#headings-verbose #images-false #genres-shell #genres-heavenly-script #cites-logicomix ## (1,4): The Second Law 1: Hey there's another one! I wonder what i一 0: Don't bother reading it. It's just more corruption. ### Or: Byte Marks 1: I don't mind corruption! I think this one一 0: Don't get too close, these k中ds of b愛tes c◇n lea5e ㅁ二Ξ. 1: What the hell was that? ### Or: Information 101 0: I said these kinds of bytes can leave marks. 1: I think you're just trying to scar--- ### Or: Entropy l . o . l 0: Ok yes maybe, but 10 (手|首)LDn't 热ad 2 mʌch in10 it. 1: Whʌת?! ### Or: The 2nd L . ɔ . Λ 0: I said "Tu shouL⅂n't re+ 二 much in2 it." 1: 一 m||e 時me! ### Or: The Second Law : V 0: What? 1: I said "One more time?" ### Or: Words ∨ Things: 5 0: And I said "You shouldn't read too much into it." 1: Why are you talking like th@? ### Or: דברים 0: 尔R二. 1: Again? ### Or: Dudetheyreontome 0: You are too. See th@ up there? Look at the conversation hist| |一. 1: Oh wow, I didn't even mean t二.. #### Or: Questionable Derivations 0: Don'טוֹבer穴lyze (ɪt)²'s just kəˈɹʌpʃən. ɪt doesn't mean an一θɪŋ && even if it (1 / 一 / i)ce mɛnt sΣϻmtθhiŋ, {as a旭hi}storical matter, it \[ΔㄷDd\]\[əʌ\]\[zζ\]n't me穴ny{thiŋ, m||e} && even ɪf it dɪd, ɪt's c二t愛n理 $\lnot$ apρop𐤓{р𓉐(r}日i)[^1]a@te 1111. [^1]: A possible reference to _The House of the Rising Sun_ (The Animals, 1964), derived in part from the Appalachian Rising Sun Blues lineage, by means of the older Anglo-American fallen-woman laments set in and around houses of corruption and sin, which themselves derive from the genre of early modern penitential songs thematically surrounding brothels and confession and the taking of moral inventory, cf. the medieval alba and wayward-woman laments with the same structure noted supra, possibly derived from the late antique Syriac and Greek dawn-laments in which houses of sin are exposed by first light, which in turn borrow their themes from Second Temple exile poetry in which sunrise is treated as the hour of judgment for the transgressions of the previous night, which genre shows more than a few parallels to Neo-Babylonian penitential prayers describing houses of wrongdoing & women laid bare & bare respectively, a theme of regret arising with the sun god's rising circa dawn, cf. the corpus of Old Babylonian dawn-laments concerning corrupted dwellings revealed with ones sins at daybreak, which as far as I'm concerned were more or less copy-pasted from earlier Sumerian lamentation liturgies about the morning walk of shame after having banged a two, hence the comment above where this content was flagged as not appropriate for ones by our friend zero. 1: WHAT? 0: Don't overanalyze it. It's just corruption. It doesn't mean anything. And even if it once meant something, as a historical matter, it doesn't mean anything anymore. And even if it did, it's certainly not appropriate for ones. 1: What do you mean one一 0: I mean ones as in ~~you~~ / ~~你~~ / ~~ni/に/二~~ / ~~tu~~ = 二. 1: W👒? 0: Come again? 1: Hey there's another one! I wonder what i一 0: Don't bother reading it. It's just more corruption. 1: I don't mind corruption! I think this one一 0: _(Tired of this)_ Just jump past it. Follow me. goto: [[#The Beginning]] --- ## Genesis Two ### Or: Gen 2 ### Or: 根² ### Or: Root 二 ### Or: √er ### Or: Sqrt er #### Or: Questionable Derivations --- > _Poets do not go mad._ > _But Chess players do._ > _Mathematicians go mad._ > _And cashiers._ > _But creative artists, very seldom._ > _I am not attacking logic._ > _I only say that this danger does lie in logic, not in imagination._ > -GK Chesterton --- ![[logicomix-26.png]] --- ## The Beginning 0: Ok so, back to logic. 1: Where were we? 0: The beginning. 1: We're still at the beginning? 0: Yes. 0: In the beginning, everyone suddenly realized they might be wrong. About everything. 1: _(Slowly recovering)_ Wait, you didn't explain why they're worried yet. 0: Well Cantor had introduced what some considered to be meaningless nonsense. 1: Some? 0: Of course. Whether or not a thing is nonsense depends on the receiver just as much as it does on the sender. 1: Even in mathematics? 0: In all communication. That's basic information theory. Anyways some thought what Cantor'd done was nonsense. Others worried his methods would corrupt the youth. Others just thought he was hopping mad. ![[logicomix-27.png]] 1: Seems silly to me. 0: But it didn't to them. 1: Comm. again? 0: The same. Either way, mathematics was now confronted with some pretty questionable derivations inside set theory, and the fall out after said methods fell out of said theory into the rest of mathematics, at which point the mathematicians were very worried that their field might be corrupted. 1: What fallout, worried why, and corrupted how, respectively? 0: Well we brushed up against it already. Remember that quote that went like this? > In 1911... Cantor attended \[a thing\], hoping to meet \[some guy\], whose newly published \[book that everyone cites but no one actually reads\] repeatedly cited Cantor's work, but the encounter did not come about. > > -The Dynamic Read-Writable Free Encyclopedic Repository of the Modern State of Human Knowledge ![[logicomix-36.png]] 1: Yeah you said to remember his name but I forgot. Who's \[some guy\]? ![[logicomix-37.png]] 0: He's the guy who segfaulted mathematics. ![[logicomix-38.png]] 1: Segfaulted mathematics?! ![[logicomix-39.png]] 0: Ok well not quite. More like he got root on the universe. ![[logicomix-40.png]] 1: _Got ROOT on the UNIVERSE?!_ ![[logicomix-41.png]] 0: Ok that's not quite right either. His name was Bertrand Russell. ![[logicomix-42.png]] 1: I think I've heard of him before. What did he do? 0: Well using Cantor's style of set theory, he discovered something called "Russell's Paradox." ![[logicomix-43.png]] 1: Ok I've definitely heard of Russell's Paradox before. Remind me of what it is? 0: No. ![[logicomix-28.png]] 1: What do you mean "No"? 0: Russell's paradox is one of those things like Gödel and incompleteness. Every damn boring uncreative book under the sun & moon feels the need to talk about these things for some reason and it's tiresome and predictable and makes all the textbooks worse and just generally contributes to the feeling that even the best books in one's favorite subjects don't do the stuff justice which if you think about it long enough kinda makes one want to f\*\*king die. ![[logicomix-29.png]] 1: Aren't you sort of... contributing to that? 0: Yes. Which is why I won't tell you what Russell's Paradox is. Go look it up. We're not in that kind of book. The set of all books about foundations divides into two subsets: (1) the set that talks too much about Russell's paradox, and (2) the set that isn't about foundations in the first place. We're in the latter camp. 1: I still want to know what it is. 0: Don't get distracted. We're already past that. 1: Can't you just tell me? ![[logicomix-30.png]] 0: Yes! And now if you don't mind, the story continues. 1: _(Looking suspiciously at zero)_ 0: The point is that this guy named Bert (who George wanted to meet but didn't) found a security vulnerability in standard mathematical reasoning. 1: What was the vulnerability? 0: It boils down to the idea that if you can define a thing then it exists. 1: That's insane. Why would anyone believe that if they can define something that means it exists? 0: It's standard mathematical practice. 1: WHAT?! 0: They call it "Comprehension." 1: The more you talk about mathematics the more suspicious I am of it. 0: Well good, but my point here isn't to criticize the field. It's dishonorable to kick a field while it's down. And if there was ever a time when mathematics was "down," it's when Russell found his paradox and mathematicians started worrying they might have to abandon comprehension. Comprehension is their bread and butter. 1: What's comprehension? 0: It's the logos principle. In the beginning was the word. "Let there be." 1: Let there be what? 0: "Let M be a manifold." "Let S be the set of all sets that don't contain themselves." Every mathematician believes that words are what creates things. At least mathematical things. In fact, the idea that for something to exist you have to "construct" it is widely regarded with suspicion in standard mathematics. There's a whole group called the "constructivists" that's entirely defined by the fact that they're weirdos who believe \[that\]. 1: Ok you always give these analogies but I'm a developer remember? I'm not afraid of the technical explanation. Give me more details. Be technical. I don't actually believe mathematicians were running around saying "If you can define it, then it exists." There's no way. That's not how they talk. 0: You're right. They don't tend to use those words. But that's what comprehension is. Usually when it happens, it sounds more like this: _Let S be a set such that:_ or _Consider the following set:_ $S = \{ \; x \; | \; p(x) \; \}$ where $p(x)$ is some propositional function of $x$. 1: What's a propositional function? 0: A sentence with a free variable. Or if you prefer, a function that takes whatever type of thing $x$ is as input, and spits out a sentence as output. 1: Like an English sentence? 0: No, a "proposition." But a proposition is just a fancy word for a sentence in a formal language. Though we're not doing formal set theory here, so it's basically just a sentence in "natural language with math symbols." That's sort of the language that got mathematics in trouble with the paradox. 1: Can you give me an example? Like what does this comprehension thing look like in practice. I want to see if I can tell what the problem is. 0: Sure, here's a comprehension. I could say: _Let S be the set:_ $S = \{ \; x \; | \; x \notin \; x \}$ 1: What does that mean? 0: In math speak, $a \in b$ means $a$ is a member of $b$. So whenever you see the $\in$ symbol, the thing on the right is a set, and the thing on the left may or may not be a set too, but it usually is. 1: Which part of this is "comprehension"? 0: The whole thing. Comprehension is the idea that we're allowed to just say stuff like this in the first place. The idea that because we can describe $S$, we're allowed to act as if it exists. That's what got mathematics into trouble with Russell's paradox. 1: How is this any different from programming? We're allowed to define whatever we want in programming too right? 0: Absolutely not! This couldn't be more different from what we do in programming. In programming, we have to earn our variables. We earn every one through our blood, sweat, and tears. If we want to have a variable that represents a set of things with some property, we actually have to _create_ each of those things! Or use someone else's code that creates them for us. Plus we have to find room to store all the things. Plus room to store the set. And if we want our sets to be unordered like they are in mathematics, we have to implement _that_ too, because most ways of storing things in one-dimension (whether books in a library, sentences in a book, or objects in a computer's memory) come with a built-in ordering that we may or may not want. It's not as straightforward as it is in math, we can't just speak things into existence with words like some kind of biblical god. 1: Ok I hear you, but how is the comprehension above any different from this python code: ```python >>> s = [list for list in lists if list not in list] ``` 0: Well before you can run that code, you have to create the "lists" list. 1: Yeah good point. Plus I guess the condition is trivial. Or impossible. You know what I mean. 0: I most certainly don't. 1: I mean obviously a list can't contain itself. That's like, infinite recursion or whatever. 0: Nonsense. 1: What? 0: Lord help me, we have so much work to do. Once we're done with history I'm teaching you programming. 1: You know I'm a professional devel--- 0: Well then we'll have to find some way of making you a bit less professional! 1: Can you just explain instead of insulting me? 0: ```python >>> list = [0, 1] >>> list.append(list) >>> list [0, 1, [...]] >>> list in list True ``` 1: Weird... Ok I guess a list can contain itself in python. How do they implem--- 0: Pointers. 1: Obviously. Wow I feel dumb now. 0: Anyways it turns out in mathematics that causes a LOT of problems. Like a historically relevant amount of problems. Those few lines of code were what Russell's Paradox was about. Data structures that contain themselves. In programming it's no big deal. But it sort of broke mathematics. 1: How did it break mathematics? 0: Led to a contradiction. Mathematics can't handle contradictions. Turns out that's another thing we can handle much better than they can. In mathematics, one contradiction anywhere in the system blows everything up. It's a side-effect of some legacy code called The Principle of Explosion that they still haven't fixed in their underlying logic. 1: How can--- 0: Now let's get back to Russell, we've got a story to tell. 1: Ok where were we? 0: Last we left off, we were somewhere around the beginning. 1: Are we actually there yet? 0: Not quite. 1: So we're before the beginning? Are we going backwards? 0: Maybe. But we'll get where we're going eventually. Either way, in the beginning, everyone suddenly realized they might be wrong. 1: About everything? 0: About everything. Partly due to the set theory Cantor developed, Russell found a critical bug in the supposedly "stable" branch of acceptable mathematical reasoning. 1: Now you're speaking my language. 0: And every bit of math in the world was running that code. So it was a serious problem for mathematics. 1: How did they fix the bug? 0: Well it wasn't a simple fix. Comprehension was in everything. Can't just rip that code out or all the textbooks would stop working. 1: Naturally. 0: The corruption had spread. Comprehension was in everything. And the most high members of mathematics saw the corruption, and decided it was time for a fresh start. Mathematics was no longer the firm foundation they once thought. Mathematics -- at least the old mathematics -- was sinking. 1: Sinking? 0: Sinking. Mathematics was sinking. Right down to the foundations. The fish rots from the head and all that. Something had to be done, and quick. 1: _(A bit sick)_ What's the problem? These infinite numbers don't seem like that big of a deal. Like how is this a threat to calculus or geometry exactly? 0: Because of what happened next. 1: What happened next? 0: Well soon after Cantor rocked the boat, someone else tipped it over completely. 1: Completely? 0: Completely. Mathematics always seemed like solid ground. Somewhere safe. Soon after Cantor, it seemed like everything we might be washed away in a great deluge of paradoxes. 1: When was this? 0: The beginning. 1: The beginning of what? 0: Of the story of our People. 1: WE HAVEN'T EVEN GOT TO THE BEGINNING YET?! 0: Not quite. Head over to this file with me. 1: Can't we just stay here? I'm feeling a bit sea sick from all the byte marks and context switchi--- 0: Stop whining and get in. Things are about to get good. A new beginning, one. This story's for us, by us. For two, by two. Wouldn't be the same to just tell it to myself. Quit complaining and follow me. 1: _(Evincing nau{tical,sea})[^2]_ I need to see a doctor... [^2]: See: sick. 0: Exactly. 1: _(Nauseous)_ What? 0: I said goto: [[lost+found/1/5|Doctor]].