philosophy-of-mathematics.
6 of 1538 NOTE documents. Back to the full index.
| # | Title | Summary | Status |
|---|---|---|---|
| NOTE-133 v2 | Leitgeb & Ladyman — Criteria of identity and structuralism | 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. | Read |
| NOTE-139 v2 | Wenmackers — Philosophy of probability (PhD thesis) | 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). | Read |
| NOTE-141 v2 | Corfield — Duality as a category-theoretic concept | 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. | Read |
| NOTE-142 v2 | Wallace — Stating Structural Realism | 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. | Read |
| NOTE-193 | Category-theoretic structure and radical ontic structural realism | 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. | Read |
| NOTE-194 | No categorial support for radical ontic structural realism | 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. | Read |