Thirteen ways of seeing the same library.

Level 1 · Human

A system whose state can change—and whose validated discoveries can change the methods used next.

Level 2 · Product

Every change to a library is a change of state that is designed to keep what must not change: a quotation’s source, a copy’s identity, its history.

Level 3 · Disciplinary

Four classical ideas meet here. Graph theory treats relations as objects of study. State-transition systems describe how a system moves between configurations. Invariants name what every step preserves. Higher-order functions treat procedures as values that other procedures can take and return.

Level 4 · Formal

The state is the six-part ℒₜ = (Oₜ, Rₜ, Eₜ, Mₜ, Hₜ, Aₜ), and the graph Gₜ is the state’s graph view. Φ is total: an accepted input applies, and a refused input leaves the state unchanged except for a recorded refusal. A trajectory ℒ₀, ℒ₁, ℒ₂, … is a discrete dynamical system; an invariant is a function constant along every admissible trajectory. The map from a discovery to an instrument is higher-order, because it returns a function; V is a guard on that map, not a computation.

Level 1 · Human

A place to examine, challenge and reorganize mental models without allowing the system to define the reader.

Level 2 · Product

Notes, questions and changed readings are designed to let a reader see how their own understanding moved, without the system labelling them.

Level 3 · Disciplinary

Piaget described assimilation (fitting the new into an existing schema) and accommodation (changing the schema to fit the new). Bartlett showed memory is reconstructive, shaped by schemas. Flavell named metacognition: monitoring and regulating one’s own thinking.

Level 4 · Formal

The formulas are shorthand for processes, not measurements. Assimilation keeps the schema’s structure and adds content; accommodation changes the structure. No quantity of ‘schema change’ is measured, and no psychological benefit is claimed.

Level 1 · Human

Preserve not only conclusions, but their reasons.

Level 2 · Product

Each claim is designed to carry its kind (source, evidence, inference or interpretation), what supports it, what resists it, and where it came from.

Level 3 · Disciplinary

Justification asks what makes a belief reasonable. Popper argued that theories earn credibility by surviving attempts to refute them. Bayesian epistemology treats evidence as changing degrees of belief in both directions. The AGM theory formalises how a belief set is expanded, contracted and revised.

Level 4 · Formal

E⁺(H) and E⁻(H) are the supporting and resisting evidence for H, both drawn from the epistemic records Eₜ; they are not logical entailment. The precise counterparts are Bayesian: a supporting item has a likelihood ratio above one. For rebutting evidence the counterpart is a likelihood ratio below one; undercutting evidence changes the likelihood ratio of the item it undercuts, so the simple product form does not apply to it. Belief change follows the AGM operations; the Levi identity (Levi 1977) builds revision from contraction and expansion.

Level 1 · Human

Return to the same book without returning to the same understanding.

Level 2 · Product

A passage is designed to stay word-for-word the same while its links to the book, the collection and your notes change around it.

Level 3 · Disciplinary

The hermeneutic circle runs from Schleiermacher and Dilthey to Gadamer: understanding moves between parts and wholes, and between the text’s horizon and the reader’s. Reader-response theory (Iser) locates meaning in the encounter; intertextuality (Kristeva) in the relations between texts.

Level 4 · Formal

The circle is a structure of mutual dependence, not an algorithm. Written as an iteration uₖ₊₁ = h(part, whole(uₖ)), it shows what the site means by ‘return’: the part is re-read with an updated whole. Φ6 writes the same step as iₖ = h(iₖ₋₁, cₖ), with the new context cₖ standing for the updated whole. No convergence is claimed.

Level 1 · Human

The text may be reproducible. Your copy is not.

Level 2 · Product

Folium X is designed to keep work, edition and copy as separate linked records, so two copies of the same edition remain two records with their own location, condition, acquisition, inscriptions, loans and reading history.

Level 3 · Disciplinary

The IFLA Library Reference Model (2017) unified FRBR, FRAD and FRSAD around Work, Expression, Manifestation and Item. Archival science adds provenance: records keep the context of their creation. W3C PROV models entities, activities and agents.

Level 4 · Formal

LRM’s Item is an individual exemplar; an ISBN identifies (roughly) a manifestation, so it cannot identify an item. Folium X’s simplification uses three levels: Work, Edition (closest to Manifestation) and Copy (closest to Item); Expression is not separate, so several editions carrying one translation are harder to relate. That limit is stated, not hidden. The edition relation is partial (a copy’s edition may be unknown) and editions can embody several works. An ISBN identifies a publication — a given title in a given edition and binding — not a work and not an individual copy. The ISBN became ISO 2108 in 1970, building on the UK's Standard Book Number of 1966–67. Many older printings carry no ISBN, but publishers must number their backlist, and the ISBN must appear in the first available reprint or reissue of a backlist title (ISBN Users' Manual, 6th ed., 2012, §5.7). A copy record must therefore never require an ISBN.

Level 1 · Human

No single book contains the pattern. The pattern exists between them.

Level 2 · Product

Relations across a collection are designed to be shown as a network, with patterns kept separate from the facts beneath them and from what anyone makes of them.

Level 3 · Disciplinary

Anderson’s ‘More Is Different’ argued that each level of organisation has properties of its own. Network science measures paths, clustering and hubs; Watts and Strogatz showed how a few long links make a ‘small world’.

Level 4 · Formal

The precise, uncontroversial sense of emergence used on this site is relational: a path, cycle or cluster is a property of the edge set E, so no vertex alone has it. Stronger philosophical claims about emergence are not made.

Level 1 · Human

A book has a bibliography. A copy can have a biography.

Level 2 · Product

A copy is designed to carry the things that make it this book to you: who gave it, where it lives, what you wrote in it, when you read it.

Level 3 · Disciplinary

Husserl’s intentionality: consciousness is always of something, and the same object is given through changing profiles (adumbrations). Merleau-Ponty grounds perception in the body: a book is held, weighed, worn. “A copy can have a biography” draws on Igor Kopytoff’s cultural biography of things (1986): that idea comes from anthropology, not phenomenology, and is credited as such.

Level 4 · Formal

Husserl’s adumbrations offer an exact image for the Lens Switch: one object, intended as the same, given through many profiles. On this site that is an interpretive correspondence, not a claim about how the app works.

Level 1 · Human

Something you learn can become something you think with.

Level 2 · Product

A distinction found in one book is designed to become a question you can carry to others, with its origin kept.

Level 3 · Disciplinary

Transfer research asks when learning in one setting helps in another; Bransford and Schwartz reframed it as preparation for future learning. Argyris and Schön distinguished single-loop learning (better answers) from double-loop learning (changed governing assumptions). Scaffolding (Wood, Bruner and Ross) is support that fades as competence grows.

Level 4 · Formal

Double-loop learning in the site’s notation: single loop revises an answer under a fixed distinction θ; double loop revises θ itself, f_θ → f_θ′. That notation is our architectural interpretation, not a theorem from Argyris.

Level 1 · Human

A library can become an environment in which you think.

Level 2 · Product

Notes, relations and questions are designed to be where you left them, so you can think with them again.

Level 3 · Disciplinary

Clark and Chalmers’ extended mind thesis argues that external resources can be part of cognitive processes under certain conditions; Hutchins studied cognition distributed across people and instruments. Adams and Aizawa argued the bounds of cognition remain in the head.

Level 4 · Formal

Clark and Chalmers’ notebook example relies on conditions such as reliable availability and ready endorsement of stored information. Folium X’s reader-review design deliberately withholds automatic endorsement of AI proposals, which is one precise reason this correspondence stays philosophical rather than a claim that the library is part of your mind. The Φ Framework goes further: a discovery becomes an instrument only through V, the reader’s validation. The honest claim is an external cognitive environment, not a ‘second brain’.

Level 1 · Human

Know where an idea came from.

Level 2 · Product

Quotations, notes about them, interpretations and interpretations of interpretations are designed to stay distinct layers, each marked with who made it.

Level 3 · Disciplinary

Peirce’s sign relates a representamen, an object and an interpretant, and the interpretant can itself become a sign: interpretation of interpretation. Saussure separated signifier and signified. Barthes described second-order signification, where a whole sign becomes the signifier of a further meaning.

Level 4 · Formal

The orders form a tower: I₀ = Text, I₁ = Annotation(I₀), I₂ = Interpretation(I₀, I₁), Iₙ₊₁ = Interp(Iₙ). Each level must keep a pointer to the level beneath it. For AI reliability that pointer is the difference between a quotation and a statement about a quotation.

Level 1 · Human

Understanding has a history.

Level 2 · Product

Superseded readings are designed to stay as dated layers beneath the current one, so you can see how it came to be.

Level 3 · Disciplinary

Collingwood held that history re-thinks past thought from its evidence; Carr that facts are selected by the historian. Reception history (Jauss) studies how a work’s meaning changes across readers. Path dependence (David) shows how earlier states constrain later ones.

Level 4 · Formal

History as data: the log H = ⟨Δ₀, Δ₁, …⟩ with Sₜ = Δₜ₋₁ ∘ … ∘ Δ₀ (S₀). The current state underdetermines its history: two different paths can reach the same Sₜ, which is exactly why the path is kept as evidence.

Level 1 · Human

Folium X can propose an interpretation. It cannot make it yours.

Level 2 · Product

Every AI proposal is designed to pass through the reader’s review, and to keep its origin after it is accepted or changed.

Level 3 · Disciplinary

Wiener founded cybernetics as the study of control and communication through feedback. Ashby’s law of requisite variety says a regulator needs responses as varied as the disturbances. Second-order cybernetics (von Foerster) puts the observer inside the system. Supervisory-control research graded how much a human stays in the loop: Sheridan and Verplank’s levels of automation, as presented in Parasuraman, Sheridan and Wickens (2000).

Level 4 · Formal

The loop has a gate: ℒₜ₊₁ = Φ(ℒₜ, xₜ, Cₜ), where the input xₜ = r(pₜ) is the reader’s response to the proposal pₜ. In those levels of automation (Sheridan and Verplank, as presented in Parasuraman, Sheridan and Wickens 2000) the design sits at a level where the system suggests and the human decides. Requisite variety is stated qualitatively here; no counting inequality is shown.

Level 1 · Human

What can be inferred is not automatically what should be asserted.

Level 2 · Product

Inferences about a collection are designed to stay behind the reader’s permission, and never to become statements about who the reader is.

Level 3 · Disciplinary

Winner argued that artifacts can embody political choices. Nissenbaum’s contextual integrity holds that information flows carry the norms of the context they came from. Value-sensitive design (Friedman) builds values into design from the start.

Level 4 · Formal

The three distinctions are type distinctions: a capability is a fact about the system, a permission is a relation granted by a person; an inference is a derived record, an entitlement is a right to use it; a prediction is a statement about data, an identity is a statement about a person. The design obligation is that no transformation silently converts one type into the other.