Topics — What a work is about. Nucleation holds whatever reading the owner files here, whether or not an Anthology of the SOTA topic could also hold it; instruction for ML practice belongs to the anthology (ADR-048). The words are seeded from the clusters the reading-time triage of 2026-09-25 found, and grow by decision.
Philosophy of mathematics — what mathematics is about and how we know it — mathematical structuralism and ontology, the nature of proof and construction, and its history from Descartes to Kant and after (group: philosophy).
5 of 1352 THEORY documents. Back to the full index.
| # | Title | Summary | Status |
|---|---|---|---|
| THEORY-032 | The Yoneda lemma determines an object only up to isomorphism, from its whole hom-functor, among objects the category already has; as a formal version of “an object is determined by its relations” it presupposes the relata | nLab, Yoneda lemma (rev. 2026-08-17), LIT-221 — y is fully faithful and y(c) ≅ y(d) iff c ≅ d, and pointwise bijections of hom-sets without naturality do not suffice. Lam & Wüthrich (LIT-222) state two of its consequences without the name. The philosophical reading, a relational criterion of identity up to isomorphism rather than a construction of objects from relations, is the record’s own. It does not say that the lemma supports eliminating objects. | Active |
| THEORY-038 | Category theory gives radical ontic structural realism no formal support: generalized elements exist in every category, morphisms relate objects the category presupposes, and Bain’s physical cases eliminate spacetime points, not relata | Lam & Wüthrich (2015), LIT-222, against Bain (2013), LIT-223. Bain’s “structure as object in a category” relabels relata rather than removing them, and his GR and TQFT cases dispose of points only. That the Yoneda lemma (LIT-221) strengthens the rebuttal is the record’s inference: no paper in the debate names it. This does not refute radical OSR. It removes one argument for it, and Eva 2016’s reply is unread. | Proposed |
| THEORY-401 | A fair lottery on the natural numbers has a uniform, regular probability defined on every set of tickets, by normalised numerosity with values in the hyperrationals; it is fixed only up to a choice of free ultrafilter, it is hyperfinitely and not countably additive, and its standard part is ordinary asymptotic density | Wenmackers (2011), LIT-163, chapter 2, read in NOTE-139. P_num(A) = num(A)/α is defined on all of 𝒫(ℕ), gives each ticket 1/α > 0, and satisfies FAIR, ALL and a hypercountable form of SUM (eqs. 2.20–2.27). Proved, given the axiom of choice. It is not a free lunch: one solution per free ultrafilter, no invariance under relabelling, no countable additivity, and by the author’s own admission it is empirically indistinguishable from the finitely additive real-valued solution. | Active |
| THEORY-604 v2 | Most mathematical dualities are adjunctions between one category and the opposite of another, which become dual equivalences on their fixed points; what makes them dualities rather than equivalences is the reversal of arrows, so many of the ‘dualities’ physicists name are, in the mathematician’s sense, equivalences or symmetries | Corfield (2017), LIT-115, read in NOTE-141. A short programmatic paper. That most dualities of interest are covered by constructions around adjunction is shown by worked exposition of standard cases, not proved. There is unlikely to be “a single monolithic account” (pp. 59–60). Reserving “duality” for structural reversal is a stated preference, and the verdicts on string dualities are a paragraph each. The proposal that philosophers of physics need homotopy type theory is an expectation. | Proposed |
| THEORY-729 | An adjunction between a category and the opposite of another restricts to a dual equivalence between the full subcategories on which its unit and counit are isomorphisms | Corfield (2017), LIT-115, §3, read in full in NOTE-141, with the proof the paper omits written out here. For F : C → D^op left adjoint to G : D^op → C, let C′ and D′ be the full subcategories on which the unit and the counit are isomorphisms. Then F and G restrict to inverse equivalences C′ ≃ D′, and D′ is a full subcategory of D^op, so C′ is dually equivalent to a full subcategory of D. Pontrjagin duality is the example. That most dualities are of this form is THEORY-604’s thesis, still Proposed. | Active |