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).
9 of 2004 LIT documents. Back to the full index.
| # | Title | Summary | Status |
|---|---|---|---|
| LIT-104 v4 | Philosophy of Mathematics from Descartes to Kant | Carson (2026), DOI-10.1017/9781108982405. The early-modern debate over whether metaphysics can be demonstrated with geometric certainty turned on how Euclidean demonstration secures certainty, and the Element traces the answers from Descartes through Locke, Leibniz, Wolff, Lambert and Mendelssohn to Kant. | Unavailable — seeded from the abstract on 2026-09-26; a close reading was attempted the same day and found the Element is not open access (Cambridge Core shows “Get access” with purchase temporarily unavailable; Crossref gives only the standard Cambridge licence). Print publication 2026-08-31. It stays Unavailable until a copy is supplied. |
| LIT-108 v3 | Philosophy of Mathematics and Natural Science | Weyl (1949), DOI-10.1515/9781400833337. Philosophy of mathematics and philosophy of physics are intertwined, with advances in philosophy led by scientific discovery. Weyl’s two-part treatment (mathematics, then natural science), drawing on Descartes, Galileo, Hume, Kant, Leibniz and Newton, is a classic mathematician-physicist’s statement of that view. | Unavailable — seeded from the abstract on 2026-09-26; a close reading was attempted the same day and found the book in copyright with no open copy: every Internet Archive scan (the 1949, 1959 and 1963 printings) is lending-only, and the De Gruyter e-book is paywalled. It stays Unavailable until a copy is supplied. |
| LIT-115 v5 | Duality as a category-theoretic concept | Corfield (2015), DOI-10.1016/j.shpsb.2015.07.004. A short programmatic paper. Its thesis is that category theory, not set theory, is the framework in which the scattered instances of mathematical duality can be drawn together. The constructions in §3 (dual adjunctions restricting to dual equivalences on fixed points, truth-value-enriched Galois connections, Isbell conjugation, dualising objects “keeping summer and winter homes”) all “revolve around” adjunction, though Corfield expects no “single monolithic account” (p. 59–60). On his usage, ‘duality’ is reserved for “involution with some form of structural reversal” (p. 57), so several physicists’ dualities (homological mirror symmetry as an A∞-equivalence, mirror symmetry’s Z2 on the Hodge diamond, S-duality as a vestige of SL(2,Z)) are better called equivalences or symmetries. He proposes, without argument beyond the examples, that philosophers of physics will need homotopy type theory rather than predicate logic for the sameness questions dualities raise. | Active — read in full 2026-09-26 (NOTE-141); worth reading, briefly, as the mathematician-philosopher’s counterweight to the physics-first accounts of duality this record holds: it is the clearest statement that what physicists call a duality is often what a mathematician calls an equivalence, and it is only seven pages. Its positive programme (HoTT for philosophy of physics) is announced, not carried out. |
| LIT-151 v2 | Stating Structural Realism: Mathematics-First Approaches to Physics and Metaphysics | Wallace (2021), DOI-10.1111/phpe.12172. Wallace argues that reading a physical theory “math-first”, as mathematical models plus representation rather than as sentences aiming at truth, already makes scientific realism structural. Math-first theoretical equivalence and reduction (as instantiation) are coarse-grained enough to dissolve the standard underdetermination and theory-change objections. On this footing epistemic and ontic structural realism differ only in whether metaphysics itself is language-first or math-first. On math-first OSR, objects are whatever the best “predicate precisification” of a theory quantifies over, real patterns are defined through instantiation plus best precisification, and ontology is scale-relative. At every non-fundamental level the two versions coincide. The paper is explicitly a statement of the position, not a defence of it. | Active — read in full 2026-09-26 (NOTE-142); worth reading as the sharpest statement of what structural realism could mean once theories are taken math-first. It gives precise definitions of math-first ESR/OSR and of real patterns, its diagnosis of why priority-thesis OSR drifted is useful, and it is candid about the undeveloped notion of mathematical equivalence it relies on. |
| LIT-154 v3 | Criteria of Identity and Structuralist Ontology | Leitgeb & Ladyman (2007), DOI-10.1093/philmat/nkm039. Graph theory counts unlabelled graphs whose nodes, or whose isomorphic connected components, cannot be told apart even by an irreflexive relation. Examples are the edgeless two-node graph G′ and any graph with two or more isolated nodes (11 of the 156 unlabelled 6-node graphs). Mathematicians treat these as determinately distinct objects, so the authors conclude that the identity or difference of places in a structure is given by the structure itself, with identity counted among its relations, and not by any criterion of identity. The argument appeals to mathematical practice; it is not a proof. | Active — read in full 2026-09-26 (NOTE-133); worth reading as the short, standard counterexample to weak-discernibility versions of the PII for mathematical structuralism. It is widely cited, and its graph cases are simple enough to check by hand. Its physics coda is explicitly left open. |
| LIT-163 v2 | Philosophy of Probability: Foundations, Epistemology, and Computation | Wenmackers (2011), https://philpapers.org/rec/WENPOP. A fair lottery on ℕ gets a uniform, regular probability on all of 𝒫(ℕ) by normalising Benci–Di Nasso numerosities, P_num(A) = num(A)/α. This is hyperrational-valued, determined only up to the choice of free ultrafilter, and “hypercountably” (really hyperfinitely) additive rather than countably additive, with st∘P_num = generalised asymptotic density (§2.5–2.6). The same “rounding-error” diagnosis gives Stratified Belief, a contextual Lockean thesis on which conjunction holds for a standard number of beliefs (Ch. 3). A separate agent-based model (Ch. 5) finds that one round of bounded-confidence averaging sends an agent to the inconsistent theory with probability at most 1.81 % in the cases computed (M = 2, 3 atomic sentences). | Active — read in full 2026-09-26 (NOTE-139); worth reading for the Ch. 1 survey of interpretations of probability and the Ch. 4 synthesis, which appear only here; for the rest, each chapter has a later journal version that is the better citation, none of them held on this record: Wenmackers & Horsten, “Fair infinite lotteries”, Synthese 190 (2013) 37–61 (Ch. 2); Wenmackers, “Ultralarge lotteries”, J. Applied Logic 11 (2013) 452–467 (Ch. 3); Wenmackers, Vanpoucke & Douven, “Probability of inconsistencies in theory revision”, Eur. Phys. J. B 85 (2012) 44 (Ch. 5); and Benci, Horsten & Wenmackers, “Non-Archimedean probability”, Milan J. Math. 81 (2013) 121–151 (the Ch. 6 outlook). |
| LIT-222 v2 | No categorial support for radical ontic structural realism | Lam & Wüthrich (2015), DOI-10.1093/bjps/axt053. Lam and Wüthrich argue that category theory gives radical OSR no support. Identifying objects with identity arrows is relabelling (§3). Generalized elements X → A exist in every category, so elements are not eliminated (§4). Categories without products also lose set-style relations, “throwing out the relations with the relata” (§4). Morphisms relate C-objects, not the elements that stand for physical relata, and they “only distinguish isomorphism classes of structures” (§5). Bain’s GR and TQFT cases eliminate at most spacetime points, not relata tout court (§§6–7). They conclude that category theory “offers little if any comfort to ROSR” (abstract) and favour “balanced” OSR. The Yoneda lemma is not named, although §§4–5 state two of its consequences. | Active — read in full 2026-09-26 (NOTE-194); Worth reading as the most careful direct refutation of Bain (yo2). Its main points (generalized elements, morphisms relate C-objects, determination only up to isomorphism, sections are not maximal ideals) are correct and are exactly what the Yoneda reading needs. It has a few category-theoretic slips of its own: Hilb said to lack products, a mis-definition of topos, and the claim that non-commutative algebras have no maximal ideals. |
| LIT-223 v2 | Category-theoretic structure and radical ontic structural realism | Bain (2013), DOI-10.1007/s11229-011-9896-6. Bain argues that the “relations without relata” objection to radical OSR depends on defining structure set-theoretically, where a relation is a subset of X × X. If a structure is instead “an object in a category”, its elements (arrows 1 → A) are “not essential”, because category-theoretic objects need not be structured sets. His examples are sheaves of Einstein algebras for GR with asymptotic boundary conditions, which may lack global sections (§3), and nCob and Hilb (§4). He concludes that ROSR “avoids the charge that it rests on an incoherent claim” (p. 1634). The paper never mentions the Yoneda lemma, representable functors or universal properties by name. | Active — read in full 2026-09-26 (NOTE-193); Worth reading as the short, canonical statement of the category-theoretic defence of radical OSR, which the later literature (yo3; Lal & Teh; Eva 2016) responds to. Read it with yo3. Its central inference is overstated, and its GR section contains technical errors that yo3 documents. |
| LIT-437 v2 | Gödel, Escher, Bach: an Eternal Golden Braid | Hofstadter (1979), Basic Books. The book that introduced the “strange loop”: per its publisher, an account of the “I”, self and consciousness built on the self-referential construction Gödel used to make a sentence assert its own unprovability. Chapters alternate with contrapuntal dialogues, chiefly between Achilles and the Tortoise, each built to exhibit the idea it discusses. Unread: no lawful full text was reachable, so it is registered from the publisher’s description and the catalogue record, and its year is the only date a source gives. | Unavailable — registered 2026-10-01 from the publisher’s description and the Library of Congress catalogue record, not read: no lawful full text could be reached. The Internet Archive scans of the 1979 Basic Books printing are lending-only, HathiTrust is search-only, and no publisher or author copy is posted. It stays Unavailable until a copy is supplied. |