even slightly the same thing. [00:24:29] Speaker B: Like, I totally, even. [00:24:30] Speaker A: I even get what he's saying from that standpoint but the making it illegal is not relevant, relevant to his point. [00:24:37] Speaker B: But anyway, I'm. You don't even know what I'm talking about in that context, so it doesn't matter. Let's get into the paradoxes and talk about how these things confuse our assumptions and end up being logical illusions. [00:24:50] Speaker A: In the same way that a visual. [00:24:52] Speaker B: Illusion or sleight of hand can trick us into believing something or making false, really bad decisions in response to something and obscuring a very simple reality or a truth that would actually have us able to respond in a positive or. [00:25:14] Speaker A: Constructive way in some way that would. [00:25:16] Speaker B: Actually lead to a better outcome. So, first is Hilbert's hotel. And this one really is just on. [00:25:23] Speaker A: The nature of infinity, and the fact. [00:25:26] Speaker B: That infinity is not a static thing, and that we as humans can only imagine a static thing. And it's even more confused by not only do we think of infinity as. [00:25:37] Speaker A: A singular thing that has some sort. [00:25:40] Speaker B: Of conclusion, even though it's literally just saying this thing doesn't end, there is no defined limit of it, but it also denotes a sense of it following it, making sense across time. Like even this whole actual. This is something that I find it interesting that he didn't point out, is the relationship of infinity and time is that you can't like everything that they denote here of, oh, well, now you can move one person to room one and two, and then room two to four, and you could just double all the room numbers, and everybody can just. [00:26:19] Speaker A: Scoot over, and then an infinity number. [00:26:21] Speaker B: Of people can come in and just take all of the remaining. [00:26:24] Speaker A: All the odd spaces that opened up. [00:26:26] Speaker B: And like, I. This is kind of true from a theoretical position. If you're trapping yourself in this, we can adjust around infinity, but infinity almost. [00:26:39] Speaker A: Necessitates that it doesn't have a sense of time. [00:26:41] Speaker B: So to, like, move things in order to fit something else doesn't really logically fit. They can't really be one in the same, in my opinion, because it suggests. [00:26:56] Speaker A: That we can squeeze something in by kind of extending the thing at the end. [00:27:01] Speaker B: But there is no extending. Like, infinity isn't even something like you want to just think of a number. [00:27:06] Speaker A: That'S counting up forever. [00:27:08] Speaker B: That's also not representative of infinity, because that's suggesting that if we give it another few more seconds, it will just be higher and bigger. [00:27:17] Speaker A: But that's not true either. [00:27:18] Speaker B: The concept of infinity is everywhere, in. [00:27:20] Speaker A: All directions, all at once. [00:27:22] Speaker B: So it includes time itself, or at least have to conceptually in order to meaningfully be infinity. [00:27:31] Speaker A: And so really, the paradox of Hilton's. [00:27:33] Speaker B: Hotel is this idea that we only think in single rooms, in static realities, and in statements that work in the real world, and that the statement every room is occupied and so the hotel cannot take any more guests. [00:27:53] Speaker A: These two things are not contradictory. [00:27:56] Speaker B: When you're looking at a hotel with. [00:27:58] Speaker A: A certain number of rooms, but in a world of infinity, when we're talking. [00:28:02] Speaker B: About, when we're referencing infinity, they make no sense. They are contradictory to the reality that they are in. So essentially, you're mixing two sets of rules. You're mixing the outcome, the conclusion of a set of rules, with a set of rules in which that conclusion doesn't actually work. And now, even though this isn't super relevant to Hilbert's hotel, this specifically got me when thinking about the whole idea of the puzzle and worldview and things that break out of people's worldviews is to understand the environment and what causes what. Like the problems that happen in the fiat world and then where those problems. [00:28:48] Speaker A: Arise or what those same conditions cause. [00:28:52] Speaker B: In the bitcoin world, when you have a different logical foundation, when your set. [00:28:57] Speaker A: Of rules, when your environment is totally. [00:29:00] Speaker B: Different, you cannot take those same conditions and then lay them out in this new thing and then say, oh, well, nothing's different. We haven't solved any problems because it's the very environment that these conditions interact. [00:29:13] Speaker A: In that has been altered, and we do not yet know the outcomes. All we can do is game theory about how these things will unfold. And the game theory suggests something very different from the fiat world. [00:29:25] Speaker B: So a great example is saying, oh, there's an evil cabal of banks who are controlling everything in fiat and infinitely rich. And then here they are over here, you know, running Coinbase or something. [00:29:38] Speaker A: Well, sure, there's always going to be. [00:29:40] Speaker B: Rich people vying for power and doing. [00:29:43] Speaker A: Whatever they can in order to do so. The question is, are the rules of the game the same now? And the whole point of bitcoin is that the rules have changed. [00:29:53] Speaker B: Another great example is when people say, oh, well, now there's just a whole bunch of people who bought into bitcoin and who are rich, and therefore nothing has changed because there's rich people and they're going to control everything. But it's a complete failure to understand. [00:30:06] Speaker A: What and how, why and how someone has achieved wealth. The problem with the fiat system is not that there are wealthy people. Wealthy people just means that someone is successful. It's like saying that the problem with football is that some people are really good at it. That's not just not relevant to the game of football. The problem is, would be if the highest paid football player is someone who literally is cheating, who can just throw out red cards against any of their components opponents and get them thrown out of the game and then just charge down the field without anybody in their way. The question is, do the rules apply the same? Because what they've done is they've taken a judgment of wealthy people that clearly have not earned it. Wealthy people who have clearly done so through collusion and corruption, who have lobbied their asses off to get political favor and contracts and subsidies, who have used malicious practices to sue their competition out of existence and to attack and steal. [00:31:06] Speaker B: Things from tons of young or unfunded. [00:31:12] Speaker A: Innovators that don't have any funding and no way to protect themselves. [00:31:16] Speaker B: And then they see that these people. [00:31:17] Speaker A: Are successful because the fiat system rewards. [00:31:20] Speaker B: That type of behavior. [00:31:22] Speaker A: It rewards that level of cheating, because the system is designed functionally a systemic process of cheating. And therefore all it knows how to do is support and defend cheaters. And so then someone builds this into their worldview and they say, look at all these wealthy people who are evil, corrupt cheaters. [00:31:41] Speaker B: And then they look at bitcoin, and. [00:31:43] Speaker A: Some people got wealthy and they say. [00:31:44] Speaker B: Look, corrupt cheaters are the ones who are still going to run everything if you do nothing. Account for the environment and the rules. [00:31:57] Speaker A: By which you draw conclusions. When that environment and those rules change, you will draw the same conclusions from. [00:32:07] Speaker B: Something that is not true. [00:32:10] Speaker A: It will seem like it aligns in all the ways that things you already know align. It will look like that same puzzle piece that fits into the pieces that you've already got there. [00:32:21] Speaker B: You can just swap it out. [00:32:23] Speaker A: Rich people in fiat, rich people in bitcoin, whatever it is. And you can project all of those same problems if you failed to account for why and how those problems arise, which is also not an easy thing to do. [00:32:37] Speaker B: I'm not even saying like, somebody should. [00:32:38] Speaker A: Feel guilty or stupid for doing it. [00:32:40] Speaker B: It just means that we have overly. [00:32:43] Speaker A: Simplistic ideas about how things work, which is also necessarily the case. The question is, what simple pieces of. [00:32:51] Speaker B: It do you understand and does it align with reality? [00:32:56] Speaker A: Are they the consequential pieces of the puzzle? [00:32:59] Speaker B: And that's a lot of things. One thing is we as humans, we. [00:33:02] Speaker A: Are extremely social beings. [00:33:05] Speaker B: And so we always over amplify and overly equate to outcomes and actions and long term consequences. And the systems that we live under to social aspects of humanity. We want to blame this evil person. [00:33:30] Speaker A: This evil company, which is just a big institution that we personified. [00:33:35] Speaker B: We want to blame groups, we want. [00:33:38] Speaker A: To blame decisions and politicians and people. [00:33:41] Speaker B: Who got in power, when, honestly, most of this stuff is the wind inside of the climate. It's. It's weather when far deeper and more. [00:33:53] Speaker A: Fundamental things are changing. And that is our technology. [00:33:57] Speaker B: That, I think, is what has really peeled back the layers of this system and drawn back the curtain to reveal the man behind the curtain who is. [00:34:09] Speaker A: Running all of the machinery. [00:34:11] Speaker B: But we get so lost in the. [00:34:12] Speaker A: Short term, and the short term is. [00:34:14] Speaker B: A lot more dictated by the social aspects and who gets elected at any one particular point. [00:34:22] Speaker A: But over the long term, it matters less and less and less. [00:34:26] Speaker B: Those things are mostly just waves and noise happening inside of shifting walls, of. [00:34:36] Speaker A: Shifting realities that are being altered by. [00:34:39] Speaker B: How our technology works, the economics of violence, and how information is able to spread and at what bandwidth. [00:34:48] Speaker A: Each type of information is able to. [00:34:51] Speaker B: Infiltrate each group, mind, network, or culture, etcetera. So, anyway, it's a really important framing, because something that you're certain is true can actually be not true at all. [00:35:05] Speaker A: If the underlying rules or the environment. [00:35:08] Speaker B: Have actually shifted and you don't recognize that that's what's being equated when they actually aren't equal. Now, the birthday paradox. [00:35:19] Speaker A: The birthday paradox is cool because it. [00:35:21] Speaker B: Has an explicit hash function relevance. And this is, as I understand it, how the. How sha one. How they proved Sha one has hash collisions. They would use a birthday attack. But I think this is even a problem with something like MD five, if I'm not mistaken. But it could be. I could be wrong. MD five is also pretty big. I don't know. I don't know of any case in which any of the other most dominant used hash functions are even, like, a. [00:35:54] Speaker A: Slight threat to this. [00:35:56] Speaker B: But it's also important to understand, like, let me get to the specificity of what causes the birthday paradox, because even though I think he did a good job of explaining it, it's. [00:36:05] Speaker A: It. [00:36:05] Speaker B: Sometimes it doesn't sink in. So, a good example is, let's take Bob, who was born on May 1, 2020. Now, the likelihood that somebody else has. [00:36:18] Speaker A: His exact birthday on May 1, 2020. [00:36:22] Speaker B: Is one in 365. And with that framing, we naturally want. [00:36:28] Speaker A: To think, oh, you've got a room. [00:36:29] Speaker B: Of 23 students or 30 students or something in a classroom room. Well, then, obviously, it's one in 365 that any one person shares a date with anybody else. [00:36:39] Speaker A: And that is true from an individual perspective, starting with an explicit birthday, a very specific date that you are then trying to match. However, if you don't care what the birthday is, and you just compare all, any set of two students, and you. [00:36:59] Speaker B: Say, okay, Bob was born on. Born on May 1, and Nancy was born on May 27, well, obviously, those aren't the same, so who cares? Well, what about May 27? Somebody else? Nope. [00:37:09] Speaker A: What about April 13, which is lisa's? [00:37:11] Speaker B: Nope. [00:37:12] Speaker A: But if you're willing to change the. [00:37:14] Speaker B: Birthday date with every person that you. [00:37:17] Speaker A: Compare to, well, then all you're looking for is a pair. You're looking for a match of any kind, which is a very different mathematical problem than looking for an explicit match to an explicit date that doesn't change. And this is actually at the heart of the problem with Dunbar's number. This is a problem of networking connections versus individual ideas. So the reason Dunbar's number is 150 to 200 people is because when we account as social creatures, social dynamics are necessarily interaction dynamics. They are relationships between people. And as they did in. [00:38:03] Speaker B: As they proved in their example, when. [00:38:06] Speaker A: You have 23 people in a classroom, there are 253 different pairs of people, 253 different groupings. It's exponential. Whereas when you have one person and. [00:38:21] Speaker B: You ask how many other people can. [00:38:23] Speaker A: They talk to or how many people. [00:38:25] Speaker B: They can connect with, it's 22. [00:38:28] Speaker A: It's linear, because there's only 23 people in the room. This is the exact same thing with the matching birthdays. Does anybody have a matching birthday with this one person? Or do any two people. Are there any two pairs, which. There are 253 different pairs possible with 23 people? Suddenly your chances are exponential versus linear. And in this exact same way, when. [00:38:52] Speaker B: You build a network or when you. [00:38:55] Speaker A: Are trying to understand relationships and complex. [00:38:58] Speaker B: Systems, like a social. A social group of people and interactions. [00:39:04] Speaker A: The problem of understanding that group is an exponential problem. And it's also important to understand. [00:39:11] Speaker B: This actually goes back to kind of a couple things we were just talking. [00:39:14] Speaker A: About with Hilbert's hotel, is to understand that systems, networks, and technologies run our. [00:39:20] Speaker B: Lives, not evil cabals and not rich. [00:39:24] Speaker A: People, because they have the exact same problem of having no idea how to understand massive, complex groups of people. [00:39:33] Speaker B: It's not a problem of some people. [00:39:37] Speaker A: Can control massive, complex social organisms and do whatever they want, and some people. [00:39:42] Speaker B: Can'T, because those others are in huge systems of power, and they control everything, and they run everybody, and they're. [00:39:50] Speaker A: They have infinite knowledge and infinite power. [00:39:53] Speaker B: And everybody else is just a slave and can't do anything and have. Has no effect whatsoever. [00:39:58] Speaker A: To the contrary. It's about people who have the resources to take advantage of a group, versus. [00:40:05] Speaker B: People who are simply subject to the. [00:40:08] Speaker A: Currents of the world, so to speak, because they either don't have the resources. [00:40:12] Speaker B: Or the knowledge, which is just resources as well. [00:40:15] Speaker A: You can be poor and still be. [00:40:17] Speaker B: Brilliant at navigating, and so many people. [00:40:20] Speaker A: Don'T have the resources to actually do anything or respond to anything, and they basically just shoved around by the waves and the current. [00:40:27] Speaker B: But I think it's very wrong to assume that there is somebody who can control or understand this thing. All they can do is basically extremely dumb and extremely simplistic things that will. [00:40:41] Speaker A: Never have the outcomes that they expect. [00:40:44] Speaker B: They can try to manipulate, they can. [00:40:47] Speaker A: Try to cause problems, and they can try to benefit or profit off of it as much as they can. [00:40:52] Speaker B: But ultimately, when establishment powers are just. [00:40:55] Speaker A: Drumming up as much hate and fear and chaos as they possibly can in creating division and just lying about stuff, they have absolutely no real concrete control over whether or not their heads end up on the spikes as well. They just hope they have a big. [00:41:12] Speaker B: Enough ship and enough resources to navigate. [00:41:15] Speaker A: Out of it after they create a storm. [00:41:17] Speaker B: So, just something to keep in mind. Social networks are massive, complicated things, and. [00:41:24] Speaker A: The outcomes and the conditions and the probabilities of things occurring or shifts happening. [00:41:31] Speaker B: In some form or fashion are not. [00:41:33] Speaker A: What they may intuitively seem. Just like it may seem like, oh, a group of 300 people should be very easy to manage, when in fact, it breaks down very, very quickly. And there is this bizarre number at. [00:41:46] Speaker B: 150 to 200, where it's just too many different pairs of people, there's too. [00:41:51] Speaker A: Many different relationships, too many different dynamics. [00:41:54] Speaker B: And the group itself, as a social. [00:41:58] Speaker A: Species, the group itself fails. [00:42:00] Speaker B: Without numerous reinforcing, scaling technologies, the group. [00:42:05] Speaker A: Begins to dissolve into division, into hate, and into fighting each other. [00:42:09] Speaker B: Now, going back to hash functions, just so you understand, the associated risk of a birthday attack, which is a common thing and is something that, you know. [00:42:23] Speaker A: You could be, quote unquote, concerned about. [00:42:25] Speaker B: But it's also important to understand, but. [00:42:28] Speaker A: How exactly this occurs. [00:42:30] Speaker B: So remember, the example they gave was version one and version two of a contract. And they try to make two completely different versions of this contract, one with wording preferable to them, and one with wording preferable to you, and they iterate. They make announce like bitcoin mining does, and they just endlessly rehash over and over and over and over and over and over and over again until they find a pair of version one. Remember, they're not trying to match the first hash, they're not trying to match the document of the original document. [00:43:03] Speaker A: They are changing the hash of the. [00:43:04] Speaker B: Original document and the hash of the. [00:43:07] Speaker A: Cheat document, so to speak, at the same time, and seeing if anything can come up. And the more and more they create. [00:43:14] Speaker B: You know, if they create a million hashes and of version one and a million hash hashes of version two, that's. [00:43:20] Speaker A: Not a million different possibilities. [00:43:22] Speaker B: That's half a trillion different possible pairs. So in other words, they're not trying to match the millionth hash of version one and the millionth hash of version two. The millionth hash of version two can. [00:43:34] Speaker A: Match with the third hash of version one. Doesn't matter which one. [00:43:38] Speaker B: They just need any two to match up. Now, just from a very naive perspective, and just to understand, this also means that this is not a problem in bitcoin. You can't. Well, I mean, it's not a problem in the cases that you're probably thinking it might be a problem in you hash your own transaction, you sign your own transactions, then you present them to the network. So if anybody's trying to make a collision, to spend your transaction or ut your UTxO, they need a flat, linear sha 256 collision. Not only is a birthday attack on SHA 256 not going to happen, a just straight up collision, a one to one collision of some specific thing, also. [00:44:24] Speaker A: Not going to happen. Definitely not going to happen, actually. [00:44:29] Speaker B: So even though it's a really cool mathematical thought experiment, it's not something that you have to worry about. Now, the last one is one that I thought was really cool because it actually alludes to how you can create logical contradictions with a kind of a form of self reflection of recursiveness in bitcoin script or in any sort of. [00:44:54] Speaker A: Logical system, a system that has very explicit and clear rules, is that you have to be able to define your sets. [00:45:02] Speaker B: And I thought this was really, really interesting because it actually, like, it applies to a lot of different things. So now we're talking about the barber Russell's antinomy. [00:45:12] Speaker A: If a barber shaves all men who do not shave themselves, does the barber shave themselves? [00:45:18] Speaker B: And this is a stupid statement that essentially virtually cannot be true. It is false if they shave themselves. [00:45:26] Speaker A: And it is false if they don't shave themselves. [00:45:29] Speaker B: And why is that? How is that possible? For something to always be false is. [00:45:35] Speaker A: Because the condition itself includes the person changing the conditions. It is a recursive piece of logic. [00:45:45] Speaker B: In which the set of people that. [00:45:48] Speaker A: The barber establishes a rule about includes himself. [00:45:56] Speaker B: And in the absence of explaining some explicit cryptography or tool or structure or malicious transaction. This is a lot of what the logical problem that I have with recursiveness. [00:46:12] Speaker A: In any sort of bitcoin script. [00:46:14] Speaker B: And when I had. I had reardencode on in the previous chat episode, and we talked a little bit about this. And this is also why I have come down on CTV as being a very useful or one of the most valuable covenants, because I think it solves that simple risk or that potential problem. [00:46:36] Speaker A: Very easily by just making it a. [00:46:39] Speaker B: Hash, by simply locking the transaction to hashes of the potential outputs and the. [00:46:45] Speaker A: Various amounts that would be involved in those outputs. [00:46:48] Speaker B: And why do I say that solves this? Is because it cannot reference itself. As soon as it has a hash. [00:46:55] Speaker A: Inside of the hash. [00:46:56] Speaker B: As soon as you take that hash. [00:46:58] Speaker A: That you've gotten from that transaction and. [00:47:00] Speaker B: Then try to put it back inside that transaction or those details, well, then it's immediately invalid because you include a. [00:47:07] Speaker A: Hash, and now the other hash is broken. It's no longer the actual hash. [00:47:10] Speaker B: You literally have to find a hash. [00:47:12] Speaker A: Collision in order for that to work. [00:47:15] Speaker B: Where the hash is included in the hash of itself, which just isn't going to happen. But that's what worries me about any other sort of kind of recursiveness or introspection in transactions, because it presents the possibility, and I'm not saying that this is. We're going to have. [00:47:32] Speaker A: Oh, my God, we're going to have. [00:47:33] Speaker B: A barber paradox immediately. I'm just saying that allowing the rules. [00:47:39] Speaker A: To do that introduces that sort of. [00:47:42] Speaker B: Edge case as now being something that. [00:47:45] Speaker A: You have to worry about because your sets are not properly defined, your limits as to what can be referred to start to blur, those lines start to crisscross each other. And now, if something ever occurred and. [00:47:57] Speaker B: There was some sort of introspective like. [00:48:00] Speaker A: Oh, the rules of this transaction can. [00:48:02] Speaker B: Only be spent if this transaction does this. And if it does it, then the rules no longer apply, and it's invalid. And if it doesn't do it, then it's not in the set, and thus it's not valid, and it doesn't apply. If something like that could be constructed. [00:48:18] Speaker A: Well, it could also just be that you just lost your coins because you did something stupid. You. You built a contradiction into your thing. [00:48:25] Speaker B: And there's no limit. There's no, like, scarcity on the number of ways that we can just lock. [00:48:30] Speaker A: Up coins forever and lose them. But it's called a paradox because it's not exactly obvious, and it introduces a potential problem in which something like that could be exploited without even realizing that that's part of the code, which I'll. [00:48:43] Speaker B: Eat without realizing that some sort of a transaction that was done could have actually been exploited in that sense, and coins could have been locked up. That was in part, in some sort of a system or network that thought that it had its security thing worked. [00:48:59] Speaker A: Out and it didn't account for some really bizarre paradoxes on exactly what the set of things is included. [00:49:05] Speaker B: And now somebody can actually add itself into the set without anyone knowing with the script inside of it. This is all just me brain farting, right? I'm just kind of thinking at an. [00:49:18] Speaker A: Extremely high level about the potential of. [00:49:21] Speaker B: Fundamentally changing a primitive in the bitcoin system. [00:49:26] Speaker A: But I think it's a legitimate concern about what could be if you start to include that sort of thing. This is the type of paradox, this is the type of problem that could. [00:49:37] Speaker B: Arise when you do not have properly. [00:49:39] Speaker A: Defined sets or when you can include. When you can make a set of rules and then include the thing that defines the rules in the set of. [00:49:46] Speaker B: Thing that you're including, you created the rules for. And so in my lack of understanding and in my lack of dealing with. [00:49:53] Speaker A: Specifics, this is what bugs me about a lot of proposals that do that. [00:49:58] Speaker B: Sort of thing or potentially enable that introspection. And I think it's also just kind of the natural thing that people have been bothered about covenants is this idea. [00:50:08] Speaker A: That, oh, it could be burdened forever. [00:50:11] Speaker B: And that's why I specifically, after digging into how all of it works, I've increasingly liked CTV because the best way, and I've gone kind of deeper down this way of thinking about it, because I used an analogy while we were talking with Reardon, Reardon code, that it actually seemed really similar. The more I dig into it seems really similar to p two sh, where. [00:50:36] Speaker A: You are paying to a script hash. [00:50:38] Speaker B: And you don't know what that hash is. And so you can have an entire hash tree of, like, a bunch of different things that are possible, but then you only pay out to. You only reveal the. And this is another part of taproot as well. Pay to script hash really didn't enable all of this, but in combination of that and taproot and how signatures and scripts break out is that you basically can hide it behind a lot of. [00:51:02] Speaker A: You can hide what you're. [00:51:04] Speaker B: What you're locking all of the things to and the conditions and the different paths and outcomes and signatures involved to just a hash and CTV is actually extremely similar, except rather than creating conditions and a script and different signatures, you're creating separate transactions. You're creating places that those coins are. [00:51:26] Speaker A: Destined to go, so to speak. [00:51:28] Speaker B: So I think it actually makes more sense. [00:51:30] Speaker A: CTV makes it more confusing. [00:51:32] Speaker B: And I think if you wanted to have something that was relevant to understand its relationship is, I think you should. [00:51:37] Speaker A: Actually, a better name for it would actually be pay two transaction hash. So you would have p two sh. [00:51:43] Speaker B: Which is obviously p two sh, and then you would have p two th. [00:51:47] Speaker A: Which is essentially CTV. [00:51:50] Speaker B: And then I feel like it's a little bit easier to mentally assess where it lies in the risk spectrum because there's actually quite a few things that seem to apply very, very similarly to the two different designs or the two different concepts. So that's something I'll probably explore a little bit more and dig into. I'll probably get, like, a real deep, deep dive on p two sh and. [00:52:17] Speaker A: See how deep that analogy goes. [00:52:19] Speaker B: And maybe that's something we explore in a future episode. But anyway, I hope you enjoyed this episode, this read. I thought this was really interesting, and I couldn't help, but I read it once and then was like, I don't know if I should do this on the show. It's like, it feels super irrelevant, but it also seems useful. And so I kind of beat around the bush about it for a couple of weeks, and then finally today, I was like, you know what? [00:52:42] Speaker A: This is just a really fun one. Let's go ahead and dig into this. [00:52:44] Speaker B: One, and hopefully people get something useful and valuable out of it. And I think. I think there's some really interesting perspectives and lessons to be learned from a lot of paradoxes that exist, especially when we deal with a lot of nuances and really difficult comparisons and restrictions and rules that have to interact with each other in bitcoin. And it's so crucial to understand how they interact and that not everything can be. [00:53:14] Speaker A: Not everything is as it seems. [00:53:16] Speaker B: It's very easy to get it wrong. [00:53:17] Speaker A: It's very easy to simplify things. It's very easy for our worldview to confuse us or for the puzzle we. [00:53:23] Speaker B: Are building to make it seem like. [00:53:25] Speaker A: This piece fits right in where we. [00:53:27] Speaker B: Think it always should and where everything else in this category of things go, when, in fact, it's actually fundamentally and shockingly different. And the. The borders of our puzzle are not even in the right place. They exist on an entirely different field. And sometimes our whole worldview just collapses and we have to figure things out anew, and figure out new people to trust and realize that everything that we thought we could depend on was a lie or was just based on a foolish misunderstanding or false assumption. And it's not easy to deal with, but it's important to recognize that it happens to everybody. It's kind of the, that is the process of life. You know, it happens all the time when you're young. It's when you get old that you, you kind of expect that you've already dealt with that and that your worldview shouldn't change. But then it still ends up for everybody who's awake and for a world that is changing and very rapidly changing. And both the political, social, and also networking and technological dynamics are shifting so. [00:54:34] Speaker A: Viciously and so quickly. [00:54:37] Speaker B: The worldview will have to break down. It will just have to, because inevitably we have oversimplified something, and it is not going to account for some chaotic, random, unexpected event or change or new social dynamic or expansion of a network that just never fit. And we are going to have to start from scratch and build ourselves back up. So I hope the discussion kind of makes, makes that make a little bit more sense and applies to something useful. All right, we will close this out for today. Thank you guys so much for listening. Shout out to everybody who boosts on fountain. Shout out to coinkite and the cold card hardware wallet. Don't forget to use my discount code right there in the show notes and I will catch you on the next episode of bitcoin Audible. I am guy Swan. And until then, everybody take it easy. Guys. [00:55:47] Speaker A: Avoid negative people. They have a problem with every solution. Albert Einstein.