#cites-dyn-enc #footnotes-unicode #headings-simple #one-skeptical #images-false #math-latex #cites-logicomix ## (1,3): Names > _Mathematics is the most unclear of all sciences._ > -Carl Johannes Thomae¹ > _I'm gonna talk a little bit about that. How programming helps you understand things that are usually written very poorly. No matter how good the authors are. It turns out that the traditional methods of writing these things are terrible._ > -Gerald Sussman² > _Having read Vinogradov's "Fundamentals of Number Theory"... In my opinion, writing a mathematics book in a textbook format is heartless: give things like this to the armchair biologists. A mathematics book should be in the form of a novel, an epic, a tragedy, a drama._ > > -Someone in Kyrgyzstan, translated from Russian, shouting into the void on social media, under the name "Абай" and the handle "@josaefal", ca Aug 28, 2025 OC³, and who later (or possibly earlier) showed up in the DMs of one of The Authors of this book (the one you're reading now, not the number theory one) and said some very nice things, and who also (note to self) We should probably ask before including this quote in the book (this one) though (We/I)'m almost sure(-ish) that he (she?) wouldn't mind, probably. Almost probably. ¹ The Search for Mathematical Roots, section 4.5, page 198. ² Gerald Sussman: Programming for the Expression of Ideas ³ Old Calendar. --- 1: Where were we? 0: You said: > 1: This math stuff seems pretty imprecise. 0: Then I said: > 0: It is. At least sometimes. But not in this case. In this case, it makes sense. It's all about names. 1: Names? 0: Names. That's why the cardinality of $\omega \cdot 2$ is the same as the cardinality of $\omega$. And "cardinality" is just a fancy word for "size." Because Georg's definition of "same size" is "If you change only the names, do they have the same number?" 1: What do you mean "If you change only the names?" 0: Well no matter what we believe about mathematics, like whether we believe that "infinite stuff exists" or whether we think that's all nonsense, one thing everyone can agree on is names aren't magic. 1: Obviously. I don't know what you mean, but I think I agree. 0: All I meant by magic is that the following situation would be completely unacceptable: Suppose you have a set of stuff, like: $0, \; 1, \dots, \; \omega, \; \omega+1, \; \dots$ Then suppose you just change the squiggles we use to write down the numbers without changing anything else about the set. For example, suppose you put hats on everything: $\hat{0}, \; \hat{1}, \dots, \; \hat{\omega}, \; \hat{\omega}+\hat{1}, \; \dots$ 0: Is that the same set? 1: I don't know. Maybe not. I could imagine the hat-numbers might be different objects. 0: Exactly. Now, do the two sets have _the same size_? 1: I don't know. I'm not convinced they do. 0: Explain? 1: Like maybe the hats weigh something. Maybe $\hat{1}$ means "The number $1s home directory." And home directories can have all sorts of stuff inside them. So $\hat{1}$ might be bigger than $1$, and that would make it a different size. Or maybe the hats represent some code that takes a nonzero amount of time to execute, and maybe by "size" we mean something related to "execution time." Words can mean all sorts of stuff man I dunno. 0: Wonderful point. Mathematicians don't usually consider that sort of thing. But in their defense, that's not what we mean by "size" here. We're not talking about weight or physical volume or execution time or anything like that. In this situation, "size" just means the number of numbers. 1: The number of numbers? Like a meta-number? 0: No, the number of numbers in the two lists. The list without hats, and the list with hats. Do those two lists have the same number of numbers? 1: Oh, got it. I'd say they do. 0: What about now? $0, \; 1, \dots, \; \omega, \; \omega+1, \; \dots$ vs $\hat{0}, \; \hat{1}, \dots, \; \hat{\omega}, \; \hat{\omega}+\hat{1}, \; \dots$ vs $0, \; 2, \; \dots, \; 1, \; 3, \; \dots$ 1: No, the third list is different. 0: Different how? 1: You changed it. 0: What did I change? 1: You got rid of the infinite numbers. 0: Did I? 1: Stop being zen. Was your third list supposed to be an abbreviation for this: $0, \; 2, \; 4, \; 6, \; 8, \; \dots , \; 1, \; 3, \; 5, \; 7, \; 9, \; \dots$ 0: Exactly. 1: But that's just all the even numbers on the left, then a dot-dot-dot, then all the odd numbers. 0: True. But in a sense it's no different than the list with the infinite numbers. 1: How is that even remotely a sane thing to say? 0: Well, didn't you mean this idea in your list up above? $0, \; 2, \; 4, \; 6, \; 8, \; \dots (\mathrm{forever, \; but \; then \; after \; forever}),\; 1, \; 3, \; 5, \; 7, \; 9, \; \dots \; (\mathrm{forever})$ 1: Yeah. 0: So your so-called list of "finite numbers" involves the concept of "after forever." 1: Oh. Yeah I guess that's pretty infinite. But none of the numbers are infinite. 0: That's just the squiggles. Don't trust the squiggles. 1: Define squiggles? 0: The squiggles on paper. The shapes of the symbols. Don't get confused by the squiggles. Suppose I'm from a strange country where I use the symbol $2$ to mean one, and the symbol $1$ to mean infinity. 1: Sounds like a pretty nice country. 0: So you agree they're the same set? 1: No, but I see the trick you did, and I see that I'm gonna have trouble arguing that it's a different size just because the shape of the squiggles changed. I still feel like the first two have infinite numbers in them and the third one doesn't though. But I wouldn't exactly expect to win a debate about this. 0: Exactly. That was Cantor's whole idea. He defined the size of two sets to be the same if you can turn one into the other just by changing the names like we did. He defined it in terms of functions, but it's the same idea. If you can write down a function that turns each element in set A into exactly one element in set B, and if we can verify somehow that everything in B gets "hit" (i.e., paired with) some element of A, so the function puts the two sets into a state of total marriage -- total monogamy, no polygamy and no loneliness, or one-to-one and onto as the math people say -- when a function like that exists between the two sets, assuming we manage to find it and write it down, we can use it to force the reader into the same position you found yourself in just now. 1: Force them into what position? 0: They might agree this is a trick, and they might feel like one set "has infinite numbers in it" while the other one "doesn't," but no matter what they think or no matter what they think they "know" about our hidden agenda in all this, they're not gonna be able to argue their way out of the idea that the two sets are the same size. 1: Yeah ok. How is this relevant to us? I thought we were learning about the Foundational People. 0: Right, this is where it all started. 1: Where what all started? 0: The story of our people. See Cantor caused a total crisis. Mathematics as a whole wasn't sure what to do with Cantor or his infinite number of infinite numbers with different sizes. A lot of people thought it was nonsense. He felt totally humiliated. > Cantor suffered his first known bout of depression in May 1884. Criticism of his work weighed on his mind: every one of the fifty-two letters he wrote to Mittag-Leffler in 1884 mentioned Kronecker. A passage from one of these letters is revealing of the damage to Cantor's self-confidence: > > _... I don't know when I shall return to the continuation of my scientific work. At the moment I can do absolutely nothing with it, and limit myself to the most necessary duty of my lectures; how much happier I would be to be scientifically active, if only I had the necessary mental freshness._ > > -The Dynamic Read-Writable Free Encyclopedic Repository of the Modern State of Human Knowledge 0: He eventually sort of lost his mind. > After Cantor's 1884 hospitalization there is no record that he was in any sanatorium again until 1899. Soon after that second hospitalization, Cantor's youngest son Rudolph died suddenly on 16 December (Cantor was delivering a lecture on his views on Baconian theory and William Shakespeare), and this tragedy drained Cantor of much of his passion for mathematics. Cantor was again hospitalized in 1903. One year later, he was outraged and agitated by a paper presented by Julius König at the Third International Congress of Mathematicians. The paper attempted to prove that the basic tenets of transfinite set theory were false. Since the paper had been read in front of his daughters and colleagues, Cantor perceived himself as having been publicly humiliated. Although Ernst Zermelo demonstrated less than a day later that König's proof had failed, Cantor remained shaken, and momentarily questioning God. > > -The Dynamic Read-Writable Free Encyclopedic Repository of the Modern State of Human Knowledge 0: The legend goes that he lost his mind from coming face to face with the absolute infinite. As if the subject itself somehow scrambled his mind. 1: Seems unlikely. 0: He was probably bipolar. But that had been a part of his life all along. Toward the end of his life though, it looks like his "mental illness" was at least partly caused by him and his work being perceived as mentally ill. 1: _(Squinting at 0.)_ He went crazy because people thought he went crazy" 0: I'd say so. At least a bit. 1: What the hell kind of quantum psychology bullsh\*t is this zero? 0: Being called crazy can drive any one a bit crazy. 1: Oh I guess that's true. 0: Especially if its your life's work being called crazy. 1: Yeah I see. 0: Even more so if it's members of a community you love doing it. 1: Ok yeah I guess quantum psychology's kind of a thing. 0: It's not quantum. It's just psychology. Every one needs a community. > Cantor suffered from chronic depression for the rest of his life, for which he was excused from teaching on several occasions and repeatedly confined to various sanatoria. The events of 1904 preceded a series of hospitalizations at intervals of two or three years. He did not abandon mathematics completely, however, lecturing on the paradoxes of set theory (Burali-Forti paradox, Cantor's paradox, and Russell's paradox) to a meeting of the Deutsche Mathematiker-Vereinigung in 1903, and attending the International Congress of Mathematicians at Heidelberg in 1904. > > -The Dynamic Read-Writable Free Encyclopedic Repository of the Modern State of Human Knowledge 0: He wasn't totally rejected by any means. A lot of pretty famous mathematicians thought his work was great. Like this guy Bertrand Russell. Remember that name, we'll run into him again soon. > In 1911, Cantor was one of the distinguished foreign scholars invited to the 500th anniversary of the founding of the University of St. Andrews in Scotland. Cantor attended, hoping to meet Bertrand Russell, whose newly published Principia Mathematica repeatedly cited Cantor's work, but the encounter did not come about. The following year, St. Andrews awarded Cantor an honorary doctorate, but illness precluded his receiving the degree in person. > > -The Dynamic Read-Writable Free Encyclopedic Repository of the Modern State of Human Knowledge 0: Either way, things didn't end well for Cantor. > Cantor retired in 1913, and lived in poverty and suffered from malnourishment during World War I. The public celebration of his 70th birthday was canceled because of the war. In June 1917, he entered a sanatorium for the last time and continually wrote to his wife asking to be allowed to go home. Georg Cantor had a fatal heart attack on 6 January 1918, in the sanatorium where he had spent the last year of his life. > > -The Dynamic Read-Writable Free Encyclopedic Repository of the Modern State of Human Knowledge 1: Well that's a depressing story. 0: Definitely. That's why he's not one of the Foundational People, as far as I'm concerned. 1: Woah that's cruel. Why exclude him? 0: No no, I meant it as a sign of respect! As far as I'm concerned, Cantor was a proper mathematician. That's how he thought. That's where he belonged. That's the community he wanted to be accepted by. It's only right to include him in among the mathematicians in any proper history that brushes up against him in any way. It's what he would have wanted. Plus he wasn't really concerned with Foundations. He was wild. A proper Platonist, using his definitions and theorems to ascend to arbitrary heights in the eternal hierarchy of abstract non-constructive existence that ends precisely never, at the exact moment that one comes face to face with God, the absolute infinite, which Cantor and everyone after him called $\Omega$. 1: Sounds like theology. 0: Exactly. That's mathematics. And Cantor was a mathematician. It would be a disservice to consider him one of the Foundational People. But Cantor is where our people's history begins. At least the official part of it. Our people existed before that, but let's call the time before Cantor "pre-history." History has to start somewhere. After all, if it doesn't start somewhere, then by definition, it never starts. 1: What do you mean we existed before that? 0: Well there had always been people like us in the world, Foundational People that is. Leibniz, that guy who invented Calculus around the same time as Newton, he was clearly our type of person. And Charles Babbage and Ada Lovelace probably had this type of mind too. But the story that starts right around and after Cantor was the first time our people all came out of the woodwork and came together as a People. 1: Why then? 0: Because this was the first time the mathematicians realized how much they needed some real Foundations. For the first time, after Cantor, mathematics realized it was in trouble. Calculus had caused a bit of an uproar about Foundations too, but nothing like this. So when Cantor rocked the boat, soon after, mathematics realized it was sinking. 1: Sinking h--- 0: No time, follow me. goto: [[lost+found/1/4]]