The organizing question isn't “is this field useful,” it's “how long does it take, and how do you count the ones that haven't happened yet.”
A field with no documented application today is not a failure of the thesis. It's a censored observation: set theory's distinctive machinery hasn't found an external use in the 152 years since Cantor's first paper, which means its true time-to-application is greater than 152 years, not infinite. Treated this way, the 33 fields with a known founding date (combinatorics is excluded here — its founding has no single event to measure from) form a standard survival-analysis dataset: 32 observed events (a year when a first application appeared) and 1 right-censored observation (still waiting, as of 2026).
\(S(t)\) is the estimated probability that a field is still without a major application \(t\) years after its founding. It starts at 1 and drops at each observed event; the censored observation is marked with a plus sign and doesn't cause a drop, since set theory hasn't failed to be applied — it just hasn't been resolved yet. Hover a step for the field that dropped there; click to open its page.
Six fields — potential theory, ODEs, PDEs, integral equations, calculus of variations, probability — drop at \(t=0\): they were tied to physics or gambling from their first page and were never really “pure” in the sense this survey is testing. That single step accounts for most of the early drop in the curve.
With one censored observation out of 33, this is a small, almost degenerate survival dataset — call it a crude, improvable empirical sketch, not a validated statistical result. The interesting question it raises, though, is real: does the hazard of “becoming useful” decline with a field's age, the way unrepaired equipment gets less likely to ever be fixed the longer it sits broken — or does each new wave of technology reopen the question for fields that looked exhausted? Thirty-three data points, nearly all of them events, can't settle that. They can at least state it as a question instead of leaving it as folklore.
The selection question underneath all of this is the same one the Methodology page raises: this dataset is the fields the 1959 Mathematics Subject Classification thought worth naming, which already excludes whatever genuinely went nowhere and left no trace before 1959. A survival analysis over a survivorship-biased sample is still useful — it's just a different, narrower question than “how often does mathematics in general become useful,” and it should be read as one.