Hannes Leitgeb, James Ladyman (2007), Philosophia Mathematica 16(3), 388–396 (2008) — DOI-10.1093/philmat/nkm039
Key takeaways
- Graph theory contains structures whose nodes are not even weakly discernible, such as the two-node edgeless graph. Mathematical practice treats them as distinct objects anyway, so the identity or difference of places in a structure is fixed by the structure itself, with identity counted among its structural relations, and not by any further criterion of identity.
Seeded from the abstract and a skim, before the reading in NOTE-133. What follows is what the work says about itself.
The debate over whether structuralist ontologies of mathematics can keep the Principle of the Identity of Indiscernibles usually assumes that individuals fall under criteria of identity that account for their identity. Most of that debate concerns structures with non-trivial automorphisms. The authors present graph-theoretic cases that violate even weak versions of PII. They argue that the identity or difference of places in a structure is accounted for by nothing beyond the structure, and that mathematical practice supports this.
Standing in the record
Filed from the 2026-09-26 sweep of the papers-feed tracker for philosophy: 5 seconds of active reading over 1 session. It was filed Deferred, unread. NOTE-133 is the close reading of 2026-09-26, and it placed the work: Active — 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.
When seeded, before the reading in NOTE-133, the entry gave a priority for a deeper reading: low — a short discussion note (t=5s) that the skim covers almost entirely.
What the seeding asked a deeper reading to check (NOTE-133 is that reading):
- A key text for the identity/individuality strand shared by OSR (LIT-173, LIT-196) and quasi-set theory (LIT-107), which takes the opposite tack by dropping identity for quantum items.
- Its closing disanalogy between mathematical and physical objects bears on whether mathematical-structuralist arguments transfer to ontic structural realism about physics (LIT-151, LIT-170).
- Check the published version’s wording, and whether “identity is a structural relation” begs the question against those who demand a criterion.
Access when seeded: The PhilPapers record was not fetched, since the brief says it is Cloudflare-blocked. The DOI, volume/issue/pages and online date 2007-09-20 are from Crossref. The batch spells the first author “Leitgib” (PhilPapers’ spelling); the paper itself and Crossref give Hannes Leitgeb. The OUP page was not opened, and Unpaywall lists no open copy. I read an author-uploaded manuscript (ResearchGate cover sheet: “uploaded by Hannes Leitgeb on 17 April 2014”; 7 manuscript pp., “Discussion Note”) from a third-party mirror at horreum.pages.dev that a web search turned up. Page numbers are the manuscript’s, and the published text may differ. Keywords are my own.