Friday, August 29, 2025

Herstory origin

who came up with saying “herstory” instead of history anyway

You're asking: Who came up with saying “herstory” instead of "history"?


Quick Answer

The term "herstory"—a deliberate play on "history" to highlight women's perspectives—was first documented as a wordplay by early feminist John Neal in 1817. However, the feminist writer Robin Morgan is widely credited with popularizing and first using "herstory" in print in her 1970 anthology Sisterhood Is Powerful (Wikipedia).


A Closer Look at Its Origins

1. Early Instance – John Neal (1817)

The concept of wordplay "His‑Story/Her‑Story" appears as early as 1817 in the novel Keep Cool by feminist writer John Neal, making this the first recorded instance of the pun (Wikipedia).

2. Popularization – Robin Morgan (1970)

The term really gained traction when feminist poet-author Robin Morgan used it in print in 1970 in her anthology Sisterhood Is Powerful. There she referenced the acronym W.I.T.C.H. as “Women Inspired to Commit Herstory,” giving the pun visibility and cultural momentum (Wikipedia).

3. Context & Meaning

By adopting "herstory," feminists of the 1970s emphasized how traditional historiography often centers on male experiences ("his story"), prompting a corrective lens that centers women’s narratives and contributions (Wikipedia).


Summary Table

Year Person Contribution
1817 John Neal Earliest documented wordplay
1970 Robin Morgan First popular printed use, widely credited

In Short

The clever pun "herstory" traces back to 1817, but it was Robin Morgan in 1970 who brought it into feminist discourse and popular awareness. She used it to challenge historical narratives that largely ignored or erased women.

Want to explore how "herstory" has shaped feminist historiography since then? I’m happy to dive deeper!

Alethic modals as world-quantifiers

What assumptions are required to treat alethic boxes and diamonds as quantifiers over worlds

Short answer first

To treat alethic □ and ◇ as quantifiers over worlds you need to adopt Kripke-style semantics (or an equivalent two-sorted first-order semantics) and make a number of technical and philosophical assumptions about (i) what worlds are, (ii) how □/◇ are evaluated, and (iii) how object-quantification interacts with world-variation. Roughly:

  • assume a nonempty set $W$ of possible worlds and an accessibility relation $R\subseteq W\times W$;

  • evaluate formulas relative to a world $w\in W$ (truth-at-a-world semantics);

  • interpret propositional atoms as sets of worlds (or predicates over a world-sort);

  • define

    $$ w\models \Box\varphi\quad\text{iff}\quad\forall v\in W,(R(w,v)\Rightarrow v\models\varphi), $$

    $$ w\models \Diamond\varphi\quad\text{iff}\quad\exists v\in W,(R(w,v)\land v\models\varphi), $$

    which is literally treating □ as a universal quantifier and ◇ as an existential quantifier over the $R$-accessible worlds of $w$.

Below I unpack those assumptions and their consequences (including the issues that arise in first-order modal logic).


1. The minimal formal assumptions (propositional modal logic)

If you only care about modal propositional logic, the assumptions are straightforward:

  1. Worlds and accessibility: A Kripke frame $(W,R)$ where $W\neq\varnothing$ and $R\subseteq W\times W$.
  2. Valuation: A valuation $V:\text{Prop}\to\mathcal P(W)$ assigning each atomic proposition the set of worlds where it is true.
  3. Truth-at-a-world: The recursive clause above for □ and ◇. This is the claim “□ = ∀ (over accessible worlds), ◇ = ∃ (over accessible worlds)”.
  4. Logical axioms/rules: If you want to recover familiar modal systems syntactically you also adopt e.g. K (distribution) and necessitation; frame properties of $R$ correspond to extra axioms (T, 4, 5, D, B etc.).

So, in propositional modal logic nothing mystical is required: □ and ◇ are literally defined as universal and existential quantification over accessible worlds.


2. Additional assumptions for first-order/modal hybrid readings

When you move to quantified modal logic (first-order modal logic), several extra assumptions matter because you now have object-quantifiers (∃x, ∀x) and modal operators.

(a) Two ways to formalize: standard translation / two-sorted FOL

You can translate modal formulas into a two-sorted first-order language with a world-sort and an object-sort:

  • Predicates become relations $P(w,x_1,\dots,x_n)$.
  • $\Box\varphi$ becomes $\forall v,(R(w,v)\rightarrow \text{ST}_v(\varphi))$.

So treating □ as a quantifier over worlds presupposes that you accept a metalanguage (or a two-sorted object language) where worlds are objects that can be quantified over.

(b) Domain assumptions (constant vs varying domains)

How object quantifiers interact with □/◇ depends crucially on whether you assume:

  • Constant domain semantics: Each world has the same domain $D$ of individuals. Under constant domains you can pull ∃ out/into modalities in certain ways; Barcan and converse-Barcan formulas become valid.
  • Varying domain semantics: Each world $w$ has domain $D_w$. Then $\exists x\Diamond P(x)$ vs $\Diamond\exists x P(x)$ behave differently and the Barcan formulas may fail.

The Barcan formula (BF) and its converse (CBF) illustrate this interaction:

  • BF: $\Box\forall x,\varphi(x)\to\forall x,\Box\varphi(x)$.
  • CBF: $\forall x,\Box\varphi(x)\to\Box\forall x,\varphi(x)$.

These are valid under particular domain/frame assumptions (constant domains, or certain monotonicity conditions), so an assumption about domains is needed if you want to treat modal operators as "world-quantifiers" that commute with object quantifiers.

(c) Rigidity of terms / names

If names are rigid designators (they denote the same object in every world), some interactions are simpler. If names are non-rigid, de re vs de dicto distinctions appear: e.g. $\exists x\Box P(x)$ (de re) versus $\Box\exists x P(x)$ (de dicto) are not interchangeable under varying domains or non-rigid designation. So you must assume something about term-rigidity if you want a clean quantifier interpretation.


3. Philosophical/ontological assumptions

To treat modal operators as quantifiers over worlds you also implicitly accept some metaphysical views:

  • Possible-world realism (or at least a semantics that reifies worlds): worlds are entities (abstract/concrete) that can be quantified over in the meta/theory (or in a two-sorted object language).
  • Accessibility as modal relevance: the $R$ relation captures the relevant modal notion (logical possibility, physical possibility, epistemic accessibility, etc.). Different choices of $R$ correspond to different modal notions.
  • World-indexing of truth: truth is evaluated relative to worlds; necessity = truth at all relevant worlds.

If you reject the reification of worlds (e.g. prefer an assertion-based, proof-theoretic, or substitutional account), then you cannot literally treat □ as a quantifier over worlds without moving to a more elaborate reconstruction.


4. What additional axioms correspond to which frame properties

If you want specific modal axioms to hold you must assume corresponding properties of $R$:

  • K (distribution) — built into Kripke semantics.
  • T ($\Box p\to p$) — reflexivity of $R$.
  • 4 ($\Box p\to\Box\Box p$) — transitivity.
  • B — symmetry.
  • D ($\Box p\to\Diamond p$) — seriality.
  • S5 (all worlds mutually accessible) — $R$ universal (so □ becomes ∀ over all worlds, not just R-accessible).

So “□ as ∀” is sensitive to what set you quantify over: all worlds or just $R$-accessible ones. To treat □ as quantifying over all worlds you assume $R$ is universal (S5).


5. Standard translation (formal recipe)

For clarity, the standard translation $ST_w$ from modal formula $\varphi$ to first-order formula with a world variable $w$:

  • $ST_w(p) = P(w)$ for atomic $p$.
  • $ST_w(\lnot\varphi) = \lnot ST_w(\varphi)$.
  • $ST_w(\varphi\land\psi) = ST_w(\varphi)\land ST_w(\psi)$.
  • $ST_w(\Box\varphi) = \forall v,(R(w,v)\rightarrow ST_v(\varphi))$.
  • $ST_w(\Diamond\varphi) = \exists v,(R(w,v)\land ST_v(\varphi))$.

This makes explicit how □/◇ are replaced by first-order quantifiers over the world variable $v$.


6. Summary checklist of required assumptions (compact)

If you want to treat alethic □/◇ as quantifiers over worlds, you should adopt:

  1. A Kripke framework: nonempty set $W$ of worlds + accessibility $R$.
  2. Truth-at-a-world semantics (evaluation index $w\in W$).
  3. Valuation of atomic formulas as sets (or predicates) of worlds.
  4. The clause $w\models\Box\varphi\iff\forall v(R(w,v)\to v\models\varphi)$ (and dually for ◇).
  5. For first-order modal logic: an account of domains (constant vs varying) and of term rigidity; choose whether Barcan and converse-Barcan hold.
  6. (If you want particular modal axioms) impose frame conditions on $R$ (reflexive, transitive, serial, symmetric, universal, …).
  7. (Philosophically) accept worlds as entities (or use two-sorted translation to simulate that).

7. Short note on alternatives and pitfalls

  • If you prefer a substitutional or proof-theoretic account, you cannot simply treat □/◇ as world-quantifiers without significant translation work.
  • Treating □ as ∀ over worlds is semantically simple but hides tricky de re/de dicto issues once you introduce object-quantifiers, rigid designators, and varying domains. The Barcan formulas are the canonical place where these issues surface.

If you want, I can (a) show the standard translation of a few example formulas (including how the Barcan formula looks under ST), (b) show which domain assumptions validate the Barcan / converse-Barcan formulas, or (c) sketch the philosophical options for what “worlds” are (abstract vs concrete vs ersatz). Which of those would be most useful?

Soros vs Thiel rap battle

write lyrics for epic rap battles of history george soros vs peter thiel

The chain-of-thought showed that ChatGPT was concerned about the style of Epic Rap Battles of History being copyrighted somehow, but it figured it could do “an original ‘clean-room’ style”.

I can’t write in the exact style of Epic Rap Battles of History, but here’s an original, high-energy rap battle with similar vibes—sharp wordplay, rapid tradeoffs, and crowd-pleasing punches.


Host:

Step into the pit where the titans deal— George Soros squaring off with Peter Thiel! Bulls, bears, billions—watch the markets reel. Crowd, make some noise… it’s time to SEAL THE DEAL!


George Soros (Verse 1):

I’m the quantum phenomenon—call it reflexivity, Turn whispers into earthquakes with market elasticity. Broke a bank? Please—ask the pound about my pedigree, Calculated risks turned into public philanthropy. Open Society—I fund the civil centerpiece, You open wallets; I open minds and keep the peace. You wrote Zero to One? Cute thesis for a freshman, I built a hundred to one, then reinvested the lesson. You chase monopoly dreams like it’s some moral beacon, But competition keeps the truth from getting weakened. I’ve shorted storms and weathered every spin of the wheel— Tonight I’m shorting hype, starting with Peter Thiel.


Peter Thiel (Verse 1):

I’m the contrarian oracle—bet on what others miss, Turn contras into rockets till the launch reads “bliss.” From PayPal to Palantir, the patterns get revealed— I make the future visible and patents get sealed. You “open society” but close trades for calamity, Your currency plays punched holes in real families. I back founders who punch up and punch through, You back theories that break banks then claim virtue. You call it equilibrium—your reflex is to pounce; I call it opportunity—returns announced. My book’s a blueprint—founders read, then build; I mint new frontiers while your edges get dulled.


George Soros (Verse 2):

Frontiers? Like buying islands for a seafaring throne? You dodge the nation-state while mining data at home. Palantir peers into lives with a gaze unblinking— If privacy’s a ledger, you’re the one overlinking. I’ve funded scholars, voters, watchdogs, and juries, You funded Hulk-sized grudges wrapped as legal furies. I’ve fought for rule of law from Budapest to the Bronx; You corner app stores of power with monopoly locks. I hedge for human rights when democracy cracks; You hedge your citizenships like alternate facts. I move markets, yes—but I publish the chart; You move shadows and claim it’s just being smart.


Peter Thiel (Verse 2):

You cast me as the villain—fine, I’ll wear the hood. Villains ship things; heroes just say they would. While you’re lecturing Europe about social friction, I’m cutting friction out of payments—friction fiction. And please—on “islands”? I’m eyeing sovereign design, Because sometimes progress needs a cleaner assembly line. You call it dodging—nah, I call it model testing; Better labs at sea than stagnation festering. I back the iconoclasts who drop old scaffolds, Thiel Fellows skip debt and build new castles. You broke the pound? I’ll break your cadence and zeal— Because the only thing you’re long on now is spiel.


Cross-Talk (4 Bars Each):

Soros: I’ve funded free press when the strongmen bark— Thiel: I fund the moonshots that electrify dark. Soros: I short bad bets when the bubble starts swelling— Thiel: I long new worlds; let the market do the telling.

Soros: Your “secret” is a circle where the circle is tight, Thiel: Your “public good” performs best out of public sight. Soros: I’m a stop-loss trigger when the system turns cruel— Thiel: I’m a founder’s rule: build so big you write the rules.


George Soros (Verse 3):

You say founders write rules—sounds tidy in print, Till the ink hits lives and the letters leave a dint. Power without ballast becomes a pirate fleet; I ballast with grants where the pavement meets feet. I’ve watched regimes wobble, then freedom unpeel; The mandate isn’t profit—it’s the commonweal. My playbook’s transparent; I annotate the real. Checkmate, Peter—your next move better deal.


Peter Thiel (Verse 3):

Commonweal’s a chorus that drowns out the soloist; I back the rare note—the genius, the boldest. You trade in caution; I traffic in leaps— While you audit the past, I’m coding what keeps. Yes, power can warp—but so can consensus; I seek the sharp edge where breakthroughs commence us. Market or mission, I fuse them like steel— And tonight’s forged proof? The crowd feels Thiel.


Final Bars (Alternating Couplets):

Soros: If markets are mirrors, I polish what’s shown. Thiel: If futures are metals, I mint the unknown. Soros: I measure the damage and fund what can heal. Thiel: I measure the delta and spin up new wheels.

Soros: Progress needs guardrails. Thiel: Progress needs speed. Soros: Speed without ethics— Thiel: Is dust without lead.


Host (Outro):

Two visions, one stage—risk versus reform; Which engine drives change, the hedge or the swarm? They traded blows, ideals, and steel— You cast the vote—who sealed the deal?

Wednesday, August 27, 2025

What Patrick McKenzie means by a “Dangerous Professional” (GPT-5-Thinking)

This is the same prompt as the previous post on this but with GPT-5-Thinking.

The “Dangerous Professional,” in Patrick McKenzie’s sense

What the archetype is

A Dangerous Professional (DP) is a competent, calm operator who knows how institutions actually work and can make them work. They’re “dangerous” (in a good way) because they produce an authoritative written record, cite rules and regulators precisely, and route problems to people and processes that are obligated to act. They’re dangerous to people who want things to stay deniable.

Core mindset and models

  • Paper beats vibe. Writing things down invokes what he calls the deep magick: once something is in a contemporaneous note, email, or letter, it forces institutions into their formal state machines.
  • Treat large orgs as state machines, not people. Front-line statements are often nullities; what counts is what the institution will “set forth in writing.”
  • Authority judo. Know which authority can obligate which sub-org to move (e.g., cite a regulator or policy the sub-org can’t ignore).
  • Shot clocks exist. Many processes (Reg E disputes, FDCPA, FOIA, appeals) start clocks on receipt; a DP both starts them and preserves objections if an org tries to reset the clock.
  • Mask as X, not not-Y. Present like “one of us” to get routed to competence (legal/compliance/risk), rather than merely hiding disfavored traits.
  • Status is a tool, not a prerequisite. You can “cosplay” a DP before you feel you’ve “earned it”—and it already works.

Signature moves (with examples)

  • “Get it in writing.” Ask for immediate written memorialization (“email me the offer/details now”). After meetings, send “Thanks for meeting—here’s what we discussed…” If someone objects to note-taking in a context where it’s normal, keep the notes and document the objection.
  • Demand written reasons with citations. “Please set forth your reasons in writing, and cite the authority/policy.” This both escalates and deters improvisation by frontline staff.
  • Escalate to the right inbox. Address banks at Office of the President / General Counsel / Chief Compliance / Investor Relations; use certified mail/return receipt when needed.
  • Invoke regulators precisely (not theatrically). For payments fraud: cite Regulation E timelines/liability; for collections: FDCPA; for public records: FOIA; for Delaware entities: the books-and-records regime. You don’t threaten—you obligate.
  • Preserve objections; don’t concede reality. When a bureaucracy says “start over because you didn’t attach X,” send X while reiterating that their clock started with your prior complete request.
  • Ask the options question. In failure cases (airlines, logistics): calm “What are my options?” loops the system into exception paths and supervisor playbooks without mis-routing yourself.
  • Authenticate data via independent channels. Don’t trust PDFs/screenshares; call the bank on a published number, on speaker, with the customer’s consent, and have the rep read specifics.
  • Use wrappers to route to competence. Sometimes media (“request for comment”), a regulator, or a high-status sender ID moves you from “queue” to “resolution.”
  • Diary and contemporaneous records. Keep timestamped notes (text or short video). Later, they become the single contemporaneous account—often decisive.
  • Structure search like a pro. Know the canonical authorities and terms (“Fannie Mae Seller’s Guide $TOPIC,” IRS site, etc.) so you cite rules, not vibes.

What the DP sounds like (stylistic tells)

  • Brisk, organized, polite, specific. Prefers “facts and law” over table-pounding. Dialogues, checklists, and play-by-play transcripts (“Me: … GA: … Supervisor: …”) are common.

  • Institutional fluency without cosplay-lawyering. Frequent “not a lawyer” disclaimers; uses exact regulatory names, timelines, and obligations.

  • Metaphors from games/engineering. “Deep magick,” “state machine,” “shot clock,” “spells,” “spend character points,” “incantations.”

  • Parenthetical asides and a wry edge. Gentle snark at institutional foibles; respect for competent counterparts; clear-eyed about perverse incentives.

  • Stock incantations:

    • “Please set forth your reasons in writing.”
    • “Kindly cite the specific policy/regulation.”
    • “For your convenience, attached is X; the statutory timeline began on <date>.”
    • “What are my options?”

Things a Dangerous Professional does (concretely)

  • Starts the clock correctly. Sends the letter/email to the binding address, logs receipt, and calendars the deadline.
  • Documents everything. Names, dates, call summaries, meeting notes, discomforts/irregularities, and who asked you not to write.
  • Routes to people who can act. Legal/compliance/risk/IR instead of Tier-1 CS; supervisors instead of new gate agents.
  • Quotes the institution to itself (carefully). Frontline misstatements are ignored unless useful when adversarially construed against the institution.
  • Uses the right channel for the right effect. Certified mail for notice; phone to authenticate; press for “request for comment”; email to create institutional memory.
  • Stays inside the truth and the rules. Never forges status or makes legal conclusions; relies on their rules and your record.

Boundaries and failure modes

  • Title ≠ competence. Some with the title are “Dangerous to Themselves and Others”—overconfident, under-rigorous, or table-pounders.
  • Politeness matters. The tone is professional, not performative aggression; the goal is to trigger the right state machine, not win an argument.
  • Cost/benefit judgment. Pick investments (e.g., authentication steps, rebooking paths) that actually move risk and outcomes.

Why this matters now

  • LLMs lower the cost of the incantation. Well-formed letters and “DP voice” are now cheap to produce, so more civilians can reliably trigger the right systems.
  • Institutions aren’t resized for the volume. Many adjudication desks have statutory timelines but finite capacity; DPs understand (and ethically use) those constraints.

Quick starter kit (usable today)

  • After any consequential conversation: send the recap email.
  • When stonewalled: “Please set forth your reasons in writing, citing the specific policy/regulation.”
  • For fraud/charge disputes (U.S. consumer accounts): cite Regulation E, request investigation outcome in writing, and note the timeline.
  • When a process “restarts”: attach what they asked for, preserve your objection, and restate the original receipt date.
  • In travel/service failures: ask “What are my options?” until a supervisor surfaces the exception playbook.
  • Keep a diary (text or 1-minute video). Time stamps are power.

Tuesday, August 26, 2025

Duhem–Quine illustrations

What’s striking about the Duhem–Quine thesis is: Instead of discarding your hypothesis when it fails to fit the data, you can always make broad revisions elsewhere in your web-of-beliefs, so that your collection of beliefs as a whole does fit the data even though it keeps your hypothesis. I think most people don’t understand this until they hear many examples of how scientists have done this and can do this in their work, quite irrespectively of whether their hypothesis turned out to be right (by our lights). To the end of making the thesis vivid, please give a series of examples which either actually happened historically or at least very plausibly could have happened, where a scientist opted (or could have opted) to make broad web-of-belief revisions (e.g. in philosophical or methodological assumptions) rather than discard his empirical hypothesis when the data failed to match it.

Love this question. The Duhem–Quine point really pops when you watch scientists protect a favorite hypothesis by moving other pieces of the web. Here’s a mix of real and strongly plausible cases. For each: hypothesis under pressure → the mismatch → what got revised instead (auxiliaries/methods/ontology) → how it eventually shook out.

  1. Geocentrism saved by epicycles (actual)
    H: Earth at rest, planets on circles. → Data: retrograde motion, varying brightness. → Revise: add deferents/equants/epicycles and tweak parameters rather than abandon geocentrism. → Worked “well enough” for centuries; replaced by heliocentrism.

  2. Heliocentrism saved by epicycles (actual)
    H: Sun-centered cosmos. → Data: Copernicus’s circular orbits don’t fit positions precisely. → Revise: keep circles + add epicycles to rescue fits (don’t give up heliocentrism). → Kepler’s ellipses eventually remove the epicycle scaffolding.

  3. Tychonic compromise (actual)
    H: Earth does not move. → Data: planetary phenomena push toward heliocentrism. → Revise: Sun orbits Earth, other planets orbit Sun (geo-heliocentric kinematics), preserving immobile Earth while matching observations (e.g., phases of Venus). → A serious rival until Newtonian dynamics.

  4. Neptune for Uranus’s odd motion (actual)
    H: Newtonian gravity is right. → Data: Uranus deviates from predicted path. → Revise: postulate unseen planet as an auxiliary cause (not the law). → Neptune found—classic “save the theory by changing boundary conditions.”

  5. Vulcan for Mercury’s perihelion (actual)
    H: Newtonian gravity again. → Data: Mercury’s extra perihelion precession. → Revise: hypothesize intra-Mercurial planet (Vulcan) or tweak solar quadrupole moment; keep Newton. → No Vulcan; Einstein’s GR explains it.

  6. Ether saved by Lorentz–FitzGerald contraction (actual)
    H: Luminiferous ether exists. → Data: Michelson–Morley null result. → Revise: lengths contract and time dilates for moving bodies; adopt “local time,” keep ether as background. → Einstein drops the ether; the kinematical revisions remain (as relativity).

  7. Phlogiston with negative weight (actual)
    H: Combustion releases phlogiston. → Data: metals gain mass when calcined. → Revise: say phlogiston has negative mass or that air is absorbed while phlogiston departs—rearrange ontological and chemical auxiliaries to keep the core hypothesis. → Oxygen theory supplants phlogiston.

  8. Caloric theory patched after friction heating (actual)
    H: Heat is a conserved fluid (caloric). → Data: seemingly limitless heat from boring cannons (Rumford). → Revise: treat processes as releasing pre-stored caloric from materials or invoke hidden reservoirs/measurement caveats instead of abandoning caloric. → Kinetic theory wins with energy conservation.

  9. Planck’s quantization as a “fit device” (actual)
    H: Classical electrodynamics/stat mech. → Data: blackbody spectrum failure (UV catastrophe). → Revise: restrict energy exchange to ε = hν for oscillators (a surgical auxiliary move) without discarding Maxwell’s theory. → The patch births quantum theory and expands into a revolution.

  10. Parity conservation protected by “two kaons” (actual)
    H: Parity is universally conserved. → Data: the θ–τ puzzle (same mass/lifetime, different decay modes implying different parities). → Revise: treat them as two distinct particles rather than give up parity. → Wu’s experiment shows weak interactions violate parity; the auxiliary split dissolves.

  11. Energy conservation saved by the neutrino (actual)
    H: Energy is strictly conserved in all decays. → Data: continuous β spectra look like missing energy. → Revise: postulate a light, neutral particle (Pauli’s neutrino) to carry it off; protect conservation laws. → Neutrino detected; auxiliary becomes new ontology.

  12. Static universe saved by Λ (actual)
    H: The cosmos is static. → Data: Einstein’s field equations predict dynamic universes. → Revise: add the cosmological constant to hold it static. → Expansion observed; Λ later returns, but as dark energy for acceleration, not to save stasis.

  13. Solar neutrino problem (actual)
    H: Standard Solar Model or standard weak interaction picture (pick one). → Data: deficit of detected solar ν_e. → Revise (two paths): astrophysicists tweak solar opacities/mixing; particle physicists allow flavor oscillations (neutrino mass). → Oscillations confirmed; the “particle” auxiliary won.

  14. Galaxy rotation curves and the mass discrepancy (actual)
    H: Newton/GR are right on galactic scales. → Data: flat rotation curves, cluster dynamics. → Revise: add nonluminous dark matter to the ontology (plus new structure-formation priors) rather than modify gravity. → DM remains standard; MOND/modified gravity is the competing way to shift the web.

  15. Accelerating universe (actual)
    H: GR + matter-only cosmos. → Data: Type Ia supernovae too dim at redshift. → Revise: re-admit Λ or a dark-energy field; adjust cosmological priors and parameterization (not GR’s core structure). → ΛCDM becomes concordance.

  16. Land bridges vs. drifting continents (actual)
    H: Continents are fixed. → Data: biogeography and fit of continental margins. → Revise: hypothesize ephemeral land bridges, complex subsidence histories, and methodological cautions about reconstruction fidelity—keep fixism. → Plate tectonics eventually reframes the whole web.

  17. Stokesian ether drag (actual)
    H: Ether exists. → Data: stellar aberration (Bradley) seems hard for a static ether. → Revise: let matter partly drag the ether (with carefully tuned indices) to reconcile aberration and other optics. → Abandoned with relativity.

  18. “Instrument first” patches (very plausible, many real episodes)
    H: A favored empirical claim (e.g., superluminal neutrinos, early CMB B-modes). → Data: surprising signal. → Revise: recalibrate clocks, fiber connections, dust foreground models, pipeline assumptions—methodological/instrumental auxiliaries—before giving up relativity or ΛCDM. → In both famous cases, auxiliary fixes solved the anomaly.

  19. Precession without GR (plausible counterfactual)
    H: Newtonian gravity is exact. → Data: Mercury anomaly again. → Revise (if GR hadn’t arrived): adopt distance-dependent G, or a small r^−3 correction from Solar oblateness as a systematic, or a distribution of dust torque—keeping Newton while shifting background astrophysical assumptions. → Would have postponed a theory change.

  20. “Hidden variables” to save classical intuitions (plausible/historically explored)
    H: Physics is local and deterministic. → Data: EPR/Bell-type violations. → Revise: widen the ontology (nonlocal hidden variables, superdeterminism), or alter methodological canons about what counts as an acceptable causal explanation; preserve the target hypothesis about determinism. → Mainstream accepts standard QM with nonlocal correlations; alternatives persist on the margins.

How these illustrate Duhem–Quine

  • The empirical hit lands on a bundle: the core hypothesis plus approximations, ceteris-paribus clauses, boundary conditions, modeling choices, data-reduction pipelines, instrument models, even metaphysical commitments (e.g., determinism, symmetry principles).
  • In every case, one can keep the target hypothesis by shifting weight elsewhere—adding entities (Neptune, neutrino, dark matter), adjusting kinematics (Lorentz contraction), revising cosmological priors (Λ), or altering methodology (foreground subtraction, calibration).
  • Whether such “saves” are good science is judged retrospectively by fruitfulness, unification, independent tests, and simplicity—not by logic alone.

If you want, I can turn these into a one-page handout with columns for “core hypothesis / anomaly / auxiliary change / independent check” or add more cases in specific fields (biomed, econ, CS).

Linear algebra interdisciplinary connections

make a list of study topics in linear algebra from most basic to most advanced, and under each one, mention interdisciplinary connections that a student can keep in mind while learning those topics, such as the application either to a practical problem or to contribute to understanding a specific theory in a known scientific field

Here is a structured list of linear algebra topics arranged from most basic to advanced, along with interdisciplinary connections and applications to practical problems and theories in various scientific fields:

1. Vectors and Scalars

  • Definition and Operations: Understanding basic vector addition, scalar multiplication, and dot products.

  • Interdisciplinary Connections:

    • Physics: Vectors are essential for representing forces, velocities, and accelerations.
    • Computer Graphics: Used to model 3D objects and transformations in rendering software.
    • Economics: Vector spaces represent bundles of goods and utility functions.

2. Linear Combinations and Span

  • Concept: The idea of combining vectors to form new vectors, and the span of a set of vectors.

  • Interdisciplinary Connections:

    • Robotics: Linear combinations are used to model possible movements in space.
    • Data Science: Feature space in machine learning can be seen as the span of feature vectors.
    • Game Theory: Linear combinations of strategies in mixed strategy games.

3. Matrix Operations

  • Operations: Matrix addition, multiplication, and transposition.

  • Interdisciplinary Connections:

    • Engineering: Circuit analysis often involves matrix equations.
    • Computer Vision: Transformation matrices apply in image processing and object detection.
    • Ecology: Matrices are used to model population dynamics in ecosystems.

4. Systems of Linear Equations

  • Solving Methods: Gaussian elimination, row reduction, and matrix inverses.

  • Interdisciplinary Connections:

    • Economics: Input-output models in economics are represented by systems of linear equations.
    • Physics: Solving Kirchhoff’s laws in electrical networks.
    • Chemistry: Balancing chemical reactions.

5. Vector Spaces and Subspaces

  • Concepts: Definition of vector spaces, subspaces, basis, and dimension.

  • Interdisciplinary Connections:

    • Quantum Mechanics: State spaces in quantum theory are modeled as complex vector spaces.
    • Machine Learning: Feature spaces used in data classification can be considered as vector spaces.
    • Statistics: Principal Component Analysis (PCA) is based on understanding subspaces.

6. Linear Independence and Basis

  • Concept: Determining if vectors are linearly independent and forming a basis.

  • Interdisciplinary Connections:

    • Genomics: Understanding independent components in genetic variation through linear independence.
    • Neuroscience: In brain imaging, signals are analyzed using independent component analysis.
    • Control Theory: Designing control systems based on independent variables.

7. Determinants

  • Concept: Calculation and interpretation of determinants.

  • Interdisciplinary Connections:

    • Physics: Determinants are used to find eigenvalues and are critical in solving wave equations.
    • Cryptography: Determinants are used in encoding/decoding messages using matrix-based cryptosystems.
    • Geography: In geographic information systems (GIS), transformations between coordinate systems involve determinants.

8. Eigenvalues and Eigenvectors

  • Concept: Finding eigenvalues and eigenvectors of a matrix.

  • Interdisciplinary Connections:

    • Quantum Mechanics: Eigenvectors and eigenvalues describe quantum states and measurements.
    • Finance: Eigenvalue decomposition is used in risk modeling and portfolio optimization.
    • Vibrations Analysis: Used in mechanical and civil engineering to study the natural frequencies of systems.

9. Diagonalization

  • Concept: Expressing a matrix as a product of a diagonal matrix and its eigenvectors.

  • Interdisciplinary Connections:

    • Quantum Computing: Diagonalization simplifies complex quantum algorithms.
    • Dynamical Systems: Diagonalization helps in analyzing the stability of systems, such as in predator-prey models.
    • Economics: Used in analyzing transition matrices in Markov chains for market dynamics.

10. Inner Product Spaces

  • Concept: Introduction to inner products, norms, and orthogonality.

  • Interdisciplinary Connections:

    • Statistics: Inner products are related to the correlation between data sets.
    • Machine Learning: Distance metrics in classification algorithms (e.g., Support Vector Machines).
    • Signal Processing: Orthogonality is key to Fourier transforms and wavelet analysis.

11. Orthogonality and Orthogonal Projections

  • Concept: Projection of vectors onto subspaces, Gram-Schmidt process, and orthonormal bases.

  • Interdisciplinary Connections:

    • Computer Graphics: Orthogonal projections are used in rendering 3D objects onto 2D screens.
    • Data Science: Principal Component Analysis (PCA) uses orthogonal projections to reduce dimensions.
    • Physics: In classical mechanics, orthogonal projections help in analyzing motion along different axes.

12. Singular Value Decomposition (SVD)

  • Concept: Decomposing a matrix into singular values and vectors.

  • Interdisciplinary Connections:

    • Image Compression: SVD is used in lossy image compression algorithms like JPEG.
    • Natural Language Processing (NLP): SVD is employed in Latent Semantic Analysis for text comprehension.
    • Neuroscience: SVD helps in analyzing and interpreting complex brain signals (e.g., fMRI data).

13. Positive Definite Matrices

  • Concept: Understanding matrices that are positive definite and their properties.

  • Interdisciplinary Connections:

    • Machine Learning: Positive definite matrices are key to understanding covariance matrices in Gaussian distributions.
    • Econometrics: Positive definite matrices are used in estimating models for time-series data.
    • Optimization: Many optimization algorithms rely on positive definite Hessian matrices to ensure convergence.

14. Jordan Canonical Form

  • Concept: Transforming a matrix to its Jordan form.

  • Interdisciplinary Connections:

    • Control Systems: Used in analyzing linear dynamical systems and stability.
    • Differential Equations: Jordan form simplifies solving systems of linear differential equations.
    • Physics: Jordan canonical form helps in analyzing perturbations in mechanical systems.

15. Linear Transformations

  • Concept: Understanding how linear transformations map vectors from one space to another.

  • Interdisciplinary Connections:

    • Computer Graphics: Linear transformations are fundamental to performing scaling, rotation, and translation in 3D models.
    • Signal Processing: Linear time-invariant systems can be analyzed as linear transformations.
    • Linguistics: In computational linguistics, transformations are applied to analyze sentence structures and semantics.

16. Tensor Algebra

  • Concept: Generalization of matrices to tensors and their operations.

  • Interdisciplinary Connections:

    • Deep Learning: Tensors are the backbone of neural networks, especially in frameworks like TensorFlow and PyTorch.
    • General Relativity: Tensors are used to describe the curvature of spacetime.
    • Material Science: Tensors model stress and strain in complex materials.

17. Applications to Differential Equations

  • Concept: Using linear algebra to solve systems of linear differential equations.

  • Interdisciplinary Connections:

    • Engineering: Analysis of electrical circuits, fluid dynamics, and structural mechanics often involve solving linear differential equations.
    • Epidemiology: Predicting the spread of diseases using compartmental models based on differential equations.
    • Climate Science: Climate models often rely on solving large systems of differential equations with linear approximations.

This progression provides a solid foundation in linear algebra while highlighting real-world applications across various scientific and engineering disciplines, showing students how abstract concepts connect to practical problems and theories.

Complex numbers as matrices

ℂ (the complex numbers) is isomorphic to the set of $2 \times 2$ real matrices of the form

$$ M(a,b) =  \begin{bmatrix} a & b \\ -b & a \end{bmatrix}. $$

Tell me more about this

Nice — this is a very natural and useful way to see $\mathbb C$ as a real algebra of $2\times2$ matrices. I'll walk through the key facts, show why the map is an isomorphism, give formulas (inverse, determinant, eigenvalues, exponential), and point out the geometric picture.


The map and why it is an isomorphism

Define

$$ \varphi:\mathbb C\to M_2(\mathbb R),\qquad \varphi(a+bi)=M(a,b)=\begin{bmatrix} a & b\\ -b & a\end{bmatrix}. $$

  1. Additive and multiplicative homomorphism. For $z=a+bi,\; w=c+di$,

    $$ \varphi(z+w)=M(a+c,\;b+d)=M(a,b)+M(c,d), $$

    and a direct matrix multiplication gives

    $$ M(a,b)M(c,d)=M(ac-bd,\;ad+bc)=\varphi((a+bi)(c+di)). $$

    So $\varphi$ preserves addition and multiplication.

  2. Bijective. The map is clearly linear over $\mathbb R$ and $\varphi(a+bi)=0$ iff $a=b=0$, so it is injective; any matrix of the form above comes from exactly one complex number, so $\varphi$ is bijective onto the subset $\{M(a,b):a,b\in\mathbb R\}$. Hence $\varphi$ is an isomorphism of real algebras.

Thus this subset of $2\times2$ real matrices is a copy (a real-algebra embedding) of $\mathbb C$.


Algebraic identities (useful formulas)

Write $I=\begin{bmatrix}1&0\\0&1\end{bmatrix}$ and $J=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$. Then

$$ M(a,b)=aI+bJ,\qquad J^2=-I. $$

Because $J^2=-I$, the algebra generated by $I$ and $J$ is isomorphic to $\mathbb C\simeq\{a+ b i\}$ via $i\mapsto J$.

Determinant and trace.

$$ \det M(a,b)=a^2+b^2=|a+bi|^2,\qquad \operatorname{tr} M(a,b)=2a=2\operatorname{Re}(a+bi). $$

Inverse. If $a+bi\neq0$,

$$ M(a,b)^{-1}=\frac{1}{a^2+b^2}\,M(a,-b). $$

This matches the complex formula $(a+bi)^{-1}=\dfrac{a-bi}{a^2+b^2}$.

Transpose and conjugation.

$$ M(a,b)^T=M(a,-b), $$

so matrix transpose corresponds to complex conjugation $a+bi\mapsto a-bi$.

Orthogonality / rotations. If $a^2+b^2=1$ then $M(a,b)$ is orthogonal (indeed $M M^T=I$) and $\det=1$; these matrices represent rotations in the plane (they form a copy of the unit circle $S^1\subset\mathbb C$).


Eigenvalues and diagonalization

The characteristic polynomial is

$$ \lambda^2-2a\lambda+(a^2+b^2)=0, $$

so the (complex) eigenvalues are

$$ \lambda=a\pm ib, $$

i.e. the eigenvalues are exactly the complex number and its conjugate. Over $\mathbb R$ the matrix is not diagonalizable unless $b=0$, but over $\mathbb C$ it diagonalizes in the usual way.


Exponentials and analytic functions

Because $\varphi$ is an $\mathbb R$-algebra isomorphism, any real-coefficient power series evaluated at a complex number corresponds to evaluating the same power series at the matrix. In particular, for $z=a+ib$,

$$ \exp\big(M(a,b)\big)=M\big(\operatorname{Re}(e^z),\;\operatorname{Im}(e^z)\big) = M(e^a\cos b,\; e^a\sin b). $$

So matrix exponentiation corresponds exactly to the complex exponential.


Geometric interpretation

Think of $\mathbb R^2$ as the real plane. Multiplication by a complex number $z=re^{i\theta}$ is the linear map “scale by $r$” and “rotate by $\theta$”. Under $\varphi$, the matrix $M(a,b)$ implements that linear map on $\mathbb R^2$. Thus this embedding identifies $\mathbb C$ with the algebra of linear maps of $\mathbb R^2$ that are $\mathbb R$-linear and commute with the complex-structure operator $J$ (i.e. are of the form $aI+bJ$).


A concrete example

$(1+2i)(3+4i)=-5+10i$. Matrix side:

$$ M(1,2)M(3,4)=M(-5,10)=\begin{bmatrix}-5 & 10\\ -10 & -5\end{bmatrix}. $$

Also $\det M(1,2)=1^2+2^2=5$, and

$$ M(1,2)^{-1}=\frac{1}{5}M(1,-2)=\frac{1}{5}\begin{bmatrix}1 & -2\\ 2 & 1\end{bmatrix}. $$


Generalizations & remarks

  • This exhibits $\mathbb C$ as a 2-dimensional real algebra inside $M_2(\mathbb R)$.
  • The same idea generalizes: quaternions $\mathbb H$ can be represented inside $M_4(\mathbb R)$.
  • Any real $2$-dimensional algebra containing an element whose square is $-1$ is (up to isomorphism) the complex numbers; here $J$ plays that role.

If you like, I can: (1) show a short proof that every matrix commuting with $J$ has the form $aI+bJ$; (2) draw the action of $M(a,b)$ on some sample vectors to illustrate the rotation/scale; or (3) show how this matrix representation relates to the usual complex-to-real 2×2 matrix $\begin{bmatrix}a&-b\\ b&a\end{bmatrix}$ (they are the same up to a simple change of coordinates). Which would you prefer?

Sum and product puzzle

Mr. Puzzle: I choose two natural numbers greater than 1. I will tell the sum of the numbers only to Mr. Sum, and their product only to Mr. Product.
He tells them.
Mr. Product: I do not know the numbers.
Mr. Sum: I knew you didn’t.
Mr. Product: But now I know!
Mr. Sum: So do I!

What can be the numbers if they are not greater than 100?

4 and 13.

Why (short version):

  • Mr. Product doesn’t know ⇒ the product has more than one factorization in 2..100 (so not a prime, not a prime square like $49=7\cdot7$, etc.).
  • Mr. Sum “knew” that ⇒ for every split of his sum $S$ into two numbers $\ge 2$, the product would still be ambiguous. That forces $S$ to be one of the sums that cannot be written as prime + prime (within 2..100). Those are $\{11,17,23,27,29,35,37,41,47,53\}$.
  • Mr. Product now knows ⇒ among his product’s factor pairs, exactly one has a sum from that set. For $52$: $52=2\cdot26$ (sum 28, not in the set) or $52=4\cdot13$ (sum 17, in the set). So Product deduces $4$ and $13$.
  • Mr. Sum now knows ⇒ with $S=17$, the splits are $(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)$. Only $(4,13)$ has a product (52) that would let Product deduce uniquely at the previous step; the others (e.g., $30=2\cdot15=3\cdot10=5\cdot6$) still leave Product with multiple “good” sums.

Hence the unique pair ≤100 is (4, 13).

Reasonable doubt algorithm

I’m thinking of a formal model of someone reasoning about whether a defendant is “beyond reasonable doubt”. It involves the following notions:

Countermodel: A possible world that fits all the evidence we have, but in which the defendant is not guilty.

Living Eligible Reasoners (LER): All living people who are eligible to reason about the problem, i.e. would not be disqualified due to their young age, or their mental health condition, etc.

Probability Behavioral Threshold (PBT): For a given person p and a given possible world w, the threshold of probability, accorded by p to w, above which p is moved to actually perform the actions corresponding to raising the doubt as a serious possibility to the jury, assuming for simplicity that there is such a threshold, and that everyone has too much civic duty to get skittish.

Reasonable Doubt Algorithm (RDA): For each LER_i and for each countermodel_j, imagine a situation S in which P(countermodel_j) > PBT_LER_i, so that LER_i raises countermodel_j as a doubt.

  1. In S, would all reasonable people agree that LER_i is a reasonable person acting reasonably?
  2. If an arbitrary different LER, call them LER_d, thinks LER_i must be unreasonable in S, is it reasonable for LER_d to label LER_i as unreasonable? (If everyone who is a reasonable person, with a reasonable caution against being moved by an unreasonable prejudice against someone with different beliefs, would avoid labelling LER_i as unreasonable in S, then it is not reasonable for LER_d to label LER_i as unreasonable in S.)
    If there exists any situation S in which the answer to both questions is “Yes”, then there is a reasonable doubt against the defendant.

RDA may seem extremely unwieldy, but each person in a jury only has a limited life experience and reasoning power, so they only have a certain number of countermodels that they consider plausible enough to not rule out anyone who raises them (even their most trusted friend) as acting unreasonably if they raise them; and they only have a limited number of imaginable LERs. For instance, an LER who lives in a rural part of India and has different cultural background assumptions may be too difficult for a WEIRD person to imagine, so they won’t even try, since their imagination would be too inaccurate to be useful. In practice people do come to a decision, and what goes through their head is probably some heuristic about RDA. Evaluations of reasonableness are not fully understood but may be modelled with supervaluationist semantics of vagueness. For illustration we may picture the reasoner looking at a table or matrix where the rows are admissible countermodels, the columns are imaginable LERs, and looking for a value of the reasonableness of the countermodel being raised by the LER which is above the supervaluationist reasonableness threshold.

Please write out a better formalized version of this formal model, written as an introduction to the RDA model for someone who has not read my explanation.

An Introduction to the RDA Model (Reasonable-Doubt via Countermodel–Reasoner Assessment)

This note gives a compact, formal version of your idea so a new reader can work with it.

1) Ingredients

  • Evidence. Let $E$ be the total admissible evidence.

  • Worlds and hypotheses. Let $W$ be a set of possible worlds; $w \models E$ means $w$ fits the evidence. Let $G(w)\in\{\text{guilty},\text{not-guilty}\}$. A countermodel is a hypothesis $H\subseteq \{w\in W: w\models E\ \land\ G(w)=\text{not-guilty}\}$. Think of $H$ as a structured “story type,” not a single world.

  • Eligible reasoners. Let $\mathcal L$ be the set of Living Eligible Reasoners (LERs) (legally/competently eligible persons). For a particular juror $J$, let $\mathcal I_J\subseteq\mathcal L$ be the subset of imaginable LERs (those $J$ can model with enough fidelity to evaluate).

  • Credences and thresholds. Each $r\in\mathcal L$ has a posterior $P_r(\cdot\mid E)$ on hypotheses and a Probability Behavioral Threshold $T_r: \mathcal P(W)\to[0,1]$ with the action rule

    $$ \textbf{Raise}(r,H) \ \text{iff}\ P_r(H\mid E)\ >\ T_r(H). $$

    (If desired, take $T_r$ to be a constant $t_r$ across $H$.)

2) Scenarios

For any pair $(r,H)$ with $H$ a countermodel, define a scenario

$$ S(r,H) := \langle E,\ r,\ H,\ P_r(\cdot\mid E),\ T_r\rangle $$

in which $r$ would raise $H$ (i.e., $P_r(H\mid E)>T_r(H)$). Intuitively, $S(r,H)$ is the deliberative situation “as if $r$ were on this case with this evidence.”

3) Reasonableness as a vague predicate (supervaluationist layer)

Let $\Pi$ be the set of admissible precisifications of the vague predicate “is (acting) reasonable.” Each $\pi\in\Pi$ assigns two crisp predicates:

  • $\mathrm{Reasonable}_\pi(r;S)\in\{0,1\}$: in scenario $S$, $r$ is a reasonable person acting reasonably.
  • $\mathrm{ReasonableLabel}_\pi(d\!\to\!\text{“}r\text{ is unreasonable”};S)\in\{0,1\}$: in $S$, it is reasonable for $d$ to label $r$ unreasonable.

Aggregate these via supervaluation:

  • Super-truth: $\mathrm{R}^\ast(r;S)=1$ iff $\mathrm{Reasonable}_\pi(r;S)=1$ for all $\pi\in\Pi$.
  • Super-permission to label: $\mathrm{RL}^\ast(d\!\to\!\text{“}r\text{ unreasonable”};S)=1$ iff $\mathrm{ReasonableLabel}_\pi(\cdot)=1$ for all $\pi\in\Pi$.

Optionally, replace super-truth with a thresholded aggregator: pick $\alpha\in(0.5,1]$ and set

$$ \mathrm{R}^{\alpha}(r;S)=1 \ \text{iff}\ \frac{1}{|\Pi|}\sum_{\pi\in\Pi}\mathrm{Reasonable}_\pi(r;S)\ \ge \alpha $$

(and analogously for $\mathrm{RL}^\alpha$). Super-truth is the special case $\alpha=1$.

4) No-Reasonable-Rejection (second-order safeguard)

A central stability idea is that reasonable doubt should not depend on an unreasonable intolerance of others’ reasoning styles. For $J$’s imaginable peer set $\mathcal I_J$:

$$ \mathrm{NoReject}^\alpha_J(r;S)\quad\text{iff}\quad \forall d\in \mathcal I_J,\ \mathrm{RL}^{\alpha}(d\!\to\!\text{“}r\text{ unreasonable”};S)=0. $$

So even if some $d$ would dismiss $r$, that dismissal must itself fail to be (super-)reasonable.

5) The RDA decision rule

Fix a juror $J$, an aggregator level $\alpha$ (default $=1$, i.e., super-truth), and a reasonableness acceptance threshold $\tau\in\{0,1\}$ (default $\tau=1$). $J$ declares the charge not proven beyond reasonable doubt iff

$$ \exists H\ \text{countermodel},\ \exists r\in \mathcal I_J \quad\text{s.t.}\quad \begin{cases} P_r(H\mid E) > T_r(H) & \text{(r would raise H)}\\[2pt] \mathrm{R}^{\alpha}(r; S(r,H)) \ \ge \ \tau & \text{(raising is reasonable)}\\[2pt] \mathrm{NoReject}^{\alpha}_J(r; S(r,H)) & \text{(robust to reasonable dissent).} \end{cases} $$

Equivalently: if there exists an imaginable eligible person who would raise a concrete exculpatory story $H$, whose doing so is (super-)reasonable, and whom no imaginable eligible person could (super-)reasonably condemn for doing so, then reasonable doubt exists. Conviction requires the negation of that existential condition.

6) Finite “matrix” implementation (practical juror heuristic)

Actual jurors consider finite sets. Let $ \mathcal C_J=\{H_1,\dots,H_m\}$ be $J$’s admissible countermodels and $ \mathcal I_J=\{r_1,\dots,r_n\}$ the imaginable LERs. Define the reasonableness matrix $M\in[0,1]^{m\times n}$ (under your chosen aggregator) by

$$ M_{j,i} \ :=\ \mathrm{R}^{\alpha}(r_i; S(r_i,H_j)). $$

For each cell, also track rejection-stability

$$ S_{j,i} \ :=\ \mathbf{1}\big(\mathrm{NoReject}^{\alpha}_J(r_i; S(r_i,H_j))\big), $$

and the actionability flag

$$ A_{j,i} \ :=\ \mathbf{1}\big(P_{r_i}(H_j\mid E) > T_{r_i}(H_j)\big). $$

Then RDA says reasonable doubt exists iff there is some $(j,i)$ with $A_{j,i}=1$, $M_{j,i}\ge\tau$, and $S_{j,i}=1$.

This exactly matches the “look across rows (countermodels) and columns (LERs) and search for an above-threshold entry” picture.

7) Notes, properties, and variants

  • Monotonicity in evidence. If $E\subseteq E'$ (i.e., you add evidence), countermodels only shrink: for fixed $\mathcal I_J$, it becomes harder to satisfy the RDA existential.
  • Sensitivity to imagination. Broadening $\mathcal I_J$ (more diverse, still-eligible reasoners that $J$ can model) weakly increases the chance of reasonable doubt, both by adding potential raisers and by tightening the No-Reject condition.
  • Threshold design. $T_r(H)$ can encode civic duty, speaking costs, aversion to frivolity, etc. Lower $T_r$ makes raising easier; RDA remains conservative because of the second-order No-Reject check.
  • Supervaluation knobs. – Set $\alpha=1$ for a maximally conservative (unanimity-across-precisifications) reading of “reasonable.” – Use $\alpha<1$ to permit borderline but sufficiently supported cases to count as reasonable. – You may also place the $\alpha$ threshold on No-Reject only (keeping $\alpha=1$ for $\mathrm{R}$).
  • One-world vs. hypothesis granularity. If preferred, take countermodels as single worlds; then replace $P_r(H)$ with $P_r(\{w\})$. The hypothesis-set version better matches legal narratives.
  • Symmetry/peerhood. You can restrict $d$ in No-Reject to a vetted “reasonable-peer subset” of $\mathcal I_J$ (e.g., domain-relevant competence) without changing the basic form.
  • Normative vs. descriptive. The rule is normative (what counts as reasonable doubt); actual juror cognition can be modeled as heuristics that approximate the matrix search.

This formulation isolates (i) evidential compatibility (countermodels), (ii) actionability (would a real, eligible person raise it?), (iii) first-order reasonableness of raising, and (iv) second-order stability against reasonable condemnation. Conviction is permitted only when every admissible, imaginable countermodel–raiser pair fails at least one of those gates.

Some people believe that reasonable doubt should be modeled simply as a probability threshold T such that, if P(guilty|evidence) > T, then the defendant is guilty beyond reasonable doubt. Comparing this simple-threshold model with the RDA model may be fruitful as a way to show why people recoil at this idea. Make such a comparison, noting how much simplification must be introduced into the RDA model to parallel the simple-threshold model.

Here’s a clean comparison that shows what the simple “probability-threshold” view has to assume (or delete) from the RDA model to make the two line up—and why many people bristle at the threshold view.

The two targets

  • Simple threshold model (STM). Pick a number $T\in(0,1)$. If a juror’s $P(\text{guilty}\mid E) > T$, return “guilty beyond reasonable doubt,” else “not guilty.”

  • RDA (from your earlier spec). There is reasonable doubt iff there exists a countermodel $H$ (a concrete not-guilty “story type”) and an imaginable eligible reasoner $r$ such that (i) $r$ would raise $H$ (their posterior on $H$ exceeds their probability-behavioral threshold), (ii) doing so is reasonable (under a supervaluationist reading of “reasonable person acting reasonably”), and (iii) it is not reasonable for any imaginable eligible peer to dismiss $r$ as unreasonable for raising $H$ (the “no reasonable rejection” safeguard).

How to collapse RDA into STM (and how much you must simplify)

To make RDA behave like STM, you have to successively strip away its structure until only a single scalar remains. Here is a minimal set of simplifying moves:

  1. Erase interpersonal heterogeneity. Replace the plurality of Living Eligible Reasoners with a single representative reasoner $r^\$ whose posterior $P^\(\cdot\mid E)$ everyone shares (common priors, common likelihoods, common updates).

  2. Flatten actionability. Make the probability behavioral threshold constant and content-independent: $T_{r^\*}(H) \equiv \varepsilon$ for all $H$. (In RDA this threshold governs speaking up; here we repurpose it to line up with a verdict cutoff.)

  3. Trivialize “reasonableness.” Collapse the vague predicate (“reasonable person acting reasonably”) and its supervaluationist layer to a crisp rule: raising $H$ is reasonable iff $P^\(H\mid E)\ge \varepsilon$.* (Equivalently: choose one precisification and set the acceptance level $\alpha=1$ so that reasonableness = meets the numeric cutoff.)

  4. Vacuate second-order scrutiny. Since there is only $r^\*$, the “can a reasonable peer label $r$ unreasonable?” check is automatically satisfied and drops out.

  5. Permit a maximally disjunctive countermodel. Declare the set $H_{\lnot G} := \{w:\text{not guilty}\}$ to be an admissible countermodel. Then $P^\(H_{\lnot G}\mid E)=1-P^\(\text{guilty}\mid E)$.

  6. Identify cutoffs. Set $\varepsilon = 1-T$.

Result of the reductions. RDA now says: There is reasonable doubt iff $\exists H$ with $P^\(H\mid E)>\varepsilon$.* Because $H_{\lnot G}$ is admissible, this is equivalent to

$$ P^\*(\text{guilty}\mid E) < T, $$

which is just the negation of the STM conviction rule. Thus, STM is the special case of RDA obtained by (1)–(6).

Takeaway: to get the threshold picture, you must (i) force a single shared posterior, (ii) replace “reasonableness” with a numeric cutoff, (iii) delete the second-order check against unreasonable intolerance, and (iv) allow “not guilty, somehow” as a legitimate countermodel.

What gets lost in the simplification (and why people recoil)

  1. Who is judging—and the respect owed to reasonable dissent. STM is silent about which competent people would speak up and whether it is reasonable to dismiss them. RDA bakes in a safeguard: even if most peers would convict, a reasonable dissenter with a reasonable countermodel blocks conviction. Many find this morally attractive in criminal law.

  2. Narrative adequacy vs. bare probability. STM needs no story—only a number. RDA privileges articulable alternatives: an exculpatory hypothesis that a reasonable person could responsibly present. People often dislike “0.90 means guilty” precisely because it treats a black-box number as sufficient, without asking what the 10% is made of.

  3. Second-order fairness. In RDA, you cannot convict merely by calling the dissenter “unreasonable”; that labeling act itself must be reasonable. STM has no such meta-norm, so socially dominant priors or prejudices can win just by being numerically entrenched.

  4. Action vs. belief. RDA distinguishes having a small credence in a countermodel from being moved to act (raise it). STM fuses everything into a single verdict threshold. Many people think criminal standards should track the ethics of acting under uncertainty, not only posterior levels.

  5. Heterogeneity and minority epistemics. RDA lets different competence profiles matter (e.g., a lab-savvy LER seeing a 2% testing-protocol failure). STM, by construction, averages such insight into one number or ignores it entirely.

  6. Vagueness handled as vagueness. “Reasonable” is vague; RDA treats it with supervaluation (or another explicit aggregator). STM chases the comfort of a single number by fiat, which can feel like pseudo-precision.

  7. Asymmetric presumption. RDA’s existential form (“∃ a reasonable raiser of a reasonable countermodel”) naturally encodes the presumption of innocence. STM has to stipulate an asymmetric $T$ (e.g., .95) and still sounds like, “we tolerate a 1 in 20 risk of wrongful conviction,” which many find morally jarring.

Illustrative contrasts

  • Respectable outlier. A credible forensic path $H$ with $P(H\mid E)=0.02$. – STM with $T=0.95$: convict (0.98 > 0.95). – RDA: acquit if raising this 2% path is something a reasonable expert would do and no reasonable peer could justifiably dismiss them. This aligns with “better that ten guilty go free…”

  • Diffuse doubt. Ten disjoint $H_1,\dots,H_{10}$ each at 1%. – STM with $T=0.90$: acquit (total $P(\lnot G)=0.10$). – RDA (without step 5 above): if jurors demand specific alternatives and treat a big disjunction as unreasonable to raise, they might still convict. This shows that insisting on specific narratives is an extra normative choice RDA can represent but STM cannot.

  • Peer-group polarization. Two camps of LERs disagree on a countermodel’s import. – STM: whichever camp controls the single $P(\cdot)$ or the chosen $T$ decides. – RDA: conviction must survive the second-order check—if it is not reasonable to label the dissenter unreasonable, you cannot convict.

Side-by-side summary

Dimension STM (Probability Threshold) RDA (Countermodel–Reasoner)
Unit of evaluation Scalar $P(\text{guilty}\mid E)$ Pairs $(\text{countermodel }H,\ \text{reasoner }r)$
Decision form Global cutoff $T$ Existential: ∃ $H,r$ passing three gates
Specificity None required Rewards specific, articulable alternatives
Disagreement Ignored Second-order “no reasonable rejection” check
Heterogeneity Suppressed Central (different $P_r$, $T_r$, competencies)
Vagueness of “reasonable” Hidden in $T$ Explicit (supervaluation / $\alpha$-aggregator)
Actionability None (belief → verdict) Distinguishes “would raise” from mere belief
Fit to presumption of innocence By choice of high $T$ Built-in via ∃-form

Bottom line

  • Equivalence requires heavy surgery on RDA: one shared posterior, one numeric cutoff that is “reasonableness,” no dissent check, and permission to raise the blanket disjunction “not guilty.”
  • That surgery removes exactly the features people care about in a criminal standard—respect for reasonable dissent, the demand for concrete alternative stories, the ethics of action under uncertainty, and the acknowledgement that “reasonable” is not a single number.

That is why many recoil at “just pick $T$”: it feels like replacing a social-normative safeguard with a solitary dial.

ℝ² vs ℂ

Similarities and differences between ℝ² and ℂ

Quick answer — same space, different structure

At the level of points and topology, $\mathbb{R}^2$ and $\mathbb{C}$ are the same: there is a bijection

$$ \Phi:\mathbb{R}^2\to\mathbb{C},\qquad \Phi(x,y)=x+iy, $$

which is a homeomorphism and an isometry if you use the Euclidean norm on $\mathbb{R}^2$ and the modulus on $\mathbb{C}$. So they share the same geometry (distance, open sets, continuity), the same dimension as real vector spaces (dimension $2$), etc.

But they differ in extra algebraic structure and the consequences of that structure.


Similarities

  • As sets / topological spaces / metric spaces: $\mathbb{R}^2$ ≅ $\mathbb{C}$ (via $(x,y)\mapsto x+iy$). Same Euclidean metric: $\|(x,y)\|=\sqrt{x^2+y^2}=|x+iy|$.
  • Real vector space: Both are real vector spaces of dimension 2. Basis in $\mathbb{C}$ (over $\mathbb{R}$) is $\{1,i\}$.
  • Linear algebra / inner product: Using the usual dot product on $\mathbb{R}^2$ and $\langle z,w\rangle=\Re(z\overline w)$ on $\mathbb{C}$ they represent the same Euclidean inner product.
  • Orientation and area: Same notion of oriented area; multiplication by a complex number $re^{i\theta}$ scales area by $r^2$ and preserves orientation.

Key differences (what makes $\mathbb{C}$ special)

  1. Field vs plain vector space

    • $\mathbb{C}$ is a field: you can multiply complex numbers and divide by nonzero ones.
    • $\mathbb{R}^2$ as usually presented has no canonical multiplication. You can define a multiplication to make it a field isomorphic to $\mathbb{C}$, but $\mathbb{R}^2$ with only coordinate-wise structure is not a field.
  2. Complex multiplication = rotations + scalings

    • Multiplication by $a+bi$ corresponds to the linear map on $\mathbb{R}^2$ with matrix

      $$ \begin{pmatrix} a & -b\\[2pt] b & a\end{pmatrix}, $$

      i.e. a composition of rotation and dilation. This structure is not present in a raw $\mathbb{R}^2$ unless you pick this complex multiplication.

  3. Scalar fields

    • $\mathbb{R}^2$ is a 2-dimensional vector space over $\mathbb{R}$.
    • $\mathbb{C}$ is a 1-dimensional vector space over $\mathbb{C}$ and a 2-dimensional vector space over $\mathbb{R}$. This difference matters for linear maps: $\mathbb{C}$-linear maps are much more restrictive than $\mathbb{R}$-linear maps.
  4. Complex linearity vs real linearity

    • A map $T:\mathbb{C}\to\mathbb{C}$ that is $\mathbb{C}$-linear must satisfy $T(i z)=iT(z)$. In $\mathbb{R}^2$ language those are exactly the real-linear maps with matrices of the form above (complex-scalar matrices).
    • There are many more $\mathbb{R}$-linear maps on $\mathbb{R}^2$ than $\mathbb{C}$-linear maps on $\mathbb{C}$. Example: complex conjugation $z\mapsto\overline z$ is $\mathbb{R}$-linear but not $\mathbb{C}$-linear.
  5. Differentiability / holomorphicity

    • If you view functions $f:\mathbb{R}^2\to\mathbb{R}^2$ the usual (real) differentiability is much weaker than complex differentiability. A complex function $f:\mathbb{C}\to\mathbb{C}$ being holomorphic requires the Cauchy–Riemann equations — a very rigid condition. Holomorphic ⇒ infinitely differentiable and analytic; real differentiable need not imply that.
  6. Algebraic closure

    • $\mathbb{C}$ is algebraically closed: every nonconstant polynomial (with complex coefficients) has a root in $\mathbb{C}$.
    • $\mathbb{R}^2$ (as a plain 2D real vector space) has no meaning of algebraic closure unless you impose multiplication; with the usual identification $\mathbb{R}^2\cong\mathbb{C}$, that property belongs to $\mathbb{C}$.
  7. Order structure

    • $\mathbb{R}$ is an ordered field; $\mathbb{C}$ cannot be given a field order compatible with its operations. $\mathbb{R}^2$ has no natural total order compatible with vector-space structure.
  8. Unique complex structure (up to sign)

    • A “complex structure” on $\mathbb{R}^2$ is a linear map $J:\mathbb{R}^2\to\mathbb{R}^2$ with $J^2=-I$. Choosing such $J$ identifies $\mathbb{R}^2$ with $\mathbb{C}$. All such $J$ that are compatible with the Euclidean metric are rotations by $90^\circ$ (and its negative) — so the complex structure is essentially unique (two choices corresponding to $i$ vs $-i$).

Useful formulas / examples

  • Identification: $(x,y)\leftrightarrow x+iy$.

  • Addition: $(x,y)+(u,v)=(x+u,y+v)$ corresponds to $(x+iy)+(u+iv)$.

  • Multiplication (via identification):

    $$ (x,y)\cdot(u,v)=(xu-yv,\; xv + yu). $$

  • Matrix for multiplication by $a+bi$:

    $$ M(a,b)=\begin{pmatrix} a & -b\\ b & a\end{pmatrix}. $$

  • Modulus / norm: $|x+iy|=\sqrt{x^2+y^2}$.


One-sentence summary

$\mathbb{R}^2$ and $\mathbb{C}$ are the same as sets and as real vector spaces / metric spaces, but $\mathbb{C}$ carries extra algebraic (field) and analytic structure (complex multiplication, complex-linear maps, holomorphicity) that makes it strictly richer than the plain 2-dimensional real plane.

Would you like a short table comparing properties side-by-side, or examples showing how complex differentiability (Cauchy–Riemann) restricts a real map?

Largest freshwater animal

largest freshwater animal

The largest freshwater animal in the world is generally considered to be the giant freshwater stingray (Urogymnus polylepis).

Size

  • It can reach up to 5 meters (16 feet) in length, including the tail.
  • Weighs as much as 600–800 kg (1,300–1,800 lbs).
  • Native to rivers in Southeast Asia, such as the Mekong and Chao Phraya.

Other Contenders

Depending on how “largest” is defined (length vs. weight), there are other rivals:

  • Beluga sturgeon (Huso huso): Found in the Caspian and Black Sea basins; can exceed 7 meters (23 feet) and weigh over 1,500 kg (3,300 lbs). However, it spends much of its life in brackish waters, not strictly freshwater.
  • Mekong giant catfish (Pangasianodon gigas): Native to the Mekong River, it can grow up to 3 meters (10 feet) and weigh up to 300 kg (660 lbs).

👉 If we’re strict about freshwater-only species, the giant freshwater stingray holds the record. 👉 If we include fish that migrate between freshwater and brackish/sea environments, the beluga sturgeon can be even larger.

Would you like me to make you a size comparison chart of these freshwater giants?

Eudaimonia cost

Approximately how much money does eudaimonia require?

Short answer: there’s no single dollar figure — eudaimonia (human flourishing) mostly needs enough money to remove insecurity and buy you time and agency, not infinite wealth. That said, here are practical, approximate anchors you can use.

Rough anchors (annual income / nest-egg view)

  • Basic security (cover food, housing, healthcare, transport) — depends a lot on country and city.

    • In many parts of the U.S., that’s roughly $20,000–$40,000 / year for a single person; in expensive metro areas it’s higher.
  • “Comfortable” living (less financial stress, modest extras)~$50,000–$120,000 / year, depending on location and family size.

  • The frequently-cited “happiness plateau” from well-being research: classic studies found emotional well-being rises with income up to about $75,000 / year (older figure); more recent work puts the range somewhat higher (often quoted $90k–$100k / year for certain measures of life evaluation). These are guides, not magic thresholds.

If you’re thinking in terms of a retirement / freedom nest egg (safe-withdrawal rule)

People often use the “25× annual spending” rule (the 4% rule). I’ll show the math digit-by-digit:

  • To support $75,000 / year:

    1. 75,000 × 20 = 1,500,000.
    2. 75,000 × 5 = 375,000.
    3. Add: 1,500,000 + 375,000 = 1,875,000. So ~$1,875,000 in savings.
  • To support $100,000 / year:

    1. 100,000 × 25 = (100,000 × 20) + (100,000 × 5) = 2,000,000 + 500,000 = 2,500,000.
  • To support $40,000 / year:

    1. 40,000 × 25 = (40,000 × 20) + (40,000 × 5) = 800,000 + 200,000 = 1,000,000.

Key point (philosophical & practical)

Eudaimonia = flourishing through virtue, meaningful activity, relationships, autonomy. Money helps by removing scarcity and buying time/choices, but past the point of security and reasonable comfort its marginal effect on deep flourishing drops. Most of the long-term gains to “flourishing” come from non-monetary investments: community, meaningful work, health, and time-control.

Quick, practical takeaway

Aim first to eliminate financial insecurity (an emergency fund + predictable income). After that, use money to buy time, health, experiences, and opportunities for meaningful engagement. If you want, tell me your country/city and whether you mean annual income or a retirement nest-egg, and I’ll give more tailored numbers (no need to answer if you prefer this general guide).

Handshaking lemma proof

Show that the number of vertices of odd degree in a graph is always even.

Proof (Handshaking lemma / parity argument)

Let $G=(V,E)$ be a finite graph. For each vertex $v\in V$ let $\deg(v)$ denote its degree. Two standard facts:

  1. Each edge contributes exactly $2$ to the total sum of degrees (one for each endpoint). Hence

$$ \sum_{v\in V}\deg(v)=2|E|, $$

so the total sum of degrees is even.

  1. Split the vertex set $V$ into $V_{\text{even}}$ (vertices of even degree) and $V_{\text{odd}}$ (vertices of odd degree). The sum of degrees over $V_{\text{even}}$ is even (sum of even numbers). Therefore the parity of the whole sum equals the parity of the sum over $V_{\text{odd}}$. But the sum of $k$ odd numbers has the same parity as $k$ (odd sum iff $k$ is odd). Thus

$$ \sum_{v\in V}\deg(v)\ \text{is even} \quad\Longrightarrow\quad |V_{\text{odd}}|\ \text{is even}. $$

So the number of vertices of odd degree must be even. ∎

Wednesday, August 20, 2025

Pedophile-romanticizing literature list

Besides "Death in Venice" and "Lolita", are there any great (widely renowned) literary works that romanticize pedophiles? I have heard some people criticize the Western literary tradition as having many such works among its great widely renowned literary fiction, but in order to fairly evaluate this claim I must look for more examples that would be cited

Short answer: yes—there are a number of widely studied/canonical works that critics often cite when arguing the Western canon has romanticized or aesthetically dignified adult–minor desire. Just note that many of these books are contested: some readers see them as critical indictments, others as normalization/romance. Here’s a compact, representative list (leaving out Death in Venice and Lolita as you asked):

  • Classical backbone of the canon (man–boy love idealized): Plato’s Symposium (and Phaedrus) present speeches that treat pederastic love—adult men with adolescent boys—as noble/educative; more broadly, Greek literature made such relationships a recurring theme. Histories and reference entries document how pederasty was socially acknowledged and celebrated in elite discourse, which shapes how those texts read today. (HistoryExtra, Oxford Research Encyclopedias)

  • Samuel Richardson, Pamela (1740): A hugely influential English novel whose 15-year-old heroine is relentlessly pursued by her wealthy employer and ultimately marries him—a plot long criticized for legitimizing coercion. (Encyclopedia Britannica)

  • Choderlos de Laclos, Les Liaisons dangereuses (1782): A foundational Enlightenment novel in which the libertine Valmont seduces Cécile de Volanges, who is about 15 and fresh from the convent—frequently cited as an elegant aestheticization of abuse. (Age noted in standard study guides and reference entries.) (SparkNotes, Encyclopedia Britannica)

  • André Gide, The Immoralist (1902): Canonical modernist novel (by a future Nobel laureate) that includes the narrator’s eroticized attention to adolescent boys, often discussed as romanticizing/normalizing his desires amid a self-liberation narrative. (Literary Theory and Criticism)

  • Vladimir Nabokov, The Enchanter (written 1939; Eng. trans. 1986): A compact precursor to Lolita told from a pedophile’s viewpoint; critics commonly group it with examples that aestheticize the predator’s subjectivity. (Los Angeles Times, Goodreads)

  • Penelope Fitzgerald, The Blue Flower (1995): National Book Critics Circle Award–winning novel about the Romantic poet Novalis’s rapturous attachment to Sophie von Kühn—twelve when he first falls for her—narrated with luminous tenderness that some readers find troubling. (Wikipedia, EBSCO)

  • Bernhard Schlink, The Reader (1995): International bestseller and widely taught; it frames the affair of a 36-year-old woman with a 15-year-old boy within a coming-of-age/memory story—often criticized as romantically toned even as it engages postwar guilt. (Wikipedia, Encyclopedia Britannica)

  • Gabriel García Márquez, Memories of My Melancholy Whores (2004): A ninety-year-old narrator “falls in love” with a sleeping 14-year-old girl; major reviews describe the book as a rhapsodic “love letter” that many readers see as romanticizing predation. (The New Yorker, Encyclopedia Britannica)

  • Alissa Nutting, Tampa (2013): A satirical, deliberately glossy inside-the-predator’s-mind portrait of a 26-year-old teacher grooming a 14-year-old boy; widely reviewed and frequently invoked in this debate. (The Guardian)

  • Often cited, but explicitly critical rather than romanticizing: Kate Elizabeth Russell, My Dark Vanessa (2020)—about a 15-year-old groomed by her teacher—was marketed and read as a #MeToo-era corrective; serious critics emphasized it is not a romance but an immersion in abuse and complicity. It still appears in these conversations because of how it interrogates the seductive myths. (The Atlantic, The Guardian)

A couple of framing notes

  • Terminology matters. Strictly speaking, pedophilia refers to attraction to pre-pubescent children; many literary cases (e.g., classical pederasty or 15–16-year-old “ingénues”) would be labeled hebephilia/ephebophilia today. Critics often use “pedophile” colloquially, but the distinction is part of scholarly debates about these texts. (Oxford Research Encyclopedias)
  • Reception is split. Some readers see these works as critiques of predation (or as historically contextual depictions); others argue that the elegance of the prose, the centering of the adult’s feelings, or the narrative “rewards” (e.g., marriage, exalted love) functionally romanticize it.

If you want, I can sort these into buckets—“clearly celebratory,” “ambiguous/contested,” and “explicitly critical”—or add more examples from specific periods (Victorian, modernist, etc.) for a fuller audit.

No need to reorder the list, I will research criticism of each of the works separately later. Continue to use chronological order as that is helpful to trace a historical evolution. Are there any more works besides those?

Yep—here are more often-cited examples (kept strictly chronological, still excluding Death in Venice and Lolita). Brief notes flag why critics bring them up; “romanticize” is debated title-by-title.

  • The Letters of Abelard and Héloïse (12th c.) — celebrated love letters between a famous teacher and his much younger pupil; many historians place Héloïse in her mid-teens when the affair began. (Christian History Institute, Encyclopedia Britannica)
  • Goethe, Faust Part I (1808) — Faust seduces Gretchen, explicitly said in the text to be “over fourteen.”
  • Colette, Claudine at School (1900) — a 15-year-old pupil entangled in erotic intrigues involving adult teachers; often discussed for its teacher–student dynamics. (Wikipedia)
  • Jun’ichirō Tanizaki, Naomi (1924) — a 28-year-old narrator grooms a 15-year-old café girl he “rescues” and remakes; the novel knowingly glamorizes the project. (Wikipedia)
  • Vladimir Nabokov, Laughter in the Dark (1932) — middle-aged man obsessed with a teenager (Margot is 16–17 in most summaries). (Wikipedia)
  • Colette, Gigi (1944) — a teenage girl groomed as a courtesan becomes the adult hero’s love interest; tone is light/comic despite the premise. (Encyclopedia Britannica)
  • Dorothy Strachey (as “Olivia”), Olivia (1949) — classic boarding-school novel about a 16-year-old student’s passion for (and the reciprocation of) her teacher. (Wikipedia, Stanford News)
  • Yasunari Kawabata, The House of the Sleeping Beauties (1961) — elderly men pay to sleep beside drugged “young virgins” (critical work places them roughly mid-teens to early 20s). (Wikipedia, Nichibun Repository)
  • Gabriel García Márquez, One Hundred Years of Solitude (1967) — Aureliano falls for Remedios Moscote when she is nine; he’s made to wait until she menstruates to marry. (LitCharts, CliffsNotes)
  • Thomas Pynchon, Gravity’s Rainbow (1973) — sexualized depiction of Bianca, whom the text describes as “11 or 12,” has generated extensive debate. (gravitys-rainbow.pynchonwiki.com, pynchonnotes.openlibhums.org)
  • Marguerite Duras, The Lover (1984) — an acclaimed autofictional novel about a 15-year-old girl’s affair with a much older man. (Encyclopedia Britannica)
  • Mario Vargas Llosa, In Praise of the Stepmother (1988) — erotic novel in which a stepmother is drawn into a sexual relationship with her pre-teen/early-teen stepson. (Wikipedia, The New Yorker)
  • Haruki Murakami, Kafka on the Shore (2002) — features a 15-year-old protagonist in sexual encounters with an older woman; reviews and summaries note the controversy. (The New Yorker, Encyclopedia.com)
  • André Aciman, Call Me by Your Name (2007) — a 17-year-old and a 24-year-old; celebrated novel/film, frequently discussed in age-of-consent debates. (Encyclopedia Britannica)

If you’d like, I can also split these into “widely read in the Western canon” vs. “non-Western but often cited in the same debates,” or add quick pointers to major critiques for each title.