Algebraic topology
Topological spaces studied via algebraic invariants: homology, cohomology.
Founded: 1895 — Poincaré's Analysis Situs
The claim
No credible dismissal found. Topology's founding self-understanding — discarding exactly the distances and coordinates data analysis depends on — is a fair characterization of the field's design, not a quote from a named critic.
What happened
Topological data analysis (2002) — Edelsbrunner–Letscher–Zomorodian's persistent-homology algorithm.
Breast-cancer subtype discovery (2011) — Nicolau–Levine–Carlsson, PNAS — 295 tumors as points in 24,479 dimensions.
107 years — founding (1895) to persistent homology (2002)
Verdict: Never dismissed, applied anyway.
Confidence: medium-high.