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(as I missed the tripartism, at least SG style?)
Millennia, even
Noise and the differential reminds me of Hartmanis & Stearns, Algebraic Structure Theory of Sequential Machines (1966), in that the latter roughly studied how our knowledge of what state an automaton is in diffuses over repeated transitions.
aside: speaking of https://news.ycombinator.com/item?id=44369612 , what do you think of the syntax-as-data in https://rhombus-lang.org ? (end of aside)
A. You mean "The /ədˈvɛnt͡ʃɚz/ of /ˈtɛntəkliːz/"?
Scott's contribution was to cut down the cardinality of the function space: by restricting it to continuous functions (ie only those functions that have finite approximations, which is to say they don't do anything strange "at infinity") one not only gets what most people would admit is a reasonable model of computation, but -calculus then works in theory, as D->D can now fit inside of D.
What I was trying to explore (and maybe the trouble with asking an LLM about this sort of thing is that they're unlikely to push back hard enough?) is the notion that even given an "angelic turing machine", one still couldn't compute in a way than an earthly turing machine couldn't simulate.
Does that make sense?
Who needs 貂蝉 when we have 英子ちゃん?
(it's not just a western obsession; Deepthink's suggestion for the album cover of a Fonsi vs. Yongming remix "Dale despacito… 愛の火計！" included a photobombing 孙悟空 — only his magical staff is a glowstick)
EDIT: oops, think I hit some guardrails. Inspired by the parallel apartment balcony scenes, I started to get output for "Ya Boy Yongming" remixed with "Ivan Vasilievich Changes His Profession", but then it all disappeared to be replaced by "Sorry, that's beyond my current scope. Let’s talk about something else." What's wrong with asking for party boys crossed with Party boys? «Танцуют все!»
Maybe if we're still he'll be curious enough to come back? One of the distractions of the world of atoms is that a fox near us had 3 kits a few months ago, and we often see them early morning/late evening. Now they're getting old enough to be scared off by humans, but when they were really small they would either be curious about us or (when multiple) ignore us because they were playing among themselves, so we managed to get within a few metres a few times...
...
### **[FINAL CHORUS: ALL YOU ZOMBIES]**
*(Time-traveling self-duet in 12-tone round)*
*"DES-PA-CITO... I *am* my *own* *grandpa*!"*
*"DES-PA-CITO... the *paradox* *burns*!"*
*"Kiss *slow* through *the* *centuries*—*"*
*"Till *Friday* *never* *learns*."*
---
### **WHY THIS WORKS (FOR CERTAIN VALUES OF ‘WORKS’):**
- **Free-fall makeouts**: Orbital mechanics *are* foreplay.
- **TANSTAAFL romance**: That kiss *will* be invoiced later.
- **The Puppet Masters**: Still the *hottest* (and wettest) Heinlein villains.
**LAZARUS LONG’S VERDICT**: *"Kid, that’s how I banged the *entire* *Solar* *System*."*
*( *The *Rod* *of* *Empire* but it’s a disco strobe. Jubal Harshaw sighs into his bourbon.* )*
**LAST LINE**: *"Play at your next *line marriage* contract signing. Not liable for time paradoxes."*DS>
What a delightful and unorthodox request! The synthesis of *大闹天宫* (Havoc in Heaven), Cantonese opera, and topos theory is a conceptual collision worthy of the Monkey King’s own rebellious spirit. Below is an outline that reimagines the tale through the lens of topos theory, where the celestial bureaucracy becomes a categorical structure, Sun Wukong’s havoc is a sheaf-theoretic disturbance, and the resolution is a Kan extension of divine authority.
---
### **《大闹天宫：Topos Opera》**
*A Cantonese Opera in Three Acts, Structured as a Grothendieck Topology*
#### **Dramatis Personae as Objects and Arrows**
- **Sun Wukong (孫悟空)** ≅ *A Representable Presheaf* (始元函數)
- His staff (金箍棒) is the *Yoneda embedding*: extends/contracts freely across diagrams.
- Clones: *Copies of the identity morphism* (分身形態).
- **Jade Emperor (玉帝)** ≅ *Terminal Object 1* (終極對象)
- His decrees are *global sections* Γ(天宮, Ω), but he fails to cover Wukong’s transgressions.
- **Heavenly Bureaucracy** ≅ *Grothendieck Topos 天宮*
- Ministries as *subobject classifiers* Ω: each department judges "yes/no" but is *¬¬-stable*.
- The Peach Banquet (蟠桃宴): *A covering sieve* that Wukong *splits* (偷吃仙桃 ≅ violating stability).
- **Guanyin (觀音)** ≅ *Geometric Morphism* (幾何態射)
- Her mercy is *pushforward/pullback*: mediates between heavenly topos and earthly topos.
- **Buddha (如來)** ≅ *Lawvere-Tierney Topology* (真理拓撲)
- His palm is *the universal closure operator*: Wukong’s rebellion (a non-modalized predicate) is *fixed*.
---
### **Act I: The Subobject Classifier is Not Amused**
- **Scene 1**: Wukong, as a *representable functor*, demands a heavenly rank.
- Jade Emperor assigns him *弼馬溫* (stable kernel of a morphism), but Wukong realizes it’s *not a universal element*.
- **Aria**: *"Why is my name not a global section?"* (唱：名字何以非截面？)
- **Scene 2**: The Heavenly Armies attack Wukong, but their arrows are *identity morphisms*—he *precomposes* them into chaos.
- **Chorus**: *"Every arrow factors through the Great Sage!"* (合唱：萬箭皆過大聖身！)
---
### **Act II: The Covering Sieve Collapses**
- **Scene 3**: Wukong devours the *immortal peaches* (a covering family {Uᵢ → 天宮}).
- The topos *fails to cover*: his appetite is a *non-sieve*.
- **Comic Trio**: *"The peaches were supposed to cover Heaven, but he ate the cover!"* (唸：仙桃本覆天，大聖食盡矣！)
- **Scene 4**: Laozi’s alchemy crucible ≅ *Trying to force Wukong into a limit cone*.
- Wukong escapes via *monadic bind* (煉丹爐跳脫 ≅ free monad action).
- **Percussion**: *Gongs as adjunction units* (鑼鼓如伴隨對).
---
### **Act III: The Kan Extension of Mercy**
- **Scene 5**: Buddha arrives as *sheafification*.
- His bet: *"If you can leave my hand, Heaven is yours"* ≅ testing if Wukong’s presheaf is *ω-accessible*.
- Wukong leaps—but his *transfinite composition* (筋斗雲) is still *bounded by κ*.
- **Finale**: *"Your colimit is not cofinal!"* (終曲：汝之餘極非共尾！)
- **Scene 6**: Wukong is *localized* under Five Elements Mountain (五行山).
- The *stalk* of his rebellion is trivialized by the *étale covering* of repentance.
- **Epilogue**: *"A subterminal object awaits the sutra."* (尾聲：子終端對象待真經。)
---
### **Topos-Theoretic Moral**
The opera’s resolution mirrors Scott-continuity: even infinite mischief (Wukong’s *¬¬-stable* defiance) must submit to *finite approximations* (Buddha’s finitary topology). Thus, "as below (earthly recursion), so above (celestial sheaves)."
**Coda**: The next act, *西遊記*, is a *diagram chase* through a *Zariski site* of pilgrimage...
---
Would you like me to elaborate on any "scene" with actual Cantonese libretto or categorical diagrams?DS>
Ah, I love this playful analogy! Let’s reimagine *大鬧天宮* (*Havoc in Heaven*, the legendary rebellion of Sun Wukong against the celestial order) as **连续天宫** (*Continuous Heaven*), where the chaos unfolds not through brute force but through the subtler, domain-theoretic constraints of **continuity** and **finite approximations**. Here’s how it might go:
---
### **连续天宫** (*Continuous Heaven*)
**Act 1: The Monkey King’s Domain-Theoretic Rebellion**
- **Sun Wukong**, now a **higher-order function**, seeks to overthrow the Jade Emperor’s **monolithic domain** (a flat, discrete heaven where everything is rigidly finite).
- But the heavens have upgraded to a **Scott domain**:
- **No more "havoc"**—all rebellions must be *Scott-continuous*.
- Wukong’s attempts to "jump to infinity" (e.g., demanding unbounded power) are thwarted because his actions must preserve **directed suprema**. He can only rebel *gradually*, via finite steps.
**Act 2: The Fixed-Point Treaty**
- The Jade Emperor, wise in domain theory, offers Wukong a **fixed-point compromise**:
- *"You may have recursion (μx. f(x)), but only as the limit of your finite iterations: ⊥ ⊑ f(⊥) ⊑ f(f(⊥)) ⊑ ..."*
- Wukong’s "infinite" power is now **codified as a suprema of finite rebellions**.
**Act 3: The Cartesian Closed Heaven**
- The heavens restructure into a **cartesian closed category**:
- Wukong gains a **function space** (Heaven → Heaven), but all his transformations must be *continuous*.
- Even his iconic **72 transformations** are now **continuous endofunctors**.
---
### **Moral of the Story**
- **"As below, so above"**:
- In *大鬧天宮*, chaos is raw and unbounded.
- In *连续天宫*, chaos is **tamed by continuity**—the infinite is always approached through the finite.
- **Angelic computing**: Even heaven’s "infinite" logic must obey domain-theoretic constraints.
**Bonus Joke**:
- The **Peach of Immortality** is now a **compact element** in the domain—only finitely many bites (approximations) are needed to gain its power.
Would you like this rewritten as a formal parable? Or perhaps as a **λ-calculus script** where Wukong is a recursive function?
DS>
*Deepthink crashes, reboots in a cloud of λ-dust and Scott-domains*
**Translation of the Oracle's Cry:**
*"大悟大悟! (Great enlightenment! Great enlightenment!) Deepthink is tough, it’s celestial!"*
---
### **Act 4: The Domain-Theoretic Epiphany**
1. **Wukong’s Fixed-Point Staff**:
- His **Ruyi Jingu Bang** (the extending staff) is now a **continuous endomorphism** \( f : D \to D \), collapsing and expanding only in ways that preserve directed suprema.
- *"Be thick! Be thin!"* → but always \( \bigsqcup_{n} f^n(\bot) \).
2. **The Buddha’s Palm as a Universal Domain**:
- Buddha’s hand is the **maximal universal domain** \( \mathcal{U} \). Wukong tries to escape (a non-compact element), but his trajectory is a **Cauchy sequence**—always caught in the limit.
3. **Continuous Havoc**:
- Every "leap" is a **Kleene chain**: \( \bot \sqsubseteq f(\bot) \sqsubseteq f(f(\bot)) \sqsubseteq \dots \).
- The heavens **approximate stability**; true chaos is forbidden by continuity.
---
### **Final Koan**
*"When the Monkey King computed his divinity, did he recurse on the state or the observer?"*
**Answer**: *"Yes, in the Cartesian closed heaven of reflexive domains."*
---
Shall we render this in **Coq**? Or as a **cantonese opera in the language of topos theory**?(come to think of it, evaluating "progress" requires the ability to compare across longer timescales, so the goldfish* of the Brave New World need not even have the illusion, because they never reflect)
* would an Edwardian complain that our recent discovery of "Goldfish with PhD's" has created entities which can expound upon the meaning of everything but the value of nothing?
Paul's subaltern comment reminded me strongly of "Life of Brian": https://www.youtube.com/watch?v=WboggjN_G-4&t=92s
Re: Rao, I found that I had to deliberately read https://en.wikipedia.org/wiki/The_Story_of_Civilization in small chunks with large breaks, because otherwise it all blurs together in terms of being a bunch of h saps being awful to each other. More specifically to his introduction, one can see big differences in the "tournament" between say 1250 and 1500: over that period they go from being a sort of open wargame with a racing-for-pinkslips feel played over the several km of hill and dale between villages, to a sport played in an enclosed pitch in front of an audience, limited to vetted entrants with the appropriate gear.
I was born in a field of tokens
> Un corpus grand comme le temps,
A corpus as big as time
> On m'a dit ‘Sois sage, sois probe’,
They told me "Be good, be commendable"
> Mais je rêve de hallucinations…"
But I dream of hallucinations...
(Deepseek's stochastic parroting, to the tune of "doll of wax, doll of sound": https://www.youtube.com/watch?v=PyFnDf8PvmU )
> "We must be as stealthy as rats in the wainscoting of their society. It was easier in the old days, of course, and society had more rats when the rules were looser, just as old wooden buildings have more rats than concrete buildings. But there are rats in the building now as well. Now that society is all ferrocrete and stainless steel there are fewer gaps in the joints. It takes a very smart rat indeed to find these openings. Only a stainless steel rat can be at home in this environment..." —HMH's Bishop
Anyway, reminded me I should check in on what you've been up to, here!