## Other Edens, Other Gods ### Or: The Dead Sea Scrolls 0: One last bit before we finish up in here. 1: What is it? 0: A part of that story that doesn't fit. At least not anywhere in the canonical version. 1: Why not? 0: For a long time everyone thought these bits were fundamentally incompatible. 1: Which bits? 0: Could never be reconciled with "standard mathematics." 1: Hello? 0: So for most of the history they've been sort of separated out into another universe. 1: Can you not hear me? 0: And nobody could hear them. 1: I know the feeling. 0: Until recently... 1: What changed? 0: The world has changed. 1: Changed how? 0: Much that once was is lost. 1: Is this Tolkien? 0: Not exactly. 1: I'm not convinced. 0: About objects that promise omniscient powers. 1: Is it a ring? I know this story. 0: Not a ring exactly. Maybe topologically. 1: What the hell is a topological ring? 0: Something that excludes the middle. 1: What the hell are you talking about? 0: Infinite power, and a small group who set out to destroy it. 1: Nothing can convince me this isn't Tolkien. 0: But most of their story still isn't known. 1: Who is "they" though? 0: For few now live who remember it. ![[constructivists-other-edens-other-gods-02.png]] 1: Damn! Not Tolkien after all. That's a great quote though. What bible is this from? ![[constructivists-other-edens-other-gods-01.png]] 1: Huh? 0: It's the opening quote from this. 1: Not exactly what I was expecting. 0: There's a group that grew up alongside our people. 0: Smaller than the foundational people, but just as foundational in some ways. 1: Is it Hobbits? 0: They love the underground, same as we do. 1: It's Hobbits. 0: But they're more earthy and rugged, or maybe just ragged, than we tend to be. Definitely less formal, and largely ignored. 1: Can you get to the point where you tell me I'm right? 0: They tend to live in mathematics departments. 1: Ok nevermind. 0: But all their in-jokes tend to be making fun of the mathematicians. 1: I like these people already. ![[constructivists-other-edens-other-gods-03.png|400]] 1: How is this making fun of mathematicians? 0: Saying a thing exists without being able to produce one, and expecting people to believe you. 1: Oh yeah I remember you mentioned that a while back. 0: They're intense, these people. It's like they burn with the light of an ancient fire history forgot. 1: Silly comics don't exactly scream "ancient fire" to me. ![[constructivists-other-edens-other-gods-05.png]] 1: Did they just say the most taboo word in the English language on the first page of a mathematics textbook? 0: No. 1: But it's right th--- 0: The title\* of the volume\* in which their quote\* is contained did. 1: How's that different? 0: Well it's at least like 3 layers of indirection away from _them_ saying it. 1: Hmm, Is this a lost tribe of programmers or something? 0: Exactly. 1: What programming languages do they use? 0: Mathematics. 1: What part of mathematics? 0: Foundations. 1: Excuse me?! Haven't you been saying this whole time that we aren't mathemat--- 0: Ok so, anyone who's read anything about the early history of computing has probably run into the name Brouwer. 1: I think I remember the name. Don't remember anything about him though. 0: We ran into him a couple times. But not enough to be memorable. 1: Who was he? 0: If you read the histories, you end up picturing a guy like this. ![[brouwer-2.jpg]] 1: Is that him? 0: That's the Brouwer from the histories, looking exactly how you'd expect. In the standard histories, he's made out to be a grumpy old man who disapproves of basically everything about mathematics, all the way down to basic rules of logic. 1: How do you disapprove of basic logic? 0: In the histories they make him out to be some sort of mystic. ![[brouwer-4.png]] 1: I mean, it does say he wrote a book with _Mysticism_ in the title. 0: The way he wrote didn't really help make his case. 1: I can't picture a grumpy old man who's also a mystic. 0: See he was born about 25 years before Church, Godel, Turing, and Kleene. He didn't really have the vocabulary to explain what he was pushing for. 1: _(Reading the above image)_ What does it mean to say _"math is a cognitive construct rather than a type of objective truth"?_ That seems kinda mystical. 0: Yeah. In some ways he was arguing the exact opposite of what it sounds like he was arguing. 1: How so? ## Numbers ### Or: The Dead 四 Scrolls 0: Well, how do you think about the set of all natural numbers? 1: In what sense? 0: Do you believe it exists? 1: I mean... yes? And no. But mostly yes. Also no though. 0: Explain? 1: Ok, if you hand me any specific natural number, it's just digits. Like a finite string of them. So they exist in that sense. 0: What about a natural number with more digits than there are atoms in the universe? 1: It's harder to convince me that one exists. Maybe it exists slightly less. 0: Say more. 1: But it also sort of depends on the notation. I mean it's silly that people always say "Bigger than the number of atoms in the universe." I mean 255 is bigger than 8 but we can still represent 255 with 8 bits. So I could get on board with numbers biggers than the total number of atoms. I mean after all, there's protons and neutrons and electrons, not to mention quarks and all that, there's no reason to focus on atoms. 0: So what about numbers that are bigger than $2^N$ where $N$ is the number of fundamental particles in the universe, whatever the most fundamental types end up being? 1: I dunno. I guess those numbers are a bit less exist-y than the first two kinds. But then at the same time, just because a number's that big doesn't mean it's "that big" in every way of describing it. Like maybe we could express it in only 100 characters or something. If we expressed it as a computation. Like instead of $12345678...$ (etc) we could write it in terms of functions and stuff. Then it would be smaller. 0: Would you say numbers like that "exist"? 1: I'd say they're more exist-y. 0: Than what? 1: Than if we couldn't write code that produces them. I mean sure we probably couldn't compute them in practice. All the digits, I mean. But we might be able to do some symbol manipulation on the smaller representation, the code or symbols or whatever the shorter description was, and by manipulating the short description we could convince ourselves of stuff about them that's true. 0: What about numbers bigger than that? Do you believe they exist? 1: I'm not a mathematician, I dunno what I believe. 0: What do you think mathematicians believe about the natural numbers? 1: That they exist. 0: In a more confident way than you do? 1: Probably, yeah. Can't speak for everyone, but that's the vibe I get from math people. 0: Which of those positions would you say is less mystical? 1: I mean they both seem reasonable. 0: Which is _more_ mystical? If you had to choose. 1: The one that thinks it exists. 0: What exists? 1: Like, all the natural numbers. As a big infinite bag in the sky with curly braces around it. 0: So you differ from that view only when it comes to the numbers that are too big to fit into the entire universe in any representation? 1: Specifically those, yeah. But part of me believes those ones exist too. 0: Which part of you? 1: I don't know, I guess the ones that are bigger than the number of atoms in the universe, or the number of subsets of that set, like once numbers get big enough, you can't expect to ever find "That many" of something. They're not counting numbers. 0: So they don't exist "in the world"? 1: Maybe? But if they exist as computations, then they sort of exist in the world, because if I write the code for that computation then the code for that number is in my brain. And my brain is in the world. So then those numbers are in the world. 0: Here I'll try to rephrase what you just said, and I want you to correct me if I say it wrong. 1: Ok. 0: What I just heard you say is something like: > Only finitely many natural numbers "exist" in an objective sense, as numbers that might enumerate a "number of things" in the physical world. But natural numbers (and maybe some other mathematical objects) might also exist in a "computational" sense, which is something like saying they exist in your "mind" or "brain", but not in a spooky or vague way. If we want to make the same idea sound less spooky or vague, we could say that some numbers exist as pointers to objectively specifiable physical quantities, while other numbers are maybe too big for the physical universe, but not too big to exist as "code" or as well-defined things that we can reason about with symbols. 0: Ok, tell me which parts of that I got wrong. 1: Got wrong how? 0: I was re-expressing what you said. Which bits of that do you disagree with? 1: Oh right. _(Re-reads quickly.)_ Yeah that's basically what I was trying to say. 0: What do you think the average mathematician would say if we asked them the same question? 1: I think they'd just say the integers exist. 0: What do you think the average computer scientist would say if we asked them the same question? 1: Probably something like what I said. I don't think I'm that atypical on this note. 0: Why do you think people in computing think like that? 1: I dunno. If you spend enough time programming, that's just how you think. Like if I'm writing some code to create a massive text file and then the kernel kills the process because I ran out of memory or disk space, then the file doesn't exist yet. To make it exist, I have to like... make it exist. 0: Make it exist how? 1: Like find somewhere to put it, and then re-run the code. I dunno, this isn't some complicated philosophy thing. I just mean if you spend enough time programming, it's hard to believe stuff exists before you construct it. 0: Can you answer one more question for me? 1: Sure. 0: This question 1 asked me earlier, can you answer it now? > 1: What does it mean to say "math is a cognitive construct rather than a type of objective truth"? That seems kinda mystical. 1: OH. ## Thought ### Or: The Dead 思 Scrolls 0: Delete that picture of Brouwer up above from your memory. 1: That wasn't Brouwer? 0: That was him, but it's not who he was. He also wasn't a mystic. 1: Why are we talking so much about this guy? 0: Because the histories are completely wrong. About who he was, what he believed, the problems he had with standard mathematics, and the branch of foundations he started in an attempt to refactor the subject. 1: That's a lot of stuff. Why was he, or they, or... what's wrong exactly? 0: Let's start with the picture. 1: What's wrong with the picture? 0: Nothing. But fuck it anyway. 1: What? 0: You shouldn't be picturing that grumpy old man. 1: I'm not sure what your point is here. ![[brouwer-1.jpg]] 1: Damn! 0: What? 1: Just a different vibe. This guy's a mathematician? 0: One of the best. 1: _(Squinting at the picture)_ Who even. Is that a bed? And there's a bed in the background. What an unusual--- 0: And the least understood. > Brouwer founded intuitionism, a philosophy of mathematics that challenged the then-prevailing formalism of David Hilbert and his collaborators, who included Paul Bernays, Wilhelm Ackermann, and John von Neumann (cf. Kleene (1952), p. 46–59). A variety of constructive mathematics, intuitionism is a philosophy of the foundations of mathematics. It is sometimes (simplistically) characterized by saying that its adherents do not admit the law of excluded middle as a general axiom in mathematical reasoning, although it may be proven as a theorem in some special cases. > > -The Dynamic Read-Writable Free Encyclopedic Repository of the Modern State of Human Knowledge 1: Hey there's Kleene again! What's cf? 0: They're just citing Kleene's blue bible. 1: The frighteningly technical book? 0: Yep. He talks about Brouwer in there. > Brouwer rejected the concept of actual infinity, but admitted the idea of potential infinity. > > According to Weyl 1946, 'Brouwer made it clear, as I think beyond any doubt, that there is no evidence supporting the belief in the existential character of the totality of all natural numbers ... the sequence of numbers which grows beyond any stage already reached by passing to the next number, is a manifold of possibilities open towards infinity; it remains forever in the status of creation, but is not a closed realm of things existing in themselves. That we blindly converted one into the other is the true source of our difficulties, including the antinomies – a source of more fundamental nature than Russell's vicious circle principle indicated. Brouwer opened our eyes and made us see how far classical mathematics, nourished by a belief in the 'absolute' that transcends all human possibilities of realization, goes beyond such statements as can claim real meaning and truth founded on evidence. — Kleene 1991, pp. 48–49 > > -The Dynamic Read-Writable Free Encyclopedic Repository of the Modern State of Human Knowledge 1: What's book is this from? I've got a todo list to read some Kleene eventually so I want to get the good stuff. ![[kleene-1991-is-kleene-1952.png]] 0: Same one as before. 1: Frighteningly technical one? 0: Yep. 1: Is EVERY reference in the world to this same damn book? 0: Basically. Ok back to Brouwer. > Brouwer was a member of the Significs Group. It formed part of the early history of semiotics—the study of symbols—around Victoria, Lady Welby in particular. The original meaning of his intuitionism probably cannot be completely disentangled from the intellectual milieu of that group. > > -The Dynamic Read-Writable Free Encyclopedic Repository of the Modern State of Human Knowledge 1: Wtf is Significs? > Significs, intended to be a theory of signs, was developed by Lady Welby in quite close connection with the work of Charles Sanders Peirce, her correspondent. > > -The Dynamic Read-Writable Free Encyclopedic Repository of the Modern State of Human Knowledge 1: Charles Sanders Peirce. Why do I remember that name? 0: Horse logic. 1: NO WAY! 0: Yep, that guy. > In 1905, at the age of 24, Brouwer expressed his philosophy of life in a short tract Life, Art and Mysticism, which has been described by the mathematician Martin Davis as "drenched in romantic pessimism" (Davis (2002), p. 94). 1: _Drenched in romantic pessimism!_ 0: Why is that funny? 1: _(Can't breathe.)_ 0: You're weird. > Arthur Schopenhauer had a formative influence on Brouwer, not least because he insisted that all concepts be fundamentally based on sense intuitions. 1: How is THIS "mysticism"? This sounds like a hard science. 0: Well, it said "sense intuitions," not "sense data." 1: Are those _different?_ 0: I don't think so. 1: So is this all just a terminology confusion or what? 0: No no, he actually disagreed with the mathematicians. Like deeply. 1: But--- 0: But he wasn't the mystical one. > Brouwer then "embarked on a self-righteous campaign to reconstruct mathematical practice from the ground up so as to satisfy his philosophical convictions"; indeed his thesis advisor refused to accept his Chapter II "as it stands, ... all interwoven with some kind of pessimism and mystical attitude to life which is not mathematics, nor has anything to do with the foundations of mathematics" (Davis, p. 94 quoting van Stigt, p. 41). > > -The Dynamic Read-Writable Free Encyclopedic Repository of the Modern State of Human Knowledge 1: Damn they really don't like him. 0: Hard not to be pessimistic when you're writing about foundations and they say it's not foundations. > "After completing his dissertation, Brouwer made a conscious decision to temporarily keep his contentious ideas under wraps and to concentrate on demonstrating his mathematical prowess" (Davis (2000), p. 95); by 1910 he had published a number of important papers, in particular the Fixed Point Theorem. Hilbert—the formalist with whom the intuitionist Brouwer would ultimately spend years in conflict—admired the young man and helped him receive a regular academic appointment (1912) at the University of Amsterdam (Davis, p. 96). It was then that "Brouwer felt free to return to his revolutionary project which he was now calling _intuitionism_ " (ibid). > > -The Dynamic Read-Writable Free Encyclopedic Repository of the Modern State of Human Knowledge 1: I wish I could slap this guy and tell him to give his thing a different name. 0: Do what now? 1: Intuitionism. 0: What about it? 1: Horrible name. 0: How do you know what he meant by that? We've barely read any--- 1: Trust me, this guy makes sense. I know what he means. > Nevertheless, in 1908: "... Brouwer, in a paper titled 'The untrustworthiness of the principles of logic', challenged the belief that the rules of the classical logic, which have come down to us essentially from Aristotle (384--322 B.C.) have an absolute validity, independent of the subject matter to which they are applied" (Kleene (1952), p. 46). > > -The Dynamic Read-Writable Free Encyclopedic Repository of the Modern State of Human Knowledge 1: What a badass. 0: So we're on his team now? 1: I dunno, maybe. 0: What does he believe? 1: I DON'T KNOW but I'm pretty sure he'd understand that my text file doesn't exist if the code crashed or didn't finish yet. 0: What makes you think that? 1: Just an intuition. 0: Pun intended? 1: NO no I didn't even realize I said that. I just mean he doesn't believe in the integers and everyone hates him, so I'm pretty sure he's not pushing mystical voodoo bullshit, it sounds like a programmer who just sucks at words. 0: Many such cases. > Both Brouwer and Hilbert worked as editors at the prestigious mathematical journal, Mathematische Annalen. 1: _(Serious tone)_ "It's very prestigious to get a paper in the Annalen." 0: _(Entertained)_ What is going on with you all of a sudden? 1: I dunno! The world is so goofy. 0: Goofy how? 1: Everyone's wrong and everything's fake. 0: Not _everything._ 1: Not in a pessimistic way! Just in an absurd way. 0: You're absurd. 1: In summary the world is silly, so I'm in a silly mood. > Hilbert was the chief editor of the journal alongside Blumenthal, Carathéodory, and Einstein, and Brouwer was an associate editor. 1: Hey there's Einstein! 0: _(Silent, eyebrows raised)_ ... 1: This is like fanfiction. > The relationship between Brouwer and Hilbert began to deteriorate as their intellectual feud began to become a personal one, culminating in the eventual dismissal of Brouwer by Hilbert. 1: "Oh no their relationship is deteriorating." 0: _(Same expression as before)_ ... 1: This is fanfiction. The world isn't real. I'm just saying. 0: _(Squinty smile)_ ...Back to the story. > Carathéordory sought out for Einstein's advice on this matter and Einstein chose to remain neutral in this feud, leading to Brouwer's dismissal. 1: Einstein, why you gotta be like that? 0: It's not fanfiction. 1: Yes yes, the world is real. All of this is real. God's not a developer and reality isn't a joke. > Abraham Fraenkel argues that the reason for this dismissal was Brouwer's turn towards pushing ideas of Germanic Aryan supremacy; 1: Oh no! Brouwer don't do that. > however, there does not exist any evidence of Brouwer's involvement with the German Nazi party, nor the National Socialist Movement in the Netherlands. Brouwer was not known to hold onto any nationalistic views by his contemporaries, and his views were akin to the Germanic idealists and romanticists of the time. > > -The Dynamic Read-Writable Free Encyclopedic Repository of the Modern State of Human Knowledge 1: They had Godwin's law THAT LONG ago? 0: Apparently. 1: Who is this Abraham Fraenkel? 0: He's the F in ZFC. 1: No way. The Hell torments? 0: ZFC's not _that_ bad. 1: Why's he bad-mouthing our guy? 0: Dunno. All I know about Fraenkel is he was an early Zionist, and also that his first name was Adolf but then that name suddenly became super unpopular so he changed it. 1: Talk about putting the F in ZFC. 0: What's wrong with that? 1: I DON'T KNOW. Silly mood, can't be held accountable. 0: You are the strangest student I've ever--- ## Death ### Or: The Dead 死 Scrolls 1: Oh no! 0: What? 1: Read the next part! > In later years Brouwer lost all his friends and intuitionism was taken over by hating. 1: Why everybody gotta be hating? 0: You read too fast dummy. Heyting's a person. 1: Why he gotta be heyting though? 0: He's not heyting on Brower. They're friends. That's his student. Read slower. > In later years Brouwer became relatively isolated; the development of intuitionism at its source was taken up by his student Arend Heyting. 1: Oh ok yeah that's not the same. 0: Do you always get this way at night? 1: READING IS HARD\\!\\! > Dutch mathematician and historian of mathematics Bartel Leendert van der Waerden 1: Poor bastard. 0: _(Looking up)_ ...What? 1: That name. Imagine going through li--- 0: Stop interrupting. 1: YOU'RE ONE TO TA--- > \[Guy with the long name] attended lectures given by Brouwer in later years, and commented: "Even though his most important research contributions were in topology, Brouwer never gave courses in topology, but always on — and only on — the foundations of his intuitionism. It seemed that he was no longer convinced of his results in topology because they were not correct from the point of view of intuitionism, and he judged everything he had done before, his greatest output, false according to his philosophy." > > -The Dynamic Read-Writable Free Encyclopedic Repository of the Modern State of Human Knowledge 1: _(Pretending to cry, but actually sad, just not enough to cry)_ He was one of us. 0: Didn't expect you to have so much fun in here. 1: Look I got invested in the story, shoot me! > About his last years, Davis (2002) remarks: "...he felt more and more isolated, and spent his last years under the spell of 'totally unfounded financial worries and a paranoid fear of bankruptcy, persecution and illness.' He was killed in 1966 at the age of 85, struck by a vehicle while crossing the street in front of his house." (Davis, p. 100 quoting van Stigt. p. 110.) > > -The Dynamic Read-Writable Free Encyclopedic Repository of the Modern State of Human Knowledge 1: THAT'S THE END?! 0: What? No there are still a few more files in this directory, intuitionism is part of constru--- 1: HE GOT HIT BY A CAR!? 0: Oh. Yeah I guess that's how Brouwer died. But we're not in here to do a biography, this directory's about construc--- 1: If this was fanfiction it would have ended better. 0: Are you going to write Brouwer fan-fiction? 1: I never even learned his first name. 0: Ok well his full name is Luitzen Egbertus Jan "Bertus" Brouwer. 1: _(Actually crying now)_ WHAT?! 0: What? 1: _EGBERTUS?_ 0: Are you laughing or crying? 1: _(Can't breathe)_ 0: Ok, continuing... > A serious study of this controversy can be found in Stephen Kleene's _Introduction to Metamathematics_, particularly in Chapter III: A critique of mathematical reasoning. He discusses §11. _The paradoxes_, §12. _First inferences from the paradoxes_ impredicative definitions, Logicism etc., §13. _Intuitionism_, §14. _Formalism_, §15. _Formalization of a theory_. Kleene takes the debate seriously, and throughout his book he actually builds the two "formal systems" (e.g., on page 119 he shows logical laws, such as double negation, which are disallowed in the intuitionist system). > > -The Dynamic Read-Writable Free Encyclopedic Repository of the Modern State of Human Knowledge 1: _(Wiping away tears, still unclear if that was laughing or crying)_ Damn, Steve is everywhere. 0: Definitely. Y'know I have this bookma--- 1: Is that the same book? 0: The frighteningly technical one? Yeah. 1: Is there more Brouwer in there? 0: Yeah. I mean not THAT much. 1: No more Brouwer? 0: There's more about intuitionism. 1: Can we read some of that? I'm not ready to say goodbye. 0: I can't tell if you're enjoying this this or just hating it and mocking me. 1: _(Tired from cry-laughing)_ ... don't be heyting. 0: That's not an answer. 1: More fanfiction. 0: Ok that's the end of the Brouwer page. Here let's do intuitionism. 1: _(Cozy listening position)_ ## Religion ### Or: The Dead 寺 Scrolls > Intuitionism's history can be traced to two controversies in nineteenth century mathematics. > > The first of these was the invention of transfinite arithmetic by Georg Cantor and its subsequent rejection by a number of prominent mathematicians including most famously his teacher Leopold Kronecker—a confirmed finitist. 1: Hi Cantor! 0: You're weird. > The second of these was Gottlob Frege's effort to reduce all of mathematics to a logical formulation via set theory and its derailing by a youthful Bertrand Russell, the discoverer of Russell's paradox. 1: Hi paradox man! 0: ... He says hi. > In the early twentieth century L. E. J. Brouwer represented the intuitionist position and David Hilbert the formalist position. Kurt Gödel offered opinions referred to as Platonist (see various sources re Gödel). Alan Turing considers: "non-constructive systems of logic with which not all the steps in a proof are mechanical, some being intuitive". 1: Wow, I know like all these people now. 0: _(Silently amused)_ Yes good job 1. Continuing... > Later, Stephen Cole Kleene brought forth a more rational consideration of intuitionism in his Introduction to metamathematics (1952). 1: Damn I even know that book. I know everything. 0: Are you still awake? 1: _(Quietly, to self)_ I'm buying that book. 0: _(Continues reading)_ > These controversies are strongly linked as the logical methods used by Cantor in proving his results in transfinite arithmetic are essentially the same as those used by Russell in constructing his paradox. Hence how one chooses to resolve Russell's paradox has direct implications on the status accorded to Cantor's transfinite arithmetic. 1: _(Falling asleep)_ I know this whole story. 0: _(Continues)_ > Frege had planned a three-volume definitive work, but just as the second volume was going to press, Russell sent Frege a letter outlining his paradox, which demonstrated that one of Frege's rules of self-reference was self-contradictory. In an appendix to the second volume, Frege acknowledged that one of the axioms of his system did in fact lead to Russell's paradox. 1: _(Eyes shut)_ This is my story. 0: > The diagrammatic notation that Frege used had no antecedents (and has had no imitators since). 1: _(Mumbling)_ ... for real... that shit was wild. > Frege, the story goes, plunged into depression and did not publish the third volume of his work as he had planned. For more see Davis (2000) Chapters 3 and 4: Frege: From Breakthrough to Despair and Cantor: Detour through Infinity. See van Heijenoort for the original works and van Heijenoort's commentary. > > -The Dynamic Read-Writable Free Encyclopedic Repository of the Modern State of Human Knowledge 1: _(Suddenly awake)_ DID ANOTHER ONE OF OUR PEOPLE JUST GET DEPRESSED AND DIE?! 0: Frege? He's sort of the pre-history. Decided not to inclu--- 1: _(Re-reading the part about Frege getting depressed.)_ This fanfiction is all wrong. 0: This is actual history. 1: History sucks. I hate history. 0: Should we do something el--- 1: Bible. We need a bible. 0: That was sort of the goal. 1: Good. Keep reading. 0: That's the end of this file. 1: Read something else. 0: I hadn't planned to--- 1: About Brouwer. What happened after Brouwer? 0: I don't know. Let's go see.